MULTIPLE-OBJECTIVE DECISION MAKING TECHNIQUE Analytical Hierarchy Process Business Intelligence and Decision Making Professor Jason Chen The analytical hierarchy process (AHP) is a systematic procedure for representing the elements of any problem, hierarchically. It organizes the basic rationality by breaking down a problem into its smaller and smaller constituent parts and then guides decision makers through a series of pairwise comparison judgments to express the relative strength or intensity of impact of the elements in the hierarchy. These judgments are then translated to numbers. The steps of creating an AHP is summarized as follows:. Define the problem and determine what you want to know.. Structure the hierarchy from the top (the objectives from a managerial viewpoint) through the intermediate levels (criteria on which subsequent levels depend) to the lowest level (which usually is a list of the alternatives).. Construct a set of pairwise comparison matrices for each of the lower levels - one matrix for each element in the level immediately above. An element in the higher level is said to be a governing element for those in the lower level since it contributes to it or affects it. In a complete simple hierarchy, every element in the lower level affects every element in the upper level. The elements in the lower level are then compared to each other based on their effect on the governing element above. This yields a square matrix of judgments. The pairwise comparisons are done in terms of which element dominates the other. These judgments are then expressed as integers (see table below for judgment values). If element A dominates element B, then the whole number (integer) is entered in row A, column B and the reciprocal (fraction) is entered in row B, column A. Of course, if element B dominates element A then the reverse occurs. The whole number is then placed in the B, A position with the reciprocal automatically being assigned to the A, B position. If A and B are judged to be equal, a one is assigned to both positions.. There are n (n-)/ judgments required to develop each matrix in step (remember, reciprocals are automatically assigned in each pairwise comparison). 5. Having made all the pairwise comparisons and entered the data, the consistency is determined using the eigenvalue. The consistency index is tested then using the departure of from ncompared with corresponding average values for random entries yielding the consistency ratio C.R. (See Random Indices Table below). 6. Steps, and 5 are performed for all levels and clusters in the hierarchy. 7. Hierarchical synthesis is now used to weight the eigenvectors by the weights of the criteria and the sum is taken over all weighted eigenvector entries corresponding to those in the next lower level of the hierarchy. 8. The consistency of the entire hierarchy is found by multiplying each consistency index by the priority of the corresponding criterion and adding them together. The result is then divided by the same type of expression using the random consistency index corresponding to the dimensions of each matrix weighted by the priorities as before. Note first the consistency ratio (C.R.) should be about 0%, or less to be acceptable. If not, the quality of the judgments should be improved, perhaps by revising the manner in which questions are asked in making the pairwise comparisons. If this should fail to improve consistency then it is likely that the max AHP Page-
problem should be more accurately structured; that is, grouping similar elements under more meaningful criteria. A return to step would be required, although only the problematic parts of the hierarchy may need revision. (The reader can calculate the consistency of the hierarchy of the house buying example and show its value to be.08 which is acceptable.) Scale of Relative Importance Intensity of Definition Explanation relative importance Equal importance Two activities contribute Moderate importance of one over another equally to the objectives Experience and judgment slightly favor one activity over another 5 Essential or strong importance Experience and judgment strongly favor one activity over another 7 Demonstrated importance An activity is strongly favored and its dominance is demonstrated in practice 9 Extreme importance The evidence favoring one activity over another is of the highest possible order of affirmation.,, 6, 8 Intermediate value between the two adjacent judgments Reciprocals of above non-zero numbers If an activity has one of the above numbers (e.g., ) compared with a second activity, then the second activity has the reciprocal value (i.e., /) when compared to the first. When compromise is needed. Random Indices (R.I.) for Consistency Check n 5 6 7 8 9 0 R.I. 0.58.90.....5.5 Reference Satty, Thomas L. & Kearns, Kevin P. (985), Analytical Planning: The Organization of Systems, Pergamon Press AHP Page-
