User:Lou Sander/Sandbox5

TOM, DICK, AND HARRY: A NEW EXAMPLE FOR THE AHP ARTICLE
The Tom, Dick, and Harry example has been moved HERE



Model the problem as a hierarchy

 * Define your problem. Discuss it very thoroughly.
 * Decide what the goal is. Be as clear as you can. Describe it in great detail.
 * Decide on your alternatives.
 * Decide what the criteria are.
 * Create the hierarchy (sometimes called a model)
 * Homogeneity
 * Step back and take a look at it (Says Lou. Maybe these should be explained for inexperienced readers. Maybe not.)
 * Within the hierarchy, identify the groups of nodes within which you will do pairwise comparisons (this is straightforward for simple tree-like hierarchies, but maybe not so simple for others) ROZANN Questions this.
 * Note that Alternatives are handled differently from Criteria
 * Decide whether you will work from the top down or from the bottom up (familiar decisions are often best from the top down, unique ones from the bottom up)

Pairwise compare the appropriate elements in the hierarchy; process the data to get local priorities
''Now we do the pairwise comparisons. Lou thinks that this process will be unfamiliar to most readers, so we owe it to them to spell it out in detail. There are really three aspects to it: 1) the mechanical process of doing the comparisons and processing them; 2) the notion that this is really a way of measuring things, esp. intangibles, and 3) the theory and math that underlie it and makes it work.''

Within each group:
 * The number of comparisons will be n(n-1)/2 where n is the number of nodes to be compared
 * Construct a square matrix of that size, with 1's in the diagonal (Does the diagonal have a name?) and the other cells (the right term?) empty

There are two aspects of what we are doing next. Maybe both need to be explained. Maybe not. First, we are making some measurements in an unusual way. Second, we are going through some mechanical steps to make those measurements. One can describe the mechanical steps without saying too much about the measurement scheme of which they are a part. This has the benefit of being clear and straightforward, but also has the drawback of leaving out some really cool and elementary information about the AHP.

For each pair in a group: When all comparisons are finished, the matrix for this group will be full. 
 * Determine which member of the pair is less dominant
 * Assign it a value of 1 (hold it as the unit)
 * Use your judgment and the Fundamental Scale to "assign a value" to the other member. How much better is the better one. There are some fine points in the terminology here.
 * Plug that value into the appropriate cell of the matrix. (NOTE: If you know what you're doing, it's not hard to determine what cell that is, but it's a challenge to explain it. We might want to find some middle ground between using the abstract mathematical a(i,j) terminology and the elementary color-coding method.)
 * Plug the reciprocal of that value into the appropriate cell of the matrix (same NOTE) ALSO NOTE: Maybe we could consider filling in the 1's and the reciprocals as something that is automatically done by the software after the human has done the important job of making the judgments and filling in the larger numbers
 * Compute its principal right eigenvector. Normalize it. This is the local priority vector for this group of nodes. We need some explanation of priorities
 * We should provide a detailed example of this computation, maybe in a sidebar. You can do it in Excel, but matrix math is a little tricky
 * Check consistency (related to the eigenvalue, which gives you a measure of the inconsistency)
 * If not consistent, think about modifying one or more of your judgments
 * Complete the above procedure for all the groups

Process (synthesize?) the local priorities to get global priorities

 * Do the simple arithmetic to convert the local priorities into global priorities (use a weight and add process and sum)

Check your work

 * Do a sensitivity analysis
 * See if the overall results seem reasonable in light of all your work
 * What do you think of Foos? They are experimental at this stage. Don't let them confuse you.

More references

 * Fooses

Image Manager
This is intended to be a list of all images in AHP articles other than this Sandbox.

The original Word or Excel files, along with the original images, should be gathered in one place so they can be shared with potential translators.

It would be helpful, but difficult, to get a master list of the files in the Wikimedia Commons Category:Analytic Hierarchy Process.

