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THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE CAR PURCHASE EXAMPLE We now consider a motivating example. After completing this example, you will have an understanding of the basics of AHP and its application through Expert Choice (www.expertchoice.com). We want to apply the AHP to help a couple decide which car they should purchase. CAR PURCHASE EXAMPLE The couple is considering three criteria: cost, safety, and appearance. They have narrowed their alternatives to three specific cars: Honda, Mazda, and Volvo. We demonstrate how to build the AHP hierarchy in Expert Choice. EXPERT CHOICE: FILE SETUP After launching Expert Choice, select the File, New option, and after selecting a destination folder, enter a file name such as CARS. (Expert Choice add the AHP file extension.) Next, enter a description for your goal, such as, “Select the best car.” EXPERT CHOICE: FILE SETUP To enter the criteria, for example, cost, safety, and appearance, use the Edit, and Insert Child of Current Node commands. Use the Esc key or hit an extra enter when finished entering the criteria. To add the alternative cars select the Edit, Alternative, and Insert commands. EXPERT CHOICE: FILE SETUP You can also use the “Add Alternative” button in the upper right hand corner of the model window. Repeat for all alternatives. Additional details can be found in the Expert Choice tutorial provided with the software. ANALYZING THE HIERARCHY 1. Determine the weights of the alternatives for each criterion. 2. Determine the priorities or weights of the criteria in achieving the goal. 3. Determine the overall weight of each alternative in achieving the goal. This is accomplished by combining the results of the first two stages and is called synthesis. ANALYZING THE HIERARCHY To complete the first stage, the couple can base their judgments on the following (hypothetical) performance information. All alternative pairwise comparisons should be based on data. HYPOTHETICAL DATA FOR CAR PURCHASE EXAMPLE Car Honda Mazda Volvo Cost $22,000 28,500 33,000 Safety* 28 39 52 Appearance Sporty Slick Dull * Safety Rating from a consumer testing service - the higher the number, the safer the car. DETERMINING PRIORITIES The couple begins by making pairwise comparison judgments between each pair of cars for the cost criterion. In our example, three judgments are needed: Honda to Mazda, Mazda to Volvo, and Honda to Volvo. The scale on the next page is the standard one. STANDARD 1 - 9 MEASUREMENT SCALE Intensity of Importance 1 3 Definition Equal importance Moderate importance 5 Strong importance 7 Very strong 9 Extreme importance 2, 4, 6, 8 1.1 - 1.9 Reciprocals of above For compromise values For tied activities If activity A has one of the above numbers assigned to it when compared with activity B, then B has the reciprocal value when compared to A. Explanation Two activities contribute equally Experience and judgment slightly favor one activity over another Experience and judgment strongly favor one activity over another An activity is favored very strongly over another The evidence favoring one activity over another is of the highest possible order of affirmation Sometimes one needs to interpolate a compromise between the above judgment numerically because there is no good word to describe it When elements are close and nearly indistinguishable; moderate is 1.3 and extreme is 1.9 For example, if the pairwise comparison of A to B is 3.0, then the pairwise comparison of B to A is 1/3 COST PAIRWISE COMPARISONS The pairwise comparisons are represented in the form of pairwise comparison matrices. The computation of the weights are also shown. Consider the pairwise comparison matrix to compare the cars for the cost criterion. Remember that the costs of the three cars are: $22000, $28500, and $33000, respectively. COST PAIRWISE COMPARISONS If we compare the Honda to the Honda, obviously they are equal. Therefore, a 1 (equal preferred) is placed in the first row, first column entry of the matrix. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 28.5K Mazda 33K Volvo Volvo COST PAIRWISE COMPARISONS The other entries along the main diagonal of the matrix are also 1. This simply means that everything is equally preferred to itself. