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

Abstract

Decisions in the real-world contexts are often made in the presence of multiple, conflicting, and incommensurate criteria. Multiobjective programming methods such as multiple objective linear programming (MOLP) are techniques used to solve such multiple-criteria decision-making (MCDM) problems. 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. Finally, a numerical example for illustration of the new method is given to clarify the main results developed in this paper.

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.

Therefore, it is often necessary to analyse each alternative in light of its determination of each of several goals. Multicriteria decision making (MCDM) refers to making decision in the presence of multiple and conflicting criteria. Problems for MCDM may range from our daily life, such as the purchase of a car, to those affecting entire nations, as in the judicious use of money for the preservation of national security. However, even with the diversity, all the MCDM problems share the following common characteristics [1]:(i)Multiple criteria: each problem has multiple criteria, which can be objectives or attributes.(ii)Conflicting among criteria: multiple criteria conflict with each other.(iii)Incommensurable unit: criteria may have different units of measurement.(iv)Design/selection: solutions to an MCDM problem are either to design the best alternative(s) or to select the best one among previously specified finite alternatives.

There are two types of criteria: objectives and attributes. Therefore, the MCDM problems can be broadly classified into two categories:(i) Multiobjective decision making (MODM).(ii) Multiattribute decision making (MADM).

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.

This paper discusses the STEM procedure. STEM or step method proposed by Benayoun et al. [3] is a reduced feasible region method for solving the MOLP max𝑓₁(𝑥),…,𝑓_𝑘(𝑥)s.t.𝑥∈𝑆, (1.1) where all objectives are bounded over 𝑆. Each iteration STEM makes a single probe of the efficient set. This is done by computing the point in the iteration’s reduced feasible region whose criterion vector is closest to ideal criterion vector. STEM is one first interactive procedure to have impact on the field of multiple-objective programming.

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.

Rest of the paper is organized as follows. In Section 2 some preliminaries about the following concept are given:(i)MODM Problems,(ii)Basic definitions,(iii)STEM Method.

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].

2.1. MODM Problems

Multiobjective decision making is known as the continuous type of the MCDM. The main characteristics of MODM problems are that decision makers need to achieve multiple objectives while these multiple objectives are noncommensurable and conflict with each other.

An MODM model considers a vector of decision variables, objective functions, and constrains. Decision makers attempt to maximize (or minimize) the objective functions. Since this problem has rarely a unique solution, decision makers are expected to choose a solution from among the set of efficient solutions (as alternatives). Generally, the MODM problem can be formulated as follows: ((𝑥)MODM)max𝑓s.t.𝑥∈𝑆, (2.1) where 𝑓(𝑥) represents 𝑛 conflicting objective functions and 𝑥 is an 𝑛-vector of decision variables, 𝑥∈ℝ^𝑛.

Multiobjective linear programming (MOLP) is one of the most important forms to describe MODM problems, which are specified by linear objective functions that are to be maximized (or minimized) subject to a set of linear constrains. The standard form of an MOLP problem can be written as follows: ((𝑥)MOLP)max𝑓=𝐶𝑥s.t.𝑥∈𝑆=𝑥∈ℝ^𝑛𝐴𝑥≤𝑏,𝑥≥0, (2.2) where 𝐶 is a 𝑘×𝑛 objective function matrix, 𝐴 is an 𝑚×𝑛 constraint matrix, 𝑏 is an 𝑚-vector of right-hand side, and 𝑥 is an 𝑛-vector of decision variables.

Example 2.1 (instance of MODM problem). For a profit-making company, in addition to earning money, it also wants to develop new products, provide job security to its employees, and serve the community. Managers want to satisfy the shareholders and, at the same time, enjoy high salaries and expense accounts; employees want to increase their take-home pay and benefits. When a decision is to be made, say, about an investment project, some of these goals complement each other while others conflict.

2.2. Basic Definitions

We have the following notion for a complete optimal solution (for more details, see [5]).

