### The Analytic Hierarchy Process

**The Analytic Hierarchy Process**

All the people are decision makers fundamentally. Everything they do consciously or unconsciously is the result of some decision. The information they gather is for helping them to understand the occurrences, in order to develop good judgements to make decisions about those occurrences. Not all the information is useful for improving the understanding and judgements. If people only make decisions intuitively, then they are inclined to believe that all kinds of information are useful and the larger the quantity, the better. But this is not true. There are a number of examples, which show that too much information is as bad as too little information. Knowing more does not guarantee that people understand better.

Decision making is one of the most important activities performed by the organizational management which is also referred to as the core of management. The decision making process can be defined as a set of activities which 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.

There are two sides to the decision-making process, namely (i) the content side of decision making which reflects the differences of individual decision making processes, and (ii) the formal-logical (procedural) side of decision making which 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 the 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 which the decision making theory deals with is the issue of multiple criteria decision making.

The basic characteristics of multiple criteria decision making are considered to be (i) multiple criteria nature of the decision making problem which means that the decision maker assesses selected alternatives on the basis of several criteria, (ii) non additivity of the criteria which means that the selected criteria are expressed in different units of measure, and (iii) mixed set of criteria which means that some criteria are quantitative and some are qualitative and the decision maker is to maximize some criteria (e.g. yield criteria) and minimize others (e.g. cost criteria).

Multi criteria decision making plays a critical role in many real life problems. It is not an exaggeration to argue that almost any industrial or organizational activity involves, in one way or the other, the evaluation of a set of alternatives in terms of a set of decision criteria. Very frequently these criteria are conflicting with each other. Even more frequently the pertinent data are very expensive to collect.

Applications of the multiple criteria decision making methods are especially popular in the last few decades. People face up with the decision making problems every day, starting from elementary problems to very complex situations. One way of making a correct decision is by application of multi-criteria optimization method. The method of AHP is one of the most used methods in decision making processes. It aims to quantify the relative priority of the given set according to the appropriate value scale. The decision is normally based on the perception of the individual who is supposed to make the final decision and to assess priorities, emphasizing the importance of consistency and correlation of the alternatives which has been compared in the complete decision making process.

Thomas Saaty developed in 1970s a strong and helpful technique for managing qualitative and quantitative multi criteria elements involving in decision making behaviour. Saaty developed this technique for solving the complex problems. This technique of Saaty is called ‘analytic hierarchy process’ (AHP) and it is based on a hierarchical structure. The principles of analytical thinking have led to the development of a useful model for the quantitative solution of problems involving multiple criteria decision making. This is also known as AHP. The AHP belongs to those methods which are with a cardinal level of information on criteria preferences based on a pair wise comparison.

The AHP is one of the most commonly exploited multi criteria decision making approach. When a decision has to be taken, it assesses a set of alternatives through pair wise comparisons by deriving priority scales. Broadly speaking, the AHP is a theory and methodology for relative measurement. This procedure uses an assortment of options in the decision and has capability to apply sensitivity analysis on the subsequent criteria and benchmarks. In addition, the procedure makes judgments and calculations easy because of paired comparisons. Further, a single decision maker is also considered to be perfectly rational and can precisely express the preferences on all pairs of independent alternatives and criteria using positive real numbers.

People have been concerned with the measurement of both the physical events as well as the psychological events since a long time. By physical it is meant the realm of what is known as the tangibles as it relates to some kind of objective reality outside the individual conducting the measurement. By contrast, the psychological is the realm of the intangibles as it relates to subjective ideas and beliefs of the people about themselves and the world of experience. The question is whether there is a coherent theory which can deal with both these worlds of reality without compromising either. The AHP is a method which can be used to establish measures in both the physical and social domains.

Although the utility of the AHP is not limited, it is safe to say that it has been especially advocated to be used with intangible criteria and alternatives, and thus used to solve multi criteria decision making problems, which are choice problems where alternatives are evaluated with respect to multiple criteria. Tangible properties of alternatives, for example the weight of different product bundles or the salary of different employees, can be measured without ambiguity and subjectivity. Hence, the machinery of the AHP becomes unnecessary. On the other hand, when the magnitude of some properties of alternatives, such as the agility of an operator towards safety or the aesthetic appeal of a building, cannot be easily grasped and measured the issue is in the domain of the intangibles, which is where the AHP gives its best.

