Access the full text.
Sign up today, get DeepDyve free for 14 days.
J. Cadenas, J. Verdegay (2000)
Using ranking functions in multiobjective fuzzy linear programmingFuzzy Sets Syst., 111
(2010)
A fuzzy optimal simplex algorithm and its application for solving flexible linear programming problems
(2012)
Interval linear programming: a survey. In: ZAMann, editor. Linear programming – new frontiers in theory and applications
M. Ghaznavi, F. Soleimani, N. Hoseinpoor (2016)
Parametric Analysis in Fuzzy Number Linear Programming ProblemsInternational Journal of Fuzzy Systems, 18
M. Inuiguchi, J. Ramík (2000)
Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problemFuzzy Sets Syst., 111
Hsien-Chung Wu (2008)
Optimality conditions for linear programming problems with fuzzy coefficientsComput. Math. Appl., 55
(2012)
Linear programming -new frontiers in theory and applications. 2. New York: Nova science publishers
(2012)
Interval linear programming : a survey
S. Nasseri, H. Zavieh (2018)
A Multi-objective Method for Solving Fuzzy Linear Programming Based on Semi-infinite ModelFuzzy Information and Engineering, 10
S. Nasseri, O. Kohpar (2015)
Pareto-optimal solutions in multi-objective linear programming with fuzzy numbers
Y. Leung (1988)
Spatial Analysis and Planning under Imprecision
H. Zimmermann (1978)
Fuzzy programming and linear programming with several objective functionsFuzzy Sets and Systems, 1
(2018)
Optimization problems as decision problems: the case of fuzzy optimization problems
(1911)
The optimal solutions set of the interval linear programming problems
N. Mahdavi-Amiri, S. Nasseri (2007)
Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variablesFuzzy Sets Syst., 158
Shaocheng Tong (1994)
Interval number and fuzzy number linear programmingsFuzzy Sets and Systems, 66
(1978)
Fuzzy sets. Decision making and expert systems
M. Allahdadi, H. Nehi (2013)
The optimal solution set of the interval linear programming problemsOptimization Letters, 7
(2016)
Parametric analysis in fuzzynumber linear programming problems
(1989)
Fuzzy optimization: an appraisal. Fuzzy Sets Syst
M. Luhandjula (1989)
Fuzzy optimization: an appraisalFuzzy Sets and Systems, 30
Amin Amid, S. Ghodsypour, C. O’Brien (2006)
Fuzzy multiobjective linear model for supplier selection in a supply chainInternational Journal of Production Economics, 104
Rafael Bello, J. Verdegay (2012)
Rough sets in the Soft Computing environmentInf. Sci., 212
M. Delgado, F. Herrera, J. Verdegay, M. Vila (1993)
Post optimality analysis on the membership functions of a fuzzy linear programming problemFuzzy Sets and Systems, 53
M. Allahdadi, H. Nehi, H. Ashayerinasab, M. Javanmard (2016)
Improving the modified interval linear programming method by new techniquesInf. Sci., 339
H. Attari, S. Nasseri (2014)
New Concepts of Feasibility and Efficiency of Solutions in Fuzzy Mathematical Programming ProblemsFuzzy Information and Engineering, 6
S. Nasseri, N. Mahdavi-Amiri (2009)
Some duality results on linear programming problems with symmetric fuzzy numbersFuzzy Information and Engineering, 1
G. Huang, R. Moore (1993)
Grey linear programming, its solving approach, and its applicationInternational Journal of Systems Science, 24
(1982)
Fuzzy mathematical programming. Fuzzy mathematical programming
M. Lamata, D. Pelta, J. Verdegay (2017)
Optimisation problems as decision problems: The case of fuzzy optimisation problemsInf. Sci., 460-461
Uday Sharma, Shashi Aggarwal (2018)
Solving Fully Fuzzy Multi-objective Linear Programming Problem Using Nearest Interval Approximation of Fuzzy Number and Interval ProgrammingInternational Journal of Fuzzy Systems, 20
Adi Ben-Israel, P. Robers (1970)
A Decomposition Method for Interval Linear ProgrammingManagement Science, 16
FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 221–238 https://doi.org/10.1080/16168658.2021.1886821 A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach a b a S. H. Nasseri , J. L. Verdegay and F. Mahmoudi a b Department of Mathematics, University of Mazandaran, Babolsar, Iran; Department of Computer Science and A.I., Universidad de Granada, Granada, Spain ABSTRACT ARTICLE HISTORY Received 21 January 2021 Recently fuzzy interval flexible linear programs have attracted many Revised 8 March 2021 interests. These models are an extension of the classical linear pro- Accepted 17 March 2021 gramming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the KEYWORDS decision-making problems are generally imprecise. Such uncertain- Multi-objective linear ties can lead to increased complexities in the related optimisation programming; fuzzy interval efforts. Simply ignoring these uncertainties is considered undesired ﬂexible linear programming; interval linear programming; as it may result in inferior or wrong decisions. Therefore, inexact interval arithmetic; ﬂexible linear programming methods are desired under uncertainty. In this constraints 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, a numerical example is included to illustrate the mentioned solving process. 