A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tfie20 Fuzzy Information and Engineering ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tfie20 A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach S. H. Nasseri, J. L. Verdegay & F. Mahmoudi To cite this article: S. H. Nasseri, J. L. Verdegay & F. Mahmoudi (2021): A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach, Fuzzy Information and Engineering, DOI: 10.1080/16168658.2021.1906154 To link to this article: https://doi.org/10.1080/16168658.2021.1906154 © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. Published online: 14 Jul 2021. Submit your article to this journal Article views: 29 View related articles View Crossmark data FUZZY INFORMATION AND ENGINEERING https://doi.org/10.1080/16168658.2021.1906154 A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach S. H. Nasseri a, J. L. Verdegayband F. Mahmoudia aDepartment of Mathematics, University of Mazandaran, Babolsar, Iran; bDepartment of Computer Science and A.I. Universidad de Granada, Granada, Spain ABSTRACT Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decisionmaking problems is generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper, we concentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, numerical example is included to illustrate the mentioned solving process. ARTICLE HISTORY Received 20 January 2021 Accepted 9 March 2021 KEYWORDS Multi-objective linear programming; fuzzy interval flexible linear programming; interval linear programming; interval arithmetic; flexible constraints 1. Introduction Fuzzy sets theory has been extensively employed in linear programming. The main objective in fuzzy linear programming is to find the best solution possible with imprecise, vague, uncertain or incomplete information. There are many sources of imprecision in fuzzy linear programming. The sources of imprecision in fuzzy linear programming vary. For example, sometimes constraint satisfaction limits are vague and other times coefficient variables are not known precisely. The research on fuzzy linear programming has risen highly since Bellman and Zadeh proposed the concept of decision making in fuzzy environment. Zimmermann [1] introduced the first formulation of fuzzy linear programming to address the impreciseness and vagueness of the parameters in linear programming problems with fuzzy constraints and objective functions. There are generally four fuzzy linear programming classifications in the literature. Zimmerman [2] has classified fuzzy linear programming problems into two categories: symmetrical and non-symmetrical models. In a symmetrical fuzzy decision, there is no difference between the weight of the objectives and constraints while in the asymmetrical CONTACT S. H. Nasseri [email protected] © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2S. NASSERI ET AL. fuzzy decision, the objectives and constraints are not equally important and have different weights [3]. Leung [4] has classified fuzzy linear programming problems into four categories: a precise objective and fuzzy constraints; a fuzzy objective and precise constraints; a fuzzy objective and fuzzy constraints; and robust programming. Luhandjula [5] has classified fuzzy linear programming problems into three categories: flexible programming; mathematical programming with fuzzy parameters and fuzzy stochastic programming. Inuiguchi et al. [6] have classified fuzzy linear programming problems into six categories: flexible programming; possibilistic programming; possibilistic linear programming using fuzzy max; robust programming; possibilistic programming with fuzzy preference relations and possibilistic linear programming with fuzzy goals. Delgado et al. [7] studied a general model for fuzzy linear programming problems which simultaneously