The AHP is a general theory of measurement. It is used to derive ratio scales from both discrete and continuous paired comparisons. These comparisons can be taken from actual measurements or from a fundamental scale which reflects the relative strength of preferences and feelings. The AHP has a special concern with departure from consistency, its measurement and on dependence within and between the groups of elements of its structure. It has found its widest applications in multi criteria decision making, planning, and resource allocation, and in conflict resolution. In its general form the AHP is a nonlinear framework for carrying out both deductive and inductive thinking without use of the reasoning by taking several factors into account simultaneously and allowing for dependence and for feedback, and making numerical tradeoffs to arrive at a synthesis or conclusion.

The AHP has attracted the interest of several organizational management mainly due to the nice mathematical properties of the method and the fact that the required input data is rather easy to get. The AHP is a decision support technique which can be used to solve complex decision problems. It uses a multi-level hierarchical structure of objectives, criteria, sub-criteria, and alternatives. The relevant data is derived by using a set of pair wise comparisons. These comparisons are used to get the weights of importance of the decision criteria, and the relative performance measures of the alternatives in terms of each individual decision criterion. If the comparisons are not perfectly consistent, then it provides a mechanism for improving consistency.

Organizational management is to make decisions several times a day. The decision making process can be defined as an essential activity of the management. Decisions can be taken 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 techniques for of decision making can be considered as a relatively young discipline. Because of the decision making techniques, decision making is no longer an intuitive process. Most decision making situations are of a multiple criteria nature. For the multiple criteria decision making, several techniques can be used. AHP is one of the technique which helps in selecting the optimal alternative among different alternatives which are available.

In the present day scenario, where complexities are rapidly growing, making the best decisions has become an increasingly demanding task for the organizational managements. In recent years, this has gone arm-in-arm with the growth of what is now known as decision analytics methodologies. Namely, decision makers are more reluctant to make gut decisions based of feelings and hunches, and instead prefer to use analytic and quantitative techniques, and base and analyze their decisions on a solid ground. Several methods stemming from applied mathematics and operational research have proved useful to help decision makers making informed decisions, and among these methods there are also those methods which need subjective judgments from a decision maker. It is in this context that the AHP has becomes a useful tool for analyzing decisions.

The AHP is a very flexible and powerful technique since the scores, and hence the final ranking, are obtained on the basis of the pair wise relative evaluations of both the criteria and the options provided by the user. The computations made by the AHP are always guided by the experience of the decision maker. Hence, The AHP can be considered as a technique which is able to translate the evaluations (both qualitative and quantitative) made by the decision maker into a multi criteria ranking. In addition, the AHP is simple since there is no need of building a complex expert system with the knowledge of the decision maker embedded in it. On the other hand, the AHP can need large number of evaluations by the user, especially for problems with many criteria and options. Although every single evaluation is very simple, since it only needs the decision maker to express how two options or criteria compare to each other, the load of the evaluation task can become unreasonable. In fact the number of pair wise comparisons grows quadratically with the number of criteria and options.

In relative measurement people are not interested in the exact measurement of some quantities, but rather on the proportions between them. The AHP can be applied to a multitude of decision making problems involving a finite number of alternatives. Formally, in this setting, in a decision process there is one goal and a finite set of alternatives, X = (x1, . . . , and xn), from which the decision maker, is normally asked to select the best one. Explaining the AHP is easier to show with an example than to explain with words. For explaining this, an example of a pair of objects with weight ratio of 2:1 is considered here. In classical measurement, people can be interested in knowing the exact weights of the objects and the pair of measurements is not correct unless the weight of the first object is 2 kilograms and the weight of the second object is 1 kilogram. On the other hand, in relative measurement the interest is confined to the knowledge of how much heavier each object is compared to another. Hence, the pair of measurements (2:1) is correct as long as the weight of the first object is double the weight of the second. It follows that, in this example, if the relative measurement theory is used then the pairs of measurements 2/5 and 1/5, 4 and 2, 6 and 3 are also correct for the two objects.

Relative measurement theory suits particularly well with the problems where the best alternative is to be chosen. In fact, in several cases people are not really interested in the precise measurements of the alternatives but it is sufficient to know their relative measurements for knowing which alternative is the best. Further, when attributes of alternatives are intangible, it is difficult to devise a measurement scale and using relative measurements simplifies the analysis. The ultimate scope of the AHP is that of using pair wise comparisons between alternatives as inputs, to produce a rating of alternatives, which is compatible with the theory of relative measurement.

