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FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 4, 435–451 https://doi.org/10.1080/16168658.2020.1750871 Iterative Solution Process for Multiple Objective Stochastic Linear Programming Problems Under Fuzzy Environment a b b Arindam Garai , Palash Mandal and Tapan Kumar Roy a b Department of Mathematics, Sonarpur Mahavidyalaya, Kolkata, India; Department of Mathematics, IIEST, Shibpur, India ABSTRACT KEYWORDS Main objective function; This article presents one interactive algorithm, and thereby deter- trade-oﬀ ratio; mines the Pareto optimal solution to multi-objective stochastic linear chance-constrained programming (MOSLP) problems in real-life oriented fuzzy environ- programming; expectation ment. Among the various objective functions, there always exists one model; fuzzy multi-objective objective function, referred to as the main objective function in this optimisation; stochastic article, to multi-objective models, whose optimal value is most vital optimisation to decision-makers. When the optimal value to main objective func- tion meets the pre-determined aspiration level, and the correspond- ing values to other objective functions are satisfactory in nature, that Pareto optimal solution is acceptable to decision-makers. Again, in several existing interactive fuzzy optimisation methods to MOSLP models, all reference membership levels of expectations to objec- tive functions are considered as a unity. However, this seems to be less rational that the expectation of each conflicting objective func- tion simultaneously attains the individual goal. So, the present article proposes to employ the trade-off ratios of membership functions to analytically determine reference membership levels in a fuzzy environment. Numerical applications further illustrate this algorithm. Finally, conclusions are drawn. 1. Introduction Multiple objective linear programming is the process of optimising simultaneously and systematically a collection of objective functions [1]. In crisp optimisation methods to solve multiple objective linear programming problems (MOLPP), usually, we are unable to employ preferences of decision-maker (DM) effectively. Therefore, on the basis of such imprecise information in real-life decision-making problems, usually, we use two approaches: fuzzy programming and stochastic programming. By assuming that the DM has imprecise aspiration levels for each of the objective func- tions, mathematicians have proposed several methods in the literature for characterising Pareto optimal solutions to MOLPP in a fuzzy environment [2–4]. In 1970, Bellman and Zadeh introduced MOLPP in fuzzy environment for such decision-making problems [5]. In 1974, Tanaka, Okuda and Asai first formulated MOLPP with fuzzy parameters [6]. In fact, in 2015, Luhandjula has correctly observed that Tanaka et al. have helped raise the CONTACT Arindam Garai fuzzy_arindam@yahoo.com © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group 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. 436 A. GARAI ET AL. intellectual respectability of fuzzy optimisation [7,8]. As pointed out by Zimmermann in 1976 and later in 1978, various kinds of impreciseness can be categorised as fuzziness and consequently he developed one mathematical method to solve MOLPP in fuzzy environ- ment [9]. In 1984 and 1986, interactive fuzzy optimisation method was first developed by Sakawa et al. [10]. In their article, DM is involved in solution steps. Subsequently, in real-life decision-making cases, interactive approaches have played an important role in deriving best-preferred compromise solutions to MOLPP. The main advantage of the inter- active approach is that DM controls the search direction during the solution procedure; consequently, such solutions can usually achieve DM’s aspiration levels. Fuzzy optimisation methods were applied in solving real-world problems by Werners [11], Lai and Hwang [12], Gupta et al. [13], Kahraman et al. [14], Sahoo et al. [15], Zhou et al. [16], Wu et al. [17], Garg et al. [18], Wei et al. [19], Salehi et al. [20], Su et al. [21], Garai et al. [22,23] and several other mathematicians. We can find extensive studies of evolutionary computing approaches in fuzzy environment, e.g. Fernandeza et al. [24]. Fuzzy set theory also found applications in structural models; many researchers like Wang and Wang [25], Yeh et al. [26], Xu [27], Shih et al. [28] made distinctive implementations of fuzzy set theory. Fuzzy set theory has been used to yield decisions in multiple objective decision-making problems among multiple choices and multiple items by Yu et al. [29], Ebrahimnejad and Verdegay [30], Chakraborty et al. [31], Garai et al. [2], etc. Studies involving genetic algorithm have been studied by Martinez-Soto et al. [32], Jongsuebsuk et al. [33]and many others in recent past. Optimal design of fuzzy tracking controllers for autonomous mobile robots under perturbed torques has been developed by Melin et al. [34]. Emer- gency transportation planning in disaster relief supply chain management was elaborately discussed by Zheng and Ling [35]. A survey on nature-inspired optimisation algorithms with fuzzy logic was performed by Valdez et al. [36]. Also study of MOLPP involving impre- cise objective functions, interactive techniques, KKT optimality conditions, etc. have been investigated by researchers