1. Introduction Fuzzy sets theory has been extensively employed in linear programming. The main objec- tive 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 coeffi- cient 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 a fuzzy envi- ronment. Zimmermann [1] introduced the first formulation of fuzzy linear programming to address the impreciseness and vagueness of the parameters in linear programming prob- lems with fuzzy constraints and objective functions. There are generally four fuzzy linear programming classifications in the literature: Zimmermann [2] has classified fuzzy linear programming problems into two categories: symmetrical and non-symmetrical models. In a symmetrical fuzzy decision, there is no dif- ference 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. 222 S. H. 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 pre- cise 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 [26] proposed a fuzzy linear programming model where a lin- ear ranking function was used to rank 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 pri- mal simplex tableau. Mahdavi-Amiri and Nasseri [8] developed some methods for solving fuzzy linear programming problems by introducing and solving certain auxiliary prob- lems. 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 [9] derived the optimality conditions for fuzzy linear programming problems by proposing two solution concepts based on similar solution con- cept, called the non-dominated solution, in the multi-objective programming problem. Inuiguchi and Ramik [6] and Peidro et al. have developed a number of fuzzy linear pro- gramming models to solve problems ranging from supply chain management to product development. Then Verdegay in [10] used the duality results to solve the original fuzzy linear programming. After that, Nasseri et al. in [11] 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 [12] introduced a concept of feasibility and efficiency of solution for the fuzzy mathematical programming prob- lems. The suggested algorithm needs to solve two classical associated linear programming problems to achieve an optimal flexible solution. Interval linear programming (ILP), based on interval analysis, was proved to be an effec- tive approach in dealing with uncertainties. Interval linear programming did not require distributional information and would not lead to complicated intermediate models. How- ever, it was to be noted that the outputs of ILP were with lower and upper bounds, and thus could not reflect the distribution of uncertainty within the lower and upper bounds [13]. In some methods, 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 the best model are positive. In the Best and Worst Cases (BWC) method pre- sented by Tong, the ILP model transformed into two sub-models [14,15], which consist of the largest and the smallest feasible regions, so the BWC method introduces exact bounds of objective function values. A given point is feasible of ILP model, if it satisfies in best model FUZZY INFORMATION AND ENGINEERING 223 constraints and it is optimal of ILP model, if it is optimal solution of arbitrary characteris- tic model of ILP model. Chinnec and Ramadan developed BWC method when ILP model includes equality constraints [16], and a new method for solving proposed by Huang and More [17]. 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 program- ming (MILP) method, by adding an extra constraint to the second sub-model. Some of the solutions which are obtained by 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 [17]. Solution space of TSM method may be included infeasible solutions. To eliminate infeasi- ble solutions from solution space of TSM method, some methods were 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 [16,18] modified and improved the Tong method, which was unable to get an optimal response on some issues In this study, we give a generalised form of these problems in two ways: in the first way, we consider the flexibility condition for the con- straints, and in the 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 defini- tions and results on interval linear