Research Article Open Access
On The Frank FREs and Its Application in Optimization Problems
Amin Ghodousian*
Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
*Corresponding author: Amin Ghodousian, Faculty of Engineering Science, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran, E-mail: @
Received: June 17, 2018; Accepted: June 23, 2018; Published: June 29, 2018
Citation: Ghodousian A (2018) On The Frank FREs and Its Application in Optimization Problems. J Comp Sci Appl Inform Technol. 3(2): 1-14. DOI: 10.15226/2474-9257/3/2/00130
Abstract
Frank t-norms are parametric family of continuous Archimedean t-norms which are also strict when the parameter is nonnegative. Very often, this family of t-norms is also called the family of fundamental t-norms because of the role it plays in several applications. In this paper, we study a nonlinear optimization problem with a special system of Fuzzy Relational Equations (FRE) as its constraints. We firstly investigate the resolution of the feasible solutions set when it is defined with max-Frank composition and present some necessary and sufficient conditions for determining the feasibility and some procedures for simplifying the problem. Since the feasible solutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem, conventional nonlinear programming methods may involve high computation complexity. Based on the obtained theoretical properties of the problem, a genetic algorithm is used, which preserves the feasibility of new generated solutions and does not need to initially find the minimal solutions. Moreover, a method is presented to generate feasible max-Frank FREs as test problems for evaluating the performance of our algorithm. The presented method has been compared with some related works. The obtained results confirm the high performance of the current method in solving such nonlinear problems.

Keywords: Fuzzy relational equations; Nonlinear optimization; Genetic algorithm;
Introduction
In this paper, we study the following nonlinear problem in which the constraints are formed as fuzzy relational equations defined by Frank t-norm:
minf(x) Aφx=b x [0,1] n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGTb GaaiyAaiaac6gacaaMc8UaaGPaVlaaykW7caaMc8UaamOzaiaacIca caWG4bGaaiykaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyqaiabeA8aQjaa dIhacqGH9aqpcaWGIbaabaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG4bGaeyic I4Saai4waiaaicdacaGGSaGaaGymaiaac2fadaahaaWcbeqaaiaad6 gaaaaaaaa@7121@
where I={1,2,...,m},J={1,2,...,n}, A= ( a ij ) m×n ,0 a ij 1 (iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 da9iaacUhacaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGG UaGaaiilaiaad2gacaGG9bGaaiilaiaaykW7caWGkbGaeyypa0Jaai 4EaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGG SaGaamOBaiaac2hacaGGSaGaaeiiaiaadgeacqGH9aqpcaGGOaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaWaaSbaaSqaaiaa d2gacqGHxdaTcaWGUbaabeaakiaacYcacaaMc8UaaGimaiabgsMiJk aadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyizImQaaGymaiaa bccacaqGOaGaeyiaIiIaamyAaiabgIGiolaadMeaaaa@680C@ and jJ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaam OAaiabgIGiolaadQeacaGGPaGaaiilaaaa@3B64@ is a fuzzy matrix, b= ( b i ) m×1 ,0 b i 1 (iI), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2 da9iaacIcacaWGIbWaa0baaSqaaiaadMgaaeaaaaGccaGGPaWaaSba aSqaaiaad2gacqGHxdaTcaaIXaaabeaakiaacYcacaaMc8UaaGimai abgsMiJkaadkgadaWgaaWcbaGaamyAaaqabaGccqGHKjYOcaaIXaGa aeiiaiaabIcacqGHaiIicaWGPbGaeyicI4SaamysaiaacMcacaGGSa aaaa@4F21@ is an m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E7@ dimensional fuzzy vector, and " φ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOgaaa@37B2@ " is the max-Frank composition, that is, φ(x,y)= T F s (x,y)= log s ( 1+ ( s x 1)( s y 1) s1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaaG PaVlaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0Jaamivamaa DaaaleaacaWGgbaabaGaam4CaaaakiaacIcacaWG4bGaaiilaiaadM hacaGGPaGaeyypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadoha aeqaaOWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaGGOaGaam4Cam aaCaaaleqabaGaamiEaaaakiabgkHiTiaaigdacaGGPaGaaiikaiaa dohadaahaaWcbeqaaiaadMhaaaGccqGHsislcaaIXaGaaiykaaqaai aadohacqGHsislcaaIXaaaaaGaayjkaiaawMcaaaaa@5A53@ in which s >0 and s≠ 1.

If a i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbaabeaaaaa@37F5@ is the i’th row of matrix A, then problem (1) can be expressed as follows:
minf(x) φ( a i ,x)= b i ,iI x [0,1] n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGTb GaaiyAaiaac6gacaaMc8UaaGPaVlaaykW7caaMc8UaamOzaiaacIca caWG4bGaaiykaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOXdOMaaiikaiaa dggadaWgaaWcbaGaamyAaaqabaGccaaMc8UaaiilaiaadIhacaGGPa Gaeyypa0JaamOyamaaDaaaleaacaWGPbaabaaaaOGaaGPaVlaaykW7 caaMc8UaaiilaiaaykW7caaMc8UaaGPaVlaadMgacqGHiiIZcaWGjb aabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWG4bGaeyicI4Saai4waiaaicdaca GGSaGaaGymaiaac2fadaahaaWcbeqaaiaad6gaaaaaaaa@8450@
where the constraints mean: φ( a i ,x)= max jJ {φ( a ij , x j )}= max jJ { T F s ( a ij , x j )}= max jJ { log s ( 1+ ( s a ij 1)( s x j 1) s1 ) }= b i ,iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaaG PaVlaacIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadIha caGGPaGaeyypa0ZaaCbeaeaaciGGTbGaaiyyaiaacIhaaSqaaiaadQ gacqGHiiIZcaWGkbaabeaakiaacUhacqaHgpGAcaaMc8Uaaiikaiaa dggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiilaiaadIhadaWgaa WcbaGaamOAaaqabaGccaGGPaGaaiyFaiabg2da9maaxababaGaciyB aiaacggacaGG4baaleaacaWGQbGaeyicI4SaamOsaaqabaGccaGG7b GaamivamaaDaaaleaacaWGgbaabaGaam4CaaaakiaaykW7caGGOaGa amyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaamiEamaaBa aaleaacaWGQbaabeaakiaacMcacaGG9bGaeyypa0ZaaCbeaeaaciGG TbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZcaWGkbaabeaakmaacm aabaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWa aeaacaaIXaGaey4kaSYaaSaaaeaacaGGOaGaam4CamaaCaaaleqaba GaamyyamaaBaaameaacaWGPbGaamOAaaqabaaaaOGaeyOeI0IaaGym aiaacMcacaGGOaGaam4CamaaCaaaleqabaGaamiEamaaBaaameaaca WGQbaabeaaaaGccqGHsislcaaIXaGaaiykaaqaaiaadohacqGHsisl caaIXaaaaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabg2da9iaadk gadaWgaaWcbaGaamyAaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaiilaiabgcGiIiaadMgacqGHiiIZcaWGjb aaaa@9A64@
The above definition can be extended for s=0, s=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9iaaicdacaGGSaGaaeiiaiaadohacqGH9aqpcaaIXaaaaa@3CB9@ and s= MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9iabg6HiLcaa@3964@ by taking limits. So, it is easy to verify that T F 0 (x,y)=min{x,y},  T F 1 (x,y)=xy MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaaGimaaaakiaacIcacaWG4bGaaiilaiaadMha caGGPaGaeyypa0JaciyBaiaacMgacaGGUbGaai4EaiaadIhacaGGSa GaamyEaiaac2hacaGGSaGaaeiiaiaadsfadaqhaaWcbaGaamOraaqa aiaaigdaaaGccaGGOaGaamiEaiaacYcacaWG5bGaaiykaiabg2da9i aadIhacaWG5baaaa@4FFF@ and T F (x,y)=max{x+y1,0}, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaeyOhIukaaOGaaiikaiaadIhacaGGSaGaamyE aiaacMcacqGH9aqpciGGTbGaaiyyaiaacIhacaGG7bGaamiEaiabgU caRiaadMhacqGHsislcaaIXaGaaiilaiaaicdacaGG9bGaaiilaaaa @49BE@ that is, Frank t-norm is converted to minimum, product and Lukasiewicz t-norm, respectively. Frank family of t-norms plays a central role in the investigation of the contraposition law for QL-implications [8].

The theory of fuzzy relational equations was firstly proposed by Sanchez, [52]. He introduced a FRE whit max-min composition and applied the model to medical diagnosis in Brouwerian logic. Nowadays, it is well-known that many issues associated with a body knowledge can be treated as FRE problems [44]. In addition to such applications, FRE theory has been applied in many fields including fuzzy control, discrete dynamic systems, prediction of fuzzy systems, fuzzy decision making, fuzzy pattern recognition, fuzzy clustering, image compression and reconstruction, and so on. Pedrycz, [45] categorized and extended two ways of the generalizations of FRE in terms of sets under discussion and various operations which are taken into account. Since then, many theoretical improvements have been investigated and many applications have been presented [2,3,5,11,24,28,32,37,38, 41,43,46,48,49,57,59,65].

The solvability and the finding of solutions set are the primary (and the most fundamental) subject concerning FRE problems. Many studies have reported fuzzy relational equations with max-min and max-product compositions. Both compositions are special cases of the max-triangular-norm (max-t-norm). Di Nola, et al. proved that the solution set of FRE defined by continuous max-t-norm composition is often a non-convex set that is completely determined by one maximum solution and a finite number of minimal solutions [6]. Over the last decades, the solvability of FRE defined with different max-t compositions has been investigated by many researches [36,47,50,51,53,55,56,60, 64,68].

Optimizing an objective function subjected to a system of fuzzy relational equations or inequalities (FRI) is one of the most interesting and on-going topics among the problems related to the FRE (or FRI) theory [1,9,19-27,25-27,33,34,39,54,61,66]. By far the most frequently studied aspect is the determination of a minimize of a linear objective function and the use of the maxmin composition [1,20]. So, it is an almost standard approach to translate this type of problem into a corresponding 0-1 integer linear programming problem, which is then solved using a branch and bound method [10,62]. The topic of the linear optimization problem was also investigated with max-product operation [19,26,40]. Moreover, some studies have determined a more general operator of linear optimization with replacement of max-min and max-product compositions with a max-t-norm composition, max-average composition or max-star composition [25,34,54,31,61,22,27].

Recently, many interesting generalizations of the linear and non-linear programming problems constrained by FRE or FRI have been introduced and developed based on composite operations and fuzzy relations used in the definition of the constraints, and some developments on the objective function of the problems [4,7,12,20,35,39,58,63,66]. For instance, the linear optimization of bipolar FRE was studied by some researchers where FRE was defined with max-min composition and max- Lukasiewicz composition [12,35,39]. Ghodousian and khorram, [21] focused on the algebraic structure of two fuzzy relational Aφx b 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabeA 8aQjaadIhacqGHKjYOcaWGIbWaaWbaaSqabeaacaaIXaaaaaaa@3CF9@ and Dφx b 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabeA 8aQjaadIhacqGHLjYScaWGIbWaaWbaaSqabeaacaaIYaaaaaaa@3D0E@ , and studied a mixed fuzzy system formed by the two preceding FRIs, where ϕ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BD@ is an operator with (closed) convex solutions. Yang, [67] studied the optimal solution of minimizing a linear objective function subject to fuzzy relational inequalities where the constraints defined as a i1 x 1 + a i2 x 2 +...+ a in x n b i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaaGymaaqabaGccqGHNis2caWG4bWaaSbaaSqaaiaa igdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaWGPbGaaGOmaaqaba GccqGHNis2caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaaiOl aiaac6cacaGGUaGaey4kaSIaamyyamaaBaaaleaacaWGPbGaamOBaa qabaGccqGHNis2caWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyyzImRa amOyamaaBaaaleaacaWGPbaabeaaaaa@520D@ for i=1,...,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad2gaaaa@3D0C@ and ab=min{a,b}. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgE IizlaadkgacqGH9aqpciGGTbGaaiyAaiaac6gacaGG7bGaamyyaiaa cYcacaWGIbGaaiyFaiaac6caaaa@4277@ In [20], the authors introduced FRI-FC problem min{ c T x:Aφxb,x [0,1] n }, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacM gacaGGUbGaai4EaiaaykW7caaMc8Uaam4yamaaCaaaleqabaGaamiv aaaakiaadIhacaaMc8UaaGPaVlaaykW7caGG6aGaaGPaVlaaykW7ca aMc8UaaGPaVlaadgeacqaHgpGAcaWG4bWefv3ySLgznfgDOjdaryqr 1ngBPrginfgDObcv39gaiuaacqWF8jcScaWGIbGaaGPaVlaaykW7ca GGSaGaaGPaVlaaykW7caWG4bGaeyicI4Saai4waiaaicdacaGGSaGa aGymaiaac2fadaahaaWcbeqaaiaad6gaaaGccaaMc8UaaiyFaiaacY caaaa@6CE1@ where ϕ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BD@ is max-min composition and “ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWF8jcSaaa@416B@ ” denotes the relaxed or fuzzy version of the ordinary inequality “ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWF8jcSaaa@416B@ ”. Another interesting generalization of such optimization problems are related to objective function. Wu, et al. [63] represented an efficient method to optimize a linear fractional programming problem under FRE with max-Archimedean t-norm composition. Dempe and Ruziyeva, [4] generalized the fuzzy linear optimization problem by considering fuzzy coefficients. Dubey, et al. [7] studied linear programming problems involving interval uncertainty modeled using intuitionistic fuzzy set [7]. On the other hand, Lu and Fang considered the single non-linear objective function and solved it with FRE constraints and max-min operator [42]. They proposed a genetic algorithm for solving the problem. In [29], the authors used the same method for max-product operator. Also, Ghodousian, et al. [15,16,18] presented GA algorithms to solve the non-linear problem with FRE constraints defined by Lukasiewicz, Dubois –Prade and Sugeno-Weber operators.

Generally, there are three important difficulties related to FRE or FRI problems. Firstly, in order to completely determine FREs and FRIs, we must initially find all the minimal solutions, and the finding of all the minimal solutions is an NP-hard problem. Secondly, a feasible region formed as FRE or FRI is often a nonconvex set [21]. Finally, FREs and FRIs as feasible regions lead to optimization problems with highly non-linear constraints. Due to the above mentioned difficulties, although the analytical methods are efficient to find exact optimal solutions, they may also involve high computational complexity for high-dimensional problems (especially, if the simplification processes cannot considerably reduce the problem).

In this paper, we use the genetic algorithm proposed in [21] for solving problem (1), which keeps the search inside of the feasible region without finding any minimal solution and checking the feasibility of new generated solutions. Since the feasibility of problem (1) is essentially dependent on the t-norm (Frank t-norm) used in the definition of the constraints, a method is also presented to construct feasible test problems. More precisely, we construct a feasible problem by randomly generating a fuzzy matrix and a fuzzy vector according to some criteria resulted from the necessary and sufficient conditions. It is proved that the max-Frank fuzzy relational equations constructed by this method are not empty. Moreover, a comparison is made between the current method and the algorithms presented in [42] and [29].

The remainder of the paper is organized as follows. Section 2 takes a brief look at some basic results on the feasible solutions set of problem (1). In Section 3, the GA algorithm is briefly described. Finally, in Section 4 the experimental results are demonstrated and a conclusion is provided in Section 5.
Basic Properties of Max-Frank FRE
Characterization of feasible solutions set
This section describes the basic definitions and structural properties concerning problem (1) that are used throughout the paper. For the sake of simplicity, let S T F s ( a i , b i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaaaaa@3FF6@ denote the feasible solutions set of i‘th equation, that is, S T F s ( a i , b i )={ x [0,1] n : max j=1 n { T F s ( a ij , x j ) }= b i } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaeyypa0ZaaiWaaeaacaWG4bGaeyic I4Saai4waiaaicdacaGGSaGaaGymaiaac2fadaahaaWcbeqaaiaad6 gaaaGccaaMc8UaaGPaVlaacQdacaaMc8UaaGPaVlaaykW7daWfWaqa aiGac2gacaGGHbGaaiiEaaWcbaGaamOAaiabg2da9iaaigdaaeaaca WGUbaaaOGaaGPaVpaacmaabaGaamivamaaDaaaleaacaWGgbaabaGa am4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaki aacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaaiykaaGaay5Eaiaa w2haaiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaaakiaawUhaca GL9baaaaa@6ACF@ . Also, let S T F s (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaaaaa@3D8E@ denote the feasible solutions set of problem (1). Based on the foregoing notations, it is clear that S T F s (A,b)= iI S T F s ( a i , b i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyypa0ZaaqbuaeaacaWGtb WaaSbaaSqaaiaadsfadaqhaaadbaGaamOraaqaaiaadohaaaaaleqa aOGaaiikaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamOyam aaBaaaleaacaWGPbaabeaakiaacMcaaSqaaiaadMgacqGHiiIZcaWG jbaabeqdcqWIPissaaaa@4D7D@ .

Definition 1: For each iI, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGSaaaaa@39E5@ we define J i ={ jJ: a ij b i }. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGPbaabeaakiabg2da9maacmaabaGaamOAaiabgIGiolaa dQeacaaMc8UaaGPaVlaacQdacaaMc8UaaGPaVlaaykW7caWGHbWaaS baaSqaaiaadMgacaWGQbaabeaakiabgwMiZkaadkgadaWgaaWcbaGa amyAaaqabaaakiaawUhacaGL9baacaGGUaaaaa@4E52@

According to definition 1, we have the following lemmas, which are easily proved by the monotonicity and identity law of t-norms, definition 1 and the definition of Frank t-norm.

Lemma 1: Let iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGUaaaaa@39E7@ If j J i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgM GiplaadQeadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@3B0D@ then T F s ( a ij , x j )< b i ,  x j [0,1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO GaaiykaiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGa aeiiaiabgcGiIiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHiiIZca GGBbGaaGimaiaacYcacaaIXaGaaiyxaiaac6caaaa@4D5B@

Lemma 2: Let iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeaaaa@3935@ and j J i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolaadQeadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@3B0D@

(a) If x j > log s ( 1+ ( s b i 1)(s1) s a ij 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbaabeaakiabg6da+iGacYgacaGGVbGaai4zamaaBaaa leaacaWGZbaabeaakmaabmaabaGaaGymaiabgUcaRmaalaaabaGaai ikaiaadohadaahaaWcbeqaaiaadkgadaWgaaadbaGaamyAaaqabaaa aOGaeyOeI0IaaGymaiaacMcacaGGOaGaam4CaiabgkHiTiaaigdaca GGPaaabaGaam4CamaaCaaaleqabaGaamyyamaaBaaameaacaWGPbGa amOAaaqabaaaaOGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaaa@5045@ and b i 0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaakiabgcMi5kaaicdacaGGSaaaaa@3B31@ then T F s ( a ij , x j )> b i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaiabg6da+iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGUaaa aa@43B1@

(b) If x j = log s ( 1+ ( s b i 1)(s1) s a ij 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbaabeaakiabg2da9iGacYgacaGGVbGaai4zamaaBaaa leaacaWGZbaabeaakmaabmaabaGaaGymaiabgUcaRmaalaaabaGaai ikaiaadohadaahaaWcbeqaaiaadkgadaWgaaadbaGaamyAaaqabaaa aOGaeyOeI0IaaGymaiaacMcacaGGOaGaam4CaiabgkHiTiaaigdaca GGPaaabaGaam4CamaaCaaaleqabaGaamyyamaaBaaameaacaWGPbGa amOAaaqabaaaaOGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaaa@5043@ and b i 0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaakiabgcMi5kaaicdacaGGSaaaaa@3B31@ then T F s ( a ij , x j )= b i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGUaaa aa@43AF@

(c) If x j < log s ( 1+ ( s b i 1)(s1) s a ij 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbaabeaakiabgYda8iGacYgacaGGVbGaai4zamaaBaaa leaacaWGZbaabeaakmaabmaabaGaaGymaiabgUcaRmaalaaabaGaai ikaiaadohadaahaaWcbeqaaiaadkgadaWgaaadbaGaamyAaaqabaaa aOGaeyOeI0IaaGymaiaacMcacaGGOaGaam4CaiabgkHiTiaaigdaca GGPaaabaGaam4CamaaCaaaleqabaGaamyyamaaBaaameaacaWGPbGa amOAaaqabaaaaOGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaaa@5041@ and b i 0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaakiabgcMi5kaaicdacaGGSaaaaa@3B31@ then T F s ( a ij , x j )< b i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO GaaiykaiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGUaaa aa@43AD@

(d) If a ij = b i =0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGIbWaaSbaaSqaaiaa dMgaaeqaaOGaeyypa0JaaGimaiaacYcaaaa@3E6F@ then T F s ( a ij , x j )= b i , x j [0,1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGa aGPaVlabgcGiIiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHiiIZca GGBbGaaGimaiaacYcacaaIXaGaaiyxaiaac6caaaa@4E45@

(e) If a ij > b i =0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH+aGpcaWGIbWaaSbaaSqaaiaa dMgaaeqaaOGaeyypa0JaaGimaiaacYcaaaa@3E71@ then T F s ( a ij , x j )= b i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaaaaa@42F3@ for x j =0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbaabeaakiabg2da9iaaicdacaGGSaaaaa@3A87@ and T F s ( a ij , x j )> b i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaiabg6da+iaadkgadaWgaaWcbaGaamyAaaqabaaaaa@42F5@ for 0< x j 1. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iaadIhadaWgaaWcbaGaamOAaaqabaGccqGHKjYOcaaIXaGaaiOl aaaa@3CF7@

Lemma 3 below gives a necessary and sufficient condition for the feasibility of sets S T F s ( a i , b i ), iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaaiilaiaabccacqGHaiIicaWGPbGa eyicI4Saamysaiaac6caaaa@460B@

Lemma 3: For a fixed iI,  S T F s ( a i , b i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGSaGaaeiiaiaadofadaWgaaWcbaGaamivamaaDaaa meaacaWGgbaabaGaam4CaaaaaSqabaGccaGGOaGaamyyamaaBaaale aacaWGPbaabeaakiaacYcacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGa aiykaiabgcMi5kabgwGigdaa@47C9@ if and only if J i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGPbaabeaakiabgcMi5kabgwGiglaac6caaaa@3BDA@

Proof: The proof is similar to the proof of Lemma 3 in [15].

