LogisticsDecisionAnalysisMethods

1、1Logistics Decision Analysis MethodsAnalytic Hierarchy Process Presented by Tsan-hwan Lin E-mail: 2Motivation - 1lIn our complex world system, we are forced to cope with more problems than we have the resources to handle.nWhat we need is not a more complicated way of thinking but a framework that wi

2、ll enable us to think of complex problems in a simple way.The AHP provides such a framework that enables us to make effective decisions on complex issues by simplifying and expediting our natural decision-making processes.3Motivation - 2lHumans are not often logical creatures.nMost of the time we ba

3、se our judgments on hazy impressions of reality and then use logic to defend our conclusions.The AHP organizes feelings, intuition, and logic in a structured approach to decision making.4Motivation - 3lThere are two fundamental approaches to solving problems: the deductive approach（演繹法）and the induc

4、tive (歸納法；or systems) approach.nBasically, the deductive approach focuses on the parts whereas the systems approach concentrates on the workings of the whole.The AHP combines these two approaches into one integrated, logic framework.5Introduction - 1lThe analytic hierarchy process (AHP) was develope

5、d by Thomas L. Saaty.nSaaty, T.L., The Analytic Hierarchy Process, New York: McGraw-Hill, 1980lThe AHP is designed to solve complex problems involving multiple criteria.lAn advantage of the AHP is that it is designed to handle situations in which the subjective judgments of individuals constitute an

6、 important part of the decision process.6Introduction - 2lBasically the AHP is a method of (1) breaking 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 rela

7、tive importance of each variable; and (4) synthesizing the judgments to determine which variables have the highest priority and should be acted upon to influence the outcome of the situation.7Introduction - 3lThe process requires the decision maker to provide judgments about the relative importance

8、of each criterion and then specify a preference for each decision alternative on each criterion.lThe output of the AHP is a prioritized ranking indicating the overall preference for each of the decision alternatives.8Major Steps of AHP1) To develop a graphical representation of the problem in terms

9、of the overall goal, the criteria, and the decision alternatives. (i.e., the hierarchy 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 d

10、ecision alternative in terms of how it contributes to each criterion.4) Given the 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 ove

11、rall preference.9Constructing HierarchieslHierarchies are a fundamental mind toollClassification of hierarchieslConstruction of hierarchies10Establishing PrioritieslThe need for prioritieslSetting prioritieslSynthesislConsistencylInterdependence11Advantages of the AHPUnityProcess RepetitionTradeoffs

12、SynthesisConsistencyMeasurementInterdependenceComplexityAHPJudgment and ConsensusHierarchic StructuringThe AHP provides a single, easily understood, flexible model for a wide range of unstructured problemsThe AHP integrates deductive and systems approaches in solving complex problemsThe AHP can deal

13、 with the interdependence of elements in a system and does not insist on linear thinkingThe AHP reflects the natural tendency of the mind to sort elements of a system into different levels and to group like elements in each levelThe AHP provides a scale for measuring intangibles and a method for est

14、ablishing prioritiesThe AHP tracks the logical consistency of judgments used in determining prioritiesThe AHP leads to an overall estimate of the desirability of each alternativeThe AHP takes into consideration the relative priorities of factors in a system and enables people to select the best alte

15、rnative based on their goalsThe AHP does not insist on consensus but synthesizes a representative outcome from diverse judgmentsThe AHP enables people to refine their definition of a problem and to improve their judgment and understanding through repetition12Q & A13Hierarchy DevelopmentlThe firs

16、t 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.Car ACar BCar CCar ACar BCar CCar ACar BCar CCar ACar BCar CPriceMPGComfortStyleSelect the Best CarOverall Goal:Criteria:Decision Alternatives:14Pairwis

17、e ComparisonslPairwise comparisons are fundamental building blocks of the AHP.lThe AHP employs an underlying scale with values from 1 to 9 to rate the relative preferences for two items.15Pairwise Comparison MatrixlElement Ci,j of the matrix is the measure of preference of the item in row i when com

18、pared to the item in column j.lAHP assigns a 1 to all elements on the diagonal of the pairwise comparison matrix.nWhen we compare any alternative against itself (on the criterion) the judgment must be that they are equally preferred.lAHP obtains the preference rating of Cj,i by computing the recipro

19、cal (inverse) of Ci,j (the transpose position).nThe preference 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.lAccording above rules, the number of entries actuall

20、y filled in by decision makers is (n2 n)/2, where n is the number of elements to be compared.16Preference Scale - 1Verbal Judgment of PreferenceNumericalRatingExtremely preferred9 Very strongly to extremely preferred8Very strongly preferred7 Strongly to very strongly preferred6Strongly preferred5 Mo

21、derately to strongly preferred4Moderately preferred3 Equally to moderately preferred2Equally preferred117Preference Scale - 2lResearch and experience have confirmed the nine-unit scale as a reasonable basis for discriminating between the preferences for two items.nEven numbers (2, 4, 6, 8) are inter