programming problem. Sections 3 give the definition of fuzzy flexible linear programming (FFLP) problem and propose 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, the conclusions are 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 ILP prob- lem. In the first time, Ben and Robbers presented the first ILP 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 [13,17,18]. − + − + Definition 2.1: Given x and x ∈ R such 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 x itself, therefore it holds that x = [x, x]. − + m×n − + Definition 2.3: Given two matrices A and A ∈ R such that A ≤ A , we define a − + m×n − + − + real interval matrix A = [A , A ] as the set {A ∈ R : A ≤ A ≤ A }. The matrices A , A 224 S. H. NASSERI ET AL. are called the lower bound matrix and the upper bound matrix of the interval matrix A, respectively. 1 1 + − + − The radius and centre of A are A = (A − A ) and A = (A + A ), respectively. Δ C 2 2 − + Thus A = [A , A ] = [A − A , A + A]. C Δ C − + n An interval vector I is introduced as the set {I : I ≤ I ≤ I }, where I, I ∈ R are crisp vector [19]. Definition 2.4: A general form of the ILP model is defined as follows: ± ± max Z = C x j j j=1 (1) ± ± ± s.t. a x ≤ b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n, ± − + ± − + ± − + − + where C [C , C ], a [a , a ]and b [b , b ] are interval numbers and x [x , x ]isan j j j ij ij ij i i i j j n-dimensional interval decision vector. Theorem 2.1: In the ILP model (1), the largest and smallest feasible regions are + − a x ≤ b , i = 1, 2, ... , m, x ≥ 0, j = 1, 2, ... , n and ij j i j j=1 − + a x ≤ b , ∀i, x ≥ 0, j = 1, 2, ... , n, respectively. ij j i j j=1 Proof: The proof is straightforward by the common interval arithmetic. Definition 2.5: A point y = (y , y , ... , y ) is said to be a feasible point of ILP model (1) if 1 2 n + − a y ≤ b , i = 1, 2, ... , m,and y ≥ 0, j = 1, 2, ... , n. ij i j j=1 There are several methods for solving ILP problems, one of which the 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 [9,16]. Mentioned method transforms the ILP problem (1) into pessimistic and optimistic sub- problems, which are summarised as follows: The pessimistic sub-problem: max Z = C x , j=1 (2) + − s.t. a x ≤ b , i = 1, 2, ... , m, ij i j=1 x ≥ 0, j = 1, 2, ... , n. j FUZZY INFORMATION AND ENGINEERING 225 The optimistic sub-problem: + + max Z = C x , j=1 (3) − + s.t. a x ≤ b , i = 1, 2, ... , m, ij i j=1 x ≥ 0, j = 1, 2, ... , n. The optimal solutions to sub-problems (2) and (3) are in box form as follows: x = ± ± ± − + (x , x , ... , x ), where for all j = 1, 2, ... , n, x = [x , x ]. This box is the solution area 1 2 j j j which is introduced by Tong. Theorem 2.2: In solving process of 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 0 0 interval [z , z ]. If x is a solution given by the above model, then we will have it a x ≥ ij j=1 b , i = 1, 2, ... , m. On the other hand, n n x ≥0 0 0 0 0 a ≤ a → a x ≤ a x ∀i → a x ≤ a x . ij ij ij ij ij ij j j j j j=1 j=1 n n 0 0 Given the above phrases and b ≥ b we have b < b ≤ a x ≤ a x , i = 1, 2, ... , m. i i i i ij ij j j j=1 j=1 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): ∗ n n x ≥0 ∗ ∗ ∗ ∗ c ≤ c → c x ≤ c x ∀j → c x ≤ c x . j j j j j j j j j j j=1 j=1 ∗ ∗ If z = c x is the objective value of model (1), then we have: z ≤ z and if z is optimal j=1 solution of model (2), then z < z and so z < z . Similarly, if x ˜ is a solution of model (3), then n n n x ˜ ≥0 a x ˜ ≥ b and a ≤ a ∀j a x ˜ ≤ a x ˜ ∀j → a x ˜ ≤ a x ˜ , and since b ≥ b .Thus, ij j i ij ij ij j ij j ij j ij j i i j=1 j=1 j=1 n n we will have: b ≤ b ≤ a x ˜ ≤ a x ˜ . i i ij j ij j j=1 j=1 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). 226 S. H. NASSERI ET AL. x ≥0 n n c ≤ c ∀j c x ≤ c x ∀j → c x ≤ c x . If z is the value of the objective j j j j j j j j j j j=1 j=1 function model (3) for the feasible solution, z < z ¯ and since x is the solution of model ∗ ∗ ∗ (2), should be z ≤ z,asaresult 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 fulfilment 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 FFLP problems with fuzzy resources can be formulated as follows (see in [20] too): max z = f (x, C) = c x j j j=1 (4) s.t.g (x) = a x b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n. In the above model, the relation is called ‘ fuzzy less than or equal to’ and it is assumed that the tolerance p for each constraint is given [21]. This means that the decision-maker can accept a violation of each constraint up to degree p . In this case, constraint g (x) b ,is i i equivalent to g (x) ≤ b + θp , (i = 