When using the AHP, the decision making process is driven by the adoption of criteria which have been easily identified and selected. However, the AHP technique has certain limitations. Among them, an important limitation is the independence of the sub-criteria, which are considered unrelated to each other. Another important limitation is its static nature which does not make it suitable for medium / long-term decisions in dynamic environments. Because of its static nature, AHP provides the optimal result at a given moment for a settled situation. Of course, when a dynamic situation is dealt with, priorities obtained at a specific time ‘t’ can change at time ‘t + dt’. This problem can be addressed by using the dynamic judgments method. In this case, judgments changing over time are represented as functions of time, but this procedure needs the prediction of future trends. The combination of AHP technique with other mathematical models allows enhancing its performance by overcoming its above mentioned limitations. The adaptation of static AHP to an environment which changes over time is normally called dynamic AHP.

Some of the industrial engineering applications of the AHP include its use in integrated manufacturing, in the evaluation of technology investment decisions, in flexible manufacturing systems, layout design, and also in other engineering issues. As an example, a case, in which there is a need to upgrade the computer system of a computer integrated manufacturing (CIM) facility, is considered here. There are a number of different configurations available to choose from. The different systems are the alternatives. The decision is also to consider issues such as cost, performance characteristics (i.e. CPU speed, memory capacity, and RAM, etc.), availability of software, maintenance, expendability, etc. These can be some of the decision criteria for this case. In this case, the interest is in determining the best alternative (i.e. computer system). In some other situations, however, people can be interested in determining the relative importance of all the alternatives under consideration. For example, if people are interested in funding a set of competing projects (which now are the alternatives), then the relative importance of these projects is needed so the budget can be distributed proportionally to their relative importance.

In using the AHP to model a problem one needs a hierarchic or a network structure to represent that problem and pair wise comparisons to establish relations within the structure. In the discrete case, these comparisons lead to dominance matrices and in the continuous case to kernels of Fredholm operators, from which ratio scales are derived in the form of principal eigenvectors, or eigenfunctions, as the case may be. These matrices, or kernels, are positive and reciprocal, e.g. aij = 1/aji. In particular, special efforts are to be made to characterize these matrices.

The structure of the typical decision problem consists of a number, say M, of alternatives and a number, say N, of decision criteria. Each alternative can be evaluated in terms of the decision criteria and the relative importance (or weight) of each criterion can be estimated as well. Let aij i=1,2,3,4,…..,M, and N=1,2,3,4,…..,N) denote the ‘performance value’ of the ‘i’th alternative (i.e., Ai) in terms of the ‘j’th criterion (i.e., Cj). Also denote as Wj the weight of the criterion Cj. Then, the core of the typical multi criteria decision making problem can be represented by the decision matrix given in Fig 1.

**Fig 1 Structure of the typical decision problem**

Given the above decision matrix, the decision problem is how to determine the best alternative. A slightly different problem is to determine the relative significance of the M alternatives when they are examined in terms of the N decision criteria combined.

In a simple multiple criteria decision making situation, all the criteria are expressed in terms of the same unit (e.g. monetary unit). However, in several real live multiple criteria decision making problems different criteria can be expressed in different dimensions. Examples of such dimensions include monetary unit figures, weight, time, quality impact, and environmental impact, etc. It is this issue of multiple dimensions which makes the typical multiple criteria decision making problem to be a complex one and the AHP, or its variants, can offer a great assistance in solving these types of the problems.

The AHP and its use of pair wise comparisons have encouraged the creation of several other decision making methods. Besides its wide acceptance, it also created some considerable criticism, both for theoretical as well as for practical reasons. Since the early days, it became apparent that there are some problems with the way pair wise comparisons are used and the way the AHP evaluates alternatives. First, Belton and Gear observed that the AHP can reverse the ranking of the alternatives when an alternative identical to one of the already existing alternatives is introduced. In order to overcome this deficiency, Belton and Gear proposed that each column of the AHP decision matrix to be divided by the maximum entry of that column. Thus, they introduced a variant of the original AHP, called the revised AHP. Later, Saaty accepted the previous variant of the AHP and now it is called the ‘ideal mode’ AHP. Besides the revised-AHP, some other variants of the original AHP have been introduced. However, the AHP (in the original or in the ideal mode) is the most widely accepted method and is considered by many as the most reliable multiple criteria decision making method.

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 technique, in order to help organizations around the world make better, more transparent, and faster decisions. ‘Expert Choice’ combines team collaboration techniques and proven mathematical techniques to enable the best decision to be made for achieving a goal.

**Procedure for AHP technique**

AHP is one of the most inclusive techniques and is considered to make decisions with multiple criteria since this technique gives to formulate the problem as a hierarchical structure. It believes a mixture of quantitative and qualitative criteria as well. In addition, AHP demonstrates the compatibility and incompatibility decisions which are the recompense of multi criteria decision making.