Application Case of AHP: Jane is about to graduate from college and is trying to determine which job offer to accept. She plans to choose between three offers by determining how well each offer meets the following criteria (objectives): High starting salary of life in city where job is located of work of job to family Assumptions: Jane has hard time in prioritizing those criteria. In other words, she needs to find one way to decide the weights for those criteria. AHP provides such a function.. Determine the problem: What job offer will give Jane possibly highest satisfaction?. Structure the hierarchy by putting the top objective (satisfaction with job), criteria, and alternatives as follows. Satisfaction with a job Starting salary Life to family. Construct a pairwise comparison matrix for the criteria level as follows. And then compare those criteria to each other with respect to the top objective. Take the first row as an example, we may ask: With respect to Satisfaction with a job, comparing Starting Salary to Life, which is more important than the other? How much more important it is than the other? Here, the author assumes that Starting Salary is strongly more important than Life. That is why 5 is entered into the Salary row and column. Compared to, Salary is just a little bit more important. That is why is entered into Salary row and column. Satisfaction with a job Salary Salary / 5 / / 5 / / / (six judgments) AHP Page-
. After pairwise comparisons, we need to determine the weights of the four criteria. First, normalizing the entries by adding entries in each column and then divide each entry in the column by the sum of the column. For example, the sum of the entries in the Salary column is.95 (=+0.++) and each entry in the Salary column is divided by.95. The same procedure is applied to other columns. A normal = Salary Salary.58.95 0..06.95.56.95.8.95 5.5000 0.000 0.000 0.000 0.5000.50.500.50.5.0667.667. Second, estimate weights by calculating averages of entries in each row. Thus, the weight of Salary is the weight of is the weight of is the weight of is.58.5000.5000.5 w w w.06.000.50.0667.56.000.500.667 w.8.000.50..55.0986..66 5. Using the same steps of and to determine the score of each alternative on each criterion. Take the first criterion Salary as an example. One pairwise matrix is constructed as follows: A SALARY / / / Using the procedure in to normalize the matrix and calculate the score of each job on the Salary criterion. SALARY A.57.857.9.57.857.9.57 (Normalized pairwise matrix).857.9 Calculate the average of entries in row to obtain the vector of scores for the three jobs on Salary. AHP Page-
S 7 0.857 0.9 6. Repeating the same procedure to calculate scores of these three jobs on criteria Life,, and. A / / / S 0.6 09 0.97 A Job 7 A / 7 / / S 0.088 0.6687 0. A Job 7 A / / 7 / S 0.08 0.605 7. Determining the overall weights by combining the relative importance matrix with scores matrices. Starting Life Overall Salary Relative.55.0986..66 Importance.57.6.088.08.857.509.6687.5 0.089.9.97..605 0.96 The computation for obtaining the overall weights is as follows:.57.6.088.857.509.6687.9.97..08.55.5.0986 0.089.605. 0.96.66 Thus, according to the overall score, the AHP suggests that Jane should take job B, which has the highest score among three jobs. AHP Page-5
8. Check for consistency. Calculating consistency index (C.I) by multiplying the comparison (judgment) matrix by the vector of priority as follows: /5 Aw / / 5 / /.55.077 /.0986 0.958. 0.989.66 9 And then calculate the largest eigenvalue of the judgment matrix A. max =.077 5.958 0.0986.989 0. max n.077 C. I. 0.059 n 9 0.66.077 (largest eigenvalue of A) and then compare the value of C.I. to the value of random index (R.I). If the ratio of C.I. to R.I. is less than 0%, then we can say the judgment process is relatively consistent and the matrix is acceptable. Otherwise, the decision maker may need to re-examine the judgment process and re-compare criteria or alternatives. The consistency ratio (C.R.) is computed as follows: C.R. = C.I. / R.I. = 0.059/0.9 = 0.076666 =.7% < 0% Therefore, the judgment process is relatively consistent and the matrix is acceptable. The software (Web-HIPRE) is available at http://www.hipre.hut.fi or http://hipre.aalto.fi (note that you may need to update java component if the software is not working properly) AHP Page-6