1, 2, ... , m), where θ ∈ [0, 1]. i i Thus, problem (4) can be equivalently considered as the following fuzzy inequality constraints (see also in [15]): max z = f (x, C) = c x j j j=1 (5) s.t.g (x) = a x ≤ b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n. In model (5), b is a fuzzy number with the following membership function: 1, x ≤ b , ⎨ i μ (x) = 1 − (x − b )/p , b ≤ x ≤ b + p , (6) ˜ i i i i i 0, x ≥ b + p . i i Verdegay [10] proved that Problem (4) is equivalent to the crisp parametric LP prob- lem when the membership functions of the fuzzy constraints are continuous and non- increasing functions. According to this non-symmetric approach, the membership function of fuzzy inequality constraints of problem (4) can be modelled as follows: 1, g (x) ≤ b , ⎨ i μ (g (x)) = 1 − (g (x) − b )/p , b ≤ g (x) ≤ b + p , (7) i i i i i i i i i 0, g (x) ≥ b + p , i i i FUZZY INFORMATION AND ENGINEERING 227 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{μ (g (x)), μ (g (x)), ... , μ (g (x))}.(8) 1 2 m 1 2 m Assuming, α = min{μ (g (x)), μ (g (x)), ... , μ (g (x))}, then Problem (4) is equivalent 1 1 2 2 m m to: max z = f (x, C) = c x j j j=1 (9) s.t. μ (g (x)) ≥ α, i = 1, 2, ... , m, x ≥ 0, j = 1, 2, ... , n, α ∈ [0, 1]. 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 multi- parameter approach, the decision-maker’s need and appeal will be easily resolved. The following is a description of this method [12]. 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) = c x j j j=1 (10) s.t. μ (g (x)) ≥ α , i = 1, 2, ... , m, i i x ≥ 0, j = 1, 2, ... , n, α ∈ [0, 1]. j i Now, by substituting membership function (7) into problem (10), the following crisp para- metric LP problem is achieved: max z = f (x, C) = c x j j j=1 (11) s.t.g (x) = (Ax) − b ≤ (1 − α )p , i = 1, 2, ... , m, i i i i x ≥ 0, α ∈ (0, 1], j = 1, 2, ... , n. j i Note that for each α ∈ (0, 1], i = 1, 2, ... , m, an optimal solution is obtained. This indicates that the solution with α grade of membership function is actually fuzzy. Let us start with the following definitions below to continue the article. Definition 3.1: Let α ¯ = (α , ... , α ) ∈ (0, 1] be a vector, and 1 m X ={x ∈ R |x ≥ 0, μ {g (x, a ) 0}≥ α , i = 1, 2, ... , m.}. α ¯ i i i Then, a vector x ∈ X is called an α ¯ -feasible solution of model (5). α ¯ Following proposition enables us to define feasible set of model (5) as an intersection of all α-cuts corresponding to fuzzy constraints. 228 S. H. NASSERI ET AL. m i i n Proposition 3.1: Let α ¯ = (α , ... , α ) ∈ (0, 1] , then X = X , where X ={x ∈ R |x 1 m α ¯ α α i i i=1 ≥ 0, μ {g (x, a ) 0}≥ α },for i ∈ I ={1, ... , m} (Namely, X is the α-cuts of the i-th con- i i i i α straint). Proof: For α ¯ = (α , ... , α ) ∈ (0, 1] ,let x ∈ X . Therefore, μ {g (x, a ) 0}≥ α and from 1 m α ¯ i i i i n i X ={x ∈ R |x ≥ 0, μ {g (x, a ) 0}≥ α },wehave x ∈ X , i ∈ I. i i i α i α i i m m i i i Therefore, x ∈ X . Also, if x ∈ X ,wehave x ∈ X , i ∈ I,thus μ {g (x, a ) 0}≥ α i i i α α α i i i i i=1 i=1 and hence, x ∈ X . α ¯ Therefore, the proof is completed. Proposition 3.2: Let α = (α , ... , α ) and α = (α , ... , α ), where α ≤ α for all i. 1 m 1 m i i Then, α -feasibility of x implies the α -feasibility of it. Proof: The proof is straightforward. For a given α ∈ (0, 1], let a solution x ∈ R be ordinary α-feasible to problem (4) a solution in which has the same satisfaction degree in all of constraints. It means that i m μ {g (x, a ) 0}≥ α ,or x ∈ X , for all i ∈ I.If α ¯ = (α , ... , α ) ∈ (0, 1] , then x ∈ X ,which i i i 1 m α ¯ i α 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 is not empty. α ¯ Definition 3.2: Let be a fuzzy extension of relation ≤ and a solution X = T n m (x , ... , x ) ∈ R be an α ¯ − feasible to problem (5), where α ¯ = (α , ... , α ) ∈ (0, 1] and 1 n 1 m let f (x, C) be an objective function in the form of maximisation. Then, X = (x , ... , x ), 1 n where x ∈ R is an α ¯ − efficient solution to problem (5), if there is no x ∈ X so that j α ¯ 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 FFLP problem which is defined in (4) with interval multi-objective functions. We first need to denote the form of mentioned problem as follows: ± ± ± ± max Z (x) ={z (x), z (x), ... , z (x)} 1 2 p s.t.g (x) = a x b , i = 1, 2, ... , m, ij j i (12) j=1 x ≥ 0, j = 1, 2, ... , n, T ± where x = (x , x , ... , x ) is a real vector of decision variables, and where Z (x) is an inter- 1 2 n val multi-objective function that is the objective coefficients is interval numbers. a shows a ij coefficient matrix as A = [a ], where A is an m × n-dimensional