This is a worked-through example showing the use of the analytic hierarchy process (AHP) in a practical decision situation.
See Analytic hierarchy process#Practical examples for context for this example.
Overview
AHP stands for analytic hierarchy process – a multi-criteria decision-making (MCDM) method. In AHP, values like price, weight, or area, or even subjective opinions such as feelings, preferences, or satisfaction, can be translated into measurable numeric relations. The core of AHP is the comparison of pairs instead of sorting (ranking), voting (e.g. assigning points) or the free assignment of priorities. Teachers and users of the AHP know that the best way to understand it is to work through an example. The example below shows how a broad range of considerations can be managed through the use of the analytic hierarchy process. The decision at hand requires a reasonably complex hierarchy to describe. It involves factors from the tangible and precisely measurable (purchase price, passenger capacity, cargo capacity), through the tangible but difficult to measure (maintenance costs, fuel costs, resale value) to the intangible and totally subjective (style). (https://bpmsg.com/ahp-introduction)
In the end, there is a clear decision whose development can be seen, traced, and understood by all concerned.
A practical example: choosing an automobile In an AHP hierarchy for a family buying a vehicle, the goal might be to choose the best car for the Jones family. The family might decide to consider cost, safety, style, and capacity as the criteria for making their decision. They might subdivide the cost criterion into purchase price, fuel costs, maintenance costs, and resale value. They might separate Capacity into cargo capacity and passenger capacity. The family, which for personal reasons always buys Hondas, might decide to consider as alternatives the Accord Sedan, Accord Hybrid Sedan, Pilot SUV, CR-V SUV, Element SUV, and Odyssey Minivan.
Constructing the hierarchy The Jones' hierarchy could be diagrammed as shown below:
As they build their hierarchy, the buyer should investigate the values or measurements of the different elements that make it up. If there are published safety ratings, for example, or manufacturer's specs for cargo capacity, they should be gathered as part of the process. This information will be needed later, when the criteria and alternatives are evaluated.
Note that the measurements for some criteria, such as purchase price, can be stated with absolute certainty. Others, such as resale value, must be estimated, so must be stated with somewhat less confidence. Still others, such as style, are really in the eye of the beholder and are hard to state quantitatively at all. The AHP can accommodate all these types of criteria, even when they are present in a single problem.
Also note that the structure of the vehicle-buying hierarchy might be different for other families (ones who don't limit themselves to Hondas, or who care nothing about style, or who drive less than a year, etc.). It would definitely be different for a 25-year-old playboy who doesn't care how much his cars cost, knows he will never wreck one, and is intensely interested in speed, handling, and the numerous aspects of style. (This book is the primary source for the sections in which it is cited.)
Pairwise comparing the criteria with respect to the goal To incorporate their judgments about the various elements in the hierarchy, decision makers compare the elements two by two. How they are compared will be shown later on. Right now, let's see which items are compared. Our example will begin with the four criteria in the second row of the hierarchy, though we could begin elsewhere if we wanted to. The criteria will be compared as to how important they are to the decision makers, with respect to the goal.
Each pair of items in this row will be compared; there are a total of six pairs (cost/safety, cost/style, cost/capacity, safety/style, safety/capacity, and style/capacity). You can use the diagram below to see these pairs more clearly.
In the next row, there is a group of four subcriteria under the cost criterion, and a group of two subcriteria under the capacity criterion.
In the Cost subgroup, each pair of subcriteria will be compared regarding their importance with respect to the Cost criterion. (As always, their importance is judged by the decision makers.) Once again, there are six pairs to compare (Purchase Price/Fuel Costs, Purchase Price/Maintenance Costs, Purchase Price/Resale Value, Fuel Costs/Maintenance Costs, Fuel Costs/Resale Value, and Maintenance Costs/Resale Value).
In the Capacity subgroup, there is only one pair of subcriteria. They are compared as to how important they are with respect to the Capacity criterion.
Things change a bit when we get to the alternatives row. Here, the cars in each group of alternatives are compared pair-by-pair with respect to the covering criterion of the group, which is the node directly above them in the hierarchy. What we are doing here is evaluating the models under consideration with respect to Purchase Price, then with respect to fuel costs, then maintenance costs, resale value, safety, style, cargo capacity, and passenger capacity. Because there are six cars in the group of alternatives, there will be fifteen comparisons for each of the eight covering criteria.
When the pairwise comparisons are as numerous as those in our example, specialized AHP software can help in making them quickly and efficiently. We will assume that the Jones family has access to such software, and that it allows the opinions of various family members to be combined into an overall opinion for the group.