For the successful application the AHP technique, four axioms namely (i) reciprocity axiom, (ii) axiom for homogeneity, (iii) axiom for dependence and (iv) axiom of expectation are required to be met. These axioms are explained below.

**Reciprocity axiom** – This axiom says that the comparison matrices which are constructed are formed of paired reciprocal comparisons. – If element A is n times more important than element B, then element B is 1/n times more important element of A. It is a simple but powerful relationship which is the basis of the AHP.

**Axiom for homogeneity** – Homogeneity is necessary for meaningful comparisons, as the mind cannot compare widely disparate elements. The comparison makes sense only if the elements are equally comparable. One element cannot be much better than the other. When the disparity is great, elements are to be placed in separate clusters of comparable size, or in different levels altogether.

**Axiom for dependence** – The process of relating elements (e.g. alternatives) in one level of the hierarchy according to the elements of the next higher level (e.g. criteria) expresses the dependence of the lower elements on the higher so that comparisons can be made between them. The elements in a level can also depend on one another with respect to a property in another level. Input-output of an industry is an example of the idea of inter dependence. Axiom of dependence allows comparing the group of elements from one level with elements of the higher level. The comparison in the lower level is depended on the elements from the higher level.

**Axiom of expectation** – Any change in the structure of the hierarchy needs a new calculation of the priorities in the new hierarchy. This axiom is merely the statement that thoughtful individuals who have reasons for their beliefs are to make sure that their ideas are adequately represented in the AHP model. All alternatives, criteria and expectations (explicit and implicit) can be and are to be represented in the hierarchy. This axiom does not assume rationality. People are known at times to harbour irrational expectations and such expectations can be accommodated.

For making a decision in an organised way to generate priorities people need to decompose the decision into the four steps namely (i) defining of the problem and determining the kind of knowledge needed, (ii) structuring of the decision hierarchy from the top with the objective of the decision, then the objectives from a broad perspective, through the intermediate levels (criteria on which subsequent elements depend) to the lowest level which normally is a set of the alternatives (Fig 2), (iii) constructing a set of pair wise comparison matrices with each element in an upper level is used to compare the elements in the level immediately below with respect to it (Fig 1), and (iv) using the priorities obtained from the comparisons to weigh the priorities in the level immediately below. This is to be done for every element. Then for each element in the level below its weighed values are added and its overall or global priority is obtained. This process of weighing and adding is continued until the final priorities of the alternatives are obtained in the bottom most level.

**Fig 2 Example of hierarchy in the AHP**

In Fig 2, Level 1 is the objective of the analysis. Level 2 is multi criteria which consist of several factors. One can also add several other levels of sub criteria and sub-sub criteria but it is not used in the figure. The last level (Level 3 in figure above) is the alternative choices. The lines between levels indicate relationship between factors, choices and objective. In Level 2, the figure has one comparison matrix corresponds to pair wise comparisons between 3 criteria with respect to the objective.

For making comparisons, people need a scale of numbers which indicates how many times more important or dominant one element is over another element with respect to the criterion or property with respect to which they are compared. Tab 1 shows the fundamental scale of absolute numbers for priority description. This scale is also known as Saaty’s scale.

Tab 1 The fundamental scale of absolute numbers | ||

Intensity of importance on absolute scale | Definition | Explanation |

1 | Equal importance | Two activities contribute equally to the objective |

3 | Moderate importance of one over another | Experience and judgment strongly favour one activity over another |

5 | Essential or strong importance | Experience and judgment strongly favour one activity over another |

7 | Very strong importance | An activity is strongly favoured and its dominance demonstrated in practice |

9 | Extreme importance | The evidence favouring one activity over another is the highest possible order of affirmation |

2,4,6, and 8 | Intermediate values between the two adjacent judgements | When compromise is needed |

Reciprocals | If activity ‘i’ has one of the above numbers assigned to It when compared with activity ‘j’, then ‘j’ has the reciprocal value when compared with ‘i’ | |

Ratios arising from the scale | If consistency were to be forced by obtaining ‘n’ numerical values to span the matrix | |

1.1–1.9 | If the activities are very close | Can be difficult to assign the best value but when compared with other contrasting activities the size of the small numbers would not be too noticeable, yet they can still indicate the relative importance of the activities |

AHP consists of the decomposition of complex multiple criteria decision making problems into partial components, which create the hierarchical structure of the problem. Then, quantitative pair wise comparison procedure of Saaty is applied to each level of the hierarchical structure. There are three stages of the AHP technique, namely (i) hierarchy, (ii) priority, and (iii) consistency.