22、mediate values for the scale. nA value of 1 is reserved for the case where the two items are judged to be equally preferred.18SynthesislThe procedure to estimate the relative priority for each decision alternative in terms of the criterion is referred to as synthesization（綜合；合成）.nOnce the matrix of

23、pairwise comparisons has been developed, 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.nThe exact mathematical procedure required to perform synthesization involves the com

24、putation of eigenvalues and eigenvectors, which is beyond the scope of this text.19Procedure for Synthesizing JudgmentslThe 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: Divid

25、e each element in the pairwise matrix by its column total.uThe 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.uThese averages provide an estimate of the relative priorities of the elements

26、being compared.Example:20Example: Synthesizing Procedure - 0Step 0: Prepare pairwise comparison matrixComfortCar ACar BCar CCar A128Car B1/216Car C1/81/6121Example: Synthesizing Procedure - 1Step 1: Sum the values in each column.ComfortCar ACar BCar CCar A128Car B1/216Car C1/81/61Column totals13/819

27、/61522Example: Synthesizing Procedure - 2Step 2: Divide each element of the matrix by its column total.nAll columns in the normalized pairwise comparison matrix now have a sum of 1.ComfortCar ACar BCar CCar A8/1312/198/15Car B4/136/196/15Car C1/131/191/1523Example: Synthesizing Procedure - 3Step 3:

28、Average the elements in each row.nThe values in the normalized pairwise comparison matrix have been converted to decimal form.nThe result is usually represented as the (relative) priority vector.0.0660.3410.59324Consistency - 1lAn important consideration in terms of the quality of the ultimate decis

29、ion relates to the consistency of judgments that the decision maker demonstrated during the series of pairwise comparisons.nIt should be realized perfect consistency is very difficult to achieve and that some lack of consistency is expected to exist in almost any set of pairwise comparisons.nExample

30、:25Consistency - 2lTo handle the consistency question, the AHP provides a method for measuring the degree of consistency among the pairwise judgments provided by the decision maker.nIf the degree of consistency is acceptable, the decision process can continue.nIf the degree of consistency is unaccep

31、table, the decision maker should reconsider and possibly revise the pairwise comparison judgments before proceeding with the analysis.26Consistency RatiolThe AHP provides a measure of the consistency of pairwise comparison judgments by computing a consistency ratio（一致性比率）.nThe ratio is designed in s

32、uch a way that values of the ratio exceeding 0.10 are indicative of inconsistent judgments.nAlthough the exact mathematical computation of the consistency ratio is beyond the scope of this text, an approximation of the ratio can be obtained.27Procedure: Estimating Consistency Ratio - 1Step 1: Multip

33、ly 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: Divide the elements of the vector of weighted sums o

34、btained 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.28Procedure: Estimating Consistency Ratio - 2Step 4: Compute the consistency index (CI):Where n is the number of items being comparedStep 5: Compute the

35、 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.Example: 1nnCImaxRICICR 29Random IndexlRandom index (RI)

36、 is the consistency index of a randomly generated pairwise comparison matrix.nRI depends on the number of elements being compared (i.e., size of pairwise comparison matrix) and takes on the following values:n12345678910RI0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.4930Example: InconsistencyPrefer

37、ences: If, A B (2); B C (6)Then, A C (should be 12) (actually 8) InconsistencyComfortCar ACar BCar CCar A128Car B1/216Car C1/81/6131Example: Consistency Checking - 1Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered

38、. Same procedures for other items. Sum the values across the rows to obtain a vector of values labeled “weighted sum.” 197. 0034. 1803. 1066. 0396. 0528. 0057. 0341. 0682. 0074. 0297. 0593. 0168 0.066 6112 0.341 81211 593. 032Example: Consistency Checking - 2Step 2: Divide the elements of the vector

39、 of weighted sums by the corresponding priority value.2.9853.0323.040 066. 0197. 0341. 0034. 1593. 0803. 1Step 3: Compute the average of the values computed in step 2 (lmax).019. 33985. 2032. 3040. 3max33Example: Consistency Checking - 3Step 4: Compute the consistency index (CI).Step 5: Compute the

40、consistency ratio (CR).010. 0133019. 31nnCImax0.10 017. 058. 0010. 0RICICRThe degree of consistency exhibited in the pairwise comparison matrix for comfort is acceptable.34Development of Priority RankinglThe overall priority for each decision alternative is obtained by summing the product of the cri

41、terion priority (i.e., weight) (with respect to the overall goal) times the priority (i.e., preference) of the decision alternative with respect to that criterion.lRanking these priority values, we will have AHP ranking of the decision alternatives.lExample:35Example: Priority Ranking 0AStep 0A: Oth

