An Interactive Procedure to Solve Multi-Objective Decision-Making Problem: An Improvment to STEM Method

1. Introduction

Managerial problems are seldom evaluated with a single or simple goal like profit maximization. Today’s management systems are much more complex, and managers want to attain simultaneous goals, in which some of them conflict. In the other words, decisions in the real-world contexts are often made in the presence of multiple, conflicting, and incommensurate criteria. Particularly, many decision problems at tactical and strategic levels, such as strategic planning problems, have to consider explicitly the models that involve multiple conflicting objectives or attributes.

The main difference between MODM and MADM is that the former concentrates on continuous decision spaces, primarily on mathematical programming with several objective functions, and the latter focuses on problems with discrete decision spaces.

For the further discussion about MODM and MADM, some basic solution concepts and terminologies are supplied by Hwang and Masud [2] and Hwang and Yoon [1].

Criteria are the standard of judgment or rules to test acceptability. In the MCDM literature, it indicates attributes and/or objectives. In this sense, any MCDM problem means either MODM or MADM, but is more used for MADM.

Objectives are the reflections of the desire of decision makers and indicate the direction in which decision makers want to work. An MODM problem, as a result, involves the design of alternatives that optimises or most satisfies the objectives of decision makers.

Goals are things desired by decision makers expressed in terms of a specific state in space and time. Thus, while objectives give the desired direction, goals give a desired (or target) level to achieve.

Attributes are the characteristics, qualities, or performance parameters of alternatives. An MADM problem involves the selection of the best alternative from a pool of preselected alternatives described in terms of their attributes.

We also need to discuss the term alternatives in detail. How to generate alternatives is a significant part of the process of MODM and MADM model building. In almost MODM models, the alternatives can be generated automatically by the models. In most MADM situations, however, it is necessary to generate alternatives manually. Multiobjective programming method such as multiple objective linear programming (MOLP) are techniques used to solve such multiple-criteria decision-making (MCDM) problems.

The future of multiple-objective programming is in its interactive application. In interactive procedures, we conduct an exploration over the region of feasible alternatives for an optimal or satisfactory near-optimal solution. Interactive procedures are characterized by phases of decision making alternating with phases of computation. At each iteration, a solution, or group of solutions, is generated for examination. As a result of examination, the decision maker inputs information to the solution procedure. One of the first interactive procedures to solve MOLP is STEM method.

In this paper we try to improve STEM method in a way that we search a point in reduced feasible region whose criterion vector is closest to positive ideal criterion vector and furthest to negative ideal criterion vector. Therefore the presented method tries to increase the rate of satisfactoriness of the obtained solution.

In Section 3, we will focus on the proposed method. In Section 4, a numerical example is demonstrated. In Section 5 some conclusions are drawn for the study.

2. Preliminaries

In this section we express the following useful concepts that are taken from [4].

3. Improved STEM Method

The procedure for improving STEM method has been given in the following steps.

In Table 1, row j corresponds to the solution vector x^j+ which maximizes the objective function f_j. A f_ij is the value taken by the ith objective f_i when the jth objective function f_j reaches its maximum $$, that is, f_ij = f_i(x^j+).

In Table 2, row j corresponds to the solution vector x^j− which minimizes the objective function f_j. A z_ij is the value taken by the ith objective f_i when the jth objective function f_j reaches its minimum $$, that is z_ij = f_i(x^j−).

Let h be iteration counter and set h = 0 and go to Step 3.

For more information about how we obtain a compromise solution, see the following example.

4. Numerical Example

In this section we investigate the capability of the proposed method.

4.2. Solve with the Proposed Method

In order to find a satisfactory solution, we carry out the following steps.

Iteration Number 1

Step 1 (first pay-off table). The first pay-off table of the problem is as shown in Table 3.

Table 3 is constructed using formula (3.1). Figure 2 shows the feasible region and objectives. Also in Figure 2 we can see the negative ideal solution and positive ideal solution.

Step 2 (second pay-off table). The second pay-off table of the problem is as shown in Table 4.

Table 4 is constructed using formula (3.3).

Step 3 (obtain the first group of weights). Since $$ and m₁ = −30 and c₁₁ = −5, c₁₂ = 2, then from formula (3.5) we have

$$ ()

Similarly, we can get π₂ = 0.728. From (3.6) there are

$$ ()

Step 4 (obtain the second group of weights). Since $$ and n₁ = 3 and c₁₁ = −5, c₁₂ = 2, then from formula (3.8) we have

$$ ()

Similarly, we can get $$. From (3.9) there are

$$ ()

Step 5 (calculation phase). Now we can start the iteration process. Therefore we have the following steps.

