1. Introduction

Compromise programming (CP) was initially proposed by Zeleny (1973) and subsequently used by many researchers. [24]. Yu (1973) and Zeleny (1974) define the ideal solution (Yu describes this solution as the "utopia point") as any solution that would simultaneously optimize each individual objective. CP assumes that any DM seeks a solution as close as possible to the ideal point, [21, 25].

The non-centralized planning has been recognized as an important decision making problem. It searches for a simultaneous compromise among the various objectives of the different departments. Multi-Level programming, a tool for modeling non-centralized decisions, consists of the objective(s) of the Manager at its first level and that is of the followers at the other levels. The decision-maker at each level seeks to optimize his individual objective Functions, which depends in part on the variables controlled by the decision makers at the other levels and their final decisions are executed sequentially where the upper-level decision-maker makes his decision firstly, [5, 7, 8, 11, 12, 16, 20].

An extended TOPSIS method for solving interactive large scale multiple objective optimization problems involving fuzzy parameters is introduced in [4].

Several algorithms for solving different kinds of large scale multiple objective optimization problems using TOPSIS approach are presented in [1].

A review on theory, applications and softwares of bi-level, multi-level multiple criteria decision making and TOPSIS approach is presented in [2].

Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems are given in [6].

A modified TOPSIS method for solving large scale two-level linear multiple objective optimization problems with fuzzy parameters in the right-hand side of the independent constraints is introduced in [3].

We extend the TOPSIS method [4, 15] to find compromise solutions [14, 22, 25, 26, 27] for the Multi-Level Multiple Objective Decision Making (MLMODM) Problems with fuzzy parameters [18, 19, 23, 27] in the objective functions and the right hand side of the constraints (FMLMODM).

In the following sections, the formulation of FMLMODM problems is given in section (2). By use of TOPSIS method, a new interactive algorithm for solving MLMODM problems is proposed in section (3). For the sake of illustration, we present an example for the extended TOPSIS method and compared the solution of proposed algorithm with the solution of traditional Global Criterion (GC) method in section (4).

2. Formulation of the Problem

Consider the following FMLMODM problem:

(1-1)

where solves the level

(1-2)

where solves the level

(1-3)

where solves the level

(1-4)

Subject to

(1-5)

where

m: the number of constraints,

n: the number of variables,

h: the number of levels

k: the number of objective functions,

The number of objective functions of the

The number of variables of the

an n-dimensional row vector of fuzzy parameters for the objective functions = 1,2,…,h.

An m-dimensional column vector of right-hand sides of constraints whose elements are fuzzy Parameters

an coefficient matrix,

R: the set of all real numbers,

X: an n-dimensional column vector of variables,

X_j: an -dimensional column vector of variables for the j^th level, j=1,2,…,h,

Throughout this paper, we assume that the column vectors of fuzzy parameters and the row vectors of fuzzy parameters are characterized as the column vectors of fuzzy numbers and row vectors of fuzzy numbers respectively [18, 19, 23, 29].

It is appropriate to recall that a real fuzzy number whose membership function is defined as, [18, 19, 23, 29]:

(1) A continuous mapping from R¹ to the closed interval [0,1],

(2)

(3) Strictly increasing on

(4) for all

(5) Strictly decreasing on

(6) for all

A possible shape of fuzzy number is illustrated in figure (1). The concept of α-cut of the vectors parameters and whose elements are fuzzy numbers is introduced as follows:

Figure (1). Membership function of fuzzy number

Definition 1.

The α-cut of is defined as the ordinary set for which the degree of their membership function exceeds the level α є [0,1]:

(2)

For a certain degree the FMLMODM problem (1) can be understood as the following nonfuzzy Multiple Objective Decision Making problem:

(3-1)

where solves the level

(3-2)

where solves the level

(3-3)

where solves the level

(3-4)

Subject to

(3-5)

(3-6)

In the problem (3), the parameters are treated as decision variables rather than constants.

Based on the definition of α-cut of the fuzzy numbers, we characterize α–efficient solution of α- MLMODM problem (3):

Definition 2:

A solution is said to be an α- efficient solution to the α- MLMODM problem (4), if and only if there does not exist another such that for maximizati for minimization) and with strictly inequality holding for at least one where the corresponding value of the parameter is called α-cut optimal parameters.