The family's first pairwise comparison is cost vs. safety. They need to decide which of these is more important in choosing the best car for them all. This can be a difficult decision. On the one hand, "You can't put a price on safety. Nothing is more important than the life of a family member." But on the other hand, the family has a limited amount of money to spend, no member has ever had a major accident, and Hondas are known as very safe cars. In spite of the difficulty in comparing money to potential injury or death, the Jones family needs to determine its judgment about cost vs. safety in the car they are about to buy. They have to say which criterion is more important to them in reaching their goal, and how much more important it is (to them) than the other one. In making this judgment, they should remember that since the AHP is a flexible process, they can change their judgment later on.
You can imagine that there might be heated family discussion about cost vs. safety. It is the nature of the AHP to promote focused discussions about difficult aspects of the decisions to which it is applied. Such discussions encourage the communication of differences, which in turn encourages cooperation, compromise, and agreement among the members of the group.
Let's say that the family decides that in this case, cost is moderately more important to them than safety. The software requires them to express this judgment by entering a number. They can use this table to determine it; in this case they would enter a 3 in favor of cost:
Continuing our example, let's say they make the following judgments about all the comparisons of criteria, entering them into the software as numbers gotten from the table: as stated, cost is moderately important (3) over safety; also, cost is very strongly important (7) over style, and is moderately important (3) over capacity. Safety is extremely more important (9) than style, and of equal importance (1) to capacity. Capacity is very strongly important (7) over style.
We could show those judgments in a table like this:
The AHP software uses mathematical calculations to convert these judgments to priorities for each of the four criteria. The details of the calculations are beyond the scope of this article, but are readily available elsewhere. The software also calculates a consistency ratio that expresses the internal consistency of the judgments that have been entered.
In this case the judgments showed acceptable consistency, and the software used the family's inputs to assign these new priorities to the criteria:
You can duplicate this analysis at this online demonstration site; use the Line by Line Method by clicking its button, and don't forget to enter a negative number if the Criterion on the left is less important than the one on the right. If you are having trouble, click here for help. IMPORTANT: The demo site is designed for convenience, not accuracy. The priorities it returns may differ somewhat from those returned by rigorous AHP calculations. Nevertheless, it is useful in showing the mechanics of the pairwise comparison process. Once you are comfortable with the demo, you can experiment by entering your own judgments for the criteria in question. If your judgments are different from those of the Jones family, your priorities will possibly be quite different from theirs.Note that the demo site expresses priorities in percentages rather than decimal fractions as we do. It also uses different numbers to represent the verbal descriptions presented here. It's only a demo, but you should use our numbers, not theirs, and you should convert the percentages to decimal fractions. IMPORTANT: The demo site is designed for convenience, not accuracy. The priorities it returns may be significantly different from those returned by rigorous AHP calculations.
Look again at the above diagram and note that the Subcriteria still show their default priorities. This is because the decision makers haven't entered any judgments about them. So next on the family's agenda is to pairwise compare the four Subcriteria under Cost, then the two Subcriteria under Capacity. They will compare them following the same pattern as they did for the Criteria.
We could imagine the result of their comparisons yielding the priorities shown here:Their comparisons under Cost were Purchase Price 2 over Fuel Cost, 5 over Maintenance Cost, and 3 over Resale Value; Fuel Cost 2 over Maintenance Cost and 2 over Resale Value; Maintenance Cost -2 vs. Resale Value. Their comparisons under Capacity were Cargo Capacity -5 vs. Passenger Capacity.
At this point, all the comparisons for Criteria and Subcriteria have been made, and the AHP software has derived the local priorities for each group at each level. One more step can be made here. We know how much the priority of each Criterion contributes to the priority of the Goal. Since we also know how much the priority of each Subcriterion contributes to the priority of its parent, we (and the AHP software) can calculate the global priority of each Subcriterion. That will show us the priority of each Subcriterion with respect to the Goal. The global priorities throughout the hierarchy will add up to 1.000, like this:
Based on the judgments entered by the family, the AHP has derived the priorities for the factors against which each of the six cars will be compared. They are shown, from highest to lowest, in the table below. Notice that Cost and Capacity will not be evaluated directly, but that each of their Subcriteria will be evaluated on its own:
Factor Priority Purchase Price 0.246 Safety 0.237 Passenger Capacity 0.181 Fuel Costs 0.127 Resale Value 0.081 Maintenance Costs 0.050 Style 0.042 Cargo Capacity 0.036
The next step is to evaluate each of the cars with respect to these factors. In the technical language of AHP, we will pairwise compare the alternatives with respect to their covering criteria.
Pairwise comparing the Alternatives with respect to the Criteria The family can evaluate alternatives against their covering criteria in any order they choose. In this case, they choose the order of decreasing priority of the covering criteria. That means Purchase Price first.
Purchase price