involved in the constraints set both fuzzy numbers and fuzzy constraints. Mahdavi-Amiri and Nasseri [8] proposed a fuzzy linear programming model where a linear ranking function was used to rank order trapezoidal fuzzy numbers. They established the dual problem of the linear programming problem with trapezoidal fuzzy variables and deduced some duality results to solve the fuzzy linear programming problem directly with the primal simplex tableau. Mahdavi-Amiri and Nasseri [9] developed some methods for solving fuzzy linear programming problems by introducing and solving certain auxiliary problems. They apply a linear ranking function to order trapezoidal fuzzy numbers and deduce some duality results by establishing the dual problem of the linear programming problem with trapezoidal fuzzy variables. Wu [10] derived the optimality conditions for fuzzy linear programming problems by proposing two solution concepts based on similar solution concept, called the non-dominated solution, in the multiobjective programming problem. Inuiguchi and Ramik [6] and Peidro et al. have developed a number of fuzzy linear programming models to solve problems ranging from supply chain management to product development. Then Verdegay in [11] used the duality results to solve the original fuzzy linear programming. After that, Nasseri et al. in [12] introduced an equivalent fuzzy linear model for the flexible linear programming problems and proposed a fuzzy primal simplex algorithm to solve these problems. Recently, Attari and Nasseri [13] introduced a concept of feasibility and efficiency of solution for the fuzzy mathematical programming problems. The suggested algorithm needs to solve two classical associated linear programming problems to achieve an optimal flexible solution. Interval linear programming, based on interval analysis, was proved to be an effective approach in dealing with uncertainties. Interval linear programming did not require distributional information and would not lead to complicated intermediate models. However, it was to be noted that the outputs of Interval linear programming were with lower and upper bounds, and thus could not reflect the distribution of uncertainty within the lower and upper bounds [14]. In some methods, Interval Linear Programming (ILP) model transformed into two sub-model whereas their optimal solutions formed a set which is called solution space of ILP model. The optimal solution set of ILP is determined by the best and worst model constraints, when the feasible solution components of best model are positive. In the Best and Worst Cases (BWC) method presented by Tong, the ILP model transformed into two sub-models [15,16], which consist of the largest and smallest feasible regions, so the BWC method introduces exact bounds of objective function values. A given point FUZZY INFORMATION AND ENGINEERING 3 is feasible of ILP model, if it is satisfies in best model constraints and it is optimal of ILP model, if it is optimal solution of arbitrary characteristic model of ILP model. Chinnec and Ramadan developed BWC method when ILP model includes equality constraints [17], and a new method for solving proposed by Huang and More [18]. Part of solution space of BWC and ILP methods may be infeasible. To ensure that solutions are absolutely feasible, Zhou et al. exhibited Modified Interval Linear Programming (MILP) method, by adding an extra constraint to the second sub-model. Some of the solutions which are obtained by the MILP method may be non-optimal. Also, among methods for solving ILP model, a Two-Step Method (TSM) had been presented by Huang et al. [18]. Solution space of the TSM method may be included infeasible solutions. To eliminate infeasible solutions from solution space of the TSM method, some methods are proposed. Wang and Huang added