Thus, the α-MLMODM problem (3) can be written as follows:

(4-1)

where solves the level

(4-2)

where solves the level

(4-3)

where solves the level

(4-4)

Subject to

(4-5)

(4-6)

(4-7)

It should be noted that the constraint (3-6) is replaced by the equivalent constraints (4-6) and (4-7), where and are lower and upper bound on respectively.

3. TOPSIS for FMLMODEMs

A modified version of TOPSIS method is introduced to find compromise solutions for the FMLMODM problems. Modified equations for the distance function [17] from the positive ideal solution (PIS) and the distance function from the negative ideal solution (NIS) are introduced to include all the objective functions of multi-level of the problem. An interactive decision making algorithm to find a compromise solution through TOPSIS approach is provided in (3-1) where the first level decision maker is asked to specify the membership function for each fuzzy parameter, the maximum negative and positive tolerance values, the power p of the distance functions, the degree α and the relative importance of the objectives. Then, the j^th Level decision maker is asked to specify the maximum negative and positive tolerance values, and the relative importance of the objectives. An illustrative numerical example for the extended TOPSIS method is given in section (4).

In order to obtain a compromise solutions to the FMLMODM problems using the modified TOPSIS approach, a generalized formulas for the distance function from the PIS and the distance function from the NIS are proposed and modeled to include all the objective functions of all the levels. Thus, we propose an interactive decision making algorithm to find a compromise solutions through TOPSIS approach where the is asked to specify the membership function for each fuzzy parameter, the maximum negative and positive tolerance values, the power p of the distance functions, the degree α and the relative importance of the objectives. Then, the is asked to specify the maximum negative and positive tolerance values, and the relative importance of the objectives.

Algorithm (I):

Phase (1):

Step 1:

(!-1): Let h = the number of the levels of the FMLMODM problem (1). Set j=1, "The 1^st level".

(1-2): Ask the to specify a membership function for each fuzzy numbers and in the FMLMODM problem (1).

For example, the fuzzy numbers can have a membership function of the following form [13]:

(5)

(1-3): Ask the to select

(1-4): Transform the FMLMODM problem (1) to the form of α-MLMODM problem (4) by using steps (1-2) and (1-3).

Step 2:

Construct the PIS payoff table of the following problem:

Subject to (6)

and obtain the individual positive ideal solutions.

where

Objective Functions for Maximization, and

Objective Functions for Minimization,

Step 3:

Construct the NIS payoff table of problem (6) and obtain the individual negative ideal solutions.

Step 4:

Use steps (2 and 3) to construct the distance function from the PIS and the distance function from the NIS:

(7-1)

and

(7-2)

Where are the relative importance (weighs) of objectives, and

Step 5:

Transfer the problem (6) into the following bi-objective problem with two commensurable (but often conflicting) objectives:

Subject to (8)

Where

Step 6:

(6-1): Ask the to select

(6-2): Ask the to select where

Step 7: Use step (6) and equation (7) to compute and

Step 8: Construct the payoff table of problem (8) and obtain:

Step 9:

(9-1): Construct the following membership functions and (figure (2)):

(9-1)

(9-2)

(9-2): By using the max-min decision model, [9], the Tchebycheff model, [10], and the membership functions (9), construct the following satisfactory level model:

(10-1)

subject to

(10-2)

(10-3)

(10-4)

where is the satisfactory level for both criteria of the shortest distance from the PIS and the farthest distance from the NIS.

(9-3): Solve problem (10) to obtain the Pareto optimal solution then is a nondominated solution of (8) and a compromise solution of the problem (6).

(9-4): If the is satisfied with then go to step (10). Otherwise, go to step (6-2).

Step 10:

(10-1): Ask the to select the maximum acceptable negative and positive tolerance (relaxation) values, [18, 20], and on the decision vector,

(10-2): Construct the linear membership functions (Figure 3) for each of the components of the decision vector controlled by the can be formulated as:

(11)

(10-3): Set j = j+1, go to the next phase.

Figure (2). The membership functions of

Phase (2):

Step 11:

Set j = 2, “The 2^nd level".

