Analytic Hierarchy Process in Multiple–Criteria Decision– Making: A Model Example

. Managers have to make decisions several times a day. The decision-making process can be defined as an essential activity realized by managers every day. Decisions can be implemented intuitively, or by using relevant decision-making methods. This depends on the nature of the decision, as well as the intensity of its possible future effects. The theory of decision-making can be defined as a relatively young discipline. It can be stated that decision-making is no longer an intuitive process. Most decision-making situations are of a multiple criteria character. In this contribution, the authors focus on multiple-criteria decision-making, to which several methods can be applied. In the practical part, the authors use Saaty's method, also known as the Analytic Hierarchy Process. Saaty is considered to be the most important researcher dealing with the issue of multiple-criteria decision-making. The set multiple-criteria decision-making problem was to choose one business partner out of eight under consideration. The decision-making criteria included selected financial indicators and non-financial criteria. The aim of the contribution is to use the Analytic Hierarchy Process to assess potential business partners and to select an optimal candidate.


Introduction
Decision-making is described as one of the most important activities performed by business managers, and is also referred to as the core of management. The decision-making process can be defined as a set of activities that leads to the resolution of a decision-making problem that involves at least two meaningful alternatives, of which the selected one offers the best result with respect to the set goal, as well as the possibility of its implementation [1]. There are two sides to the decision-making process, namely [2]: 1. The content side of decision-making -this reflects the differences of individual decision-making processes.
2. The formal-logical (procedural) side of decision-making -this expresses that individual decision-making processes have certain common features, regardless of their content.
The formal-logical side of decision-making is a precondition for claiming that there is a framework procedure for resolving a decision-making situation with different content. It is the basis for decision-making theory, which aims to create appropriate procedures (methods) for solving decision-making situations with the highest degree of universality. One of the biggest issues that decision-making theory deals with is the issue of multiple-criteria decision-making. The beginnings of multiple-criteria decision-making (hereinafter "MCDM") can be seen in Benjamin Franklin's decisionmaking process. He used a simple paper system for the decision-making process for important issues. On one side of the sheet of paper, he wrote the arguments in favour of a decision; on the other side, he wrote the arguments against the decision. Gradually, he deleted the arguments on each side which were of relatively equal importance. When all the arguments on one side were deleted, the side which had remaining arguments was the side of the argument that he would support. In 1785, Marie-Jean-Antoine-Nicolas de Caritat published "Essay on the Application of Analysis to the Probability in Majority Decisions". The most famous results of this study is the Condorcet (voting) Paradox. The development of studies dealing with multiple-criteria decision-making accelerated during the 1980s and early 1990s. Francis Edgeworth, a philosopher and political economist, significantly contributed to the development of statistical methods. He was the first who applied formal mathematical concepts to decision-making. The basis of modern MCDM theory is Pareto-optimality, also known as the concept of efficiency by Vilfredo Pareto. Frank Ramsey published an essay "Truth and Probability" dealing with the decision-making process with uncertain outcomes. It was the basis for later published study "Foundation of Statistics" by Leonard Savage. Studies by John von Neumann, Oskar Morgenstern, Gerard Debreu, as well as John Nash have become famous. They dealt with utility theory and influenced modern economics. Ward Edwards is also known as the father of behavioural decision-making. His study focused on the process of how people make decisions or how they are able to improve the decision-making process. Another author, Herbert Simon, dealt with the rationality of the decision-making process. His study stated that people are not utility "maximisers", rather "satisfiers". They prefer decision-making based on a set of aspirational level criteria [3].
At present, the issue is being addressed by many researchers in various scientific fields. Liao and Wu [4] developed a comprehensive algorithm for multi-expert multiple-criteria decision-making problems, whereby quantitative and qualitative criteria in the form of benefits, costs or target types are considered. In another study, they focus on the integrated method for cognitive complex multi-expert multiple-criteria decision-making based on ELECTRE III with weighted Borda rule [5]. Fu, Chang and Yang [6] deal with multiple-criteria group decision-making based on group satisfaction. As mentioned above, there are a number of fields in which MCDM methods are used. Costache and Bui [7] used MCDM to identify areas prone to flash-flood phenomenon. Haddad, Sanders and Tewkesbury [8] evaluated four global market regions suitable for Boeing.
The definitions of MCDM vary and depend on the author. For the purposes of this contribution, the basic characteristics of multiple-criteria decision-making are considered to be as follows [9]: x Multiple criteria nature of the decision-making problem -the decision maker assesses selected alternatives on the basis of several criteria.
x Non-additivity of criteria -the selected criteria are expressed in different units of measure.
x Mixed set of criteria -this means that some criteria are quantitative and some qualitative; the decision-maker wants to maximize some criteria (yield criteria) and minimize others (cost criteria).
As was mentioned above, the main characteristic of MCDM is the multiple criteria nature thereof. MCDM problems can be solved only on the basis of quantitative criteria of the yield type. However, although non-additivity and/or a mixed set of criteria are not a condition, these characteristics occur very often. They can be defined as complementary features of MCDM [10].
The general procedure for solving MCDM problems can be divided into three phases, namely: 1. Creation of a decision matrix for the MCDM problem. Formula 1 shows the general decision matrix for a MCDM problem. As previously stated, there are several methods used within MCDM. The different methods that can be used for determining criteria weights and for evaluating alternative solutions are presented in Figures 1 and 2. The classification was developed by the authors on the basis of published research [11,12].  In this contribution, the authors focus on the application of the Analytic Hierarchy Process (hereinafter "AHP") on a model example of a MCDM problem. The AHP method is a pairwise comparison based methodology.

