Uncertainty in real-world fractional transportation problem is the major issue. In this paper, a multi-choice fractional stochastic multi-objective transportation problem (MCFS-MOTP) is investigated. Due to the uncertainty, in the introduced model, the coefficients of the fractional objective functions are a multi-choice type also, the parameters of the constraints are treated as multi-choice independent normally distributed random variables. Firstly, an interpolating polynomial is detailed utilizing practical qualities at nonnegative integer nodes to deal with any multi-choice parameters based on Newton divided difference method. Secondly, stochastic programming approach is applied to transform the probabilistic constraints into crisp ones. Moreover, we introduced a linearization methodology to work out the linear form of the problem. Finally, a fuzzy goal programming approach and the ϵ -constrain method were utilized to think through the issue. Applicability of the proposed model was verified through a numerical example.

In this paper, a modified TOPSIS (techniques for order preference by similarity to ideal solution) approach for solving bi-level multi-objective programming (BL-MOP) problems with fuzzy parameters is presented. These fuzzy parameters are assumed to be characterized by fuzzy numerical data, reflecting the experts' imprecise or fuzzy understanding of the nature of the parameters in the problem formulation process. Firstly, the corresponding non-fuzzy bi-level programming model is introduced based on the α-level set. Secondly, a modified TOPSIS approach is developed, in which the fuzzy goal programming (FGP) approach is used to solve the conflicting bi-objective distance functions instead of max-min operator. As the FGP approach utilized to achieve the highest degree of each membership goal by minimizing the sum of the unwanted deviational variables. Finally, an algorithm to clarify the modified TOPSIS approach, as well as Illustrative numerical example and comparison with the existing methods, are presented.

I n this paper, TOPSIS (technique for order preference by similarity to ideal solution) approach for solving bi-level multi-objective programming problems (BL-MOPP) with fuzzy parameters is proposed. These fuzzy parameters are assumed to be characterized as fuzzy numbers, reflecting the experts' imprecise or fuzzy understanding of the nature of parameters in the problem formulation process. Using the level sets of fuzzy parameters, the corresponding non fuzzy bi-level programming problem is introduced. The proposed approach for obtaining the satisfactory solution of the     BL-MOPP with fuzzy parameters includes the membership functions of the distance function from the positive ideal solution (PIS), the membership functions of the distance function from the negative ideal solution (NIS) and the membership functions of the upper level decision variables vector with possible tolerances. Also, a modified TOPSIS approach is presented in this paper. Illustrative numerical example is given to demonstrate the proposed TOPSIS and modified TOPSIS approach. Also, a comparison between the proposed TOPSIS and the modified TOPSIS approaches with existing Algorithms is given to clarify the powerful of the proposed approaches.

In this paper, a fuzzy goal programming (FGP) algorithm for solving bi-level multi-objective programming problems with fuzzy demands is presented. These fuzzy demands reflect the experts' imprecise or fuzzy understandings of the nature of parameters in the problem formulation process are assumed to be characterized as fuzzy numbers. Using the level sets of fuzzy parameters, the corresponding non fuzzy bi-level programming problem is introduced. In the proposed algorithm, the membership functions for the defined fuzzy goals of all objective functions at the two levels, as well as the membership functions for the vector of decision variables controlled by FLDM are developed in the model formulation of the problem. Then FGP algorithm is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers. Illustrative numerical example is given to demonstrate the proposed algorithm.

This paper extended the concept of the technique for order preference by similarity to ideal solution (TOPSIS) to develop a methodology for solving multi-level non-linear multi-objective decision-making (MLN-MODM) problems of maximization-type. Also, two new interactive algorithms are presented for the proposed TOPSIS approach for solving these types of mathematical programming problems. The first proposed interactive TOPSIS algorithm includes the membership functions of the decision variables for each level except the lower level of the multi-level problem. These satisfactory decisions are evaluated separately by solving the corresponding single-level MODM problems. The second proposed interactive TOPSIS algorithm lexicographically solves the MODM problems of the MLN-MOLP problem by taking into consideration the decisions of the MODM problems for the upper levels. To demonstrate the proposed algorithms, a numerical example is solved and compared the solutions of proposed algorithms with the solution of the interactive algorithm of Osman et al. (2003) [4]. Also, an example of an application is presented to clarify the applicability of the proposed TOPSIS algorithms in solving real world multi-level multi-objective decision-making problems.