in fuzzy environment in recent years [2]. E.g. an improved fuzzy self-adaptive learning multiple objective particle swarm optimisation algorithm has been developed for dynamic economic emission dispatch problem by applying interac- tive fuzzy optimisation method under stochastic environment by Bahman Bahmani-Firouzi et al. [37]. But in existing interactive fuzzy optimisation methods by Sakawa et al. [38], although DM may consider goals and tolerances of fuzzy objective functions, we may observe that DM cannot effectively specify any objective function as the main objective function. In other words, we may find that each of the objective functions is treated with equal importance in existing interactive solution method for solving MOLPP in fuzzy environment. Although DM can assign weights, etc. [39] to each of the fuzzy objective functions, such methods have several disadvantages. We may further note that DM usually prefers the optimal solution in which optimal value of specified main objective function is most preferable and optimal values of other objec- tive functions are acceptable to DM [3]. In other words, till date, mathematicians have used fuzziness of DM’s aspirations to construct and solve optimisation models in fuzzy environ- ment; but time is ripe to go beyond. In fuzzy environment, we need one iterative process in which most preferable Pareto optimal solutions to MOLPP may be generated based on specified main objective function [3]. FUZZY INFORMATION AND ENGINEERING 437 Further, we may find that interaction in each step of decision process may satisfy plan of DM to actively participate in solution processes; but in long run, it may waste precious time and money of DM. In several recent cases, we may find that DM may not be available at each stage of decision-making activities. In 1979, Wierzbicki correctly raised one funda- mental issue: ‘what is more valuable: the perfection of a compromise, which is based on a never-perfect model, or the time of a top-rank decision maker?’ [40]. If confronted with a multitude of questions (e.g. ‘would you prefer this alternative to the other one?’) at each and every stage of decision-making activity, DM may simply send the analyst back to where he/she belongs [40]! Consequently, it may be more rational to look for minimal yet efficient interaction with DM [3]. Further, we may observe that concept of reference membership levels, which first appeared in Sakawa et al. [41], can be viewed as one natural extension of the idea of ref- erence point by Wierzbicki. In existing interactive fuzzy optimisation techniques, unity is taken as an initial reference membership level of each fuzzy objective function. But we bet- ter not expect each conflicting fuzzy objective function to attain the corresponding goal simultaneously [3]. Accordingly, we may propose that the initial reference membership level of each fuzzy objective function need to be based on analytically derived results. On the other hand, in most real-life situations, it seems natural to consider that impre- ciseness should be expressed by a fusion of fuzziness and randomness rather than by either fuzziness or randomness [42]. To effectively handle DM’s ambiguous judgements in MOLPP as well as the randomness of parameters involved in objective functions and/or constraints, Sakawa and his colleagues incorporated interactive fuzzy satisfying methods associated with deterministic problems into multiple objective stochastic programming problems [10,41]. In last few decades, many other scientists have introduced several stochastic pro- gramming models such as expectation optimisation, variance minimisation, probability maximisation, fractile criterion optimisation together with chance-constrained program- ming methods to derive satisficing solutions for DM from a set of Pareto optimal solutions. As Sakawa and other fellow researchers have described, in chance-constrained problems, for random data variations, a mathematical model can be formulated so that the violation of constraints is permitted up to specified probability levels [38]. In this article, we propose one iterative process to determine most preferable Pareto optimal solutions based on the specified main objective function and thereby using ref- erence membership levels obtained analytically from trade-off ratios to multiple objective stochastic linear programing problems (MOSLPP) in fuzzy environment. If DM is not sat- isfied with this Pareto optimal solution, he/she may alter his/her aspiration levels and/or priority. Based on the updated information, another set of Pareto optimal solution may be yielded to MOSLPP in fuzzy environment. The iterative process may be continued until DM is satisfied with the latest Pareto optimal solutions. Here, for the sake of simplicity alone, we choose expectation optimisation model in a stochastic environment so that readers can easily digest the proposed theme of our manuscript without unnecessarily deviating into other details. The rest of the article is organised as follows. In Section 2, we introduce a mathematical model of MOSLPP in fuzzy environment. In Section 3, we discuss the solution method to find preferable Pareto optimal solutions to MOSLPP based on specified main objective function in fuzzy environment. Related theorems are discussed here. In Section 4, we develop one iterative process to find preferable Pareto optimal solutions to MOSLPP based on specified 438 A. GARAI ET AL. main objective function in fuzzy environment. In Section 5, numerical examples are consid- ered to further illustrate our proposed iterative process. Here, we compare results by the proposed process with several other existing solution techniques. In Section 6, conclusions are drawn and scopes for future research are given. 