Document Sample

Why Barzilai’s Criticisms of the AHP are Incorrect: Validation Rozann Whitaker Creative Decisions Foundation 4922 Ellsworth Avenue Pittsburgh, PA USA rozann@creativedecisions.net Abstract AHP has been claimed to have flaws on the basis of examples that have been offered. We show that such an AHP model was incorrectly formulated and offer many examples of AHP giving correct answers. 1. Introduction Jonathan Barzilai has made several criticisms of the AHP (Analytic Hierarchy Process) in various papers he has written. One such paper, brought to our attention by a Canadian friend who asked us to write a commentary on it, is “On the decomposition of value functions,” Operations Research Letters 22 (1998) 159-170. The flaws he thinks he has found are due solely to the incorrect way he formulates his AHP models. We use his same examples to show they give the right results when modeled according to AHP principles. The main thrust of Barzilai’s paper is that AHP gives different results for different decision hierarchy structures, and that therefore it must be flawed. But his different structures give the same results when they are set up properly. In the AHP the criteria inescapably depend on the alternatives when data are known for the alternatives. One of Barzilai’s mistakes is that he assigned equal default values to the criteria. This is not correct when the criteria depend on the outcomes or alternatives because they have known data, as they do in his marketing example. Truth often takes time to surface and it is no different in this case. Perhaps there are critics of the AHP because the AHP offers such new ideas that people are unable to reconcile them with their traditionally acquired knowledge, and mixing old and new knowledge does not work. An important point in the AHP is that there can be decisions where the criteria weights depend on the alternatives and decisions where they do not. In Section 2 we give a small instructional example to show how to determine the priorities of the criteria and sub- criteria when data are given for the alternatives. Then we turn to Barzilai’s marketing example and apply the same process to his example to get the right weights for his criteria and subcriteria. When the weights are right all his structures give the same outcome, and it is the outcome he himself expects: that the two marketing strategies result in the same overall income and are equally preferred, regardless of the hierarchical structure. 1 The same marketing problem can be formulated as an ANP (Analytic Network Process) network with feedback that leads to the right priorities for the alternatives. We show how to structure the ANP network for our simple example of Section 2 and for Barzilai’s marketing example. The limit supermatrix of the ANP automatically produces the correct weights for both the criteria and alternatives when the data are entered as priority vectors in the supermatrix. It is simpler in a way because no thinking is required on the part of the user. Just plug in the known data and the supermatrix will crank out the priorities for all the nodes in the model. Criteria may depend on alternatives even for intangibles. If one is dealing with intangibles, the priorities for the criteria are determined by using judgments to pairwise compare the criteria, but judgments on the criteria should be made only after carefully studying the alternatives. For example, in a car purchase decision where one of the criteria is style and another is cost, one might have a pre-conceived idea that cost is the most important criterion. But after actually visiting the auto showroom and seeing the stylish cars side-by-side with the cheap cars, it is surprising how many people revise the importance of their criteria and walk out with a beautiful but costly car. This is a case of careful study of the properties exhibited by the alternatives influencing the weights of the criteria. The Analytic Network Process (ANP) makes it clear how to determine the priorities of criteria, tangible and intangible, that depend on alternatives. 2. Determining Priorities for the Criteria when Alternatives have known Outcomes expressed as Data Priorities like probabilities are relative numbers. Numbers from a scale can be transformed to relative priorities, but not conversely, unless there is some link to the actual values. Consider the case of two objects A and B worth 25 dollars and 50 dollars respectively. One converts from dollars to relative numbers by normalizing as follows: Object A = 25/75 = 1/3 of the total dollars Object B = 50/75 = 2/3 of the total dollars If we are told only that A = 1/3 and B = 2/3, we can say B is worth twice A, but we cannot say whether A and B originally were 25 dollars and 50 dollars or 1000 dollars and 2000 dollars. In passing we note that the AHP with its homogeneity requirement requires elements of different orders of magnitude to be grouped together and appropriately linked for comparison purposes. One would not directly pairwise compare $25 and $2000 using judgments from the Fundamental Scale of the AHP because the largest number should be no more than 10 times the smallest number. When using data it is all right to lump them together. Though the ratio of 2000/25 is 80, is a much bigger number than the maximum 9 of the Fundamental Scale, when using direct data for the entire comparisons set it is okay. The following example will make clear how one determines the criteria weights. The investments A and B, for the same period of time, have both interest and capital gains 2 returns as shown in Table 1. The capital gains returns are assumed to be known as well as the interest returns (certainly not true in real life!). The first criterion C1 is the interest return and the second C2 is the capital gains return. Note that the capital gains returns are much heftier than the interest returns. We can see from Table 1 that investment A’s gain of 13 would be 0.382 of the total dollars ($34) that might be brought in by the two together while investment B’s would be 0.618. Table 1 Investments with Returns Measured in Dollars using Ordinary Arithmetic to give the Right Answer Values (in dollars) C1 C2 Total Normalized Interest Capital Gains Return Total Return Return Return Investment A 3 10 13 13/34 = 0.382 Investment B 6 15 21 21/34 = 0.618 Totals 9 25 34 This is, however, not the result we get if we normalize first before adding. In Table 2 we first converted the dollars in each column to relative values by dividing by the total, and then summed the row. The numbers in the “Sum of Normalized Return” column do not sum to 1, so they must be normalized to obtain the final column, and it does not correspond to the final column of Table 1. Table 2 The Wrong Outcome is Obtained by Normalizing then Adding Relative Values (normalize each column by dividing by its sum) C1 C2 Sum of Normalized Interest Capital Gains Normalized Sum Return Return Return (normalized) (normalized) Investment A 3/9 10/25 11/15 11/30=.367 Investment B 6/9 15/25 19/15 19/30=.633 Totals 1 1 2 To make the values in Table 2 correspond to those in Table 1, the correct relative values, we need to determine weights for the criteria. The weight of each criterion is obtained by dividing the sum of the values in its column in Table 1 by the total value of 34 for both criteria. The priority of C1 is 9/34, or about 0.265, and that of C2 is 25/34 or 0.735. We then multiply the normalized returns of the alternatives in each column in Table 2 by these criteria weights and add. This yields Table 3 whose final column is identical with the final column of Table 1. Thus we see here that the criteria in a sense “inherit” their 3 priorities from the measurements of the alternatives under them and cannot be arbitrarily assigned. This is always the process in multi-criteria decisions when one has measurements – in the same scale – that need to be converted to relative values. Table 3 Relative Values Weighted by the Criteria then Added Relative Values (weighted) C1 C2 Total (9/34)=0 .264706 (25/34)=0.735294 A 3/9×9/34=3/34 10/25×25/34=10/34 13/34 =0 .382 B 6/9×9/34=6/34 15/25×25/34=15/34 21/34 = 0.618 ANP Solution to the Investment Problem The same problem can be solved by structuring it as an ANP network with feedback and dependence with two clusters: Criteria and Alternatives as shown in Figure 1 from the SuperDecisions ANP software. The Criteria cluster contains the elements Interest and Capital Gains and the Alternatives cluster contains the elements Investment A and Investment B. Compare both alternatives with respect to Interest and Capital Gains, then invert the process and compare both criteria with respect to Alternative A and Alternative B. The question to be answered in the latter case is, for example: “Which criterion is more dominant for Alternative A? For Alternative B? By how much?” Since we have data, for Investment A, for example, this question is answered by normalizing the Capital Gains and Interest return, obtaining 10/13 = 0.769 for Capital Gains, and 3/13 = 0.231 for Interest Return. This normalized data is entered in the column headed Investment A in the supermatrix, shown in Table 5 . Criteria Cluster Interest Capital Gains Return Return Alternatives Cluster Alternative A Alternative B Figure 1The ANP Model for the Investment Example 4 We start with the raw data shown in Table 4 below. We then normalize the data so it sums to 1 for each cluster in each column as shown in Table 5. We raise this matrix to powers until it converges to obtain the limit supermatrix which has all its columns the same, as shown in Table 6. Finally, normalize for each cluster in the column to obtain the overall priorities as shown in the rightmost column of Table 6. Note that we now have overall or synthesized priorities for all the nodes in the entire structure. Table 4 Data for the Investment Problem Criteria Alternatives C1 C2 A B Interest Capital First Second Gains Investment Investment 1Criteria C1 - Interest 0 0 3 6 C2 - Capital Gains 0 0 10 15 2Alternatives A - First inv. 3 10 0 0 B - Second inv. 6 15 0 0 Table 5 Supermatrix with Data Normalized Criteria Alternatives C1 C2 A B Interest Capital First Second Gains Investment Investment 1Criteria C1 - Interest 0 0 0.230769 0.285714 C2 - Capital Gains 0 0 0.769231 0.714286 2Alternatives A - First inv. 0.333333 0.400012 0 0 B - Second inv. 0.666667 0.599988 0 0 Table 6 Limit Supermatrix Yields Priorities of Alternatives and Criteria OVERALL Criteria Alternatives PRIORITIES C1 C2 A B (Normalize to Interest Capital First Second 1.0 for Gains Investment Investment each cluster) 1Criteria C1 - Interest 0.132353 0.132353 0.132353 0.132353 0.264706 C2 - Capital Gains 0.367647 0.367647 0.367647 0.367647 0.735294 2Alternatives A - First inv. 0.191181 0.191181 0.191181 0.191181 0.382362 B - Second inv. 0.308819 0.308819 0.308819 0.308819 0.617638 The overall priorities of the nodes in the model are given in the rightmost column in Table 6. They are correct and are what we computed they should be in Table 3 using ordinary arithmetic: A1 (Investment A) = 0.382 B1 (Investment B) = 0.618 5 C1 (Interest Criterion) = 0.265 C1 (Capital Gains Criterion) = 0.735 The final priorities fall out of the ANP model in a very natural way by normalizing the data to get priorities and entering the priorities in the ANP supermatrix. There is no need to do any of the side calculations that were necessary to produce the required priorities for the criteria to get the correct answer with the AHP model. The ANP model produces the correct priorities for all the nodes in one fell swoop. This is a simple case involving an ANP framework, but ANP models generally deal with far greater complexity and interdependence than this trivial example shows. 