Substep 5.6 (find the closest criterion vector to positive ideal). We can obtain a criterion vector which is closest to the positive ideal one by solving model (3.12) as follows:

$$ ()

The optimal solution of the problem is w⁽⁰⁾ = (0,0.452) with criterion vector f(w⁽⁰⁾) = {f₁(w⁽⁰⁾), f₂(w⁽⁰⁾)} = {−0.904, −1.808}.

Substep 5.7 (find the furthest criterion vector to negative ideal). We can obtain a criterion vector which is furthest to the negative ideal one by solving model (3.15) as follows:

$$ ()

The optimal solution of the problem is y⁽⁰⁾ = (0,3) with criterion vector f(y⁽⁰⁾) = {f₁(y⁽⁰⁾), f₂(y⁽⁰⁾)} = {6, −12}.

Substep 5.8 (obtain a compromise solution). We can obtain a criterion vector which is closest to positive ideal and furthest to negative ideal by solving model (3.18) as follows:

$$ ()

The optimal solution of the problem is x⁽⁰⁾ = (0,2.013) with criterion vector f(x⁽⁰⁾) = {f₁(x⁽⁰⁾), f₂(x⁽⁰⁾)} = {4.026, −8.052}.

Step 6 (decision phase). The results x⁽⁰⁾ = (0,2.013) and f(x⁽⁰⁾) = {4.026, −8.052} are shown to the decision maker. Suppose the solution is not satisfied as f₂(x⁽⁰⁾) = −8.052 is too small. Suppose f₁(x) can be sacrificed by 2 units, or Δf₁ = 2. Then the new search space is given by

$$ ()

We set $$ and begin iteration 2.

In Figure 3 we can see the obtained solutions from iteration 1. Specially we can see the compromise solution obtained from iteration 1 that is denoted by x⁽⁰⁾.

Table 3. First pay-off table of Section 4.

	f1	f2	Solution	Vector
f1	$$	f12 = −12	$$	$$
f2	f21 = −30	$$	$$	$$

Table 4. Second pay-off table of Section 4.

	f1	f2	Solution	Vector
f1	$$	z12 = 6	$$	$$
f2	z21 = 3	$$	$$	$$

Iteration Number 2 It is obvious that $$, and we go to Step 5. It is done in three steps.

Substep 6.6 (find the closest criterion vector to positive ideal). In this step we solve model (3.7) as follows:

$$ ()

The optimal solution of the problem is w⁽¹⁾ = (0,1.013) with criterion vector f(w⁽¹⁾) = {f₁(w⁽¹⁾), f₂(w⁽¹⁾)} = {2.026, −4.052}.

Substep 6.7 (find the furthest criterion vector to negative ideal). Solve model (3.7) as follows:

$$ ()

The optimal solution of the problem is y⁽¹⁾ = (0,1.013) with criterion vector f(y⁽¹⁾) = {f₁(y⁽¹⁾), f₂(y⁽¹⁾)} = {2.026, −4.052}.

Substep 6.8 (obtain a compromise solution). In order to obtain a criterion vector which is closest to the positive ideal and furthest to the negative ideal, we solve model (3.7) as follows:

$$ ()

The optimal solution of the problem is x⁽¹⁾ = (0,1.013) with criterion vector f(x⁽¹⁾) = {f₁(x⁽¹⁾), f₂(x⁽¹⁾)} = {2.026, −4.052}.

Note that x⁽¹⁾ is the point in feasible region whose criterion vector is closet to criterion vector w⁽¹⁾ and y⁽¹⁾ and therefore whose criterion vector has minimum distance to positive ideal and has maximum distance to negative ideal one.

In Figure 4 we can see the obtained solutions from iteration 2. Specially we can see the compromise solution and other solutions obtained from iteration 1 coincide. The compromise solution that is denoted by x⁽¹⁾ is also the preferred solution.

According to the behavioral assumptions of the STEM method (discussed in decision phase), the decision maker should be satisfied with the solution x⁽¹⁾; otherwise, there would be no best compromise solution.

For this two-objective problem, this conclusion may be acceptable as f₁(x) can be sacrificed by as much as 2 units from f₁(x⁽⁰⁾), and his sacrifice has been fully used to benefit the objective f₂(x). In general, such conclusion may not be rational for problems having more than two objectives. In such circumstances, whether the decision maker is satisfied with a solution depends on the range of solutions he has investigated. Also, the sacrifices of multiple objectives should also be investigated in addition to the sacrifice of a single objective at each iteration.

5. Conclusion

The suggested method in this paper improves the STEM method by finding a point in reduced feasible region whose criterion vector is closest to the positive ideal criterion vector and furthest to the negative ideal criterion vector. Therefore the presented method increases the rate of satisfactoriness of the obtained solution.