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1、Logistics Decision Analysis Methods,Analytic Hierarchy Process TIAN Hong The Institute of Aeronautical Eng. T,Thomas L. Saaty,UNIVERSITY CHAIR, QUANTITATIVE GROUP Office: 322 Mervis Hall Phone: 412-648-1539 E-mail: saatykatz.pitt.edu Degrees PhD in Mathematics, Yale University (1953) Postgraduate St
2、udy, University of Paris (195253),Prior to coming to the University of Pittsburgh, Thomas L. Saaty was professor at the Wharton School, University of Pennsylvania for 10 years and before that was for seven years in the Arms Control and Disarmament Agency at the U.S. State Department. He is a member
3、of the National Academy of Engineering.,He is the architect of the decision theory, the Analytic Hierarchy Process (AHP) and its generalization to decisions with dependence and feedback, the Analytic Network Process (ANP). He has published numerous articles and more than 12 books on these subjects.
4、His nontechnical book on the AHP, Decision Making for Leaders, has been translated to more than 10 languages. His book, The Brain: Unraveling the Mystery of How It Works, generalizing the ANP further to neural firing and synthesis, appeared in the year 2000.,He is currently involved in extending his
5、 mathematical multicriteria decision-making theory to how to synthesize group and societal influences. He is also developing the Super Decisions software that implements the ANP and it is available free at http:/ AHP is used in both individual and group decision-making by business, industry, and gov
6、ernments and is particularly applicable to complex large-scale multiparty multicriteria decision problems The ANP has been applied to a variety of decisions involving benefits, costs, opportunities, and risks and is particularly useful in predicting outcomes. At the Katz School he teaches Decision M
7、aking in Complex Environments, using both the AHP and the ANP and Creativity and Problem Solving. He has recently completed a book on the subject of creativity and problem solving that includes a CD of more than 140 colorful specially designed PowerPoint slides.,Motivation 1（动机之一）,In our complex wor
8、ld system, we are forced to cope with more problems than we have the resources to handle. What we need is not a more complicated way of thinking but a framework that will enable us to think of complex problems in a simple way. The AHP provides such a framework that enables us to make effective decis
9、ions on complex issues by simplifying and expediting our natural decision-making processes.,Motivation 2（动机之二）,Humans are not often logical creatures. Most of the time we base our judgments on hazy impressions （模糊的感觉）of reality and then use logic to defend（坚持） our conclusions. The AHP organizes feel
10、ings, intuition, and logic in a structured approach to decision making.,Motivation 3（动机之三）,There are two fundamental approaches to solving problems: the deductive approach（演绎法）and the inductive (归纳法；or systems) approach. Basically, the deductive approach focuses on the parts whereas the systems appr
11、oach concentrates on the workings of the whole. The AHP combines these two approaches into one integrated, logic framework.,Introduction 1（介绍之一）,The analytic hierarchy process （AHP） was developed by Thomas L. Saaty. Saaty, T.L., The Analytic Hierarchy Process, New York: McGraw-Hill, 1980 The AHP is
12、designed to solve complex problems involving multiple criteria. An advantage of the AHP is that it is designed to handle situations in which the subjective judgments of individuals constitute an important part of the decision process.,Introduction 2（介绍之二）,Basically the AHP is a method of (1) breakin
13、g down a complex, unstructured situation into its component parts; (2) arranging these parts, or variables into a hierarchic order; (3) assigning numerical values to subjective judgments on the relative importance of each variable; and (4) synthesizing the judgments to determine which variables have
14、 the highest priority and should be acted upon to influence the outcome of the situation.,Introduction 3（介绍之三）,The process requires the decision maker to provide judgments about the relative importance of each criterion and then specify a preference for each decision alternative on each criterion. T
15、he output of the AHP is a prioritized ranking （优先顺序排序）indicating the overall preference for each of the decision alternatives.,Major Steps of AHP（主要步骤）,1) To develop a graphical representation of the problem in terms of the overall goal, the criteria, and the decision alternatives. (i.e., the hierar
16、chy of the problem) 2) To specify his/her judgments about the relative importance of each criterion in terms of its contribution to the achievement of the overall goal. 3) To indicate a preference or priority for each decision alternative in terms of how it contributes to each criterion. 4) Given th
