Decision Making Lab1

Laboratory work №1

Topic: Learning the models of decision selection under conditions of certainty

Goal of the work:

- study the possibilities of applying models of decision selection in conditions of certainty;

- learn how to perform. a graphic presentation of a decision-making task;

- learn how to search for solutions using the language of binary relations.

The order of work

1. Study theoretical information about the decision-making under conditions of certainty. To study the essence and features of decision-making models.

2. Choose a problem area for decision-making. Perform. a verbal formulation of the decision-making task in the selected problem area and identify 6 alternatives, evaluating them according to 5 criteria.

3. Create income matrices in MS Excel. Give a graphic representation of your problem.

4. Based on various choice models, search for solutions using the language of binary relations and constructed graphs. Compare the obtained solutions. Explain the results. Demonstrate the process of finding solutions to the teacher.

5. Make a report on laboratory work.

6. Defend laboratory work.

Content of the report

1. Topic of the work.

2. Goal of the work.

3. Individual task.

4. Description of the work execution.

5. Interpretation of the obtained results.

6. Conclusions.

Individual tasks

Select the problem area according to the ID in the group list. Search for solutions for the given problem area (Table 1).

Table 1

BRIEF THEORETICAL INFORMATION

1. Binary relations

An important assumption in the language of binary relations is that the preference of two alternatives is independent of any third. Binary relations can be established on multiple alternatives and multiple criteria. In both cases, for each pair of compared objects x_i, x_j ∈ X in a certain way it is possible to establish that one of them is better than the other or that they are equal or incomparable.

In general, to specify a binary relation R on a set X, it is necessary to specify in one way or another all pairs (x_i, x_j) of the set X for which the relation R is fulfilled.

There are four ways to specify relationships:

1) a direct list of pairs,

2) matrix,

3) graph,

4) intersection.

Consider an example of relationships in a student group consisting of three people. On the set X = (x₁, x₂, x₃) of students, let's put the ratio R – "study better". Let the ratio R be given in the first way as follows: : x₁ R x₂ ; x₁ R x₃. Then you can make a matrix A of relations R, consisting of zeros and ones, in which

The relationship graph, in which the arrows are directed towards the student with a smaller advantage, is shown in fig. 1.

Fugure 1 − A graphical method of specifying relations

The intersections are given for each element of the set X. The upper intersection is R⁺(x) and the lower one is R⁻(x). The upper intersection of x is the set of elements from X that exceed x. The lower intersection for x is the set of elements of X that are less dominant for x.

In the given example, the relations R are not specified on the entire set X. If not all elements can be compared with respect to the relation R, then it is called incomplete (imperfect, nonlinear, partial).

Equivalence relations, strict order and non-strict order can be established on the entire set of objects X. The relation of equivalence is meaningfully interpreted as interchangeability, sameness of objects. The concepts of equivalence, equivalence and incomparability are often equated. The relation of equivalence gives rise to the division of a set of objects into classes that unite objects that cannot be distinguished by one or a group of criteria. In the given example, x₂ and x₃ are equivalent: x₂ ~ x₃.

The relation of strict order can be interpreted as the superiority of one object compared to another, for example, "better", "more important", "older", etc. In the given example, x₁ studies better than x₂ and x₃ , x₁ x₂ and x₁ x₃. A strict order relation gives rise to a strict ordering by preference. If we added, for example, the relationship x₃ x₂, we would get a strict order x₁ x₂ x₃.

P. Fishburn proved the theorem that in the case of a strict ordering of objects by preference, it is possible to construct a utility function U(x) such that for

x_i x_j ⇒U(x_i) > U(x_j).

The definition of the function U(x) allows us to move from the language of binary relations to the criterion language, taking U(x) as a criterion function.

A relation of non-strict order is a combination of relations of strict order and equivalence, it is interpreted as superiority or equivalence of x_i ≥ x_j objects (x_i is not worse than x_j). The relation of complete non-strict ordering gives rise to a strict ordering of equivalence classes of objects. If we add the relation x₂≥ x₃, and x₃ ≥ x₂, then we get the order x₁ (x₂ ∼x₃).

An alternative in a decision-making problem can be represented by a description in the criterion space. Through the criterion space, binary relations can be established on a set of alternatives.

We denote:

x = (x₁, x₂, x₃) – vector of alternative х scores;

y = (y₁, y₂, y₃)  – vector of alternative y scores.

Let's introduce the relations of strict superiority (Pareto relation), equality and inequality for equivalent criteria on the alternatives x and y.