Definition 2.2 (satisfactory solution). A satisfactory solution is a reduced subset of the feasible set that exceeds all of the aspiration levels of each attribute. A set of satisfactory solutions is composed of acceptable alternatives. Satisfactory solutions do not need to be nondominated.

Definition 2.3 (preferred solution). A preferred solution is a nondominated solution selected as the final choice through decision maker’s involvement in the information processing.

In the presented method (and in traditional STEM method), in order to measure the distance between two vectors we use the following metric.

Definition 2.4. Consider the weight vector 𝜃 where ∑^𝑘_𝑖=1𝜃_𝑖=1 and 𝜃_𝑖≥0. These weights define the weighted Tchebychev metric. 𝑓^∗−𝑓(𝑥)^𝜃_∞=max_{𝑖=1,…,𝑘}𝜃_𝑖𝑓^∗_𝑖−𝑓_𝑖(𝑥). (2.3)

2.3. STEM Method

The STEM method [3] is based on minimizing the Tchebychev distance from the ideal point to the criterion space. The parameters of the distance formula and the feasible space can be changed by a normalized weighting method based on the DM’s preferences in the previous solution. The procedure of STEM allows the DM to recognize good solutions and the relative importance of the objectives.

At each iteration, the DM is able to improve some objectives, by sacrificing others. In addition, the DM must provide the maximum amount by which the objective functions can be sacrificed, although it is not necessary to provide tradeoffs on objectives. To carry out an iteration in the STEM method, given a solution 𝑥^(ℎ−1), the DMs must provide their preferences for objective functions to be improved {𝑓_𝑖,𝑖∈{1,…,𝑠}−𝐽^(ℎ)}, as well as the objective functions to be relaxed 𝑓_𝑖,𝑖∈𝐽^(ℎ) with corresponding maximal amounts to relax Δ𝑓_𝑖^(ℎ),𝑖∈𝐽^(ℎ).

The following problem can be solved using the above preferences: min𝛼s.t.𝑤_𝑖𝑓^∗_𝑖−𝑓_𝑖(𝑥)≤𝛼,𝑖∈{1,…,𝑠}−𝐽^(ℎ),𝑓_𝑗(𝑥)≥𝑓_𝑗𝑥^(ℎ−1)−Δ𝑓_𝑗^(ℎ),𝑗∈𝐽^(ℎ),𝑓_𝑗(𝑥)≥𝑓_𝑗𝑥^(ℎ−1),𝑗∈{1,…,𝑠}−𝐽^(ℎ),𝑥∈𝑆^(ℎ)0≤𝛼∈ℝ, (2.4) where 𝑓^∗_𝑖=max_{𝑥∈𝑆(ℎ)}𝑓_𝑖(𝑥),𝑖=1,…,𝑠 are the ideal values, resulting from maximizing the objective functions individually.

3. Improved STEM Method

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

Step 1 (construct the first pay-off table). In this step we first maximize each objective function and construct a pay-off table to obtain the positive ideal criterion vector 𝑓⁺∈ℝ^𝑘.
Let 𝑓⁺_𝑗,𝑗=1,…,𝑘, be the solutions of the following 𝑘 problems, namely, positive ideal solution: 𝑓⁺_𝑗=max𝑓_𝑗(𝑥)s.t.𝑥∈𝑆. (3.1) The first pay-off table is of the form of Table 1.

Table 1 

First pay-off table.

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 𝑓⁺_𝑗, that is, 𝑓_𝑖𝑗=𝑓_𝑖(𝑥^𝑗+).

Then the positive ideal criterion can be defined as follows: 𝑓⁺=𝑓⁺₁,…,𝑓⁺_𝑘=𝑓₁𝑥¹⁺,…,𝑓_𝑘𝑥^𝑘+. (3.2) And consider that 𝑥⁺ is the inverse image of 𝑓⁺. Generally, we know maybe 𝑥⁺ dose not belong to 𝑆^(ℎ).