2. Expectation Model of Multiple Objective Chance-Constrained Programming in Fuzzy Environment 2.1. Formation of MOSLPP Assuming that the coefficients in the objective functions and right-hand side constants of the constraints are random variables, we may formulate one general MOSLPP as follows: minimize (z (x)... z (x)) 1 k subject to the constraints (1) Ax ≤ b, x ≥ 0, Here x is an n-dimensional column vector of decision variables, A is an m × n coefficient matrix, z (x), i = 1 ... k are k conflicting objective functions with z (x) = c ¯ x, ∀i = 1 ... k, i i i c ¯ , ∀i = 1 ... k are n-dimensional random variable row vectors with finite means E(c ¯ ) and i i n × n positive definite variance covariance matrices V = [v ] = [Cov{c , c }], ∀i = 1 ... k i ij ih jh and b is n-dimensional column vector whose elements are mutually independent random variables. Since coefficients of model (1) are taken as random variables; we have to apply solution methods appropriate for stochastic events. Consequently, to deal with con- straints associated with model (1), we may employ chance-constrained conditions that permit constraint violations up to specified probability limits. Suppose that a is the ¯ ¯ qth row vector of A and b is the qth element of b. Here we assume that each ran- ¯ ¯ dom variable b has a continuous probability distribution function F (r) = P(b ≤ r). q q q −1 Hence ∀q = 1 ... m,P(a x ≤ b ) ≥ β implies that a x ≤ F (1 − β ). If we denote X(β) q q q q q −1 asX(β) ={x|a x ≤ F (1 − β ), q = 1 ... m, x ≥ 0} and replace constraints in model (1) q q q with chance-constrained conditions with satisfying probability levels β , ∀i = 1 ... k,we may reformulate model (1) as chance-constrained problem as follows: minimize (z (x)... z (x)) 1 k (2) subject to the constraints x ∈ X(β) 2.2. Expectation Model for MOSLPP In expectation model, DM wishes to minimise expected values of objective functions subject to constraints in his/her attempt to deal with MOSLPP. Therefore by replacing objec- tive functions z (x) = c ¯ x, ∀i = 1 ... k with corresponding expectations, MOSLPP may be i i FUZZY INFORMATION AND ENGINEERING 439 reformulated as follows: E E T minimize (z (x)... z (x)) subject to the constraints (3) x ∈ X(β) ¯ ¯ where z (x) = E[z (x)] = E[c ]x denotes expectation of z (x) = c x, ∀i = 1 ... k. i i i i i 2.3. MOSLPP in Fuzzy Environment and Corresponding Existing Solution Techniques, Including Interactive Technique It is well known that fundamental to MOSLPP (3) is the concept of Pareto optimal solutions. Considering the imprecise nature inherent in our judgements in MOSLPP (3), linguistic statements can be quantified by generating membership functions μ (z (x)), i = 1 ... k i i from goals and tolerances (supplied by DM) for expectations z (x), i = 1 ... k of corre- sponding fuzzy objective functions [38,43]. Consequently, in fuzzy environment, MOSLPP may be converted into optimisation problems having membership functions μ (z (x)), i = i i 1 ... k as follows: max min{μ (z (x)), i = 1 ... k} i i (4) subject to x ∈ X(β) We may note that the approach to model (4) is preferable only when fuzzy decision by Bellman and Zadeh [43] is a proper representation of fuzzy preferences of DM. However as Sakawa and other researchers have described several times, such situations seem to occur rarely in practice [38]. Consequently in existing interactive fuzzy optimisation techniques, initially, DM speci- fies aspiration levels of achievements for membership functions (i.e. reference membership levels μ ˆ , μ ˆ ... μ ˆ ) of expectations z (x), i = 1 ... k of fuzzy objective functions z (x), i = 1 2 i i 1 ... k. Next for DM’s reference membership levels μ ˆ = (μ ˆ , μ ˆ ... μ ˆ ) , corresponding M- 1 2 Pareto optimal solutions nearest to μ ˆ = (μ ˆ , μ ˆ ... μ ˆ ) in minimax sense or even better 1 2 k than that, if attainable at all, maybe obtained by solving following minimax problem: min max (μ ˆ − μ (z (x))) i i i i=1...k (5) subject to the constraints x ∈ X(β). Or equivalently if v = max (μ ˆ − μ (z (x))),wehave: i i i i=1...k minimize v subject to the constraints (6) μ ˆ − μ (z (x)) ≤ v, i = 1 ... k, x ∈ X(β). i i i 440 A. GARAI ET AL. 3. Finding Pareto Optimal Solutions to MOSLPP in Fuzzy Environment Based on Specified Main Objective Function We may observe in real-life decision-making problems that DM may like to suggest one spe- cific objective function as a main objective function. And in the existing decision-making process, especially in the interactive process, DM’s focus may lie essentially in the opti- mal value of the main objective function. And DM may get satisfaction only if the optimal value of the specified main objective function meets the aspiration level and values of other objective functions are satisfactory. But we may find that in existing interactive fuzzy opti- misation methods to solve MOSLPP, each objective function is equally treated. As a result, DM cannot effectively specify any one objective function as a main objective function in existing interactive fuzzy