2. Decision making models under conditions of certainty

When making decisions, one of the main issues is the issue of choice. The choice of decision is made taking into account the choice model.

Let A be a given set. R is an arbitrary binary relation A, i.e. system of a decision maker benefits. Then the pair <A,R> is called a model of choice.

Three main formal models of choice can be:

- choose the best;

- choose the maximum;

- choose the optimal one.

2.1 Model for selecting the best element

Let the given model is <А,R>. Element аº ∈ А is the best by R in А, if

(аº,а) ∈ R given that а ∈ А\аº.

The concept of the best element is shown in fig. 2. In fig. 2(a), the best elements are а₁ and а₂ and you can make a conclusion, that element а₃ is not the best. In fig. 2(a), there is no essential elements.

With graphs, the concept of the best element indicates the existence of a vertex connected by arrows that go from it to all other vertices of the graph. In this case there may be some other additional connections.

Figure 2 – Concept of the best element

2.2 Model for selecting the maximum element

Let the given model is <А,R>. Element а^о ∈ А is maximum by R in А, if

а є А: (а,а^о) ∈ R à (а^о,а) є R.

The set of all maximum elements <А,R> let’s denote as Мах_R А.

The relationship graph with the maximum elements, should have the vertices, in which for each incoming arrow from a vertex (as such), exists the “compensating” outcoming arrow, which goes to the same vertex.

For example, maximum elements by R are а₁, а₂ (fig. 2, (а)) and а₂, а₃ (fig. 2, (b)).

The best element by R in А is also maximum.

The reverse is correct only if the relationship R has the feature of weak connected graph:

а₁, а₂ : (а₁ ≠ а₂) à ((а₁, а₂)) ∈ R) \/ ((а₂, а₁) ∈ R) .

2.3 Model for selecting the optimal element

An element а^о ∈ А is R-optimal by А, if

а ∈ А, а ≠ а^о à (а^о,а) ∈ R.

The optimal vertex does not have any incoming arrow.

In fig. 2(а) R-optimal elements are absent, in fig. 2(б) а₃ is the R-optimal element.

For a problem of decision making, the definition of external stability is important:

A set Мах_R А is externally stable if for any element а ∈ А \ Мах_R А exists such one а^о ∈ Мах_R А, for which is fair (а^о,а) ∈ R.

If the set Мах_R А is externally stable, then the next selection of the optimal element (based on, for example, the involvement of additional information) can be carried out only within the set of Мах_R А. Otherwise, when there is no stability, such a conclusion will no longer have a reasonable justification.

The externally stable set Мах_R А is the core of relation R in А. Sometimes the term "kernel" of the set Мах_R А is used without the requirement of external stability.

In the fig. 2(а) and 2(б) the sets Мах_R А are externally stable. In the fig. 3 you can see the set Мах_R А = {a₁, a₂}, which is not externally stable.

Thus, the problem of making a decision, formulated in the language of binary relations, is understood as the problem of extracting a kernel, that is, the set of maximal elements from A according to some binary relation R: А * = Мах_R А.

Figure 3 – No external stability

The following special types of binary relations are important from a practical point of view:

o quasi-order (R is reflexive and transitive);

o strict order (R is anti-reflexive and transitive);

o equivalence (R is reflexive, symmetric and transitive).

3. An example of work execution

Is given:

- a set of alternatives "laptop" - Toshiba Satellite A660- 10X, HP Pavilion dv7-1253ca, Asus G73JHTY031;

- criteria for evaluating alternatives: the number of processor cores, weight, screen diagonal, hard disk capacity, cost.

Using the language of binary relations, determine the most acceptable alternative according to the models of choosing the best, maximum, optimal elements.

The main characteristics of the "laptop" system of the given alternatives are presented in Table 2.

Table 2 – Parameters of alternatives according to the given criteria

Let's set the matrices of the advantages for alternatives and perform. a graphical presentation of the advantages of alternatives according to the given criteria:

Let’s analyze the obtained data and build Table 3 with the results of selection according to various models of choice applied.

Conclusions.

1. According to the model of choosing the best element, taking into account all criteria, it is advisable to choose alternative A.

2. According to the model of choosing the maximum element, taking into account all the criteria, it is advisable to choose alternative A.

3. According to the model of choosing the optimal element, taking into account all criteria, it is advisable to choose alternative A.

That is, it is advisable to choose alternative A based on the totality of evaluations by all criteria.