Definition 2: Suppose that iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeaaaa@3935@ and S T F s ( a i , b i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaeyiyIKRaeyybIymaaa@4336@ (hence, J i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGPbaabeaakiabgcMi5kabgwGigdaa@3B28@ from lemma 3). Let x ^ i =[ ( x ^ i ) 1 , ( x ^ i ) 2 ,..., ( x ^ i ) n ] [0,1] n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaai4waiaacIcaceWG4bGb aKaadaWgaaWcbaGaamyAaaqabaGccaGGPaWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaacIcaceWG4bGbaKaadaWgaaWcbaGaamyAaaqabaGc caGGPaWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaai OlaiaacYcacaGGOaGabmiEayaajaWaaSbaaSqaaiaadMgaaeqaaOGa aiykamaaBaaaleaacaWGUbaabeaakiaac2facqGHiiIZcaGGBbGaaG imaiaacYcacaaIXaGaaiyxamaaCaaaleqabaGaamOBaaaaaaa@5345@ where the components are defined as follows:
( x ^ i ) k ={ log s ( 1+ ( s b i 1)(s1) / ( s a ik 1) ) k J i , b i 0 0 k J i , a ik > b i =0 1 otherwise , kJ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadI hagaqcamaaBaaaleaacaWGPbaabeaakiaacMcadaWgaaWcbaGaam4A aaqabaGccqGH9aqpdaGabaqaauaabeqadiaaaeaaciGGSbGaai4Bai aacEgadaWgaaWcbaGaam4CaaqabaGcdaqadaqaaiaaigdacqGHRaWk daWcgaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWGIbWaaSbaaWqaai aadMgaaeqaaaaakiabgkHiTiaaigdacaGGPaGaaiikaiaadohacqGH sislcaaIXaGaaiykaaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWGHb WaaSbaaWqaaiaadMgacaWGRbaabeaaaaGccqGHsislcaaIXaGaaiyk aaaaaiaawIcacaGLPaaaaeaacaWGRbGaeyicI4SaamOsamaaBaaale aacaWGPbaabeaakiaaykW7caGGSaGaaGPaVlaaykW7caWGIbWaaSba aSqaaiaadMgaaeqaaOGaeyiyIKRaaGimaaqaaiaaicdaaeaacaWGRb GaeyicI4SaamOsamaaBaaaleaacaWGPbaabeaakiaaykW7caGGSaGa aGPaVlaaykW7caWGHbWaaSbaaSqaaiaadMgacaWGRbaabeaakiabg6 da+iaadkgadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIWaaabaGa aGymaaqaaiaad+gacaWG0bGaamiAaiaadwgacaWGYbGaam4DaiaadM gacaWGZbGaamyzaaaaaiaawUhaaiaacYcacaqGGaGaeyiaIiIaam4A aiabgIGiolaadQeaaaa@8489@
Also, for each j J i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolaadQeadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@3B0B@ we define x i (j)=[ x i (j) 1 , x i (j) 2 ,..., x i (j) n ] [0,1] n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadQgacaGGPaGaeyypa0Ja ai4waiqadIhagaafamaaBaaaleaacaWGPbaabeaakiaacIcacaWGQb GaaiykamaaBaaaleaacaaIXaaabeaakiaacYcaceWG4bGbaqbadaWg aaWcbaGaamyAaaqabaGccaGGOaGaamOAaiaacMcadaWgaaWcbaGaaG OmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiqadIhagaaf amaaBaaaleaacaWGPbaabeaakiaacIcacaWGQbGaaiykamaaBaaale aacaWGUbaabeaakiaac2facqGHiiIZcaGGBbGaaGimaiaacYcacaaI XaGaaiyxamaaCaaaleqabaGaamOBaaaaaaa@5886@ such that
x i (j) k ={ log s ( 1+ ( s b i 1)(s1) / ( s a ik 1) ) b i 0andk=j 0 otherwise , kJ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadQgacaGGPaWaaSbaaSqa aiaadUgaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGacaaabaGaciiBai aac+gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGa ey4kaSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBa aameaacaWGPbaabeaaaaGccqGHsislcaaIXaGaaiykaiaacIcacaWG ZbGaeyOeI0IaaGymaiaacMcaaeaacaGGOaGaam4CamaaCaaaleqaba GaamyyamaaBaaameaacaWGPbGaam4AaaqabaaaaOGaeyOeI0IaaGym aiaacMcaaaaacaGLOaGaayzkaaaabaGaamOyamaaBaaaleaacaWGPb aabeaakiabgcMi5kaaicdacaaMc8UaaGPaVlaadggacaWGUbGaamiz aiaaykW7caaMc8Uaam4Aaiabg2da9iaadQgaaeaacaaIWaaabaGaam 4BaiaadshacaWGObGaamyzaiaadkhacaWG3bGaamyAaiaadohacaWG LbaaaaGaay5EaaGaaiilaiaabccacqGHaiIicaWGRbGaeyicI4Saam Osaaaa@755B@
The following theorem characterizes the feasible region of the i‘th relational equation (iI). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadM gacqGHiiIZcaWGjbGaaiykaiaac6caaaa@3B40@

Theorem 1: Let iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGUaaaaa@39E7@ If S T F s ( a i , b i ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaeyiyIKRaeyybIySaaiilaaaa@43E6@ then S T F s ( a i , b i )= j J i [ x i (j), x ^ i ] . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaeyypa0ZaambuaeaacaGGBbGaaGPa VlqadIhagaafamaaBaaaleaacaWGPbaabeaakiaacIcacaWGQbGaai ykaiaaykW7caGGSaGaaGPaVlqadIhagaqcamaaBaaaleaacaWGPbaa beaakiaac2faaSqaaiaadQgacqGHiiIZcaWGkbWaaSbaaWqaaiaadM gaaeqaaaWcbeqdcqWIQisvaOGaaiOlaaaa@559A@

Proof: For a more general case, see Corollary 2.3 in [21].

From theorem 1, x ^ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja WaaSbaaSqaaiaadMgaaeqaaaaa@381C@ is the unique maximum solution and x i (j) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadQgacaGGPaaaaa@3A79@ ‘s (j J i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadQ gacqGHiiIZcaWGkbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3BB4@ are the minimal solutions of S T F s ( a i , b i ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaaiOlaaaa@40A8@

Definition 3: Let x ^ i  (iI) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja WaaSbaaSqaaiaadMgaaeqaaOGaaeiiaiaacIcacaWGPbGaeyicI4Sa amysaiaacMcaaaa@3D62@ be the maximum solution of S T F s ( a i , b i ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaamyAaaqabaGccaGGPaGaaiOlaaaa@40A8@ We define X ¯ = min iI { x ^ i }. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabg2da9maaxababaGaciyBaiaacMgacaGGUbaaleaacaWG PbGaeyicI4SaamysaaqabaGccaGG7bGaaGPaVlqadIhagaqcamaaBa aaleaacaWGPbaabeaakiaac2hacaGGUaaaaa@44AC@

Definition 4: Let e:I J i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiaacQ dacaWGjbGaeyOKH4QaamOsamaaBaaaleaacaWGPbaabeaaaaa@3C41@ so that e(i)=j J i , iI, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiaacI cacaWGPbGaaiykaiabg2da9iaadQgacqGHiiIZcaWGkbWaaSbaaSqa aiaadMgaaeqaaOGaaiilaiaabccacqGHaiIicaWGPbGaeyicI4Saam ysaiaacYcaaaa@44A5@ and let E MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36BF@ be the set of all vectors e MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36DF@ . For the sake of convenience, we represent each eE MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabgI Giolaadweaaaa@392D@ as an m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E7@ –dimensional vector e=[ j 1 , j 2 ,..., j m ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2 da9iaacUfacaWGQbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadQga daWgaaWcbaGaaGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaai ilaiaadQgadaWgaaWcbaGaamyBaaqabaGccaGGDbaaaa@43A3@ in which j k =e(k). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaWGRbaabeaakiabg2da9iaadwgacaGGOaGaam4AaiaacMca caGGUaaaaa@3CF5@

Definition 5: Let e=[ j 1 , j 2 ,..., j m ]E. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2 da9iaacUfacaWGQbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadQga daWgaaWcbaGaaGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaai ilaiaadQgadaWgaaWcbaGaamyBaaqabaGccaGGDbGaeyicI4Saamyr aiaac6caaaa@46A3@ We define X _ (e)=[ X _ (e) 1 , X _ (e) 2 ,..., X _ (e) n ] [0,1] n , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaiabg2da9iaacUfadaadaaqaaiaa dIfaaaGaaiikaiaadwgacaGGPaWaaSbaaSqaaiaaigdaaeqaaOGaai ilamaamaaabaGaamiwaaaacaGGOaGaamyzaiaacMcadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilamaamaaaba GaamiwaaaacaGGOaGaamyzaiaacMcadaWgaaWcbaGaamOBaaqabaGc caGGDbGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaac2fadaahaa Wcbeqaaiaad6gaaaGccaGGSaaaaa@53F0@ where X _ (e) j = max iI { x i (e(i)) j }= max iI { x i ( j i ) j }, jJ. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykamaaBaaaleaacaWGQbaabeaakiab g2da9maaxababaGaciyBaiaacggacaGG4baaleaacaWGPbGaeyicI4 SaamysaaqabaGcdaGadaqaaiqadIhagaafamaaBaaaleaacaWGPbaa beaakiaacIcacaWGLbGaaiikaiaadMgacaGGPaGaaiykamaaBaaale aacaWGQbaabeaaaOGaay5Eaiaaw2haaiabg2da9maaxababaGaciyB aiaacggacaGG4baaleaacaWGPbGaeyicI4SaamysaaqabaGcdaGada qaaiqadIhagaafamaaBaaaleaacaWGPbaabeaakiaacIcacaWGQbWa aSbaaSqaaiaadMgaaeqaaOGaaiykamaaBaaaleaacaWGQbaabeaaaO Gaay5Eaiaaw2haaiaacYcacaqGGaGaeyiaIiIaamOAaiabgIGiolaa dQeacaGGUaaaaa@6235@

From the relation S T F s (A,b)= iI S T F s ( a i , b i ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyypa0ZaaqbuaeaacaWGtb WaaSbaaSqaaiaadsfadaqhaaadbaGaamOraaqaaiaadohaaaaaleqa aOGaaiikaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamOyam aaBaaaleaacaWGPbaabeaakiaacMcaaSqaaiaadMgacqGHiiIZcaWG jbaabeqdcqWIPissaaaa@4D7D@ and Theorem 1, the following theorem is easily attained.

Theorem 2: S T F s (A,b)= eE [ X _ (e), X ¯ ] . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyypa0ZaambuaeaacaGGBb GaaGPaVpaamaaabaGaamiwaaaacaGGOaGaamyzaiaacMcacaaMc8Ua aiilaiaaykW7daqdaaqaaiaadIfaaaGaaiyxaaWcbaGaamyzaiabgI GiolaadweaaeqaniablQIivbGccaGGUaaaaa@4F6B@

As a consequence, it turns out that X ¯ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaaaa@36E3@ is the unique maximum solution and X _ (e) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaaaa@3925@ 's (eE) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw gacqGHiiIZcaWGfbGaaiykaaaa@3A86@ are the minimal solutions of S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaaiOlaaaa@3E40@ Moreover, we have the following corollary that is directly resulted from theorem 2.

Corollary 1(first necessary and sufficient condition): S T F s (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyiyIKRaeyybIymaaa@40CE@ if and only if X ¯ S T F s (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabgIGiolaadofadaWgaaWcbaGaamivamaaDaaameaacaWG gbaabaGaam4CaaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaai ykaaaa@4000@ .

Example 1: Consider the problem below with Frank t-norm
[ 0.9  0.4   0.3  0.7  0.4  0.6 0.5  0.1   0.4  0.3  0.5  0.2 0.9  0.3   0.4  0.7  0.7  0.3 0.6  0.7   0.3  0.8  0.8  0.5 0.0  0.01 0.0  0.2  0.0  0.9 ]φx=[ 0.7 0.5 0.7 0.8 0.0 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaqaabe qaaiaabcdacaqGUaGaaeyoaiaabccacaqGGaGaaeimaiaab6cacaqG 0aGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaeiiaiaabc cacaqGWaGaaeOlaiaabEdacaqGGaGaaeiiaiaabcdacaqGUaGaaein aiaabccacaqGGaGaaeimaiaab6cacaqG2aaabaGaaeimaiaab6caca qG1aGaaeiiaiaabccacaqGWaGaaeOlaiaabgdacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabsdacaqGGaGaaeiiaiaabcdacaqGUaGaae 4maiaabccacaqGGaGaaeimaiaab6cacaqG1aGaaeiiaiaabccacaqG WaGaaeOlaiaabkdaaeaacaqGWaGaaeOlaiaabMdacaqGGaGaaeiiai aabcdacaqGUaGaae4maiaabccacaqGGaGaaeiiaiaabcdacaqGUaGa aeinaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaaeiiaiaabccaca qGWaGaaeOlaiaabEdacaqGGaGaaeiiaiaabcdacaqGUaGaae4maaqa aiaabcdacaqGUaGaaeOnaiaabccacaqGGaGaaeimaiaab6cacaqG3a GaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaeiiaiaabcca caqGWaGaaeOlaiaabIdacaqGGaGaaeiiaiaabcdacaqGUaGaaeioai aabccacaqGGaGaaeimaiaab6cacaqG1aaabaGaaeimaiaab6cacaqG WaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqGXaGaaeiiaiaabc dacaqGUaGaaeimaiaabccacaqGGaGaaeimaiaab6cacaqGYaGaaeii aiaabccacaqGWaGaaeOlaiaabcdacaqGGaGaaeiiaiaabcdacaqGUa GaaeyoaaaacaGLBbGaayzxaaGaeqOXdOMaamiEaiabg2da9maadmaa baqbaeqabuqaaaaabaGaaGimaiaac6cacaaI3aaabaGaaGimaiaac6 cacaaI1aaabaGaaGimaiaac6cacaaI3aaabaGaaGimaiaac6cacaaI 4aaabaGaaGimaiaac6cacaaIWaaaaaGaay5waiaaw2faaaaa@AA3A@
where φ(x,y)= T F 2 (x,y)= log 2 ( 1+( s x 1)( s y 1) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaaG PaVlaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0Jaamivamaa DaaaleaacaWGgbaabaGaaGOmaaaakiaacIcacaWG4bGaaiilaiaadM hacaGGPaGaeyypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaaikda aeqaaOWaaeWaaeaacaaIXaGaey4kaSIaaiikaiaadohadaahaaWcbe qaaiaadIhaaaGccqGHsislcaaIXaGaaiykaiaacIcacaWGZbWaaWba aSqabeaacaWG5baaaOGaeyOeI0IaaGymaiaacMcaaiaawIcacaGLPa aaaaa@572B@ (i.e., s=2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9iaaikdaaaa@38AF@ ). By definition 1, we have J 1 ={ 1,4 },  J 2 ={ 1,5 },  J 3 ={ 1,4,5 },  J 4 ={ 4,5 } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaaIXaaabeaakiabg2da9maacmaabaGaaGymaiaaykW7caGG SaGaaGPaVlaaisdaaiaawUhacaGL9baacaGGSaGaaeiiaiaadQeada WgaaWcbaGaaGOmaaqabaGccqGH9aqpdaGadaqaaiaaigdacaaMc8Ua aiilaiaaykW7caaI1aaacaGL7bGaayzFaaGaaiilaiaabccacaWGkb WaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaiWaaeaacaaIXaGaaGPa VlaacYcacaaMc8UaaGinaiaacYcacaaI1aaacaGL7bGaayzFaaGaai ilaiaabccacaWGkbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaiWa aeaacaaI0aGaaGPaVlaacYcacaaMc8UaaGynaaGaay5Eaiaaw2haaa aa@6440@ and J 5 ={ 1,2,3,4,5,6 }. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaaI1aaabeaakiabg2da9maacmaabaGaaGymaiaaykW7caGG SaGaaGPaVlaaikdacaaMc8UaaiilaiaaykW7caaIZaGaaGPaVlaacY cacaaMc8UaaGinaiaaykW7caGGSaGaaGPaVlaaiwdacaaMc8Uaaiil aiaaykW7caaI2aaacaGL7bGaayzFaaGaaiOlaaaa@52F1@ The unique maximum solution and the minimal solutions of each equation are obtained by definition 2 as follows, where 0.783316030456544 has been rounded to 0.7833:

x ^ 1 = x ^ 3 =[0.7833 , 1 , 1 , 1 , 1 , 1],  x ^ 2 = x ^ 4 =[1 , 1 , 1 , 1 , 1 , 1], x ^ 5 =[1 , 0 , 1 , 0 , 1 , 1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JabmiEayaajaWaaSbaaSqa aiaaiodaaeqaaOGaeyypa0Jaai4waiaabcdacaqGUaGaae4naiaabI dacaqGZaGaae4maiaabccacaqGSaGaaeiiaiaabgdacaqGGaGaaeil aiaabccacaqGXaGaaeiiaiaabYcacaqGGaGaaeymaiaabccacaqGSa GaaeiiaiaabgdacaqGGaGaaeilaiaabccacaqGXaGaaiyxaiaacYca caqGGaGabmiEayaajaWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jabm iEayaajaWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0Jaai4waiaabgda caqGGaGaaeilaiaabccacaqGXaGaaeiiaiaabYcacaqGGaGaaeymai aabccacaqGSaGaaeiiaiaabgdacaqGGaGaaeilaiaabccacaqGXaGa aeiiaiaabYcacaqGGaGaaeymaiaac2facaGGSaGaaGPaVlqadIhaga qcamaaBaaaleaacaaI1aaabeaakiabg2da9iaacUfacaqGXaGaaeii aiaabYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabgdacaqGGa GaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGaaeymaiaabcca caqGSaGaaeiiaiaabgdacaGGDbGaaiOlaaaa@7C25@

x 1 (1)=[0.7833 , 0 , 0 , 0 , 0 , 0], x 1 (4)=[0 , 0 , 0 , 1 , 0 , 0] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaaigdacaGGPaGaeyypa0Ja ai4waiaabcdacaqGUaGaae4naiaabIdacaqGZaGaae4maiaabccaca qGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaa bYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaae ilaiaabccacaqGWaGaaiyxaiaacYcacaaMc8UabmiEayaauaWaaSba aSqaaiaaigdaaeqaaOGaaiikaiaaisdacaGGPaGaeyypa0Jaai4wai aabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGa aeimaiaabccacaqGSaGaaeiiaiaabgdacaqGGaGaaeilaiaabccaca qGWaGaaeiiaiaabYcacaqGGaGaaeimaiaac2faaaa@6582@

x 2 (1)=[1 , 0 , 0 , 0 , 0 , 0],  x 2 (5)=[0 , 0 , 0 , 0 , 1 , 0] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacaGGPaGaeyypa0Ja ai4waiaabgdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcaca qGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaa bccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaac2facaGGSaGaae iiaiqadIhagaafamaaBaaaleaacaaIYaaabeaakiaacIcacaaI1aGa aiykaiabg2da9iaacUfacaaIWaGaaeiiaiaabYcacaqGGaGaaeimai aabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGa aeiiaiaabYcacaqGGaGaaeymaiaabccacaqGSaGaaeiiaiaabcdaca GGDbaaaa@6113@

x 3 (1)=[0.7833 , 0 , 0 , 0 , 0 , 0],  x 3 (4)=[0 , 0 , 0 , 1 , 0 , 0],  x 3 (5)=[0 , 0 , 0 , 0 , 1 , 0] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaiodaaeqaaOGaaiikaiaaigdacaGGPaGaeyypa0Ja ai4waiaabcdacaqGUaGaae4naiaabIdacaqGZaGaae4maiaabccaca qGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaa bYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaae ilaiaabccacaqGWaGaaiyxaiaacYcacaqGGaGabmiEayaauaWaaSba aSqaaiaaiodaaeqaaOGaaiikaiaaisdacaGGPaGaeyypa0Jaai4wai aabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGa aeimaiaabccacaqGSaGaaeiiaiaabgdacaqGGaGaaeilaiaabccaca qGWaGaaeiiaiaabYcacaqGGaGaaeimaiaac2facaGGSaGaaeiiaiqa dIhagaafamaaBaaaleaacaaIZaaabeaakiaacIcacaaI1aGaaiykai abg2da9iaacUfacaaIWaGaaeiiaiaabYcacaqGGaGaaeimaiaabcca caqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiai aabYcacaqGGaGaaeymaiaabccacaqGSaGaaeiiaiaabcdacaGGDbaa aa@7ADD@

x 4 (4)=[0 , 0 , 0 , 1 , 0 , 0],  x 4 (5)=[0 , 0 , 0 , 0 , 1 , 0] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaisdaaeqaaOGaaiikaiaaisdacaGGPaGaeyypa0Ja ai4waiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcaca qGGaGaaeimaiaabccacaqGSaGaaeiiaiaabgdacaqGGaGaaeilaiaa bccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaac2facaGGSaGaae iiaiqadIhagaafamaaBaaaleaacaaI0aaabeaakiaacIcacaaI1aGa aiykaiabg2da9iaacUfacaqGWaGaaeiiaiaabYcacaqGGaGaaeimai aabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGa aeiiaiaabYcacaqGGaGaaeymaiaabccacaqGSaGaaeiiaiaabcdaca GGDbaaaa@6113@

x 5 (j)=[0 , 0 , 0 , 0 , 0 , 0],j{ 1,2,3,4,5,6 } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaiwdaaeqaaOGaaiikaiaadQgacaGGPaGaeyypa0Ja ai4waiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcaca qGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaa bccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaac2facaaMc8UaaG PaVlaacYcacaaMc8UaaGPaVlaadQgacqGHiiIZdaGadaqaaiaaigda caGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaaisdacaGGSaGaaGynai aacYcacaaI2aaacaGL7bGaayzFaaaaaa@5E6C@

Therefore, by theorem 1 we have

S T F s ( a 1 , b 1 )=[ x 1 (1), x ^ 1 ][ x 1 (4), x ^ 1 ], MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaaGymaaqabaGccaGGPaGaeyypa0Jaai4waiaaykW7ceWG4bGb aqbadaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGymaiaacMcacaaMc8 UaaiilaiaaykW7ceWG4bGbaKaadaWgaaWcbaGaaGymaaqabaGccaGG DbGaeSOkIuLaai4waiaaykW7ceWG4bGbaqbadaWgaaWcbaGaaGymaa qabaGccaGGOaGaaGinaiaacMcacaaMc8UaaiilaiaaykW7ceWG4bGb aKaadaWgaaWcbaGaaGymaaqabaGccaGGDbGaaiilaaaa@5CD3@

S T F s ( a 2 , b 2 )=[ x 2 (1), x ^ 2 ][ x 2 (5), x ^ 2 ], MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaaGOmaaqabaGccaGGPaGaeyypa0Jaai4waiaaykW7ceWG4bGb aqbadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaaGymaiaacMcacaaMc8 UaaiilaiaaykW7ceWG4bGbaKaadaWgaaWcbaGaaGOmaaqabaGccaGG DbGaeSOkIuLaai4waiaaykW7ceWG4bGbaqbadaWgaaWcbaGaaGOmaa qabaGccaGGOaGaaGynaiaacMcacaaMc8UaaiilaiaaykW7ceWG4bGb aKaadaWgaaWcbaGaaGOmaaqabaGccaGGDbGaaiilaaaa@5CDA@

S T F s ( a 3 , b 3 )=[ x 3 (1), x ^ 3 ][ x 3 (4), x ^ 3 ][ x 3 (5), x ^ 3 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaaG4maaqabaGccaGGPaGaeyypa0Jaai4waiaaykW7ceWG4bGb aqbadaWgaaWcbaGaaG4maaqabaGccaGGOaGaaGymaiaacMcacaaMc8 UaaiilaiaaykW7ceWG4bGbaKaadaWgaaWcbaGaaG4maaqabaGccaGG DbGaeSOkIuLaai4waiaaykW7ceWG4bGbaqbadaWgaaWcbaGaaG4maa qabaGccaGGOaGaaGinaiaacMcacaaMc8UaaiilaiaaykW7ceWG4bGb aKaadaWgaaWcbaGaaG4maaqabaGccaGGDbGaeSOkIuLaai4waiaayk W7ceWG4bGbaqbadaWgaaWcbaGaaG4maaqabaGccaGGOaGaaGynaiaa cMcacaaMc8UaaiilaiaaykW7ceWG4bGbaKaadaWgaaWcbaGaaG4maa qabaGccaGGDbaaaa@6A95@ and

S T F s ( a 4 , b 4 )=[ x 4 (4), x ^ 4 ][ x 4 (5), x ^ 4 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaaisdaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaaGinaaqabaGccaGGPaGaeyypa0Jaai4waiaaykW7ceWG4bGb aqbadaWgaaWcbaGaaGinaaqabaGccaGGOaGaaGinaiaacMcacaaMc8 UaaiilaiaaykW7ceWG4bGbaKaadaWgaaWcbaGaaGinaaqabaGccaGG DbGaeSOkIuLaai4waiaaykW7ceWG4bGbaqbadaWgaaWcbaGaaGinaa qabaGccaGGOaGaaGynaiaacMcacaaMc8UaaiilaiaaykW7ceWG4bGb aKaadaWgaaWcbaGaaGinaaqabaGccaGGDbaaaa@5C39@ and

S T F s ( a 5 , b 5 )=[ 0 1×6 , x ^ 5 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGHbWaaSbaaSqaaiaaiwdaaeqaaOGaaiilaiaadkgadaWgaa WcbaGaaGynaaqabaGccaGGPaGaeyypa0Jaai4waiaaykW7ieqacaWF WaWaaSbaaSqaaiaaigdacqGHxdaTcaaI2aaabeaakiaaykW7caGGSa GaaGPaVlqadIhagaqcamaaBaaaleaacaaI1aaabeaakiaac2faaaa@4E34@ where 0 1×6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeGaa8hmam aaBaaaleaacaaIXaGaey41aqRaaGOnaaqabaaaaa@3A6E@ is a zero vector. From definition 3, X ¯ =[0.7833 , 0 , 1 , 0 , 1 , 1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabg2da9iaacUfacaqGWaGaaeOlaiaabEdacaqG4aGaae4m aiaabodacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGa GaaeymaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabcca caqGXaGaaeiiaiaabYcacaqGGaGaaeymaiaac2facaGGUaaaaa@4BEB@ It is easy to verify that X ¯ S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabgIGiolaadofadaWgaaWcbaGaamivamaaDaaameaacaWG gbaabaGaam4CaaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaai ykaiaac6caaaa@40B2@ Therefore, the above problem is feasible by corollary 1. Finally, the cardinality of set E MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36BF@ is equal to 24 (definition 4). So, we have 24 solutions X _ (e) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaaaa@3925@ associated to 24 vectors e MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36DF@ . For example, for e=[1,5,1,5,6], MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2 da9iaacUfacaaIXaGaaGPaVlaacYcacaaMc8UaaGynaiaaykW7caGG SaGaaGPaVlaaigdacaaMc8UaaiilaiaaykW7caaI1aGaaGPaVlaacY cacaaMc8UaaGOnaiaac2facaGGSaaaaa@4D21@ we obtain X _ (e)=max{ x 1 (1), x 2 (5), x 3 (1), x 4 (5), x 5 (6) } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaiabg2da9iGac2gacaGGHbGaaiiE aiaaykW7daGadaqaaiqadIhagaafamaaBaaaleaacaaIXaaabeaaki aacIcacaaIXaGaaiykaiaacYcacaaMc8UabmiEayaauaWaaSbaaSqa aiaaikdaaeqaaOGaaiikaiaaiwdacaGGPaGaaGPaVlaacYcacaaMc8 UabmiEayaauaWaaSbaaSqaaiaaiodaaeqaaOGaaiikaiaaigdacaGG PaGaaGPaVlaacYcacaaMc8UabmiEayaauaWaaSbaaSqaaiaaisdaae qaaOGaaiikaiaaiwdacaGGPaGaaiilaiqadIhagaafamaaBaaaleaa caaI1aaabeaakiaacIcacaaI2aGaaiykaaGaay5Eaiaaw2haaaaa@5FDA@ from definition 5 that means X _ (e)=[0.7833 , 0 , 0 , 0 , 1 , 0]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaiabg2da9iaacUfacaqGWaGaaeOl aiaabEdacaqG4aGaae4maiaabodacaqGGaGaaeilaiaabccacaqGWa GaaeiiaiaabYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcda caqGGaGaaeilaiaabccacaqGXaGaaeiiaiaabYcacaqGGaGaaeimai aac2facaGGUaaaaa@4E2B@
Simplification Processes
In practice, there are often some components of matrix A MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BB@ that have no effect on the solutions to problem (1). Therefore, we can simplify the problem by changing the values of these components to zeros. For this reason, various simplification processes have been proposed by researchers. We refer the interesting reader to [21] where a brief review of such these processes is given. Here, we present two simplification techniques based on the Frank t-norm.