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 28.5K Mazda 1 33K Volvo Volvo 1 COST PAIRWISE COMPARISONS Suppose we believe the Honda ($22000) is equally to moderately preferred to the Mazda ($28500). Place a 2 in the row 1, column 2 entry. Some might argue that the Honda should be 1.295 times better than the Mazda (28,500/22,000). COST PAIRWISE COMPARISONS Do you agree? It depends! For some, $28,500 is significantly greater than $22,000, implying a judgments greater than 1.295. Others with a lot of money may perceive virtually no difference between the two costs, implying a judgment somewhere between 1 and 1.295. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1 33K Volvo Volvo 1 COST PAIRWISE COMPARISONS If the Honda is 2 times better than the Mazda, this implies that the Mazda ($28500) is one half as good as the Honda ($22000). The reciprocal judgment, (1/2), should be placed in the row 2, column 1 entry of the matrix. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo Volvo 1 COST PAIRWISE COMPARISONS Suppose that we judge the Mazda ($28500) to be equally to moderately preferred to the Volvo ($33000). The following judgments would be entered in the matrix. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/2 Volvo 2 1 COST PAIRWISE COMPARISONS Assuming perfect consistency of judgments, we would expect that the Honda ($22000) is 4 times (that is, moderately to strongly) preferred to the Volvo ($33000). We will relax this assumption later. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 Volvo 4 2 1 COST PAIRWISE COMPARISONS The matrix is now complete and the weights for each car (for the cost criterion) can be computed. The exact computational procedure is implemented in Expert Choice. For details see Expert Choice homepage and download AHPDEMO.EXE. COST PAIRWISE COMPARISONS A simple three step procedure can be used to approximate the weights for each alternative. Essentially, this procedure normalizes the ratios of the judgments between any pair of alternatives. COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS Volvo 4 2 1 ------- COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS 7/4 7/2 Volvo 4 2 1 ------7 COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS 7/4 7/2 Volvo 4 2 1 ------7 COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS 7/4 7/2 Volvo 4 2 1 ------7 B. ADJUSTED COST PAIRWISE COMPARISON MATRIX Honda Mazda Honda 4/7* 4/7 Mazda 2/7 2/7 Volvo 1/7 1/7 Volvo 4/7 2/7 1/7 * This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). COST PAIRWISE COMPARISONS Notice that no variation is seen across the rows because the judgments are perfectly consistent. For the third column, judgments totaling 7 were awarded. The Honda received 4 of 7 (57.1%), the Mazda 2 of 7 (28.6%), and the Volvo 1 of 7 (14.3%) of the weight. Similar comparisons can be made for the other two columns. COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS 7/4 7/2 Volvo 4 2 1 ------7 B. ADJUSTED COST PAIRWISE COMPARISON MATRIX Honda Mazda Honda 4/7* 4/7 Mazda 2/7 2/7 Volvo 1/7 1/7 Volvo 4/7 2/7 1/7 * This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/2 ------------COLUMN TOTALS 7/4 7/2 B. ADJUSTED COST PAIRWISE COMPARISON MATRIX Honda Mazda Honda 4/7* 4/7 Mazda 2/7 2/7 Volvo 1/7 1/7 Volvo 4 2 1 ------7 Volvo 4/7 2/7 1/7 TOTAL WEIGHTS (ROW AVG.) 0.571 0.286 0.143 --------1.000 * This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). EXPERT CHOICE: Entering Judgments Expert Choice offers a variety of modes for entering the judgments. Highlight the cost node and select the Pairwise Numerical comparison button (3:1). This button appears on the top left-hand side of the toolbar to the right of the model view button. EXPERT CHOICE: Entering Judgments Sliding the bar between Honda and Mazda to the left so that it rests on the 2 means that the Honda is two times better than the Mazda when considering cost. If the Mazda were 2 times better than the Honda, the bar would be slid to the 2 on the right. The other comparisons are entered in a similar fashion. EXPERT CHOICE: Entering Judgments For our problem, Expert Choice only displays three judgments. 