Multiple objective **stochastic** **linear** **programming** is a relevant topic. As a matter of fact, many practical **problems** rang- ing from portfolio selection to water resource management may be cast into this framework. Severe limitations on ob- jectivity are encountered in this field because of the simultaneous presence of randomness and conflicting goals. In such a turbulent environment, the mainstay of rational choice cannot hold and it is virtually impossible to provide a truly scientific foundation for an optimal decision. In this paper, we resort to the bounded rationality principle to introduce satisfying solution for **multiobjective** **stochastic** **linear** **programming** **problems**. These solutions that are based on the chance-constrained paradigm are characterized under the assumption of normality of involved random variables. Ways for singling out such solutions are also discussed and a numerical example provided for the sake of illustration.

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hierarchical **multiobjective** **linear** **programming** **problems**. On the other hand, in the actual decision making situations, the decision makers often encounter difficulties to deal with vague information or uncertain data. Sakawa et al.[5],[6] for- mulated **multiobjective** **stochastic** **linear** **programming** prob- lems through a probability maximization model and a fractile optimization model, and proposed interactive algorithm to obtain a satisfactory solution from among a Pareto optimal solution set. Using a probability maximization model or a fractile optimization model, it is required for the decision maker to specify parameters called permissible objective levels or permissible probability levels in advance. However, it seems to be very difficult to specify such values in advance. In order to cope with such difficulties, Yano et al.[10] proposed fuzzy approaches to **multiobjective** **stochastic** **linear** **programming** **problems**, where the decision maker has fuzzy goals for permissible objective levels and permissible proba- bility levels, and such fuzzy goals are quantified by eliciting the membership functions. Unfortunately, in the proposed method, it is assumed that the decision maker adopts the fuzzy decision [4] to obtain the satisfactory solution.

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In this paper, we have proposed an interactive decision making method for hierarchical **multiobjective** **stochastic** **linear** **programming** **problems** to obtain a satisfactory solution from among a Pareto optimal solution set. In the proposed method, by considering the conflict between permissible objective levels and and permissible probability levels, the corresponding membership functions are integrated through the fuzzy decision. In the integrated membership space, the candidate of a satisfactory solution is obtained from among Pareto optimal solution set by updating the reference mem- bership values and/or the decision powers. In our proposed method, it is expected to obtain the satisfactory solution, in which the proper balance between permissible objective values and permissible probability levels are attained.

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Abstract—In this paper, two kinds of fuzzy approaches are proposed for not only **multiobjective** **stochastic** **linear** program- ming **problems**, but also **multiobjective** fuzzy random **linear** **programming** **problems** through a probability maximization model. In a probability maximization model, it is necessary for the decision maker to specify permissible values of objective functions in advance, which have a great influence on the corresponding distribution function values. In our proposed methods, the decision maker does not specify permissible values of objective functions, but sets his/her membership functions for permissible values. By assuming that the decision maker adopts the fuzzy decision as an aggregation operator of fuzzy goals for the original objective functions and distribution functions, a satisfactory solution of the decision maker is easily obtained based on **linear** **programming** technique.

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had been considered by Mahdavi-Amiri and Nasseri (2007), where b G (F (R ))m, A G R (m,n), cT G R n are given and x G (F(R ))n is to be determined , and a **linear** ranking function on TPFN; a = (aL, a u , a , / 3 ) defined as: R ( F ) ( a ) = The dual problem on FVLP was established to deduce duality results, those results are then used to develop a dual algorithm to solve the problem by using the primal simplex tableau. It should be noted all these studies did not consider the MFLP **problems** in the optimization **problems** but contented themselves with single FLP problem. They also did not use the **linear** ranking function R ( F ) as a tool to transform the FLP problem to its corresponding deterministic LP problem. R (F )s were used only as a tool to compare FVs.