3. Barzilai’s Marketing Example Consider the following problem. Barzilai uses it to illustrate his proposition that the AHP gives different results depending on the structure of the decision hierarchy. He uses data in his exercise so that it will not be clouded by the issue of deriving priorities from pairwise comparisons. “The president and three vice-presidents of a company are analyzing their marketing options. The company produces and sells a single product for a fixed price through five stores in the city. Stores 1 and 2 are in the city’s West Side, store 3 is at City Centre, and stores 4 and 5 are in the East Side. They all agree to define the company’s value function v( x) v( x1 ,..., x5 ) as its total annual revenue where xi represents annual sales in millions of dollars in store i, i = 1,…,5. The company needs to choose between marketing strategies A and B. These strategies will result in annual revenue of P = (3, 3, 1, 1, 1)(i.e., x1 3, x2 3, x3 1, etc.) if strategy A is chosen and Q = (1, 1, 1, 3, 3) if strategy B is implemented. In our terminology, the criteria are xi , i 1,...,5 , the alternatives are marketing strategies A and B and the coordinates of the alternatives in the evaluation space are given by the points P and Q. The four executives agree that the criteria are identical: xi differs from x j only in its index. Therefore, the president concludes that v( x) x1 x2 x3 x4 x5 and since v( P) v(Q) 9, the two alternatives are equally preferred.” Barzilai is somewhat confused in the above paragraph when he says: “The four executives agree that the criteria are identical: xi differs from x j only in its index.” Earlier he had defined the xi as: “ xi represents annual sales in millions of dollars in store i, i = 1,…,5.” Perhaps this was a simple grammatical error that makes it seem as though he is referring to the xi as criteria when what he really means is that the stores aggregated into territories, si in Figure 2, are criteria, but it may have contributed to his misunderstanding. 6 In any case, the four executives cannot simply agree that “ xi differs from x j only in its index.” They are the annual sales from the stores and their values are either 1 or 3 depending on which of the two marketing strategies is employed. This confusion of using the same symbol to denote both a node and a value for the node leaves his whole work mathematically inaccurate and highly questionable. Even when one tries with the best will to interpret it in the most positive light, one is able to see much confusion in Barzilai’s logic about working with value functions and the way he assigns values to the variables. Barzilai’s value function v( x) x1 x2 x3 x4 x5 is a common sense arithmetic solution to the problem that everyone would agree gives the correct answer. This value function uses data, the annual sales for each store, in terms of dollars and the answer happens to be the same, 9 million dollars, whether one applies it to strategy A or strategy B. He then structures three different hierarchies for this decision problem and obtains a different value function for each hierarchy, on page 168. But the value functions this time are weights, not dollars, though he compares the results with his original value function of 9 million dollars. He incorrectly concludes that the AHP is flawed, rather than that the flaw is in how he went about calculating his value functions. In fact, all three hierarchical structures do give the same result when the priorities of the nodes in the three structures are correctly calculated along the lines of the measurement example of Section 2 above. His basic reference is a value function, shown above, which expresses the total annual sales in dollars as a function of the xi . But writing a single value function is incorrect. The 9M dollars is computed as the outcome of a strategy and there should be two value functions, one for each strategy. It is a coincidence that his original value function happens to give the same result, with annual sales totaling 9 million dollars, for both strategies: vA ( x) x1 x2 x3 x4 x5 3 3 1 1 1 9M , and vB ( x) x1 x2 x3 x4 x5 1 1 1 3 3 9M According to Barzilai the three vice-presidents use the AHP to decompose the problem as follows. The first, whose territory is City Centre, decomposes the criteria into the tree in Figure 2. The variables xi are defined as above with s1 representing total revenue from sales in stores 1 and 2, s2 sales in store 3 and s3 sales in stores 4 and 5, that is, this vice president groups the stores the same way they are grouped into sales territories. As before, he assumes that equal weights must be attached to the edges at each node (essentially a criterion) of the tree “Therefore, according to the AHP procedure, in an obvious notation, the intermediate weights (“local” priorities in AHP terminology) are given by ws1 ws2 ws3 1/ 3 , wx1 wx2 1/ 2 , wx3 1 , and wx4 wx5 1/ 2 .” This casual and unjustifiable assumption is why Barzilai’s AHP formulations do not give the correct result. The priorities of the criteria inescapably depend on data when it is given for the alternatives. One would have to calculate what those priorities must be in order for the model to result in the given data. It is a case of dependence not independence that Barzilai assumes. Usually in an AHP model the priorities of the alternatives are derived, not given, but Barzilai takes this approach in an effort to prove the AHP is faulty. 7 v1 s1 s2 s3 x1 x2 x3 x4 x5 Figure 2 Hierarchy by VP for “Center” v2 t1 t2 x1 x2 x3 x4 x5 Figure 3 Hierarchy by VP for “East” v3 u1 u2 x1 x2 x3 x4 x5 8 Figure 4 Hierarchy by VP for “West” Starting with these three structures and arbitrarily assuming default equal priorities for the criteria at the same time that he assumes data for the alternatives that contradict the assumed priorities Barzilai ends up with three different answers. Furthermore, he computes a value function on the structures that is in terms of priorities, but he then compares it to his original value function that was in terms of dollars. The value functions for the three structures, his measure of whether the AHP composition works correctly or not, are computed without any input concerning the known annual sales data of the two strategies for the stores. How could a value function applied to some hierarchical structure with arbitrary priorities be expected to give back known answers when there is no information supplied to the function about the assumed data? It puts one in mind of giving a Chinese student without knowledge of English a question written in English and when he does not reply properly concluding that he does not know the correct answer. The central issue in this example, and the wrong thinking that may be at the root of many of Barzilai’s criticisms, is that he mixes up how he is deals with data and with priorities and furthermore he does not know that though it is possible to go from data to priorities it is not possible to go the other way without outside information to link the priorities to the data. One would rarely if ever set up a practical decision problem in the AHP the way he does. If there are two criteria with outcomes in the same scale, one would usually first combine them into a single criterion using whatever arithmetic is used in practice to combine the outcomes. One would then use that criterion in the AHP problem. In any case, the AHP always gives the correct answer if the model is set up correctly. 4. Barzilai’s Wrong Assumption To formulate his problem correctly with AHP, Barzilai should have followed his two strategies, A and B, with corresponding outcomes (alternatives) Pi and Qi, for the ith store. Let us show that when his hierarchies are set up properly for the data that are given, the AHP composition is correct and gives the same results, in relative terms, for v A ( x) and vB ( x) . This problem can also be formulated as an ANP network with dependence and it gives the correct answer as well but we demonstrate it here only for the first of Barzilai’s three cases. Correct AHP Model for “Center” VP’s Structure The first vice president, whose territory is the center of the city comes up with the structure shown in Figure 5. In Barzilai’s notation the strategies P and Q for the alternatives are given in terms of millions of dollars for each store, with P=(3M,3M,1M,1M,1M) the result of applying strategy A and Q=(1M,1M,1M,3M,3M) 9 the result of applying strategy B, having aggregated the data upward through the hierarchy for all nodes above the bottom level alternative nodes. The value xi associated with the ith store Si is the sum of the ith component of the strategies P and Q immediately below it and is written in the second line of each box. Normalizing the data for the nodes directly under each parent node one obtains the relative values, the priorities, shown in the first line of each box. V1 s1=8/18=.444 s2=2/18=.111 s3=8/18=.444 8M 2M 8M Store S1 Store S2 Store S3 Store S4 Store S5 x1=4/8=.5 x2=4/8=.5 x3=2/2=1 x4=4/8=.5 x5=4/8=.5 4M 4M 2M 4M 4M P1=3/4=.75 P2=3/4=.75 P3=1/2=.5 P4=1/4=.25 P5=1/4=.25 3M 3M 1M 1M 1M Q1=1/4=.25 Q2=1/4=.25 Q3=1/2=.5 Q4=3/4=.75 Q5=3/4=.75 1M 1M 1M 3M 3M Figure 5 Marketing Decision