1’s along the main diagonal and reciprocal judgments do not appear. EXPERT CHOICE: Entering Judgments There are different modes for entering judgments. The Pairwise Verbal Comparisons (ABC) and the Pairwise Graphical Comparisons (the button that looks like a bar graph) are available. The only difference between these modes is how the pairwise comparison questions are displayed. EXPERT CHOICE: Entering Judgments A 1-9 scale is used for numerical comparisons. The verbal comparisons are: equal, moderate, strong, very strong, and extreme. The graphical mode makes comparisons based on the length of two bars. The user selects the desired mode. EXPERT CHOICE: Entering Judgments After entering all pairwise comparisons, record judgments by clicking Yes. The model view will be displayed with alternative weights for the cost criterion now appearing. INCONSISTENCY OF JUDGMENTS Since our pairwise comparisons were perfectly consistent, Expert Choice reports Incon: 0.00. If this ratio is greater than 0.1 some revision of judgments is required. Select Inconsistency (within any Pairwise Comparison mode) to identify the most inconsistent judgments. INCONSISTENCY OF JUDGMENTS Inconsistency of judgments may result from: problems of estimation; errors between the comparisons; or, the comparisons may be naturally inconsistent. INCONSISTENCY OF JUDGMENTS One example of natural inconsistency is in a sporting contest. If team A is twice as likely to beat team B, and if team B is three times as likely to beat team C, this does not necessarily imply that team A is six times as likely to beat team C. This inconsistency may result because of the way that the teams “match-up” overall. INCONSISTENCY OF JUDGMENTS The point is not to stop inconsistency from occurring. Make sure that the level of inconsistency remains within some reasonable limit. INCONSISTENCY OF JUDGMENTS How does a judgment change affect the car weights? Suppose the Mazda to Volvo changes from 2 to 3. This obviously changes the comparison for Volvo to Mazda from (1/2) to (1/3). The judgments are now somewhat inconsistent. COST PAIRWISE COMPARISONS A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/3 Volvo 4 3 1 COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/3 ------------COLUMN TOTALS 7/4 10/3 Volvo 4 3 1 ------8 COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/3 ------------COLUMN TOTALS 7/4 10/3 Volvo 4 3 1 ------8 B. ADJUSTED COST PAIRWISE COMPARISON MATRIX Honda Mazda Honda 4/7* 6/10 Mazda 2/7 3/10 Volvo 1/7 1/10 Volvo 4/8 3/8 1/8 * This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). COST PAIRWISE COMPARISONS 1. 2. 3. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. THIS RESULTS IN THE ADJUSTED MATRIX. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda 22K Honda 1 2 28.5K Mazda 1/2 1 33K Volvo 1/4 1/3 ------------COLUMN TOTALS 7/4 10/3 B. ADJUSTED COST PAIRWISE COMPARISON MATRIX Honda Mazda Honda 4/7* 6/10 Mazda 2/7 3/10 Volvo 1/7 1/10 Volvo 4 3 1 ------8 Volvo 4/8 3/8 1/8 TOTAL WEIGHTS (ROW AVG.) 0.557 0.320 0.123 -------1.000 * This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). INCONSISTENCY OF JUDGMENTS The new weights are: 0.557, 0.320, and 0.123. The inconsistency resulted in some change in the original weights of 0.571, 0.286, and 0.143. As expected, the weight for the Mazda increased while the weight for the Volvo decreased. The weights now vary across each row. Essentially, inconsistency measures the degree of variation across the rows. EXPERT CHOICE: Revising Judgments To make this change in Expert Choice, highlight cost node and select any Pairwise Comparison mode. Within the numerical mode, slide the comparison bar to the left from 2 to 3, select the Model View, and record the judgments to see the new weights. The weights of 0.558, 0.320, and 0.122 are slightly different from the three-step procedure weights. This is not due to rounding -- Expert Choice gives the exact results. INCONSISTENCY OF JUDGMENTS The inconsistency ratio is now 0.02. The weights can also be used to measure the effectiveness of the alternatives. For example, based on all comparisons, the Honda is 1.74 (0.558/0.320) times better than the Mazda. INCONSISTENCY