extra constraints to the second sub-model of TSM to ensure feasibility of solutions (namely ITSM). Part of solution space of ITSM is not optimal. Recently, Mishmast Nehi and Allahdadi [17,19] modified and improved the Tong method, which was unable to get optimal response on some issues. In this study, we give a generalised form of these problems in two ways: in first way, we consider the flexibility condition for the constraints, and in second way we consider the multi-objective case for the objective. In this sense, we introduce a new extended model and then propose a method for solving the proposed model. The rest of this paper is organised as follows: In Section 2, we review the basic definitions and results on interval linear programming problem. Section 3 gives the definition of FFLP problem and proposes parametric approach to solve it. We give a new method for solving FFLP problem with multi-objective and interval objective function in Section 4. Section 5 is assigned to the illustrated example. Finally, conclusion is discussed in Section 6. 2. Interval Linear Programming Problems In many real-word models, these coefficients are uncertain, so that they are bounded between upper and lower bounds. Therefore, in the formulation of research question in operations, if the data are in form of interval numbers, then the problem is an interval linear programming problem. In first time, Ben and Robbers presented the first interval linear programming model for interval constraints. Subsequently Huang and Moore introduced a new linear programming model in which all parameters and variables were interval. Generally, the solution method in these cases is the application of concepts that can turn the interval problem into problems with ordinary coefficients [14,18,19]. Definition 2.1: Given x−and x+∈Rsuch that x−≤x+, we define a closed interval x=[x−,x+] as the set {x∈R:x−≤x≤x+}. The values x−and x+are called the lower bound and upper bound of the interval x, respectively. Definition 2.2: An interval [x,¯ x] with x−=x+is said to be degenerate. Since a degenerate interval [x−,x+] only contains a single number, it is often identified with the number xitself, therefore it holds that x=[x,x]. Definition 2.3: Given two matrices A−and A+∈Rm×nsuch that A−≤A+, we define a real interval matrix A=[A−,A+] as the set {A∈Rm×n:A−≤A≤A+}. The matrices A−,A+ 4S. NASSERI ET AL. are called the lower bound matrix and the upper bound matrix of the interval matrix A, respectively. The radius and centre of Aare AΔ=1 2(A+−A−)and AC=1 2(A++A−), respectively. Thus A=[A−,A+]=[AC−AΔ,AC+A]. An interval vector Iis introduced as the set {I:I−≤I≤I+}where I,¯ I∈Rnare crisp vector [20]. Definition 2.4: A general form of the Interval Linear Programming (ILP) model is defined as follows: max Z±= n  j=1 C± jx± j s.t. n  j=1 a± ij x± j≤b± i,i=1, 2, ...,m, x± j≥0, j=1, 2, ...,n, (1) where C± j[C− j,C+ j], a± ij [a− ij ,a+ ij ]andb± i[b− i,b+ i] are interval numbers and xj[x− j,x+ j]isan n-dimensional interval decision vector. Theorem 2.1: In the ILP model (1), the largest and smallest feasible regions are n  j=1 aij+xj≤bi−,i=1, 2, ...,m,xj≥0, j=1, 2, ...,nand n  j=1 aij−xj≤bi+,∀i,xj≥0, j=1, 2, ...,n, respectively. Proof: The proof is straightforward by the common interval arithmetic.  