 These images are in the MAIN AHP ARTICLE AND in the Wikimedia Commons Category:Analytic Hierarchy Process as of 9/30/10:
 * File:AHPDevice.jpg
 * File:AHPFundamentalScale.png
 * File:AHPHierarchyAlternativesOnly.png
 * File:AHPHierarchy1.png
 * File:AHPHierarchy1Labeled.png
 * File:AHPHierarchy1.1.png
 * File:AHPHierarchy3.0.png
 * File:AHPHierarchy4.0.png
 * File:AHPJones01.png
 * File:AHPJones03.png
 * File:AHPJonesCriteria01.png
 * File:AHPLeadImage.png

 These images are in the MORE HIERARCHIES PAGE AND in the Wikimedia Commons Category:Analytic Hierarchy Process as of 9/30/10:


 * File:AHPDamHierarchy.png
 * File:AHPJones01.png
 * File:AHPHierarchyAlternativesOnly.png

 These images are in the JONES EXAMPLE AND in the Wikimedia Commons Category:Analytic Hierarchy Process as of 9/30/10:
 * File:AHPFundamentalScale.png
 * File:AHPHierarchyAlternativesOnly.png
 * File:AHPJones01.png
 * File:AHPJones03.png
 * File:AHPJones04.png
 * File:AHPJones05.png
 * File:AHPJonesCargoCapacity01.png
 * File:AHPJonesCargoCapacity02.png
 * File:AHPJonesCriteria01.png
 * File:AHPJonesFinalPriorities.png
 * File:AHPJonesFuelCost01.png
 * File:AHPJonesFuelCost02.png
 * File:AHPJonesMaintenanceCosts01.png
 * File:AHPJonesMaintenanceCosts02.png
 * File:AHPJonesPassengerCapacity01.png
 * File:AHPJonesPassengerCapacity02.png
 * File:AHPJonesPurchasePrice02.png
 * File:AHPJonesPurchasePrice03.png
 * File:AHPJonesPurchasePrice04REV01.png
 * File:AHPJonesResaleValue01.png
 * File:AHPJonesResaleValue02.png
 * File:AHPJonesResaleValue03.png
 * File:AHPJonesSafety01.png
 * File:AHPJonesSafety02.png
 * File:AHPJonesStyle01.png

 These images in the Wikimedia Commons are UNUSED, but should be kept, as of 9/30/10:
 * File:AHPHierarchy1.3.png 1-4-3 leader with labels only

 These images in the Wikimedia Commons are TO BE DELETED, and we are trying, as of 9/30/10:
 * File:AHPHierarchy02.png 1-3-2-3 generic.
 * File:AHPHierarchy02.1.png restaurant example.

 These images in the Wikimedia Commons DO NOT APPLY to the Analytic Hierarchy Process:
 * File:AhpFront.jpg is a house, not our AHP

Some more complex hierarchies
This section shows two AHP hierarchies that are more complex than those in the main article.

Buying an automobile
In an AHP hierarchy for a decision about 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.

The hierarchy for this decision could be diagrammed like this:





Note that the covering criteria for the alternatives consist of the four subcriteria under Cost, plus the Safety and Style criteria, plus the two subcriteria under Capacity. The covering criteria are always at the lowest level of criteria (and subcriteria, sub-subcriteria, etc.) directly above the alternatives.

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 5000 mi 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.

Water height in a dam


This was an actual decision. Look at all the factors! Put the More Hierarchy stuff from CZ here, please. HERE IT IS!

More old stuff from 2009:
I'm consolidating stuff here. Lou Sander (talk) 14:59, 5 July 2015 (UTC)

Model the problem as a hierarchy
The first step in the Analytic Hierarchy Process is to model the problem as a hierarchy. In doing this, participants explore the aspects of the problem at levels from general to detailed, then express it in the multileveled way that the AHP requires. As they work to build the hierarchy, they increase their understanding of the problem, of its context, and of each other's thoughts and feelings about both.

Hierarchies defined
A hierarchy is a system of ranking and organizing people, things, ideas, etc., where each element of the system, except for the top one, is subordinate to one or more other elements. Diagrams of hierarchies are often shaped roughly like pyramids, but other than having a single element at the top, there is nothing necessarily pyramid-shaped about a hierarchy.

Human organizations are often structured as hierarchies, where the hierarchical system is used for assigning responsibilities, exercising leadership, and facilitating communication. Familiar hierarchies of "things" include a desktop computer's tower unit at the "top," with its subordinate monitor, keyboard, and mouse "below."

In the world of ideas, we use hierarchies to help us acquire detailed knowledge of complex reality: we structure the reality into its constituent parts, and these in turn into their own constituent parts, proceeding down the hierarchy as many levels as we care to. At each step, we focus on understanding a single component of the whole, temporarily disregarding the other components at this and all other levels. As we go through this process, we increase our global understanding of whatever complex reality we are studying.