matrix of interval technical ij FUZZY INFORMATION AND ENGINEERING 229 coefficients. Objective functions and constraints where i ∈{1, ... , m} possess continuous property up to the second derivatives. Also, denote a fuzzy extension of ≤ onR which is used to compare the left and right side of fuzzy constraints [12]. In general, model (12) is not well-defined due to the following reasons: We cannot maximise the interval and multi-objective quantity Z (x). The constraint g (x) = a x b , i = 1, 2, ... , m do not result in a crisp feasible set. ij j i j=1 We first need to solve FFLP problem with multi-objectives. We show that this problem will reduce to one objective function by use of weighted technique for objective function. In the weighted method as well as used in [8], we assign k-th value function equal to w that these w should be positive [22]. In other words, to find efficient solutions to the following multi-objective issues. ± ± max Z = w Z (x) k K k=1 (13) s.t.g (x) = a x b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n, where 0 ≤ w ,w , ... ,w ≤ 1suchthatw + w + ... + w = 1 are the weights of the 1 2 p 1 2 p mentioned functions, which are determined by the decision-maker. It is important to have a few points in weighting: The weight of each target w is between zero and one and the total weight must be one. All target functions are Max or Min. The coefficients of the decision variables in each objective function with the other objective function must be both scalable and therefore of a large category. Now, by using BWC method transformed the ILP problem (13) into pessimistic and optimistic sub-problems, which are summarised as follow (see in [23] for more details): The optimistic sub-problem: + ± max Z = w Z (x) k K k=1 (14) s.t.g (x) = a x b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n, The pessimistic sub-problem: max Z = w Z (x) k K k=1 (15) s.t.g (x) = a x b , i = 1, 2, ... , m, ij j i j=1 x ≥ 0, j = 1, 2, ... , n. Now, since constraints of problem are flexible, first, we select one of the above sub- prob- lems and describe the solving process, and then solve the following problem using the approach outlined below. Let us select sub-problem (14) for solving process. Therefore, in 230 S. H. NASSERI ET AL. order to obviate those mentioned restrictions, we introduce the following problem: + ± max Z = w Z (x) k=1 (16) s.t. μ {g (x) b }≥ α , i = 1, 2, ... , m, i i i x ≥ 0, 0 ≤ α ≤ 1, j = 1, 2, ... , n, j i + ± where Z means the corresponding crisp value of interval function Z . To motivate for a meaningful choice of membership function for each fuzzy constraints, it is argued that if g (x) ≤ b , then the i-th constraint is absolutely satisfied, where as if g (x) ≥ b , where i i i i p the predefined maximum tolerance from zero, as determined by the decision-marker, then the i-th constraint is absolutely violated. For g (x) ∈ (0, p ), the membership function is monotonically decreasing. If this decrease is along a linear function, then it makes sense to choose the membership function of the i-th constraint (i = 1, 2, ... , m)as: 1, g (x) ≤ b , ⎨ i μ (g (x)) = 1 − (g (x) − b )/p , b ≤ g (x) ≤ b + p , (17) i i i i i i i i i 0, g (x) ≥ b + p , i i Now, in order to find maximum efficient solution, i.e. an α ¯ -efficient solution with α ¯ ≥ α, i = 1, 2, ... , m, we perform the following two-phase approach. To express this two-phase approach to the above problem, let us consider the problem (13) and implement a two- phase approach for this sub-problem, and then, with the resumption of the approach discussed below, we solve the second problem. In the two-phase approach, Equation (7) is solved in Phase I, while in Phase II, a solution is obtained which has higher satisfaction degrees than the previous solution. Thus by using this two-Phase approach, we achieve a better utilisation of available resources. Further the solution resulting by this approach is always an α ¯ -efficient solution. Let us consider the definition and substituting in the problem (13) achieve the parametric linear programming that solved by linear techniques. by substituting membership function (7) into problem (14), the following crisp paramet- ric LP problem is achieved: + + max Z = w Z (x) k K k=1 (18) s.t.g (x) = (Ax) − b ≤ (1 − α )p , i = 1, 2, ... , m, i i i i x ≥ 0, 0 ≤ α ≤ 1, j = 1, 2, ... , n. j i Let us call the problem (15) as Phase I problem. 