Definition 2.5: A point y=(y1,y2,...,yn)is said to be a feasible point of ILP model (1) if n  j=1 aij+y≤bi−,i=1, 2, ...,m,andyj≥0, j=1, 2, ...,n. There are several methods for solving interval linear programming problems, one of which the Best and Worst Cases (BWC) method. The BWC method for solving linear interval programming problems in such a way that in general the linear programming problem with interval parameters turns into two optimistic and pessimistic linear programming models, where their solutions are the optimal interval of the main problem. This method examines the answers to the linear programming problems derived from the standard form [10,17]. Mentioned method transforms the ILP problem (1) into pessimistic and optimistic subproblems, which are summarised as follows: FUZZY INFORMATION AND ENGINEERING 5 The pessimistic sub-problem: max Z−= n  j=1 C− jxi, s.t. n  j=1 a+ ij xi≤b− i,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n. (2) The optimistic sub-problem: max Z+= n  j=1 C+ jxi, s.t. n  j=1 a− ij xi≤b+ i,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n. (3) The optimal solutions to sub-problems (2) and (3) are in box form as follows: x±= (x± 1,x± 2,...,x± n), where for all j=1, 2, ...,n,x± j=[x− j,x+ j]. This box is the solution area which is introduced by Tong. Theorem 2.2: In solving process of the ILP model, if Z∗is the optimal objective value of model (1), and Z−∗,Z+∗ are the optimal objective value of the model (2) and model (3), respectively, then Z∗∈[Z−∗,Z+∗]. Proof: Let us consider the problem (1), we prove that the solution of this model is in the interval [z,¯ z]. If x0is a solution given by the above model, then we will have it n j=1aijx0≥ bi,i=1, 2, ...,m. On the other hand, aij ≤aij x0 j≥0 →aijx0 j≤aijx0 j∀i→ n  j=1 aijx0 j≤ n  j=1 aijx0 j. Given the above phrases and bi≥biwe have bi<bi≤n j=1aijx0 j≤n j=1aijx0 j, i=1, 2, ...,m. Therefore, every solution to the problem (3) is a solution to the model (1). So, the feasible area for the problem (1) includes the feasible area of problem (3). We now prove that the optimal value of the model (3) is less than the optimal value of the model (1). If x∗is the optimal solution for the model (3): cj≤cj x∗ j≥0 →cjx∗ j≤cjx∗ j∀j→ n  j=1 cjx∗ j≤ n  j=1 cjx∗ j. If z=n j=1cjx∗ jis the objective value of model (1), then we have z≤z∗and if zis optimal solution of model (2), then z<zand so z<z∗. Similarly, if ˜ xis a solution of model 6S. NASSERI ET AL. (3), then n j=1aij ˜ xj≥biand aij ≤aij∀j→˜ xj≥0aij ˜ xj≤aij ˜ xj∀j→n j=1aij ˜ xj≤n j=1aij ˜ xj,and since bi≥bi.Thuswewillhavebi≤bi≤n j=1aij ˜ xj≤n j=1aij ˜ xj. Therefore, each feasible solution of model (2) is a feasible solution of model (3), or, in other words, the feasible area of the model (3) including the feasible area of the model (2). Now, we prove that the optimal value of the model (2) is greater than the optimal model (3). Now consider x be the optimal solution of problem (2). cj≤cj∀j→x j≥0cjx j≤cjx j∀j→ n j=1cjx j≤n j=1cjx j.Ifzis the value of the objective function model (3) for the feasible solution, z<¯ zand since x∗is the solution of model (2), should be z∗≤z, as a result z∗≤ ¯ z. So we’ll have it z≤z∗≤¯ zand the theorem is completed. 3. Fuzzy Flexible Linear Programming Let us consider a case where the decision maker assumes that there is a certain tolerance in the fulfillment of constraints. In other word, a certain degree of violation is allowed and this is created by the decision makers. The general form of the Fuzzy Flexible Linear Programming (FFLP) problems with fuzzy resources can be formulated as follows (see in [21] too): max z=f(x,C)= n  j=1 cjxj s.t.g i(x)= n  j=1 aijxjbi,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n. (4) In the above model, the relation is called ‘ fuzzy less than or equal to’ and it is assumed that the tolerance pifor each constraint is given [22]. This means that the decision maker can accept a violation of each constraint up to degree pi. In this case, constraint gi(x)bi is equivalent to gi(x)≤bi+θpi,(i=1, 2, ...,m), where θ∈[0, 1]. Thus problem (4) can be equivalently considered as the following fuzzy inequality constraints (see also in [16]): max z=f(x,C)= n  j=1 cjxj s.t.g i(x)= n  j=1 aijxj≤˜ bi,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n. (5) In model (5), ˜ biis a fuzzy number