The hierarchy serves the purpose of representing the structure of the problem and is compounded by (i) the objective, (ii) the set of criteria, (iii) the set of alternatives, and (iv) a relation of the objective, the criteria and the alternatives. The hierarchy of the AHP method normally consists of the objective, the criteria which apply to the individual alternatives of the objective, and the alternatives for achieving the objective. It is a so-called three-level hierarchy, whereby the peak is the objective (the first level), followed by the criteria (the second level), and lastly, the alternative solutions to the problem (the third level). In the 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 pair wise comparison of criteria, as well as for the pair wise comparison of the alternatives.

The procedure for AHP technique is based on a three level hierarchical structure (Fig 2). This procedure occupies an assortment of options in the decision and is capable to apply sensitivity analysis on the subsequent criteria and benchmarks. In addition, it makes judgments and calculations easy because of paired comparisons. The first step is to create a hierarchy of the problem. The second step is to give a nominal value to each level of the hierarchy and create a matrix of pair wise comparison judgment. During the first step, the issue and objective of the decision making is brought hierarchically into the scene of the related decision elements. Decision making elements are decision indicators and decision choices. The objective is required to reflect the understudy problem.

In second step and in order to conduct pair comparison, a questionnaire is to be designed and distributed among the respondents (can be managers, experts, and users etc.) to collect their opinion. It is noteworthy that each decision maker is to enter the desired amount for each member and then individual judgments (of each respondents) have been converted into group judgments (for each one of the pair comparison) using the geometrical average. The scale ranges from one to nine where one implies that the two elements are the same or are equally important. On the other hand, number nine implies that one element is extremely more important than the other one in a pair wise matrix. The pair wise scale and the importance value attributed to each number are given in Tab 1.

When many pair wise comparisons are performed, some inconsistencies can typically arise. Hence, in the next step, in order to validate the results of the AHP, the consistency ratio is calculated. Consistency is described within the definition of the three stages of the AHP technique. The consistency ratio expresses the level of consistency. Consistency ratio (CR) is calculated using the formula, CR = CI/RI in which the CI is the consistency index and RI is the random index. The random index is obtained by generating a large number of reciprocal random matrices (matrix elements come from a cardinal scale) and then averaging the eigenvalues of the matrices. It is to be noted that consistency ratio lower than 0.10 verifies that the results of comparison are acceptable. However, sometimes the judgements which are at the limit of consistency though CR > 0.1 (but not too much more) have to be accepted. But a CR with high values means that the pair wise judgements are just about random and are completely untrustworthy.

**Advantages and disadvantages**

The AHP has revolutionized the way how the complex decision problems are resolved. It has been by far the most studied and applied multi criteria decision making technique. It has been applied worldwide to help decision makers in every conceivable decision context across both the public sector and private sector organizations, with literally thousands of reported applications.

The strength of the AHP technique is that no special knowledge is needed to apply the technique, unlike some other methods of multiple -criteria decision making. Quantitative pair wise 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 the Saaty’s scale.

A big benefit of the technique, 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 analyzing the hierarchy, it is possible to see if all the important aspects of the problem have been included.

The AHP technique can work with different types of criteria. Hence, the technique can combine criteria defined both verbally and numerically without any additional modifications. The technique 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 makers being able to find new information in the reassessment, which can lead to their determining a different intensity of importance.

The advantage of the AHP technique is also the fact that it has a broad spectrum of applications. For example, the technique can also be used for problems where there is a risk.

The disadvantage of the AHP can be the number of pair wise comparisons. This is since 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 can begin to increase due to the loss of attention and lack of concentration of the subject.

A minor disadvantage is also the basic scale (nine point scale). When comparing the criteria, respectively alternatives, the decision maker is required to determine the strength of the preference between the two criteria and respective alternatives. With a large number, whether criteria or alternatives, a given scale cannot be enough if it is necessary to sufficiently differentiate the individual elements.

Another shortcoming is in the form of consistency, which is a relatively common problem when using the AHP technique. 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 AHP has a major disadvantage in that if the decision maker uses the technique 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 pair wise comparisons with regards to individual criteria or their preferences. Long analysis has shown that the order of the alternatives can change due to the lack of consistency of the intensities. Although it can sound nonsense to introduce or remove alternatives from the evaluated model, in practice such a need can arise.

Despite the above shortcomings of the AHP technique, it is one of the most popular and one of the most objective methods of multiple criteria decision-making.

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