TOPSIS Approach for Solving Bi-Level Non-Linear Fractional MODM Problems

TOPSIS (technique for order preference similarity to ideal solution) is considered one of the known classical multiple criteria decision making (MCDM) methods to solve bi-level non-linear fractional multi-objective decision making (BL-NFMODM) problems, and in which the objective function at each level is considered nonlinear and maximization type fractional functions. The proposed approach presents the basic terminology of TOPSIS approach and the construction of membership function for the upper level decision variable vectors, the membership functions of the distance functions from the positive ideal solution (PIS) and of the distance functions from the negative ideal solution (NIS). Thereafter a fuzzy goal programming model is adopted to obtain compromise optimal solution of BL-NFMODM problems. The proposed approach avoids the decision deadlock situations in decision making process and possibility of rejecting the solution again and again by lower level decision makers. The presented TOPSIS technique for BL-NFMODM problems is a new fuzzy extension form of TOPSIS approach suggested by Baky and Abo-Sinna (2013) (Applied Mathematical Modelling, 37, 1004-1015, 2013) which dealt with bi -level multi-objective decision making (BL-MODM) problems. Also, an algorithm is presented of the new fuzzy TOPSIS approach for solving BL-NFMODM problems. Finally, an illustrative numerical example is given to demonstrate the approach.


Introduction
Technique for order preference by similarity to ideal solution (TOPSIS), one of the known classical multiple criteria decision making (MCDM) methods, based upon the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). It was first developed by Hwang and Yoon [1] for solving a multiple attribute decision making problem. Generally, TOPSIS provides a broader principle of compromise for solving multiple criteria decision-making problems. It transfers m-objectives (criteria), which are conflicting and non-commensurable, into two objectives (the shortest distance from the PIS and the longest distance from the NIS). They are commensurable and most time conflicting. Then, the bi-objective problem can be solved by using membership functions of fuzzy set theory to represent the satisfaction level for both criteria and obtain TOPSIS's compromise solution by a second-order compromise. The max -min operator is then considered as a suitable one to resolve the conflict between the new criteria (the shortest distance from the PIS and the longest distance from the NIS) [2,3,4]. Hwang and Yoon [1] used both PIS and NIS to normalize the distance family and obtain the form of distance family equations. Lia et al. [5] extended the concept of TOPSIS to develop a methodology for solving multiple objective decision making (MODM) problems. A similar concept has also been pointed out by Zeleny [6]. Recently, Baky and Abo-Sinna [7] proposed a fuzzy TOPSIS algorithm to solve bi-level multi-objective decision making (BL-MODM) problems. Abo -Sinna [2] extended TOPSIS approach to solve multi-objective dynamics programming (MODP) problems. As he showed that using the fuzzy max-min operator with non-linear membership functions, the obtained solutions are always non-dominated by the original MODP problems. Further extension of TOPSIS for large scale multi-objective non-linear programming problems with block angular structure was presented by Abo-Sinna et al. in [3,4]. Deng et al. [8] formulated the inter-company comparison process as a multi-criteria analysis model, and presented an effective approach by modifying TOPSIS for solving such a problem. Chen [9] extended the concept of TOPSIS to develop a methodology for solving multi-person multi -criteria decision-making problems in a fuzzy environment and he defined the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS).
Bi-level programming problems (BLPs) concern with decentralized programming problems with two decision makers (DMs) in bilevel where decisions have interacted with each other were studied. A bibliography of the related references on bi-level programming problems in both linear and non-linear cases, which is updated biannually, can be found in [10]. In brief, the basic concept of the bilevel programming problem (BLPP) is that the upper-level decision makers (ULDMs) set their goals and/or decision, and then ask subordinate levels of the hierarchy for their optima, calculated in isolation. The lower level decision maker's (LLDMs) are then submitted and modified by the ULDM in consideration of the overall benefit for the organization or hierarchy. This process continues until a satisfactory solution is reached. Bi -level organization has the following common characteristics: Interactive decision-making units exist within a predominantly hierarchical structure; the execution of decisions is sequential from upper-level to lower-level; each decision-making unit independently controls a set of decision variables and is interested in maximizing its own objective but is affected by the reaction of lower-level DMs due to their dissatisfaction with the decision of the upper-level DMs. So, the decision deadlock arises frequently in the decision-making situation.
Over the last three decades, tremendous amount of research efforts has been made on multi-level programming problems (MLPPs) for hierarchical decentralized planning problems leading to the publication of many interesting results in literature [11,12,13,14,15,16,17,18,19] and many methodologies have been proposed to solve MLPP's which potentially arise in various fields such as Agriculture, Bio fuel production, Economic systems, Finance Government policy, Network designs etc. Candler and Townsley [20] have suggested applications of multi-level programming in governmental problems involving issues such as the setting of penalties for illegal drug import, the fixing of import quotas and the development of transportation and communications infrastructure. Applications to strategic weapons exchange problems and to the distribution of federal budgets among states have been described In this paper, we propose the new fuzzy TOPSIS (technique for order preference by similarity to ideal solution) approach to solve bi-level non-linear fractional multi-objective decision-making (BL-NFMODM) problems, and in which the objective function at each level are considered maximization type non-linear functions. The proposed approach presents the basic terminology of TOPSIS approach and the construction of membership function for the upper-level decision variable vectors, the membership functions of the distance functions from the positive ideal solution (PIS) and of the distance functions from the negative ideal solution (NIS) and there after fuzzy goal programming model is adopted to obtain compromise optimal solution of ML-NFMODM problems. The proposed approach avoids the decision deadlock situations in decision making process and possibility of rejecting the solution again and again by lower-level decision makers. The proposed TOPSIS technique for BL-NFMODM problems is an extension form of TOPSIS approach suggested by Baky and Abo-Sinna [7] which dealt with bi-level multi-objective decision making (BL-MODM) problems. An illustrative numerical example is given to demonstrate the approach.