with the following membership function: μ˜ bi(x)=⎧ ⎨ ⎩ 1, x≤bi, 1−(x−bi)/pi,bi≤x≤bi+pi, 0, x≥bi+pi. (6) FUZZY INFORMATION AND ENGINEERING 7 Verdegay [11] proved that Problem (4) is equivalent to the crisp parametric LP problem when the membership functions of the fuzzy constraints are continuous and nonincreasing functions. According to this non-symmetric approach, the membership function of fuzzy inequality constraints of problem (4) can be modelled as follows: μi(gi(x)) =⎧ ⎨ ⎩ 1, gi(x)≤bi, 1−(gi(x)−bi)/pi,bi≤gi(x)≤bi+pi, 0, gi(x)≥bi+pi, (7) In this case, the membership function of all constraints of the problem (4) according to the Bellman and Zadeh operator is given by μ(g(x)) =min{μ1(g1(x)),μ2(g2(x)),...,μm(gm(x))}.(8) Assuming, α=min{μ1(g1(x)),μ2(g2(x)),...,μm(gm(x))}, then Problem (4) is equivalent to max z=f(x,C)= n  j=1 cjxj s.t.μi(gi(x)) ≥α,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n,α∈[0, 1]. (9) Consider the circumstances that the decision maker seeks to achieve the optimal answer with different degrees of validity in different constraints, according to a priority among the constraints. Clearly, the Verdegay’s approach or single-parameter method is rejected in this case. By introducing various parameters for different constraints and using this multiparameter approach, the decision-maker’s need and appeal will be easily resolved. The following is a description of this method [13]. Consider the linear programming problem (9), the general form of the fuzzy linear programming problem is modified in this way: max z=f(x,C)= n  j=1 cjxj s.t.μi(gi(x)) ≥αi,i=1, 2, ...,m, xj≥0, j=1, 2, ...,n,αi∈[0, 1]. (10) Now, by substituting membership function (7) into problem (10), the following crisp parametric LP problem is achieved: max z=f(x,C)= n  j=1 cjxj s.t.g i(x)=(Ax)i−bi≤(1−αi)pi,i=1, 2, ...,m, xj≥0, αi∈(0, 1], j=1, 2, ...,n. (11) Note that for each αi∈(0, 1], i=1, 2, ...,m, an optimal solution is obtained. This indicates that the solution with αgrade of membership function is actually fuzzy. 8S. NASSERI ET AL. Let’s start with the following definitions below to continue the article. Definition 3.1: Let ¯α=(α1,...,αm)∈(0, 1]mbe a vector, and X¯α={x∈Rn|x≥0, μi{gi(x,ai)0}≥αi,i=1, 2, ...,m.}. Then, a vector x∈X¯αis called an ¯α-feasible solution of model (5). Following proposition enables us to define feasible set of model (5) as an intersection of all α-cuts corresponding to fuzzy constraints. Proposition 3.1: Let ¯α=(α1,...,αm)∈(0, 1]m, then X¯α= m  i=1 Xi αi, where Xi αi={x∈Rn|x≥0, μi{gi(x,ai)0}≥αi},fori∈I={1, ...,m}(namely, Xi αis the αcuts of the ith constraint). Proof: For ¯α=(α1,...,αm)∈(0, 1]m,, let x∈X¯α. Therefore, μi{gi(x,ai)0}≥αiand from Xi αi={x∈Rn|x≥0, μi{gi(x,ai)0}≥αi},wehavex∈Xi αi,i∈I. Therefore, x∈ m  i=1 Xi αi. Also, if x∈ m  i=1 Xi αi,wehavex∈Xi αi,i∈I,thusμi{gi(x,ai)0}≥αi and hence, x∈X¯α. Therefore, the proof is completed.  Proposition 3.2: Let α=(α1,...,αm)and α =(α1,...,αm), where α i≤α ifor all i. Then, α -feasibility of ximplies the α-feasibility of it. Proof: The proof is straightforward. For a given α∈(0, 1], let a solution x∈Rnbe ordinary α−feasible to problem (4) a solution in which has the same satisfaction degree in all of constraints. It means that μi{gi(x,ai)0}≥αi,orx∈Xi α, for all i∈I.If¯α=(α1,...,αm)∈(0, 1]m, then x∈X¯α,which implies that the ¯α−feasibility of problem (5) can be understood as a special case of the ¯α− feasibility. Therefore, we have the next result.  Remark 3.1: If problem (5) is not infeasible, we immediately conclude that X¯αis not empty. Definition 3.2: Let be a fuzzy extension of relation ≤and a solution X=(x1,...,xn)T∈Rnbe an ¯α−feasible to problem (5), where ¯α=(α1,...,αm)∈(0, 1]m and let f(x,C)be an objective function in the form of maximisation. Then, X=(x1,...,xn), where xj∈Rnis an ¯α−efficient solution to problem (5), if there is no x∈X¯αso that f(x,C)<f(x,C). Clearly, any ¯α−efficient solution to the FFLP is a ¯α−feasible solution to the FFLP with some additional properties. 