17、e information on relative importance and preferences, a mathematical process is used to synthesize the information (including consistency checking) and provide a priority ranking of all alternatives in terms of their overall preference.,Constructing Hierarchies,Hierarchies are a fundamental mind too
18、l Classification of hierarchies Construction of hierarchies,Establishing Priorities,The need for priorities Setting priorities Synthesis Consistency Interdependence,Advantages of the AHP,The AHP provides a single, easily understood, flexible model for a wide range of unstructured problems,The AHP in
19、tegrates deductive and systems approaches in solving complex problems,The AHP can deal with the interdependence of elements in a system and does not insist on linear thinking,The AHP reflects the natural tendency of the mind to sort elements of a system into different levels and to group like elemen
20、ts in each level,The AHP provides a scale for measuring intangibles and a method for establishing priorities,The AHP tracks the logical consistency of judgments used in determining priorities,The AHP leads to an overall estimate of the desirability of each alternative,The AHP takes into consideratio
21、n the relative priorities of factors in a system and enables people to select the best alternative based on their goals,The AHP does not insist on consensus but synthesizes a representative outcome from diverse judgments,The AHP enables people to refine their definition of a problem and to improve t
22、heir judgment and understanding through repetition,Hierarchy Development MPG（油耗）,The first step in the AHP is to develop a graphical representation of the problem in terms of the overall goal, the criteria, and the decision alternatives.,Pairwise Comparisons,Pairwise comparisons are fundamental buil
23、ding blocks of the AHP. The AHP employs an underlying scale with values from 1 to 9 to rate the relative preferences for two items.,Pairwise Comparison Matrix,Element Ci,j of the matrix is the measure of preference of the item in row i when compared to the item in column j. AHP assigns a 1 to all el
24、ements on the diagonal of the pairwise comparison matrix. When we compare any alternative against itself (on the criterion) the judgment must be that they are equally preferred. AHP obtains the preference rating of Cj,i by computing the reciprocal (inverse) of Ci,j (the transpose position). The pref
25、erence value of 2 is interpreted as indicating that alternative i is twice as preferable as alternative j. Thus, it follows that alternative j must be one-half as preferable as alternative i. According above rules, the number of entries actually filled in by decision makers is (n2 n)/2, where n is t
26、he number of elements to be compared.,Preference Scale 1（优先的尺度）,Preference Scale 2（优先顺序2）,Research and experience have confirmed the nine-unit scale as a reasonable basis for discriminating between the preferences for two items. Even numbers (2, 4, 6, 8) are intermediate values for the scale. A valu
27、e of 1 is reserved for the case where the two items are judged to be equally preferred.,Synthesis（合成）,The procedure to estimate the relative priority for each decision alternative in terms of the criterion is referred to as synthesization（綜合；合成）. Once the matrix of pairwise comparisons has been deve
28、loped, priority（優先次序；相對重要性）of each of the elements (priority of each alternative on specific criterion; priority of each criterion on overall goal) being compared can be calculated. The exact mathematical procedure required to perform synthesization involves the computation of eigenvalues and eigenv
29、ectors, which is beyond the scope of this text.,Procedure for Synthesizing Judgments,The following three-step procedure provides a good approximation of the synthesized priorities. Step 1: Sum the values in each column of the pairwise comparison matrix. Step 2: Divide each element in the pairwise ma
30、trix by its column total. The resulting matrix is referred to as the normalized pairwise comparison matrix. Step 3: Compute the average of the elements in each row of the normalized matrix. These averages provide an estimate of the relative priorities of the elements being compared. Example:,Example
31、: Synthesizing Procedure - 0,Step 0: Prepare pairwise comparison matrix,Example: Synthesizing Procedure - 1,Step 1: Sum the values in each column.,Example: Synthesizing Procedure - 2,Step 2: Divide each element of the matrix by its column total. All columns in the normalized pairwise comparison matr
32、ix now have a sum of 1.,Example: Synthesizing Procedure - 3,Step 3: Average the elements in each row. The values in the normalized pairwise comparison matrix have been converted to decimal form. The result is usually represented as the (relative) priority vector.,Consistency - 1,An important conside
33、ration in terms of the quality of the ultimate decision relates to the consistency of judgments that the decision maker demonstrated during the series of pairwise comparisons. It should be realized perfect consistency is very difficult to achieve and that some lack of consistency is expected to exis
