The Analytic Hierarchy Process. M. En C. Eduardo Bustos Farías

Transcription

1 The Analytic Hierarchy Process M. En C. Eduardo Bustos Farías

2 Outline of Lecture Summary MADM ranking methods Examples Analytic Hierarchy Process (AHP) Examples pairwise comparisons normalization consistency scores Métodos Cuantitativos M. En C. Eduardo Bustos Farías 2

3 Summary Discrete decision problems involving two or more conflicting criterion or objectives: Choosing Among Job Offers: salary, location, career potential, etc. Selecting a Camcorder: price, warranty, zoom, weight, lighting, etc. Choosing Among Job Applicants: education, experience, personality, etc. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 3

4 MADM Methods We ll consider one MADM technique for these types of problems: The Analytic Hierarchy Process (AHP) Métodos Cuantitativos M. En C. Eduardo Bustos Farías 4

5 The Multicriteria Scoring Model Score (or rate) each alternative on each criterion. Assign weights the criterion reflecting their relative importance. For each alternative i, compute a weighted average score as: n V = i w j = weight for criterion j j=1 wr j ij r ij = transformed score for alternative i on criterion j Métodos Cuantitativos M. En C. Eduardo Bustos Farías 5

6 The Analytic Hierarchy Process A formalization of our intuitive understanding of a complex problem using a hierachical structure Enables DM to structure a MADM problem visually in an attribute hierarchy A hierarchy has at least three levels: focus or overall goal (top) multiple criteria (attributes) that define alternatives (middle) competing alternatives at the bottom Métodos Cuantitativos M. En C. Eduardo Bustos Farías 6

7 Analytic Hierarchy Process Analytic Hierarchy Process Métodos Cuantitativos M. En C. Eduardo Bustos Farías 7

8 Analytic Analytic is a form of the word analysis, which means the separating of any material or abstract entity into its constituent elements. financial analysis marketing analysis operations analysis Analysis is the opposite of synthesis, which involves putting together or combining parts into a whole In a sense, AHP should really be called the Synthesis Hierarchy Process because at its core, AHP helps us measure and synthesize the multitude of factors involved in complex decisions. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 8

9 Hierarchy Large organizations are almost universally hierarchical in structure They are divided into units which are subdivided into smaller units, which are, in turn, subdivided and so on. Hierarchical subdivision is not a characteristic that is peculiar to human organizations. It is common to virtually all complex systems of which we have knowledge The near universality of hierarchy in the composition of complex systems suggest that there is something fundamental in this structural principle that goes beyond the peculiarities of human organization An organization will tend to assume hierarchical form whenever the task environment is complex relative to the problem-solving and communicating powers of the organization members and their tools Hierarchy is the adaptive form for finite intelligence to assume in the face of complexity. L.L. Whyte expressed this thought as follows: The immense scope of hierarchical classification is clear. It is the most powerful method of classification used by the human brain-mind in ordering experience, observations, entities and information. The use of hierarchical ordering must be as old as human thought, conscious and unconscious. L.L. Whyte, Hierarchical Structures, American Elsevier, New York, N.Y., 1969 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 9

10 Process A process is a series of actions, changes, or functions that bring about an end or result The Analytic Hierarchy Process (AHP) is not a magic formula or model that finds the right answer Rather it is a process that helps decision-makers to find the best answer Métodos Cuantitativos M. En C. Eduardo Bustos Farías 10

11 Multi-Criteria Evaluation (MCE) Evaluates a number of alternatives in the light of multiple criteria / factors. Factor scores of the alternatives Factor weights Relative importance of the factors Ranking scores of the alternatives Weighted combination of factors Factor 1 (1/3) Factor 2 (1/3) Factor 3 (1/3) Score A A A Métodos Cuantitativos M. En C. Eduardo Bustos Farías 11

12 Analytic Hierarchy Process Organizes factors into a tree structure Innate model of operation of human mind Helps complex decisions by decomposing the problem Determines factor weights Difficult to determine factor weights (numbers) directly Derives weights by comparing the relative importance between two factors Métodos Cuantitativos M. En C. Eduardo Bustos Farías 12

13 AHP Factor Weights Determination Factor relative importance pairwise comparison (9- point scale) 1 equally importance 3 moderately more importance 5 strongly more importance 7 very strongly more importance 9 extremely more importance The best weights fit into the pairwise comparisons. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 13