4. Solve FFLP Problem with Interval Multi-objective Function In this section, we will present a new approach to solve the Fuzzy Flexible Linear Programming (FFLP) problem which is defined in (4) with interval multi-objective functions. We first FUZZY INFORMATION AND ENGINEERING 15 Table 5. Some typical ¯α-efficient solution of sub-problem1. ab c d e f ¯α(0.5,0.6) (0.8,0.2) (0.5,0.8) (0.2,0.8) (0.5,0.2) cx 44,997 41,657 44,997 47,224 44,997 x133333 x26.3333 5.33 6.3333 7 6.3333 x355555 Take w1=0.4, w2=0.3 and w3=0.3. Then the interval weighted sum secularisation of the MOILP problem with respect to is as follows: max Z=[1349, 1448]x1+[2443, 3340]x2+[3213, 3900]x3 s.t. [2000, 2100]x1+[3000, 3200]x2+[4000, 5000]x340000, [8000, 9000]x1+[1000, 1200]x2+[4000, 4600]x350000, [4000, 4500]x1+[2000, 2400]x2≤50000, x1≥3, x3≥5, x1≥0, x2≥0, x3≥0, (27) Now, we solve the problem (27) by using the solving technique of interval problems and solving algorithm in this paper and the given tolerances p1=10000, p2=5000. We will simplify the first sub-problem based on mentioned solving algorithm steps in (4.1) as follows: max Z+=1448x1+3340x2+3900x3 s.t. 2000x1+3000x2+4000x3≤40000 +10000(1−α1), 8000x1+1000x2+4000x3≤50000 +5000(1−α2), 4000x1+2000x2≤50000, x1≥3, x3≥5, x1≥0, x2≥0, x3≥0, α1,α2∈[0, 1], (28) Some ¯α-efficient solution with satisfaction degrees which decision maker can be found in Table 5. Let x∗=(3, 6.3333, 5)be (0.5, 0.6)-efficient solution with CTx∗=44997 as an optimal value of problem (28). In step 6, we need to solve the following linear problem: max 2  i=1 αi s.t. 1448x1+3340x2+3900x3≥44997, 2000x1+3000x2+4000x3≤40000 +10000(1−α1) 16 S. NASSERI ET AL. 8000x1+1000x2+4000x3≤50000 +5000(1−α2) 4000x1+2000x2≤50000, 0.5 ≤α1≤1, 0.6 ≤α2≤1, x1≥3, x3≥5, x1≥0, x2≥0, x3≥0, (29) An optimal solution to the above problem is x∗∗ =(3, 6.3332, 5),alsoCTx∗=CTx∗∗ = 44997 we have μ1(g1(x∗∗,a1)) =0.5, μ2(g2(x∗∗,a2)) =1. Using the approach, we can get an optimal solution x∗which not only achieves the optimal objective value but also give a higher value in μ2. Now, we use all these steps to solve the second sub-problem. Finally, by solving the second sub-problem obtain that if x∗=(3, 3.33, 5)be (0.5, 0.4)-efficient solution with CTx∗=CTx∗∗ =26219 and x∗=(3, 0, 6.2784), μ1(g1(x∗∗,a1)) =1, μ2(g2(x∗∗,a2)) =1. Finally, with regard to Theorem 2.2 optimal objective value of problem (27) is Z∗=[18595, 44997]. 6. Conclusion In this paper, two main contributions are appeared. First, considering the feasibility for the constraints and second, an extra condition for the objective function where we assumed a multi-objective cases. Based on the generalised form of the problem, we suggested a new two-phase method. We saw that it was observed that using this concept as a generalisation of parametric the approach in linear programming provides a more appropriate tool for modelling real problems and improving the solving process. Also, in the process of solving a weighty technique for the multi-objective linear programming problem, it was suggested. This approach will be useful in obtaining flexible responses with a degree of satisfaction determined by the decision maker for fuzzy mathematical programming. There are still other approaches, such as for instance Rough Sets (see in [25]), to deal with the problem approached in the research. The second to indicate that readers interested in new Fuzzy Optimisation problems could consult that paper (see in [26]). Acknowledgments The first author would like to appreciate from the research grant of University of Mazandaran. Also, Prof. José Luis Verdegay is supported in