Satisficing Decision-Making

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.

1. Introduction

Decision support systems are a very broad field, including theoretical approaches and methods of their application [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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 decision-making 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, 16, 17]. Simon’s definition of this process includes four stages:

1. Problem intelligence activity.

2. Problem design activity.

3. Choice activity.

4. Implementation and supervision activity.

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:

1. The decision maker determines aspiration levels for each decision outcome. These aspiration levels are determined adaptively, through a learning process.

2. The choice of decision is not a single act of optimization, but a dynamic process of solution search; in it, the decision maker also learns and may change preferences and aspirations.

3. The process ends when the decision maker finds a decision that achieves an outcome that meets his aspirations (hence the name satisfactory decision) or is in some sense closest to the aspirations.

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

$$\underset{x}{max}\left\{\left(f_1\left(x\right),\dots ,f_m\left(x\right)\right):x\in X_0\right\}$$E1

where

$f=\left(f{}_1,\dots ,f_m\right)$ is a (vector) function that transforms the decision (implementation) space $X=R^n$ into the evaluation space $Y=R^m$; individual coordinates $f{}_i$ represent scalar evaluation functions; and $I=\left\{1,2,\dots m\right\}$ is a set of evaluation indices.

$X_o\subset X$ is the set of feasible solutions.

$x\in X_o$ is the vector of decision variables.

The function $f$ assigns an evaluation vector $y=f\left(x\right)$, 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 $x\in X_{o}f=\left(f{}_1,\dots ,f_m\right)$. 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 $X_o$ for the function $f$ is the set of achievable evaluation vectors $Y_0=\left\{y:y=f\left(x\right),x\in X_0\right\}$.

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:

$$\underset{x}{max}\left\{y=\left(y_1,\dots ,{\mathrm{y}}_{\mathrm{m}}\right):y_i=f_i\left(x\right)\forall i,x\in X_0\right\}$$E2

where

$x_o$ is a vector of decision variables.

$y=\left(y_1,\dots ,{\mathrm{y}}_{\mathrm{m}}\right)$ is a vector of achievable evaluation vectors; the first coordinate is the evaluation function $f{}_1$ and the last coordinate is the evaluation function $f_m$.

$Y_0=f\left(X_0\right)$ is the set of achievable evaluation vectors.

The set of achievable vectors $Y_0$ is given in an implicit form, i.e., through the set of admissible decisions $X_o$ and the model mapping $f=\left(f_1,\dots ,f_m\right)$. A simulation of the model $y=f\left(x\right)\begin{array}{cc}\mathit{\text{for}}& x\in X_0\end{array}$ is required to determine $y$.

Each vector $x\in X_o$ corresponds to a vector $y\in Y_0$. The decision maker selects a vector from the set $Y_0$ and chooses for implementation the decision corresponding to that vector from the set $X_0$ [4, 10, 11, 14, 18, 19, 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 $y^1,y^2\in R^m$ is better? This is the following relation:

$$y^1\succ y^2\iff y_i^1\ge y_i^2\forall i=1,\dots ,m\wedge \exists \mathrm{j}y{}_j{}^1>y_j^2$$E3

An evaluation vector $\widehat{y}\in Y_0$ is called a non-dominated vector if there is no $y\in Y_0$ that the vector $\widehat{y}$ which is dominated by the vector $y$ [10, 13, 14, 21, 22, 23]. The dominance decision structure in $R^2$ is shown in Figure 1.

Figure 1.

Dominance structure in $R^2$.

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

$${\widehat{Y}}_0=\left\{\widehat{y}\in Y_0:\left(\widehat{y}+\tilde{D}\right)\cap Y_0=\varnothing \right)\}$$E4

where

$\tilde{D}$ is a positive cone without a vertex. This positive cone can be as follows: $\tilde{D}=R_{+}^m$.

The set of non-dominated vectors ${\widehat{Y}}_0$ is shown in Figure 2.

Figure 2.

Non-dominated evaluation vectors${\widehat{Y}}_0$.

The corresponding admissible decisions are defined in the decision space. A decision $\widehat{x}\in X_0$ is referred to as an efficient decision (Pareto optimal) if the corresponding evaluation vector $\widehat{y}=f\left(\widehat{x}\right)$ 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 multi-criteria problem [10, 11, 14, 21]:

$$\underset{x}{max}\{s\left(p,f\left(x\right)\right):\begin{array}{cc}& x\in X_0\}\end{array},\begin{array}{cc}& p\in P\end{array}$$E5

where

$p$ is a vector of control parameters.

$s:P\times Y\to R$ is a scalarizing function.

The scalarizing should satisfy the following conditions:

• efficiency condition—for each vector of control parameters $p\in P$, the optimal solution of the scalar problem was an efficient solution of the original multi-criteria problem;

• condition of completeness of the set of efficient solutions, so that for each non-dominated evaluation vector, there is a set of values of control parameters at which the system determines the efficient solution generating this vector of grades.