optimisation methods to solve MOSLPP. Consequently, DM may not be able to find optimal values as per his/her specifications. In this article, we have taken these issues under consideration and have developed one iterative process, in which spec- ified main objective function is prioritised over other objective functions to solve MOSLPP in fuzzy environment. Another problem is to determine reference membership levels μ ˆ , μ ˆ ... μ ˆ of expecta- 1 2 k tions (z (x), i = 1 ... k) of fuzzy objective functions. We may note that DM usually specifies these reference membership levels μ ˆ , μ ˆ ... μ ˆ of expectations of fuzzy objective func- 1 2 k tions from past experiences, intuition, etc. In existing interactive fuzzy optimisation meth- ods to solve MOSLPP, initial reference membership levels are taken as unity. But we should not always expect that each of such conflicting objective functions shall attain respective goals simultaneously. Hence in this article, we may propose to compute initial reference membership levels of expectations of fuzzy objective functions analytically from trade-off ratios. Consequently, we may choose expectation of any one objective function, say z (x), t ∈ {1 ... k} arbitrarily. Next using the chain rule, we may compute trade-off ratios π between tj E E E membership function μ (z (x)) of z (x) and membership functions μ (z (x)), j = t t t j j 1 ... k, j = t of expectations of other objective functions z (x), j = 1 ... k, j = t, one by one, as follows [10,41,42]: −1 E E E E ∂μ (z (x)) ∂μ (z (x)) ∂z (x) ∂μ (z (x)) t t t t t j j π =− =− , j = 1 ... k, j = t tj E E E E ∂μ (z (x)) ∂z (x) ∂z (x) ∂z (x) j j t j j Next,wemay deriveasetofnumbers μ ¯ ... μ ¯ by using the following formula [38]: −1 ∂μ (z (x)) t t μ ¯ = − μ ¯ , j = 1 ... k, j = t j t ∂μ (z (x)) j j Next, we may take μ ¯ = ξ, any positive non-zero real number and μ ˆ = max{¯ μ ... μ ¯ }. t 1 Clearly μ ˆ = 0. (0) Finally, the initial reference membership level μ ˆ , i = 1 ... k of expectation of ith objec- tive function may be determined by the following formula [38]: (0) μ ˆ =¯ μ /μ ˆ , ∀i = 1 ... k(∵ μ ˆ = 0) (7) Finally, we may employ these reference membership levels of expectations of fuzzy objec- tive functions to determine M-Pareto optimal solutions to generalised MOSLPP (2). FUZZY INFORMATION AND ENGINEERING 441 The relationship between M-Pareto optimal solution to model (6) and Pareto optimal solution to MOSLPP (1 maybe characterised by the following theorems) Theorem 3.1: If x ∈ X is a unique optimal solution to model (6), x is Pareto optimal solution to MOSLPP (1). Proof: Let x ∈ X is a unique optimal solution to minimax problem (6) and v be the corre- sponding optimal value of v. Without loss of generality, we may assume that all constraints of model (6) are active constraints. Therefore, we have E E v = min{|μ ˆ − μ (z (x ))|, μ ˆ − μ (z (x ))} or |ˆ μ − μ (z (x ))|= v =ˆ μ − μ (z (x )) i i i α α α i i i α α α If possible, suppose x is not Pareto optimal solution to MOSLPP (1). Then ∃ at least E E E E one x ∈ X such that z (x)) ≤ z (x ), ∀m = 1 ... k, m = n and z (x)< z (x ) for some m m n n n.Since ∀i = 1 ... k, μ (z (x)) is a strictly monotonic decreasing function for the i i E E expectation of minimising type of objective function z (x),wehave: μ (z (x )) ≤ i m m E E E μ (z (x)) ∀m = 1 ... k, m = n and μ (z (x )) < μ (z (x)). Hence ∀j,wehave: μ ˆ − m m n n n n j E E μ (z (x)) ≤ˆ μ − μ (z (x )) = v (since all constraints are active). Consequently, we may j j j j j E E get: min{|μ ˆ − μ (z (x))|, μ ˆ − μ (z (x))}≤ v , which is the desired contradiction to i i i α α α the fact that x is a unique optimal solution to the minimax problem (6). Therefore, our assumption is wrong. Hence x is Pareto optimal solution to MOSLPP (1) Theorem 3.2: If x ∈ X is a Pareto optimal solution of the MOSLPP (1), x is an optimal solution to model (6) for some μ ˆ ,i = 1 ... k. Proof: Let x ∈ X be Pareto optimal solution to MOSLPP (1); also assume that ∀i = 1 ... kz (x ) is the corresponding optimal value of expectation of objective function z (x). i i Next, ∀i = 1 ... k, we may construct a membership function μ (z (x)) of expectation i i (z (x)) of fuzzy objective function z (x). Therefore ∀i = 1 ... k, we may choose refer- i i ence membership levels μ ˆ such that μ ˆ − μ (z (x )) = v and v ≥ 0. Consequently, all i i i i constraints of MOSLPP (1) become active constraints for x ∈ X. Since x ∈ X is Pareto optimal solution to MOSLPP (1), E E E E ∃ no y ∈ X such that ∀i = 1 ... k, i = j, z (y) ≤ z (x ) and z (y)< z (x ), for some j i i j j E E E E i.e. ∀i = 1 ... k, i = j, μ (z (y)) ≥ μ (z (x )) and μ (z (y)) > μ (z (x )), for some j i i i i j j j j [∵ ∀i = 1 ... k, μ is monotonically decreasing function] E E i.e. ∀i = 1 ... k, i = j, μ ˆ − μ (z (y)) ≤ˆ μ − μ (z (x )) and i i i i i i E E μ ˆ − μ (z (y)) < μ ˆ − μ (z (x )), for some j j j j j j j E E i.e. ∀i = 1 ... k, μ ˆ − μ (z (y)) ≤ˆ μ − μ (z (x )) i i i i i i E E i.e. ∀y ∈ X, ∃ at least one α ∈{1 ... k} such that μ ˆ − μ (z (y)) > μ ˆ − μ (z (x )) = v α α α α α α Hence it implies that x is optimal solution to single objective optimisation model (6) for some μ ˆ , i = 1 ... k. Hence the result is proved. For generating Pareto optimal solution using theorems 3.1 and 3.2, we have to verify uniqueness of this M-Pareto optimal solution. This can be performed by considering Pareto 442 A. GARAI ET AL. T T optimality test problem with decision variables x = (x , x , ... , x ) , ε = (ε , ε ...