part by the project TIN2017-86647-P (Spanish Ministry of Economy and Competitiveness) which includes FEDER funds from the European Union. Disclosure statement No potential conflict of interest was reported by the author(s). FUZZY INFORMATION AND ENGINEERING 17 Notes on contributors S.H. Nasseri received his PhD degree in 2007 on Fuzzy Mathematical Programming from Sharif University of Technology, and since 2007 he has been a faculty member at the Faculty of Mathematical Sciences in University of Mazandaran, Babolsar, Iran. During this program, he also got JASSO Research Scholarship from Japan (Department of Industrial Engineering and Management, Tokyo Institute and Technology (TIT), Tokyo, 2006-2007). Recently, in 2018, he also completed a postdoctoral program at the Department of Industrial Engineering, Sultan Qaboos University, Muscat, Oman on Logistic on Uncertainty Conditions. Also, he collaborated with Foshan University (Department of Mathematics and Big Data), Foshan, China as a visiting professor, since 2018. He serves as the Editor-in-Chief (MiddleEast Area) of Journal of Fuzzy Information and Engineering since 2014 and the Editorial board member of five reputable academic journals. He is a council member of the International Association of Fuzzy Information and Engineering, a Standing Director of the International Association of Grey Systems and Uncertainty Analysis since 2016, and vice-president of Iranian Operations Research Society. International Center of Optimization and Decision Making is established by him in 2014. His research interests are in the areas of Fuzzy Mathematical Models and Methods, Fuzzy Arithmetic, Fuzzy Optimization and Decision Making, Operations Research, Gray Systems, Logistics and Transportation. J.L. Verdegay received the MS degree in mathematics and the PhD degree in sciences from the University of Granada (Spain) in 1975 and 1981, respectively. He is a full Professor at Department of Computer Science and Artificial Intelligence (DECSAI), University of Granada, Spain and director of the Models of Decision and Optimization (MODO) Research Group. He has published twenty nine books and almost 400 scientific and technical papers in leading scientific journals and has been Advisor of 21 Ph.D. dissertations. He has been member and President of a number of committees with the European Training Foundation and the Spanish Ministry of Education. He also is a member of the Editorial Board of several international leading journals and, among other responsibilities he has served as Chairman of DECSAI (1990-1994), President (founder) of the Spanish Association for Fuzzy Logic and Technologies (1990-1996), Advisor for Intelligent Technologies of the Spanish Science Inter-Ministry Commission (1995-1996), Director of International Affairs at the University of Granada (1996-2000) and Delegate of the Rector for ICT in University of Granada (2008-2015). In July 2015 he was appointed Regional Director of the Postgrade Iberoamerican Universities Association. Professor Verdegay is an IFSA fellow, IEEE Senior member and Honorary member of the Cuban Society of Mathematics and Computation. Besides he has the Featured Position of “Invited Professor” at the Technical University of Havana (CUJAE, Cuba), Central University of Las Villas (Santa Clara, Cuba) and University of Holguín (Cuba). He is also a “Distinguished Guest” of the National University of Trujillo (Perú). His current scientific interests are on Soft Computing, fuzzy sets and systems, decision support systems, metaheuristic algorithms, nature systems and all their applications to real world problems. F. Mahmoudi received his BSc degree in Applied Mathematics, Department of Mathematics, University of Mazandaran (2011-2015). She obtained his MSc degree in Applied Mathematics, Operations Research, Department of Mathematics, University of Mazandaran under supervision of Prof. Hadi Nasseri (2015-2017). Now, she is a PhD student in Applied Mathematics, Operations Research, University of Mazandaran, Babolsar, Iran. ORCID S. H. Nasseri http://orcid.org/0000-0002-4821-7191 References [1] Zimmermann HJ. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1978;1:45–55. [2] Zimmermann HJ. Fuzzy sets. Decision making and expert systems. Norwell (MA): Kluwer Academic Publishers; 1978. 18 S. NASSERI ET AL. [3] Amid A, Ghodsypour SH, O’Brien C. Fuzzy multiobjective linear model for supplier selection in a supply chain. Int J Prod Econ. 2006;104:394–407. [4] Leung Y. Spatial analysis and planning under imprecision. Amsterdam: North-Holland; 1998. [5] Luhandjula MK. Fuzzy optimization: an appraisal. Fuzzy Sets Syst. 1989;30:257–282. [6] Inuiguchi M, Ramik J. Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst. 2000;111:3–28. [7] Delgado M, Herrera F, Verdegay JL, et al. Post-optimality analysis on the membership functions of a fuzzy linear programming problem. Fuzzy Sets Syst. 1993;53:289–297. [8] Mahdavi-Amiri N, Nasseri SH. Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets Syst. 2007;158:1961–1978. [9] Nasseri SH, Mahdavi-Amiri N. Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Inform Eng. 2009;1:59–66. [10] Wu HC. Optimality conditions for linear programming problems with fuzzy coefficients. Comp Math Appl. 2008;55:2807–2822. [11] Verdegay JL. Fuzzy Mathematical Programming. In Gupta MM and Sanchez E, editors. Fuzzy Information and Decision Processes. North Holland; 1982. P. 231–237. [12] Nasseri SH, Ebrahimnejad A. A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems. European J Ind Eng. 2010;4(3):372–389. [13] Attari H, Nasseri SH. New concepts of feasibility and efficiency of solutions in fuzzy mathematical programming problems. Fuzzy Inform Eng. 2014;6(2):203–221. [14] Hladik M. Interval linear programming: A survey. In: ZA Mann, editor. Linear programming-New frontiers in theory and applications. 2. Ct: Nova Science Publishers; 2012. p. 85–120. [15] Nasseri SH, Zavieh H. A multi-objective method for solving fuzzy linear programming based on semi-infinite model. Fuzzy Inform Eng. 2018;10(1):91–98. [16] Sharma U, Shashi A. Solving fully fuzzy multi-objective linear programming problem using nearest interval approximation of fuzzy number and interval programming. Int J Fuzzy Syst. 2018;20(2):488–499. [17] Allahdadi M, Mishmast Nehi H, Ashsyerinasab HA. and Javanmard, M.: improving the modified interval linear programming method by new techniques. Info Sci. 2016;339:224–236. [18] Huang GT, Moore RD. Grey linear programming, its solving approach and its application. Int J Syst Sci. 1993;24:159–172. [19] Allahdadi M, Mishmast Nehi H. The optimal solution set of the interval linear programming problems. Optim Lett. 2013;7:1893–1911. [20] BenIsrael A, Robers PD. A decomposition method for interval linear programming. Manage Sci. 1970;16(5):374–387. [21] Cadenas JM, Verdegay JL. Using ranking functions in multi-objective fuzzy linear programming. Fuzzy Sets Syst. 2000;111(1):47–53. [22] Ghaznavi M, Soleimani F, Hoseinpoor N. Parametric analysis in fuzzy number linear programming problems. Int J Fuzzy Syst. 2016;18(3):463–477. [23] Nasseri SH, Khazaee Kohpar O. Paretooptimal solutions in multi-objective linear programming with fuzzy numbers. Ann Fuzzy Math Inform. 2015;5:823–833. [24] Tong SC. Interval number and fuzzy number linear programmings. Fuzzy Sets Syst. 1994;66: 301–306. [25] Bello R, Verdegay JL. Rough sets in the soft computing environment. Inf Sci (Ny). 2012;212:1–14. [26] Lamata MT, Pelta D, Verdegay JL. Optimisation problems as decision problems: The case of fuzzy optimisation problems. Inf Sci (Ny). 2018;460-461:377–388.