Tree by VP of “Center” Below the Goal are the territories, below them the stores and below the latter the two alternative strategies. v1 ( A) = 0.444×0.5×0.75+0.444×0.5×0.75+ 0.333×1×0.5+0.444×0.5×0.25+0.444×0.5×0.25 = 0.50 v1 ( B ) = 0.444×0.5×0.25+0.444×0.5×0.25+ 0.333×1×0.5+0.444×0.5×0.75+0.444×0.5×0.75 = 0.50 We see that the two value functions, for strategy A and strategy B, have correct equal priorities of 0.50, and not those Barzilai calculated incorrectly. The priorities of the criteria need to be calculated from the priorities of the alternatives in the part of the tree beneath each criterion. Barzilai does not calculate his value function for the two strategies. Had he done it correctly he would have had to develop priorities for the criteria above the strategies as shown in Figure 5. All three of his examples would give the correct answer when done correctly. One cannot simply assign arbitrary weights 10 to the criteria and hope that these weights will somehow give the correct answer for value functions at whatever level in the tree they are computed. The ANP Formulation of the “Center” VP’s Structure We will formulate the AHP structure above as an ANP model. Here again we get the precise answer we obtained before. It too is at variance with what Barzilai obtained. We start by putting the known data in the supermatrix and showing that the resulting priorities for each of the alternative strategies are 0.5. The same results would be obtained if the ANP model were done for the other two structures. Table 7 The Data* for Center VP’s Structure Districts Stores Alternative Strategies S1 S2 S3 X1 X2 X3 X4 X5 P Q Districts S1 0.000 0.000 0.000 4 4 0.000 0.000 0.000 0.000 0.000 S2 0.000 0.000 0.000 0.000 0.000 2 0.000 0.000 0.000 0.000 S3 0.000 0.000 0.000 0.000 0.000 0.000 4 4 0.000 0.000 Stores X1 4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 1 X2 4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 1 X3 0.000 2 0.000 0.000 0.000 0.000 0.000 0.000 1 1 X4 0.000 0.000 4 0.000 0.000 0.000 0.000 0.000 1 3 X5 0.000 0.000 4 0.000 0.000 0.000 0.000 0.000 1 3 Alternative P 0.000 0.000 0.000 3 3 1 1 1 0.000 0.000 Strategies Q 0.000 0.000 0.000 1 1 1 3 3 0.000 0.000 * With data in terms of millions of dollars. Table 8 Data Normalized* by Cluster to become the Supermatrix for Center VP’s Structure Districts Stores Alternative Strategies S1 S2 S3 X1 X2 X3 X4 X5 P Q Districts S1 0.000 0.000 0.000 1 1 0.000 0.000 0.000 0.000 0.000 S2 0.000 0.000 0.000 0.000 0.000 1 0.000 0.000 0.000 0.000 S3 0.000 0.000 0.000 0.000 0.000 0.000 1 1 0.000 0.000 Stores X1 .5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 .333 .111 X2 .5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 .333 .111 X3 0.000 1 0.000 0.000 0.000 0.000 0.000 0.000 .111 .111 X4 0.000 0.000 .5 0.000 0.000 0.000 0.000 0.000 .111 .333 X5 0.000 0.000 .5 0.000 0.000 0.000 0.000 0.000 .111 .333 Alternative P 0.000 0.000 0.000 .75 .75 .5 .25 .25 0.000 0.000 Strategies Q 0.000 0.000 0.000 .25 .25 .5 .75 .75 0.000 0.000 *With normalized data in the form of priorities Table 9 Limit Supermatrix for Center VP’s Structure 11 Districts Stores Alternative Strategies S1 S2 S3 X1 X2 X3 X4 X5 P Q Districts S1 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 S2 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 S3 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 Stores X1 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 X2 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 X3 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 X4 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 X5 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 Alternative P 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 Strategies Q 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 Table 10 Final Normalized Priorities for Center VP Structure Districts S1 0.444 S2 0.111 S3 0.444 Stores X1 0.222 X2 0.222 X3 0.111 X4 0.222 X5 0.222 Alternative P 0.500 Strategies Q 0.500 Again, as we see, the priorities of P and Q (for strategies A and B) are shown using the ANP to be 0.5 and 0.5, the correct answer. A side benefit is that we also get the priorities of all the other nodes in the model. Note that S2 with a priority of 0.111 is the least desirable district. It has only one store while S1 and S3 have two stores providing income. This matches our common sense that the less stores in a territory the less the income for that territory and the less desirable it is regardless of marketing strategy. Correct AHP Model for “East” VP’s Structure Let us now create the right AHP model for the second VP’s structure. The second vice president whose East side territory contains stores 4 and 5 lumps together all sales outside his territory and uses the structure shown in Figure 6. 12 V2 t1=10/18=.555 t2=8/18=.444 10M 8M X1=4/10=.40 X2=4/10=.40 X3=2/10=.20 X4=4/8=.5 X5=4/8=.5 4M 4M 2M 4M 4M P1=3/4=.75 P2=3/4=.75 P3=1/2=.50 P4=1/4=.25 P5=1/4=.25 3M 3M 1M 1M 1M Q1=1/4=.25 Q2=1/4=.25 Q3=1/2=.50 Q4=3/4=.75 Q5=3/4=.75 1M 1M 1M 3M 3M Figure 6 Marketing Decision Tree by VP of “East” (Stores 4 and 5) Here we have v2 ( A) = .555×.40×.75+.555×.40×.75+.555×.20×.5+.444×.5×.25+.444×.5×.25 = .50 v2 ( B) = .555×.40×.25+.555×.40×.25+.555×.20×.5+.444×.5×.75+.444×.5×.75 = .50 and again the two value functions for strategy A and strategy B have equal priority of 0.5. Correct AHP Model for “West” VP’s Structure The third vice president whose territory is the West side with stores 1 and 2 and 3 lumps together all sales outside his territory and arrives at the structure shown in Figure 7. V3 u1=8/18=.444 u2=10/18=.555 10M 8M X2=4/8=.50 X3=2/10=.2 X4=4/10=.4 X5=4/10=.4 X1=4/8=.50 4M 4M 2M 4M 4M P1=3/4=.75 P2=3/4=.75 P4=1/2=.50 P5=1/4=.25 P5=1/4=.25 3M 3M 1M 1M 1M Q1=1/4=.25 Q2=1/4=.25 Q4=1/2=.50 Q5=3/4=.75 Q5=3/4=.75 1M 1M 1M 3M 3M Figure 7 Marketing Decision Tree by VP of “West” (Stores 1 and 2) Here we have: v3 ( A) = .444×.50×.75 + .444×.50×.75+.555×.2×.5+.555×.4×.25+.555×.4×.25 = .50 v3 ( B) = .444×.50×.25 + .444×.50×.25+.555×.2×.5+.555×.4×.75+.555×.4×.75 = .50 and again the value functions for the two strategies are equal. 13 Finally, we have outside information so in this case we can convert the strategies back into dollars. We know the total sales are 18M, by adding all the dollars for both strategies at the bottom of the model. Multiplying the priority value of 0.50 for either strategy by 18M we obtain 9M. This corresponds to the original value function of the president of the company, and the actual annual sales from the five stores under both scenarios. AHP is known to be particularly appropriate for handling intangibles, so one might ask how would the criteria be weighted when dealing with intangibles? Criteria, whether they are tangible or intangible, need to be weighted when they depend on their alternatives. With intangibles one needs to be very familiar with the alternatives before pairwise comparing the criteria. In Barzilai’s marketing example, in Figure 5, when making pairwise comparisons of the importance of the S1 and S2 groupings of stores, but not directly using the information about the incomes from the stores under the different strategies, the decision maker would observe that there are two stores in S1 and only one in S2, and therefore would know that S1 should be about twice as important as S2. But the annual sales from the store in the middle grouping are known to be less under than the other districts under either strategy, so the decision maker might raise his or her judgment from 2 to 3 when comparing the two to take this into account. And as S1 and S2 have the same number of stores, without precisely calculating the income under the two strategies, S1 would be judged to be equally as important as S3. The pairwise comparison matrix and vector of priorities using judgment rather than data might sensibly be: Table 11 Pairwise Comparison of Groupings of Stores based on Perceived Importance Priorities Priorities S1 S2 S3 derived earlier (from judgment) from data S1 1 3 1 .43 .444 S2 1/3 1 1/3 .14 .111 S3 1 3 1 .43 .444 The priorities derived from the decision maker’s “soft” judgments about the importance of the territories are not very different from the values computed using the data. Similar kinds of analyses can be done throughout the structure, one would see that the priorities obtained based on judgments, which one would have to use for intangibles, would be fairly similar to those obtained from data. 5. On Barzilai’s So-Called Linearity Proof Barzilai assumes what he is trying to prove in his Theorem 2 on page 161. He shows that the marginal substitution rate of xi with respect to x j for a linear value function is constant and goes on to enunciate the following theorem on page 162 of his paper, without proof, that the AHP gives rise to linear value functions. He writes, incorrectly, that: “Theorem 3. The value functions generated by the AHP are always linear.” 