34、t in almost any set of pairwise comparisons. Example:,Consistency - 2,To handle the consistency question, the AHP provides a method for measuring the degree of consistency among the pairwise judgments provided by the decision maker. If the degree of consistency is acceptable, the decision process ca
35、n continue. If the degree of consistency is unacceptable, the decision maker should reconsider and possibly revise the pairwise comparison judgments before proceeding with the analysis.,Consistency Ratio,The AHP provides a measure of the consistency of pairwise comparison judgments by computing a co
36、nsistency ratio（一致性比率）. The ratio is designed in such a way that values of the ratio exceeding 0.10 are indicative of inconsistent judgments. Although the exact mathematical computation of the consistency ratio is beyond the scope of this text, an approximation of the ratio can be obtained.,Procedur
37、e: Estimating Consistency Ratio - 1,Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered. Same procedures for other items. Sum the values across the rows to obtain a vector of values labeled “weighted sum.” Step 2: Di
38、vide the elements of the vector of weighted sums obtained in Step 1 by the corresponding priority value. Step 3: Compute the average of the values computed in step 2. This average is denoted as lmax.,Procedure: Estimating Consistency Ratio - 2,Step 4: Compute the consistency index (CI): Where n is t
39、he number of items being compared Step 5: Compute the consistency ratio (CR): Where RI is the random index, which is the consistency index of a randomly generated pairwise comparison matrix. It can be shown that RI depends on the number of elements being compared and takes on the following values. E
40、xample:,Random Index,Random index (RI) is the consistency index of a randomly generated pairwise comparison matrix. RI depends on the number of elements being compared (i.e., size of pairwise comparison matrix) and takes on the following values:,Example: Inconsistency,Preferences: If, A B (2); B C (
41、6) Then, A C (should be 12) (actually 8) Inconsistency,Example: Consistency Checking - 1,Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered. Same procedures for other items. Sum the values across the rows to obtain
42、a vector of values labeled “weighted sum.”,Example: Consistency Checking - 2,Step 2: Divide the elements of the vector of weighted sums by the corresponding priority value.,Step 3: Compute the average of the values computed in step 2 (lmax).,Example: Consistency Checking - 3,Step 4: Compute the cons
43、istency index (CI).,Step 5: Compute the consistency ratio (CR).,The degree of consistency exhibited in the pairwise comparison matrix for comfort is acceptable.,Development of Priority Ranking,The overall priority for each decision alternative is obtained by summing the product of the criterion prio
44、rity (i.e., weight) (with respect to the overall goal) times the priority (i.e., preference) of the decision alternative with respect to that criterion. Ranking these priority values, we will have AHP ranking of the decision alternatives. Example:,Example: Priority Ranking 0A,Step 0A: Other pairwise
45、 comparison matrices,Example: Priority Ranking 0B,Step 0B: Calculate priority vector for each matrix.,Example: Priority Ranking 1,Step 1: Sum the product of the criterion priority (with respect to the overall goal) times the priority of the decision alternative with respect to that criterion.,Step 2
46、: Rank the priority values.,Hierarchies: A Tool of the Mind,Hierarchies are a fundamental tool of the human mind. They involve identifying the elements of a problem, grouping the elements into homogeneous sets, and arranging these sets in different levels. Complex systems can best be understood by b
47、reaking them down into their constituent elements, structuring the elements hierarchically, and then composing, or synthesizing, judgments on the relative importance of the elements at each level of the hierarchy into a set of overall priorities.,Classifying Hierarchies,Hierarchies can be divided in
48、to two kinds: structural and functional. In structural hierarchies, complex systems are structured into their constituent parts in descending order according to structural properties (such as size, shape, color, or age). Structural hierarchies relate closely to the way our brains analyze complexity
49、by breaking down the objects perceived by our senses into clusters, subclusters, and still smaller clusters. (more descriptive) Functional hierarchies decompose complex systems into their constituent parts according to their essential relationships. Functional hierarchies help people to steer a system toward a desired goal like conflict resolution, efficient performance, or overall happiness. (more normative) For the purposes of the study, functional hierarchies are the only link that need be considered.,H