0 0 +∗ ± +∗ Let α = (α , ... , α ) and (x , C x ) be the optimal solution of pessimistic sub- 1 m ∗ ∗ 0 problem of Phase I with α degree of efficiency. Set α = μ {g (x , a ) 0}≥ α , i = i i i i 1, 2, ... , m. In Phase II, we solve the following problem, max α i=1 p p (19) + + +∗ s.t. w Z (x) ≥ w Z (x ) k K k K k=1 k=1 x ≥ 0, α ≤ α ≤ 1, j = 1, 2, ... , n. j i i FUZZY INFORMATION AND ENGINEERING 231 ∗∗ Theorem 4.1: The optimal solution x to problem (19) is a maximum α ¯ -efficient solution to problem (16). The process of the parametric approach for implementing the second sub-problem and the final solution are obtained. Due to theorem, and optimal value and optimal solution of two sub-problems, interval solution of problem (12), is equal to: Algorithm 4.1: Assumption 1: Consider a FFLP problem is given to solve and obtain optimise Z such as problem (12). Step 1: Using weighted method for problem (13) that transformed the multi-objectives into one target for objective function. Step 2: By use of the BWC method, obtained the corresponding crisp objective function for the objective function of model (13), and achieved two sub-problems. Step 3: Consider one of the two sub-problems and go to Step 4. Step 4: In Phase I, obtain the corresponding Multi-Parametric Linear Programming (MPLP) problem for problem (16) based on Equation (7). Step 5: Solve the MPLP problem (18) and obtain optimal solution of problem. Step 6: Based on the optimal solution of MPLP problem in Step 5, obtain the MPLP problem of Phase II such as problem (19), and solve it. Step 7: Consider second sub-problem (15) and go to Step 4. Now, we are a place to illustrate our suggested algorithm in the next section. 5. Numerical Examples In this section, we solve the FFLP problem which is multi-objective and has interval coeffi- cients in objective function by use of the proposed approach which is introduced in the last section. Example 5.1: Consider the following interval multi-objective linear programming prob- lems with flexible constraints. max z = [1, 3]x + [−1, 1.5]x 1 1 2 max z = [0.5, 2]x + [−1.5, −1]x 2 1 2 s.t. 1.5x + 2x 4, (20) 1 2 2x + 3x 1, 1 2 x ≥ 0, x ≥ 0, 1 2 where p = 2and p = 5 are predefined maximum tolerance. 1 2 1 1 Step 1: By considering the weights as w = and w = for the objective function, 1 2 2 2 where w = 1, and then by use of weighted method reduce above multi-objective in one i=1 3 5 −5 1 objective function in form of Z = w z + w z = , x + , x , and we can rewrite 1 1 2 2 1 2 4 2 4 4 232 S. H. NASSERI ET AL. problem (20) as follows: 3 5 −5 1 max Z = , x + , x 1 2 4 2 4 4 s.t. 1.5x + 2x 4, 1 2 (21) 2x + 3x 12, 1 2 x ≥ 0, x ≥ 0, 1 2 Step 2: By use of the BWC method convert the interval linear problem (21) into two sub- problems as follows: 5 1 max Z = x + x 1 2 2 4 s.t. 1.5x + 2x 4, 1 2 (22) 2x + 3x 12, 1 2 x ≥ 0, x ≥ 0, 1 2 3 −5 max Z = x + x 1 2 4 4 s.t. 1.5x + 2x 4, 1 2 (23) 2x + 3x 12, 1 2 x ≥ 0, x ≥ 0, 1 2 Step 3: consider the sub-problem (22) and continue Step 4: obtain the corresponding multi-parametric linear programming (MPLP) problem for problem (22) based on Equation (7) 5 1 max Z = x + x 1 2 2 4 s.t. 1.5x + 2x ≤ 4 + 2(1 − α ), 1 2 1 (24) 2x + 3x ≤ 12 + 5(1 − α ), 1 2 2 x ≥ 0, x ≥ 0, α , α ∈ [0, 1], 1 2 1 2 Step 5: By solve above problem achieve x = (3.333, 0) be an (0.5, 0.4)− efficient solution T ∗ with C x = 8.333 as an optimal value of problem (24). Step 6: Obtain the MPLP problem of Phase II such as problem (25) based on optimal solution of problem (24). max α + α 1 2 5 1 s.t. x + x ≥ 8.333, 1 2 2 4 (25) 1.5x + 2x ≤ 4 + 2(1 − α ), 1 2 1 2x + 3x ≤ 12 + 5(1 − α ), 1 2 2 ∗ ∗ x ≥ 0, x ≥ 0, 0.5 ≤ α ≤ 1, 0.4 ≤ α ≤ 1, 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 233 ∗∗ T ∗ T ∗∗ An optimal solution to the above problem is x = (3.332, 0),also C x = C x = 8.333, andwehave ∗∗ ∗∗ μ (g (x , a )) = 1, μ (g (x , a )) = 0.5. 1 1 2 2 1 2 Step 7: consider second sub-problem (23) and solve it. Then we achieve that an ∗∗ T ∗ T ∗∗ optimal solution to the above problem is x = (3.332, 0),also C x = C x = 2.5, and ∗∗ ∗∗ μ (g (x , a )) = 1, μ (g (x , a )) = 0.5. 1 1 2 2 1 2 Finally, with regard to Theorem 2.2, we can obtain Z = [2.5, 8.333] that is an interval optimal value of problem (20), and higher satisfaction in membership function in μ . Example 5.2: An automobile factory produces three models A , A and A . Three types of 1 2 3 raw materials M , M and M are required to manufacture them. The amounts (in kg) of the 1 2 3 materials are given in Table 1. Based on market analysis, the expected unit profits of A , A and A are given in Table 2. 