Problem Formulation
Assume that there are two levels in a hierarchy structure with upper-level decision maker (ULDM) and lower-level decision maker (LLDM). Let the vector of decision variables are the upper-level and lower-level vector of non-linear objective functions, respectively. So, the BL-NFMODM problem of maximization type may be formulated as follows: where 2 x solves , (

Some Basic Concepts of Distance Measures
This section briefly surveys some basic concepts of distance measures, for more details see [2,3,4,5,7,39,40]. To obtain a compromise solution of MODM problems of the form: As the measure of "closeness", Lpmetric is used, the Lp-metric defines the distance between two points F (x) and F * as: .
The global criterion method, goal programming, fuzzy programming, and interactive approaches use the distance family of (3) and (4) when the ideal vector of objective functions ) ,..., , ( . ,..., The value chosen for p reflects the way of achieving a compromise by minimizing the weight sum of the divisions of objective from their respective reference point (ideal solution). The parameter p plays the role of the "balancing factor" between the group utility and maximal individual regret. As p increases, the group utility (distance p and greater emphasis is given to the largest deviation in forming the total. Specifically, p = 1 implies an equal importance (weights) for all these deviations, while p = 2 implies that these deviations are weighted proportionately with the largest deviation having the largest weight [5]. Finally for ,   p the largest deviation completely dominates the distance determination, the  L -metric is of the form:

TOPSIS for BL-NMODM Problems
In most practical situations, we might like to have a decision, which not only makes as much profit as possible, but also avoids as much risk as possible. This concept has been developed by Hwang and Yoon [1]. They provided a new approach, TOPSIS, for solving a multiple attribute decision making (MADM) problems. It is based upon the principle that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). Hwang and Yoon used both PIS ) ( * F and NIS ) (  F to normalize the distance family and obtain the form of distance family of Eq. (4). Lia et al. [5] extended the concept of TOPSIS to develop a methodology for solving multiple objective decision making (MODM) problems. In this paper, the researchers further extended the concept of TOPSIS [5] for BLNMODM problems.