Construct the PIS payoff table of the following problem:

Subject to (12)

and obtain the individual positive ideal solutions.

where

Objective Functions for Maximization, and

Objective Functions for Minimization,

Step 12:

Construct the NIS payoff table of problem (12) and obtain theindividual negative ideal solutions.

Step 13:

Use steps (11 and 12) to construct the distance function from the PIS and the distance function from the NIS:

Figure (3). The membership function of the decision variable

(13-1)

and

(13-2)

where are the relative importance (weighs) of objectives, and

Step 14:

Transfer the problem (12) into the following bi-objective problem with two commensurable (but often conflicting) objectives:

Subject to (14)

where

Step 15:

(15-1): Ask the to select

(15-2): Ask the to select where

Step 16: Use step (12) and equation (13) to compute and

Step 17: Construct the payoff table of problem (14) and obtain:

Step 18:

(18-1): Construct the following membership functions and

(15-1)

(15-2)

Where

(18-2): By using the max-min decision model [9], the Tchebycheff model, [10], and the membership functions (15), construct the following satisfactory level model:

(16-1)

Subject to

(16-2)

(16-3)

(16-4)

(16-5)

(16-6)

Where is the satisfactory level for both criteria of the shortest distance from the PIS and the farthest distance from the NIS.

(18-3): Solve problem (16) to obtain the Pareto optimal solution , then is a non-dominated solution of (14) and a compromise solution of the problem (12).

(18-4): If the is satisfied with then go to step (19). Otherwise, go to step (15-2).

Step 19: If h =j, stop. Otherwise, go to step (20).

Step 20:

(20-1): Ask the to select the maximum acceptable negative and positive tolerance (relaxation) values and on the decision vector,

(20-2): Construct the linear membership functions for each of the components of the decision vector controlled by the can be formulated as:

(17)

(20-3): Set j = j+1, go to the next phase.

4. Illustrative Numerical Example for Algorithm (I)

Consider the following FMLMODM problem:

where and solves the second level

where and solves the third level

Subject to

where

Use the introduced algorithm in subsection (3-1) to solve the above problem.

Solution:

- h= 3, j=1,

- Use the membership function (5) to convert the above problem to the following α-MLMODM problem,

where and solves the second level

where and solves the third level

Subject to

- Obtain PIS and NIS payoff tables for the of the Problem:

Table (1). PIS payoff table for the problem

Table (2). NIS payoff table for the problem

- Next, construct equation and obtain the following equations:

- Thus, problem is obtained. In order to get numerical solutions, assume that and p=2,

Table (3). PIS payoff table of problem when p=2

- Now, it is easy to compute problem (15):

Subject to

- The maximum “satisfactory level” is achieved for the solution

- Let the decide with positive tolerance and and with positive tolerance and

- j=2. Obtain PIS and NIS payoff tables for the Problem.

Table (4). PISpayoff table for the problem

Table (5). NIS payoff table for the problem

- Next, compute and obtain the following equations:

- Thus, problem is obtained. In order to get numerical solutions, assume that and p=2,

Table (6). PIS payoff table of problem when p=2

- Now, it is easy to compute:

subject to

- The maximum “satisfactory level” is achieved for the solution Let the decide with positive tolerance and

- j=3. Obtain PIS and NIS payoff tables for the Problem:

Table (7). PIS payoff table for the problem

Table (8). NIS payoff table for the problem

- Next, compute and obtain the following equations:

- Thus, problem is obtained. In order to get numerical solutions, assume that and p=2

Table (9). PIS payoff table of problem when p=2

- Now, it is easy to compute:

Subject to

- The “satisfactory level” is achieved for the solution

The comparison between the proposed TOPSIS method and the traditional GC method is given in Table (10). In general, the results show that the proposed interactive modified TOPSIS method is introducing better results than (or closer results to) the traditional GC method.

Table (10)

5. Conclusions

This paper extended TOPSIS approach to find compromise solutions for the FMLMODM of mixed (Maximize/Minimize)-type. Anew interactive algorithm is presented for the proposed TOPSIS approach for solving these type of mathematical programming problems. Also, an illustrative numerical example is solved and compared the solution of proposed algorithm with the solution of the traditional GC method. In general, the results show that the proposed TOPSIS method is introducing better results than (or closer results to) the traditional GC method.