User:Louis F. Sander/Sandbox

Sections

• Lead/Introduction
• Where it is used
• How it works
• Criticisms (keep it short and objective)

Notes

Aspects of AHP

• Math
• Psychological
• Computerized now, with gadgets
• Disciplined look at a decision (focus on objectives, alternatives, more than voting)
• Handles rational, intuitive, irrational, all at the same time
• Lets you compare the alternatives and fiddle with them, vs. just giving the best one

Miscellaneous

Selecting a spouse is important and has long-term consequences.

This is like logarithms, the LaPlace transform, etc., in that it replaces something hard to deal with with something much easier to deal with.

Deleted Material for Reinsertion

and has been successfully applied to many complex planning, resource allocation, and priority setting problems

Good Stuff

Problems with high stakes, involving human perceptions and judgements, and whose resolutions have long-term repercussions, call for a rational approach to their solution. (Bhushan promo)

You might also want to read the paper "The Analytic Hierarchy Process - An Exposition," E.J. Forman and S. I. Gass, Operations Research, 49, 4, July-August, 2001, pp. 469-486. (Saul Gass, U of MD)

Drafts

Pairwise Comparisons

In AHP, we rank a group of items by comparing them to each other in pairs. If the group has more than a very few items, this can be very much easier than trying to rank all the items at once.

As we consider the pairs of items, we express the importance of one vs. the other by assigning it a weight from -9 to +9. To facilitate the assignment of weights, we can use a verbal scale ranging from "Much Less Important" to "Equally Important," to "Much More Important."

When we have finished all the comparisons, AHP's mathematical algorithm evaluates our work and assigns the appropriate overall weight to each of the items in the group. The greater the weight, the more important to us is the item to which it is assigned.

The algorithm also gives a numeric indicator of the consistency of our rankings. (If we greatly prefer apples to oranges, and greatly prefer oranges to turnips, it would be inconsistent to say that we also slightly prefer turnips to apples. When we say such a thing, AHP notices it and prompts us to reconsider. We may react to the prompting or not; after all we, not AHP, are making the judgments here.)

Example

To illustrate the AHP technique of pairwise comparisons, let's use it on something that AHP is very well suited for: a problem with high stakes, involving human perceptions and judgments, whose resolution has long-term effects. To keep things simple, we'll choose a problem that is universally familiar, and we'll limit our comparisons to only three criteria.

Imagine that you want to evaluate, on a rational, disciplined basis, how important each of three criteria is to you in selecting a prospective spouse: Brains, Looks, and Personality. If we compare the criteria two by two, AHP can use our comparisons to assign a numerical weight to each of the three.

You can make such an evaluation on THIS WEB SITE from the Canadian Conservation Institute. Here's what to do after opening the site:

1. On the first screen, enter the number 3, to specify how many criteria you will be comparing.
2. Click Continue, then enter the names of the criteria: Brains, Looks, and Personality. (To facilitate our discussion, please enter them in that order; normally, the order doesn't matter.)
3. Scroll down and select the Line-by-Line method to facilitate entering the data.
4. Click Continue and begin your pairwise comparisons. (Look at the bar chart and note that before you make any comparisons, all three criteria have equal importance.)
5. For each of the three pairs of items, you will compare the first to the second by entering a number between -9 and 9. You can use the verbal scale to help you choose each number. (Example: If Brains were a little less important to you than Looks, you might enter -3 for this pair. If Brains were very much more important, you might enter a 9.) To facilitate our discussion, imagine that Brains are moderately more important to you than either Looks or Personality, and that you have no preference for Looks vs. Personality. Indicate these preferences by entering 5, 5, and 1 for the pairwise comparisons.
6. Click Calculate to see the results of your work. The higher the number assigned to each criterion, the more important it is to you, based on your own pairwise comparisons. In our example, Brains accounts for 71.43% of your preference, and each of the others for 14.29%. (These numbers are a bit too precise to apply to the human situation involved, so we might just rely on the bar chart to compare them.)
7. Look at the Consistency Ratio below the bar chart. If you've entered the numbers we asked you to, your consistency ratio will be zero, indicating that your comparisons were perfectly consistent. The Consistency Ratio is a measurement of the consistency of the data you entered (remember the apples, oranges, and turnips above). The higher the ratio, the less consistent were your entries. Whenever it is greater than 0.1 or so, you may want to review your entries to see if they really reflect your thinking.

The web site allows for easy modification of your judgments, and it can be instructive to change them and see how that affects the outcome. (The modification feature "times out" after a few minutes, after which you must start your evaluation from scratch.)

By experimenting with this small number of criteria, you can develop a feel for what AHP is doing. Keeping your first two comparisons the same, express a slight preference for Looks or Personality, and notice what happens to the results. Change your preferences in other ways, and see if you can follow the changes in results. (Don't worry too much about inconsistencies unless the ratio is larger than 0.1 or so -- inconsistency is a part of human nature, and we are dealing fairly closely with human nature here.)

To experiment at a more challenging level, enter a new problem with four criteria: Brains, Looks, Personality, and Wealth. With the larger number of criteria, it is harder to "see" how they interact, but the technique is every bit as valid. Imagine how difficult it would be for you, without the aid of AHP, to handle the six criteria of Brains, Looks, Personality, Wealth, Age, and Religion. Try it with AHP and see how much easier it becomes.

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Diverse perceptions and judgments

VERY ROUGH DRAFT OF THIS PART...

• AHP is a method of measuring perceptions and judgments in relative terms.
• Its inputs depend on the judgments of the decision makers.
• It allows the consideration and combination of many kinds of judgments about the matter at hand, e.g., rational, intuitive, irrational, other(?).
• If the judgments are just irrational, won't we likely get a clue to that from the Consistency Ratio?

Example

Consider the case of purchasing an automobile, where part of the decision is based on price. The prices of the alternatives can be gotten from formal price quotations, and can be known precisely to the penny. A typical example might be:

1. $18,285.00 - Chevrolet Cobalt
2. $28,240.73 - Audi A4
3. $34,885.50 - Chrysler 300
4. $35,001.67 - Saab 9-5
5. $43,548.27 - BMW 5 Series

As each decision maker does pairwise comparisons of these numbers, he or she will apply their personal judgment to the task. Those judgments can be of various kinds, and whatever they are, AHP will capture them. Imagine different decision makers using these different thought processes as they compare a pair of prices:

"We're comparing many other factors about these automobiles, but now that we are focusing on price,...

• ... prices are numbers, and numbers never lie. Two items should be preferred in exact proportion to their price, with the lower price being better. A $20,000 price is exactly twice as preferable as a $40,000 price, and I will make my comparisons accordingly."
• ... price is no object with me. Since we have enough money to buy whatever car we want, I'll rank all prices as though they were exactly equal."
• ... you should always go with the lowest responsible bidder. The lower price should always be the one you take, with the higher one not even considered."
• ... you get what you pay for. The higher price should always be the one you take, with the lower one not even considered."
• ... no automobile is worth over $35,000. I'll do my best in comparing the less expensive ones, but I'll reject anything over $35K."
• ... I know that U.S. car companies are always flexible on price. I'll give extra preference to the Chevrolet and Chrysler whenever they're part of a comparison."
• ... I've never been comfortable with numbers. I'll just guess."
• ... I may not know cars, but I know my numerology. I'll just add up the digits in each price, then keep adding the digits in the results until I'm down to a single digit. The closer it is to a seven, the better is the price."
• ... I'll do the best I can, but any car under $25K is a piece of junk, and I won't even consider it."