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The fuzzy concept, fuzzy objective functions and constraints and what related it introduced for the first time by Bellman and Zadeh (1970), then others studied in the fuzzy environment.Zimmermann (1978) was presented early works of fuzzy **programming** (Zimmermann, 1983) and Zhong et al. (1994) studied fuzzy random **linear** programming.Cadenas and Verdegay (2000) considered a **multiobjective** **linear** **programming** **problems**, when coefficients of the objective functions are given as a fuzzy numbers, and used ranking functions in it. An interactive method has been provided for solving **multiobjective** fuzzy **stochastic** **programming** **problems** (Mohan and Nguyen, 2001). Iskander (2003) utilized weighted objective function to convert a multiobjectivestochastic fuzzy **linear** **programming** problem. On the other hand he studied another type of fuzzy **stochastic** fractional **linear** **programming** **problems** (Iskander, 2004b), furthermore a fuzzy weighted additive approach used to **stochastic** fuzzy goal **programming** and possibilityand necessity suggested in **stochastic** fuzzy **linear** **programming** too (Iskander, 2004a; Iskander, 2005).

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In the real world decision making situations, we often have to make a decision under uncertainty. In order to deal with decision **problems** involving uncertainty, **stochastic** **programming** approaches [1], [2], [3], [6] and fuzzy pro- gramming approaches [12], [14], [25] have been developed. Recently, mathematical **programming** **problems** with fuzzy random variables [11] have been formulated [13], [15], [17], whose concept includes both probabilistic uncertainty and fuzzy one simultaneously. Extensions to **multiobjective** fuzzy random **linear** **programming** **problems** (MOFRLP) have been done and interactive methods to obtain a satisfactory solu- tion for the decision maker have been proposed [7], [9], [15]. In their methods, it is required in advance for the decision maker to specify permissible possibility levels in a probability maximization model or permissible probability levels in a fractile optimization model [16]. However, it seems to be very difficult for the decision maker to specify such permissible levels appropriately. From such a point of view, a fuzzy approach to MOFRLP has been proposed [21], in which the decision maker specifies not the values of permissible levels but the membership functions for the

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In the real world decision making situations, we often have to make a decision under uncertainty. In order to deal with decision **problems** involving uncertainty, **stochastic** **programming** approaches [1], [2], [3], [7] and fuzzy pro- gramming approaches [13], [17], [18] have been developed. Recently, in order to deal with mathematical **programming** **problems** involving the randomness and the fuzziness, ran- dom fuzzy **programming** has been developed [8], in which the coefficients of the objective functions and/or the con- straints are represented with random fuzzy variables [14], [15]. As a natural extension, a random fuzzy **multiobjective** **programming** problem (RFMOLP) was formulated and the interactive decision making methods were proposed to obtain the satisfactory solution of the decision maker from among the Pareto optimal solution set [9], [10], [11], [12]. Moreover, in order to show the efficiency of random fuzzy **programming** techniques, real-world decision making **problems** under ran- dom fuzzy environments were formulated as random fuzzy **programming** **problems**, and the corresponding algorithms to obtain the optimal solutions were proposed [5], [6], [16].

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Many Optimization **problems** in engineering and economics involve the challenging task of pondering both conflicting goals and random data. In this paper, we give an up-to-date overview of how important ideas from optimization, probability theory and multicriteria decision analysis are interwoven to address situations where the presence of several objective functions and the **stochastic** nature of data are under one roof in a **linear** optimization context. In this way users of these models are not bound to caricature their **problems** by arbitrarily squeezing different objective functions into one and by blindly accepting fixed values in lieu of imprecise ones.

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Backward **stochastic** partial differential equations (BSPDEs) have recently received a lot of attentions. The existence, uniqueness, and regularity of solutions to the Cauchy problem of BSPEs is fairly complete nowadays. See, among oth- ers, Du et al. (2012) for an L p theory for non-degenerate BSPEs, Du and Zhang (2013) for the existence and uniqueness of degenerate parabolic BSPEs, and the relevant references therein. The previous research usually assumes that the genera- tor f is uniformly Lipschitz in the unknown variables. Du and Chen (2012) study the Cauchy − Dirichlet problem of a super-parabolic quadratic BSPDE in a simply connected bounded domain D :

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In this paper, we first formulate a pair of nondifferentiable **multiobjective** fractional pro-gramming **problems**. For a differentiable function h: R n ×R n ® R, we introduce the definition of higher-order (F, a, r, d)-convexity, which extends some kinds of general- ized convexity, such as second order F-convexity in [15] and higher-order F -convexity in [14]. Under the higher-order (F, a, r, d)- convexity assumptions, we prove the higher- order weak, higher-order strong and higher-order converse duality theorems.

