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1 DECISION MODELING WITH MICROSOFT EXCEL Chapter 12 Copyright 2001 Prentice Hall Multi-Objective Decision Making

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2 ANALYTICAL HIERARCHY PROCESS This section deals with the real-world topic of making a decision when there are multiple objectives or criteria to consider. For example: Choosing which employment offer to accept. Picking which computer (or car, etc.) to buy. Deciding which new product to launch first. Selecting a site for a new restaurant, hotel, etc. Rating the best cities in which to live. Choosing a new software package for your company.

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3 A simple way to attack such a decision would be to assign weights to each of the criteria that were to be considered in making the decision. Then, rank each decision alternative on a scale from 1 (worst) to 10 (best). Finally, you would multiply the weights times the rankings for each criterion and sum them up. The alternative with the highest score would be the most preferred.

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4 For example, you are in charge of purchasing the next computer for the office. You have to choose between the following three computers: 1. Model A runs an AMD K6-II chip at 400 MHz 2. Model B runs a Celeron chip at 333 MHz 3. Model C runs a Pentium II chip at 450 MHz The important criteria and their weights are: Criteria Weight Price 50% Price 50% Speed 15% Speed 15% Hard-disk Size 20% Hard-disk Size 20% Warranty/Support 15%

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5 Now, rank each of the three models on these four criteria. Rank them on a scale from 1 to 10 as described earlier. =SUM(C4:C7) =SUM(C4:C7) =SUMPRODUCT($C$4:$C$7,E4:E7) =SUMPRODUCT($C$4:$C$7,E4:E7) Model B has the highest weighted score and thus would be the best computer to purchase.

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6 This approach is quite simplistic and there are difficulties in setting the ranking scales on such different criteria. Analytic hierarchy process (AHP) also uses a weighted average approach idea, but it uses a method for assigning ratings (or rankings) and weights that is considered more reliable and consistent. (AHP) is based on pairwise comparisons between the decision alternatives on each of the criteria. Then, a similar set of comparisons are made to determine the relative importance of each criterion and thus produces the weights.

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7 Analytic Hierarchy Process Multiple criteria –quantitative –qualitative, “intangible”, subjective provides measures of judgement consistency derives priorities among criteria and alternatives “user-friendly” pair-wise comparisons

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8 Using AHP 1. Decompose the problem into a hierarchy 2. Make pairwise comparisons and establish priorities among the elements in the hierarchy overall ranking 3. Synthesise the results (to obtain the overall ranking of alternatives w.r.t. goal) 4. Evaluate the consistency of judgement

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9 The basic procedure is as follows: 1. Develop the ratings for each decision alternative for each criterion by developing a pairwise comparison matrix for each criteriondeveloping a pairwise comparison matrix for each criterion normalizing the resulting matrixnormalizing the resulting matrix averaging the values in each row to get the corresponding ratingaveraging the values in each row to get the corresponding rating calculating and checking the consistency ratiocalculating and checking the consistency ratio

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10 2. Develop the weights for the criteria by developing a pairwise comparison matrix for each criteriondeveloping a pairwise comparison matrix for each criterion normalizing the resulting matrixnormalizing the resulting matrix averaging the values in each row to get the corresponding ratingaveraging the values in each row to get the corresponding rating calculating and checking the consistency ratiocalculating and checking the consistency ratio 3. Calculate the weighted average rating for each decision alternative. Choose the one with the highest score.

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11 Consider the following example: Sleepwell Hotels is looking for some help in selecting the “best” revenue management software package from among several vendors. The director of revenue management for this chain of hotels has been given this task. Three vendors have been identified whose software meets the following basic needs: Revenue Technology Corporation (RTC) PRAISE Strategic Solutions (PSS) El Cheapo (EC)

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12 The important criteria are: 1. The total cost of the installed system 2. The follow-up service provided over the coming year 3. The sophistication of the underlying math engines 4. The amount of customization for Sleepwell

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13 The first step in the AHP procedure is to make pairwise comparisons between the vendors for each criterion. Here is the standard scale for making these comparisons: DESCRIPTION 1Equally preferred 3Moderately preferred 5Strongly preferred 7Very strongly preferred 9Extremely strongly preferred RATING Values 2, 4, 6, or 8 may also be assigned and represent preferences halfway between the integers on either side.

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14 Start with the total cost criterion and generate the following data in a spreadsheet: The vendor in the row is being compared to the vendor in the column. =1/C4 =1/D4 =1/D5 A value between 1 and 9 indicates that the vendor in the row is preferred to the vendor in the column. If the vendor in the column is preferred to the vendor in the row, then the inverse of the rating is given.

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15 The next step is to normalize the matrix. This is done by totaling the numbers in each column. Each entry in the column is then divided by the column sum to yield its normalized score. =SUM(B4:B6) =B4/B$8 =AVERAGE(B12:D12) The average is calculated for the “Total Cost” criterion. Highest average score

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16 Now, calculate the consistency ratio and check its value. The purpose for doing this is to make sure that the original preference ratings were consistent. There are 3 steps to arrive at the consistency ratio: 1. Calculate the consistency measure for each vendor. 2. Calculate the consistency index (CI). 3. Calculate the consistency ratio (CI/RI where RI is a random index). To calculate the consistency measure, we can take advantage of Excel’s matrix multiplication function =MMULT().