Think of the hierarchy that medical students use while learning anatomy—they separately consider the musculoskeletal system (including parts and subparts like the hand and its constituent muscles and bones), the circulatory system (and its many levels and branches), the nervous system (and its numerous components and subsystems), etc., until they've covered all the systems and the important subdivisions of each. Advanced students continue the subdivision all the way to the level of the cell or molecule. In the end, the students understand the "big picture" and a considerable number of its details. Not only that, but they understand the relation of the individual parts to the whole. By working hierarchically, they've gained a comprehensive understanding of anatomy.

Similarly, when we approach a complex decision problem, we can use a hierarchy to integrate large amounts of information into our understanding of the situation. As we build this information structure, we form a better and better picture of the problem as a whole.

AHP hierarchies explained
An AHP hierarchy is a structured means of modeling the problem at hand. It consists of an overall goal, a group of options or alternatives for reaching the goal, and a group of factors or criteria that relate the alternatives to the goal. The criteria can be further broken down into subcriteria, sub-subcriteria, and so on, in as many levels as the problem requires.

The hierarchy can be visualized as a diagram like the one below, with the goal at the top, the alternatives at the bottom, and the criteria, subcriteria, etc. in the middle. There are useful terms for describing the parts of such diagrams: Each box is called a node. The boxes descending from any node are called its children. The node from which a child node descends is called its parent. The parents of an Alternative, which are often from different comparison groups, are called its covering criteria. In this diagram, groups of related children are called comparison groups.

Applying these definitions to the diagram, the three Criteria are children of the Goal, and the Goal is the parent of each of the three Criteria. The two Subcriteria are the children of Criterion 3, which is their parent. Each Alternative is the child of its four covering criteria: Criterion 1, Criterion 2, Subcriterion 1, and Subcriterion 2. There are three comparison groups: the group of three Criteria, the group of two Subcriteria, and the group of three Alternatives.



The design of any AHP hierarchy will depend not only on the nature of the problem at hand, but also on the knowledge, judgments, values, opinions, needs, wants, etc. of the participants in the process. Published descriptions of AHP applications often include diagrams and descriptions of their hierarchies. These have been collected and reprinted in at least one book.

As the AHP proceeds through its other steps, the hierarchy can be changed to accommodate newly-thought-of criteria or criteria not originally considered to be important; alternatives can also be added, deleted, or changed.

As a by-product of building and optimizing the hierarchy, those who work on it increase their understanding of the decision process that it models. This can be useful to them in the subsequent stages of the AHP.

Take measurements within the model
Once the hierarchy has been constructed, it serves as a detailed model of the decision to be made. It clearly identifies the goal of the decision, the alternative ways of reaching it, and the criteria to be used in evaluating those alternatives. It also explicitly shows the relationships between the goal, the decision criteria, and the alternatives.

What remains is to assess the importance of the model's various nodes and groups as they relate to reaching the goal. This is done with a special measuring system that elicits information from the decision participants, then processes it mathematically. The system is applied separately to each comparison group in the hierarchy. Within each group, it determines the importance of each node with respect to the others, expressing the results as numbers called local priorities.

AHP priorities have values between zero and 1.000, and within a comparison group they always sum to 1.000. Each node’s priority is a measure of its relative weight or importance within its comparison group. A node whose priority is 0.500 has twice the weight of one whose priority is 0.250, for example, and five times the weight of one whose priority is 0.100.

Importantly, priorities are not directly assigned from scales like dollars, kilograms, "a scale from one to five," etc. Instead, they are derived from a process of comparative measurement. This allows the AHP and its users to deal rationally with the mix of incommensurable factors that arise in complex decision making.

Comparisons and dominance
The AHP’s measurement system relies on our natural ability to compare things and judge their relative dominance. Dominance refers to the strength of one item over the other with respect to a given property. Dominance is a general term that applies to all comparisons. Other words are sometimes used for specific comparisons: "importance," "preference," and "likelihood," and so forth.

If we hold a penny and a dime in the palm of our hand, we can easily and naturally compare the two with respect to any of their properties. For example we can easily tell which coin is dominant in size, or redness, or shininess.

With a little more effort, we can judge the magnitude of the differences, and express that magnitude in words. It’s easy to say that the penny's size slightly dominates the dime's, or that the penny's redness extremely dominates that of the dime. If we look at the specific coins we’re holding, it’s easy to say whether both are equally shiny, or whether the penny is strongly shinier than the dime, or whether the dime is moderately shinier than the penny, or whatever the situation happens to be.