ε ) as 1 2 n 1 2 follows [38,42]: maximize ε i=1 subject to the constraints (8) E E for minimizing type of objective functions: z (x ) − ε ≥ z (x), i = 1 ... k, i i i E E (for maximizing type of objective functions: z (x) − ε ≥ z (x ), i = 1 ... k) i i i x ∈ X(β). Theorem 3.3: Let xa ¯ nd ε ¯ are optimal solutions to model (8). Then: (1) If ε ¯ = 0, i = 1 ... k, x’ is Pareto optimal solution to MOSLPP (1). (2) If at least one ε ¯ > 0, x’ is not Pareto optimal solution of MOSLPP (1). Instead of x’, x ¯ is Pareto optimal solution to MOSLPP (1). We may skip its proof as proof of the analogous theorem is already recorded in the literature. 4. Proposed Iterative Process to Find Pareto Optimal Solutions to Given MOSLPP in Fuzzy Environment The above ideas can be further integrated into a general framework and an algorithm may be developed to find preferable Pareto optimal solutions to MOSLPP (1) based on specified main objective function in fuzzy environment. The steps of our proposed algorithm may be synthesised as follows Step 1: Request DM to specify satisficing probability levels (β , q = 1 ... m) for each constraint. Compute individual maximum and minimum of expectation of each objec- tive function (z (x), i = 1 ... k) under given constraints. If unbounded solution(s) is (are) found for expectations of objective function(s), we may assign suitable large number(s) as extremum(s). Based on the information, elicit goals and tolerances for the expectation of each fuzzy objective functions from DM. Step 2: Using these goals and tolerances, construct membership functions μ (z (x)), i = i i 1 ... k of expectations (z (x), i = 1 ... k) of fuzzy objective functions (z (x), i = 1 ... k).If i i DM is not available or DM is unable to specify goal(s) and/or tolerance(s) for expecta- tion(s) of fuzzy objective function(s), corresponding individual minimum and/or maximum value(s) or any suitable value(s) may be employed. (0) Step 3: Next, we have to determine initial reference membership levels μ ˆ , i = 1 ... k for expectations of fuzzy objective functions analytically. First, expectation (z (x), t ∈{1 ... k}) of any one objective function may be chosen arbitrarily. Using chain rule, we may com- E E pute trade-off ratios π between membership function μ (z (x)) of z (x) and member- tj t t t E E ship functions μ (z (x)), j = 1 ... k, j = t of expectations (z (x), j = 1 ... k, j = t) of other j j j objective functions one by one as follows [10,38,41,42]: −1 E E E E ∂μ (z (x)) ∂μ (z (x)) ∂z (x) ∂μ (z (x)) t t t t t j j π =− =− , j = 1 ... k, j = t tj E E E E ∂μ (z (x)) ∂z (x) ∂z (x) ∂z (x) j j t j j FUZZY INFORMATION AND ENGINEERING 443 Step 4:Next,wemayderiveasetofnumbers μ ¯ , μ ¯ ... μ ¯ by using the formula [10,38,41,42] 1 2 k −1 ∂μ (z (x)) t t μ ¯ = − μ ¯ , j = 1 ... k, j = t j t ∂μ (z (x)) j j Here, we may choose μ ¯ = ξ (any positive non-zero real number) and suppose that μ ¯ = ξ t t μ ˆ = max{¯ μ , μ ¯ ... μ ¯ } (so, μ ˆ = 0). Finally, in fuzzy environment, initial reference member- 1 2 (0) ship levels μ ˆ , i = 1 ... k of expectations of objective functions may be determined by the (0) μ ¯ formula: μ ˆ = / , ∀i = 1 ... k (∵ μ ˆ = 0). μ ˆ Step 5: Now we may request DM to specify the main objective function. Suppose that DM chooses z (x), for some α ∈{1 ... k}, as the main objective function. Next in fuzzy environ- ment, by using initial reference membership levels of expectations of objective functions, following optimisation model may be solved. minimize v subject to the constraints (0) E μ ˆ − μ (z (x)) ≤ v, α α (0) |ˆ μ − μ (z (x))|≤ v, ∀i = 1 ... k, i = α i i v ≥ 0, x ∈ X. (1) (1) Let x , v are optimal values of x and v to the above model after the first iter- E (1) E (1) ation; also suppose that z (x ), μ (z (x )) are corresponding optimal values of i i i E E z (x), μ (z (x)), ∀i = 1 ... k, respectively. This completes the first iteration. i i i Step 6: Next, to determine the updated reference membership levels of expectations of objective functions after n iterations (n being natural number) in fuzzy environment, we may apply the method of bisection as follows [10,42] (n−1) (n−1) E (n) E (n) (μ (z (x )) +ˆ μ )/2, if μ (z (x )) < μ ˆ i i (n) i i i i ∀i = 1 ... k, μ ˆ = (9) E (n) (1 + μ (z (x )))/2, otherwise (n) Step 7: Finally, employing these updated reference membership levels μ ˆ , we may solve the following optimisation model minimize v subject to the constraints (n) E μ ˆ − μ (z (x)) ≤ v, α α (n) E |ˆ μ − μ (z (x))|≤ v, ∀i = 1 ... k, i = α i i v ≥ 0, x ∈ X. (n+1) (n+1) Let x , v are optimal values of x and v to the above model after (n + 1) iter- E (n+1) E (n+1) ations; also suppose z (x ), μ (z (x )) are corresponding optimal values of i i i E E z (x), μ (z (x)), ∀i = 1 ... k, respectively, after (n + 1) iterations. i i i Step 8: Proceeding in this way, we may stop when the iterative process converges or the main objective function z (x) has attained its goal. Suppose that the iteration stops α 444 A. GARAI ET AL. (m) E (m) E (m) after m steps and x , z (x ), μ (z (x )) are corresponding M-Pareto optimal values j j j E E of x, z (x), μ (z (x)), j = 1 ... k, respectively. To test uniqueness of the M-Pareto optimal j j j solution, we may solve the following model [38,42]: maximize ε i=1 subject to the constraints E (m) E for minimizing type of objective functions: z (x ) − ε ≥ z (x), i = 1 ... k, i i i E E (m) (for maximizing type of objective functions: z (x) − ε ≥ z (x ), i = 1 ... k,) i i i ε ≥ 0, i = 1 ... k, x ∈ X. Let ∀i = 1 ... k, ε ¯ and x ¯ are optimal values of ε and x, respectively. If ε ¯ = 0, ∀i = 1 ... k, i i i (m) solution after m iterations, i.e. the solution x is Pareto optimal solution to the given MOSLPP. Otherwise, the solution x ¯ to the above model is Pareto optimal solution to given MOSLPP. Step 9: If the DM is satisfied with the Pareto optimal values, stop. Otherwise request DM to update goal(s) or tolerance(s) or even main objective function and go to Step 2. Finally, the iterative process is complete. 