14 In the AHP with its relative measurement, the priorities of the alternatives are derived from paired comparisons and the resulting values are not linearly related, therefore substituting different values of the variables for different alternatives generally produce nonlinear values for the value function. Later in the appendix we give several validation examples to show that the priorities in the AHP are non-linear functions of their attributes. In addition, the AHP is a special case of the ANP with its dependencies in which composition yields multilinear forms of infinite order whose entries are functions, not linear variables. The AHP derived function is not a linear Cartesian function involving orthogonal axes nor is it affine involving non-orthogonal axes, but curvilinear possibly with locally affine coordinate systems from which it is well known one cannot read off incremental changes from point to point by simply taking the affine derivatives. One must consider not only the increments of the components due to the transition from a point to an infinitesimally close point, but also to the change in the local coordinate system. Counterexamples are well-known. In other words one must not identify the components of the increments of a tensor with the increments of its components. Nor are the coefficients in the AHP arbitrarily determined. When one performs comparisons of the attributes that depend on the alternatives, one either uses the ANP directly for accuracy, or approximates with a hierarchy by thinking about what the attributes mean through the alternatives. Their meaning is not independent of the concrete alternatives which they characterize. Barzilai, rather haphazardly, although I am certain that he knows better, concludes that the AHP has a linear value function and proceeds to arrive at a conclusion having nothing to do with the AHP. In his heart example discussed in the next section it appears that he thinks that the criteria may depend in their priorities on the alternatives, and admits that it is a subject for the ANP but then gallantly dismisses the ANP by, for example, referring to himself and to Kamenetzky, a student of Saaty’s, who knew the subject of hierarchies well and was correct to observe that hierarchies can only be used when the criteria are assumed to be independent of the alternatives. At the time Kamenetzky did not know about the ANP that deals with this type of dependence. Hierarchic composition is given by: i i i (1) x11 x22 x pp i1 , ,i p where the xi do not belong to a linear system of coordinates, but, as we said before, are generally non-linear functions derived in the AHP through paired comparisons. Recall that the components of a priority vector are not linear variables, but are readings from non-linear functions. The richer the structure of a hierarchy in breadth and depth, for example that shown in Figure 8, the more complex are the multilinear forms, which of course are non-linear, derived from it. Let us show that AHP composition gives rise to multilinear forms whose marginal rate of substitution is not constant. The composite vector for the entire hth level is shown in (1) above and it has covariant tensorial components. Similarly, the left 15 eigenvector approach to a hierarchy gives rise to a vector with contravariant tensor components. Consider a hierarchy of two criteria, the first with three subcriteria and the second with two subcriteria and finally with two alternatives as shown in the figure below together with hierarchic composition and marginal rate of substitution. One can see that neither is the first linear nor is the second constant contradicting Barzilai’s incorrect claims. 16 Focus: At what level should the Dam be kept: Full or Half-Full? Decision Financial Political x11 x12 Criteria: Decision Congress Dept. of Interior Courts State Lobbies Makers: x21 x22 x23 x24 x25 Alternatives: Half-Full Dam Full Dam x31 x32 v=(x11 x21 x31+ x11 x22 x31+ x11 x23 x31+ x12 x24 x31+ x12 x25x31, x11 x21 x32+ x11 x22 x32+ x11 x23 x32+ x12 x24 x32+ x12 x25 x32) Figure 8 A Simple Hierarchy of Three Levels to Demonstrate Hierarchic Composition Consider a hierarchy whose goal is to find the relative cost of three alternatives. Let the criteria be area (A) and volume (V) and let alternatives be three spheres ( S1 , S2 , S3 ). The criteria weights are the relative cost of unit of area versus a unit of volume. In general, the area and volume of a sphere with radius r is equal to 4 r 2 and 4 r 3 , respectively. 3 Let the three spheres S1 , S2 and S3 have radii equal to r1 , r2 and r3 , respectively. Thus, the priorities of the spheres with respect to their area is given by 4 r12 4 r22 4 r32 f1 (r1 ) , f1 (r2 ) , f1 ( r3 ) 2 3 3 3 4 ri 2 4 ri 2 4 ri i 1 i 1 i 1 and their relative volume is given by 4 r 3 4 r 3 4 r 3 f 2 (r1 ) 3 3 1 , f 2 (r2 ) 3 3 2 , f 2 (r3 ) 3 3 3 . 3 3 ri 3 ri 3 ri 4 3 4 3 4 i 1 i 1 i 1 17 Note that the relative areas and volumes are non-linear functions. Let c1 be the cost per unit area and let c2 be the cost per unit volume. Then, the costs of the spheres are given by c1 4 ri2 c2 4 ri3 , 3 i=1,2,3, and their relative values are given by c1 4 ri c2 4 ri 3 2 3 3 , i=1,2,3. (c 4 r i 1 1 i 2 c2 4 ri ) 3 3 To obtain the relative costs of the spheres from the relative areas and volumes we would have to find values 1 and 2 such that 4 ri 2 4 ri3 c1 4 ri 2 c2 4 ri3 1 3 2 3 3 3 3 , i=1,2,3. Clearly, 1 and 2 are given 4 ri 2 4 3 ri 3 (c 4 r 1 i 2 c 4 2 3 ri ) 3 i 1 i 1 i 1 by 3 3 c1 4 ri 2 c2 4 ri3 3 1 3 i 1 , 2 3 i 1 . 3 (c1 4 ri c2 3 ri ) i 1 2 4 3 (c1 4 ri c2 3 ri ) i 1 2 4 In practice the criteria weights would be estimates of these values but so would also be the functions f1 and f 2 . We give an example of three spheres with different radii in Table 12. A sphere costs $1 for each unit of its surface area plus $5 for each unit of its volume. Think of hot air balloons that have to be purchased by the square unit of area, but filled with hot gas to fly them according to their volume. Volume and surface area are not linearly related. Table 12 Hot Air Balloon Example Showing Non-linear Relationship f1(x) f2(x) Area Volume 18 Spheres Radius $1 $5 Area Cost Volume Cost Total Cost Relative Cost S1 1 12.56637 4.1887902 $12.57 $20.94 $33.51 0.036036036 S2 2 50.26548 33.510322 $50.27 $167.55 $217.82 0.234234234 0.72972973 S3 3 113.0973 113.09734 $113.10 $565.49 $678.58 Total 175.9292 150.79645 $175.93 $753.98 $929.91 $929.91 Criteria $175.93/$929.91 $753.98/$925.91 Weights =0.189189 =0.8108108 Estimated Spheres Relative Area Volume Final Priorities Final Priorities S1 0.071429 0.0277778 0.036036036 0.0351 S2 0.285714 0.2222222 0.234234234 0.2328 S3 0.642857 0.75 0.72972973 0.7321 1 0.166667 0.8333333 Barzilai assumes incorrectly that the AHP approximates hierarchic composition by a linear value function in his statement on page 165: “It is tempting to assume that when the components fk of v( x) are replaced by their first-order approximations, these components may be converted to normalized linear value functions as well. We show in Section 5.4 that this implicit assumption which is utilized in AHP decompositions is incorrect.” In Section 5.4 of his paper Barzilai assumes that the priorities of the alternatives are given but those of the criteria are arbitrarily determined and from that concludes that his value functions will be the same under linear transformations but not under affine transformations. Here again he misses the fundamental point that once the weights of the alternatives are given, those of the criteria are automatically determined in relative terms and that composing with these normalized weights of the criteria automatically gives the correct relative values of the alternatives which he ignores by assuming that the priorities of the criteria are arbitrarily given. He is correct to conclude that when the weights of the criteria are independent from those of the alternatives, the final value functions would not lead to the same relative measurement of the alternatives under affine transformations but one never does that when one has measurements for the alternatives. Let Ai , i 1,..., n be the alternatives and let C j , j 1,..., m be the criteria. Let xij be the m actual measurement of alternative Ai with respect to criterion C j then x j 1 ij is the total m n m value of alternative Ai with respect to all the criteria C j and xij / xij is its overall j 1 i 1 j 1 19 relative value. Let us show now how to obtain this by normalizing the alternatives with respect to each criterion. With that we have for alternative Ai with respect to criterion C j n m the normalized value xij / xij . Now we assign criterion C j the relative value of the i 1 j 1 sum of the alternative values under it to the sum of the values of all the alternatives with n n m respect to all the criteria. We have xij / xij for the priority of criterion C j . i 1 i 1 j 1 Finally we multiply the normalized weight of alternative Ai under each criterion by the corresponding weight of that criterion and take the sum over all the criteria. We obtain n m n xij x ij x ij for the relative value of Ai , n i 1 n m j 1 n m which is precisely the overall j 1 x x i 1 ij i 1 j 1 ij x i 1 j 1 ij relative value of alternative Ai obtained above from measurements. It is clear that there is no other meaningful way to obtain this answer without the particular weights assigned to the criteria. Again, we repeat that when given the weights of the alternatives the AHP does how we just described and not the way Barzilai does it by assigning arbitrary weights to the criteria. When measurements are not given one either proceeds from the bottom up or the top down to derive weights for the alternatives in terms of the criteria and the criteria in terms of higher level criteria. problems are not ever set up this way in practice by starting with the weights of the alternatives. In the usual AHP problem the alternative weights are unknown and we need the judgments of a knowledgeable person to derive them. The many validation examples in the Appendix provide evidence that the AHP correctly reflects measurable events in the real world. 6. The Heart Transplant Example – A Poorly Drawn Non-conclusion by Barzilai Not only does Barzilai draw incorrect conclusions and prove false theorems about the AHP but then also draws misleading attention to examples that he does not show any fault with. He takes a heart transplant example and says see I have shown with my examples that the AHP does not work, so how can one trust it with an application like the heart transplant example. He not only does his mathematics poorly but regrettably spreads his erroneous thinking and conclusions to groups where he has no justification whatever to doubt the intelligence and understanding of the medical experts who did the exercise and who understand the AHP and its fundamentals very well. In the heart transplant example Barzilai simply shows that hierarchic composition is a product of matrices of priorities, a multilinear mapping that involves sums of products of nonlinear variables (functions) linearly. That is, each is raised to the first power. Thus neither the variables nor the composition are linear. Barzilai incorrectly assumes it to be linear. 20 In its simplest hierarchic form with only a goal, criteria and alternatives, the outcome of the AHP is a convex combination of the weights of the alternatives i fi , i where i 1 . These priorities ( f i ) are themselves non-linear functions and not i ordinary linear Cartesian variables. 