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This note gives the characterization of the stability set of the first kind for convex **multiobjective** nonlinear **programming** (MONLP) **problems** with fuzzy parameters in the constraints and for convex MONLP **problems** with fuzzy parameters in the objective functions. These fuzzy parameters are characterized by TFNs and treatment under the concept of α-Pareto optimality. In this note, no di ﬀ erentiability assumptions are needed and the KTSP is used in the derivation of the proposed results.

This paper is divided into five sections. The background of this paper is introduced in Section 1. Section 2 introduces some basic definitions and recalls some results that regard to crisp multi-objective matrix games and the fuzzy relations. In Section 3, a new generalized model for a multi-objective matrix game with fuzzy goals and fuzzy payoffs via fuzzy relation approach is established. Section 4 presents a kind of multi-objective **linear** **programming** **problems**. In Section 5, the results of this paper are illustrated with a small numerical example.

proposed method is provided to solve this model and then implemented and analyzed on an actual example. Data of Milan stock market showed that the new model can be solved in shorter time. Whereas, expected return rate of 12% in based model cannot solve a problem with more than 14 types of stocks. The new model can solved a problem with 20 different stocks easily and in a short time. Furthermore, a new model greatly reduced undesirable risk very much compared to the base model. So that the process continued decreasingly with increasing the number of shares. Another study by Kuhpaei & others (2007) as a program to determine the optimal transportation of wheat was done using **linear** **programming**. Every year, millions tonnes of wheat from domestic shopping centers and import origins was transported to storage center and then distributed among regions based on consumer demand. In 2000, wheat allocated the largest volume of goods transported after cement. Using a logical model is necessary to reduce the cost of transportation. This study presents a mathematical model for determining the optimal plan of wheat transport from provincial centers and import origins to storage centers and thence to applied regions. Based on the data from 2000 and Lingo software package, studies was done separately for each month. The proposed program reduced the cost to 138 billion rials (13.5 percent) compared to the program performed in 2000. After solving the problem and determining the optimal schedule by using the concept of shadow price, time and place schedule and the method of direct or indirect transportation of wheat from shopping centers to storage centers was identified according to priorities determined by time and place. With implementation of this program, cost of transportation is reduced to 45 billion rials.

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[10] Yan, C.L. and Feng, B.C. (2015) Sufficiency and Duality for Nonsmooth Multiob- jective **Programming** **Problems** Involving Generalized ( , ) ϕ ρ -V-Type I Functions. Journal of Mathematical Modelling and Algorithms in Operations Research , 14, 159-172. https://doi.org/10.1007/s10852-014-9264-x

1. Introduction. The concept of symmetric dual programs, in which the dual of the dual equals the primal, was introduced and developed in, e.g., [2, 4, 5]. Recently, Chandra, Craven, and Mond [1] formulated a pair of symmetric dual programs with a square root term. Weir and Mond [7] discussed symmetric duality in **multiobjective** **programming**. Mond, Husain, and Prasad gave symmetric duality result for nondif- ferentiable **multiobjective** programs in [6]. In this paper, a pair of symmetric dual nondiﬀerentiable **multiobjective** **programming** **problems** is formulated and appropri- ate duality theorems are established under suitable generalized invexity assumptions. These results include duality results for **multiobjective** programs given in [6, 7] as spe- cial cases.

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In this paper, we formulate a **multiobjective** fuzzy ran- dom simple recourse **programming** problem, in which fuzzy random variables coefficients are involved in equality con- straints. In the proposed method, equality constraints with fuzzy random variables are defined on the basis of a possi- bility measure and and a two-stage **programming** method. For a given permissible possibility level and reference objective values specified by the decision maker, corresponding min- imax problem is solved to obtain a Pareto optimal solution. The proposed method is applied to a farm planning problem in the Philippines, in which it is assumed that an amount

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The use of the proposed approach to real-world decision **problems** is an emerging area for study in future. The proposed approach may be extended to solve the hierarchical decentralized decision problem with interval parameter sets. However, it is hoped that approach presented in this paper will open up a new vistas of research on interval **programming** for its actual implementation of real-world problem in inexact environment.

We introduced a symmetric dual for **multiobjective** fractional variational programs in second order. Under invexity assumptions, we established weak, strong and converse duality as well as self duality relations .We work with properly efficient solutions in strong and converse duality theorems. The weak duality theorems involves efficient solutions .

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