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 so-called 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. $y_i^m=\widehat{f}{}_i,i=1,\dots ,m$. The utopia vector is the upper bound of all achievable evaluation vectors, i.e. $y\le y^u\text{for each}\mathrm{y}\in {\mathrm{Y}}_0$. It is normally unachievable $y^u\notin {\mathrm{Y}}_0$, i.e., there is no admissible solution with such values of evaluation functions. If there exists such an admissible vector $x_0\in X_0\mathrm{so}\text{that}f\left(x_o\right)=y^u$, then $x_0$ 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.

5. Reference point method

The reference point method combines the simplicity and openness of controlling the interactive analysis process with strict adherence to the principle of efficiency of the generated solutions and complete parameterization of the set of efficient solutions. The reference point method uses aspiration levels as control parameters and always generates efficient solutions.

The preference model used in the reference point method satisfies the following two postulates:

1. P1—efficient solutions dominate inefficient solutions, i.e., that the decision maker’s preferences are consistent with choosing efficient solutions.

2. P2—the decision maker prefers evaluations that achieve all aspiration levels than those that do not achieve one or more aspiration levels.

In this model, it is assumed that when solving a decision problem, the decision maker defines aspiration levels as the desired values of individual evaluations. If the values of the evaluations do not achieve the aspiration levels, the decision maker tries to find a better solution. If the values of some evaluations achieve their respective aspiration levels, the decision maker focuses attention on improving the values of those evaluations that have not achieved their aspiration levels. When all evaluations have achieved their aspiration levels, the decision maker is interested in further improving the evaluations if possible.

The reference point method relies on the minimization of a suitably defined achievement scalarizing function that generates a preference relation satisfying postulates P1 and P2. For that reason, it always determines efficient solutions. It is also required that the achievement scalarizing function ensures the completeness of the parameterization of the set of efficient solutions by aspiration levels. This requirement means that for each achievable evaluation vector $y\in Y_0$, there should be aspiration levels that allow for determining the efficient solution that generates this evaluation vector.

The achievement scalarizing function in the reference point method is as follows [10, 11, 14, 21]:

$$s\left(y,\overline{y}\right)=\underset{1\le i\le m}{min}\left(y_i-{\overline{y}}_i\right)+\epsilon \cdot \sum _{i=1}^m\left(y_i-{\overline{y}}_i\right)$$E6

where

$y=\left(y_1,y_2,\dots ,y_m\right)$ is an evaluation vector.

$\overline{y}=\left({\overline{y}}_1,{\overline{y}}_2,\dots ,{\overline{y}}_k\right)$ is a vector of aspiration levels.

$\epsilon$—an arbitrarily small regularization parameter.

The maximization of the function $s\left(y,\overline{y}\right)$ due to $y\in Y_0$ determines the non-dominated evaluation vector$\widehat{y}$ and the generating efficient solution $\widehat{x}$. The determined efficient solution depends on the values of the aspiration levels $\overline{y}$. The aspiration levels ${\overline{y}}_i,i=1,\dots ,m$ are the parameters that control the interactive analysis process. The parameter $\epsilon$ is used to introduce a regularization component to guarantee the efficiency of the solution in case of ambiguity of the minimum of the first component of the function $s\left(y,\overline{y}\right)$.

The optimization problem solved by the reference point method does not introduce significant complications into the structure of the original problem. The process of interactive analysis by the reference point method is consistent with the concept of decision support systems. It implements an open process of searching for a satisficing efficient solution on the basis of current preferences determined by aspiration levels. It is easy for the decision maker to understand the expression of current preferences in terms of aspiration levels.

In the reference point method, the scalarizing function $s\left(y,\overline{y}\right)$ is called the achievement function. This name is related to the fact that the values of this function are zero for $\mathrm{y}=\overline{\mathrm{y}}$, positive for $\mathrm{y}\in \overline{\mathrm{y}}+\tilde{D}$, and negative for $\mathrm{y}\notin \overline{\mathrm{y}}+\tilde{D}$. Therefore, the maximum values of this function can be used not only to calculate efficient outcomes but also to assess the achievability of a given aspiration point $\overline{\mathrm{y}}$:

• If the maximum of the achievement function $s\left(y,\overline{y}\right)$ relative to $\mathrm{y}\in {\mathrm{Y}}_0$ is negative, then the aspiration point $\overline{\mathrm{y}}$ is not achievable, while the maximum point $\widehat{y}$ of this function is the non-dominated outcome in some sense uniformly closest to the aspiration point $\overline{\mathrm{y}}$;

• If the maximum of the achievement function $s\left(y,\overline{y}\right)$ relative to $\mathrm{y}\in {\mathrm{Y}}_0$ is zero, then the aspiration point $\overline{\mathrm{y}}$ is an achievable and non-dominated outcome and is (perhaps one of many) the maximum point of this function;

• If the maximum of the achievement function $s\left(y,\overline{y}\right)$ relative to $\mathrm{y}\in {\mathrm{Y}}_0$ is positive, then the aspiration point $\overline{\mathrm{y}}$ is achievable, while the maximum point $\widehat{y}$ of this function is a non-dominated outcome, in a sense uniformly improved relative to the aspiration point $\overline{\mathrm{y}}$.