5. Numerical Examples To illustrate our proposed iterative process further, we may modify the production plan- ning problem, as given in the book titled Linear and Multi-objective Programming with Fuzzy Stochastic Extensions by Sakawa et al. [38] and consider the following three-objective stochastic linear programming problem in fuzzy environment. ¯ ¯ fuzzy maximize z (x) = 5x + 5x 1 1 2 ¯ ¯ fuzzy minimize z (x) = 5x + 1x 2 1 2 ¯ ¯ fuzzy maximize z (x) = 3x − 8x 3 1 2 (10) subject to the constraints 5x + 7x ≤ 16.2, 9x + x ≤ 12.525, −5x + 3x ≤ 3.85, 1 2 1 2 1 2 x ≥ 0, x ≥ 0. 1 2 In this problem, we may consider that the coefficients of objective functions are repre- ¯ ¯ ¯ ¯ sented as random variables with associated means as E(5) = 5; E(1) = 1; E(3) = 3; E(−8) = −8 and right-hand side constants of constraints are represented by normal random vari- 2 2 2 ables defined by 16.2 ∼ N(16.2, 5 ); 12.525 ∼ N(12.525, 3 ), 3.85 ∼ N(3.85, 1 ). Also, sup- pose that DM specifies the satisficing probability levels as 0.8 for each constraint. Next, we may apply the proposed iterative process on the above model to yield preferable Pareto optimal solutions based on specified main objective function in fuzzy environment. First, we may compute individual maximum values and individual minimum values of expectations (z (x), i = 1, 2, 3) of objective functions under given constraints, as given in Table 1. Based on these individual maximum and minimum values, DM may be asked to FUZZY INFORMATION AND ENGINEERING 445 Table 1. Individual maximum and minimum values of expectations of objective functions. Expectations ofobjective functions Individual maximum values Individual minimum values z (x) 10 0 z (x) 60 z (x) 3.3 −11.1 Table 2. Goals and tolerances to objective functions. Expectations ofobjective functions Goals Tolerances z (x) 51.5 z (x) 41 z (x) −22 specify goals and tolerances of expectations (z (x), i = 1, 2, 3) of objective functions in fuzzy environment, as given in Table 2. Using these goals and tolerances of expectations of objective functions in fuzzy envi- ronment, we may construct corresponding membership functions μ (z (x)), i = 1, 2, 3 of i i expectations of objective functions in fuzzy environment as follows. 1, if z (x) ≥ 5 z (x)−3.5 E μ (z (x)) = , if 3.5 ≤ z (x) ≤ 5 1 1 1.5 ⎩ E 0, if z (x) ≤ 3.5 1, if z (x) ≤ 4 5−z (x) E μ (z (x)) = ,if4 ≤ z (x) ≤ 5 2 2 ⎩ E 0, if z (x) ≥ 5 1, if z (x) ≥−2 z (x)+4 E μ (z (x)) = ,if − 4 ≤ z (x) ≤−2 3 3 ⎩ E 0, if z (x) ≤−4 Next, we may choose expectation of any one objective function, say z (x), arbitrarily. By using chain rule, we may compute trade-off ratios between membership function E E E μ (z (x)) of expectation of (z (x)) and membership functions μ (z (x)), j = 2, 3 of 1 1 1 j j expectations of other objective functions (z (x), j = 2, 3)one by one as follows: −1 E E E E ∂μ (z (x)) ∂μ (z (x)) ∂z (x) ∂μ (z (x)) 110 1 1 1 1 1 2 2 − =− = ; E E E E ∂μ (z (x)) ∂z (x) ∂z (x) ∂z (x) 129 2 2 1 2 2 −1 E E E E ∂μ (z (x)) ∂μ (z (x)) ∂z (x) ∂μ (z (x)) 80 1 1 1 1 1 3 3 − =− = E E E E ∂μ (z (x)) ∂z (x) ∂z (x) ∂z (x) 129 3 3 1 3 3 (0) In order to determine initial reference membership levels μ ˆ , i = 1, 2, 3 of expectations of objective functions z (x), i = 1, 2, 3 in fuzzy environment, we may set μ ¯ = 1. Thus, i 1 we may obtain: μ ¯ = 1.173 and μ ¯ = 1.612. And we may have μ ˆ = 1.612. Finally, using 2 3 (0) the formula: μ ˆ =¯ μ /μ ˆ , i = 1, 2, 3, initial reference membership levels of expectations i 446 A. GARAI ET AL. (0) (0) of objective functions in fuzzy environment may be obtained as: μ ˆ = 0.6202, μ ˆ = 1 2 (0) 0.7273 and μ ˆ = 1. Next suppose that DM specifies z (x) as the main objective function. Proceeding as per our proposed iterative process, as discussed in Section 4, we may find M-Pareto optimal solution and consequently Pareto optimal solution to given MOSLPP as follows (¯denotes Pareto optimality): x ¯ = 0.4623, x ¯ = 0.4235, z (x ¯) = 4.43, z (x ¯) = 2.73, z (x ¯) =−2. 