Iterative Solution Process for Multiple Objective Stochastic Linear Programming Problems Under Fuzzy Environment

This article presents one interactive algorithm, and thereby determines the Pareto optimal solution to multi-objective stochastic linear programming (MOSLP) problems in real-life oriented fuzzy environment. Among the various objective functions, there always exists one objective function, referred to as the main objective function in this article, to multi-objective models, whose optimal value is most vital to decision-makers. When the optimal value to main objective function meets the pre-determined aspiration level, and the corresponding 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 objective functions are considered as a unity. However, this seems to be less rational that the expectation of each conflicting objective function 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.


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 functions, mathematicians have proposed several methods in the literature for characterising Pareto optimal solutions to MOLPP in a fuzzy environment [2][3][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 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 environment [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 interactive 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]. Emergency 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 imprecise 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 interactive 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 objective 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 environment; 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].
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 fundamental 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 reference 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 better 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 impreciseness 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 programming models such as expectation optimisation, variance minimisation, probability maximisation, fractile criterion optimisation together with chance-constrained programming 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 reference membership levels obtained analytically from trade-off ratios to multiple objective stochastic linear programing problems (MOSLPP) in fuzzy environment. If DM is not satisfied 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 main objective function in fuzzy environment. In Section 5, numerical examples are considered 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.

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: Here x is an n-dimensional column vector of decision variables, A is an m × n coefficient matrix, . . k andb 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 constraints associated with model (1), we may employ chance-constrained conditions that permit constraint violations up to specified probability limits. Suppose that a q is the qth row vector of A andb q is the qth element ofb. Here we assume that each random variableb q has a continuous probability distribution function F q (r) = P(b q ≤ r).
and replace constraints in model (1) with chance-constrained conditions with satisfying probability levels β i , ∀i = 1 . . . k, we may reformulate model (1) as chance-constrained problem as follows:

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 objective functions z i (x) =c i x, ∀i = 1 . . . k with corresponding expectations, MOSLPP may be reformulated as follows:

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 i (z i E (x)), i = 1 . . . k from goals and tolerances (supplied by DM) for expectations z i E (x), i = 1 . . . k of corresponding fuzzy objective functions [38,43]. Consequently, in fuzzy environment, MOSLPP may be converted into optimisation problems having membership functions 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 specifies aspiration levels of achievements for membership functions (i.e. reference membership levelsμ 1 Next for DM's reference membership levelsμ = (μ 1 ,μ 2 . . .μ k ) T , corresponding M-Pareto optimal solutions nearest toμ = (μ 1 ,μ 2 . . .μ k ) T in minimax sense or even better than that, if attainable at all, maybe obtained by solving following minimax problem: subject to the constraints minimize v subject to the constraintŝ

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 specific 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 optimal 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 optimisation 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 specified main objective function is prioritised over other objective functions to solve MOSLPP in fuzzy environment.
Another problem is to determine reference membership levelsμ 1 of fuzzy objective functions. We may note that DM usually specifies these reference membership levelsμ 1 ,μ 2 . . .μ k of expectations of fuzzy objective functions from past experiences, intuition, etc. In existing interactive fuzzy optimisation methods 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 t E (x), t ∈ {1 . . . k} arbitrarily. Next using the chain rule, we may compute trade-off ratios π tj between membership function one by one, as follows [10,41,42]: Next, we may derive a set of numbersμ 1 . . .μ k by using the following formula [38]: Next, we may takeμ t = ξ , any positive non-zero real number andμ = max{μ 1 . . .μ k }. Clearlyμ = 0. Finally, the initial reference membership levelμ (0) i , i = 1 . . . k of expectation of ith objective function may be determined by the following formula [38]: Finally, we may employ these reference membership levels of expectations of fuzzy objective functions to determine M-Pareto optimal solutions to generalised MOSLPP (2).
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 corresponding optimal value of v. Without loss of generality, we may assume that all constraints of model (6) are active constraints. Therefore, we have If possible, suppose x is not Pareto optimal solution to MOSLPP (1).
) is a strictly monotonic decreasing function for the expectation of minimising type of objective function z i = v (since all constraints are active). Consequently, we may get: which is the desired contradiction to 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 someμ i , i = 1 . . . k.
Proof: Let x ∈ X be Pareto optimal solution to MOSLPP (1); also assume that ∀i = 1 . . . kz i E (x ) is the corresponding optimal value of expectation of objective function z i (x). Next, ∀i = 1 . . . k, we may construct a membership function μ i (z i E (x)) of expectation (z i E (x)) of fuzzy objective function z i (x). Therefore ∀i = 1 . . . k, we may choose reference membership levelsμ i such thatμ i − μ i (z i E (x )) = v and v ≥ 0. Consequently, all constraints of MOSLPP (1) become active constraints for x ∈ X.
Since x ∈ X is Pareto optimal solution to MOSLPP (1), Hence it implies that x is optimal solution to single objective optimisation model (6) for 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 optimality test problem with decision variables x = (x 1 , x 2 , . . . , x n ) T , ε = (ε 1 , ε 2 . . . ε k ) T as follows [38,42]: subject to the constraints for minimizing type of objective functions:

x' is Pareto optimal solution to MOSLPP (1). (2) If at least oneε i > 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.

