### Theory behind the AHP method (Analytic Hierarchy Process)

Analytic Hierarchy Process can be useful as a decision-support method in project management in instances where a few options (be it requirements, risks or other "alternatives" need to be prioritized or selected. Let's take a look at some of the key theory concepts behind it.

There are two fundamental principles used in the general decision-making theory: that of deduction and that of induction (sometimes referred to as a system approach).  Deduction arrives at particulars from the general by applying logic. In other words it goes top-down, narrowing down more general truths into detailed conclusions.
The system approach on the other hand, is based on the premise that particulars are not as important. You are going bottom-up, as if trying to seize general truth.
The essence of the Analytic Hierarchy Process is based on employing both approaches. It is a method that first decomposes a complex problem in single components, and the components (sometimes referred to as variables), are put in a form of a hierarchy. They are then given numerical values. Each variable gets a value according to its importance in relation to other variables. (That is depends on whether it is quantitative or qualitative.) What follows is a synthesis of the values which will determine what weight each variable has in influencing the overall evaluation of the problem. Eventually all evaluated outcomes (alternatives/variants) will receive its total numerical value to form a ranking.
Many decision-making situations entail both physical as well as psychological aspect. Physical aspect can be regarded as objective, as it is from “tangible” realm, something that can be taken hold of or at least measured. Price as a criterion, for example can be quantified by money units, size or distance is quantifiable by units of measurement etc. On the other hand, the psychological aspect of the decision-making problem is more tricky. It is “intengible” in essence and there is no scale or range that might sufficiently, universally and unambiguously express it. They are often a product of subjective ideas, gut-feeling  or assumptions of an individual, a group or the whole society. Let’s take design qualities of a product as an example. The AHP deals with both of those aspects and is able to incorporate them as equal inputs of a unique decision-making support system.
Breaking down a seemingly complex problem into a clear hierarchy and only then focusing on different aspects of the decision, substantially expands possibilities of those who make the decision.
Analytic Hierarchy Process has been developed in Pittsburgh, USA in 1984, originally by Dr. Thomas L. Saaty - internationally recognized scholar and innovator of the decision-making theory.  It has since become one of the most successful and widely used decision support systems of today. It has grown into a comprehensive software tool used in collaborative, teamwork corporate decision-making.

Stages of AHP Using Expert Choice Software

As mentioned above, the largest contribution of AHP is its support in decision-making process by employing both subjective and objective factors when it comes to evaluating different alternatives of outcomes. Unlike in other methods, both quantitative data (that are clearly represented by numbers) and qualitative data (that often regarded as subjective) can be fed into the process. They are then assessed depending on importance that the decision maker has given to them but also in what layer of hierarchy they were placed. In a few steps, apparently advanced and complex decision-making problems can be solved in a relatively simple way.
The whole process can be divided into 5 stages:

1. Breaking down the problem into a hierarchy (analysis)
2. Evaluating criteria and decision alternatives on different levels of the hierarchy (setting priorities)
3. Measuring consistency of evaluation (finding consistency ratio)
4. Synthesis - generating overall weight for each evaluated decision alternative and their ranking
5. Sensitivity analysis

Breaking Down the Problem Into a Hierarchy (Analysis)

Decomposing the problem into a hierarchy is the first basic step of the Analytic Hierarchy Process. A hierarchy means a system of several levels, each including a finite number of elements. There is a mutual relationship between each two vertically-neighbouring levels. The higher the level is, the more general role it plays. Elements placed higher in the hierarchy controlled and managed by elements immediately underneath them. The element at the very top of the hierarchy is always the Goal of the decision-making process. The Goal has a weight that equals 1. 1 is than divided among the elements of the second level of the hierarchy, evaluation of elements in the second level of the hierarchy are then “dissolved” into the third level etc.
Hierarchy chosen depends on the character of the decision-making problem. There are a few types of the hierarchy:

• Goal - Criteria - Alternatives
• Goal - Criteria - Subcriteria - Alternatives
• Goal - Experts - Criteria - Alternatives
• Goal - Criteria - Intensity Levels - Multiple Alternatives
In most decision-making problems we will make do with the first mentioned type of hierarchy, that is Goal - Criteria - Alternatives but it can always be extended.

Evaluating Criteria and Decision Alternatives on Hierarchy Levels (Setting Priorities)

To set priorities for individual elements of the hierarchy, one must first know whether the data has a quantitative or qualitative nature. If quantitative, they may be ruled either by maximizing or minimizing. The maximizing rule will regard the highest value to be the best, while minimizing rule will regard the lowest value to be the best.