Methodology
AHP belongs to those method with a cardinal level of information on criteria preferences based on a pairwise comparison. In the 1970s, the principles of analytical thinking led to the development of a useful model for the quantitative solution of MCDM problems, also known as Analytical Hierarchy Process, by American professor Thomas Saaty. Saaty is considered to be the most important researcher dealing with the issue of MCDM. AHP consists of the decomposition of complex MCDM problems into partial components, which create the hierarchical structure of the problem. Subsequently, Saaty's quantitative pairwise comparison procedure is applied to each level of the hierarchical structure. Saaty and Kearns [13] distinguish three stages of the AHP method, namely: Hierarchy Priority

Consistency
The hierarchy of the AHP method usually consists of the goal, the alternatives for achieving the goal and the criteria that apply to the individual alternatives of the goal. It is a so-called three-level hierarchy, whereby the peak is the goal (the 1 st level), followed by the criteria (the 2 nd level), and lastly, the alternative solutions to the problem (the 3 rd level). In specific cases, it is possible to divide the criteria into sub-criteria, i.e. a so-called four-level hierarchy.
In the AHP, priorities are used for the pairwise comparison of criteria, as well as for the pairwise comparison of the alternatives. Professor Saaty set the following scale for priority description.
"1" Both compared elements (criteria/alternatives) contribute equally to achieving the goal. "3" Opinions and experiences gently prefer one element over another.
"5" Opinions and experiences strongly prefer one element over another.
"7" Opinions and experiences very strongly prefer one element over another.
"9" Opinions and experiences completely prefer one element over another.
The consistency can be defined as follow. If the element ݇ is ‫ݏ‬ -times more important than the element ݇ , and the element ݇ is ‫ݏ‬ -times more important than the element ݇ , it should be able to state that ݇ is ‫ݏ‬ = ‫ݏ‬ * ‫ݏ‬ -times more important than ݇ . In practice, it is very difficult to meet this condition [14].
The AHP method can be divided into the following steps: Define the problem to be solved Choosing a business partner was defined as the MCDM problem. A selected company operating in the construction industry is looking for a new business partner. Eight business partners are under consideration. For the purposes of this contribution, the company, as well as potential business partners, are identified as follows: the company choosing the business partner is marked as Company 1, potential business partners are marked as PBP 1 -8. In making its decision, Company 1 takes into account several financial indicators and non-financial criteria.
The selected and calculated financial indicators are presented in Table 1. The selected non-financial criteria were previous experience with a business partner, the length of time the business partner has been operating on the market, the costs of delivering materials, entries in the credit register or register of debtors.

EBT in three years
Earnings before taxes

Return on assets
Earnings after taxes Total assets

Earnings after taxes Sales
Indicator characterizing the imminent decline of the company

Equity Liabilities
Source: Authors.