Step 2 (construct the second pay-off table). Now, we maximize each objective function and construct a second pay-off table to obtain the negative ideal criterion vector 𝑓⁻∈ℝ^𝑘.
Let 𝑓⁻_𝑗,𝑗=1,…,𝑘, be the solutions of the following 𝑘 problems, namely, negative ideal solution: 𝑓⁻_𝑗=min𝑓_𝑗(𝑥)s.t.𝑥∈𝑆. (3.3) The second pay-off table is of the form of Table 2.

Table 2 

Second pay-off table.

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 𝑓⁻_𝑗, that is 𝑧_𝑖𝑗=𝑓_𝑖(𝑥^𝑗−).

Then the negative ideal criterion can be defined as follows:𝑓⁻=𝑓⁻₁,…,𝑓⁻_𝑘=𝑓₁𝑥¹⁻,…,𝑓_𝑘𝑥^𝑘−. (3.4) And consider that 𝑥⁻ is the inverse image of 𝑓⁻. Generally, we know maybe 𝑥⁻ dose not belong to 𝑆^(ℎ).

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

Step 3 (first group of weights). Let 𝑚_𝑖 be the minimum value in the 𝑖th column of the first pay-off table (Table 1).
Calculate 𝜋_𝑖 values, where 𝜋_𝑖=𝑓⁺_𝑖−𝑚_𝑖𝑓⁺_𝑖^𝑛_𝑗=1𝑐²_𝑖𝑗^−1/2,if𝑓⁺_𝑖𝑚>0,_𝑖−𝑓⁺_𝑖𝑚_𝑖^𝑛_𝑗=1𝑐²_𝑖𝑗^−1/2,if𝑓⁺_𝑖≤0, (3.5) where 𝑐_𝑖𝑗 are the coefficients of the 𝑖th objective.

Then, the weighting factors can be calculated as follows: 𝜆_𝑖=𝜋_𝑖∑^𝑛_𝑗=1𝜋_𝑗,𝑖=1,…,𝑘. (3.6) The weighting factors defined as above are normalized; that is, they satisfy the following conditions: 0≤𝜆_𝑖≤1,𝑖=1,…,𝑘,^𝑘_𝑖=1𝜆_𝑖=1. (3.7) The weights defined above reflects the impact of the differences of the objective values on decision analysis. If the value (𝑓⁺_𝑖−𝑚_𝑖) is relatively small, then the objective 𝑓_𝑖(𝑥) will be relatively insensitive to the changes of solution 𝑥. In other words, 𝑓_𝑖(𝑥) will not play an important role in determining the best compromise solution.

Step 4 (Second group of weights). Let 𝑛_𝑖 be the maximum value in the 𝑖th column of the second pay-off table (Table 2).
Calculate 𝜋^′_𝑖 values, where 𝜋^′_𝑖=𝑛_𝑖−𝑓⁻_𝑖𝑛_𝑖^𝑛_𝑗=1𝑐²_𝑖𝑗^−1/2,if𝑓⁻_𝑖𝑓>0,_−𝑖−𝑛_𝑖𝑓⁻_𝑖^𝑛_𝑗=1𝑐²_𝑖𝑗^−1/2,if𝑓⁻_𝑖≤0, (3.8) where 𝑐_𝑖𝑗 are the coefficients of the 𝑖th objective.
Then, the weighting factors can be calculated as follows: 𝜆^′_𝑖=𝜋^′_𝑖∑^𝑛_𝑗=1𝜋^′_𝑗,𝑖=1,…,𝑘. (3.9) Also, the weighting factors defined as above are normalized; that is, they satisfy the following conditions: 0≤𝜆^′_𝑖≤1,𝑖=1,…,𝑘,^𝑘_𝑖=1𝜆^′_𝑖=1. (3.10) In a manner similar to that of the previous step, the weights defined above reflect the impact of the differences of the objective values on decision analysis.