14 Analytic Hierarchy Process It is often difficult to conceptualize all the different elements of a problem, or there is not enough cognitive energy to prioritize those elements. The AHP was formulated to counter those situations, and is a mathematically-based theory. It employs two key aspects: (1) data from the various variables that make up the decision, and (2) judgments about those variables. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 14

15 Analytic Hierarchy Process (continued) The AHP requires taking the following steps: 1. Structuring the decision into a hierarchical model 2. Pairwise comparison of all objects and alternative solutions. The form of the model has four elements: 1. Goal the desired outcome 2. Criteria elements that comprise the goal 3. Subcriteria elements inside the criteria 4. Alternatives solutions or choices available This format allows decision makers to examine every part of a complex problem. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 15

16 The Analytic Hierarchy Process (AHP) Example: A company wants to purchase a new payroll and personnel records information system. Three systems are being considered (X, Y and Z). Three criteria are relevant: Price User support Ease of use Métodos Cuantitativos M. En C. Eduardo Bustos Farías 16

17 Pairwise Comparisons Step 1: In AHP is to create a pairwise comparison matrix for each alternative on each criterion using the following values. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 17

18 Pairwise Comparisons Value Preference 1 Equally Preferred 2 Equally to Moderately Preferred 3 Moderately Preferred 4 Moderately to Strongly Preferred 5 Strongly Preferred 6 Strongly to Very Strongly Preferred 7 Very Strongly Preferred 8 Very Strongly to Extremely Preferred 9 Extremely Preferred Métodos Cuantitativos M. En C. Eduardo Bustos Farías 18

19 Pairwise comparisons P ij = extent to which we prefer alternative i to alternative j on a given criterion. We assume P ji = 1/P ij Métodos Cuantitativos M. En C. Eduardo Bustos Farías 19

20 Normalization & Scoring STEP 2: To normalize a pairwise comparison matrix: 1) Compute the sum of each column, 2) Divide each entry in the matrix by its column sum. The score (s i ) for each alternative is given by the average of each row in the normalized comparison matrix. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 20

21 Consistency Step 3: We can check to make sure the decision maker was consistent in making the comparisons. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 21

22 Consistency The consistency measure for alternative i is: where C i = P s j ij j P ij = pairwise comparison of alternative i to j s i = score for alternative i Métodos Cuantitativos M. En C. Eduardo Bustos Farías 22 s i

23 Consistency If the decision maker was perfectly consistent each C i should equal to the number of alternatives in the problem. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 23

24 Consistency (cont d) Typically, some inconsistency exists. This can be a problem with pairwise elicitation methods. Not usually a problem as long as the consistency ratio (CR) is no more than 10% Métodos Cuantitativos M. En C. Eduardo Bustos Farías 24

25 Consistency (cont d) CR = CI 0.10 RI RI = for n = Métodos Cuantitativos M. En C. Eduardo Bustos Farías 25

26 Obtaining Remaining Scores & Weights This process is repeated to obtain scores for the other criteria as well as the criterion weights. The scores and weights are then used as inputs to a multicriteria scoring model in the usual way. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 26

27 Obtaining Remaining Scores & Weights This process is repeated to obtain scores for the other criteria as well as the criterion weights. The scores and weights are then used as inputs to a multicriteria scoring model in the usual way. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 27

28 Analytic Hierarchy Process The Analytic Hierarchy Process (AHP), is a procedure designed to quantify managerial judgments of the relative importance of each of several conflicting criteria used in the decision making process. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 28

29 Analytic Hierarchy Process For each criterion, perform steps 2 through Step 1: List the Overall Goal, Criteria, and Decision Alternatives Step 2: Develop a Pairwise Comparison Matrix Rate the relative importance between each pair of decision alternatives. The matrix lists the alternatives horizontally and vertically and has the numerical ratings comparing the horizontal (first) alternative with the vertical (second) alternative. Ratings are given as follows:... continued Métodos Cuantitativos M. En C. Eduardo Bustos Farías 29

30 Analytic Hierarchy Process Step 2: Pairwise Comparison Matrix (continued) Compared to the second alternative, the first alternative is: Numerical rating extremely preferred 9 very strongly preferred 7 strongly preferred 5 moderately preferred 3 equally preferred 1 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 30

31 Analytic Hierarchy Process Step 2: Pairwise Comparison Matrix (continued) Intermediate numeric ratings of 8, 6, 4, 2 can be assigned. A reciprocal rating (i.e. 1/9, 1/8, etc.) is assigned when the second alternative is preferred to the first. The value of 1 is always assigned when comparing an alternative with itself. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 31