Develop hierarchical framework
The three-level hierarchical framework was defined (see Figure 3). Note: 1 -4 describe the non-financial criteria in the following order: previous experience with a business partner (1); length of time the business partner has been operating on the market (2); costs of delivering materials (3); entries in the credit register or register of debtors (4); 5 -13 describe the financial criteria in the same order as in Table 1.

Construct pairwise comparison matrix
Preferences between individual criteria are shown in the so-called pairwise comparison matrix (Saaty's matrix). Saaty's matrix ܵ = ൛‫ݏ‬ ൟ is a matrix that contains elements ‫ݏ‬ . ܵ expresses the intensity of the preference between the objects ݇ and ݇ . It holds that if the object ݇ (the object in the row) is more important than ݇ (the object in the column), then ‫ݏ‬ ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, on the other hand ‫ݏ‬ = ଵ ௦ ೕ . The following formula is used to construct Saaty's matrix [15].
C CHOICE OF BUSINESS PARTNER P PBP P 1 1 P PBP P 2 2 P PBP P 3 3 P PBP P 4 4 P PBP P 5 5 P PBP P 6 6 P PBP P 7 7 PBP P 8 ‫ݏ‬ ; ‫ݏ/1‬ expresses the intensity of the preference between the objects ݇ and ݇ .
Saaty's scale is used to express the intensity of the preference between the objects.

Perform judgement for pairwise comparison
The result of this step determines the weights of individual criteria (if criteria are compared) or the utility of the individual alternatives with respect to a given criterion (if alternatives are compared). Saaty defined a procedure which is based on the calculation of the eigenvector of a matrix according to the following formula. Where: ܵ Saaty's matrix; ‫ݒ‬ eigenvector of Saaty's matrix; ߣ ௫ the largest eigenvalue of Saaty's matrix.
The values of the eigenvector must be standardized. This method of calculation is the most complex of all. Precisely because of its complexity, it is used only in specialized decision support programs. The geometric mean method is considered to be a simpler method that produces almost identical results, and for which specialized programs are not needed.

Check for consistency
Consistency was described within the definition of the three stages of the AHP method. The consistency ratio expresses the level of consistency [16]. Where: ‫ܴܥ‬ the consistency ratio; ݇ the number of objects -criteria/alternatives.
The random index was obtained by generating a large number of reciprocal random matrices (matrix elements came from a cardinal scale) and then averaging the eigenvalues of the matrices. A value of 1.56 was set for the number of 13 objects (the number of criteria). A value of 1.40 was set for the number of 8 objects (the number of alternatives -potential business partners) [16].
Steps 3 -6 are performed for all levels of the hierarchy. By multiplying the utility of the individual alternatives with respect to the given criteria by the weights of these criteria, the total utility of the given alternative is determined. The final step is the selection of the optimal alternative based on the value of total utility.

Results and Discussion
A multiple-criteria matrix containing a description of the MCDM problem was created.

