*DECISION MODELING WITH MICROSOFT EXCELChapter 12Copyright 2001Prentice HallMulti-Objective Decision Making

*This section deals with the real-world topic of making a decision when there are multiple objectives or criteria to consider. For example:

*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.

*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 MHz2. Model B runs a Celeron chip at 333 MHz3. Model C runs a Pentium II chip at 450 MHzThe important criteria and their weights are:

*Now, rank each of the three models on these four criteria. Rank them on a scale from 1 to 10 as described earlier.Model B has the highest weighted score and thus would be the best computer to purchase.

*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.

*Analytic Hierarchy ProcessMultiple criteriaquantitativequalitative, intangible, subjectiveprovides measures of judgement consistencyderives priorities among criteria and alternativesuser-friendly pair-wise comparisons

*Using AHP1. Decompose the problem into a hierarchy2. Make pairwise comparisons and establish priorities among the elements in the hierarchy3. Synthesise the results (to obtain the overall ranking of alternatives w.r.t. goal)4. Evaluate the consistency of judgement

*The basic procedure is as follows:1. Develop the ratings for each decision alternative for each criterion bydeveloping a pairwise comparison matrix for each criterionnormalizing the resulting matrixaveraging the values in each row to get the corresponding ratingcalculating and checking the consistency ratio

*2. Develop the weights for the criteria bydeveloping a pairwise comparison matrix for each criterionnormalizing the resulting matrixaveraging the values in each row to get the corresponding ratingcalculating and checking the consistency ratio3. Calculate the weighted average rating for each decision alternative. Choose the one with the highest score.

*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)

*The important criteria are:1. The total cost of the installed system2. The follow-up service provided over the coming year3. The sophistication of the underlying math engines4. The amount of customization for Sleepwell

*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:Values 2, 4, 6, or 8 may also be assigned and represent preferences halfway between the integers on either side.

*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.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.

*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.

*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 Excels matrix multiplication function =MMULT().

*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.

*Approximation of the Consistency Index1. 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

*Random Index (RI)the CI of a randomly-generated pairwise comparison matrix

*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.

*Consistency ratio for Service.

*Consistency ratio for Sophistication.

*Consistency ratio for Customization.

*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.

*Consistency ratio for weights on criterion.

*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.

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

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

*

This equation is of the form: A w = l wSo w is an eigenvector of matrix A corresponding to eigenvalue l.(In fact, l 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.

*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 Aand the weights for the criteria are given by the eigenvector corresponding to eigenvalue l.Back to AHP ...

*Computing the weights for AHPEigenvector Method:1. Find largest eigenvalue of the pairwise comparison matrix2. Find corresponding eigenvectorApproximate 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.

*Consistency Indexreflects the consistency of ones judgement

CI = . lmax - n . n - 1

Random Index (RI)the CI of a randomly-generated pairwise comparison matrixTabulated by size of matrix:. n RI .20.030.5840.9051.1261.2471.3281.4191.45101.51

*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.

*AHP - SummaryIssues:InterdependenceInterval JudgementsIs a multiplicative scale appropriate?Easy to use Intuitive?General frameworkWidely usedCost/Benefit AnalysisVendor SelectionStrategic Planning