Definition 6: If a value changing in an element, say a ij MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E4@ , of a given fuzzy relation matrix A MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BB@ has no effect on the solutions of problem (1), this value changing is said to be an equivalence operation.

Corollary 2: Suppose that T F s ( a i j 0 , x j 0 )< b i , x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaacYcacaWG4b WaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiyk aiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaeiiai abgcGiIiaadIhacqGHiiIZcaWGtbWaaSbaaSqaaiaadsfadaqhaaad baGaamOraaqaaiaadohaaaaaleqaaOGaaiikaiaadgeacaGGSaGaam OyaiaacMcacaGGUaaaaa@51CE@ In this case, it is obvious that max j=1 n { T F s ( a ij , x j ) }= b i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbmaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaamOB aaaakiaaykW7daGadaqaaiaadsfadaqhaaWcbaGaamOraaqaaiaado haaaGccaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGG SaGaamiEamaaBaaaleaacaWGQbaabeaakiaacMcaaiaawUhacaGL9b aacqGH9aqpcaWGIbWaaSbaaSqaaiaadMgaaeqaaaaa@4D88@ is equivalent to max j=1 j j 0 n { T F s ( a ij , x j ) }= b i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbmaeaaci GGTbGaaiyyaiaacIhaaSabaeqabaGaamOAaiabg2da9iaaigdaaeaa caWGQbGaeyiyIKRaamOAamaaBaaameaacaaIWaaabeaaaaWcbaGaam OBaaaakiaaykW7daGadaqaaiaadsfadaqhaaWcbaGaamOraaqaaiaa dohaaaGccaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGcca GGSaGaamiEamaaBaaaleaacaWGQbaabeaakiaacMcaaiaawUhacaGL 9baacqGH9aqpcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@52E0@ that is, “resetting a i j 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAamaaBaaameaacaaIWaaabeaaaSqabaaaaa@39D6@ to zero” has no effect on the solutions of problem (1) (since component a i j 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAamaaBaaameaacaaIWaaabeaaaSqabaaaaa@39D6@ only appears in the i‘th constraint of problem (1)). Therefore, if T F s ( a i j 0 , x j 0 )< b i ,x S T F s (A,b), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaacYcacaWG4b WaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiyk aiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaGPaVl abgcGiIiaadIhacqGHiiIZcaWGtbWaaSbaaSqaaiaadsfadaqhaaad baGaamOraaqaaiaadohaaaaaleqaaOGaaiikaiaadgeacaGGSaGaam OyaiaacMcacaGGSaaaaa@52B4@ then “resetting a i j 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAamaaBaaameaacaaIWaaabeaaaSqabaaaaa@39D6@ to zero” is an equivalence operation.

Lemma 4 (first simplification): Suppose that j 0 J i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgMGiplaadQeadaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3BFD@ for some iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeaaaa@3935@ and j 0 J. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadQeacaGGUaaaaa@3AD9@ Then, “resetting a i j 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAamaaBaaameaacaaIWaaabeaaaSqabaaaaa@39D6@ to zero” is an equivalence operation.

Proof: From corollary 2, it is sufficient to show that T F s ( a i j 0 , x j 0 )< b i , x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaacYcacaWG4b WaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiyk aiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaeiiai abgcGiIiaadIhacqGHiiIZcaWGtbWaaSbaaSqaaiaadsfadaqhaaad baGaamOraaqaaiaadohaaaaaleqaaOGaaiikaiaadgeacaGGSaGaam OyaiaacMcacaGGUaaaaa@51CE@ But, from lemma 1 we have T F s ( a i j 0 , x j 0 )< b i ,  x j 0 [0,1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaacYcacaWG4b WaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiyk aiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaeiiai abgcGiIiaadIhadaWgaaWcbaGaamOAamaaBaaameaacaaIWaaabeaa aSqabaGccqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiaac6 caaaa@5031@ Thus T F s ( a i j 0 , x j 0 )< b i , x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaacYcacaWG4b WaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiyk aiabgYda8iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaeiiai abgcGiIiaadIhacqGHiiIZcaWGtbWaaSbaaSqaaiaadsfadaqhaaad baGaamOraaqaaiaadohaaaaaleqaaOGaaiikaiaadgeacaGGSaGaam OyaiaacMcacaGGUaaaaa@51CE@

Lemma 5 (second simplification): Suppose that j 0 J i 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadQeadaWgaaWcbaGaamyAamaa BaaameaacaaIXaaabeaaaSqabaaaaa@3C34@ and b i 1 0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiabgcMi5kaa icdacaGGSaaaaa@3C24@ where i 1 I MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaaIXaaabeaakiabgIGiolaadMeaaaa@3A26@ and j 0 J. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadQeacaGGUaaaaa@3AD9@ If at least one of the following conditions hold, then “resetting a i 1 j 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaamOAamaaBaaameaa caaIWaaabeaaaSqabaaaaa@3AC9@ to zero” is an equivalence operation:

(a) There exists some i 2 I ( i 1 i 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaaIYaaabeaakiabgIGiolaadMeacaqGGaGaaiikaiaadMga daWgaaWcbaGaaGymaaqabaGccqGHGjsUcaWGPbWaaSbaaSqaaiaaik daaeqaaOGaaiykaaaa@41A9@ such that j 0 J i 2 ,  b i 2 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadQeadaWgaaWcbaGaamyAamaa BaaameaacaaIYaaabeaaaSqabaGccaGGSaGaaeiiaiaadkgadaWgaa WcbaGaamyAamaaBaaameaacaaIYaaabeaaaSqabaGccqGHGjsUcaaI Waaaaa@4312@ and log s ( 1+ ( s b i 2 1)(s1) / ( s a i 2 j 0 1) )< log s ( 1+ ( s b i 1 1)(s1) / ( s a i 1 j 0 1) ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGPbWaaSbaaeaacaaIYaaabeaaaeqaaaaakiabgkHiTiaaigda caGGPaGaaiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcaca WGZbWaaWbaaSqabeaacaWGHbWaaSbaaWqaaiaadMgadaWgaaqaaiaa ykW7caaIYaGaaGPaVdqabaGaamOAamaaBaaabaGaaGimaaqabaaabe aaaaGccqGHsislcaaIXaGaaiykaaaaaiaawIcacaGLPaaacqGH8aap ciGGSbGaai4BaiaacEgadaWgaaWcbaGaam4CaaqabaGcdaqadaqaai aaigdacqGHRaWkdaWcgaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWG IbWaaSbaaWqaaiaadMgadaWgaaqaaiaaigdaaeqaaaqabaaaaOGaey OeI0IaaGymaiaacMcacaGGOaGaam4CaiabgkHiTiaaigdacaGGPaaa baGaaiikaiaadohadaahaaWcbeqaaiaadggadaWgaaadbaGaamyAam aaBaaabaGaaGPaVlaaigdacaaMc8oabeaacaWGQbWaaSbaaeaacaaI WaaabeaaaeqaaaaakiabgkHiTiaaigdacaGGPaaaaaGaayjkaiaawM caaiaac6caaaa@7409@

(b) There exists some i 2 I ( i 1 i 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaaIYaaabeaakiabgIGiolaadMeacaqGGaGaaiikaiaadMga daWgaaWcbaGaaGymaaqabaGccqGHGjsUcaWGPbWaaSbaaSqaaiaaik daaeqaaOGaaiykaaaa@41A9@ such that a i 2 j 0 >0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaikdaaeqaaSGaaGPaVlaadQgadaWg aaadbaGaaGimaaqabaaaleqaaOGaeyOpa4JaaGimaiaac6caaaa@3ED3@

Proof: (a) Similar to the proof of lemma 4, we show that T F s ( a i 1 j 0 , x j 0 )< b i , x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgadaWgaaadbaGaaGymaaqabaWccaWGQbWaaSbaaWqaaiaaicdaae qaaaWcbeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgadaWgaaadbaGa aGimaaqabaaaleqaaOGaaiykaiabgYda8iaadkgadaWgaaWcbaGaam yAaaqabaGccaGGSaGaaeiiaiabgcGiIiaadIhacqGHiiIZcaWGtbWa aSbaaSqaaiaadsfadaqhaaadbaGaamOraaqaaiaadohaaaaaleqaaO GaaiikaiaadgeacaGGSaGaamOyaiaacMcacaGGUaaaaa@52C1@ Consider an arbitrary feasible solution x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI GiolaadofadaWgaaWcbaGaamivamaaDaaameaacaWGgbaabaGaam4C aaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaaiykaiaac6caaa a@40C1@ Since x S T F s (A,b), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI GiolaadofadaWgaaWcbaGaamivamaaDaaameaacaWGgbaabaGaam4C aaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaaiykaiaacYcaaa a@40BF@ it turns out that T F s ( a i 1 j 0 , x j 0 )> b i 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgadaWgaaadbaGaaGymaaqabaWccaWGQbWaaSbaaWqaaiaaicdaae qaaaWcbeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgadaWgaaadbaGa aGimaaqabaaaleqaaOGaaiykaiabg6da+iaadkgadaWgaaWcbaGaam yAamaaBaaameaacaaIXaaabeaaaSqabaaaaa@46BF@ never holds. So, assume that T F s ( a i 1 j 0 , x j 0 )= b i 1 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgadaWgaaadbaGaaGymaaqabaWccaWGQbWaaSbaaWqaaiaaicdaae qaaaWcbeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgadaWgaaadbaGa aGimaaqabaaaleqaaOGaaiykaiabg2da9iaadkgadaWgaaWcbaGaam yAamaaBaaameaacaaIXaaabeaaaSqabaGccaGGSaaaaa@4777@ that is log s ( 1+ ( s a i 1 j 0 1)( s x j 0 1) / (s1) )= b i 1 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamyyamaaBaaame aacaWGPbWaaSbaaeaacaaMc8UaaGymaiaaykW7aeqaaiaadQgadaWg aaqaaiaaicdaaeqaaaqabaaaaOGaeyOeI0IaaGymaiaacMcacaGGOa Gaam4CamaaCaaaleqabaGaamiEamaaBaaameaacaWGQbWaaSbaaeaa caaIWaaabeaaaeqaaaaakiabgkHiTiaaigdacaGGPaaabaGaaiikai aadohacqGHsislcaaIXaGaaiykaaaaaiaawIcacaGLPaaacqGH9aqp caWGIbWaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaaaleqaaO GaaiOlaaaa@58EF@ Since b i 1 0, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiabgcMi5kaa icdacaGGSaaaaa@3C24@ from lemma 2 we conclude that x j 0 = log s ( 1+ ( s b i 1 1)(s1) / ( s a i 1 j 0 1) ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiabg2da9iGa cYgacaGGVbGaai4zamaaBaaaleaacaWGZbaabeaakmaabmaabaGaaG ymaiabgUcaRmaalyaabaGaaiikaiaadohadaahaaWcbeqaaiaadkga daWgaaadbaGaamyAamaaBaaabaGaaGymaaqabaaabeaaaaGccqGHsi slcaaIXaGaaiykaiaacIcacaWGZbGaeyOeI0IaaGymaiaacMcaaeaa caGGOaGaam4CamaaCaaaleqabaGaamyyamaaBaaameaacaWGPbWaaS baaeaacaaMc8UaaGymaiaaykW7aeqaaiaadQgadaWgaaqaaiaaicda aeqaaaqabaaaaOGaeyOeI0IaaGymaiaacMcaaaaacaGLOaGaayzkaa GaaiOlaaaa@58EF@ So, by the assumptionwe have log s ( 1+ ( s b i 2 1)(s1) / ( s a i 2 j 0 1) )< x j 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGPbWaaSbaaeaacaaIYaaabeaaaeqaaaaakiabgkHiTiaaigda caGGPaGaaiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcaca WGZbWaaWbaaSqabeaacaWGHbWaaSbaaWqaaiaadMgadaWgaaqaaiaa ykW7caaIYaGaaGPaVdqabaGaamOAamaaBaaabaGaaGimaaqabaaabe aaaaGccqGHsislcaaIXaGaaiykaaaaaiaawIcacaGLPaaacqGH8aap caWG4bWaaSbaaSqaaiaadQgadaWgaaadbaGaaGimaaqabaaaleqaaO GaaiOlaaaa@58EF@ Therefore, lemma 2 (part (a)) implies T F s ( a i 2 j 0 , x j 0 )> b i 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcacaWGHbWaaSbaaSqaaiaa dMgadaWgaaadbaGaaGOmaaqabaWccaWGQbWaaSbaaWqaaiaaicdaae qaaaWcbeaakiaacYcacaWG4bWaaSbaaSqaaiaadQgadaWgaaadbaGa aGimaaqabaaaleqaaOGaaiykaiabg6da+iaadkgadaWgaaWcbaGaam yAamaaBaaameaacaaIYaaabeaaaSqabaaaaa@46C1@ that contradicts x S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI GiolaadofadaWgaaWcbaGaamivamaaDaaameaacaWGgbaabaGaam4C aaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaaiykaiaac6caaa a@40C1@

(b) By the assumption, we have j 0 J i 2 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadQeadaWgaaWcbaGaamyAamaa BaaameaacaaIYaaabeaaaSqabaGccaGGUaaaaa@3CF1@ Now, the result similarly follows by a simpler argument.

Example 2: Consider the problem presented in example 1. From the first simplification (lemma 4), “resetting the following components a ij MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E4@ to zeros” are equivalence operations: a 12 ,  a 13 , a 15 , a 16 ;  a 22 , a 23 ,  a 24 , a 26 ;  a 32 ,  a 33 ,  a 36 ;  a 41 ,  a 42 ,  a 43 ,  a 46 ; MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIXaGaaGOmaaqabaGccaGGSaGaaeiiaiaadggadaWgaaWc baGaaGymaiaaiodaaeqaaOGaaiilaiaaykW7caWGHbWaaSbaaSqaai aaigdacaaI1aaabeaakiaacYcacaaMc8UaamyyamaaBaaaleaacaaI XaGaaGOnaaqabaGccaGG7aGaaeiiaiaadggadaWgaaWcbaGaaGOmai aaikdaaeqaaOGaaiilaiaaykW7caWGHbWaaSbaaSqaaiaaikdacaaI ZaaabeaakiaacYcacaqGGaGaamyyamaaBaaaleaacaaIYaGaaGinaa qabaGccaGGSaGaaGPaVlaadggadaWgaaWcbaGaaGOmaiaaiAdaaeqa aOGaai4oaiaabccacaWGHbWaaSbaaSqaaiaaiodacaaIYaaabeaaki aacYcacaqGGaGaamyyamaaBaaaleaacaaIZaGaaG4maaqabaGccaGG SaGaaeiiaiaadggadaWgaaWcbaGaaG4maiaaiAdaaeqaaOGaai4oai aabccacaWGHbWaaSbaaSqaaiaaisdacaaIXaaabeaakiaacYcacaqG GaGaamyyamaaBaaaleaacaaI0aGaaGOmaaqabaGccaGGSaGaaeiiai aadggadaWgaaWcbaGaaGinaiaaiodaaeqaaOGaaiilaiaabccacaWG HbWaaSbaaSqaaiaaisdacaaI2aaabeaakiaacUdaaaa@73D6@ in all of these cases, a ij < b i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH8aapcaWGIbWaaSbaaSqaaiaa dMgaaeqaaOGaaiilaaaa@3CAD@ that is j J i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgM GiplaadQeadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@3B0F@ Also, from the second simplification (lemma 5, part (a)), we can change the value of component a 21 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIYaGaaGymaaqabaaaaa@387E@ to zero; because a 21 = b 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIYaGaaGymaaqabaGccqGH9aqpcaWGIbWaaSbaaSqaaiaa ikdaaeqaaaaa@3B5D@ (i.e., 1 J 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgI GiolaadQeadaWgaaWcbaGaaGOmaaqabaaaaa@39EB@ ), b 2 0,  a 11 > b 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaaIYaaabeaakiabgcMi5kaaicdacaGGSaGaaeiiaiaadgga daWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOpa4JaamOyamaaBaaale aacaaIXaaabeaaaaa@410A@ (i.e. 1 J 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgI GiolaadQeadaWgaaWcbaGaaGymaaqabaaaaa@39EA@ ), b 1 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaaIXaaabeaakiabgcMi5kaaicdaaaa@3A4E@ and 0.7833= log s ( 1+ ( s b 1 1)(s1) / ( s a 11 1) )< log s ( 1+ ( s b 2 1)(s1) / ( s a 21 1) )=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeimaiaab6 cacaqG3aGaaeioaiaabodacaqGZaGaeyypa0JaciiBaiaac+gacaGG NbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4kaSYaaS GbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaameaacaaI XaaabeaaaaGccqGHsislcaaIXaGaaiykaiaacIcacaWGZbGaeyOeI0 IaaGymaiaacMcaaeaacaGGOaGaam4CamaaCaaaleqabaGaamyyamaa BaaameaacaaIXaGaaGymaaqabaaaaOGaeyOeI0IaaGymaiaacMcaaa aacaGLOaGaayzkaaGaeyipaWJaciiBaiaac+gacaGGNbWaaSbaaSqa aiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4kaSYaaSGbaeaacaGGOa Gaam4CamaaCaaaleqabaGaamOyamaaBaaameaacaaIYaaabeaaaaGc cqGHsislcaaIXaGaaiykaiaacIcacaWGZbGaeyOeI0IaaGymaiaacM caaeaacaGGOaGaam4CamaaCaaaleqabaGaamyyamaaBaaameaacaaI YaGaaGymaaqabaaaaOGaeyOeI0IaaGymaiaacMcaaaaacaGLOaGaay zkaaGaeyypa0JaaGymaaaa@6DDD@ . Moreover, from lemma 5 (part (b)), we can also change the values of components a 14 ,  a 34 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIXaGaaGinaaqabaGccaGGSaGaaeiiaiaadggadaWgaaWc baGaaG4maiaaisdaaeqaaaaa@3C6A@ and a 44 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaI0aGaaGinaaqabaaaaa@3883@ to zeros with no effect on the solutions set of the problem (since 4 J 1 J 3 J 4 ,  b i 0 (i=1,3,4), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgI GiolaadQeadaWgaaWcbaGaaGymaaqabaGccqWIPisscaWGkbWaaSba aSqaaiaaiodaaeqaaOGaeSykIKKaamOsamaaBaaaleaacaaI0aaabe aakiaacYcacaqGGaGaamOyamaaBaaaleaacaWGPbaabeaakiabgcMi 5kaaicdacaqGGaGaaeikaiaadMgacqGH9aqpcaaIXaGaaiilaiaaio dacaGGSaGaaGinaiaacMcacaGGSaaaaa@4DDA@ and b 5 =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaaI1aaabeaakiabg2da9iaaicdaaaa@3991@ a 54 >0). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaI1aGaaGinaaqabaGccqGH+aGpcaaIWaGaaiykaiaac6ca aaa@3BAF@

In addition to simplifying the problem, a necessary and sufficient condition is also derived from lemma 5. Before formally presenting the condition, some useful notations are introduced. Let A ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia aaaa@36CA@ denote the simplified matrix resulted from A MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BB@ after applying the simplification processes (lemmas 4 and 5). Also, similar to definition 1, assume that J ˜ i ={ jJ: a ˜ ij b i } (iI) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaiWaaeaacaWGQbGaeyic I4SaamOsaiaaykW7caaMc8UaaiOoaiaaykW7caaMc8UaaGPaVlqadg gagaacamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHLjYScaWGIbWa aSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaGaaeiiaiaacIcaca WGPbGaeyicI4SaamysaiaacMcaaaa@52FA@ where a ˜ ij MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@38F3@ denotes (i,j) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadM gacaGGSaGaamOAaiaacMcaaaa@39DB@ ‘th component of matrix A ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia aaaa@36CA@ . The following theorem gives a necessary and sufficient condition for the feasibility of problem (1).