7. Observations and Conclusions Of course, a person who knows well how the AHP works and has used it in practice would never actually set up the kind of problem Barzilai does from which he draws incorrect answers and conclusions to criticize the AHP. If there is more than one criterion with an existing scale and alternatives under them with actual data, in practice one would first combine everything into a single criterion using whatever standard arithmetic is required. These criteria are then usually put alongside intangible criteria and the usual process of paired comparisons is applied to determine the importance of the criteria with respect to the goal. Barzilai set up the problem in this way in an attempt to show that the AHP does not work. But it does work, both on his very artificially conceived and wrongly developed example, and in general as a sound theory. In his paper “On MAUT, AHP, PFM” in the Proceedings of the ISAHP meeting in Kobe in 1999, Barzilai wrote that Belton and Gear and Dyer “have misidentified the problem, and proposed incorrect revisions and failed to address the fundamental issues” yet he frequently uses them to support his arguments against the AHP. He also says in that paper about measurement theory “Basic concepts such as scale type and meaningfulness are not fully understood… they will be fully resolved in a forthcoming paper.” That was six years ago, and we await with anticipation a constructive and complete exposition of Barzilai’s ideas about measurement in multicriteria decision-making and how to use them in practice to make decisions. There are numerous validation examples developed by many people using pairwise comparison matrices, hierarchies and networks for which the answers are already known that show how robust and accurate the AHP/ANP is. References Barzilai, Jonathan, On the Decomposition of Value Functions, Operations Research Letters 22 (1998) 159-170. Saaty, Thomas L., Decision-making with the AHP: Why is the principal eigenvector necessary, European Journal of Operations Research 145 (2003) 85-9. 21 Saaty, Thomas L., Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, Vol. VI, AHP Series, RWS Publications, Pittsburgh, PA 2000 Saaty, Thomas L., The Analytic Network Process: Decision Making with Dependence and Feedback, RWS Publications, Pittsburgh, PA 2001. APPENDIX SINGLE MATRIX VALIDATION EXAMPLES Relative sizes of Areas: Here is a validation exercise we have often used with individuals when introducing them to the Analytic Hierarchy Process. We ask the individual to estimate the relative areas of the five geometric figures below using the AHP. The person makes pairwise comparison judgments as to how much bigger in area the first figure is than the second as shown in the matrix below. For example, an area of 12 square units is 3 times bigger than an area of 4 square units. So the judgment 3 is used. Use the numbers 1 to 9 (or their reciprocals if the second of the pair is bigger than the first) from the Fundamental Scale of the AHP; decimals in between are allowed. The judgments are entered in the upper triangular half of the matrix. The entire exercise requires the individual to make n(n-1)/2 judgments, where n is the number of things being compared, in this case n = 5, so for this exercise 10 judgments are required. Enter the reciprocals of those judgments in the bottom triangular half of the matrix. The relative sizes of the figures are then given by the principal eigenvector of the matrix or, in the absence of a computer, approximately by normalizing each column to one and taking the average of corresponding entries in the columns. This is the priority vector of relative values. It is remarkable how close one can come to the actual relative sizes in this way. Often verbal judgments are made first particularly when dealing with intangibles. The actual relative values of these areas are approximately A = 0.47, B = 0.05, C = 0.24, D = 0.14, and E = 0.09. Here A is 47% of the overall area of the five figures, B is 5% 22 and so on. Or, to put it another way, A/B=0.47/0.05=9.4, and A is about 9 times as large as B, and the relative sizes of the other figures can be determined in the same way. One Individual’s Judgment Matrix Circle Triangle Square Diamond Rec- Eigenvector tangle (priority vector of relative sizes) Circle 1 9 2.5 3 6 0.488 Triangle 1/9 1 1/5 1/3.5 1/1.5 0.049 Square 1/2.5 5 1 1.7 3 0.233 Diamond 1/3 3.5 1/1.7 1 1.5 0.148 Rectangle 1/6 1.5 1/3 1/1.5 1 0.082 *Only the judgments in bold must be made, the others are automatically determined If there is no computer available, a shortcut for computing the principal eigenvector is to assume the matrix is consistent, normalize each column and then take the average of the corresponding entries in the columns to obtain the priority vector. The priority vector shown in the right column of the matrix is the actual eigenvalue solution, but using the shortcut calculation gives a very similar result: A=0.486, B=0.049, C=0.233, D=0.148, E=0.082. It is easy to make consistent judgments in this exercise because we are dealing with objects with measures that are quite familiar to most people. If one were to measure the figures, compute the ratios of their areas exactly and enter these ratios instead of using judgments, the priority vector would give back the exact relative areas of the figures. The point here is to show that judgments can give accurate results. Sometimes people try to measure the figures and use formulas dimly remembered from high school geometry to verify the results. The errors associated with physically measuring the figures often give worse results than eyeballing the figures and making judgments. An interesting thing often happens when the exercise is tried with a group. In the Analytic Hierarchy Process the judgments of the members of the group on a pair are combined using the geometric mean. When the combined judgments are used in the matrix often the answer is closer to the actual relative sizes of the figures than most of the individuals’ answers are. Relative Weights of Objects: The matrix below gives the estimated pairwise comparisons of the weights of the five objects lifted by hand, made by the then President of the Diners Club, a friend of the author. The two vectors appear to be very close but are they really close? To determine closeness of two priority vectors one must use the Saaty Compatibility Index. Table Pairwise Comparisons of the Weights of Five Objects 23 Large Small Actual Radio Type- Attache Projec- Attache Eigen- Relative Weight writer Case tor Case vector Weights Radio 1 1/5 1/3 1/4 4 .09 .10 Typewriter 5 1 2 2 8 .40 .39 Large Attache Case 3 1/2 1 1/2 4 .18 .20 Projector 4 1/2 2 1 7 .29 .27 Small Attache Case 1/4 1/8 1/4 1/7 1 .04 .04 Relative Electric Consumption of Household Appliances: In the following table we have paired comparisons done by students in Electrical Engineering estimating the consumption of electricity of common household appliances. How compatible are the derived and actual vectors? Table Relative Electricity Consumption (Kilowatt Hours) of Household Appliances Annual Actual Electric Elec. Dish Hair Eigen- Refrig TV Iron Radio Relative Range Wash Dryer vector Weights Consumpti on Electric Range 1 2 5 8 7 9 9 .393 .392 Refrig- erator 1/2 1 4 5 5 7 9 .261 .242 TV 1/5 1/4 1 2 5 6 8 .131 .167 Dish- washer 1/8 1/5 1/2 1 4 9 9 .110 .120 Iron 1/7 1/5 1/5 1/4 1 5 9 .061 .047 Radio 1/9 1/7 1/6 1/9 1/5 1 5 .028 .028 Hair-dryer 1/9 1/9 1/8 1/9 1/9 1/5 1 .016 .003 The hairdryer is of such a small magnitude that it probably should have been left out of the other homogeneous comparisons. 24 This exercise was done on an airplane in 1973 by Thomas Saaty and Mohammad Khoja by simply using their common knowledge about the relative power and standing of these countries in the world and without referring to any specific economic data related to GNP values U .S U .S .S .R China France U .K Japan W .Germany U .S 1 4 9 6 6 5 5 U .S .S .R 1/ 4 1 7 5 5 3 4 China 1/ 9 1/ 7 1 1/ 5 1/ 5 1/ 7 1/ 5 France 1/ 6 1/ 5 5 1 1 1/ 3 1/ 3 U .K 1/ 6 1/ 5 5 1 1 1/ 3 1/ 3 Japan 1/ 5 1/ 3 7 3 3 1 2 W .Germany 1/ 5 1/ 4 5 3 3 1/ 2 1 Normalized Actual GNP (1972) Normalized GNP Eigenvector Values U.S .427 1,167 .413 U.S.S.R .23 635 .225 China .021 120 .043 France .052 196 .069 U.K .052 154 .055 Japan .123 294 .104 W. Germany .094 257 .091 HIERARCHIC VALIDATION EXAMPLES To make good applications needs expert knowledge of the subject, a structure that represents the pertinent issues, and a little time to do justice to the subject. In this part we give three hierarchic examples that gave results close to what the values actually were and all the works were published in refereed journals. World Chess Championship Outcome Validation – Karpov-Korchnoi Match The following criteria and hierarchy were used to predict the outcome of world chess championship matches using judgments of ten grandmasters in the then Soviet Union and the United States who responded to questionnaires they were mailed. The predicted outcomes that included the number of games played, drawn and won by each player either was exactly as they turned out later or adequately close to predict the winner. The outcome of this exercise was notarized before the match took place. The notarized statement was mailed to the editor of the Journal of Behavioral Sciences along with the paper later. See the coauthored book with Luis Vargas: Prediction, Projection and Forecasting, Kluwer, 1991. The prediction was that Karpov would win by 6 to 5 games over Korchnoi, which he did. 25 Table Definitions of Chess Factors T (1) Calculation (Q): The ability of a player to evaluate different alternatives or strategies in light of prevailing situations. B (2) Ego (E): The image a player has of himself as to his general abilities and qualification and his desire to win. T (3) Experience (EX): A composite of the versatility of opponents faced before, the strength of the tournaments participated in, and the time of exposure to a rich variety of chess players. B (4) Gamesmanship (G): The capability of a player to influence his opponent's game by destroying his concentration and self-confidence. T (5) Good Health (GH): Physical and mental strength to withstand pressure and provide endurance. B (6) Good Nerves and Will to Win (GN): The attitude of steadfastness that ensures a player's health perspective while the going gets tough. He keeps in mind that the situation involves two people and that if he holds out the tide may go in his favor. T (7) Imagination (IW: Ability to perceive and improvise good tactics and strategies. T (8) Intuition (IN): Ability to guess the opponent's intentions. T (9) Game Aggressiveness (GA): The ability to exploit the opponent's weaknesses and mistakes to one's advantage. Occasionally referred to as "killer instinct." T (10) Long Range Planning (LRP): The ability of a player to foresee the outcome of a certain move, set up desired situations that are more favorable, and work to alter