**Abstract**

The chapter presents a decision support system. The decision-making process is modeled by a multi-criteria optimization problem. The decision support method is an interactive decision-making process. The choice is made by solving the problem depending on the control parameters that define the aspirations of the decision makers for each criteria function, then it evaluates the obtained solution by accepting or rejecting it. In another case, the decision maker selects a new value and the problem is solved again for the new parameter. In this chapter, an example of a decision support system is presented.

**Keywords:** multi-criteria optimization, efficient decision, scalarizing function, method of decision selection, decision support system

## **1. Introduction**

Decision support systems are a very broad field, including theoretical approaches and methods of their application [1–14]. Decision support involves the automation of certain steps in the decision-making process. The extent of such automation is an important issue. Methods that provide a high degree of automation of the decisionmaking process are optimization methods of decision support based on value and utility theory that use analytical forms of decision situation models and expert systems in decision support, related to artificial intelligence and knowledge engineering and using logical forms of models. The practice and psychology of decision support prefer a different approach based on emphasizing the sovereign role of the decision maker, assuming that he can be assisted by automation of some stages of the decision-making process but should sovereignly and fully consciously make the final choice of decision.

A decision is usually called a choice between multiple possibilities. The person making the decision is usually referred to as the decision maker. The issue of preparing and making a decision is usually much more complex than, as the above definition of the term decision would suggest, the mere problem of choosing between some options. Initially, we usually do not know the decision options, thus, we have to prepare or generate them on our own; the very issue of preparing decision options is often complex and usually more time-consuming than the issue of choice. However, before we start preparing options, we often do not even know our exact point of interest.

Herbert Simon introduced the concept of a decision-making process [15–17]. Simon's definition of this process includes four stages:


In the fourth stage, we may also modify the decision according to feedback, i.e. observation of its effects. The advantage of Simon's approach, however, is that he was the first to pay adequate attention to the role of learning, adaptation, and changing views in the decision-making process.

Herbert Simon formulated a model of satisficing decisions, describable as follows:


In this chapter, we discuss the use of vector optimization for decision support.

## **2. Decision-making process model**

Most decision-making processes are multi-criteria in nature, that is, they include no single indicator to be optimized so the best decisions are provided. For example, in the design process, an engineer usually tries to find a trade-off between a few indicators, such as reliability and other quality attributes, and on the other hand cost, weight, device volume, etc.

We consider a decision problem defined as a multi-criteria optimization problem with *m* scalar evaluation functions

$$\max\_{\mathbf{x}} \left\{ \left( f\_1(\mathbf{x}), \dots, f\_m(\mathbf{x}) \right) : \ \mathbf{x} \in \mathbf{X}\_0 \right\} \tag{1}$$

where

*f* ¼ *f* 1, … , *f <sup>m</sup>* is a (vector) function that transforms the decision (implementation) space *<sup>X</sup>* <sup>¼</sup> *<sup>R</sup><sup>n</sup>* into the evaluation space *<sup>Y</sup>* <sup>¼</sup> *<sup>R</sup><sup>m</sup>*; individual coordinates *fi* represent scalar evaluation functions; and *I* ¼ f g 1, 2, … *m* is a set of evaluation indices.

*Xo* ⊂ *X* is the set of feasible solutions.

*x*∈*Xo* is the vector of decision variables.

The function *f* assigns an evaluation vector *y* ¼ *f x*ð Þ, which measures the quality of the decision *x* from the point of view of a fixed set of evaluation functions to each vector of decision variables *x*∈*Xof* ¼ *f* 1, … , *f <sup>m</sup>* . The formulation of a multi-criteria optimization problem is expressed in decision space. It is a natural representation of

the decision problem; its target is the choice of the correct decision. The image of the admissible set *Xo* for the function *f* is the set of achievable evaluation vectors *Y*<sup>0</sup> ¼ f g *y* : *y* ¼ *f x*ð Þ, *x*∈*X*<sup>0</sup> .

The multi-criteria optimization model may be written in an equivalent form in the evaluation space. This leads to a multi-criteria model in the evaluation space:

$$\max\_{\mathbf{x}} \left\{ \mathbf{y} = \begin{pmatrix} \mathbf{y}\_1, \ \dots \ \mathbf{y}\_m \end{pmatrix} : \ \mathbf{y}\_i = f\_i(\mathbf{x}) \ \forall i, \ \mathbf{x} \in \mathbf{X}\_0 \right\} \tag{2}$$

where

*xo* is a vector of decision variables.

*y* ¼ *y*1, … , ym is a vector of achievable evaluation vectors; the first coordinate is the evaluation function *f* <sup>1</sup> and the last coordinate is the evaluation function *f <sup>m</sup>*.

*Y*<sup>0</sup> ¼ *f X*ð Þ<sup>0</sup> is the set of achievable evaluation vectors.

The set of achievable vectors *Y*<sup>0</sup> is given in an implicit form, i.e., through the set of admissible decisions *Xo* and the model mapping *f* ¼ *f* <sup>1</sup>, … , *f <sup>m</sup>* . A simulation of the model *y* ¼ *f x*ð Þ *for x*∈*X*<sup>0</sup> is required to determine *y*.