Theorem 3 (second necessary and sufficient condition): S T F s (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyiyIKRaeyybIymaaa@40CE@ if and only if J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4Saamysaiaac6caaaa@414C@

Proof: Since S T F s (A,b)= S T F s ( A ˜ ,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyypa0Jaam4uamaaBaaale aacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaacIca ceWGbbGbaGaacaGGSaGaamOyaiaacMcaaaa@463C@ from lemmas 4 and 5, it is sufficient to show that S T F s ( A ˜ ,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcaceWGbbGbaGaacaGGSaGaamOyaiaacMcacqGHGjsUcqGHfiIXaa a@40DD@ if and only if J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4Saamysaiaac6caaaa@414C@ Let S T F s ( A ˜ ,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcaceWGbbGbaGaacaGGSaGaamOyaiaacMcacqGHGjsUcqGHfiIXca GGUaaaaa@418F@ Therefore, S T F s ( a ˜ i , b i ), iI, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcaceWGHbGbaGaadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamOyam aaBaaaleaacaWGPbaabeaakiaacMcacqGHGjsUcqGHfiIXcaGGSaGa aeiiaiabgcGiIiaadMgacqGHiiIZcaWGjbGaaiilaaaa@4958@ where a ˜ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgaaeqaaaaa@3804@ denotes i ‘th row of matrix A ˜ . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia GaaiOlaaaa@377C@ Now, lemma 3 implies J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4Saamysaiaac6caaaa@414C@ Conversely, suppose that J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4Saamysaiaac6caaaa@414C@ Again, by using lemma 3 we have J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4Saamysaiaac6caaaa@414C@ By contradiction, suppose that S T F s ( A ˜ ,b)=. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcaceWGbbGbaGaacaGGSaGaamOyaiaacMcacqGH9aqpcqGHfiIXca GGUaaaaa@40CE@ Therefore, X ¯ S T F s ( A ˜ ,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabgMGiplaadofadaWgaaWcbaGaamivamaaDaaameaacaWG gbaabaGaam4CaaaaaSqabaGccaGGOaGabmyqayaaiaGaaiilaiaadk gacaGGPaaaaa@4011@ from corollary 1, and then there exists i 0 I MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaaIWaaabeaakiabgIGiolaadMeaaaa@3A25@ such that X ¯ S T F s ( a ˜ i 0 , b i 0 ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabgMGiplaadofadaWgaaWcbaGaamivamaaDaaameaacaWG gbaabaGaam4CaaaaaSqabaGccaGGOaGabmyyayaaiaWaaSbaaSqaai aadMgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiilaiaadkgadaWg aaWcbaGaamyAamaaBaaameaacaaIWaaabeaaaSqabaGccaGGPaGaai Olaaaa@450F@ Since max j J ˜ i { T F s ( a ˜ i 0 j , X ¯ j ) }< b i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHjiYZceWGkbGbaGaadaWg aaadbaGaamyAaaqabaaaleqaaOGaaGPaVpaacmaabaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaaacaGL 7bGaayzFaaGaeyipaWJaamOyamaaBaaaleaacaWGPbWaaSbaaWqaai aaicdaaeqaaaWcbeaaaaa@5021@ (from lemma 1) , we must have either max j J ˜ i { T F s ( a ˜ i 0 j , X ¯ j ) }> b i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZceWGkbGbaGaadaWg aaadbaGaamyAaaqabaaaleqaaOGaaGPaVpaacmaabaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaaacaGL 7bGaayzFaaGaeyOpa4JaamOyamaaBaaaleaacaWGPbWaaSbaaWqaai aaicdaaeqaaaWcbeaaaaa@5023@ or max j J ˜ i { T F s ( a ˜ i 0 j , X ¯ j ) }< b i 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZceWGkbGbaGaadaWg aaadbaGaamyAaaqabaaaleqaaOGaaGPaVpaacmaabaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaaacaGL 7bGaayzFaaGaeyipaWJaamOyamaaBaaaleaacaWGPbWaaSbaaWqaai aaicdaaeqaaaWcbeaakiaac6caaaa@50DB@ Anyway, since X ¯ x ^ i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabgsMiJkqadIhagaqcamaaBaaaleaacaWGPbWaaSbaaWqa aiaaicdaaeqaaaWcbeaaaaa@3BB1@ (i.e., X ¯ j ( x ^ i 0 ) j , jJ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaamaaBaaaleaacaWGQbaabeaakiabgsMiJkaacIcaceWG4bGb aKaadaWgaaWcbaGaamyAamaaBaaameaacaaIWaaabeaaaSqabaGcca GGPaWaaSbaaSqaaiaadQgaaeqaaOGaaiilaiaabccacqGHaiIicaWG QbGaeyicI4SaamOsaiaacMcacaGGSaaaaa@4620@ we have max j J ˜ i 0 { T F s ( a ˜ i 0 j , X ¯ j ) } max j J ˜ i 0 { T F s ( a ˜ i 0 j , ( x ^ i 0 ) j ) }= b i 0 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZceWGkbGbaGaadaWg aaadbaGaamyAamaaBaaabaGaaGimaaqabaaabeaaaSqabaGccaaMc8 +aaiWaaeaacaWGubWaa0baaSqaaiaadAeaaeaacaWGZbaaaOGaaiik aiqadggagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaaicdaaeqaaS GaamOAaaqabaGccaGGSaWaa0aaaeaacaWGybaaamaaBaaaleaacaWG QbaabeaakiaacMcaaiaawUhacaGL9baacqGHKjYOdaWfqaqaaiGac2 gacaGGHbGaaiiEaaWcbaGaamOAaiabgIGiolqadQeagaacamaaBaaa meaacaWGPbWaaSbaaeaacaaIWaaabeaaaeqaaaWcbeaakiaaykW7da GadaqaaiaadsfadaqhaaWcbaGaamOraaqaaiaadohaaaGccaGGOaGa bmyyayaaiaWaaSbaaSqaaiaadMgadaWgaaadbaGaaGimaaqabaWcca WGQbaabeaakiaacYcacaGGOaGabmiEayaajaWaaSbaaSqaaiaadMga daWgaaadbaGaaGimaaqabaaaleqaaOGaaiykamaaBaaaleaacaWGQb aabeaakiaacMcaaiaawUhacaGL9baacqGH9aqpcaWGIbWaaSbaaSqa aiaadMgadaWgaaadbaGaaGimaaqabaaaleqaaOGaaiilaaaa@6E07@ and then the former case (i.e., max j J ˜ i { T F s ( a ˜ i 0 j , X ¯ j ) }> b i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZceWGkbGbaGaadaWg aaadbaGaamyAaaqabaaaleqaaOGaaGPaVpaacmaabaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaaacaGL 7bGaayzFaaGaeyOpa4JaamOyamaaBaaaleaacaWGPbWaaSbaaWqaai aaicdaaeqaaaWcbeaaaaa@5023@ ) never holds. Therefore, max j J ˜ i { T F s ( a ˜ i 0 j , X ¯ j ) }< b i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaadQgacqGHiiIZceWGkbGbaGaadaWg aaadbaGaamyAaaqabaaaleqaaOGaaGPaVpaacmaabaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaaacaGL 7bGaayzFaaGaeyipaWJaamOyamaaBaaaleaacaWGPbWaaSbaaWqaai aaicdaaeqaaaWcbeaaaaa@501F@ that implies b i 0 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiabgcMi5kaa icdaaaa@3B73@ and T F s ( a ˜ i 0 j , X ¯ j )< b i 0 , j J ˜ i 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4CaaaakiaacIcaceWGHbGbaGaadaWgaaWc baGaamyAamaaBaaameaacaaIWaaabeaaliaadQgaaeqaaOGaaiilam aanaaabaGaamiwaaaadaWgaaWcbaGaamOAaaqabaGccaGGPaGaeyip aWJaamOyamaaBaaaleaacaWGPbWaaSbaaWqaaiaaicdaaeqaaaWcbe aakiaacYcacaqGGaGaeyiaIiIaamOAaiabgIGiolqadQeagaacamaa BaaaleaacaWGPbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaac6caaa a@4D1B@ Hence, by lemma 2, we must have X ¯ j < log s ( 1+ ( s b i 0 1)(s1) / ( s a ˜ i 0 j 1) ), j J ˜ i 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaamaaBaaaleaacaWGQbaabeaakiabgYda8iGacYgacaGGVbGa ai4zamaaBaaaleaacaWGZbaabeaakmaabmaabaGaaGymaiabgUcaRm aalyaabaGaaiikaiaadohadaahaaWcbeqaaiaadkgadaWgaaadbaGa amyAamaaBaaabaGaaGPaVlaaicdaaeqaaaqabaaaaOGaeyOeI0IaaG ymaiaacMcacaGGOaGaam4CaiabgkHiTiaaigdacaGGPaaabaGaaiik aiaadohadaahaaWcbeqaaiqadggagaacamaaBaaameaacaWGPbWaaS baaeaacaaMc8UaaGimaiaaykW7aeqaaiaadQgaaeqaaaaakiabgkHi TiaaigdacaGGPaaaaaGaayjkaiaawMcaaiaacYcacaqGGaGaeyiaIi IaamOAaiabgIGiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqa aiaaicdaaeqaaaWcbeaakiaac6caaaa@6033@ On the other hand, log s ( 1+ ( s b i 0 1)(s1) / ( s a ˜ i 0 j 1) )1, j J ˜ i 0 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGPbWaaSbaaeaacaaMc8UaaGimaaqabaaabeaaaaGccqGHsisl caaIXaGaaiykaiaacIcacaWGZbGaeyOeI0IaaGymaiaacMcaaeaaca GGOaGaam4CamaaCaaaleqabaGabmyyayaaiaWaaSbaaWqaaiaadMga daWgaaqaaiaaykW7caaIWaGaaGPaVdqabaGaamOAaaqabaaaaOGaey OeI0IaaGymaiaacMcaaaaacaGLOaGaayzkaaGaeyizImQaaGymaiaa cYcacaqGGaGaeyiaIiIaamOAaiabgIGiolqadQeagaacamaaBaaale aacaWGPbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiaac6caaaa@5F8C@ Therefore, X ¯ j <1, j J ˜ i 0 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaamaaBaaaleaacaWGQbaabeaakiabgYda8iaaigdacaGGSaGa aeiiaiabgcGiIiaadQgacqGHiiIZceWGkbGbaGaadaWgaaWcbaGaam yAamaaBaaameaacaaIWaaabeaaaSqabaGccaGGSaaaaa@4201@ and then from definitions 2 and 3, for each j J ˜ i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaaicdaaeqa aaWcbeaaaaa@3B52@ there must exists i j I MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaWGQbaabeaakiabgIGiolaadMeaaaa@3A5A@ such that either j J ˜ i j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaadQgaaeqa aaWcbeaaaaa@3B87@ and X ¯ j = ( x ^ i j ) j = log s ( 1+ ( s b i j 1)(s1) / ( s a ˜ i j j 1) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaamaaBaaaleaacaWGQbaabeaakiabg2da9iaacIcaceWG4bGb aKaadaWgaaWcbaGaamyAamaaBaaameaacaWGQbaabeaaaSqabaGcca GGPaWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaciiBaiaac+gacaGG NbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4kaSYaaS GbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaameaacaWG PbWaaSbaaeaacaWGQbaabeaaaeqaaaaakiabgkHiTiaaigdacaGGPa GaaiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcacaWGZbWa aWbaaSqabeaaceWGHbGbaGaadaWgaaadbaGaamyAamaaBaaabaGaam OAaiaaykW7aeqaaiaadQgaaeqaaaaakiabgkHiTiaaigdacaGGPaaa aaGaayjkaiaawMcaaaaa@5C29@ or j J ˜ i j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaadQgaaeqa aaWcbeaaaaa@3B87@ and a ˜ i j j > b i j =0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgadaWgaaadbaGaamOAaaqabaWccaaMc8UaamOA aaqabaGccqGH+aGpcaWGIbWaaSbaaSqaaiaadMgadaWgaaadbaGaam OAaaqabaaaleqaaOGaeyypa0JaaGimaiaac6caaaa@425B@ Until now, we proved that b i 0 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiabgcMi5kaa icdaaaa@3B73@ and for each j J ˜ i 0 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaaicdaaeqa aaWcbeaakiaacYcaaaa@3C0C@ there exist i j I MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaWGQbaabeaakiabgIGiolaadMeaaaa@3A5A@ such that either j J ˜ i j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbWaaSbaaWqaaiaadQgaaeqa aaWcbeaaaaa@3B87@ and log s ( 1+ ( s b i j 1)(s1) / ( s a ˜ i j j 1) )< log s ( 1+ ( s b i 0 1)(s1) / ( s a ˜ i 0 j 1) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGPbWaaSbaaeaacaWGQbaabeaaaeqaaaaakiabgkHiTiaaigda caGGPaGaaiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcaca WGZbWaaWbaaSqabeaaceWGHbGbaGaadaWgaaadbaGaamyAamaaBaaa baGaamOAaiaaykW7aeqaaiaadQgaaeqaaaaakiabgkHiTiaaigdaca GGPaaaaaGaayjkaiaawMcaaiabgYda8iGacYgacaGGVbGaai4zamaa BaaaleaacaWGZbaabeaakmaabmaabaGaaGymaiabgUcaRmaalyaaba GaaiikaiaadohadaahaaWcbeqaaiaadkgadaWgaaadbaGaamyAamaa BaaabaGaaGPaVlaaicdaaeqaaaqabaaaaOGaeyOeI0IaaGymaiaacM cacaGGOaGaam4CaiabgkHiTiaaigdacaGGPaaabaGaaiikaiaadoha daahaaWcbeqaaiqadggagaacamaaBaaameaacaWGPbWaaSbaaeaaca aMc8UaaGimaiaaykW7aeqaaiaadQgaaeqaaaaakiabgkHiTiaaigda caGGPaaaaaGaayjkaiaawMcaaaaa@7223@ (because, log s ( 1+ ( s b i j 1)(s1) / ( s a ˜ i j j 1) )= X ¯ j < log s ( 1+ ( s b i 0 1)(s1) / ( s a ˜ i 0 j 1) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGPbWaaSbaaeaacaWGQbaabeaaaeqaaaaakiabgkHiTiaaigda caGGPaGaaiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcaca WGZbWaaWbaaSqabeaaceWGHbGbaGaadaWgaaadbaGaamyAamaaBaaa baGaamOAaiaaykW7aeqaaiaadQgaaeqaaaaakiabgkHiTiaaigdaca GGPaaaaaGaayjkaiaawMcaaiabg2da9maanaaabaGaamiwaaaadaWg aaWcbaGaamOAaaqabaGccqGH8aapciGGSbGaai4BaiaacEgadaWgaa WcbaGaam4CaaqabaGcdaqadaqaaiaaigdacqGHRaWkdaWcgaqaaiaa cIcacaWGZbWaaWbaaSqabeaacaWGIbWaaSbaaWqaaiaadMgadaWgaa qaaiaaykW7caaIWaaabeaaaeqaaaaakiabgkHiTiaaigdacaGGPaGa aiikaiaadohacqGHsislcaaIXaGaaiykaaqaaiaacIcacaWGZbWaaW baaSqabeaaceWGHbGbaGaadaWgaaadbaGaamyAamaaBaaabaGaaGPa VlaaicdacaaMc8oabeaacaWGQbaabeaaaaGccqGHsislcaaIXaGaai ykaaaaaiaawIcacaGLPaaaaaa@753C@ ) or b i j =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbWaaSbaaWqaaiaadQgaaeqaaaWcbeaakiabg2da9iaa icdaaaa@3AE7@ a ˜ i j j >0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgadaWgaaadbaGaamOAaaqabaWccaaMc8UaamOA aaqabaGccqGH+aGpcaaIWaGaaiOlaaaa@3E23@ But in both cases, we must have a ˜ i 0 j =0 (j J ˜ i 0 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgadaWgaaadbaGaaGimaaqabaWccaWGQbaabeaa kiabg2da9iaaicdacaqGGaGaaiikaiabgcGiIiaadQgacqGHiiIZce WGkbGbaGaadaWgaaWcbaGaamyAamaaBaaameaacaaIWaaabeaaaSqa baGccaGGPaaaaa@43E2@ from the parts (a) and (b) of lemma 5, respectively. Therefore, a ˜ i 0 j < b i 0 0 (j J ˜ i 0 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyyayaaia WaaSbaaSqaaiaadMgadaWgaaadbaGaaGimaaqabaWccaWGQbaabeaa kiabgYda8iaadkgadaWgaaWcbaGaamyAamaaBaaameaacaaIWaaabe aaaSqabaGccqGHGjsUcaaIWaGaaeiiaiaacIcacqGHaiIicaWGQbGa eyicI4SabmOsayaaiaWaaSbaaSqaaiaadMgadaWgaaadbaGaaGimaa qabaaaleqaaOGaaiykaaaa@48A4@ that is a contradiction.

Remark 1: Since S T F s (A,b)= S T F s ( A ˜ ,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyypa0Jaam4uamaaBaaale aacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaacIca ceWGbbGbaGaacaGGSaGaamOyaiaacMcaaaa@463C@ (from lemmas 4 and 5), we can rewrite all the previous definitions and results in a simpler manner by replacing J ˜ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaaaa@37ED@ with J i  (iI). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGPbaabeaakiaabccacaGGOaGaamyAaiabgIGiolaadMea caGGPaGaaiOlaaaa@3DD6@
A summary of the GA
In this section, the genetic algorithm proposed in [15] is briefly discussed. Since the feasible region of problem (1) is non-convex, a convex subset of the feasible region is firstly introduced. Consequently, the proposed GA can easily generate the initial population by randomly choosing individuals from this convex feasible subset. At the last part of this section, a method is presented to generate random feasible max-Yager fuzzy relational equations.
Initialization
The initial population is given by randomly generating the individuals inside the feasible region. For this purpose, we firstly find a convex subset of the feasible solutions set, that is, we find set F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C0@ such that F S T Y p (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabgA OinlaadofadaWgaaWcbaGaamivamaaDaaameaacaWGzbaabaGaamiC aaaaaSqabaGccaGGOaGaamyqaiaacYcacaWGIbGaaiykaaaa@406A@ and F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C0@ is convex. Then, the initial population is generated by randomly selecting individuals from set F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C0@ .

Definition 7: Suppose that S T F s ( A ˜ ,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcaceWGbbGbaGaacaGGSaGaamOyaiaacMcacqGHGjsUcqGHfiIXca GGUaaaaa@418F@ For each iI, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGSaaaaa@39E5@ let x i =[ ( x i ) 1 , ( x i ) 2 ,..., ( x i ) n ] [0,1] n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaai4waiaacIcaceWG4bGb aqbadaWgaaWcbaGaamyAaaqabaGccaGGPaWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaacIcaceWG4bGbaqbadaWgaaWcbaGaamyAaaqabaGc caGGPaWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaai OlaiaacYcacaGGOaGabmiEayaauaWaaSbaaSqaaiaadMgaaeqaaOGa aiykamaaBaaaleaacaWGUbaabeaakiaac2facqGHiiIZcaGGBbGaaG imaiaacYcacaaIXaGaaiyxamaaCaaaleqabaGaamOBaaaaaaa@5371@ where the components are defined as follows:
( x i ) k ={ log s ( 1+ ( s b i 1)(s1) / ( s a ik 1) ) b i 0andk J ˜ i 0 otherwise , kJ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadI hagaafamaaBaaaleaacaWGPbaabeaakiaacMcadaWgaaWcbaGaam4A aaqabaGccqGH9aqpdaGabaqaauaabeqaciaaaeaaciGGSbGaai4Bai aacEgadaWgaaWcbaGaam4CaaqabaGcdaqadaqaaiaaigdacqGHRaWk daWcgaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWGIbWaaSbaaWqaai aadMgaaeqaaaaakiabgkHiTiaaigdacaGGPaGaaiikaiaadohacqGH sislcaaIXaGaaiykaaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWGHb WaaSbaaWqaaiaadMgacaWGRbaabeaaaaGccqGHsislcaaIXaGaaiyk aaaaaiaawIcacaGLPaaaaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaO GaeyiyIKRaaGimaiaaykW7caaMc8Uaamyyaiaad6gacaWGKbGaaGPa VlaaykW7caWGRbGaeyicI4SabmOsayaaiaWaaSbaaSqaaiaadMgaae qaaaGcbaGaaGimaaqaaiaad+gacaWG0bGaamiAaiaadwgacaWGYbGa am4DaiaadMgacaWGZbGaamyzaaaaaiaawUhaaiaacYcacaqGGaGaey iaIiIaam4AaiabgIGiolaadQeaaaa@75FD@
Also, we define X _ = max iI { x i }. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiabg2da9maaxababaGaciyBaiaacggacaGG4baaleaacaWG PbGaeyicI4SaamysaaqabaGcdaGadaqaaiqadIhagaafamaaBaaale aacaWGPbaabeaaaOGaay5Eaiaaw2haaiaac6caaaa@435E@

Remark 2: According to definition 2 and remark 1, it is clear that for a fixed iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeaaaa@3935@ and j J ˜ i ,  x i (j) k ( x i ) k  (kJ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolqadQeagaacamaaBaaaleaacaWGPbaabeaakiaacYcacaqGGaGa bmiEayaauaWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadQgacaGGPa WaaSbaaSqaaiaadUgaaeqaaOGaeyizImQaaiikaiqadIhagaafamaa BaaaleaacaWGPbaabeaakiaacMcadaWgaaWcbaGaam4AaaqabaGcca qGGaGaaiikaiabgcGiIiaadUgacqGHiiIZcaWGkbGaaiykaiaac6ca aaa@4E98@ Therefore, from definitions 5 and 7 we have X _ (e) k = max iI { x i (e(i)) k }= max iI { x i ( j i ) k } max iI { ( x i ) k }= X _ k , kJ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykamaaBaaaleaacaWGRbaabeaakiab g2da9maaxababaGaciyBaiaacggacaGG4baaleaacaWGPbGaeyicI4 SaamysaaqabaGcdaGadaqaaiqadIhagaafamaaBaaaleaacaWGPbaa beaakiaacIcacaWGLbGaaiikaiaadMgacaGGPaGaaiykamaaBaaale aacaWGRbaabeaaaOGaay5Eaiaaw2haaiabg2da9maaxababaGaciyB aiaacggacaGG4baaleaacaWGPbGaeyicI4SaamysaaqabaGcdaGada qaaiqadIhagaafamaaBaaaleaacaWGPbaabeaakiaacIcacaWGQbWa aSbaaSqaaiaadMgaaeqaaOGaaiykamaaBaaaleaacaWGRbaabeaaaO Gaay5Eaiaaw2haaiabgsMiJoaaxababaGaciyBaiaacggacaGG4baa leaacaWGPbGaeyicI4SaamysaaqabaGcdaGadaqaaiaacIcaceWG4b GbaqbadaWgaaWcbaGaamyAaaqabaGccaGGPaWaaSbaaSqaaiaadUga aeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaWaaaeaacaWGybaaamaaBa aaleaacaWGRbaabeaakiaacYcacaqGGaGaeyiaIiIaam4AaiabgIGi olaadQeaaaa@7398@ and eE. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaam yzaiabgIGiolaadweacaGGUaaaaa@3AAF@ Thus X _ (e) X _ , eE. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiaacIcacaWGLbGaaiykaiabgsMiJoaamaaabaGaamiwaaaa caGGSaGaaeiiaiabgcGiIiaadwgacqGHiiIZcaWGfbGaaiOlaaaa@41D4@