Step 5 (calculation phase). The weights defined by formula (3.6) and (3.9) are used to apply the weighted Tchebycheff metric, Definition 2.4, to obtain a compromise solution. This is done in three steps.
Substep 5.1 (find the nearest criterion vector to positive ideal). We can obtain a criterion vector which is closest to the positive ideal one by solving the following model:𝑓min𝛼s.t.⁺−𝑓(𝑥)^𝜆_∞≤𝛼𝑥∈𝑆^(ℎ)0≤𝛼∈ℝ. (3.11) This model can be converted to the following model: min𝛼s.t.𝛼≥𝜆_𝑖𝑓⁺_𝑖−𝑓_𝑖(𝑥),1≤𝑖≤𝑘𝑥∈𝑆^(ℎ)0≤𝛼∈ℝ. (3.12) We solve the weighted minimax model (3.12) and obtain the solution 𝑤^(ℎ).
In this step, we solve for the point in the reduced feasible region 𝑆^(ℎ) whose criterion vector is closest to 𝑓⁺.Substep 5.2 (find the furthest criterion vector to negative ideal). We can obtain a criterion vector which is furthest to the negative ideal one by solving the following model:𝑓max𝛽s.t.⁻−𝑓(𝑥)^𝜆′_∞≥𝛽𝑥∈𝑆^(ℎ)0≤𝛽∈ℝ. (3.13) This model can be converted to the following model: max𝛽s.t.max_{1≤𝑖≤𝑘}𝜆^′_𝑖𝑓_𝑖(𝑥)−𝑓⁻_𝑖≥𝛽,𝑥∈𝑆^(ℎ)0≤𝛽∈ℝ. (3.14) The above model can be converted to the following mixed integer linear programming model: max𝛽s.t.𝜆^′_𝑖𝑓_𝑖(𝑥)−𝑓⁻_𝑖+𝑀𝑦_𝑖≥𝛽,𝑖=1,…,𝑘_𝑘𝑖=1𝑦_𝑖=𝑘−1,𝑥∈𝑆^(ℎ)𝑦_𝑖∈{0,1},𝑖=1,…,𝑘0≤𝛽∈ℝ. (3.15) By solving MILP model (3.15) we obtain the solution 𝑦^(ℎ).
In this step, we solve the problem for the point in the reduced feasible region 𝑆^(ℎ) whose criterion vector is furthest to 𝑓⁻.Substep 5.3 (obtain a compromise solution). We can obtain a criterion vector which is closest to the positive ideal and furthest to the negative ideal by solving the following model: 𝑓(𝑥)𝑤min𝛾s.t.−𝑓^(ℎ)𝜆_∞𝑓(𝑥)𝑦≤𝛾−𝑓^(ℎ)𝜆′_∞≤𝛾𝑥∈𝑆^(ℎ)0≤𝛾∈ℝ. (3.16) This model can be converted to the following model: 𝜆min𝛾s.t._𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑤^(ℎ)𝜆≤𝛾,𝑖=1,…,𝑘_′𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑦^(ℎ)≤𝛾,𝑖=1,…,𝑘𝑥∈𝑆^(ℎ)0≤𝛾∈ℝ. (3.17) Model (3.17) can be converted to the following linear programming model: min𝛾s.t.𝜆_𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑤^(ℎ)𝜆≤𝛾,𝑖=1,…,𝑘_𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑤^(ℎ)𝜆≥−𝛾,𝑖=1,…,𝑘_′𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑦^(ℎ)𝜆≤𝛾,𝑖=1,…,𝑘_′𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑦^(ℎ)≥−𝛾,𝑖=1,…,𝑘𝑥∈𝑆^(ℎ)0≤𝛾∈ℝ. (3.18) By solving model (3.18), we obtain a compromise solution as 𝑥^(ℎ). In the other words, we obtain a compromise solution 𝑥^(ℎ) in the reduced feasible region 𝑆^(ℎ) whose criterion vector is closest to 𝑓(𝑤^(ℎ)) and 𝑓(𝑦^(ℎ)). That is, we obtain a compromise solution 𝑥^(ℎ) whose criterion vector is closest to positive ideal criterion vector 𝑓⁺ and is furthest to the negative ideal criterion vector 𝑓⁻.