1 2 1 2 3 Here we may observe that optimal value of expectation of the main objective function z (x) is more preferable than corresponding aspiration level of DM and optimal values of expec- tations of all other objective functions z (x), i = 1, 3 in a proposed iterative process in fuzzy environment, are acceptable to DM (since each of them attained respective goals). Comparison of Results We may compare among optimal values obtained by the proposed iterative process and those obtained by other existing fuzzy optimisation methods in Table 3. Advantages of the proposed iterative process over several other existing solution methods are evident from Table 3. Here we may note that no two different Pareto optimal solutions can be similar or one Pareto optimal solution cannot be better over another. It is upon the DM to find out most preferable optimal solutions to MOSLPP in fuzzy environment. Moreover, completely different philosophies are ingrained in some of those existing techniques. Hence it may not be justifiable to compare elaborately. Now, we may primarily focus on advantages of proposed iterative process over goal programming techniques. The goal programming method has weakness in at least three ways: 1. Because of the insistence on lexicographic optimality, some models may com- pletely ignore lower priority objectives; 2. Some models disallow trade-off of small losses in high priority objectives for large gains in low priority objectives; 3. Some goal program- ming models ignore the non-constancy of the rates at which benefits from the objective attainments increase and they ignore non-constancy of the rates at which DM may trade- off attainments, etc. [44]. In our case, if we apply goal programming method having z as the main objective function, corresponding optimal solution, as given in Table 3,shows that optimal expected values of z and z are very poor since optimal membership values 1 3 of expectations are zero in both z and z . Hence it may be reasonable to confess that such a 1 3 solution maybe not acceptable to DM. Further, in the goal programming technique, usually, all objective functions are to be arranged in some order. Although we may consider only the main objective function as objective function and others as constraints, it may defeat the very purpose of goal programming. Moreover, in case of a large number of conflicting objective functions, it may not be feasible to rank all objective functions. But in a proposed iterative process, at one hand, we intrinsically employ trade-off rates among membership functions of expectations of fuzzy objective functions and on other hand, only one main objective function is necessary to run the process. It is said that goal programming’s great virtue is that DM understand it easily; also, many of the alternate approaches are difficult to understand. This may be true in some instances, but should not be a guiding principle in the evolution of this field. FUZZY INFORMATION AND ENGINEERING 447 Table 3. Comparison of results. Pareto optimal solutions in Existing techniques Zimmermann’s Wu et al. Weighted technique Max-product Sakawa’s technique sum Goal pro- Proposed (Max-min operator) operator technique [17] approach gramming iterative process x ¯ = 0.5455, x ¯ = 0.94, x ¯ = 0.6583, x ¯ = 0.6583, Weights: x ¯ = 0.145, x ¯ = 0.4623, 1 1 1 1 1 1 x ¯ = 0.4545, x ¯ = 1.04, x ¯ = 0.4263, x ¯ = 0.4263, 0.25,0.5,0.25. x ¯ = 0.555, x ¯ = 0.4235, 2 2 2 2 2 2 x ¯ = 0, Optimal values, ¯ z (x ¯) = 5, z (x ¯) = 9.92, z (x ¯) = 5.42, z (x ¯) = 5.42, z (x ¯) = 3.5, z (x ¯) = 4.43, 1 1 1 1 1 1 1 x ¯ = 0, denotes optimality z (x ¯) = 3.18, z (x ¯) = 5.76, z (x ¯) = 3.72, z (x ¯) = 3.72, z (x ¯) = 1.28, z (x ¯) = 2.73, 2 2 2 2 2 2 2 z (x ¯) = 0, z (x ¯) =−2. z (x ¯) =−5.49. z (x ¯) =−1.44. z (x ¯) =−1.44. 1 z (x ¯) =−4. z (x ¯) =−2. 3 3 3 3 3 3 z (x ¯) = 0, z (x ¯) = 0. In the proposed iterative process, DM may eﬀectively specify the main objective function; whereas in existing stochastic optimisation techniques under fuzzy environment, no such facility is available. And the optimal value of the main objective function z (x) is more preferable to DM in the proposed iterative process than other existing techniques; moreover optimal values of other objective functions are also acceptable to DM (in our example, each objective function has attained its goal in Results the proposed iterative process). Under fuzzy environment, the more eﬀective and appropriate solution to MOSLPP may be obtained by applying our proposed iterative process than other existing fuzzy optimisation techniques. Ability to specify one main objective function may make Remarks the proposed iterative process more worthy as well as satisfactory to DM. 448 A. GARAI ET AL. Table 4. Pareto optimal solution to MOSLP model corresponding to the speciﬁed main objective func- tions. Speciﬁed main objective functions Pareto optimal solutions Remarks x ¯ = 0.7485, x ¯ = 0.5305, 1 2 z (x) More preferable optimal value for expectation E E E z (x ¯) = 6.39, z (x ¯) = 4.27, z (x ¯) =−2. 1 2 3 of main objective function z (x) than corresponding goal. x ¯ = 0.4623, x ¯ = 0.4235, 1 2 z (x) More preferable optimal value for expectation E E E z (x ¯) = 4.43, z (x ¯) = 2.73, z (x ¯) =−2. 1 2 3 of main objective function z (x) than corresponding goal. x ¯ = 0.8467, x ¯ = 0.0394, 1 2 z (x) More preferable optimal value for expectation E E E z (x ¯) = 4.43, z (x ¯) = 4.27, z (x ¯) = 2.23. 