When we make those judgments, we don’t need any measurements or measuring instruments. Regarding size, we don’t need to think in terms of inches or millimeters, and we don’t need a ruler or a caliper. When comparing redness, we don’t need to know about colorimetry, and we don’t need any of its instruments or units of measurement. The same is true of shininess and photometry. In making these comparisons, all we need is our natural ability to compare, an understanding of the property we are comparing, and a vocabulary that lets us express degrees of difference.

The AHP uses comparisons like these to measure the judgments of the decision maker.

 ALL THAT FOLLOWS IS UNDER CONSTRUCTION

EDITOR'S NOTE: Somehow I want to bring in "pairwise comparisons using the law of comparative judgment."

ROZANN - Check out Measurement and Measure (mathematics). Should we maybe link to one or both of them? 

LFS - Here is a simple scale for expressing dominance. It's similar to the Fundamental Scale referred to in much of the AHP literature.  

Pairwise comparisons
'Splain them. See Pairwise comparison. See Law of comparative judgment.

A simple example will serve to make this clear. Consider this AHP hierarchy. It models a decision about where to go for dinner:



Our alternatives are Wendy's, a fast food restaurant, The Olive Garden, a mid-range Italian restaurant with an extensive menu, and Ruth's Chris Steakhouse, an upscale place specializing in top-quality steaks. Our decision criteria are Cost, Location, and Food. Food has two subcriteria, Variety and Quality.

It's important to note that the AHP is not telling us where to go for dinner, but is helping us, given our criteria and alternatives, determine the best place to go. Other decision makers, even in the same room as we are, might choose totally different alternatives and criteria. Their decision process and their model could be totally different from ours. But in their case as well as ours, the AHP would be useful in making the decision.

It's also important to recognize that this sample decision, chosen for simplicity in explaining a measurement process, is far from the "complex problems with high stakes, whose resolutions have long-term repercussions" for which the AHP is best suited.

Lou's notes -- Maybe we need different headings. Forman & Gass say the three primary functions of AHP are "structuring complexity, measurement of ratio scales, and providing synthesis." IMHO that's a good way to organize headings in this section.

For the next paragraphs...


 * 1) Each comparison group is considered with respect to its parent(s) (Need a better description of what is going on here)
 * 2) Participants/decision makers are asked to evaluate the members of the group, usually by providing judgments. They can provide measurements, though. (Avoid mention of pairwise stuff here. That's for later.)
 * 3) The evaluations are processed and expressed as numbers called priorities.
 * 4) The priorities for all the members of the CG will sum to 1.000, and the value of a member's priority will indicate its weight with respect to (something about the parent). If the group's parent is the Goal, for example, a criterion whose priority is 0.500 will be twice as important in reaching it as one whose priority is 0.250, and five times as important as one whose priority is 0.100.

For now, forget the stuff that follows, since it will change depending on what is put here.  A system of relative measurement is applied to the nodes of each comparison group, then those measurements are synthesized into numerical priorities for each node in the hierarchical model. The priority of each criterion, sub-criterion, etc. is a synthesis of the decision maker’s judgments about its importance in reaching the goal. The priority of each alternative is a synthesis of the decision maker’s judgments about its relationship to the criteria, and thereby its importance in reaching the goal.

Although it has wide applicability, the axiomatic foundation of the AHP carefully delimits the scope of the problem environment. It is based on the well defined mathematical structure of consistent matrices and their associated right-eigenvector's ability to generate true or approximate weights.

The AHP’s measurement system relies on the natural human ability to compare and judge dominance. It produces ratio scales (RW - Absolute scales) for the nodes to which it is applied. Because it produces ratio scales (RW - the AHP uses absolute scales), measurements of totally different aspects of the problem can be synthesized (RW - weighted and added - the weighting process makes them so they can be added) into quantities that apply to all aspects of the problem as a whole.

When the measurement and synthesis (RW - judgment and synthesis) are complete, a numerical priority will have been derived (RW - established. Derivation is for those local priorities) for each alternative. The priorities for all the alternatives will sum to 1.000, and the value of an alternative’s priority will indicate its weight (RW - ?) or effectiveness in reaching the goal. For example, an alternative whose priority is 0.500 will be twice as effective in reaching the goal as one whose priority is 0.250, and five times as effective as one whose priority is 0.100. (Absolute scales have ratio property)

The criteria, sub-criteria, etc. will also have been prioritized. Decision makers can examine these priorities to see the weight of each criterion in relating the alternatives to the goal.

Synthesize the measurements over the entire model
Put some good stuff here. Do the simple math to allocate the local priorities over the entire model. This is probably a good place for the Jones charts that show the default priorities, the local priorities, and the global ones.