32 Analytic Hierarchy Process Step 3: Develop a Normalized Matrix Divide each number in a column of the pairwise comparison matrix by its column sum. Step 4: Develop the Priority Vector Average each row of the normalized matrix. These row averages form the priority vector of alternative preferences with respect to the particular criterion. The values in this vector sum to 1. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 32

33 Analytic Hierarchy Process Step 5: Calculate a Consistency Ratio The consistency of the subjective input in the pairwise comparison matrix can be measured by calculating a consistency ratio. A consistency ratio of less than.1 is good. For ratios which are greater than.1, the subjective input should be re-evaluated For each criterion, perform steps 2 through Métodos Cuantitativos M. En C. Eduardo Bustos Farías 33

34 Analytic Hierarchy Process Step 6: Develop a Priority Matrix After steps 2 through 5 has been performed for all criteria, the results of step 4 are summarized in a priority matrix by listing the decision alternatives horizontally and the criteria vertically. The column entries are the priority vectors for each criterion. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 34

35 Analytic Hierarchy Process Step 7: Develop a Criteria Pairwise Development Matrix This is done in the same manner as that used to construct alternative pairwise comparison matrices by using subjective ratings (step 2). Similarly, normalize the matrix (step 3) and develop a criteria priority vector (step 4). Step 8: Develop an Overall Priority Vector Multiply the criteria priority vector (from step 7) by the priority matrix (from step 6). Métodos Cuantitativos M. En C. Eduardo Bustos Farías 35

36 Determining the Consistency Ratio Step 1: For each row of the pairwise comparison matrix, determine a weighted sum by summing the multiples of the entries by the priority of its corresponding (column) alternative. Step 2: For each row, divide its weighted sum by the priority of its corresponding (row) alternative. Step 3: Determine the average, λ max, of the results of step 2. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 36

37 Determining the Consistency Ratio Step 4: Compute the consistency index, CI, of the n alternatives by: CI = (λ max - n)/(n -1). Step 5: Determine the random index, RI, as follows: Number of Random Number of Random Alternative (n) Index (RI) Alternative (n) Index (RI) Step 6: Compute the consistency ratio: CR = CR/RI. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 37

38 Example: Gill Glass Designer Gill Glass must decide which of three manufacturers will develop his "signature toothbrushes. Three factors are important to Gill: (1) his costs; (2) reliability of the product; and, (3) delivery time of the orders. The three manufacturers are Cornell Industries, Brush Pik, and Picobuy. Cornell Industries will sell toothbrushes to Gill Glass for $100 per gross, Brush Pik for $80 per gross, and Picobuy for $144 per gross. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 38

39 Example: Gill Glass Hierarchy for the Manufacturer Selection Problem Overall Goal Select the Best Toothbrush Manufacturer Criteria Cost Reliability Delivery Time Decision Alternatives Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Métodos Cuantitativos M. En C. Eduardo Bustos Farías 39

40 Pairwise Comparison Matrix: Cost Gill has decided that in terms of price, Brush Pik is moderately preferred to Cornell and very strongly preferred to Picobuy.. In turn Cornell is strongly to very strongly preferred to Picobuy. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 40

41 Pairwise Comparison Matrix: Cost Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 1/3. Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is 1/7. Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 41

42 Pairwise Comparison Matrix: Cost Cornell Brush Pik Picobuy Cornell 1 1/3 6 Brush Pik Picobuy 1/6 1/7 1 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 42

43 Normalized Matrix: Cost Divide each entry in the pairwise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = /6 = 25/6. This gives: Cornell Brush Pik Picobuy Cornell 6/25 7/31 6/14 Brush Pik 18/25 21/31 7/14 Picobuy 1/25 3/31 1/14 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 43

44 Priority Vector: Cost The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: ( 6/25 + 7/31 + 6/14)/3 =.298 Brush Pik: (18/ /31 + 7/14)/3 =.632 Picobuy: ( 1/25 + 3/31 + 1/14)/3 =.069 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 44

45 Checking Consistency Multiply each column of the pairwise comparison matrix by its priority: 1 1/ = /6 1/ Divide these number by their priorities to get:.923/.298 = /.632 = /.069 = Métodos Cuantitativos M. En C. Eduardo Bustos Farías 45

46 Checking Consistency Average the above results to get λ max. λ max = ( )/3 = Compute the consistence index, CI, for two terms..051 CI = (λ max - n)/(n - 1) = ( )/2 = Compute the consistency ratio, CR, by CI/RI, where RI =.58 for 3 factors: CR = CI/RI =.051/.58 =.088 Since the consistency ratio, CR, is less than.10, this is well within the acceptable range for consistency. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 46