6. Example of application

To illustrate finding a satisficing solution, the following example of a bicriteria problem [24] is presented.

$$\begin{array}{l}max\left\{\left(f_1\left(x\right)=10\cdot x_1,{\mathrm{f}}_2\left(x\right)=x_1+5\cdot x_2\right)\right\}\\ 10\cdot x_1\ge 50\\ x_1\le 8\\ x_1+x_2\le 14\\ x_1\ge 0,x_2\ge 0\end{array}$$E7

The first step of multi-criteria analysis is the single-criteria optimization of each evaluation function is a pay-off matrix containing the values of all functions obtained when solving two single-criteria problems. This matrix allows us to estimate the extent of change of each evaluation function on the possible set, and also provides some information about the conflicting nature of the evaluation function. The objective matrix generates a utopia vector representing the best value of each of the separate criteria (Table 1).

Function	Solution
	$f_1\left(x\right)=y1$	$f_2\left(2\right)=y2$
Function $f_1$	140	14
Function $f_2$	60	46
Utopia vector	140	46

Table 1.

Pay-off matrix with utopia vector.

Source: Own calculations.

The multi-criteria analysis is shown in Table 2.

Iteration	Solution
	$y1$	$y2$
1. Aspiration point $\overline{y}$	140	46
Solution $\widehat{y}$	117.4	23.14
Values	−22.86
2. Aspiration point $\overline{y}$	125	42
Solution $\widehat{y}$	109.28	26.28
Values	−15.71
3. Aspiration point $\overline{y}$	115	40
Solution $\widehat{y}$	103.57	28.57
Values	−11.43
4. Aspiration point $\overline{y}$	100	35
Solution $\widehat{y}$	96.42	31.42
Values	−3.573
5. Aspiration point $\overline{y}$	90	35
Solution $\widehat{y}$	89.28	34.28
Values	−0.71
6. Aspiration point $\overline{y}$	85	32
Solution $\widehat{y}$	87.85	34.85
Values	2.85
7. Aspiration point $\overline{y}$	110	40
Solution $\widehat{y}$	100	30
Values	−10.00

Table 2.

Interactive analysis of finding a satisfactory solution.

Source: Own calculations.

At the beginning of the analysis, the decision maker defines his preference as an aspiration point equal to the utopia vector. The resulting value of the function s is negative. The aspiration point is not achievable. The decision maker’s requirements are too high. The obtained solution prefers the first function. To improve the solution for the second function in the next iteration, the decision maker explicitly reduces his requirements for the first function and reduces the requirements for the second function. The value of function s is still negative. The aspiration point is not achieved. The decision maker’s requirements are too high. The result is that the solution for the first function deteriorates and the solution for the second function improves. In the third iteration, the decision maker reduces the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision maker’s requirements are too high. The solution continues to deteriorate for the first function and improves for the second function. In the fourth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision maker’s requirements are still too high. The solution continues to deteriorate for the first function and improves for the second function. In the fifth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision maker’s requirements are too high. The solution continues to deteriorate for the first function and improves for the second function. In the sixth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is now positive. The aspiration point is exceeded. The decision maker’s requirements are too small. The solution continues to deteriorate for the first function and improves for the second function. In the seventh iteration, the decision maker increases the requirements for both functions. The value of function s becomes negative. The aspiration point is not achieved. The decision maker’s requirements are too high. The solution improves for the first function and deteriorates for the second function. For the seventh iteration, the corresponding decisions are as follows: ${\widehat{x}}^7=\left(10,00,4,00\right)$. The analysis shows that the solution depends heavily on the first function and affects the solution more.

The final choice of a particular solution depends on the preferences of the decision maker. The example shows that the method allows the decision maker to explore decision choices during interactive analysis and search for a satisfactory solution.

7. Conclusions

This paper presents a decision support system as a multi-criteria optimization problem. The model of the decision problem as a multi-criteria optimization problem allows for generating decision variants and tracking their consequences.

The interactive analysis is based on the reference point method. It allows the decision maker to determine solutions well suited to his preferences. A numerical example shows that the right computational problem can be solved efficiently using standard optimization software.

This type of decision support does not prejudge the final solution but supports and informs the decision maker on the specific decision problem. The final decision is to be made by the decision maker.