Example 3: Consider the problem presented in example 1, where X ¯ =[0.7833 , 0 , 1 , 0 , 1 , 1]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiabg2da9iaacUfacaqGWaGaaeOlaiaabEdacaqG4aGaae4m aiaabodacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGa GaaeymaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabcca caqGXaGaaeiiaiaabYcacaqGGaGaaeymaiaac2facaGGUaaaaa@4BEB@ Also, according to example 2, the simplified matrix A ˜ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia aaaa@36CA@ is
A ˜ =[ 0.9     0      0     0        0     0    0     0      0     0     0.5     0 0.9     0      0     0     0.7     0    0     0      0     0     0.8     0    0   0.01   0     0.2     0   0.9 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia Gaeyypa0ZaamWaaqaabeqaaiaabcdacaqGUaGaaeyoaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeimaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcda aeaacaqGGaGaaeiiaiaabccacaqGWaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGWaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG1aGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGWaaabaGaaeimaiaab6caca qG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeimaiaab6cacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGWaaabaGaaeiiaiaabccacaqGGaGaaeimaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeimaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bcdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae ioaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeimaaqaaiaabcca caqGGaGaaeiiaiaabcdacaqGGaGaaeiiaiaabccacaqGWaGaaeOlai aabcdacaqGXaGaaeiiaiaabccacaqGGaGaaeimaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeimaiaab6cacaqGYaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGWaGaaeiiaiaabccacaqGGaGaaeimaiaa b6cacaqG5aaaaiaawUfacaGLDbaaaaa@B0C0@
From definition 7, we have x 1 =[0.7833 , 0 , 0 , 0 , 0 , 0],  x 2 =[0 , 0 , 0 , 0 , 1 , 0],  MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaai4waiaabcdacaqGUaGa ae4naiaabIdacaqGZaGaae4maiaabccacaqGSaGaaeiiaiaabcdaca qGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaa bccacaqGSaGaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGaai yxaiaacYcacaqGGaGabmiEayaauaWaaSbaaSqaaiaaikdaaeqaaOGa eyypa0Jaai4waiaaicdacaqGGaGaaeilaiaabccacaqGWaGaaeiiai aabYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGa aeilaiaabccacaqGXaGaaeiiaiaabYcacaqGGaGaaeimaiaac2faca GGSaGaaeiiaaaa@61CA@ x 3 =[0.7833 , 0 , 0 , 0 , 1 , 0],  x 4 =[0 , 0 , 0 , 0 , 1 , 0],  x 5 =[0 , 0 , 0 , 0 , 0 , 0] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaua WaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaai4waiaabcdacaqGUaGa ae4naiaabIdacaqGZaGaae4maiaabccacaqGSaGaaeiiaiaabcdaca qGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaa bccacaqGSaGaaeiiaiaabgdacaqGGaGaaeilaiaabccacaqGWaGaai yxaiaacYcacaqGGaGabmiEayaauaWaaSbaaSqaaiaaisdaaeqaaOGa eyypa0Jaai4waiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiai aabYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGGaGa aeilaiaabccacaqGXaGaaeiiaiaabYcacaqGGaGaaeimaiaac2faca GGSaGaaeiiaiqadIhagaafamaaBaaaleaacaaI1aaabeaakiabg2da 9iaacUfacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaabccacaqGSa GaaeiiaiaabcdacaqGGaGaaeilaiaabccacaqGWaGaaeiiaiaabYca caqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaGGDbaaaa@7496@ , and then X _ = max i=1 5 { x i }=[0.7833 , 0 , 0 , 0 , 1 , 0]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WGybaaaiabg2da9maaxadabaGaciyBaiaacggacaGG4baaleaacaWG PbGaeyypa0JaaGymaaqaaiaaiwdaaaGcdaGadaqaaiqadIhagaafam aaBaaaleaacaWGPbaabeaaaOGaay5Eaiaaw2haaiabg2da9iaacUfa caqGWaGaaeOlaiaabEdacaqG4aGaae4maiaabodacaqGGaGaaeilai aabccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaabccacaqGSaGa aeiiaiaabcdacaqGGaGaaeilaiaabccacaqGXaGaaeiiaiaabYcaca qGGaGaaeimaiaab2facaqGUaaaaa@57FD@ Therefore, set F=[ X _ , X ¯ ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 da9iaacUfadaadaaqaaiaadIfaaaGaaGPaVlaacYcacaaMc8+aa0aa aeaacaWGybaaaiaac2faaaa@3F27@ is obtained as a collection of intervals:
F=[ X _ , X ¯ ]=[0.7833 , 0 , [0,1] , 0 , 1 , [0,1]] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 da9iaacUfadaadaaqaaiaadIfaaaGaaGPaVlaacYcacaaMc8+aa0aa aeaacaWGybaaaiaac2facqGH9aqpcaGGBbGaaeimaiaab6cacaqG3a GaaeioaiaabodacaqGZaGaaeiiaiaabYcacaqGGaGaaeimaiaabcca caqGSaGaaeiiaiaabUfacaqGWaGaaeilaiaabgdacaqGDbGaaeiiai aabYcacaqGGaGaaeimaiaabccacaqGSaGaaeiiaiaabgdacaqGGaGa aeilaiaabccacaqGBbGaaeimaiaabYcacaqGXaGaaeyxaiaaykW7ca qGDbaaaa@5B47@
By generating random numbers in the corresponding intervals, we acquire one initial individual: x=[0.7833 , 0 , 0.6 , 0 , 1 , 0.2]. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaacUfacaqGWaGaaeOlaiaabEdacaqG4aGaae4maiaabodacaqG GaGaaeilaiaabccacaqGWaGaaeiiaiaabYcacaqGGaGaaeimaiaab6 cacaqG2aGaaeiiaiaabYcacaqGGaGaaeimaiaabccacaqGSaGaaeii aiaabgdacaqGGaGaaeilaiaabccacaqGWaGaaeOlaiaabkdacaaMc8 Uaaeyxaiaab6caaaa@5051@

The algorithm for generating the initial population is simply obtained as follows:

Algorithm 1 (Initial Population):
1. Get fuzzy matrix A, fuzzy vector band population size  S pop . 2. If  X ¯ S T F s (A,b), then stop; the problem is infeasible (corollary1). 3. For i=1,2,..., S pop Generate a random ndimensional solution pop(i)in the interval [ X _ , X ¯ ]. End MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGXa GaaeOlaiaabccacaqGhbGaaeyzaiaabshacaqGGaGaaeOzaiaabwha caqG6bGaaeOEaiaabMhacaqGGaGaaeyBaiaabggacaqG0bGaaeOCai aabMgacaqG4bGaaeiiaiaadgeacaqGSaGaaeiiaiaabAgacaqG1bGa aeOEaiaabQhacaqG5bGaaeiiaiaabAhacaqGLbGaae4yaiaabshaca qGVbGaaeOCaiaabccacaWGIbGaaGPaVlaabggacaqGUbGaaeizaiaa bccacaqGWbGaae4BaiaabchacaqG1bGaaeiBaiaabggacaqG0bGaae yAaiaab+gacaqGUbGaaeiiaiaabohacaqGPbGaaeOEaiaabwgacaqG GaGaam4uamaaBaaaleaacaWGWbGaam4BaiaadchaaeqaaOGaaeOlaa qaaiaabkdacaqGUaGaaeiiaiaabMeacaqGMbGaaeiiaiaaykW7daqd aaqaaiaadIfaaaGaeyycI8Saam4uamaaBaaaleaacaWGubWaa0baaW qaaiaadAeaaeaacaWGZbaaaaWcbeaakiaacIcacaWGbbGaaiilaiaa dkgacaGGPaGaaeilaiaabccacaqG0bGaaeiAaiaabwgacaqGUbGaae iiaiaabohacaqG0bGaae4BaiaabchacaqG7aGaaeiiaiaabshacaqG ObGaaeyzaiaabccacaqGWbGaaeOCaiaab+gacaqGIbGaaeiBaiaabw gacaqGTbGaaeiiaiaabMgacaqGZbGaaeiiaiaabMgacaqGUbGaaeOz aiaabwgacaqGHbGaae4CaiaabMgacaqGIbGaaeiBaiaabwgacaqGGa GaaeikaiaabogacaqGVbGaaeOCaiaab+gacaqGSbGaaeiBaiaabgga caqGYbGaaeyEaiaabgdacaqGPaGaaeOlaaqaaiaabodacaqGUaGaae iiaiaabAeacaqGVbGaaeOCaiaabccacaqGPbGaeyypa0Jaaeymaiaa bYcacaqGYaGaaeilaiaab6cacaqGUaGaaeOlaiaabYcacaWGtbWaaS baaSqaaiaadchacaWGVbGaamiCaaqabaaakeaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8Uaae4raiaabwgacaqGUbGaaeyzaiaabkhacaqG HbGaaeiDaiaabwgacaqGGaGaaeyyaiaabccacaqGYbGaaeyyaiaab6 gacaqGKbGaae4Baiaab2gacaqGGaGaamOBaiabgkHiTiaabsgacaqG PbGaaeyBaiaabwgacaqGUbGaae4CaiaabMgacaqGVbGaaeOBaiaabg gacaqGSbGaaeiiaiaabohacaqGVbGaaeiBaiaabwhacaqG0bGaaeyA aiaab+gacaqGUbGaaeiiaiaadchacaWGVbGaamiCaiaacIcacaWGPb GaaiykaiaaykW7caaMc8UaaeyAaiaab6gacaqGGaGaaeiDaiaabIga caqGLbGaaeiiaiaabMgacaqGUbGaaeiDaiaabwgacaqGYbGaaeODai aabggacaqGSbGaaeiiaiaacUfadaadaaqaaiaadIfaaaGaaGPaVlaa cYcacaaMc8+aa0aaaeaacaWGybaaaiaac2facaGGUaaabaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caqGfbGaaeOBaiaabsgaaaaa@1CE4@
Selection Strategy
Suppose that the individuals in the population are sorted according to their ranks from the best to worst, that is, individual pop(r) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaad+ gacaWGWbGaaiikaiaadkhacaGGPaaaaa@3B23@ has rank r. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaac6 caaaa@379E@ The probability P r MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGYbaabeaaaaa@37ED@ of choosing the r MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EC@ ‘th individual is given by the following formulas:
P r = W r k=1 S pop W k ,  W r = 1 2π q S pop e 1 2 ( r1 q S pop ) 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGYbaabeaakiabg2da9maalaaabaGaam4vamaaBaaaleaa caWGYbaabeaaaOqaamaaqadabaGaam4vamaaBaaaleaacaWGRbaabe aaaeaacaWGRbGaeyypa0JaaGymaaqaaiaadofadaWgaaadbaGaamiC aiaad+gacaWGWbaabeaaa0GaeyyeIuoaaaGccaGGSaGaaeiiaiaadE fadaWgaaWcbaGaamOCaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa daGcaaqaaiaaikdacqaHapaCaSqabaGccaaMc8UaamyCaiaaykW7ca WGtbWaaSbaaSqaaiaadchacaWGVbGaamiCaaqabaaaaOGaamyzamaa CaaaleqabaGaeyOeI0IaaGPaVpaalaaabaGaaGymaaqaaiaaikdaaa WaaeWaaeaadaWcaaqaaiaadkhacqGHsislcaaIXaaabaGaaGPaVlaa dghacaaMc8UaaGPaVlaadofadaWgaaadbaGaamiCaiaad+gacaWGWb aabeaaaaaaliaawIcacaGLPaaadaahaaadbeqaaiaaikdaaaaaaaaa @6923@
where the weight to be a value of the Gaussian function with argument r, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@379C@ mean 1 , and standard deviation q S pop , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiaayk W7caWGtbWaaSbaaSqaaiaadchacaWGVbGaamiCaaqabaGccaGGSaaa aa@3D12@ where q MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36EB@ is a parameter of the algorithm.
Mutation Operator
As usual, suppose that S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaeyiyIKRaeyybIySaaiOlaa aa@4180@ So, from theorem 3 we have J ˜ i , iI, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaeyybIySaaiilaiaabcca cqGHaiIicaWGPbGaeyicI4SaamysaiaacYcaaaa@414A@ Where J ˜ i ={ jJ: a ˜ ij b i }, iI MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaiWaaeaacaWGQbGaeyic I4SaamOsaiaaykW7caaMc8UaaiOoaiaaykW7caaMc8UaaGPaVlqadg gagaacamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHLjYScaWGIbWa aSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaGaaiilaiaabccacq GHaiIicaWGPbGaeyicI4Saamysaaaa@5321@ (see definition1 and remark 1).

Definition 8: Let I + ={ iI: b i 0 }. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaCa aaleqabaGaey4kaScaaOGaeyypa0ZaaiWaaeaacaWGPbGaeyicI4Sa amysaiaaykW7caaMc8UaaiOoaiaaykW7caaMc8UaamOyamaaBaaale aacaWGPbaabeaakiabgcMi5kaaicdacaaMc8oacaGL7bGaayzFaaGa aiOlaaaa@4C06@ So, we define D={ jJ:if  i I + such thatj J ˜ i | J ˜ i |>1 }, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maacmaabaGaamOAaiabgIGiolaadQeacaaMc8UaaGPaVlaacQda caaMc8UaaGPaVlaaykW7caqGPbGaaeOzaiaabccacaqGGaGaey4aIq IaamyAaiabgIGiolaadMeadaahaaWcbeqaaiabgUcaRaaakiaaykW7 caaMc8Uaae4CaiaabwhacaqGJbGaaeiAaiaabccacaqG0bGaaeiAai aabggacaqG0bGaaGPaVlaaykW7caWGQbGaeyicI4SabmOsayaaiaWa aSbaaSqaaiaadMgaaeqaaOGaeyO0H49aaqWaaeaacaaMc8UabmOsay aaiaWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVdGaay5bSlaawIa7aiab g6da+iaaigdaaiaawUhacaGL9baacaGGSaaaaa@6DBF@ where | J ˜ i | MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaace WGkbGbaGaadaWgaaWcbaGaamyAaaqabaaakiaawEa7caGLiWoaaaa@3B19@ denotes the cardinality of set J ˜ i . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaia WaaSbaaSqaaiaadMgaaeqaaOGaaiOlaaaa@38A9@

The mutation operator is defined as follows:
Algorithm 2 (Mutation operator):
1. Get the matrix  A ˜ , vector band a selected solution  x ˙ =[ x ˙ 1 ,..., x ˙ n ]. 2. While  D      2.1. Set  x x.      2.2. Randomly choose   j 0 D,and set  x j 0 =0. 2.3. IF   x  is feasible, go to Crossover operator; otherwise, set D=D{ j 0 } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGXa GaaeOlaiaabccacaqGhbGaaeyzaiaabshacaqGGaGaaeiDaiaabIga caqGLbGaaeiiaiaab2gacaqGHbGaaeiDaiaabkhacaqGPbGaaeiEai aabccaceWGbbGbaGaacaqGSaGaaeiiaiaabAhacaqGLbGaae4yaiaa bshacaqGVbGaaeOCaiaabccacaWGIbGaaGPaVlaaykW7caqGHbGaae OBaiaabsgacaqGGaGaaeyyaiaabccacaqGZbGaaeyzaiaabYgacaqG LbGaae4yaiaabshacaqGLbGaaeizaiaabccacaqGZbGaae4BaiaabY gacaqG1bGaaeiDaiaabMgacaqGVbGaaeOBaiaabccaceWG4bGbaiaa cqGH9aqpcaGGBbGabmiEayaacaWaaSbaaSqaaiaaigdaaeqaaOGaai ilaiaac6cacaGGUaGaaiOlaiaacYcaceWG4bGbaiaadaWgaaWcbaGa amOBaaqabaGccaGGDbGaaeOlaaqaaiaabkdacaqGUaGaaeiiaiaabE facaqGObGaaeyAaiaabYgacaqGLbGaaeiiaiaabccacaWGebGaeyiy IKRaeyybIymabaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGYa GaaeOlaiaabgdacaqGUaGaaeiiaiaabofacaqGLbGaaeiDaiaabcca ceWG4bGbauaacqGHqgcRcaWG4bGaaeOlaaqaaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeOmaiaab6cacaqGYaGaaeOlaiaabccacaqG sbGaaeyyaiaab6gacaqGKbGaae4Baiaab2gacaqGSbGaaeyEaiaabc cacaqGJbGaaeiAaiaab+gacaqGVbGaae4CaiaabwgacaqGGaGaaeii aiaadQgadaWgaaWcbaGaaGimaaqabaGccqGHiiIZcaWGebGaaiilai aaykW7caqGHbGaaeOBaiaabsgacaqGGaGaae4CaiaabwgacaqG0bGa aeiiaiqadIhagaqbamaaBaaaleaacaWGQbWaaSbaaWqaaiaaicdaae qaaaWcbeaakiabg2da9iaaicdacaGGUaaabaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaabkdacaqGUaGaae4maiaab6 cacaqGGaGaaeysaiaabAeacaqGGaGaaeiiaiqadIhagaqbaiaabcca caqGPbGaae4CaiaabccacaqGMbGaaeyzaiaabggacaqGZbGaaeyAai aabkgacaqGSbGaaeyzaiaabYcacaqGGaGaae4zaiaab+gacaqGGaGa aeiDaiaab+gacaqGGaGaae4qaiaabkhacaqGVbGaae4Caiaabohaca qGVbGaaeODaiaabwgacaqGYbGaaeiiaiaab+gacaqGWbGaaeyzaiaa bkhacaqGHbGaaeiDaiaab+gacaqGYbGaae4oaiaabccacaqGVbGaae iDaiaabIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4CaiaabwgacaqG SaGaaeiiaiaabohacaqGLbGaaeiDaiaabccacaWGebGaeyypa0Jaam iraiabgkHiTiaacUhacaWGQbWaaSbaaSqaaiaaicdaaeqaaOGaaiyF aiaab6cacaqGGaaaaaa@0047@
Crossover operator
In section 2, it was proved that X ¯ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaaaa@36E3@ is the unique maximum solution of S T F s (A,b). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaGaaiOlaaaa@3E40@ By using this result, the crossover operator is stated as follows:
Algorithm 3 (Crossover operator):
1. Get the maximum solution  X ¯ , the new solution  x (generated by algorith 2)      and one parent pop(k)(for some k=1,2,..., S pop ). 2. Generate a random number  λ 1 [0,1]. Set   x new1 = λ 1 x +(1 λ 1 ) X ¯ . 3.Let  λ 2 = min j=1 jk S pop pop(k)pop(j)  and  d= X ¯ pop(k).     Set   x new2 =pop(k)+min{ λ 2 ,1 }d. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGXa GaaeOlaiaabccacaqGhbGaaeyzaiaabshacaqGGaGaaeiDaiaabIga caqGLbGaaeiiaiaab2gacaqGHbGaaeiEaiaabMgacaqGTbGaaeyDai aab2gacaqGGaGaae4Caiaab+gacaqGSbGaaeyDaiaabshacaqGPbGa ae4Baiaab6gacaqGGaWaa0aaaeaacaWGybaaaiaabYcacaqGGaGaae iDaiaabIgacaqGLbGaaeiiaiaab6gacaqGLbGaae4DaiaabccacaqG ZbGaae4BaiaabYgacaqG1bGaaeiDaiaabMgacaqGVbGaaeOBaiaabc caceWG4bGbauaacaaMc8UaaGPaVlaabIcacaqGNbGaaeyzaiaab6ga caqGLbGaaeOCaiaabggacaqG0bGaaeyzaiaabsgacaqGGaGaaeOyai aabMhacaqGGaGaaeyyaiaabYgacaqGNbGaae4BaiaabkhacaqGPbGa aeiDaiaabIgacaqGGaGaaeOmaiaabMcacaqGGaaabaGaaeiiaiaabc cacaqGGaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaqGVbGaaeOB aiaabwgacaqGGaGaaeiCaiaabggacaqGYbGaaeyzaiaab6gacaqG0b GaaeiiaiaadchacaWGVbGaamiCaiaacIcacaWGRbGaaiykaiaaykW7 caGGOaGaaeOzaiaab+gacaqGYbGaaeiiaiaabohacaqGVbGaaeyBai aabwgacaqGGaGaam4Aaiabg2da9iaaigdacaGGSaGaaGOmaiaacYca caGGUaGaaiOlaiaac6cacaGGSaGaam4uamaaBaaaleaacaWGWbGaam 4BaiaadchaaeqaaOGaaiykaiaab6caaeaacaqGYaGaaeOlaiaabcca caqGhbGaaeyzaiaab6gacaqGLbGaaeOCaiaabggacaqG0bGaaeyzai aabccacaqGHbGaaeiiaiaabkhacaqGHbGaaeOBaiaabsgacaqGVbGa aeyBaiaabccacaqGUbGaaeyDaiaab2gacaqGIbGaaeyzaiaabkhaca qGGaGaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeyicI4Saai4waiaa icdacaGGSaGaaGymaiaac2facaqGUaGaaeiiaiaabofacaqGLbGaae iDaiaabccacaqGGaGaamiEamaaBaaaleaacaWGUbGaamyzaiaadEha caaIXaaabeaakiabg2da9iabeU7aSnaaBaaaleaacaaIXaaabeaaki aaykW7ceWG4bGbauaacqGHRaWkcaGGOaGaaGymaiabgkHiTiabeU7a SnaaBaaaleaacaaIXaaabeaakiaacMcacaaMc8+aa0aaaeaacaWGyb aaaiaac6caaeaacaaIZaGaaiOlaiaaykW7caaMc8Uaaeitaiaabwga caqG0bGaaeiiaiabeU7aSnaaBaaaleaacaaIYaaabeaakiabg2da9m aaxadabaGaciyBaiaacMgacaGGUbaalqaabeqaaiaadQgacqGH9aqp caaIXaaabaGaamOAaiabgcMi5kaadUgaaaqaaiaadofadaWgaaadba GaamiCaiaad+gacaWGWbaabeaaaaGccaaMc8+aauWaaeaacaaMc8Ua amiCaiaad+gacaWGWbGaaiikaiaadUgacaGGPaGaeyOeI0IaamiCai aad+gacaWGWbGaaiikaiaadQgacaGGPaGaaGPaVdGaayzcSlaawQa7 aiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaeiiaiaadsgacqGH9a qpdaqdaaqaaiaadIfaaaGaeyOeI0IaamiCaiaad+gacaWGWbGaaiik aiaadUgacaGGPaGaaeOlaaqaaiaabccacaqGGaGaaeiiaiaabccaca qGtbGaaeyzaiaabshacaqGGaGaaeiiaiaadIhadaWgaaWcbaGaamOB aiaadwgacaWG3bGaaGOmaaqabaGccqGH9aqpcaWGWbGaam4Baiaadc hacaGGOaGaam4AaiaacMcacqGHRaWkciGGTbGaaiyAaiaac6gadaGa daqaaiabeU7aSnaaBaaaleaacaaIYaaabeaakiaacYcacaaIXaaaca GL7bGaayzFaaGaaGPaVlaaykW7caWGKbGaaiOlaaaaaa@39E8@
Construction of Test Problems
There are usually several ways to generate a feasible FRE defined with different t-norms. In what follows, we present a procedure to generate random feasible max-Frank fuzzy relational equations:
Algorithm 4 (construction of feasible Max-Frank FRE):
1. Randomly select m columns { j 1 j 2 ,... j m }from J={ 1,2,...,n }. 2. Generate vector b whose elements are random numbers from [0,1]. 3. For i{ 1,2,...,m } Assign a random number from [ b i ,1] to  a i j i . End 4. For i{ 1,2,...,m }                For each k{ 1,2,...,m }{i} If   b k =0                          Set  a k j i =0.                      Else                          Assign a random number from [0 ,  log s ( 1+ ( s b k 1)( s a i j i 1) / ( s b i 1) )] to  a k j i .                End           End      End 5. For each i{ 1,2,...,m }and each j{ j 1 j 2 ,... j m }                    Assign a random number from [0,1] to  a ij .                End MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGXa GaaeOlaiaabccacaqGsbGaaeyyaiaab6gacaqGKbGaae4Baiaab2ga caqGSbGaaeyEaiaabccacaqGZbGaaeyzaiaabYgacaqGLbGaae4yai aabshacaqGGaGaamyBaiaabccacaqGJbGaae4BaiaabYgacaqG1bGa aeyBaiaab6gacaqGZbGaaeiiaiaabUhacaaMc8UaamOAamaaBaaale aacaaIXaaabeaakiaabYcacaqGGaGaamOAamaaBaaaleaacaaIYaaa beaakiaabYcacaqGUaGaaeOlaiaab6cacaqGSaGaaeiiaiaadQgada WgaaWcbaGaamyBaaqabaGccaqG9bGaaGPaVlaabAgacaqGYbGaae4B aiaab2gacaqGGaGaamOsaiaab2dadaGadaqaaiaaigdacaqGSaGaaG OmaiaabYcacaqGUaGaaeOlaiaab6cacaqGSaGaamOBaaGaay5Eaiaa w2haaiaab6caaeaacaqGYaGaaeOlaiaabccacaqGhbGaaeyzaiaab6 gacaqGLbGaaeOCaiaabggacaqG0bGaaeyzaiaabccacaqG2bGaaeyz aiaabogacaqG0bGaae4BaiaabkhacaqGGaGaamOyaiaabccacaqG3b GaaeiAaiaab+gacaqGZbGaaeyzaiaabccacaqGLbGaaeiBaiaabwga caqGTbGaaeyzaiaab6gacaqG0bGaae4CaiaabccacaqGHbGaaeOCai aabwgacaqGGaGaaeOCaiaabggacaqGUbGaaeizaiaab+gacaqGTbGa aeiiaiaab6gacaqG1bGaaeyBaiaabkgacaqGLbGaaeOCaiaabohaca qGGaGaaeOzaiaabkhacaqGVbGaaeyBaiaabccacaqGBbGaaeimaiaa bYcacaqGXaGaaeyxaiaab6caaeaacaqGZaGaaeOlaiaabccacaqGgb Gaae4BaiaabkhacaqGGaGaamyAaiabgIGiopaacmaabaGaaeymaiaa bYcacaqGYaGaaeilaiaab6cacaqGUaGaaeOlaiaabYcacaWGTbaaca GL7bGaayzFaaaabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaabg 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aabccaciGGSbGaai4BaiaacEgadaWgaaWcbaGaam4CaaqabaGcdaqa daqaaiaaigdacqGHRaWkdaWcgaqaaiaacIcacaWGZbWaaWbaaSqabe aacaWGIbWaaSbaaWqaaiaadUgaaeqaaaaakiabgkHiTiaaigdacaGG PaGaaiikaiaadohadaahaaWcbeqaaiaadggadaWgaaadbaGaamyAai aaykW7caWGQbWaaSbaaeaacaWGPbaabeaaaeqaaaaakiabgkHiTiaa igdacaGGPaaabaGaaiikaiaadohadaahaaWcbeqaaiaadkgadaWgaa adbaGaamyAaaqabaaaaOGaeyOeI0IaaGymaiaacMcaaaaacaGLOaGa ayzkaaGaaeyxaiaabccacaqG0bGaae4BaiaabccacaWGHbWaaSbaaS qaaiaadUgacaaMc8UaamOAamaaBaaameaacaWGPbaabeaaaSqabaGc caGGUaaabaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeyraiaab6gacaqGKbaabaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabweacaqGUbGa aeizaaqaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeyraiaab6 gacaqGKbaabaGaaeynaiaab6cacaqGGaGaaeOraiaab+gacaqGYbGa aeiiaiaabwgacaqGHbGaae4yaiaabIgacaqGGaGaamyAaiabgIGiop aacmaabaGaaeymaiaabYcacaqGYaGaaeilaiaab6cacaqGUaGaaeOl aiaabYcacaWGTbaacaGL7bGaayzFaaGaaGPaVlaaykW7caqGHbGaae OBaiaabsgacaqGGaGaaeyzaiaabggacaqGJbGaaeiAaiaabccacaaM c8UaamOAaiabgMGiplaabUhacaaMc8UaamOAamaaBaaaleaacaaIXa aabeaakiaabYcacaqGGaGaamOAamaaBaaaleaacaaIYaaabeaakiaa bYcacaqGUaGaaeOlaiaab6cacaqGSaGaaeiiaiaadQgadaWgaaWcba GaamyBaaqabaGccaqG9bGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaaabaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabgeacaqG ZbGaae4CaiaabMgacaqGNbGaaeOBaiaabccacaqGHbGaaeiiaiaabk hacaqGHbGaaeOBaiaabsgacaqGVbGaaeyBaiaabccacaqGUbGaaeyD aiaab2gacaqGIbGaaeyzaiaabkhacaqGGaGaaeOzaiaabkhacaqGVb GaaeyBaiaabccacaqGBbGaaGimaiaabYcacaqGXaGaaeyxaiaabcca caqG0bGaae4BaiaabccacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabe aakiaac6cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaaeaacaqGGaGaaeiiai aabccacaqGfbGaaeOBaiaabsgaaaaa@68AE@

By the following theorem, it is proved that algorithm 4 always generates random feasible max-Frank fuzzy relational equations.

