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 We have to make decisions on a daily basis. There are many decisions we make every day which are not very important, but about some of them we think more thoroughly because they are more important than others. In the decision making process we have our own criteria. For some decisions comparative process is simple and it can be expressed in units of measurement. For example, price, weight, height and many other values can be expressed in units of measurement. What about the criteria that can not be expressed in such way? For example, quality, design, reliability, suitableness, pleasure etc. Moreover, what about those criteria depending on our own belief, taste or standards?

Have you ever been in a situation when A is much better than B, B is slightly better than C  

and C is better than A on the one hand, but on the other hand the situation is opposite? Or A is two times better then B, B is three times better than C, and A and C are equally good? If not, than this is not a web site for you.

Decision making is an evaluation process including alternatives which are all satisfying a certain set of criteria. The problem appears when one has to choose only one alternative which satisfies the entire set of our personal criteria.  

Did you know that there is one simple method that can help people make that choice and which takes into consideration things like your perception, intuition, rational and irrational, and the inconsistency of choosing among several options?

The method is called Analytic Hierarchy Process (or Analytical Hierarchy Process) - AHP

It is based on the comparison of pairs of alternative solutions during which all alternatives are compared to one another and you, as a decision maker, express intensity and the level of preference towards one alternative in relation to the other according to the criteria you find important. In the same way, you compare criteria according to your own preferences and their intensity.

AHP is a strong and flexible decision making technique which helps in setting priorities and reaching optimal decisions in situations when quantitative and qualitative aspects have already been taken into consideration. By reducing complex decision making to comparisons between pairs of alternatives and by synthesizing results AHP helps not only in decision making but leads to a rational decision. Created in a way to reflect the way people think, AHP was developed by Dr. Thomas Saaty in the 1970ies while he was a professor at the Wharton School of Business. The method is still one of the most appreciated and widely used methods. Numerous institutions and companies use it in decision making process. Why not you?

AHP - Analytic Hierarchy Process (or Analytical Hierarchy Process) is a mathematical method.

Compared to other decision making methods and techniques AHP enables you, as the decision maker, to compare the significance of each alternative in relation to another one individually and within a criterion you find relevant. This preference-based method shows the best option.  The value of the method is not only in finding the optimal result, but intermediate steps are clearly distinguishable, as well as the elements that contribute to the result the most.

Theoretical and mathematical description of the method

The first step is to determine a set of elements that consists of alternatives and criteria we wish to consider. The next step is to form the set into a hierarchical structure consisting of the mentioned criteria and alternatives.  

Upon defining that set, we begin developing the mathematical model by which we calculate priorities (weight, importance) of the elements on the same level in the hierarchical structure.

The entire process of the AHP method can be described in several steps:  

•    The development of the hierarchical model of the decision making problem by defining the goal, criteria and alternative solutions.  

•    On each level of the hierarchical model elements of the model are compared with one another in pairs, and the preferences of the decision maker are expressed with the use of the Saaty’s scale. In scientific literature that scale is more precisely described as a scale of five levels and four intermediate levels of verbally described intensities and corresponding numerical values for them on the scale from 1 to 9. The following table shows the values and their description used for the comparison of relevant values of the elements of the AHP model.


Intensity of Importance




Equal importance

Two activities contribute equally to the objective


Moderate importance

Experience and judgment slightly favor one activity over another


Strong importance

Experience and judgment strongly favor one activity over another


Very strong or demonstrated importance

An activity is favorad very strongly over another; its dominance demonstrated in practice


Extreme importance

The evidence favoring one activity over another is of the highest possible order of affirmation


Intermediate values


Let us describe it in more detail.

The first step would be to define a set in which we list the elements of the selection - the set of alternatives from which we wish to choose the best one for ourselves. Then we define the criteria we will use to compare those alternatives. It is clear that you, as the decision maker, determine all of this. That fact alone guarantees that the decision will be based on your preferences. 

For the explanation of the following steps we will use mathematical language.

If n is the number of criteria or alternatives whose weight (priority, importance) wi should be defined based on the assessment of the values of their ratios aij = wi/wj. If we form a matrix A from the ratio of their relevant importance aij, in case of consistent assessments equaling to aij= aik*akj , it will correspond to the equation Aw=nw

Matrix A has special characteristics (all her rows are proportional to the first row, all are positive and aij=1/aji is accurate resulting in only one of its eigenvalues being different from 0 and equal to n. If A matrix has inconsistent changes (in praxis that is always the case) the importance vector w can be calculated by solving the equation (A- Lamda I)w=0

The condition: SUMAwi=1 is true, where LAMDAmax is the biggest eigenvalue of A matrix.

Due to the characteristics of the matrix LAMDAmax ≥ n, and the subtraction LAMDAmax – n is used in the measuring of the assessment consistency. With a consistency index CI = (LAMDAmax -n)/(n-1) we calculate the consistency ratio CR=CI/RI where RI is a random index (consistency index for the n row matrixes of randomly generated comparisons in pairs – a table with calculated values applies.

Value of the random index RI




































If CR≤ 0,1000 is true for matrix A the assessments of the relative importance of the criteria ( alternative priorities) are considered as acceptable. To the contrary, the reasons why the assessment inconsistency is acceptably high must be investigated.

It will often happen that the consistency ratio exceeds 0,1000. That should only be taken into account as an indicator of the inconsistency level of your selection. Despite the inconsistency, you will get a suggestion of the best alternative. That is the value of this method.  You can always revise the chosen importance intensities and check which alternative is the best and to what extent compared to the following one. 
ahp examples - collection