47 Pairwise Comparison Matrix: Reliability Gill Glass has determined that for reliability, Cornell is very strongly preferable to Brush Pik and equally preferable to Picobuy. Also, Picobuy is strongly preferable to Brush Pik. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 47

48 Pairwise Comparison Matrix: Reliability Cornell Brush Pik Picobuy Cornell Brush Pik 1/7 1 5 Picobuy 1/2 1/5 1 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 48

49 Normalized Matrix: Reliability Divide each entry in the pairwise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = 1 + 1/7 + 1/2 = 23/14. This gives: Cornell Brush Pik Picobuy Cornell 14/23 35/41 2/8 Brush Pik 2/23 5/41 5/8 Picobuy 7/23 1/41 1/8 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 49

50 Priority Vector: Reliability The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: (14/ /41 + 2/8)/3 =.571 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 =.278 Picobuy: ( 7/23 + 1/41 + 1/8)/3 =.151 Checking Consistency Gill Glass responses to reliability could be checked for consistency in the same manner as was cost. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 50

51 Pairwise Comparison Matrix: Delivery Time Gill Glass has determined that for delivery time, Cornell is equally preferable to Picobuy. Both Cornell and Picobuy are very strongly to extremely preferable to Brush Pik. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 51

52 Pairwise Comparison Matrix: Delivery Time Cornell Brush Pik Picobuy Cornell Brush Pik 1/8 1 1/8 Picobuy Métodos Cuantitativos M. En C. Eduardo Bustos Farías 52

53 Normalized Matrix: Delivery Time Divide each entry in the pairwise comparison matrix by its corresponding column sum. Cornell Brush Pik Picobuy Cornell 8/17 8/17 8/17 Brush Pik 1/17 1/17 1/17 Picobuy 8/17 8/17 8/17 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 53

54 Priority Vector: Delivery Time The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: (8/17 + 8/17 + 8/17)/3 =.471 Brush Pik: (1/17 + 1/17 + 1/17)/3 =.059 Picobuy: (8/17 + 8/17 + 8/17)/3 =.471 Checking Consistency Gill Glass responses to delivery time could be checked for consistency in the same manner as was cost. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 54

55 Pairwise Comparison Matrix: Criteria The accounting department has determined that in terms of criteria, cost is extremely preferable to delivery time and very strongly preferable to reliability, and that reliability is very strongly preferable to delivery time. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 55

56 Pairwise Comparison Matrix: Criteria Cost Reliability Delivery Cost Reliability 1/7 1 7 Delivery 1/9 1/7 1 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 56

57 Normalized Matrix: Criteria Divide each entry in the pairwise comparison matrix by its corresponding column sum. Cost Reliability Delivery Cost 63/79 49/57 9/17 Reliability 9/79 7/57 7/17 Delivery 7/79 1/57 1/17 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 57

58 Priority Vector: Criteria The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cost: (63/ /57 + 9/17)/3 =.729 Reliability: ( 9/79 + 7/57 + 7/17)/3 =.216 Delivery: ( 7/79 + 1/57 + 1/17)/3 =.055 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 58

59 Overall Priority Vector The overall priorities are determined by multiplying the priority vector of the criteria by the priorities for each decision alternative for each objective. Priority Vector for Criteria [ ] Cost Reliability Delivery Cornell Brush Pik Picobuy Métodos Cuantitativos M. En C. Eduardo Bustos Farías 59

60 Overall Priority Vector Thus, the overall priority vector is: Cornell: (.729)(.298) + (.216)(.571) + (.055)(.471) =.366 Brush Pik: (.729)(.632) + (.216)(.278) + (.055)(.059) =.524 Picobuy: (.729)(.069) + (.216)(.151) + (.055)(.471) =.109 Brush Pik appears to be the overall recommendation. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 60

61 References Saaty, T. 1980, The Analytic Hierarchy Process, New York, McGraw-Hill. Saaty, T. 1990, Multicriteria Decision Making: the analytic hierarchy process, Pittsburgh, PA, RWS Publications. Schmoldt, D., Peterson, D. and R. Smith 1994, The Analytic Hierarchy Process and Participatory Decision Making, Proceedings Decision Support 2001, Vol. 1, pp , September, Toronto. Sprague, R. and E. Carlson 1982, Building Effective Decision Support Systems, Englewood Cliffs, NJ, Prentice-Hall Métodos Cuantitativos M. En C. Eduardo Bustos Farías 61