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

Example 3.1 (Graphical Example). Consider Figure 1 in which 𝑥⁺ is the inverse image of 𝑓⁺ and 𝑥⁻ is the inverse image of 𝑓⁻.
Also, consider that 𝑤 and 𝑦 are the optimal solutions of models (3.12) and (3.15), respectively.
𝑤 is a solution whose criterion vector is closest to 𝑓⁺ and 𝑦 is a solution whose criterion vector is furthest to 𝑓⁻.
Point 𝑥 which is obtained by model (3.18) is a solution whose criterion vector is closest to 𝑓⁺ and furthest to 𝑓⁻ and therefore is a compromise solution.

Figure 1 

Graph of Example 3.1.

Step 6 (decision phase). The compromise solution 𝑥^(ℎ) is presented to the decision maker, who compares objective vector 𝑓(𝑥^(ℎ)) with the positive ideal criterion vector 𝑓⁺ and negative ideal criterion vector 𝑓⁻. This decision phase has the following steps.
Substep 6.1. If all components of 𝑓(𝑥^(ℎ)) are satisfactory, stop with (𝑥^(ℎ),𝑓(𝑥^(ℎ))) as the final solution and 𝑥^(ℎ) is the best compromise solution. Otherwise go to Substep 6.2.Substep 6.2. If all components of 𝑓(𝑥^(ℎ)) are not satisfactory, then terminate the interactive process and use other methods to search for the best compromise solutions. Otherwise go to Substep 6.3.Substep 6.3. If some components of 𝑓(𝑥^(ℎ)) are satisfactory and others are not, the DM must relax an objective 𝑓_𝑗(𝑥) to allow an improvement of the unsatisfactory objectives in the next iteration. If the decision maker cannot find an objective to sacrifice, then the interactive process will be terminated and other methods have to be used for identifying the best compromise solution, otherwise, the DM gives Δ𝑓_𝑗 as the amount of acceptable relaxation. Δ𝑓_𝑗 is the maximum amount of 𝑓_𝑗(𝑥) we are willing to sacrifice. Now go to Substep 6.4.Substep 6.4. Define a new reduced feasible region as follows: 𝑆^(ℎ+1)=𝑥∈𝑆^(ℎ)𝑓_𝑗(𝑥)≥𝑓_𝑗𝑥^(ℎ)−Δ𝑓_𝑗𝑓_𝑖(𝑥)≥𝑓_𝑖𝑥^(ℎ),𝑖≠𝑗,𝑖=1,…,𝑘. (3.19) And the weights 𝜋_𝑗 and 𝜋^′_𝑗 are set to zero. Set ℎ=ℎ+1 and go to Step 3.

4. Numerical Example

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

4.1. The Problem Description

Consider a firm that manufactures two products: 𝑥₁ and 𝑥₂. The firm’s overall objective functions have been estimated as follows: 𝑓₁(𝑥)=−5𝑥₁+2𝑥₂,𝑓₂(𝑥)=𝑥₁−4𝑥₂. (4.1) The following describes the limitations on the firm’s operating environment. −𝑥₁+𝑥₂𝑥≤3,₁+𝑥₂𝑥≤8,₁𝑥≤6,₂𝑥≤4,₁,𝑥₂≥0. (4.2) Then the MODM problem can be formulated as follows: max𝑓₁(𝑥)=−5𝑥₁+2𝑥₂max𝑓₂(𝑥)=𝑥₁−4𝑥₂s.t.−𝑥₁+𝑥₂𝑥≤3₁+𝑥₂𝑥≤8₁𝑥≤6₂𝑥≤4₁,𝑥₂≥0. (4.3) We set ℎ=0 and 𝑆⁽⁰⁾=𝑥=𝑥₁,𝑥₂−𝑥₁+𝑥₂≤3,𝑥₁+𝑥₂𝑥≤8,₁≤6,𝑥₂≤4,𝑥₁,𝑥₂≥0, (4.4)