1 2 3 According to the monthly production report, the per unit harmful pollutant is given in Table 3. The unit cost of the automobile is listed in Table 4. The decision-makers of the factory attempt to achieve three goals on a weekly basis as follows: To maximise the profit; To minimise the generation of the harmful pollutant; To minimise the production cost. The three goals are constrained by the following capacities on a weekly basis: Table 1. Requirement per automobile. M M M 1 2 3 A [2000,2100] [8000,9000] [4000,4500] A [3000,3200] [1000,1200] 0 A [4000,5000] [4000,4600] [2000,2400] Table 2. Unit proﬁt. A A A 1 2 3 Proﬁt/unit [5000,5120] [10,000,12,100] [12,000,13,500] Table 3. Per unit harmful pollutant. A A A 1 2 3 Pollutant/unit [1000,1050] [2000,2100] [1000,1150] Table 4. Unit cost. A A A 1 2 3 Cost / unit [1000,1120] [3000,3090] [4000,4140] 234 S. H. NASSERI ET AL. Considering the cost of M is very high, the usage of M is required to be less than 1 1 40,000 kg and this amount is ultimately allowed in a rate of 50,000 kg. Since the shortage of M is often a concern, the usage of M only is required to be less 2 2 than 50,000 kg and the maximum amount can be increased to 5000 kg. The usage of M cannot be more than 50,000 kg. By the demand of automobiles in the market, it is required to produce at least 3 units of A and 5 units of A per week. 1 3 In order to find the optimal quantities of A , A and A per week, this problem is modelled 1 2 3 as the following multi-objective interval linear programming (MOILP) problem: max z = [5000, 5120]x + [10, 000, 12, 100]x + [12, 000, 13, 500]x 1 1 2 3 min z = [1000, 1050]x + [2000, 2100]x + [1000, 1150]x 2 1 2 3 min z = [1000, 1120]x + [3000, 3090]x + [4000, 4140]x 3 1 2 3 s.t. [2000, 2100]x + [3000, 3200]x + [4000, 5000]x 40, 000, 1 2 3 [8000, 9000]x + [1000, 1200]x + [4000, 4600]x 50, 000, (26) 1 2 3 [4000, 4500]x + [2000, 2400]x ≤ 50, 000, 1 2 x ≥ 3, x ≥ 5, x ≥ 0, x ≥ 0, x ≥ 0, 1 2 3 Take w = 0.4, w = 0.3 and w = 0.3. Then the interval weighted sum secularisation of the 1 2 3 MOILP problem with respect to is as follows: max Z = [1349, 1448]x + [2443, 3340]x + [3213, 3900]x 1 2 3 s.t. [2000, 2100]x + [3000, 3200]x + [4000, 5000]x 40, 000, 1 2 3 [8000, 9000]x + [1000, 1200]x + [4000, 4600]x 50, 000, 1 2 3 [4000, 4500]x + [2000, 2400]x ≤ 50, 000, (27) 1 2 x ≥ 3, x ≥ 5, x ≥ 0, x ≥ 0, x ≥ 0, 1 2 3 Now, we solve the problem (27) by using the solving technique of interval problems and solving algorithm in this paper and the given tolerances p = 10, 000, p = 5000. We will 1 2 simplify the first sub-problem based on mentioned solving algorithm steps in (4.1) as follows: max Z = 1448x + 3340x + 3900x 1 2 3 s.t. 2000x + 3000x + 4000x ≤ 40, 000 + 10, 000(1 − α ), 1 2 3 1 8000x + 1000x + 4000x ≤ 50, 000 + 5000(1 − α ), 1 2 3 2 4000x + 2000x ≤ 50, 000, (28) 1 2 x ≥ 3, x ≥ 5, x ≥ 0, x ≥ 0, x ≥ 0, α , α ∈ [0, 1], 1 2 3 1 2 some α ¯ -efficient solution with satisfaction degrees which decision-maker can be found in the Table 5. FUZZY INFORMATION AND ENGINEERING 235 Table 5. Some typical α ¯ -eﬃcient solution of sub-problem 1. 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 41657 44,997 47,224 44,997 x 33333 x 6.3333 5.33 6.3333 7 6.3333 x 55555 ∗ T ∗ Let x = (3, 6.3333, 5) be (0.5, 0.6)-efficient solution with C x = 44997 as an optimal value of problem (28). In Step 6, we need to solve the following linear problem: max α i=1 s.t. 1448x + 3340x + 3900x ≥ 44, 997, 1 2 3 2000x + 3000x + 4000x ≤ 40, 000 + 10, 000(1 − α ), 1 2 3 1 8000x + 1000x + 4000x ≤ 50, 000 + 5000(1 − α ), 1 2 3 2 (29) 4000x + 2000x ≤ 50000, 1 2 0.5 ≤ α ≤ 1, 0.6 ≤ α ≤ 1, 1 2 x ≥ 3, x ≥ 5, x ≥ 0, x ≥ 0, x ≥ 0, 1 2 3 ∗∗ T ∗ T ∗∗ An optimal solution to the above problem is x = (3, 6.3332, 5),also C x = C x = 44997, we have ∗∗ ∗∗ μ (g (x , a )) = 0.5, μ (g (x , a )) = 1. 1 1 2 2 1 2 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 μ . Now, we use all this 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 T ∗ T ∗∗ ∗ C x = C x = 26219 and x = (3, 0, 6.2784), ∗∗ ∗∗ μ (g (x , a )) = 1, μ (g (x , a )) = 1. Finally, with regards to Theorem 2.2 optimal 1 1 2 2 1 2 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 generalisa- tion of the parametric approach in linear programming provides a more appropriate tool for modelling real problems and improving the solving process. Also, in the solving pro- cess for the MOILP model, a weighted technique 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 236 S. H. NASSERI ET AL. instance Rough Sets (see in [24]), 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 [25]). Acknowledgments The first author would like to appreciate from the research grant of University of Mazandaran. The research of 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). Funding The first author would like to appreciate from the research grant of University of Mazandaran. The research of 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. Notes on contributors S. H. Nasseri received his Ph.D degree in 2007 on Fuzzy Mathematical Programming from Sharif Uni- versity 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 (Middle East 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. Interna- tional 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 M.S. degree in mathematics and the Ph.D. 