TOPSIS (technique for order preference by similarity to ideal solution) is a multiple criteria method to identify solutions from a finite set of alternatives based upon simultaneous minimization of distance from an ideal point and maximization of distance from a nadir point. This paper proposes a fuzzy TOPSIS algorithm to solve bi-level multi-objective decision-making (BL-MODM) problems, and in which the objective function at each level are non-linear functions which are to be maximized. The proposed model for getting the satisfactory solution of the BL-MODM problems includes the membership functions for the upper level decision variables vector with possible tolerances, the membership function of the distance function from the positive ideal solution (PIS) and the membership function of the distance function from the negative ideal solution (NIS). A numerical illustrative example is given to clarify the proposed TOPSIS approach of this paper.

This paper presents two fuzzy goal programming (FGP) procedures for solving multi-level multiobjective linear fractional programming (ML-MOLFP) problems.In the proposed procedures, the membership functions for the defined fuzzy goals of all objective functions at all levels as well as the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels, are developed in the model formulation of the problem.Then fuzzy goal programming approach is used to achieve highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers.The first proposed FGP procedure makes an extension work of Pramanik and Roy [23] and of B.B. Pal et al. [17,20].The second proposed procedure my be seem as lexicographic methods for solving multiobjective programming problems that follows Shih et al. concept [28] but by using the FGP approach.The method of variable change on the under-and over-deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology.Illustrative numerical example is given to demonstrate the algorithms.

This paper presents a fuzzy goal programming (FGP) procedure for solving bilevel multiobjective linear fractional programming (BL-MOLFP) problems. It makes an extension work of Moitra and Pal (2002) and Pal et al. (2003). In the proposed procedure, the membership functions for the defined fuzzy goals of the decision makers (DMs) objective functions at both levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by first-level decision maker are developed first in the model formulation of the problem. Then a fuzzy goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree (unity) of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The method of variable change on the under- and over-deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. Illustrative numerical example is given to demonstrate the procedure.

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem: so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.The first suggested algorithm groups the membership functions for the defined fuzzy goals of the objective functions at all levels and the decision variables for each level except the lower level of the multi-level problem. The second proposed algorithm lexicographically solves MOLP problems of the ML-MOLP problem by taking into consideration the decisions of the MOLP problems for the upper levels. An illustrative numerical example is given to demonstrate the algorithms. (C) 2009 Elsevier Inc. All rights reserved.

This paper studies a multi-level multi-objective decision-making (ML-MODM) problems with linear or non-linear constraints. The objective functions at each level are non-linear functions, which are to be maximized or minimized. This paper presents a three-level multi-objective decision-making (TL-MODM) model and an interactive algorithm for solving such a model. The algorithm simplifies three-level multi-objective decision-making problems by transforming them into separate multi-objective decision making problems at each level, thereby avoiding the difficulty associated with non-convex mathematical programming. Our algorithm is an extension of the work of Shi and Xia [X. Shi, H. Xia, Interactive bi-level multi-objective decision making, Journal of the Operational Research Society 48 (1997) 943–949], which dealt with interactive bi-level multi-objective decision-making problems, with some modifications in assigning satisfactoriness to each objective function in all the levels of the TL-MODM problem. Also, we solve each separate multi-objective decision making problem of the TL-MODM problem by the balance space approach. A new formula is introduced to interconnect the satisfactoriness and the proportions of deviation needed to reflect the relative importance of each objective function. Thus, we have the proportions of deviation including satisfactoriness. In addition, we present new definitions for the satisfactoriness and the preferred solution in view of singular-level multi-objective decision making problems that corresponds to the η-optimal solution of the balance space approach. Also, new definitions for the feasible solution and the preferred solution (η-optimal point) of the TL-MODM problem are presented. An illustrative numerical example is given to demonstrate the algorithm.