the outcome. T (1 1) Memory M: Ability to remember previous games. B (12) Personality (P): Manners and emotional strength, and their effects on the opponent in playing the game and on the player in keeping his wits. T (13) Preparation (PR): Study and review of previous games and ideas. T (14) Quickness (Q): The ability of a player to see clearly the heart of a complex problem. T (15) Relative Youth (RY): The vigor, aggressiveness, and daring to try new ideas and situations, a quality usually attributed to young age. T (16) Seconds (S): The ability of other experts to help one to analyze strategies between games. B (17) Stamina (ST): Physical and psychological ability of a player to endure fatigue and pressure. T (18) Technique M: Ability to use and respond to different openings, improvise middle game tactics, and steer the game to a familiar ground to one's advantage. Monetary Exchange Rate – Dollar versus the Yen In the late 1980’s three economists at the University of Pittsburgh, Professors A. Blair, R. Nachtmann, and J. Olson, worked with Thomas Saaty on predicting the yen/dollar 26 exchange rate. The paper was published in Socio-Economic Planning Sciences 31, 6(1987). The predicted value was fairly close to the average value for a considerable number of months after that. Value of Yen/Dollar Exchange Rate in 90 Days Relative Forward Official Rel. Deg. of Size/Direction of Past Behavior of Interest Exchange Exchg. Mkt. Confid. in US Current Acct. Exchange Rates Rate Rate Bias Intervention the US Econ. Balance .035 .423 .023 .164 .103 .252 Federal Size of Bank of Forward Rate Size of Con- erratic Relative Relative Relative Size of Antici- Rele- Irrele- Reserve Mon. Federal Japan Monet. Premium/ Forward rate sistent .027 Infaltion Real Political Deficit or pated vant vant Policy Deficit Policy Discount Differential .137 Rates Growth Stability Surplus Changes .004 .031 .294 .032 .097 .007 .016 .019 .008 .075 .032 .221 Tighter Contract Tighter High Premium Strong Strong Higher Higher More Large Decrease High High .191 .002 .007 .002 .008 .026 .009 .013 .003 .048 .016 .090 .001 .010 Steady No Steady Medium Discount Moderate Moderate Equal Equal Equal Small No Charge Medium Med. .082 Change .027 .002 .008 .100 .009 .006 .003 .022 .016 .106 .001 .010 Easier .009 Easier Low Weak Weak Lower Lower Less Increase Low Low .191 Expand .063 .002 .011 .009 .001 .003 .006 .025 .001 .010 .021 Probable Impact of Each Fourth Level Fctor 119.99 and below 119.99-134.11 134.11-148.23 148.23-162.35 162.35 and above Sharp Moderate No Moderate Sharp Decline Decline Change Increase Increase .1330 .2940 .2640 .2280 .0820 Expected Value is 139.90 yen/$ (in the late 1980’s) Number of Children in Rural Indian Families In a hierarchy whose goal is the optimal family size in India (from a study published in the Journal of Mathematical Sociology, 1983, Vol.9 pp. 181-209, there were four major criteria of Culture (with subcriteria: Religion, Women Status, Manlihood), Economic factors (with subcriteria: Cost of child Rearing, Old Age security, Labor, Economic Improvement, Prestige and Strength),Demographic factors (with subcriteria: Short Life Expectancy, High Infant Mortality) and the Availability and acceptance of Contraception (with subcriteria: High Level of Availability and Acceptance of contraception, Medium level of Availability and Acceptance of contraception, low level of Availability and Acceptance of contraception. At the bottom three alternatives were considered: Families with 3 or Less Children, Families with 4 to 7 Children, and Families with 8 or More Children. The outcome of this example for reasons explained in the research paper had two projections of 5.6 and 6.5 children per family (due to regional differences.) The actual value we obtained from the literature after the study was done were 6.8 births per woman in 1972 and 5.6 in 1978. 27 Decision by the US Congress on China Joining the World Trade Organization (WTO) in May 2000? Study Done in 1999 (see Socio-Economic Planning Sciences 35 (2001) Briefly, the alternatives of the decision are: 1. Passage of a clean PNTR bill: Congress grants China Permanent Normal Trade Relations status with no conditions attached. This option would allow implementation of the November 1999 WTO trade deal between China and the Clinton administration. China would also carry out other WTO principles and trade conditions. 2. Amendment of the current NTR status bill: This option would give China the same trade position as other countries and disassociate trade from other issues. As a supplement, a separate bill may be enacted to address other matters, such as human rights, labor rights, and environmental issues. 3. Annual Extension of NTR status: Congress extends China’s Normal Trade Relations status for one more year, and, thus, maintains the status quo. Four hierarchies shown in the figure below were considered whose outcomes were combined as briefly outlined to derive the final priorities that show how Congress was going to vote and in fact China was later admitted to the WTO. Benefits to US (0.25) Increased US Exports to China Improved Rule of Law China's Promise to Respect Increased Employment in US Benefits to Lower Income Consumers 0.44 Intellectual Property Rights, Anti-Dumping and 0.07 0.05 Improved Investment Environment Section 201 Provisions 0.26 0.18 PNTR:0.59(1) PNTR:0.58(1) PNTR:0.65(1) PNTR:0.54(1) PNTR:0.58(1) Amend NTR:0.28(0.47) Amend NTR:0.31(0.53) Amend NTR:0.23(0.53) Amend NTR:0.30(0.55) Amend NTR:0.31(0.53) Annual Extension:0.13(0.22) Annual Extension:0.11(0.19) Annual Extension:0.12(0.19) Amend NTR:0.30(0.30) Annual Extension:0.11(0.19) Benefits Synthesis (Ideal): PNTR 1.00 Amend NTR 0.51 Annual Extension 0.21 Opportunities for US (0.20) Improve Promote Democracy Improve Environment Improve Human and Labor Rights US-Sino Relations 0.23 0.14 0.08 0.55 PNTR:0.65 (1) PNTR:0.57 (1) PNTR:0.57 (1) PNTR:0.54 (1) Amend NTR:0.23 (0.35) Amend NTR:0.33 (0.58) Amend NTR:0.29 (0.51) Amend NTR:0.30 (0.44) Annual Extension:0.12 (0.18) Annual Extension:0.10 (0.18) Annual Extension:0.14 (0.25) Annual Extension:0.16 (0.20) Opportunities Synthesis (Ideal): PNTR 1 Amend NTR 0.43 Annual Extension 0.13 28 Costs to US (0.31) Loss of US Access Workers in Some Sectors to China's Market of US Economy May Lose Jobs 0.83 0.17 PNTR :0.10 (0.17) PNTR :0.57 (1) Amned NTR :0.30 (0.5) Amned NTR :0.29 (0.50) Annual Extension :0.60 (1) Annual Extension :0.14 (0.25) Costs Synthesis (which is more costly, Ideal): PNTR 0.31 Amend NTR 0.50 Annual Extension 0.87 Risks for US (0.24) Loss of Trade as US-China Conflict China Violating Regional Stability China's Reform Retreat Leverage over Other Issues 0.25 0.25 0.07 0.43 PNTR : 0.59 PNTR : 0.09 PNTR : 0.09 PNTR : 0.09 Amend NTR : 0.36 Amend NTR : 0.29 Amend NTR : 0.28 Amend NTR : 0.24 Annual Extension: 0.05 Annual Extension: 0.62 Annual Extension: 0.63 Annual Extension: 0.67 Risks Synthesis (more risky, Ideal): PNTR 0.54 Amend NTR 0.53 Annual Extension 0.58 Figure Hierarchies for Rating Benefits, Costs, Opportunities, and Risks. Factors for Evaluating the Decision Economic: 0.56 Security: 0.32 Political:0.12 -Growth (0.33) -Regional Security (0.09) -Domestic Constituencies (0.80) -Equity (0.67) -Non-Proliferation (0.24) -American Values (0.20) -Threat to US (0.67) . Figure Prioritizing the Strategic Criteria to be used in Rating the BOCR How to derive the priority shown next to the goal of each of the four hierarchies shown in the last figure is outlined in the table below. We rated each of the four merits: benefits, costs, opportunities and risks of the dominant PNTR alternative, as it happens to be in this case, in terms of intensities for each assessment criterion. The intensities, Very High, High, Medium, Low, and Very Low were themselves prioritized in the usual pairwise comparison matrix to determine their priorities. We then assigned the appropriate intensity for each merit on all assessment criteria. The outcome is as found in the bottom row of table. 29 Table Priority Ratings for the Merits: Benefits, Costs, Opportunities, and Risks Intensities: Very High (0.42), High (0.26), Medium (0.16), Low (0.1), Very Low (0.06) Benefits Opportunities Costs Risks Economic Growth (0.19) High Medium Very Low Very Low (0.56) Equity (0.37) Medium Low High Low Regional (0.03) Low Medium Medium High Security Non-Proliferation Medium High Medium High (0.32) (0.08) Threat to US (0.21) High High Very High Very High Political Constituencies (0.1) High Medium Very High High (0.12) American Values Very Low Low Medium (0.02) Low Priorities 0.25 0.20 0.31 0.24 We are now able to obtain the overall priorities of the three major decision alternatives listed earlier, given as columns in the table below which gives three ways of synthesize for the ideal mode, we see in bold that PNTR is the dominant alternative any way we synthesize as in the last two columns. Table Two Methods of Synthesizing BOCR Using the Ideal Mode bB + oO - cC - rR Opportunities Alternatives Benefits BO/CR Costs Risks (0.25) (0.20) (0.31) (0.24) PNTR 1 1 0.31 0.54 5.97 0.22 Amend 0.51 0.43 0.50 0.53 0.83 -0.07 NTR Annual 0.21 0.13 0.87 0.58 0.05 -0.33 Exten. NETWORK VALIDATION EXAMPLES Turn Around of the U.S Economy Let us consider the problem of the turn around of the US economy and introduce 3, 6, 12, 24 month time periods at the bottom (see Blair et. al). Decomposing the problem hierarchically, the top level consists of the primary factors that represent the forces or major influences driving the economy: “Aggregate Demand” factors, “Aggregate Supply” factors, and “Geopolitical Context.” Each of these primary categories was then decomposed into subfactors represented in the second level. Under Aggregate Demand, we identified consumer spending, exports, business capital investment, shifts in consumer and business investment confidence, fiscal policy, monetary policy, and expectations 30 with regard to such questions as the future course of inflation, monetary policy and fiscal policy. (We make a distinction between consumer and business investment confidence shifts and the formation of expectations regarding future economic developments.) Under Aggregate Supply, we identified labor costs (driven by changes in such underlying factors as labor productivity and real wages), natural resource costs (e.g., energy costs), and expectations regarding such costs in the future. With regard to Geopolitical Context, we identified the likelihood of changes in major