Each vector *x*∈*Xo* corresponds to a vector *y*∈*Y*0. The decision maker selects a vector from the set *Y*<sup>0</sup> and chooses for implementation the decision corresponding to that vector from the set *X*<sup>0</sup> [4, 10, 11, 14, 18–20].

The purpose of problem (1) is to help the decision maker choose a decision that is satisfactory to the decision maker.

## **3. Efficient decisions**

The solution to a multi-criteria optimization problem is a set of efficient decisions. Non-dominated solutions (Pareto optimal) are defined by a preference relation that provides an answer to the following question: Which of a given pair of evaluation vectors *y*1,*y*<sup>2</sup> ∈*R<sup>m</sup>* is better? This is the following relation:

$$\{\mathbf{y}^1 \succ \mathbf{y}^2 \Leftrightarrow \mathbf{y}\_i^1 \succeq \mathbf{y}\_i^2 \forall i = 1, \dots, m \land \exists \mathbf{j} \; \mathbf{y}\_j^1 \succ \mathbf{y}\_j^2\}\tag{3}$$

An evaluation vector ^*y*∈*Y*<sup>0</sup> is called a non-dominated vector if there is no *y*∈*Y*<sup>0</sup> that the vector ^*y* which is dominated by the vector *y* [10, 13, 14, 21–23]. The dominance decision structure in *R*<sup>2</sup> is shown in **Figure 1**.

**Figure 1.** *Dominance structure in R*<sup>2</sup>*.*

The set of non-dominated vectors is defined as follows [10, 14].

$$\hat{Y}\_0 = \left\{ \hat{y} \in Y\_0 : (\hat{y} + \tilde{D}) \cap Y\_0 = \mathcal{Q} \right\} \tag{4}$$

where

*<sup>D</sup>*<sup>~</sup> is a positive cone without a vertex. This positive cone can be as follows: *<sup>D</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup><sup>m</sup>* þ. The set of non-dominated vectors *Y*^<sup>0</sup> is shown in **Figure 2**.

The corresponding admissible decisions are defined in the decision space. A decision *x*^ ∈*X*<sup>0</sup> is referred to as an efficient decision (Pareto optimal) if the corresponding evaluation vector ^*y* ¼ *f*ð Þ *x*^ is a non-dominated vector.

## **4. Decision support system**

The solution to a multi-criteria optimization problem is the entire set of efficient solutions generating a set of all non-dominated evaluation vectors. In the general case, this set may be infinite. In order to solve the decision problem, a single solution must be chosen for implementation. Thus, the set of efficient solutions to a multi-criteria problem may not be regarded as the final solution to the corresponding decision problem.

In multi-criteria decision problems, the decision maker's preference relation is not known a priori, and, therefore, the final choice of the solution may only be made by the decision maker. Due to the size of the set of efficient solutions, even if the entire set of efficient solutions is determined by computational methods, the decision maker may not make the choice of solution without the help of an appropriate interactive information system. Such a system—the decision support system—allows for a controlled review of the set of efficient solutions. Based on the values of certain control parameters given by the decision maker, the system presents different efficient solutions for analysis. Thus, the control parameters determine a certain parameterization of the set of efficient solutions. The parametric analysis of the set of efficient solutions obviates the need to directly determine the entire set of efficient solutions. Instead, the system may each time determine one efficient solution corresponding to the

current values of the control parameters. Multi-criteria decision problems are solved by interactive decision support systems using parametric scalarizing of the multicriteria problem [10, 11, 14, 21]:

$$\max\_{\mathbf{x}} \left\{ s(p, f(\mathbf{x})) : \quad \mathbf{x} \in X\_0 \right\}, \quad p \in P \tag{5}$$

where

*p* is a vector of control parameters. *s* : *P* � *Y* ! *R* is a scalarizing function. The scalarizing should satisfy the following conditions:


The parametric scalarization is then a complete parameterization of the set of efficient solutions to the multi-criteria problem.

The control parameters should represent real quantities that are easily understood by the decision maker and that characterize his preferences. A parametric scalarization that satisfies all of the above postulates makes it possible to implement a decision support system that allows for determination of an efficient solution consistent with the decision maker's preferences.

As the first step of multi-criteria analysis, single-criteria optimization is applied to each evaluation function separately. As a result of single-criteria optimization, a socalled pay-off matrix is created, which allows for estimating the scope of changes of particular evaluation functions on the set of efficient solutions. This matrix also provides some information about the so-called conflict of the evaluation functions. The pay-off matrix is an array containing values of all evaluation functions obtained while solving particular single-criteria problems. The pay-off matrix also generates a utopia vector representing the best values of each evaluation function considered separately, i.e. *y<sup>m</sup> <sup>i</sup>* <sup>¼</sup> ^*fi* , *i* ¼ 1, … ,*m*. The utopia vector is the upper bound of all achievable evaluation vectors, i.e. *y*≤ *y<sup>u</sup>* for each y∈ Y0. It is normally unachievable *y<sup>u</sup>* ∉ Y0, i.e., there is no admissible solution with such values of evaluation functions. If there exists such an admissible vector *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>X</sup>*<sup>0</sup> so that *f x*ð Þ¼ *<sup>o</sup> <sup>y</sup><sup>u</sup>*, then *<sup>x</sup>*<sup>0</sup> is the optimal solution to the multi-criteria problem in the sense of any preference model. This situation can happen only if there is no conflict between the evaluation functions.