Source: Authors.
For the purposes of this contribution, the company, as well as potential business partners, are identified as follows: the company choosing the business partner is marked as Company 1, potential business partners are marked as PBP 1 -8. In making its decision, Company 1 takes into account several financial indicators and non-financial criteria.
The selected and calculated financial indicators are presented in Table 1. The selected non-financial criteria were previous experience with a business partner, the length of time the business partner has been operating on the market, the costs of delivering materials, entries in the credit register or register of debtors.
The pairwise comparison matrix for the criteria was created. In the matrix, the criteria are numbered as in Figure 3. Saaty's matrix ܵ = ൛‫ݏ‬ ൟ is a matrix that contains elements ‫ݏ‬ . ܵ expresses the intensity of the preference between the objects -criteria. Formula 2 was used to construct Saaty's matrix. Saaty's scale was used to express the intensity of the preference between the objects.
According to Formula 4, the weights of the individual criteria were calculated. According to Formula 3, the largest eigenvalue of Saaty's matrix ߣ ௫ was calculated, the consistency of which was subsequently checked according to Formulas 5 and 6. The same procedure was chosen for the evaluation of the preferences between the individual alternatives in terms of criteria. By multiplying the utility of the individual alternatives with respect to the given criteria by the weights of these criteria, the total utility of the given alternative is determined. The final step is the selection of the optimal alternative based on the value of total utility. Based on the results of the AHP method, PBP1 was recommended to the company as the optimal business partner.
What follows is a summary of advantages and disadvantages of the AHP method: • The strength of the AHP method is that no special knowledge is required to apply the method, unlike some other methods of multiple-criteria decision-making. Quantitative pairwise comparisons made by the decision-maker are relatively simple and acceptable, as only two elements are always used, where the strength of the preference is relatively easy to determine. The intensity of importance is determined verbally by the decision-maker, who then assigns the appropriate numerical degree to the verbal preference according to Saaty's scale. A great benefit of the method, in the first phase of the decision-making process, is the creation of the hierarchical structure. This makes the problem clearer, making the individual parts of the problem more understandable, which contributes to a more comprehensive understanding of the problem as a whole. By creating and analysing the hierarchy, it is possible to see if all the important aspects of the problem have been included in the model. The AHP method can work with different types of criteria. Therefore, the method can combine criteria defined both verbally or numerically without any additional modifications. The method offers verification of the consistency of the decision-maker's judgments. The positive thing about consistency is that a certain degree of inconsistency is also accepted because it is not always possible to achieve full consistency. In the case of a large inconsistency of the matrix, it is possible to rethink the individual comparisons. This leads to better concentration, as well as the decision-maker being able to find new information in the reassessment, which may lead to them determining a different intensity of importance. The advantage of the AHP method is also the fact that it has a broad spectrum of applications. For example, the method can also be used for problems where there is a risk.
• The disadvantage of the analytical hierarchical process may be the number of pairwise comparisons. This is because it is necessary to compare the individual criteria and then compare the alternatives with respect to a specific criterion. If the decision-making problem contains a large number of criteria as well as alternatives, the decision-making process can be time consuming, and over time, the inconsistency of the matrices may begin to increase due to the loss of attention and lack of concentration of the subject. A minor drawback is also the basic scale (nine-point scale). When comparing the criteria, respectively alternatives, the decision-maker must determine the strength of the preference between the two criteria, respectively alternatives. With a large number, whether criteria or alternatives, a given scale may not be enough if it is necessary to sufficiently differentiate the individual elements. This shortcoming is followed by another weakness in the form of consistency, which is a relatively common problem when using the AHP method. If the nine-point scale is not enough, the full consistency of the matrix cannot be achieved either. On the other hand, the method also respects a certain degree of inconsistency, which can be considered an advantage. The analytical hierarchical process has a major drawback in that if the decision-maker uses the method to produce the results, but decides to add a new alternative to the model or remove the old alternative, the preferential order of other alternatives can change without changing the values of pairwise comparisons with regards to individual criteria or their preferences. Long analysis has shown that the order of the alternatives may change due to the lack of consistency of the intensities. Although it may sound nonsense to introduce or remove alternatives from the evaluated model, in practice such a need may arise.

Conclusion
The aim of the contribution was to use the Analytic Hierarchy Process to assess potential business partners and select the optimal alternative. MCDM is the most common type of decision-making process in business practice. AHP is an MCDM method originally developed by Prof. Thomas Saaty. The method derives ratio scales from paired comparisons. The ratio scales are derived from the principal eigenvectors and the consistency index is derived from the principal eigenvalue. The AHP method can be defined as one of the most commonly used MCDM methods in practice. The problem to be solved, as well as the steps of AHP method, are described in the methodology part of contribution. The problem to be solved was defined as follows: a selected company operating in the construction industry is looking for a new business partner. Eight business partners are under consideration. Using the AHP method, PBP1 was recommended to the company as the optimal business partner.
There are a variety of software applications available for the AHP method, of which Expert Choice was the first to be developed. Expert Choice was created in the 1980s by Thomas Saaty, the creator of the AHP method, in order to help organizations around the world make better, more transparent, and faster decisions. Expert Choice combines team collaboration tools and proven math techniques to enable the best decision to be made to achieve a goal.
Despite the shortcomings of the AHP method, it is one of the most popular and one of the most objective methods of multiple-criteria decision-making.