The TOPSIS Approach for the Upper NMODM Problem
Consider the upper level multi-objective of maximization type problem of the BL-NMODM problem (1 ): The TOPSIS approach of Lia et al. [5] that solves single level MODM problems is considered. In this paper, to solve the upper-level NMO problem, the TOPSIS model formulation of this approach can be briefly stated as following, for more details see [5]: Also, we propose in this paper that, can be obtained as (see: Fig.1 Applying the max-min decision model, which is proposed by Bellmann and Zadeh [41] and extended by Zimmermann [42,43] (8) is equivalent to the form of Tchebycheff model (see [7,44], which is equivalent to the following model. If the feasible region is empty, the negative and positive tolerance must be increased to give the lower level decision makers an extent feasible region to search for the satisfactory solution [15,19].  (17) It may be noted that, the decision maker may desire to shift the range of k t y 1 ) , ( . Following Pramanik and Roy [15] and Sinha [16], this shift can be achieved.

The Proposed TOPSIS Approach for BL-NMODM Problems
In order to obtain a compromise solution (satisfactory solution) to the BL -NMODM problems using the TOPSIS approach, the distance family of (4) to represent the distance function from the positive ideal solution,     [5,44] for the general form of the distance functions that can be applied to the proposed TOPSIS approach for solving BL-NMODM problems.
In order to obtain a compromise solution, we transfer problem (1 ) into the following bi-objective problem with two commensurable (but often conflicting) objectives as [ Also, assume that . Then, based on the preference concept, we assign a larger degree to the one with shorter distance from the PIS for Applying the max-min decision model, which is proposed by Bellmann and Zadeh [41] and extended by Zimmermann [42,43] (20) is equivalent to the form of Tchebycheff model [2,3,4,5,7,43,45,46], which is equivalent to the following model:  Max (28) subject to

The TOPSIS Algorithm for BL -NMOSM Problems
The TOPSIS model (29) provides a satisfactory decision for the two DMs at the two levels. Following the above discussion, the algorithm for the proposed TOPSIS approach, in this paper, for solving BL-NMODM problems is given as follows: Step0. Use the transformation method (T1) in sec.2 to transform BL-NFMOM problems into BL-NMODM problems.
Step18. Solve model (29) Step19. If the DM is satisfied with the candidate solution in step 18, go to step 20, or else go to step 21.
Step20. Satisfactory solution is to the BL-NMODM problem. The solution procedure is straightforward and illustrated via the numerical example in the following section.

Illustrative Numerical Example
The following numerical example is considered to illustrate the proposed fuzzy TOPSIS algorithm for solving BL-FMODM problems  (32) Using the transformation method (T1), the problem (30)(31)(32) is equivalent to the following BL-NMODM problem as:   Table 1 summarizes minimum and maximum individual optimal solutions, of all objectives functions for the two levels of the BL-NMODM problem, subjected to given constraints M.

The BL-NMODM problem:
We first obtain PIS and NIS payoff tables for the lower-level NMODM problem (Tables 5 and 6 . (36)

Conclusion
The technique for order preference by similarity to ideal solution (TOPSIS) is considered the advantage approach for NMODM problems. In this paper, a TOPSIS approach is proposed for solving bi-level fractional non-linear multi-objective decision making (BL-FNMODM) problems. A compromise solution (satisfactory solution) can be obtained to the BL-FNMODM problems by using the transformation (T1) to convert BL-FNMODM problem into BL-NMODM problem and using the concept of TOPSIS approach which represents the family distance function from the positive ideal solution (PIS) and the distance function from the negative ideal Then, the bi-Level problem can be solved by using membership functions of fuzzy set theory to represent the satisfaction level for both criteria and obtain TOPSIS's compromise solution by a second -order compromise. The max-min operator is then considered as a suitable one to resolve the conflict between the new criteria (the shortest distance from PIS and the longest distance from NIS). Finally, an illustrative numerical example is given to demonstrate the proposed TOPSIS approach for BL-FNMODM problems.