42、er pairwise comparison matricesComfortCar ACar BCar CCar A128Car B1/216Car C1/81/61PriceCar ACar BCar CCar A11/3Car B31Car C421MPGCar ACar BCar CCar A11/41/6Car B411/3Car C631StyleCar ACar BCar CCar A11/34Car B317Car C1/41/71CriterionPriceMPGComfortStylePrice1322MPG1/311/41/4Comfort1/2411/2Style1/24

43、2136Example: Priority Ranking 0BStep 0B: Calculate priority vector for each matrix.557. 0320. 0123. 0639. 0274. 0087. 0066. 0341. 0593. 0080. 0655. 0265. 0299. 0218. 0085. 0398. 037Example: Priority Ranking 1Step 1: Sum the product of the criterion priority (with respect to the overall goal) times t

44、he priority of the decision alternative with respect to that criterion.Step 2: Rank the priority values.0.265 (0.265) 0.299 (0.593) 0.218 (0.087) 0.085 (0.123) 0.398 priorityA car OverallAlternativePriorityCar B0.421Car C0.314Car A0.265Total1.00038Hierarchies: A Tool of the MindlHierarchies are a fu

45、ndamental tool of the human mind.nThey involve identifying the elements of a problem, grouping the elements into homogeneous sets, and arranging these sets in different levels.nComplex systems can best be understood by breaking them down into their constituent elements, structuring the elements hier

46、archically, 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.39Classifying HierarchieslHierarchies can be divided into two kinds: structural and functional.lIn structural hierarchies, complex sys

47、tems are structured into their constituent parts in descending order according to structural properties (such as size, shape, color, or age).nStructural hierarchies relate closely to the way our brains analyze complexity by breaking down the objects perceived by our senses into clusters, subclusters

48、, and still smaller clusters. (more descriptive)lFunctional hierarchies decompose complex systems into their constituent parts according to their essential relationships.nFunctional hierarchies help people to steer a system toward a desired goal like conflict resolution, efficient performance, or ov

49、erall happiness. (more normative)lFor the purposes of the study, functional hierarchies are the only link that need be considered.40HierarchylEach set of elements in a functional hierarchy occupies a level of the hierarchy.nThe top level, called the focus, consists of only one element: the broad, ov

50、erall objective.nSubsequent levels may each have several elements, although their number is usually small between five and nine.uBecause the elements in one level are to be compared with one another against a criterion in the next higher level, the elements in each level must be of the same order of

51、 magnitude. (Homogeneity)To avoid making large errors, we must carry out clustering process. By forming hierarchically arranged clusters of like elements, we can efficiently complete the process of comparing the simple with the very complex.Because a hierarchy represents a model of how the brain ana

52、lyzes complexity, the hierarchy must be flexible enough to deal with that complexity.41Types of Functional HierarchylSome functional hierarchies are complete, that is, all the elements in one level share every property in the nest higher level.lSome are incomplete in that some elements in a level do

53、 not share properties.42Constructing Hierarchies - 1lOnes approach to constructing a hierarchy depends on the kind of decision to be made.nIf it is a matter of choosing among alternatives, we could start from the bottom by listing the alternatives. (decision alternatives = criteria = overall goal)lO

54、nce we construct the hierarchy, we can always alter parts of it later to accommodate new criteria that we may think of or that we did not consider important when we first designed it. (AHP is flexible and time-adaptable)nSometimes the criteria themselves must be examined in details, so a level of su

55、bcriteria should be inserted between those of the criteria and the alternatives.43Constructing Hierarchies - 2nIf one is unable to compare the elements of a level in terms of the elements of the next higher level, one must ask in what terms they can be compared and then seek an intermediate level th

56、at should amount to a breakdown of the elements of the next higher level.uThe basic principle in structuring a hierarchy is to see if one can answer the question: “Can you compare the elements in a lower level in terms of some all all the elements in the next higher level?”nThe depth of detail (in l

57、evel construction) depends on how much knowledge one has about the problem and how much can be gained by using that knowledge without unnecessarily tiring the mind.The analytic aspects of the AHP serve as a stimulus to create new dimensions for the hierarchy. It is a process for inducing cognitive a

58、wareness. A logically constructed hierarchy is a by-product of the entire AHP approach.44Constructing Hierarchies II - 1lWhen constructing hierarchies one must include enough relevant detail to depict the problem as thoroughly as possible.nConsider environment surrounding the problem.nIdentify the i

59、ssues or attributes that you feel contribute to the solution.nIdentify the participants associated with the problem.lArranging the goals, attributes, issues, and stakeholders in a hierarchy serves two purposes:nIt provides an overall view of the complex relationships inherent in the situation.nIt pe

60、rmits the decision maker to assess whether he or she is comparing issues of the same order of magnitude in weight or impact on the solution.（我們無法直接比較蘋果與橘子；卻可以根據它們的甜度、營養、價格來決定誰是比較好的水果。）45Constructing Hierarchies II - 2lThe elements should be clustered into homogeneous groups of five to nine so they can b