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17 Multiply the average rating for each vendor times the scores in the first row one-at-a-time, sum these products up and divide this sum by the average rating for the first vendor. =MMULT(B4:D4,$E$12:$E$14)/E12 =(AVERAGE(F12:F14)-3)/2 =F16/F18) Provided by AHP (see next slide)

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18 Approximation of the Consistency Index 1. Multiply each column of the pairwise comparison matrix by the corresponding weight. 2. Divide of sum of the row entries by the corresponding weight. 3. Compute the average of the values from step 2, denote it by Lmax. 4. The approximate CI is

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19 RANDOM INDEX 20.00 20.00 30.58 30.58 40.90 40.90 51.12 51.12 61.24 61.24 71.32 71.32 81.41 81.41 91.45 91.45 101.51 N Random Index (RI) the CI of a randomly-generated pairwise comparison matrix

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20 If we are perfectly consistent, then the consistency measures will equal n and therefore, the CIs will be equal to zero and so will the consistency ratio. If this ratio is very large (Saaty suggests > 0.10), then we are not consistent enough and the best thing to do is go back and revise the comparisons. Now, continue for the other three criteria. You can easily do this by copying the “Total Cost” sheet into three other sheets (“Service,” “Sophistication,” and “Custom”) and then simply changing the pairwise comparisons.

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21 Consistency ratio for “Service.”

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22 Consistency ratio for “Sophistication.”

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23 Consistency ratio for “Customization.”

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24 In all three cases, the CR value ranges from 0.0 to 0.047 which means that we are being consistent. Note also that PSS is the winner on the Service criterion, RTC and PSS are tied for the best in terms of Sophistication, and PSS is considered the best on Customization. All of this work concludes the first step in the procedure. The next step is to use similar pairwise comparisons to determine the appropriate weights for each of the criteria. The process is the same in that we make comparisons, except that now we make the comparisons between the criteria not the vendors.

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25 =1/C4 =1/D4 =1/D5 =1/E4 =1/E5=1/E6 =SUM(B4:B7) =B4/B$8 =AVERAGE(B12:E12) =MMULT(B4:E4,$F$12:$F$15)/F12 =(AVERAGE(G12:G15)-4)/3 =G16/G18) Consistency ratio for weights on criterion.

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26 =SOPHISTICATION!E12 =WEIGHTS!F12 =TOTAL COST!E12 =TOTAL COST!E13 =TOTAL COST!E14 =SERVICE!E12=SERVICE!E13 =SERVICE!E14 =CUSTOM!E12=CUSTOM!E13=CUSTOM!E14 =SUMPRODUCT($B$3:$B$6,C3:C6) =SOPHISTICATION!E13=SOPHISTICATION!E14 The final step is to calculate the weighted average ratings of each decision alternative and use the results to decide from which vendor to purchase the software. These results are pulled from all the other worksheets. From these results, we find that RTC barely edges out PSS for the software contract.

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27 The mathematics of AHP Suppose we already know the weights [w1, w2, w3,... wn] of the n criteria and we form the following n x n pairwise-ratio matrix:

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28 This pairwise-ratio matrix A and the vector of weights satisfy the following equation:

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29 This equation is of the form: A w = w So w is an eigenvector of matrix A corresponding to eigenvalue (In fact, is the only non-zero eigenvalue, and w the unique eigenvector.) Now, if we only know A, but not w, we can find what w is by solving for the eigenvalues and eigenvectors of A.

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30 If : we use a continuous scale instead of a 9-point scale, and, more importantly, our judgement is consistent, then the pairwise comparison matrix is exactly of the form A and the weights for the criteria are given by the eigenvector corresponding to eigenvalue. Back to AHP...

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31 Computing the weights for AHP Eigenvector Method: 1. Find largest eigenvalue of the pairwise comparison matrix 2. Find corresponding eigenvector Approximate Method: 1. Normalise each column (i.e. divide each entry by its column total) 2. The average values of row i in the normalised matrix is the estimate for weight i.

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32 Consistency Index reflects the consistency of one’s judgement CI =. max n. n 1 Random Index (RI) the CI of a randomly- generated pairwise comparison matrix Tabulated by size of matrix:. n RI. 20.0 30.58 40.90 51.12 61.24 71.32 81.41 91.45 101.51

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33 Consistency Ratio CR = CI / RI In practice, a CR of 0.1 or below is considered acceptable. Any higher value at any level indicate that the judgements warrant re-examination.

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34 AHP - Summary Issues: –Interdependence –Interval Judgements –Is a multiplicative scale appropriate? Easy to use Intuitive? General framework Widely used –Cost/Benefit Analysis –Vendor Selection –Strategic Planning

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