international political relationships and major international economic relationships as the principal subfactors. With regard to the subfactors under Aggregate Demand and Aggregate Supply, we recognized that they are, in some instances, interdependent. For example, a lowering of interest rates as the result of a monetary policy decision by the Federal Reserve should induce portfolio rebalancing throughout the economy. In turn, this should reduce the cost of capital to firms and stimulate investment, and simultaneously reduce financial costs to households and increase their disposable incomes. Any resulting increase in disposable income stimulates consumption and, at the margin, has a positive impact on employment and GNP. This assumes that the linkages of the economy are in place and are well understood. This is what the conventional macroeconomic conceptual models are designed to convey. The third level of the hierarchy consists of the alternate time periods in which the resurgence might occur as of April 7, 2001: within three months, within six months, within twelve months, and within twenty-four months. Because the primary factors and associated subfactors are time-dependent, their relative importance had to be established in terms of each of the four alternative time periods. Thus, instead of establishing a single goal as one does for a conventional hierarchy, we used the bottom level time periods to compare the two factors at the top. This entailed creation of a feedback hierarchy known as a "holarchy" in which the priorities of the elements at the top level are determined in terms of the elements at the bottom level, thus creating an interactive loop. The figure below provides a schematic representation of the hierarchy we used to forecast the timing of the economic resurgence. 31 Figure Overall View of the “2001” Model To obtain our forecast, we subsequently multiplied each priority by the midpoint of its corresponding time interval and added the results (as one does when evaluating expected values): Time Period Midpoint of Time Priority of Time Midpoint x Period Period Priority (Expressed in months from present, with the current month as 0.) Three months 0 + (3 – 0)/2 = 1.5 0.30581 0.45871 Six months 3 + (6 – 3)/2 = 4.5 0.20583 0.92623 Twelve months 6 + (12 – 6)/2 = 9.0 0.18181 1.63629 Twenty-four 12 + (24 – 12)/2 = 0.30656 5.51808 months 18.0 TOTAL 8.53932 We interpret this to mean that the recovery would occur 8.54 months from the time of the forecasting exercise, or in the fall. The Wall Street Journal of July 18, 2003, had the following to say about the turnaround date: 32 The Wall Street Journal Friday, July 18, 2003 Despite Job Losses, the Recession Is Finally Declared Officially Over JON E. HILSENRATH The National Bureau of Economic Research When calling the end to a recession, said the U.S. economic recession the NBER focuses heavily on two economic that began in March 2001 ended eight indicators: the level of employment months later, not long after the Sept. 11 and gross domestic product, or the terrorist attacks. total value of the nation's goods and services. Most economists concluded more than Since the fourth quarter of 2001, GDP has a year ago that the recession ended in late expanded slowly but consistently-rising 4 % 2001. But yesterday's declaration by the through March of 2003. NBER-a private, nonprofit economic research Employers, however, have eliminated group that is considered the official 938,000 payroll jobs since November 2001. arbiter of recession timing-came after a In addition, 150,000 people have dropped lengthy internal debate over whether out of the labor force because they are there can be an economic recovery if the discouraged about their job prospects, labor market continues to contract. according to the government. The bureau's answer: a decisive yes. Market Shares for the Cereal Industry (2002) The following is one of numerous validation examples done by my graduate students in business most of whom work at some company. Many of the examples are done in class in about one hour and without access to data. The answer is only found later on the Internet. The example below was developed by Stephanie Gier and Florian John in March 2002. They write: To become familiar with the Super Decision software we have chosen to estimate the market shares for the Ready-to Eat breakfast cereal industry. This idea was born after and delicious breakfast with Post’s OREO O’s. To see how good our assumptions were, we compare our calculated results with the market shares of 2001. First we created the model. We identified 6 major competitors in the ready to eat cereal market, Kellogg, General Mills, Post, Quaker, Nabisco and Ralston as our alternatives. There were more companies in this market having an actual cumulative market share of roughly about 6% that it turned out later that we had left out. Since we were only concerned with deriving relative values, the relative shares of other residual companies do not matter. Major impacts on the companies’ market shares are: Price of the products offered (named cost for the consumer) Advertising / Sales Ratio (how much money is spend for advertising) Shelf Space (places where the products are located in the stores) Tools (Selling Tools used to increase sales and market shares) Distribution / Availability (major distribution channels used to sell the product) These five major impacts (clusters) are further divided in the following nodes: 33 Tools: (Coupons, trade dealing, in-pack premiums, vitamin fortifications) Distribution: (Supermarket Chains, Food Stores, Mass Merchandiser) Shelf Space: (Premium Space, Normal Space, Bad Space) Cost: (Expensive, Normal, Cheap) Advertising: (<15%,<14%,<13%,<12%,<11%,<5%) Their interactions are depicted in the figure below. Second we carried out comparisons and performed calculations to obtain the final result (see later). Third we compared our calculated market shares with the real market shares for 2001. The table that follows lists estimated market share values and the actual ones taken from the website of the International Data Corporation. SHELF SPACE Premium space Normal space Bad space TOOLS COST Coupons Trade dealing Expensive In-pack premium Normal Vitamin-fortification Cheap DISTRIBUTION/ ADVERTISING AVAILABILITY < 15 % Supermarket chains < 14 % Food stores < 13 % Mass merchandiser <12 % < 11 % <5% ALTERNATIVES Kellogg General Miles Post Quker Nabisko Ralston Figure Cereal Industry Market Share Table Overall-Results, Estimated and Actual Alternatives Kellogg General Mills Post Quaker Nabisco Ralston Estimated 0.324 0.255 0.147 0.116 0.071 0.087 Actual 0.342 0.253 0.154 0.121 0.057 0.073 Compatibility index value: 1.01403 (very good). It is obtained by multiplying element- wise the matrix of ratios of ones set of data, by the transpose of the matrix of ratios of the other set, adding all the resulting entries and dividing by n2 and requiring that this ratio not be more than 1.1. Let us describe the calculations needed to derive the result in the “Estimated” column of the table. From the pairwise comparison judgments we constructed a super matrix, done automatically by the software Super Decisions. Then we weighed blocks of the 34 supermatrix by the corresponding entries from the matrix of priority vectors of paired comparisons of the influence of all the clusters on each cluster with respect to market share shown in the following table. This yielded the weighted supermatrix that is now stochastic as its columns add to one. We raised this matrix to limiting powers to obtain the overall priorities of all the elements in the figure. Table Cluster Priority Matrix Distrib./ Advertising Alternatives Cost Shelf Space Tools Availability Advertising 0.000 0.184 0.451 0.459 0.000 0.000 Alternatives 0.000 0.000 0.052 0.241 0.192 0.302 Cost 0.000 0.575 0.000 0.064 0.044 0.445 Distribution / 0.000 0.107 0.089 0.000 0.364 0.159 Availability Shelf Space 0.000 0.071 0.107 0.084 0.297 0.000 Tools 0.000 0.062 0.302 0.152 0.103 0.095 Market Shares for the Airline Industry (2001) James Nagy did the following study of the market share of eight US airlines. Nowhere did he use numerical data, but only his knowledge of the airlines and how good each is relative to the others on the factors mentioned below. Note that in three of the clusters there is an inner dependence loop that indicates that the elements in that cluster depend on each other with respect to market share. The table gives the final estimated and the actual relative values that are again very close. The figure below shows the network model of clusters and their inner and outer dependence connections that produced these results. Nagy writes: “I initially chose the airline industry for the assignment because I was a frequent traveler. My study group at Katz helped me make the comparisons between airlines that I did not have first hand experience as a passenger. Otherwise, I used my personal experience and perception of consumer sentiment towards the airlines to make the comparison. I was equally surprised at the results. In fact, I initially questioned how they could be so close. I would like to see the results of a study using today’s consumer perception. A lot has changed in the industry since the 9/11 tragedy in the year 2001. You could divide the class up into 4 to 5 small groups and let them do the comparisons as individual groups and compare the results.” Table Market Share of Airlines, Actual and Predicted Actual (yr 2000) Model Estimate American 23.9 24.0 United 18.7 19.7 Delta 18.0 18.0 Northwest 11.4 12.4 Continental 9.3 10.0 US Airways 7.5 7.1 Southwest 5.9 6.4 American West 4.4 2.9 35 Figure Airline Model from the ANP Super Decisions Software The Adjacency and Path Matrices The question often arises as to why raising the matrix of judgments or sometimes the supermatrix to a power determines for its entries the number of paths between the corresponding vertices, whose length is equal to that power. The vertex matrix is a first step for understanding this concept. We define a vertex (or adjacency) matrix for both directed and undirected graphs. The element in the (i,j) position of the matrix is equal to the number of edges incident with both vertex i and vertex j (or directed from vertex i to vertex j in the directed case). Thus for the directed graph of a network we might have: v1 v2 v3 v4 v5 v6 v1 0 0 0 0 0 0 v 2 0 0 1 0 0 0 v 3 0 1 0 0 0 0 V v 4 0 0 0 0 0 0 v5 1 1 0 1 1 1 v6 1 0 0 0 0 0 In