Evaluating by Quantitative Criteria

Price is a kind of a quantitative criterion that typically has a minimizing ruling when considered from the consumer’s point of view. The less it costs, the better. Let’s evaluate 4 products by price: A, B, C, and D, with the goal of assigning each one a numeric weight. The prices are:

Product            Price

A                $190 B$ 230
C                $320 D$ 290

Because the criterion is minimizing (lower values are considered better), the first step to calculate the weights is converting the values by using the following formula to get a coefficient kj:

### $$$k_{j}=\frac{1}{Price}*100$$$

This coefficient actually converts a minimizing criterion into a maximizing one (the higher the better because with price it's the other way round):
 Product kj A 0.526 B 0.436 C 0.313 D 0.349

The resulting quantitative pj weights are calculated by a normalization formula:

$$$p_{j}=\frac{k_{j}}{\sum_k_{j}}$$$
 Product kj pj A 0.526 0.324 B 0.436 0.268 C 0.313 0.193 D 0.349 0.215

Evaluating by Qualitative Criteria - Pairwise Comparisons

Pairwise comparisons belong to one of the most basic concepts of Analytical Hierarchy  Process. It is for evaluation of the criteria that are not clearly quantifiable but in the overall decision-making process play a crucial part. It is very difficult to assign weights to qualitative assessments by guessing and intuition, the AHP derives the information from comparing all the alternatives among themselves on every level of the hierarchy. In other words it slices the overall information into pairs of information. It is then used as a base for calculating numerical weights of each alternative.
Each pair of criteria being compared is assessed by 9-degree numerical scale that was developed specifically for this purpose:

 Numeric Scale Description Explanation 1 Equal Both elements having same importance 3 Moderate Moderate importance of one over another 5 Strong Strong/essential importance of one over another 7 Very strong Very strong or demonstrated importance 9 Extreme Extreme importance of one over another

Besides the ones mentioned, there are also half-grades of 2, 4, 6, 8.

The relationship between elements in pairwise comparisons is called ‘importance’ of one over another but it can just as well be referred to as ‘preference’ or ‘likelihood’ of their occurrence. It always depends on the type of the problem being solved.

After pairwise comparisons of k-number of elements, pairwise comparison matrix (also known as Saaty’s matrix) is constructed. It is basically a reciprocal matrix consisting of k2 elements with 1’s on its diagonal and inverted values on each side. Typical pairwise comparison of 3 evaluated "Options" by a qualitative criterion can look as follows:
 Option A Option B Option C Option A 1 2 8 Option B ½ 1 6 Option C ⅛ ⅙ 1

For better clarity usually only the values in bold are shown. The number of those values can be calculated by the following formula:

### $\frac{n * (n-1)}{2}$

While AHP appears to be rather straightforward at first sight, the background mathwork needed for calculating numerical weights out of pairwise comparisons is not as straightforward. Eigenvalues and eigenvectors are involved and computer software such as Expert Choice does the hard work.
There is however an approximation method - an algorithm that can calculate rough weights in three steps without using a computer.

Algorithm for Calculating Approximate Weights

Step 1: Add up values in each column of pairwise comparison matrix

 Option A Option B Option C Option A 1 2 8 Option B ½ 1 6 Option C ⅛ ⅙ 1 Total 13/8 19/6 15

Step 2: Each item is divided by the total of its column thus getting a normalized matrix
 Option A Option B Option C Option A 8/13 12/19 8/15 Option B 4/13 6/19 6/15 Option C 1/13 1/19 1/15 Total 1 1 1

Step 3: Total of each row will be divided by the number of items in the row
 Option A Option B Option C Option A (8/13 + 12/19 + 8/15) / 3 Option B (4/13 + 6/19 + 6/15) / 3 Option C (1/13 + 1/19 + (1/15) / 3 Total 13/8 19/6 15

The calculated means will then serve as the approximate weights of each alternative (called Options here). From the numbers below we can see that Option A "won".

Approximate weight

Option A        0.593

Option B        0.341

Option C        0.066

Total            1

Measuring Consistency of the Evaluation

Consistency of the measurements is way of expressing a certain ‘compactness’ of the preferences created during pairwise comparisons. It shows to what extent the data fed into the computer is logically cohesive. If for example, Alternative 1 is twice as important as Alternative 2 and Alternative 2 is three times as important as Alternative 3, then Alternative 1 must be 6 times (2 x3) as important as Alternative 3. This would be a case of a perfect consistency, in other words, the consistency coefficient would equal 0.
Inconsistency of judgements is at the background of human thinking. Humans don’t just use logic when drawing conclusions but rather stick to intuition, emotions, experience that all influence their attitudes and swing their decisions. If someone prefers apples to oranges and at the same time the person likes oranges better than bananas, shouldn’t it automatically be assumed that apples will be preferred over bananas? And yet the same people will still go for bananas rather than apples because there are other things to consider, like say, time of the day, season, etc. all eventually causing that they illogically, or ‘inconsistently’ choose this alternative and not the other.