Theorem 4: The solutions set S T F s (A,b) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGubWaa0baaWqaaiaadAeaaeaacaWGZbaaaaWcbeaakiaa cIcacaWGbbGaaiilaiaadkgacaGGPaaaaa@3D8E@ of FRE (with Frank t-norm) constructed by algorithm 4 is not empty. Proof. According to step 3 of the algorithm, j i J i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaWGPbaabeaakiabgIGiolaadQeadaWgaaWcbaGaamyAaaqa baGccaGGSaGaaeiiaiabgcGiIiaadMgacqGHiiIZcaWGjbGaaiOlaa aa@4194@ Therefore, J i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGPbaabeaakiabgcMi5kabgwGiglaacYcacaqGGaGaeyia IiIaamyAaiabgIGiolaadMeacaGGUaaaaa@413D@ To complete the proof, we show that j i J ˜ i , iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaWGPbaabeaakiabgIGiolqadQeagaacamaaBaaaleaacaWG PbaabeaakiaacYcacaqGGaGaeyiaIiIaamyAaiabgIGiolaadMeaca GGUaaaaa@41A3@ By contradiction, suppose that the second simplification process reset a i j i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAamaaBaaameaacaWGPbaabeaaaSqabaaaaa@3A0A@ to zero, for some iI. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadMeacaGGUaaaaa@39E7@ So, b i 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaakiabgcMi5kaaicdaaaa@3A81@ and there must exists some kI (ki) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadMeacaqGGaGaaiikaiaadUgacqGHGjsUcaWGPbGaaiykaaaa @3ED8@ such that either j i J k ,  b k 0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaBa aaleaacaWGPbaabeaakiabgIGiolaadQeadaWgaaWcbaGaam4Aaaqa baGccaGGSaGaaeiiaiaadkgadaWgaaWcbaGaam4AaaqabaGccqGHGj sUcaaIWaaaaa@4162@ and log s ( 1+ ( s b k 1)(s1) / ( s a k j i 1) )< log s ( 1+ ( s b i 1)(s1) / ( s a i j i 1) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaaIXaGaey4k aSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqabaGaamOyamaaBaaame aacaWGRbaabeaaaaGccqGHsislcaaIXaGaaiykaiaacIcacaWGZbGa eyOeI0IaaGymaiaacMcaaeaacaGGOaGaam4CamaaCaaaleqabaGaam yyamaaBaaameaacaWGRbGaaGPaVlaadQgadaWgaaqaaiaadMgaaeqa aaqabaaaaOGaeyOeI0IaaGymaiaacMcaaaaacaGLOaGaayzkaaGaey ipaWJaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadohaaeqaaOWaaeWa aeaacaaIXaGaey4kaSYaaSGbaeaacaGGOaGaam4CamaaCaaaleqaba GaamOyamaaBaaameaacaWGPbaabeaaaaGccqGHsislcaaIXaGaaiyk aiaacIcacaWGZbGaeyOeI0IaaGymaiaacMcaaeaacaGGOaGaam4Cam aaCaaaleqabaGaamyyamaaBaaameaacaWGPbGaaGPaVlaadQgadaWg aaqaaiaadMgaaeqaaaqabaaaaOGaeyOeI0IaaGymaiaacMcaaaaaca GLOaGaayzkaaaaaa@6D3B@ or b k =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGRbaabeaakiabg2da9iaaicdaaaa@39C2@ and a k j i >0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbGaaGPaVlaadQgadaWgaaadbaGaamyAaaqabaaaleqa aOGaeyOpa4JaaGimaiaac6caaaa@3E15@ In the former case, we note that a k j i > log s ( 1+ ( s b k 1)( s a i j i 1) / ( s b i 1) ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbGaaGPaVlaadQgadaWgaaadbaGaamyAaaqabaaaleqa aOGaeyOpa4JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadohaaeqaaO WaaeWaaeaacaaIXaGaey4kaSYaaSGbaeaacaGGOaGaam4CamaaCaaa leqabaGaamOyamaaBaaameaacaWGRbaabeaaaaGccqGHsislcaaIXa GaaiykaiaacIcacaWGZbWaaWbaaSqabeaacaWGHbWaaSbaaWqaaiaa dMgacaaMc8UaamOAamaaBaaabaGaamyAaaqabaaabeaaaaGccqGHsi slcaaIXaGaaiykaaqaaiaacIcacaWGZbWaaWbaaSqabeaacaWGIbWa aSbaaWqaaiaadMgaaeqaaaaakiabgkHiTiaaigdacaGGPaaaaaGaay jkaiaawMcaaiaac6caaaa@5AB5@ Anyway, both cases contradict step 4.
Experimental Results and Comparison with Related Works
In this section, we present the experimental results for evaluating the performance of our algorithm. Firstly, we apply our algorithm to 8 test problems described in Appendix A. The test problems have been randomly generated in different sizes by algorithm 4 given in section 3. Since the objective function is an ordinary nonlinear function, we take some objective functions from the well-known source: Test Examples for Nonlinear Programming Codes [30]. In section 4.2, we make a comparison against the algorithms proposed in [42] and [29]. To perform a fair comparison, we follow the same experimental setup for the parameters θ=0.5, ξ=0.01,λ=0.995 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaaGimaiaac6cacaaI1aGaaiilaiaabccacqaH+oaEcqGH9aqp caaIWaGaaiOlaiaaicdacaaIXaGaaiilaiaaykW7cqaH7oaBcqGH9a qpcaaIWaGaaiOlaiaaiMdacaaI5aGaaGynaaaa@4A7F@ and γ=1.005 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0JaaGymaiaac6cacaaIWaGaaGimaiaaiwdaaaa@3C42@ as suggested by the authors in [29] and [42]. Since the authors did not explicitly report the size of the population, we consider S pop =50 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGWbGaam4BaiaadchaaeqaaOGaeyypa0JaaGynaiaaicda aaa@3C60@ for all the three GAs. As mentioned before, we set q=0.1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2 da9iaaicdacaGGUaGaaGymaaaa@3A18@ in relation (2) for the current GA. Moreover, in order to compare our algorithm with max-min GA (max-product GA [29]), we modified all the definitions used in the current GA based on the minimum t-norm (product t-norm) [42]. For example, we used the simplification process presented in [42] for minimum, and the simplification process given in [19,29] for product. Finally, 30 experiments are performed for all the GAs and for eight test problems reported in Appendix B, that is, each of the preceding GA is executed 30 times for each test problem. All the test problems included in Appendix A, have been defined by considering s=2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9iaaikdaaaa@38AF@ in T F s . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaWGgbaabaGaam4Caaaakiaac6caaaa@397A@ Also, the maximum number of iterations is equal to 100 for all the methods.
Performance of the Max-Frank GA
To verify the solutions found by the max-Frank GA, the optimal solutions of the test problems are also needed. Since is formed as the union of the finite number of convex closed cells (theorem 2), the optimal solutions are also acquired by the following procedure:

1. Computing all the convex cells of the Frank FRE.
2. Searching the optimal solution for each convex cell.
3. Finding the global optimum by comparing these local optimal solutions.

The computational results of the eight test problems (see Appendix A) are shown in Table 1 and Figures 1-8. In Table 1, the results are averaged over 30 runs and the average best-so-far solution; average mean fitness function and median of the best solution in the last iteration are reported.

Table 2 includes the best results found by the max-Frank GA and the above procedure. According to Table 2, the optimal solutions computed by the max-Frank GA and the optimal solutions obtained by the above procedure match very well. Tables 1 and 2, demonstrate the attractive ability of the max- Frank GA to detect the optimal solutions of problem (1). Also, the good convergence rate of the max- Frank GA could be concluded from Table 1 and figures 1-8.
Table 1: Results of applying the max-Frank GA to the eight test problems of Appendix A. The results have been averaged over 30 runs. Maximum number of iterations=100.

Test problems

Average best-so-far

Median best-so-far

Average mean fitness

A.1

1.73375

1.73375

1.74575

A.2

-2.5903

-2.5907

-2.5885

A.3

-0.10266

-0.10266

-0.10249

A.4

2.677598

2.677598

2.677661

A.5

68.60251

68.60251

68.60253

A.6

-0.46007

-0.46313

-0.45953

A.7

0.001365

0.001365

0.001372

A.8

105.996

105.996

105.996

Table 2: Comparison of the solutions found by Max-Frank GA and the optimal values of the test problems described in Appendix A

Test problems

Solutions of max-Frank GA

Optimal values

A.1

1.73375

1.73372

A.2

-2.5907

-2.5908

A.3

-0.10266

-0.10266

A.4

2.677598

2.677598

A.5

68.60251

68.60251

A.6

0.001365

0.001364

A.7

-1.19221

-1.19221

A.8

105.996

105.9953

Figure 1: The performance of the max-Frank GA on test problem A.1.
Figure 2: The performance of the max-Frank GA on test problem A.2.
Figure 3: The performance of the max-Frank GA on test problem A.3.
Figure 4: The performance of the max-Frank GA on test problem A.4.
Figure 5: The performance of the max-Frank GA on test problem A.5.
Figure 6: The performance of the max-Frank GA on test problem A.6.
Figure 7: The performance of the max-Frank GA on test problem A.7.
Figure 8: performance of the max-Frank GA on test problem A.8.
Comparisons with Other Works
As mentioned before, we can make a comparison between the current GA, max-min GA and max-product GA [42,29]. For this purpose, all the test problems described in Appendix B have been designed in such a way that they are feasible for both the minimum and product t-norms.

The first comparison is against max-min GA, and we apply our algorithm (modified for the minimum t-norm) to the test problems by considering as the minimum t-norm. The results are shown in Table 3 including the optimal objective values found by the current GA and max-min GA. As is shown in this table, the current GA finds better solutions for test problems 1, 5 and 6, and the same solutions for the other test problems.

Table 4 shows that the current GA finds the optimal values faster than max-min GA and hence has a higher convergence rate, even for the same solutions. The only exception is testing problem 8 in which all the results are the same. In all the cases, results marked with “*” indicate the better cases.

The second comparison is against the max-product GA. In this case, we apply our algorithm (modified for the product t-norm) to the same test problems by considering as the product t-norm (Tables 5 and 6).

The results, in Tables 5 and 6, demonstrate that the current GA produces better solutions (or the same solutions with a higher convergence rate) when compared against max-product GAs for all the test problems.
Table 3: Best results found by our algorithm and max-min GA

Test problems

Lu and Fang

Our algorithm

B.1

8.429676

8.4296754*

B.2

-1.3888

-1.3888

B.3

0

0

B.4

5.0909

5.0909

B.5

71.1011

71.0968*

B.6

-0.3291

-0.4175*

B.7

-0.6737

-0.6737

B.8

93.9796

93.9796

Table 4: A Comparison between the results found by the current GA and max-min GA

Test problems

 

Max-min GA

Our GA

B.1

Average best-so-far

8.429701

8.4296796*

Median best-so-far

8.429676

8.429676

Average mean fitness

8.430887

8.4298745*

B.2

Average best-so-far

-1.3888

-1.3888

Median best-so-far

-1.3888

-1.3888

Average mean fitness

-1.3877

-1.3886*

B.3

Average best-so-far

0

0

Median best-so-far

0

0

Average mean fitness

7.15E-07

0*

B.4

Average best-so-far

5.0909

5.0909

Median best-so-far

5.0909

5.0909

Average mean fitness

5.091

5.0908*

B.5

Average best-so-far

71.1011

71.0969*

Median best-so-far

71.1011

71.0968*

Average mean fitness

71.1327

71.1216*

B.6

Average best-so-far

-0.3291

-0.4175*

Median best-so-far

-0.3291

-0.4175*

Average mean fitness

-0.3287

-0.4162*

B.7

Average best-so-far

-0.6737

-0.6737

Median best-so-far

-0.6737

-0.6737

Average mean fitness

-0.6736

-0.6737*

B.8

Average best-so-far

93.9796

93.9796

Median best-so-far

93.9796

93.9796

Average mean fitness

93.9796

93.9796

Table 5: Best results found by our algorithm and max-product GA

Test problems

Hassanzadeh et al.

Our algorithm

B.1

13.6174

13.61740246*

B.2

-1.5557

-1.5557

B.3

0

0

B.4

5.8816

5.8816

B.5

45.065

45.0314*

B.6

-0.3671

-0.4622*

B.7

-2.47023

-2.47023

B.8

38.0195

38.0150*

Table 6: A Comparison between the results found by the current GA and max-product GA

Test problems

 

Max-product GA

Our GA

B.1

Average best-so-far

13.61745

13.61740502*

Median best-so-far

13.6174

13.61740260*

Average mean fitness

13.61786

13.61781613*

B.2

Average best-so-far

-1.5557

-1.5557

Median best-so-far

-1.5557

-1.5557

Average mean fitness

-1.5524

-1.5557*

B.3

Average best-so-far

0

0

Median best-so-far

0

0

Average mean fitness

1.54E-05

0*

B.4

Average best-so-far

5.8816

5.8816

Median best-so-far

5.8816

5.8816

Average mean fitness

5.8823

5.8816*

B.5

Average best-so-far

45.065

45.0315*

Median best-so-far

45.065

45.0314*

Average mean fitness

45.1499

45.0460*

B.6

Average best-so-far

-0.3671

-0.4622*

Median best-so-far

-0.3671

-0.4622*

Average mean fitness

-0.3668

-0.4614*

B.7

Average best-so-far

-2.47023

-2.47023

Median best-so-far

-2.47023

-2.47023

Average mean fitness

-2.47018

-2.470213*

B.8

Average best-so-far

38.0195

38.0150*

Median best-so-far

38.0195

38.0150*

Average mean fitness

38.0292

38.0171*

In [42], the proposed mutation operator decreases one variable of vector x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F2@ to a random number between [0, x j ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaic dacaaMc8UaaiilaiaaykW7caWG4bWaaSbaaSqaaiaadQgaaeqaaOGa aiykaaaa@3E23@ each time (the same mutation operator has been used in [29]). In this mutation operator, a decreasing variable often followed by increasing several other variables to guarantee the feasibility of a new solution. However, in the current GA, the feasibility of the new solution x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafa aaaa@36FE@ is simultaneously obtained by decreasing a proper variable to zero. Therefore, we have no need to revise the new solution to make it feasible. Moreover, since the proposed mutation operator decreases the selected variables to zeros, the new individuals are more likely to have greater distances from the maximum solution X ¯ , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiaacYcaaaa@3793@ especially x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafa aaaa@36FE@ may be even a minimal solution (see remark 4). This strategy increases the ability of the algorithm to expand the search space for finding new individuals.

Finally, authors in both [29] and [42] used the same “threepoint” crossover operator. The three-point crossover is defined by three points (two parents x 1 ,  x 2 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiaacYcacaqGGaGaamiEamaaBaaaleaacaaI YaaabeaakiaacYcaaaa@3BD5@ and the maximum solution X ¯ ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiaacMcaaaa@3790@ and two operators called “contraction” and “extraction”. Both contraction and extraction operators are employed between x 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaaaaa@37D9@ and x 2 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIYaaabeaakiaacYcaaaa@3894@ and between x i  (i=1,2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiaabccacaGGOaGaamyAaiabg2da9iaaigda caGGSaGaaGOmaiaacMcaaaa@3E2D@ and X ¯ . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiaac6caaaa@3795@ However, from the four mentioned cases, only one case certainly results in a feasible offspring (i.e., the contraction between x i  (i=1,2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiaabccacaGGOaGaamyAaiabg2da9iaaigda caGGSaGaaGOmaiaacMcaaaa@3E2D@ and X ¯ ). MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiaacMcacaGGUaaaaa@3842@ Therefore, for the other three cases, the feasibility of the new generated solutions must be checked by substituting them into the fuzzy relational equations as well as the constraints x j [0,1], jJ. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGQbaabeaakiabgIGiolaacUfacaaIWaGaaiilaiaaigda caGGDbGaaiilaiaabccacqGHaiIicaWGQbGaeyicI4SaamOsaiaac6 caaaa@4397@ In contrast, the current crossover operator uses only one parent each time. Offspring x new1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIXaaabeaaaaa@3AB2@ is obtained as a random point on the line segment between x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafa aaaa@36FE@ and X ¯ . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca WGybaaaiaac6caaaa@3795@ But, offspring x new2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIYaaabeaaaaa@3AB3@ lies close to its parent. This difference between x new1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIXaaabeaaaaa@3AB2@ and x new2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIYaaabeaaaaa@3AB3@ provides a suitable tradeoff between exploration and exploitation. Also, as is stated in remark 6, the new solutions x new1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIXaaabeaaaaa@3AB2@ and x new2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamyzaiaadEhacaaIYaaabeaaaaa@3AB3@ are always feasible.
Conclusion
In this paper, we investigated the resolution of FRE defined by Frank family of t-norms and introduced a nonlinear problem with the max-Frank fuzzy relational equations. In order to determine the feasibility of the problem, two necessary and sufficient conditions were presented. Also, two simplification approaches (depending on the Frank t-norm) were proposed to simplify the problem. A genetic algorithm was designed for solving the nonlinear optimization problems constrained by the max-Frank FRE. Moreover, we presented a method for generating feasible max-Frank FREs as test problems for the performance evaluation of the proposed algorithm. Experiments were performed with the proposed method in the generated feasible test problems. We conclude that the proposed GA can find the optimal solutions for all the cases with a great convergence rate. Moreover, a comparison was made between the proposed method and maxmin and max-product GAs, which solve the nonlinear optimization problems subjected to the FREs defined by max-min and maxproduct compositions, respectively. The results showed that the proposed method (modified by minimum and product t-norms) finds better solutions compared with the solutions obtained by the other algorithms.