general, we have the following theorem regarding the vertex matrix V of a graph: Theorem: The matrix V n gives the number of arc progressions of length n between any two vertices of a directed graph. Proof: If aik is the number of arcs joining vi to vk and akj is the number of arcs joining vk to vj, then a ik a kj is the number of different paths each consisting of two arcs joining vi to 36 vj and passing through vk. If this is summed over all values of k, that is, over all the intermediate vertices, one obtains the number of paths of length 2 between vi and vj. If we now use aij to form a ija jm, we have the number of different paths of length 3 between vi and vm passing through vj, and so on. Thus if we assume the theorem is true for V n-1 , then the coefficients of V n = V n-1 V give the number of paths of length n between corresponding vertices. This completes the proof. A similar theorem holds for undirected graphs. Relationship Between the Supermatrix and the ANP and Input-Output Econometric Analysis The following proof of the relationship between the supermatrix and Leontieff’s Input- Output Model is due to Luis G. Vargas. The figure below depicts interdependence in economic input-output models. Sectors Value External Added Demand External Markets Figure Input-Output Network Let A be the matrix of relative input-output coefficients. Let v and d be the relative value added and the relative final demand vectors. Let xij the amount of resource that sector j receives from sector i. Let v j be the value added corresponding to sector j and let d i be the final demand of sector i. The value added of a sector includes wages and salaries, profit-type income, interest, dividends, rents, royalties, capital consumption allowances and taxes. The final demand of a sector includes imports, exports, government purchases, changes in inventory, private investment and sometimes, household purchases. Thus, the input-output matrix is given by: Sectors S1 S2 Sn S1 x11 x12 x1n d1 S2 x21 x22 x2 n d2 Sn xn1 xn 2 xnn dn v 0 1 v2 vn 37 Let I1 , I 2 , , I n be the total input to the sectors and let O1 , O2 , , On be the total output of n n the sectors, i.e., x j 1 ij di Oi and x i 1 ij vj I j . xij Let the relative input-output coefficients be given by aij . Ij The relative final demand of a sector with respect to the other sectors is given by d vj ai ,n 1 n i . On the other hand, the relative value added is given by an 1,1 . Thus, dh Ij h 1 the matrix of interactions represented by the network in Figure 1 is give by: a11 a12 a1n a1,n+1 a21 a22 a2 n a2,n 1 A a,n 1 W an 1, 0 an1 an 2 ann an ,n 1 a 0 n 1,1 an 1,2 an 1,n Because W is a stochastic irreducible matrix, lim W k is given by weT where w is a k (n 1) 1 vector that is the principal right eigenvector of W, and eT (1,1,...,1) is the 1 (n 1) unit vector. Thus, we have Ww w or n a w j 1 ij j wn 1ai ,n 1 wi , i 1, 2,..., n n a j 1 n 1, j w j wn 1 In matrix notation we have: ( I A) wn wn 1a,n 1 an 1, wn wn 1 where wn (w1 ,..., wn )T . Thus, we have wn wn 1 ( I A)1 a,n1 and hence, we can write w ( I A) 1 a,n 1 T lim W k n 1 e . k wn 1 Note that wn wn 1 ( I A)1 a,n1 is the relative output of the economy as given by Leontieff’s model. 38 Example of an Input-Output Matrix Consider the following input-output matrix shown in the table below. Table Input-Output Matrix Agriculture Manufacturing Services Other Final Demand Total Output Agriculture 10 65 10 5 10 100 Manufacturing 40 25 35 75 25 200 Services 15 5 5 5 90 120 Other 15 10 50 50 100 225 Value Added 20 95 20 90 0 Total Input 100 200 120 225 225 645 The supermatrix corresponding to this input-output model is given in the table below: Table The Supermatrix for the Input-Output Matrix 0.1 0.325 0.083333333 0.022222222 0.044444444 0.4 0.125 0.291666667 0.333333333 0.111111111 W= 0.15 0.025 0.041666667 0.022222222 0.4 0.15 0.05 0.416666667 0.222222222 0.444444444 0.2 0.475 0.166666667 0.4 0 and the limiting matrix lim W k is given in the table below: k Table Limiting Supermatrix of the Input-Output Table 0.114942529 0.114942529 0.114942529 0.114942529 0.114942529 0.229885057 0.229885057 0.229885057 0.229885057 0.229885057 0.137931034 0.137931034 0.137931034 0.137931034 0.137931034 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.25862069 0.114942529 wn 0.229885057 and by normalizing to unity we can see that 0.137931034 0.25862069 0.15503876 100 1 wn (1 wn 1 ) 0.310077519 1 200 0.186046512 645 120 0.348837209 225 which coincides with the normalized values of the total output of the economy. The input-output table (supermatrix) of the Sudan economy (1976) with eigenvector values as the estimates and the actual values in parentheses was computed by the Nobel Laureate Lawrence Klein who participated in the study with his Wharton Forecasting 39 Associates. The results of this fairly complex exercise using paired comparison judgments are generally close to those of the econometric forecasting model. Table Input-Output Table of Sudan Economy (1976) by Wharton Forecasting Associates Agriculture Public Manufacturing Transportation Construction Services Utilities and Mining and Distribution Agriculture .0079 0 .2331 .0008 .0699 0 (.00737) (0) (.21953) (.00042) (.06721) (0) Public .0009 0 .0130 .0075 0 .0033 Utilities (.00024) (0) (.01159) (.00618) (0) (.00283) Manufacturing .0041 0 0 .0089 .0379 .0037 and Mining (.00393) (0) (0) (.00857) (.04216) (.00322) Transportation .0691 .1694 .1281 0 .1115 .0153 and (.06993) (.145360 (.12574) (0) (.09879) (.00641) Distribution Construction 0 0 0 0 0 .0546 (0) (0) (0) (0) (0) (.05402) Services 0 .0117 .0224 .0224 .0039 .0004 (0) (.01030) (.02549) (.02422) (.00520) (.000210 Desirability of Drilling for Oil in Alaska – the ANWR Model In late 2002 a study was done [Emanuel, Cefalu, 2002] to find out whether drilling for oil should be allowed in the Artic National Wildlife Reserve (ANWR) in northern Alaska. Environmentalists, mostly living in the lower 48 US states have been blocking drilling in the region. In the ANWR study the alternatives were: Do Drill, Do Not Drill. And the study results were compared against a poll of Alaskan residents asking the question: “Do you think we should drill, or not drill in ANWR?” ANWR-Arctic National Wildlife Refuge covers 19 million acres on the Northern coast of Alaska north of the Arctic Circle and 1,300 miles south of the North Pole. The consensus of the geologic community is that the Coastal Plain of ANWR represents the highest petroleum potential onshore area yet to be explored in North America. If explored, it is estimated that it will take 15 years or more before oil and gas will reach the market. Legislation was passed in the 1980’s that created a majority of the National Parks in Alaska and expanded ANWR to its current size. The Reagan Administration was ready to drill but was derailed by the Exxon Valdez catastrophe. The first Bush Administration likewise wanted to drill, but was unsuccessful. The Clinton Administration designated it as a protected area and it has been that way ever since then. The second Bush Administration, in response to ongoing Middle East violence and the 9/11 terrorist 40 attacks, sees drilling in ANWR as vital not only for economic but national security reasons. Several environmental groups consider ANWR a great American natural treasure and one of the last places on the earth where an intact expanse of arctic and sub- arctic lands remain protected. They feel the habitat, the wildlife, and the culture need to be protected from the exploration of gas and oil. The top- level main network is shown in Figure below. Figure The Main Network of the ANWR Decision Model The strategic criteria here are General Public Opinion, International Politics, and Amount of Oil. They are first pairwise compared for importance, then used to rate the importance of the top rated alternative for each of the Benefits, Opportunities, Costs and Risks, called the merits of the decision, as shown in the table. The ratings categories themselves are pairwise compared to establish priorities for High, Medium and Low. To select the appropriate rating, keep in mind the highest priority alternative under the merit being evaluated: Risks for example as shown in the figure. Do Not Drill is the highest priority alternative under risk, meaning it is the most risky, so keeping that in mind the Risk merit under General Public Opinion is evaluated as being low. The results of rating the merits are also shown in the figure and the driving merit in this decision is Benefits at .407, closely followed by Opportunities at .364. In this decision, Costs are found to be unimportant, and Risks are about half as important as Benefits and Opportunities. These values for the merits nodes are used to weight the values for the alternatives as determined in the subnets they control to give the overall results for the alternatives. Connected to each of the BOCR merit nodes is the subnet for Benefits that contains a hierarchy of control criteria: Economic, Political, Social. The control criteria are pairwise compared for importance. 41 Figure Rating the BOCR under Strategic Criteria Figure Subnet Containing Control Criteria under Benefits The subnet for Benefits containing a hierarchy of control criteria is shown in the figure. And connected to each control criterion node under benefits is a decision network containing the alternatives of the problem. The decision subnet for the economic control criterion under benefits is shown below in the figure. The final results are shown in the figure where the results from the subnets are combined in the main network using the formula Bb + Oo - Cc – Rr where b, o, c, and r are the values for the decision alternatives from the control subnets, and B, O, C, and R are the priorities of the BOCR as determined by rating them under the strategic criteria. 42 Figure The Decision Subnet under Economic Benefits Figure Final Results of ANWR Study . In a recent poll conducted among native Alaskans they support opening ANWR to oil and gas exploration 75% to 19% with 6% undecided. The question asked was “Do you believe oil and gas exploration should or should not be allowed within the ANWR Coastal Plain?” Assigning the 6% equally yields 78% to 22%, the results of the poll are very close to the results of the model. 43

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 75 |

posted: | 6/24/2011 |

language: | English |

pages: | 43 |

OTHER DOCS BY chenmeixiu

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.