In practical applications perfect consistency is rare because new and new information is constantly added in evaluation and it changes the previous relationships. Therefore pairwise comparisons permit a certain amount of inconsistency of preferences. The AHP works with the so called Consistency Ratio, with the rule of thumb that, if the inconsistency be more than 10 per cent, the evaluation should be revisited.

High inconsistency (bad consistency ratio) implies one of the following problems:
• Ill logic in pairwise comparisons
• Errors/typos during data input

Expert Choice calculates consistency ratios automatically with each pairwise comparison. In case that the inconsistency is too high, it even has a feature that discloses the elements where inconsistency is highest. It could be repeated until inconsistency goes back to an acceptable level.
For illustration let’s look at how approximate inconsistency can be calculated without using a computer. The previous example is used:
 Option A Option B Option C Option A 1 2 8 Option B ½ 1 6 Option C ⅛ ⅙ 1

If Option A is preferred twice over Option B and Option B is preferred 6 times over Option C, then Option A should be preferred 12 times over Option C. That would be an ideal situation or perfectly consistent evaluation. However, because Option A is only 8 times more preferred than Option C, consistency ratio needs to be found.
Working out the approximate consistency ratio is dealt with in the next chapter.

Algorithm of Calculating Approximate Consistency Ratio

Step 1: Each column element of the original pairwise comparison matrix is multiplied by the resulting weight of their alternative and then rows are summarised:
 Option A Option B Option C (0.593) (0.341) (0.066) Option A 1 2 8 Option B 0.5 1 6 Option C 0.125 0.167 1

will become

 Option A Option B Option C Option A 0.593 0.682 0.528 Option B 0.297 0.341 0.396 Option C 0.074 0.057 0.066

Resulting totals then are:

Option A        1.803

Option B        1.034

Option C         0.197

Step 2: Each resulting total is divided by its weight:

Option A        1.803 / 0.593 = 3.04

Option B        1.034 / 0.341 = 3.032

Option C         0.197 / 0.066 = 2.985

Step 3: Mean is calculated:
$$$L_{max}=\frac{3.04 + 3.032 + 2.985}{3}=3.019$$$

Step 4: Consistency Index $$$CI=\frac{L_{max}-n}{n-1}$$$ is then calculated, where is number of elements being compared:

## $$$CI=\frac{3.019-3}{2}=0.0095$$$

Step 5: Consistency ratio is calculated using the so called Random Index - which is an average consistency index of a randomly generated n x n - size matrix:

## $$$CR=\frac{CI}{RI}$$$

Random Index (RI) doesn't need to be calculated as it is already provided in the following chart:

n        RI
2        0
3        0.58
4        0.9
5        1.12
6        1.24
7        1.32
8        1.41

For n = 3 RI is 0.58 and the Consistency Ratio is:

## $$$CR=\frac{0.0095}{0.58}=0.016$$$

The approximate Consistency Ratio of the three evaluated Options equals 0.016 and meets the previously stated expectation of CR < 0.1.

The evaluations are therefore considered to be sufficiently consistent.

Practical Application of Expert Choice

Analytical Hierarchy Process and Expert Choice has a wide application in real-life business-related situations. Generally, it has large usage in marketing in product comparisons but it can just as well be used to support decision-making or in planning, investing, conflict resolution, forecasting or risk management to name a few.
IBM used Expert Choice when applying for Malcolm Baldridge National Quality Award. General Motors used it in its design projects when evaluating prototypes of its new products. Xerox used it for the portfolio management, evaluating new technologies and as a support tool in marketing decisions. It has been used by government in rating of buildings by historic significance, or in assessing the condition of highways so the engineers could determine the optimum scope of the project and justify the budget to lawmakers.
In project management it can  be used in the scope management knowledge area, in estimating cost of work packages through control account level, then aggregating them into the overall project cost estimates.
It has a large usage in Human Resources in Acquire Team process to evaluate employees or potential team members from large number of applicants against the set of defined criteria. They are quickly rated and scored to select the ones that best suit the criteria.
Portfolio management is another ara of application where it helps decision-makers rate the business value of their potential projects. AOL project portfolio management can be an example.
AHP can also be used in risk management, identification and prioritization where both subjective inputs (qualitative risk analysis) and quantitative data (quantitative risk analysis) need to be assessed.