OF JUDGMENTS We knew that a $22,000 car is better than a $28,500 car, but now we know how much better. Why is this ratio 1.74 and not the pairwise comparison of 2? Inconsistency in the judgments! REMAINING COMPUTATIONS Next, the cars must be pairwise compared for the safety criterion and then for the appearance criterion. These judgments are shown on the next page. The safety comparisons are all inverted, that is, for each comparison, the top bar was moved to the left. This means that the Mazda is 2 times more preferred than the Honda, with respect to safety. SAFETY & APPEARANCE JUDGMENTS Safety Pairwise Comparison Matrix Honda Mazda 28 Honda 1 1/2 39 Mazda 2 1 52 Volvo 5 4 Appearance Pairwise Comparison Matrix Honda Mazda SportyHonda 1 5 Slick Mazda 1/5 1 Dull Volvo 1/9 1/2 Volvo 1/5 1/4 1 Volvo 9 2 1 REMAINING COMPUTATIONS Next, the criteria must be pairwise compared. These judgments are shown on the next page. There are no data to support these judgments since they are purely a reflection of your preferences. CRITERIA JUDGMENTS Original Criteria Pairwise Comparison Matrix Cost Safety Appearance Cost 1 1/2 3 Safety 2 1 5 Appearance 1/3 1/5 1 REMAINING COMPUTATIONS The last stage computes the final weights for each car. Multiply the criteria weight by the car weight for each criterion and then sum over all criteria. This is nothing more than a weighted average. The computational results are shown next. FINAL CAR WEIGHTS COST 0.309 CARS Honda Mazda Volvo 0.558 0.320 0.122 CRITERIA WEIGHTS SAFETY APPEARANCE 0.582 0.109 FINAL WEIGHTS 0.117 0.761 0.200 0.158 0.683 0.082 FINAL CAR WEIGHTS COST 0.309 CARS Honda Mazda Volvo 0.558 0.320 0.122 CRITERIA WEIGHTS SAFETY APPEARANCE 0.582 0.109 FINAL WEIGHTS 0.117 0.761 0.324 0.200 0.158 0.683 0.082 Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 0.173 0.068 0.083 FINAL CAR WEIGHTS COST 0.309 CARS Honda Mazda Volvo 0.558 0.320 0.122 CRITERIA WEIGHTS SAFETY APPEARANCE 0.582 0.109 FINAL WEIGHTS 0.117 0.761 0.324 0.200 0.158 0.232 0.683 0.082 Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 0.173 0.068 0.083 Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232 0.099 0.116 0.017 FINAL CAR WEIGHTS COST 0.309 CARS Honda Mazda Volvo 0.558 0.320 0.122 CRITERIA WEIGHTS SAFETY APPEARANCE 0.582 0.109 FINAL 0.117 0.761 0.200 0.158 0.683 0.082 WEIGHTS 0.324 0.232 0.444 Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 0.173 0.068 0.083 Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232 0.099 0.116 0.017 Volvo: (0.122)(0.309) + (0.683)(0.582) + (0.082)(0.109) = 0.444 0.038 0.397 0.009 LOCAL VS GLOBAL WEIGHTS For cost, the local weights for the cars are 0.558, 0.320, and 0.122 and sum to 1.000. The global weights are computed by multiplying the cost criterion weight by the local car weights. The global weights are 0.173, 0.099, and 0.038 and sum to the cost criterion weight of 0.309. EXPERT CHOICE: Synthesis The final weights are shown in Expert Choice after all comparisons are entered and when the Model View is displayed and the goal is highlighted. Choose Distributive Mode. The difference between the Distributive and Ideal modes will be discussed later. INTERPRETING THE RESULTS The final weights provide a measure of the relative performance of each alternative. It is important to properly interpret the meaning of these numbers. The Volvo is ranked first, the Honda second, and Mazda third. The Volvo is preferred 1.37 (0.444/0.324) times more than the Honda. INTERPRETING THE RESULTS Should we buy the Volvo? The output is a decision-making aid and cannot replace the decision-maker. The results can be used to support discussion and possibly the judgments will be revised. This iterative process is quite normal. AHP can help to facilitate communication and generate consensus between different groups. SYNTHESIS MODES The process used to compute the final weights is called distributive synthesis. This method works well when there is a fixed amount of resources that must be distributed to a fixed set of alternatives. SYNTHESIS MODES In some cases after completing an AHP analysis, an additional alternative may need to be considered. It is possible that a rank reversal could occur. Our rankings are: Volvo, Honda, and Mazda. If another Volvo is added that is similar to the original Volvo, it is possible that the Honda will be ranked higher than the original Volvo. SYNTHESIS MODES In some cases this is acceptable, in others it is not. Distributive synthesis should not be used if preservation of rank is important. Ideal Synthesis should be used to prevent rank reversal. IDEAL MODE The ideal mode gives the full weight of the criterion to the alternative that ranks highest under that criterion. The other alternatives are given a portion of the criterion weight based on their local weight. IDEAL MODE The local weights for the three cars with respect to cost are: 0.558, 0.320, and 0.122, respectively. The cost criterion weight is 0.309. Since the Honda has the highest cost weight it is initially assigned the full cost weight of 0.309. Mazda would be (0.320 / 0.558)*(0.309) = 0.177. Volvo would be (0.122 / 0.558)*(0.309) = 0.068. IDEAL MODE Using the same approach, the weights for the three cars with respect to safety are: 0.100, 0.170, and 0.582, respectively. The weights for the three cars with respect to appearance are: 0.109, 0.023, and 0.012, respectively. IDEAL MODE For each car, add the three criteria weights: Honda Cost 0.309 Safety 0.100 Appearance 0.109 Total 0.518 Mazda 0.177 0.170 0.023 0.370 Volvo 0.068 0.582 0.012 0.662 IDEAL MODE For each car, add the three criteria weights: Honda Cost 0.309 Safety 0.100 Appearance 0.109 Total 0.518 Since the sum Mazda Volvo of the three 0.177 0.068 weights is 0.170 0.582 1.550, we 0.023 0.012 divide each 0.370 0.662 weight by 1.550 to normalize the results. IDEAL MODE For each car, add the three criteria weights: Since the sum Honda Mazda Volvo of the three Cost 0.309 0.177 0.068 weights is Safety 0.100 0.170 0.582 1.550, we Appearance 0.109 0.023 0.012 divide each Total 0.518 0.370 0.662 weight by 1.550 to normalize the Total/1.550 0.335 0.239 0.427 results. These are the ideal weights reported in Expert Choice. SENSITIVITY ANALYSIS Sensitivity analysis is an important aspect of any decision-making process. Sensitivity analysis determines whether small changes in judgments affects the final weights and rankings of the alternatives. If so, the decision-maker may want to review the sensitive judgments. EXPERT CHOICE: Sensitivity Analysis In Expert Choice sensitivity analysis from the GOAL shows how the weights and the rankings of the alternatives change if some or all of the criteria weights change. There are five graphical sensitivity analysis modes available: Performance, Dynamic, Gradient, Two-Dimensional, and Difference. The first three show how a change in a criterion weight affects the final weights of the alternatives. EXPERT CHOICE: Sensitivity Analysis The last two show how the alternatives perform with respect to any two criteria. Performance: places all sensitivity information on a single chart with horizontal line graphs for the alternatives linked to vertical bars for the criteria. Dynamic: two sets of dynamically linked horizontal bar graphs: one for criteria and one for alternatives. EXPERT CHOICE: Sensitivity Analysis Gradient: a line graph that shows how the weights of the alternatives vary according to the weight assigned to a specific criterion. (Use the X-Axis to change the selected criterion.) Two-Dimensional: shows how well the alternatives perform with respect to any two criteria. Difference: a graph that shows the differences between any two alternatives for any criterion. EXPERT CHOICE: Sensitivity Analysis An important use of sensitivity analysis is to determine how much a given criterion weight must change before there is a change in the rankings of the two highest alternatives. This type of breakeven analysis can be easily done in Expert Choice. EXPERT CHOICE: Sensitivity Analysis Choose Dynamic from the Sensitivity-Graphs option. Drag the cost criterion bar 30.9% to approximately 45.9%, and see that the Volvo and Honda have the same highest final weight. The final rankings are relatively insensitive to a change in the cost weight since it had to be increased by almost 50% to get a change in the final rankings. The sensitivity results are different for the ideal mode.