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 Euro- pean 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 FUZZY INFORMATION AND ENGINEERING 237 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 scien- tific 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 B.S. degree in Applied Mathematics, Department of Mathematics, Univer- sity of Mazandaran (2011–2015). She obtained his M.Sc. degree in Applied Mathematics, Operations Research, Department of Mathematics, University of Mazandaran under supervision of Prof. Hadi Nasseri (2015–2017). Now, she is a Ph.D student in Applied Mathematics, Operations Research, University of Mazandaran, Babolsar, Iran. 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. MA: Kluwer Academic Pub- lishers; 1978. [3] Amid A, Ghodsypour SH, O’Brien C. Fuzzy multi objective 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 pro- gramming 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 function of a fuzzy linear programming problem. Fuzzy Sets Syst. 1993;53:289–297. [8] Nasseri SH, Mahdavi-Amiri N. Some duality results on linear programming problems with sym- metric fuzzy numbers. Fuzzy Inf Eng. 2009;1:59–66. [9] Wu HC. Optimality conditions for linear programming problems with fuzzy coefficients. Comput Math Appl. 2008;55:2807–2822. [10] Verdegay JL. Fuzzy mathematical programming. Fuzzy mathematical programming. In: MM Gupta, E Sanchez, editords. Fuzzy information and decision processes. Amsterdam: North Hol- land; 1982. p. 231–237. [11] Nasseri SH, Ebrahimnejad A. A fuzzy optimal simplex algorithm and its application for solving flexible linear programming problems. Eur J Ind Eng. 2010;4(3):372–389. [12] Attari H, Nasseri SH. New concepts of feasibility and efficiency of solutions in fuzzy mathematical programming problems. Fuzzy Inf Eng. 2014;6(2):203–221. [13] Hladik M. Interval linear programming: a survey. In: ZA Mann, editor. Linear programming – new frontiers in theory and applications. 2. New York: Nova science publishers; 2012. p. 85–120. [14] Nasseri SH, Zavieh H. A multi-objective method for solving fuzzy linear programming based on semi-infinite model. Fuzzy Inf Eng. 2018;10(1):1–8. [15] Sharma U, Shashi A. Solving fully fuzzy multi-objective linear programming problem using interval approximation of fuzzy number and interval programming. Int J Fuzzy Syst. 2018;20(2):488–499. [16] Allahdadi M, Mishmast Nehi H, Ashsyerinasab HA, et al. Improving the modified interval linear programming method by new techniques. Inf Sci (Ny). 2016;339:224–236. [17] Huang GT, Moore RD. Grey linear programming, its solving approach and its application. Int J Syst Sci. 1993;24:159–172. [18] Allahdadi M, Mishmast Nehi H. The optimal solutions set of the interval linear programming problems. Optim Lett. 2013;7:893–1911. [19] Ben-Israel A, Robers PD. A Composition method for interval linear programming. Manage Sci. 1970;16(5):334–374. [20] Cadenas JM, Verdegay JL. Using ranking functions in multi-objective fuzzy linear programming. Fuzzy Sets Syst. 2000;111(1):47–53. 238 S. H. NASSERI ET AL. [21] Ghaznavi M, Soleimani F, Hoseinpoor N. Parametric analysis in fuzzy number linear programming problems. Int J Fuzzy Syst. 2016;18(3):463–477. [22] Nasseri SH, Khazaee Kohpar O. Pareto- optimal solutions in multi-objective linear programming with fuzzy numbers. Ann Fuzzy Math Inform. 2015;5:823–833. [23] Tong SC. Interval number and fuzzy number linear programing. Fuzzy Sets Syst. 1994;66:301–306. [24] Bello R, Verdegay JL. Rough sets in the soft computing environment. Inf Sci (Ny). 2012;212:1–14. [25] Lamata MT, Pelta D, Verdegay JL. Optimization problems as decision problems: the case of fuzzy optimization problems. Inf Sci (Ny). 2018;460–461:377–388. [26] 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.
Fuzzy Information and Engineering – Taylor & Francis
Published: Apr 3, 2019
Keywords: Multi-objective linear programming; fuzzy interval flexible linear programming; interval linear programming; interval arithmetic; flexible constraints
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.