As future works, we aim at testing our algorithm in other type of nonlinear optimization problems whose constraints are defined as FRE or FRI with other well-known t-norms.
Acknowledgment
We are very grateful to the anonymous referees and the editor in chief for their comments and suggestions, which were very helpful in improving the paper.
Appendix A
Test Problem A.1:
f(x)= ( x 1 +10 x 2 ) 2 +5 ( x 3 x 4 ) 2 + ( x 2 2 x 3 ) 4 +10 ( x 1 x 4 ) 4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaaGymaiaaicdacaWG4bWaaSbaaSqaaiaaikdaae qaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdacaGG OaGaamiEamaaBaaaleaacaaIZaaabeaakiabgkHiTiaadIhadaWgaa WcbaGaaGinaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaaIYa GaamiEamaaBaaaleaacaaIZaaabeaakiaacMcadaahaaWcbeqaaiaa isdaaaGccqGHRaWkcaaIXaGaaGimaiaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaI0aaabeaakiaa cMcadaahaaWcbeqaaiaaisdaaaaaaa@5D96@ b T =[0.4158 , 0.1076 , 0.4595] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaein aiaabgdacaqG1aGaaeioaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeymaiaabcdacaqG3aGaaeOnaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeinaiaabwdacaqG5aGaaeynaiaac2faaaa@4B63@ A=[ 0.6709    0.7777    0.9789    1.9722 0.0882    0.4192    0.2305    0.8316 0.7977    0.6639    0.9837    2.2341 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabAdacaqG3aGaaeimaiaa bMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaae 4naiaabEdacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeyoaiaabEdacaqG4aGaaeyoaiaabccacaqGGaGaaeiiaiaabc cacaqGXaGaaeOlaiaabMdacaqG3aGaaeOmaiaabkdaaeaacaqGWaGa aeOlaiaabcdacaqG4aGaaeioaiaabkdacaqGGaGaaeiiaiaabccaca qGGaGaaeimaiaab6cacaqG0aGaaeymaiaabMdacaqGYaGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabodacaqGWaGaae ynaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqG ZaGaaeymaiaabAdaaeaacaqGWaGaaeOlaiaabEdacaqG5aGaae4nai aabEdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG2aGa aeOnaiaabodacaqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdaca qGUaGaaeyoaiaabIdacaqGZaGaae4naiaabccacaqGGaGaaeiiaiaa bccacaqGYaGaaeOlaiaabkdacaqGZaGaaeinaiaabgdaaaGaay5wai aaw2faaaaa@83EA@
Test Problem A.2:
f(x)= x 1 x 2 x 3 x 1 x 3 + x 1 x 4 + x 2 x 3 x 2 x 4 x 3 x 5 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc cqGHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamiEam aaBaaaleaacaaIZaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGym aaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiEam aaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGaaGinaaqabaGc cqGHRaWkcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaale aacaaIZaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGc caWG4bWaaSbaaSqaaiaaisdaaeqaaOGaeyOeI0IaamiEamaaBaaale aacaaIZaaabeaakiaadIhadaWgaaWcbaGaaGynaaqabaGccaGGSaaa aa@5A7C@ b T =[0.9554 , 0.8724 , 0.4773] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeyo aiaabwdacaqG1aGaaeinaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeioaiaabEdacaqGYaGaaeinaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeinaiaabEdacaqG3aGaae4maiaac2faaaa@4B6D@ A=[ 0.5613    1.0452    0.4046    0.9749    0.1845 0.2927    0.3767    0.9619    0.3497    0.0209 0.8060    0.6808    0.1306    0.0814    0.3811 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabwdacaqG2aGaaeymaiaa bodacaqGGaGaaeiiaiaabccacaqGGaGaaeymaiaab6cacaqGWaGaae inaiaabwdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeinaiaabcdacaqG0aGaaeOnaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabMdacaqG3aGaaeinaiaabMdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqGXaGaaeioaiaabsdacaqG1a aabaGaaeimaiaab6cacaqGYaGaaeyoaiaabkdacaqG3aGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaae4maiaabEdacaqG2aGaae 4naiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabMdacaqG 2aGaaeymaiaabMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqGZaGaaeinaiaabMdacaqG3aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeimaiaabkdacaqGWaGaaeyoaaqaaiaabcdaca qGUaGaaeioaiaabcdacaqG2aGaaeimaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabAdacaqG4aGaaeimaiaabIdacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaab6cacaqGXaGaae4maiaabcdacaqG 2aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabI dacaqGXaGaaeinaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabodacaqG4aGaaeymaiaabgdaaaGaay5waiaaw2faaaaa@9821@
Test Problem A.3:
f(x)= x 1 x 2 Ln(1+ x 4 x 5 )+ x 3 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaamitaiaad6gaca GGOaGaaGymaiabgUcaRiaadIhadaWgaaWcbaGaaGinaaqabaGccaWG 4bWaaSbaaSqaaiaaiwdaaeqaaOGaaiykaiabgUcaRiaadIhadaWgaa WcbaGaaG4maaqabaGccaGGSaaaaa@4B25@ b T =[0.8405 , 0.1722 , 0.2230 , 0.1664] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeio aiaabsdacaqGWaGaaeynaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeymaiaabEdacaqGYaGaaeOmaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeOmaiaabkdacaqGZaGaaeimaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeymaiaabAdacaqG2aGaaeinaiaac2faaaa@5186@ A=[ 0.1777    0.8427    0.4792    0.8871    0.9428 0.3737    0.3182    0.3945    0.1814    0.9498 0.9811    0.2967    0.0693    0.0771    0.5799 0.4287    0.7059    0.2500    0.0037    0.0515 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabgdacaqG3aGaae4naiaa bEdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG4aGaae inaiaabkdacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeinaiaabEdacaqG5aGaaeOmaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabIdacaqG4aGaae4naiaabgdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeinaiaabkdacaqG4a aabaGaaeimaiaab6cacaqGZaGaae4naiaabodacaqG3aGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaae4maiaabgdacaqG4aGaae OmaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabodacaqG 5aGaaeinaiaabwdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqGXaGaaeioaiaabgdacaqG0aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeyoaiaabsdacaqG5aGaaeioaaqaaiaabcdaca qGUaGaaeyoaiaabIdacaqGXaGaaeymaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabkdacaqG5aGaaeOnaiaabEdacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeOnaiaabMdacaqG ZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabE dacaqG3aGaaeymaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabwdacaqG3aGaaeyoaiaabMdaaeaacaqGWaGaaeOlaiaabsdaca qGYaGaaeioaiaabEdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaa b6cacaqG3aGaaeimaiaabwdacaqG5aGaaeiiaiaabccacaqGGaGaae iiaiaabcdacaqGUaGaaeOmaiaabwdacaqGWaGaaeimaiaabccacaqG GaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqGWaGaae4maiaabE dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeyn aiaabgdacaqG1aaaaiaawUfacaGLDbaaaaa@B7BF@
Test Problem A.4:
f(x)= x 1 +2 x 2 +4 x 5 + e x 1 x 4 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc cqGHRaWkcaaIYaGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRi aaisdacaWG4bWaaSbaaSqaaiaaiwdaaeqaaOGaey4kaSIaamyzamaa CaaaleqabaGaamiEamaaBaaameaacaaIXaaabeaaliaadIhadaWgaa adbaGaaGinaaqabaaaaOGaaiilaaaa@49D4@ b T =[0.3655 , 0.8803 , 0.1247 , 0.4423] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaae4m aiaabAdacaqG1aGaaeynaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeioaiaabIdacaqGWaGaae4maiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeymaiaabkdacaqG0aGaae4naiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeinaiaabsdacaqGYaGaae4maiaac2faaaa@5192@ A=[ 0.2714    0.7588    0.2497    0.6697    1.2873    0.2635 0.4336    0.2319    0.9441    0.5572    0.6589    0.1893 0.1520    0.0627    0.0946    0.7278    0.8340    0.2704 0.6816    0.3394    0.1664    0.9352    1.2527    0.1483 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabkdacaqG3aGaaeymaiaa bsdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaae ynaiaabIdacaqG4aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeOmaiaabsdacaqG5aGaae4naiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabAdacaqG2aGaaeyoaiaabEdacaqGGaGaaeii aiaabccacaqGGaGaaeymaiaab6cacaqGYaGaaeioaiaabEdacaqGZa GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabAda caqGZaGaaeynaaqaaiaabcdacaqGUaGaaeinaiaabodacaqGZaGaae OnaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabkdacaqG ZaGaaeymaiaabMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqG5aGaaeinaiaabsdacaqGXaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeynaiaabwdacaqG3aGaaeOmaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabAdacaqG1aGaaeioaiaabMda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGXaGaaeioai aabMdacaqGZaaabaGaaeimaiaab6cacaqGXaGaaeynaiaabkdacaqG WaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabA dacaqGYaGaae4naiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabcdacaqG5aGaaeinaiaabAdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG3aGaaeOmaiaabEdacaqG4aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeioaiaabodacaqG0aGaaeimai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabkdacaqG3aGa aeimaiaabsdaaeaacaqGWaGaaeOlaiaabAdacaqG4aGaaeymaiaabA dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaae4m aiaabMdacaqG0aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUa GaaeymaiaabAdacaqG2aGaaeinaiaabccacaqGGaGaaeiiaiaabcca caqGWaGaaeOlaiaabMdacaqGZaGaaeynaiaabkdacaqGGaGaaeiiai aabccacaqGGaGaaeymaiaab6cacaqGYaGaaeynaiaabkdacaqG3aGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeymaiaabsdaca qG4aGaae4maaaacaGLBbGaayzxaaaaaa@D2ED@
Test Problem A.5:
f(x)= k=1 5 [100 ( x k+1 x k 2 ) 2 + (1 x k ) 2 ] , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9maaqahabaGaai4waiaaigdacaaIWaGa aGimaiaacIcacaWG4bWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabe aakiabgkHiTiaadIhadaqhaaWcbaGaam4AaaqaaiaaikdaaaGccaGG PaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiikaiaaigdacqGHsi slcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykamaaCaaaleqabaGa aGOmaaaakiaac2faaSqaaiaadUgacqGH9aqpcaaIXaaabaGaaGynaa qdcqGHris5aOGaaiilaaaa@5569@ b T =[0.1412 , 0.3305 , 0.2189 , 0.0391 , 0.2017] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeym aiaabsdacaqGXaGaaeOmaiaabccacaqGSaGaaeiiaiaabcdacaqGUa Gaae4maiaabodacaqGWaGaaeynaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeOmaiaabgdacaqG4aGaaeyoaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeimaiaabodacaqG5aGaaeymaiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaaeOmaiaabcdacaqGXaGaae4naiaac2faaa a@57B4@ A=[ 0.0164    0.0189    1.4529    0.3365    0.9700    0.0418 0.3328    0.8476    2.3133    0.9393    1.5146    0.3614 0.0042    0.5227    2.2140    0.5775    1.2917    0.3642 0.0163    0.0729    0.8260    0.2225    0.0124    0.0663 0.2192    0.4332    0.1229    0.0621    0.9475    0.2630 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabcdacaqGXaGaaeOnaiaa bsdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaae ymaiaabIdacaqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabgdacaqG UaGaaeinaiaabwdacaqGYaGaaeyoaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabodacaqGZaGaaeOnaiaabwdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG5aGaae4naiaabcdacaqGWa GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabsda caqGXaGaaeioaaqaaiaabcdacaqGUaGaae4maiaabodacaqGYaGaae ioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqG 0aGaae4naiaabAdacaqGGaGaaeiiaiaabccacaqGGaGaaeOmaiaab6 cacaqGZaGaaeymaiaabodacaqGZaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeyoaiaabodacaqG5aGaae4maiaabccacaqGGa GaaeiiaiaabccacaqGXaGaaeOlaiaabwdacaqGXaGaaeinaiaabAda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaeOnai aabgdacaqG0aaabaGaaeimaiaab6cacaqGWaGaaeimaiaabsdacaqG YaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeynaiaabk dacaqGYaGaae4naiaabccacaqGGaGaaeiiaiaabccacaqGYaGaaeOl aiaabkdacaqGXaGaaeinaiaabcdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG1aGaae4naiaabEdacaqG1aGaaeiiaiaabcca caqGGaGaaeiiaiaabgdacaqGUaGaaeOmaiaabMdacaqGXaGaae4nai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabodacaqG2aGa aeinaiaabkdaaeaacaqGWaGaaeOlaiaabcdacaqGXaGaaeOnaiaabo dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaae4n aiaabkdacaqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUa GaaeioaiaabkdacaqG2aGaaeimaiaabccacaqGGaGaaeiiaiaabcca caqGWaGaaeOlaiaabkdacaqGYaGaaeOmaiaabwdacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqGWaGaaeymaiaabkdacaqG0aGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabAdaca qG2aGaae4maaqaaiaabcdacaqGUaGaaeOmaiaabgdacaqG5aGaaeOm aiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabsdacaqGZa Gaae4maiaabkdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6ca caqGXaGaaeOmaiaabkdacaqG5aGaaeiiaiaabccacaqGGaGaaeiiai aabcdacaqGUaGaaeimaiaabAdacaqGYaGaaeymaiaabccacaqGGaGa aeiiaiaabccacaqGWaGaaeOlaiaabMdacaqG0aGaae4naiaabwdaca qGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGYaGaaeOnaiaa bodacaqGWaaaaiaawUfacaGLDbaaaaa@F8D1@
Test Problem A.6:
f(x)=0.5( x 1 x 4 x 2 x 3 + x 2 x 6 x 5 x 6 + x 5 x 4 x 6 x 3 ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iabgkHiTiaaicdacaGGUaGaaGynaiaa cIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaaca aI0aaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGccaWG 4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiEamaaBaaaleaaca aIYaaabeaakiaadIhadaWgaaWcbaGaaGOnaaqabaGccqGHsislcaWG 4bWaaSbaaSqaaiaaiwdaaeqaaOGaamiEamaaBaaaleaacaaI2aaabe aakiabgUcaRiaadIhadaWgaaWcbaGaaGynaaqabaGccaWG4bWaaSba aSqaaiaaisdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaI2aaabe aakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaiilaaaa@5B33@ b T =[0.0509 , 0.0145 , 0.7912 , 0.0331 , 0.0593] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeim aiaabwdacaqGWaGaaeyoaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeimaiaabgdacaqG0aGaaeynaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaae4naiaabMdacaqGXaGaaeOmaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeimaiaabodacaqGZaGaaeymaiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaaeimaiaabwdacaqG5aGaae4maiaac2faaa a@57B9@ A=[ 0.6170    0.9057    0.8152    0.6013    0.3819    0.0228 0.1390    0.0366    0.0472    0.1229    0.2530    0.0046 1.7889    4.1166    0.4044    0.9866    0.6617    0.8761 0.1927    0.3448    0.8200    0.4613    0.3322    0.0193 0.9655    0.4549    0.4374    0.3960    0.7810    0.0379 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabAdacaqGXaGaae4naiaa bcdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG5aGaae imaiaabwdacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeioaiaabgdacaqG1aGaaeOmaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabAdacaqGWaGaaeymaiaabodacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaeioaiaabgdacaqG5a GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabkda caqGYaGaaeioaaqaaiaabcdacaqGUaGaaeymaiaabodacaqG5aGaae imaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqG ZaGaaeOnaiaabAdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqGWaGaaeinaiaabEdacaqGYaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeymaiaabkdacaqGYaGaaeyoaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaae4maiaabcda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeimai aabsdacaqG2aaabaGaaeymaiaab6cacaqG3aGaaeioaiaabIdacaqG 5aGaaeiiaiaabccacaqGGaGaaeiiaiaabsdacaqGUaGaaeymaiaabg dacaqG2aGaaeOnaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabsdacaqGWaGaaeinaiaabsdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG5aGaaeioaiaabAdacaqG2aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeOnaiaabAdacaqGXaGaae4nai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqG3aGa aeOnaiaabgdaaeaacaqGWaGaaeOlaiaabgdacaqG5aGaaeOmaiaabE dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaein aiaabsdacaqG4aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUa GaaeioaiaabkdacaqGWaGaaeimaiaabccacaqGGaGaaeiiaiaabcca caqGWaGaaeOlaiaabsdacaqG2aGaaeymaiaabodacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqGZaGaae4maiaabkdacaqGYaGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabgdaca qG5aGaae4maaqaaiaabcdacaqGUaGaaeyoaiaabAdacaqG1aGaaeyn aiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabsdacaqG1a GaaeinaiaabMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6ca caqG0aGaae4maiaabEdacaqG0aGaaeiiaiaabccacaqGGaGaaeiiai aabcdacaqGUaGaae4maiaabMdacaqG2aGaaeimaiaabccacaqGGaGa aeiiaiaabccacaqGWaGaaeOlaiaabEdacaqG4aGaaeymaiaabcdaca qGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaae4maiaa bEdacaqG5aaaaiaawUfacaGLDbaaaaa@F903@
Test Problem A.7:
f(x)= e x 1 x 2 x 3 x 4 x 5 0.5 ( x 2 3 + x 6 3 + x 7 3 +1) 2 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiaadIhadaWg aaadbaGaaGymaaqabaWccaWG4bWaaSbaaWqaaiaaikdaaeqaaSGaam iEamaaBaaameaacaaIZaaabeaaliaadIhadaWgaaadbaGaaGinaaqa baWccaWG4bWaaSbaaWqaaiaaiwdaaeqaaaaakiabgkHiTiaaicdaca GGUaGaaGynaiaacIcacaWG4bWaa0baaSqaaiaaikdaaeaacaaIZaaa aOGaey4kaSIaamiEamaaDaaaleaacaaI2aaabaGaaG4maaaakiabgU caRiaadIhadaqhaaWcbaGaaG4naaqaaiaaiodaaaGccqGHRaWkcaaI XaGaaiykamaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@5691@ b T =[0.5840 , 0.0098 , 0.4494 , 0.4891 , 0.7293 , 0.4567] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeyn aiaabIdacaqG0aGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeimaiaabcdacaqG5aGaaeioaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeinaiaabsdacaqG5aGaaeinaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeinaiaabIdacaqG5aGaaeymaiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaae4naiaabkdacaqG5aGaae4maiaabccaca qGSaGaaeiiaiaabcdacaqGUaGaaeinaiaabwdacaqG2aGaae4naiaa c2faaaa@5E13@ A=[ 0.5767    0.3665    0.6105    0.5165    0.9941    0.2509    0.8566 0.0102    0.0108    0.2054    0.0002    0.0093    0.0055    0.4356 0.1377    0.4889    0.9474    0.0434    0.2976    0.4919    1.6486 0.5335    0.4963    0.5755    0.1347    0.8491    0.2401    2.2493 0.5819    0.4266    0.1413    0.8192    0.7834    0.9808    4.5920 0.1953    0.3644    0.3268    0.1456    0.0479    0.7150    1.2340 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabwdacaqG3aGaaeOnaiaa bEdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaae OnaiaabAdacaqG1aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeOnaiaabgdacaqGWaGaaeynaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabwdacaqGXaGaaeOnaiaabwdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeyoaiaabsdacaqGXa GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabwda caqGWaGaaeyoaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlai aabIdacaqG1aGaaeOnaiaabAdaaeaacaqGWaGaaeOlaiaabcdacaqG XaGaaeimaiaabkdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqGWaGaaeymaiaabcdacaqG4aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeOmaiaabcdacaqG1aGaaeinaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqGWaGaaeimaiaabkda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeimai aabMdacaqGZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGa aeimaiaabcdacaqG1aGaaeynaiaabccacaqGGaGaaeiiaiaabccaca qGWaGaaeOlaiaabsdacaqGZaGaaeynaiaabAdaaeaacaqGWaGaaeOl aiaabgdacaqGZaGaae4naiaabEdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG0aGaaeioaiaabIdacaqG5aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeyoaiaabsdacaqG3aGaaeinai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqG0aGa ae4maiaabsdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6caca qGYaGaaeyoaiaabEdacaqG2aGaaeiiaiaabccacaqGGaGaaeiiaiaa bcdacaqGUaGaaeinaiaabMdacaqGXaGaaeyoaiaabccacaqGGaGaae iiaiaabccacaqGXaGaaeOlaiaabAdacaqG0aGaaeioaiaabAdaaeaa caqGWaGaaeOlaiaabwdacaqGZaGaae4maiaabwdacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqG0aGaaeyoaiaabAdacaqGZaGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeynaiaabEdaca qG1aGaaeynaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaa bgdacaqGZaGaaeinaiaabEdacaqGGaGaaeiiaiaabccacaqGGaGaae imaiaab6cacaqG4aGaaeinaiaabMdacaqGXaGaaeiiaiaabccacaqG GaGaaeiiaiaabcdacaqGUaGaaeOmaiaabsdacaqGWaGaaeymaiaabc cacaqGGaGaaeiiaiaabccacaqGYaGaaeOlaiaabkdacaqG0aGaaeyo aiaabodaaeaacaqGWaGaaeOlaiaabwdacaqG4aGaaeymaiaabMdaca qGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaaeOmaiaa bAdacaqG2aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae ymaiaabsdacaqGXaGaae4maiaabccacaqGGaGaaeiiaiaabccacaqG WaGaaeOlaiaabIdacaqGXaGaaeyoaiaabkdacaqGGaGaaeiiaiaabc cacaqGGaGaaeimaiaab6cacaqG3aGaaeioaiaabodacaqG0aGaaeii aiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeyoaiaabIdacaqGWa GaaeioaiaabccacaqGGaGaaeiiaiaabccacaqG0aGaaeOlaiaabwda caqG5aGaaeOmaiaabcdaaeaacaqGWaGaaeOlaiaabgdacaqG5aGaae ynaiaabodacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG ZaGaaeOnaiaabsdacaqG0aGaaeiiaiaabccacaqGGaGaaeiiaiaabc dacaqGUaGaae4maiaabkdacaqG2aGaaeioaiaabccacaqGGaGaaeii aiaabccacaqGWaGaaeOlaiaabgdacaqG0aGaaeynaiaabAdacaqGGa GaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeinaiaabEda caqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae4nai aabgdacaqG1aGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGXaGa aeOlaiaabkdacaqGZaGaaeinaiaabcdaaaGaay5waiaaw2faaaaa@4829@
Test Problem A.8:
f(x)= ( x 1 1) 2 + ( x 7 1) 2 +10 k=1 6 (10k) ( x k 2 x k+1 ) 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaaikdaaaGccq GHRaWkcaGGOaGaamiEamaaBaaaleaacaaI3aaabeaakiabgkHiTiaa igdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaiaaic dadaaeWbqaaiaacIcacaaIXaGaaGimaiabgkHiTiaadUgacaGGPaGa aiikaiaadIhadaqhaaWcbaGaam4AaaqaaiaaikdaaaGccqGHsislca WG4bWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiaacMcadaah aaWcbeqaaiaaikdaaaaabaGaam4Aaiabg2da9iaaigdaaeaacaaI2a aaniabggHiLdaaaa@5D61@ b T =[0.8390 , 0.5471 , 0.0128 , 0.2467 , 0.0120 , 0.2738] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeio aiaabodacaqG5aGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeynaiaabsdacaqG3aGaaeymaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeimaiaabgdacaqGYaGaaeioaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeOmaiaabsdacaqG2aGaae4naiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaaeimaiaabgdacaqGYaGaaeimaiaabccaca qGSaGaaeiiaiaabcdacaqGUaGaaeOmaiaabEdacaqGZaGaaeioaiaa c2faaaa@5DF5@ A=[ 0.7281    0.7902    0.9953    3.7319    3.2348    0.4420    0.2382 0.6692    1.2252    0.5667    0.3056    3.0033    0.0970    0.3301 0.0061    0.0441    0.0069    0.0758    0.5072    0.0054    0.5111 0.2067    0.3460    0.0664    2.9037    1.7853    0.5744    0.9343 0.0030    0.0199    0.0082    0.6132    0.1881    0.0010    0.3507 0.0901    0.9337    0.2309    1.6215    3.1454    0.4806    0.9565 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabEdacaqGYaGaaeioaiaa bgdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaae yoaiaabcdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeyoaiaabMdacaqG1aGaae4maiaabccacaqGGaGaaeiiaiaabc cacaqGZaGaaeOlaiaabEdacaqGZaGaaeymaiaabMdacaqGGaGaaeii aiaabccacaqGGaGaae4maiaab6cacaqGYaGaae4maiaabsdacaqG4a GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeinaiaabsda caqGYaGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlai aabkdacaqGZaGaaeioaiaabkdaaeaacaqGWaGaaeOlaiaabAdacaqG 2aGaaeyoaiaabkdacaqGGaGaaeiiaiaabccacaqGGaGaaeymaiaab6 cacaqGYaGaaeOmaiaabwdacaqGYaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeynaiaabAdacaqG2aGaae4naiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabodacaqGWaGaaeynaiaabAda caqGGaGaaeiiaiaabccacaqGGaGaae4maiaab6cacaqGWaGaaeimai aabodacaqGZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGa aeimaiaabMdacaqG3aGaaeimaiaabccacaqGGaGaaeiiaiaabccaca qGWaGaaeOlaiaabodacaqGZaGaaeimaiaabgdaaeaacaqGWaGaaeOl aiaabcdacaqGWaGaaeOnaiaabgdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqGWaGaaeinaiaabsdacaqGXaGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabcdacaqG2aGaaeyoai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqG3aGa aeynaiaabIdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6caca qG1aGaaeimaiaabEdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaa bcdacaqGUaGaaeimaiaabcdacaqG1aGaaeinaiaabccacaqGGaGaae iiaiaabccacaqGWaGaaeOlaiaabwdacaqGXaGaaeymaiaabgdaaeaa caqGWaGaaeOlaiaabkdacaqGWaGaaeOnaiaabEdacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqGZaGaaeinaiaabAdacaqGWaGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabAdaca qG2aGaaeinaiaabccacaqGGaGaaeiiaiaabccacaqGYaGaaeOlaiaa bMdacaqGWaGaae4maiaabEdacaqGGaGaaeiiaiaabccacaqGGaGaae ymaiaab6cacaqG3aGaaeioaiaabwdacaqGZaGaaeiiaiaabccacaqG GaGaaeiiaiaabcdacaqGUaGaaeynaiaabEdacaqG0aGaaeinaiaabc cacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabMdacaqGZaGaaein aiaabodaaeaacaqGWaGaaeOlaiaabcdacaqGWaGaae4maiaabcdaca qGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaaeymaiaa bMdacaqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae imaiaabcdacaqG4aGaaeOmaiaabccacaqGGaGaaeiiaiaabccacaqG WaGaaeOlaiaabAdacaqGXaGaae4maiaabkdacaqGGaGaaeiiaiaabc cacaqGGaGaaeimaiaab6cacaqGXaGaaeioaiaabIdacaqGXaGaaeii aiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabcdacaqGXa GaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaaboda caqG1aGaaeimaiaabEdaaeaacaqGWaGaaeOlaiaabcdacaqG5aGaae imaiaabgdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG 5aGaae4maiaabodacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabc dacaqGUaGaaeOmaiaabodacaqGWaGaaeyoaiaabccacaqGGaGaaeii aiaabccacaqGXaGaaeOlaiaabAdacaqGYaGaaeymaiaabwdacaqGGa GaaeiiaiaabccacaqGGaGaae4maiaab6cacaqGXaGaaeinaiaabwda caqG0aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeinai aabIdacaqGWaGaaeOnaiaabccacaqGGaGaaeiiaiaabccacaqGWaGa aeOlaiaabMdacaqG1aGaaeOnaiaabwdaaaGaay5waiaaw2faaaaa@47D2@
