Analytic Hierarchy Process (AHP) - Step by Step

Analytic Hierarchy Process (AHP) proposed by [Saaty, 1980] is very popular and has been applied in wide variety of areas including planning, selecting a best alternative, resource allocation and resolving conflict. 

AHP Applied in

• Engineering
• Management
• Social sciences
• Strategic decisions like facility location
• Merger and acquisition
• Selection of projects in research and development
• Information technology outsourcing decisions
• Operational decisions like software selection
• Supplier selection
• Maintenance
• Logistics
• Engineering education

 

Various steps in AHP Process

1. Develop a model for the business
2. Derive priorities (weights) for the Criteria
3. Consistency check (Weights assigned correct or not ?)
4. Derive overall priorities (Model Synthesis) and Final decision

Step 1: Developing a model 
 Here, in this example the goal is to purchase a car, with Cost, Safety, Comfort and Mileage as the criteria and I have three alternatives for the car.

 

Figure: Car Selection problem

The Fundamental Scale [Saaty, 1980]

Intensity of Importance on an 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 of the highest possible order of affirmation
2,4,6,8	Intermediate values between the two adjacent judgements	When compromise is needed

Step 2: Derive the properties of weight criteria  

Table: Pair-wise Comparison matrix for the Car selection problem attributes

Now need to verify this pair-wise matrix is consistency or not, for this the following procedure need to be adapted.

 

The values obtained from the cost, safety, comfort and mileage are need to be estimated and their sum also calculated as shown above. 

Step 3: Consistency Check 

The consistency ration should be less than 0.1, then only the pair wise comparison satisfies. Then the weights will be takes as the values from the A2 matrix. 

Step 4: Derive overall priorities and Final decision  

Weight (Cost) = 0.059246; Weight (Safety) = 0.4828; Weight (Comfort) = 0.31385;

Weight (Mileage) = 0.1441, these are the finalized weights for the problem and the weights overall sum should be equal to one. 

Now, for the car selection based on the criteria a decision matrix need to be framed, as shown below.

Generally, the Cost of car is a non-beneficial attribute (lower the better) the remaining Safety, comfort and mileage are beneficial attributes (higher the better). Need to perform the calculations as shown here. 

The final synthesis of model is shown here, the weights and the normalized values later need to estimate the final value for ranking.

The weights need to multiply with the corresponding cost value and then all value need to sum to get the final value,

CAR 1  = (0.75*0.059246)+(1*0.4828)+(0.93*0.31385)+(0.81*0.1441) =  0.9368

CAR 2  = (1.0*0.059246)+(0.75*0.4828)+(1.00*0.31385)+(0.93*0.1441)  =  0.8690

CAR 3  = (0.86*0.059246)+(0.63*0.4828)+(0.87*0.31385)+(1.0*0.1441)  =  0.7686

Finally, CAR 1 is selected as the best alternative. (Based on the high value) 

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