1 2 3 of main objective function z (x) than corresponding goal. Again, suppose that DM specifies z (x) as the main objective function. Applying the proposed iterative process with same goals and tolerances of expectations of objective functions, we may find Pareto optimal solutions to MOSLPP (10) in fuzzy environment as given in Table 4. Here, we may observe that optimal value of expectation (z (x)) of the main objective function is more preferable than the corresponding goal; and optimal val- ues of other objective functions are also acceptable to DM in a proposed iterative process (since both of these objective functions z (x), i = 2, 3 attain their respective goals). Again, suppose that DM specifies z (x) as main objective function. Applying the pro- posed iterative process with same goals and tolerances of expectations of objective func- tions, we may find Pareto optimal solutions to MOSLPP (10) in fuzzy environment as given in Table 4. Here we may observe that optimal value of expectation (z (x)) of main objective function is more preferable than corresponding goal; and optimal values of other objective functions are also acceptable to DM in the proposed iterative process (since both of these objective functions z (x), i = 1, 2 attain their respective goals). On the other hand, if the DM does not specify (or unable to specify) main objective func- tion or DM is not available at all, we may generate Pareto optimal solutions by considering each objective function as main objective function one by one; it may help in finding most suitable Pareto optimal solution to given MOSLPP in fuzzy environment. Conclusions In this article, one general iterative process is developed to obtain preferable Pareto opti- mal solutions to MOSLPP based on specified main objective function in fuzzy environment. In several existing crisp and/or fuzzy optimisation methods to solve MOSLPP, no objective function can be effectively specified as the main objective function. Even if one can assign weights, goals, priorities, utility functions etc. to objective functions, those methods have major disadvantages (as were discussed in Section 5). But, in our proposed iterative process, we may obtain preferable Pareto optimal solutions corresponding to the specified main objective function. Moreover, we show through numerical examples in Section 5 that the optimal solutions obtained by our proposed iterative process are always Pareto optimal and may be more preferable to DM in comparison with solutions obtained by several other exist- ing optimisation methods in fuzzy stochastic environment. Also, if DM alters main objective function, other sets of Pareto optimal solutions to MOSLPP may be quickly yielded based FUZZY INFORMATION AND ENGINEERING 449 on the new main objective function. Further, the proposed iterative process may be appli- cable even if DM is unable to choose the main objective function or no DM is available at all. Further, in existing fuzzy optimisation methods to solve MOSLPP, initial reference mem- bership levels are arbitrarily set at unity. But we do not find it realistic to expect that expectations of each of the conflicting objective functions shall attain their respective goals simultaneously. Here, we have noted that trade-off rates among membership functions play one main role; consequently, in the proposed iterative process, we determine initial reference membership levels analytically by making use of trade-off ratios of membership functions of expectations of objective functions in fuzzy environment. Future Research Directions There are ample scopes for research involving proposed iterative process in future. We may extend it to other stochastic programming models. We may also employ various lin- ear programming methods to solve final models obtained within this process. In future, we have scope to develop analogous iterative processes that can be applied to solve multiple objective stochastic non-linear optimisation models in fuzzy environment as well. As Wierzbicki [40] said, we may note that real-life decision-making exercises may be looked upon as tools to handle complex problems of modern society. And such tools must be checked against real-life problems always. If there are complaints about efficiency of tools, an analyst must re-examine and redesign the model. When a new tool is found, ana- lyst should be satisfied, but not to the extent of forgetting that he/she is constructing tools, which must again be checked in practice and further developed [40]. And proposed itera- tive process that is also interactive in nature may be one such optimal tool for DM to handle complex real-life problems under uncertainty. This is especially important for us not to be influenced by only past successes and fail- ures in explaining multi-objective methods to DM. If we bring into the classroom only what comes out of the boardroom, then the limitations of today’s DM’s will be imposed on tomorrow’s. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Arindam Garai, born and brought up at Burdwan. He is currently an Assistant Professor in the Department of Mathematics, Sonarpur Mahavidyalaya, Kolkata, India. He was awarded Ph.D. in Math- ematics from IIEST, Howrah in 2018. The idea of T-set is his main contribution, till date. The other research interests include multi-objective optimization, inventory, and supply chain modelling in imprecise environments. He left his job at IBM to follow his passion towards teaching and research. Palash Mandal, born and brought up at Kachukhali, Gosaba. He is currently an Assistant Teacher at Kalikapur Ramkamal Vidyapith (H.S.). He has been awarded the Ph.D. in Mathematics from IIEST, Howrah in 2018. He has published more than ten research articles in many reputed journals. His area of interest is optimization, fuzzy set theory, T-set and many more. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Oct 2, 2018
Keywords: Main objective function; trade-off ratio; chance-constrained programming; expectation model; fuzzy multi-objective optimisation; stochastic optimisation
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