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 𝑓⁺₁=6 and 𝑚₁=−30 and 𝑐₁₁=−5,𝑐₁₂=2, then from formula (3.5) we have 𝜋₁=6−(−30)6(−5)²+2^2−1/2=1.114. (4.5) Similarly, we can get 𝜋₂=0.728. From (3.6) there are 𝜆₁=1.1141.842=0.605,𝜆₂=0.7281.842=0.395. (4.6)Step 4 (obtain the second group of weights). Since 𝑓⁻₁=−30 and 𝑛₁=3 and 𝑐₁₁=−5,𝑐₁₂=2, then from formula (3.8) we have 𝜋^′₁=−30−(3))−30(−5²+2^2−1/2=0.204. (4.7) Similarly, we can get 𝜋^′₂=0.34. From (3.9) there are 𝜆^′₁=0.2040.544=0.375,𝜆^′₂=0.340.544=0.625. (4.8)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: min𝛼s.t.0.6056+5𝑥₁−2𝑥₂≤𝛼,0.3956−𝑥₁+4𝑥₂≤𝛼,𝑥∈𝑆⁽⁰⁾0≤𝛼∈ℝ. (4.9) 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: max𝛽s.t.0.375−5𝑥₁+2𝑥₂−(−30)+𝑀𝑦₁𝑥≥𝛽,0.625₁−4𝑥₂−(−15)+𝑀𝑦₂𝑦≥𝛽,₁+𝑦₂=1𝑥∈𝑆⁽⁰⁾𝑦₁,𝑦₂∈{0,1}0≤𝛽∈ℝ. (4.10) 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: min𝛾s.t.0.6050.904+5𝑥₁−2𝑥₂≤𝛾,0.6050.904+5𝑥₁−2𝑥₂≥−𝛾,0.395−1.808−𝑥₁+4𝑥₂≤𝛾,0.395−1.808−𝑥₁+4𝑥₂≥−𝛾,0.375−5𝑥₁+2𝑥₂−(6)≤𝛾,0.375−5𝑥₁+2𝑥₂−(6)𝑥≥−𝛾,0.625₁−4𝑥₂−()𝑥−12≤𝛾,0.625₁−4𝑥₂−(−12)≥−𝛾,𝑥∈𝑆⁽⁰⁾0≤𝛽∈ℝ. (4.11) 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 𝑆⁽¹⁾=𝑥∈𝑆⁽⁰⁾𝑓₁(𝑥)=−5𝑥₁+2𝑥₂≥4.026−Δ𝑓₁𝑓₂(𝑥)=𝑥₁−4𝑥₂≥−8.052. (4.12) We set 𝜋₁=𝜋^′₁=0 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.

Table 4 

Second pay-off table of Section 4.

Figure 2 

Positive and negative ideal solutions.

Figure 3 

Compromise solution in iteration 1.

Iteration Number 2
It is obvious that 𝜆₂=𝜆^′₂=1, 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: min𝛼s.t.6−𝑥₁+4𝑥₂≤𝛼,𝑥∈𝑆⁽¹⁾0≤𝛼∈ℝ. (4.13) 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: 𝑥max𝛽s.t.₁−4𝑥₂−(−15)≥𝛽,𝑥∈𝑆⁽¹⁾0≤𝛽∈ℝ. (4.14) 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: min𝛾s.t.−4.052−𝑥₁+4𝑥₂≤𝛾,−4.052−𝑥₁+4𝑥₂𝑥≥−𝛾,₁−4𝑥₂−()𝑥−4.052≤𝛾,₁−4𝑥₂−(−4.052)≥−𝛾,𝑥∈𝑆⁽¹⁾0≤𝛾∈ℝ. (4.15) 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.

Figure 4 

Compromise solution in iteration 2.

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.

4.3. Solving with the Classic STEM Method

Suppose we want to solve the above problem with the classic STEM method. Therefore the iterations of solving this problem are as shown in Table 5.

Table 5 

The results of classic STEM method.

In iteration 2, we set Δ𝑓₁=2. By comparing with the results of proposed method, we can see that the optimal objective is further to the negative ideal objective.

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.

Acknowledgment

The author is grateful to the helpful comments and suggestions made by anonymous referees.