Appendix B
Test Problem B.1:
f(x)= ( x 1 +10 x 2 ) 2 +5 ( x 3 x 4 ) 2 + ( x 2 2 x 3 ) 4 +10 ( x 1 x 4 ) 4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaaGymaiaaicdacaWG4bWaaSbaaSqaaiaaikdaae qaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdacaGG OaGaamiEamaaBaaaleaacaaIZaaabeaakiabgkHiTiaadIhadaWgaa WcbaGaaGinaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaaIYa GaamiEamaaBaaaleaacaaIZaaabeaakiaacMcadaahaaWcbeqaaiaa isdaaaGccqGHRaWkcaaIXaGaaGimaiaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaI0aaabeaakiaa cMcadaahaaWcbeqaaiaaisdaaaaaaa@5D96@ b T =[0.2077 , 0.4709 , 0.8443] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeOm aiaabcdacaqG3aGaae4naiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeinaiaabEdacaqGWaGaaeyoaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeioaiaabsdacaqG0aGaae4maiaac2faaaa@4B63@ A=[ 0.4302    0.4464    0.0741    0.0751 0.1848    0.1603    0.4628    0.5929 0.9049    0.1707    0.8746    0.4210 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabsdacaqGZaGaaeimaiaa bkdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaae inaiaabAdacaqG0aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeimaiaabEdacaqG0aGaaeymaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabcdacaqG3aGaaeynaiaabgdaaeaacaqGWaGa aeOlaiaabgdacaqG4aGaaeinaiaabIdacaqGGaGaaeiiaiaabccaca qGGaGaaeimaiaab6cacaqGXaGaaeOnaiaabcdacaqGZaGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaaeinaiaabAdacaqGYaGaae ioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabwdacaqG 5aGaaeOmaiaabMdaaeaacaqGWaGaaeOlaiaabMdacaqGWaGaaeinai aabMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGXaGa ae4naiaabcdacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdaca qGUaGaaeioaiaabEdacaqG0aGaaeOnaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabsdacaqGYaGaaeymaiaabcdaaaGaay5wai aaw2faaaaa@83AC@
Test Problem B.2:
f(x)= x 1 x 2 x 3 x 1 x 3 + x 1 x 4 + x 2 x 3 x 2 x 4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc cqGHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamiEam aaBaaaleaacaaIZaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGym aaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiEam aaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGaaGinaaqabaGc cqGHRaWkcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaale aacaaIZaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGc caWG4bWaaSbaaSqaaiaaisdaaeqaaaaa@54F3@ b T =[0.4228 , 0.9427 , 0.9831] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaein aiaabkdacaqGYaGaaeioaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeyoaiaabsdacaqGYaGaae4naiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeyoaiaabIdacaqGZaGaaeymaiaac2faaaa@4B67@ A=[ 0.1280    0.7390    0.2852    0.2409 0.9991    0.7011    0.1688    0.9667 0.1711    0.6663    0.9882    0.6981 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabgdacaqGYaGaaeioaiaa bcdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaae 4maiaabMdacaqGWaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeOmaiaabIdacaqG1aGaaeOmaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabkdacaqG0aGaaeimaiaabMdaaeaacaqGWaGa aeOlaiaabMdacaqG5aGaaeyoaiaabgdacaqGGaGaaeiiaiaabccaca qGGaGaaeimaiaab6cacaqG3aGaaeimaiaabgdacaqGXaGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaaeymaiaabAdacaqG4aGaae ioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabMdacaqG 2aGaaeOnaiaabEdaaeaacaqGWaGaaeOlaiaabgdacaqG3aGaaeymai aabgdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG2aGa aeOnaiaabAdacaqGZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdaca qGUaGaaeyoaiaabIdacaqG4aGaaeOmaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabAdacaqG5aGaaeioaiaabgdaaaGaay5wai aaw2faaaaa@83CF@
Test Problem B.3:
f(x)= x 1 x 2 x 3 x 4 x 5 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaaleaacaaIZa aabeaakiaadIhadaWgaaWcbaGaaGinaaqabaGccaWG4bWaaSbaaSqa aiaaiwdaaeqaaOGaaiilaaaa@449C@ b T =[0.6714 , 0.5201 , 0.1500] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeOn aiaabEdacaqGXaGaaeinaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeynaiaabkdacaqGWaGaaeymaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeymaiaabwdacaqGWaGaaeimaiaac2faaaa@4B4C@ A=[ 0.4424    0.3592    0.6834    0.6329    0.9150 0.6878    0.7363    0.7040    0.6869    0.2002 0.6482    0.3947    0.4423    0.0769    0.0175 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabsdacaqG0aGaaeOmaiaa bsdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaae ynaiaabMdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeOnaiaabIdacaqGZaGaaeinaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabAdacaqGZaGaaeOmaiaabMdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeymaiaabwdacaqGWa aabaGaaeimaiaab6cacaqG2aGaaeioaiaabEdacaqG4aGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaae4naiaabodacaqG2aGaae 4maiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabEdacaqG WaGaaeinaiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqG2aGaaeioaiaabAdacaqG5aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeOmaiaabcdacaqGWaGaaeOmaaqaaiaabcdaca qGUaGaaeOnaiaabsdacaqG4aGaaeOmaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabodacaqG5aGaaeinaiaabEdacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaaeinaiaabkdacaqG ZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabE dacaqG2aGaaeyoaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabcdacaqGXaGaae4naiaabwdaaaGaay5waiaaw2faaaaa@982B@
Test Problem B.4:
f(x)= x 1 +2 x 2 +4 x 5 + e x 1 x 4 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGc cqGHRaWkcaaIYaGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRi aaisdacaWG4bWaaSbaaSqaaiaaiwdaaeqaaOGaey4kaSIaamyzamaa CaaaleqabaGaamiEamaaBaaameaacaaIXaaabeaaliaadIhadaWgaa adbaGaaGinaaqabaaaaOGaaiilaaaa@49D4@ b T =[0.6855 , 0.5306 , 0.5975 , 0.2992] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeOn aiaabIdacaqG1aGaaeynaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeynaiaabodacaqGWaGaaeOnaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeynaiaabMdacaqG3aGaaeynaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeOmaiaabMdacaqG5aGaaeOmaiaac2faaaa@51A7@ A=[ 0.1025    0.7780    0.3175    0.9357    0.7425 0.0163    0.2634    0.5542    0.4579    0.9213 0.7325    0.2481    0.8753    0.2405    0.4193 0.1260    0.2187    0.6164    0.7639    0.2962 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabgdacaqGWaGaaeOmaiaa bwdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG3aGaae 4naiaabIdacaqGWaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaae4maiaabgdacaqG3aGaaeynaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabMdacaqGZaGaaeynaiaabEdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG3aGaaeinaiaabkdacaqG1a aabaGaaeimaiaab6cacaqGWaGaaeymaiaabAdacaqGZaGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabAdacaqGZaGaae inaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabwdacaqG 1aGaaeinaiaabkdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqG0aGaaeynaiaabEdacaqG5aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeyoaiaabkdacaqGXaGaae4maaqaaiaabcdaca qGUaGaae4naiaabodacaqGYaGaaeynaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabkdacaqG0aGaaeioaiaabgdacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaab6cacaqG4aGaae4naiaabwdacaqG ZaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabs dacaqGWaGaaeynaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabsdacaqGXaGaaeyoaiaabodaaeaacaqGWaGaaeOlaiaabgdaca qGYaGaaeOnaiaabcdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaa b6cacaqGYaGaaeymaiaabIdacaqG3aGaaeiiaiaabccacaqGGaGaae iiaiaabcdacaqGUaGaaeOnaiaabgdacaqG2aGaaeinaiaabccacaqG GaGaaeiiaiaabccacaqGWaGaaeOlaiaabEdacaqG2aGaae4maiaabM dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGYaGaaeyo aiaabAdacaqGYaaaaiaawUfacaGLDbaaaaa@B790@
Test Problem B.5:
f(x)= k=1 6 [100 ( x k+1 x k 2 ) 2 + (1 x k ) 2 ] , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9maaqahabaGaai4waiaaigdacaaIWaGa aGimaiaacIcacaWG4bWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabe aakiabgkHiTiaadIhadaqhaaWcbaGaam4AaaqaaiaaikdaaaGccaGG PaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiikaiaaigdacqGHsi slcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykamaaCaaaleqabaGa aGOmaaaakiaac2faaSqaaiaadUgacqGH9aqpcaaIXaaabaGaaGOnaa qdcqGHris5aOGaaiilaaaa@556A@ b T =[0.5846 , 0.8277 , 0.4425 , 0.8266] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeyn aiaabIdacaqG0aGaaeOnaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeioaiaabkdacaqG3aGaae4naiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeinaiaabsdacaqGYaGaaeynaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeioaiaabkdacaqG2aGaaeOnaiaac2faaaa@51A5@ A=[ 0.1187    0.4147    0.8051    0.3876    0.3643    0.7031 0.4761    0.8606    0.4514    0.0311    0.5323    0.1964 0.6618    0.2715    0.3826    0.0302    0.7117    0.1784 0.9081    0.1459    0.7896    0.9440    0.8715    0.1265 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabgdacaqGXaGaaeioaiaa bEdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaae ymaiaabsdacaqG3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeioaiaabcdacaqG1aGaaeymaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabodacaqG4aGaae4naiaabAdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqGZaGaaeOnaiaabsdacaqGZa GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae4naiaabcda caqGZaGaaeymaaqaaiaabcdacaqGUaGaaeinaiaabEdacaqG2aGaae ymaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqG 2aGaaeimaiaabAdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqG0aGaaeynaiaabgdacaqG0aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeimaiaabodacaqGXaGaaeymaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabwdacaqGZaGaaeOmaiaaboda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGXaGaaeyoai aabAdacaqG0aaabaGaaeimaiaab6cacaqG2aGaaeOnaiaabgdacaqG 4aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabE dacaqGXaGaaeynaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabodacaqG4aGaaeOmaiaabAdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqGWaGaae4maiaabcdacaqGYaGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaae4naiaabgdacaqGXaGaae4nai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabgdacaqG3aGa aeioaiaabsdaaeaacaqGWaGaaeOlaiaabMdacaqGWaGaaeioaiaabg dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGXaGaaein aiaabwdacaqG5aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUa Gaae4naiaabIdacaqG5aGaaeOnaiaabccacaqGGaGaaeiiaiaabcca caqGWaGaaeOlaiaabMdacaqG0aGaaeinaiaabcdacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqG4aGaae4naiaabgdacaqG1aGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeymaiaabkdaca qG2aGaaeynaaaacaGLBbGaayzxaaaaaa@D2BF@
Test Problem B.6:
f(x)=0.5( x 1 x 4 x 2 x 3 + x 2 x 6 x 5 x 6 + x 5 x 4 x 6 x 7 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iabgkHiTiaaicdacaGGUaGaaGynaiaa cIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaaca aI0aaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGccaWG 4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiEamaaBaaaleaaca aIYaaabeaakiaadIhadaWgaaWcbaGaaGOnaaqabaGccqGHsislcaWG 4bWaaSbaaSqaaiaaiwdaaeqaaOGaamiEamaaBaaaleaacaaI2aaabe aakiabgUcaRiaadIhadaWgaaWcbaGaaGynaaqabaGccaWG4bWaaSba aSqaaiaaisdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaI2aaabe aakiaadIhadaWgaaWcbaGaaG4naaqabaGccaGGPaaaaa@5A87@ b T =[0.9879 , 0.6321 , 0.8082 , 0.6650] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeyo aiaabIdacaqG3aGaaeyoaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeOnaiaabodacaqGYaGaaeymaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeioaiaabcdacaqG4aGaaeOmaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeOnaiaabAdacaqG1aGaaeimaiaac2faaaa@51A1@ A=[ 0.0832    0.3312    0.4580    0.7001    0.8287    0.9978    0.1876 0.3904    0.4277    0.2302    0.1373    0.4850    0.3495    0.8831 0.2393    0.8619    0.2734    0.8265    0.6598    0.4328    0.9315 0.4863    0.3787    0.6748    0.9301    0.4564    0.5893    0.8943 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabcdacaqG4aGaae4maiaa bkdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGZaGaae 4maiaabgdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeinaiaabwdacaqG4aGaaeimaiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabEdacaqGWaGaaeimaiaabgdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG4aGaaeOmaiaabIdacaqG3a GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeyoaiaabMda caqG3aGaaeioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlai aabgdacaqG4aGaae4naiaabAdaaeaacaqGWaGaaeOlaiaabodacaqG 5aGaaeimaiaabsdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqG0aGaaeOmaiaabEdacaqG3aGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeOmaiaabodacaqGWaGaaeOmaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabgdacaqGZaGaae4naiaaboda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaaeioai aabwdacaqGWaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGa ae4maiaabsdacaqG5aGaaeynaiaabccacaqGGaGaaeiiaiaabccaca qGWaGaaeOlaiaabIdacaqG4aGaae4maiaabgdaaeaacaqGWaGaaeOl aiaabkdacaqGZaGaaeyoaiaabodacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG4aGaaeOnaiaabgdacaqG5aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabEdacaqGZaGaaeinai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqGYaGa aeOnaiaabwdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6caca qG2aGaaeynaiaabMdacaqG4aGaaeiiaiaabccacaqGGaGaaeiiaiaa bcdacaqGUaGaaeinaiaabodacaqGYaGaaeioaiaabccacaqGGaGaae iiaiaabccacaqGWaGaaeOlaiaabMdacaqGZaGaaeymaiaabwdaaeaa caqGWaGaaeOlaiaabsdacaqG4aGaaeOnaiaabodacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqGZaGaae4naiaabIdacaqG3aGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOnaiaabEdaca qG0aGaaeioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaa bMdacaqGZaGaaeimaiaabgdacaqGGaGaaeiiaiaabccacaqGGaGaae imaiaab6cacaqG0aGaaeynaiaabAdacaqG0aGaaeiiaiaabccacaqG GaGaaeiiaiaabcdacaqGUaGaaeynaiaabIdacaqG5aGaae4maiaabc cacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabIdacaqG5aGaaein aiaabodaaaGaay5waiaaw2faaaaa@EE33@
Test Problem B.7:
f(x)= e x 1 x 2 x 3 x 4 x 5 0.5 ( x 1 3 + x 2 3 + x 6 3 +1) 2 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiaadIhadaWg aaadbaGaaGymaaqabaWccaWG4bWaaSbaaWqaaiaaikdaaeqaaSGaam iEamaaBaaameaacaaIZaaabeaaliaadIhadaWgaaadbaGaaGinaaqa baWccaWG4bWaaSbaaWqaaiaaiwdaaeqaaaaakiabgkHiTiaaicdaca GGUaGaaGynaiaacIcacaWG4bWaa0baaSqaaiaaigdaaeaacaaIZaaa aOGaey4kaSIaamiEamaaDaaaleaacaaIYaaabaGaaG4maaaakiabgU caRiaadIhadaqhaaWcbaGaaGOnaaqaaiaaiodaaaGccqGHRaWkcaaI XaGaaiykamaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@568B@ b T =[0.9521 , 0.0309 , 0.8627 , 0.8343 , 0.6290] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaaeyo aiaabwdacaqGYaGaaeymaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeimaiaabodacaqGWaGaaeyoaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeioaiaabAdacaqGYaGaae4naiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeioaiaabodacaqG0aGaae4maiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaaeOnaiaabkdacaqG5aGaaeimaiaac2faaa a@57CD@ A=[ 0.9869    0.0805    0.8373    0.1417    0.9988    0.6320 0.0139    0.0169    0.0182    0.4379    0.0295    0.5095 0.2497    0.6914    0.8961    0.3504    0.8225    0.2433 0.9691    0.6170    0.5921    0.4785    0.5994    0.5714 0.6197    0.6298    0.2372    0.5874    0.2560    0.9817 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGWaGaaeOlaiaabMdacaqG4aGaaeOnaiaa bMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqGWaGaae ioaiaabcdacaqG1aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqG UaGaaeioaiaabodacaqG3aGaae4maiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabgdacaqG0aGaaeymaiaabEdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeyoaiaabIdacaqG4a GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOnaiaaboda caqGYaGaaeimaaqaaiaabcdacaqGUaGaaeimaiaabgdacaqGZaGaae yoaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqG XaGaaeOnaiaabMdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6 cacaqGWaGaaeymaiaabIdacaqGYaGaaeiiaiaabccacaqGGaGaaeii aiaabcdacaqGUaGaaeinaiaabodacaqG3aGaaeyoaiaabccacaqGGa GaaeiiaiaabccacaqGWaGaaeOlaiaabcdacaqGYaGaaeyoaiaabwda caqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG1aGaaeimai aabMdacaqG1aaabaGaaeimaiaab6cacaqGYaGaaeinaiaabMdacaqG 3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOnaiaabM dacaqGXaGaaeinaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabIdacaqG5aGaaeOnaiaabgdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqGZaGaaeynaiaabcdacaqG0aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaaeioaiaabkdacaqGYaGaaeynai aabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabkdacaqG0aGa ae4maiaabodaaeaacaqGWaGaaeOlaiaabMdacaqG2aGaaeyoaiaabg dacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG2aGaaeym aiaabEdacaqGWaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUa GaaeynaiaabMdacaqGYaGaaeymaiaabccacaqGGaGaaeiiaiaabcca caqGWaGaaeOlaiaabsdacaqG3aGaaeioaiaabwdacaqGGaGaaeiiai aabccacaqGGaGaaeimaiaab6cacaqG1aGaaeyoaiaabMdacaqG0aGa aeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeynaiaabEdaca qGXaGaaeinaaqaaiaabcdacaqGUaGaaeOnaiaabgdacaqG5aGaae4n aiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabAdacaqGYa GaaeyoaiaabIdacaqGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6ca caqGYaGaae4maiaabEdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiai aabcdacaqGUaGaaeynaiaabIdacaqG3aGaaeinaiaabccacaqGGaGa aeiiaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaaeOnaiaabcdaca qGGaGaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeioaiaa bgdacaqG3aaaaiaawUfacaGLDbaaaaa@F94A@
Test Problem B.8:
f(x)= ( x 1 1) 2 + ( x 7 1) 2 +10 k=1 6 (10k) ( x k 2 x k+1 ) 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaaikdaaaGccq GHRaWkcaGGOaGaamiEamaaBaaaleaacaaI3aaabeaakiabgkHiTiaa igdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaiaaic dadaaeWbqaaiaacIcacaaIXaGaaGimaiabgkHiTiaadUgacaGGPaGa aiikaiaadIhadaqhaaWcbaGaam4AaaqaaiaaikdaaaGccqGHsislca WG4bWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiaacMcadaah aaWcbeqaaiaaikdaaaaabaGaam4Aaiabg2da9iaaigdaaeaacaaI2a aaniabggHiLdaaaa@5D61@ b T =[0.7840 , 0.4648 , 0.8864 , 0.8352 , 0.9839] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaaaeaacaWGubaaaOGaeyypa0Jaai4waiaabcdacaqGUaGaae4n aiaabIdacaqG0aGaaeimaiaabccacaqGSaGaaeiiaiaabcdacaqGUa GaaeinaiaabAdacaqG0aGaaeioaiaabccacaqGSaGaaeiiaiaabcda caqGUaGaaeioaiaabIdacaqG2aGaaeinaiaabccacaqGSaGaaeiiai aabcdacaqGUaGaaeioaiaabodacaqG1aGaaeOmaiaabccacaqGSaGa aeiiaiaabcdacaqGUaGaaeyoaiaabIdacaqGZaGaaeyoaiaac2faaa a@57E8@ A=[  0.8522    0.2376    0.3586    0.7260    0.8891    0.2771    0.1316  0.4673    0.8176    0.1173    0.5350    0.1426    0.0020    0.2892  0.9707    0.4058    0.7248    0.1826    0.6193    0.8108    0.9630  0.8412    0.4663    0.7011    0.1124    0.6848    0.9434    0.4656  0.0785    0.9515    0.9997    0.0028    0.4982    0.6384    0.3852 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maadmaaeaqabeaacaqGGaGaaeimaiaab6cacaqG4aGaaeynaiaa bkdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae OmaiaabodacaqG3aGaaeOnaiaabccacaqGGaGaaeiiaiaabccacaqG WaGaaeOlaiaabodacaqG1aGaaeioaiaabAdacaqGGaGaaeiiaiaabc cacaqGGaGaaeimaiaab6cacaqG3aGaaeOmaiaabAdacaqGWaGaaeii aiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeioaiaabIdacaqG5a GaaeymaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabkda caqG3aGaae4naiaabgdacaqGGaGaaeiiaiaabccacaqGGaGaaeimai aab6cacaqGXaGaae4maiaabgdacaqG2aaabaGaaeiiaiaabcdacaqG UaGaaeinaiaabAdacaqG3aGaae4maiaabccacaqGGaGaaeiiaiaabc cacaqGWaGaaeOlaiaabIdacaqGXaGaae4naiaabAdacaqGGaGaaeii aiaabccacaqGGaGaaeimaiaab6cacaqGXaGaaeymaiaabEdacaqGZa GaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeynaiaaboda caqG1aGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlai aabgdacaqG0aGaaeOmaiaabAdacaqGGaGaaeiiaiaabccacaqGGaGa aeimaiaab6cacaqGWaGaaeimaiaabkdacaqGWaGaaeiiaiaabccaca qGGaGaaeiiaiaabcdacaqGUaGaaeOmaiaabIdacaqG5aGaaeOmaaqa aiaabccacaqGWaGaaeOlaiaabMdacaqG3aGaaeimaiaabEdacaqGGa GaaeiiaiaabccacaqGGaGaaeimaiaab6cacaqG0aGaaeimaiaabwda caqG4aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaae4nai aabkdacaqG0aGaaeioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGa aeOlaiaabgdacaqG4aGaaeOmaiaabAdacaqGGaGaaeiiaiaabccaca qGGaGaaeimaiaab6cacaqG2aGaaeymaiaabMdacaqGZaGaaeiiaiaa bccacaqGGaGaaeiiaiaabcdacaqGUaGaaeioaiaabgdacaqGWaGaae ioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabMdacaqG 2aGaae4maiaabcdaaeaacaqGGaGaaeimaiaab6cacaqG4aGaaeinai aabgdacaqGYaGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGa aeinaiaabAdacaqG2aGaae4maiaabccacaqGGaGaaeiiaiaabccaca qGWaGaaeOlaiaabEdacaqGWaGaaeymaiaabgdacaqGGaGaaeiiaiaa bccacaqGGaGaaeimaiaab6cacaqGXaGaaeymaiaabkdacaqG0aGaae iiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeOnaiaabIdacaqG 0aGaaeioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOlaiaabM dacaqG0aGaae4maiaabsdacaqGGaGaaeiiaiaabccacaqGGaGaaeim aiaab6cacaqG0aGaaeOnaiaabwdacaqG2aaabaGaaeiiaiaabcdaca qGUaGaaeimaiaabEdacaqG4aGaaeynaiaabccacaqGGaGaaeiiaiaa bccacaqGWaGaaeOlaiaabMdacaqG1aGaaeymaiaabwdacaqGGaGaae iiaiaabccacaqGGaGaaeimaiaab6cacaqG5aGaaeyoaiaabMdacaqG 3aGaaeiiaiaabccacaqGGaGaaeiiaiaabcdacaqGUaGaaeimaiaabc dacaqGYaGaaeioaiaabccacaqGGaGaaeiiaiaabccacaqGWaGaaeOl aiaabsdacaqG5aGaaeioaiaabkdacaqGGaGaaeiiaiaabccacaqGGa Gaaeimaiaab6cacaqG2aGaae4maiaabIdacaqG0aGaaeiiaiaabcca caqGGaGaaeiiaiaabcdacaqGUaGaae4maiaabIdacaqG1aGaaeOmaa aacaGLBbGaayzxaaaaaa@1E55@
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