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 , q = 1 . . . m) for each constraint. Compute individual maximum and minimum of expectation of each objective function (z i (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 If DM is not available or DM is unable to specify goal(s) and/or tolerance(s) for expectation(s) of fuzzy objective function(s), corresponding individual minimum and/or maximum value(s) or any suitable value(s) may be employed.
Step 3: Next, we have to determine initial reference membership levelsμ (0) i , i = 1 . . . k for expectations of fuzzy objective functions analytically. First, expectation (z t E (x), t ∈ {1 . . . k}) of any one objective function may be chosen arbitrarily. Using chain rule, we may compute trade-off ratios π tj between membership function μ t (z t E (x)) of z t E (x) and membership functions μ j (z j E (x)), j = 1 . . . k, j = t of expectations (z j E (x), j = 1 . . . k, j = t) of other objective functions one by one as follows [10,38,41,42]: Step 4: Next, we may derive a set of numbersμ 1 ,μ 2 . . .μ k by using the formula [10,38,41,42] Here, we may chooseμ t = ξ (any positive non-zero real number) and suppose thatμ t = ξ μ = max{μ 1 ,μ 2 . . .μ k } (so,μ = 0). Finally, in fuzzy environment, initial reference membership levelsμ Step 5: Now we may request DM to specify the main objective function. Suppose that DM chooses z α (x), for some α ∈ {1 . . . k}, as the main objective function. Next in fuzzy environment, by using initial reference membership levels of expectations of objective functions, following optimisation model may be solved.
minimize v subject to the constraintŝ Let x (1) , v (1) are optimal values of x and v to the above model after the first iteration; also suppose that z i E (x (1) respectively. This completes the first iteration.
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] Step 7: Finally, employing these updated reference membership levelsμ (n) i , we may solve the following optimisation model minimize v subject to the constraintŝ are optimal values of x and v to the above model after (n + 1) iterations; also suppose z i E (x (n+1) ), μ i (z i E (x (n+1) )) are corresponding optimal values of z i E (x), μ i (z i E (x)), ∀i = 1 . . . k, respectively, after (n + 1) iterations.
Step 8: Proceeding in this way, we may stop when the iterative process converges or the main objective function z α (x) has attained its goal. Suppose that the iteration stops after m steps and x (m) , z j E (x (m) ), μ j (z j E (x (m) )) are corresponding M-Pareto optimal values of x, z j E (x), μ j (z j E (x)), j = 1 . . . k, respectively. To test uniqueness of the M-Pareto optimal solution, we may solve the following model [38,42]: subject to the constraints for minimizing type of objective functions: (for maximizing type of objective functions: Let ∀i = 1 . . . k,ε i andx are optimal values of ε i and x, respectively. Ifε i = 0, ∀i = 1 . . . k, solution after m iterations, i.e. the solution x (m) is Pareto optimal solution to the given MOSLPP. Otherwise, the solutionx 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.

Numerical Examples
To illustrate our proposed iterative process further, we may modify the production planning 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.
subject to the constraints 5x 1 + 7x 2 ≤ 16.2, 9x 1 + x 2 ≤ 12.525, −5x 1 + 3x 2 ≤ 3.85, In this problem, we may consider that the coefficients of objective functions are represented as random variables with associated means as E(5)  N(3.85, 1 2 ). Also, suppose 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 i E (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  Table 2. Goals and tolerances to objective functions.
Expectations ofobjective functions Goals Tolerances specify goals and tolerances of expectations (z i E (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 environment, we may construct corresponding membership functions μ i (z i E (x)), i = 1, 2, 3 of expectations of objective functions in fuzzy environment as follows.
Next, we may choose expectation of any one objective function, say z 1 E (x), arbitrarily. By using chain rule, we may compute trade-off ratios between membership function μ 1 (z 1 E (x)) of expectation of (z 1 E (x)) and membership functions μ j (z j E (x)), j = 2, 3 of expectations of other objective functions (z j E (x), j = 2, 3)one by one as follows: In order to determine initial reference membership levelsμ Next suppose that DM specifies z 2 (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): Here we may observe that optimal value of expectation of the main objective function z 2 (x) is more preferable than corresponding aspiration level of DM and optimal values of expectations of all other objective functions z i (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 completely 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 programming 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 tradeoff attainments, etc. [44]. In our case, if we apply goal programming method having z 2 as the main objective function, corresponding optimal solution, as given in Table 3, shows that optimal expected values of z 1 and z 3 are very poor since optimal membership values of expectations are zero in both z 1 and z 3 . Hence it may be reasonable to confess that such a 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.
More preferable optimal value for expectation of main objective function z 1 (x) than corresponding goal.
More preferable optimal value for expectation of main objective function z 2 (x) than corresponding goal.
More preferable optimal value for expectation of main objective function z 3 (x) than corresponding goal.
Again, suppose that DM specifies z 1 (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 1 E (x)) of the main objective function is more preferable than the corresponding goal; and optimal values of other objective functions are also acceptable to DM in a proposed iterative process (since both of these objective functions z i (x), i = 2, 3 attain their respective goals).
Again, suppose that DM specifies z 3 (x) as 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 3 E (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 i (x), i = 1, 2 attain their respective goals).
On the other hand, if the DM does not specify (or unable to specify) main objective function 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 optimal 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 existing 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 on the new main objective function. Further, the proposed iterative process may be applicable 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 membership 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 linear 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, analyst 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 iterative 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 failures 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.
Dr. Tapan Kuma Roy, is currently a Professor in the Department of Mathematics, IIEST, Howrah, India. He completed the Doctor of Philosophy from the Vidyasagar University. He has published good number of articles on fuzzy and stochastic optimization, reliability optimization, inventory, supply chain in imprecise environments. Under his supervision, 23 students have been awarded Ph.Ds. He is much passionate about collecting rare books in regional languages.