# Analytic Network Process (ANP) slides

The Analytic Network Process (ANP) for Decision Making and Forecasting with Dependence and Feedback With feedback the alternatives depend on the criteria as in a hierarchy but may also depend on each other. The criteria themselves can depend on the alternatives and on each other as well. Feedback improves the priorities derived from judgments and makes prediction much more accurate. 1 Linear Hierarchy Goal Criteria component, cluster (Level) Subcriteria element Alternatives A loop indicates that each element depends only on itself. 2 Feedback Network with components having Inner and Outer Dependence among Their Elements C4 C1 Feedback Arc from component C4 to C2 indicates the outer dependence of the elements in C2 on the elements in C4 with respect to a common property. C2 C3 Loop in a component indicates inner dependence of the elements in that component with respect to a common property. 3 Inner and Outer Dependence and the Control Hierarchy In a network, the elements in a component may be people (e.g., individuals in the White House) and those in another component may also be people (e.g., individuals in Congress). The elements in a component may influence other elements in the same component (inner dependence) and those in other components (outer dependence) with respect to each of several properties. We want to determine the overall influence of all the elements. In that case we must organize the properties or criteria and prioritize them in the framework of a control hierarchy (or a network), perform comparisons and synthesize to obtain the priorities of these properties. We then derive the influence of elements in the feedback system with respect to each of these properties. Finally, we weight the resulting influences by the importance of the properties and add to obtain the overall influence of each element. 4 Main Operations of the ANP Relative measurement: Reciprocal relation Judgments: Homogeneity Hierarchy or Network: Structure of problem; the control hierarchy Priorities, Dominance and Consistency: Eigenvector Composition, Additive to also handle dependence through the supermatrix Supermatrix: Dependence Neural Firing: Fredholm Kernel and Eigenfunctions 5 Inner and Outer Dependence and the Control Hierarchy cont. Control hierarchies fall in four groups: Benefits, Costs, Risks, & Opportunities. Benefits and costs measure the positive and negative contributions or importance of the alternatives if they happen, but will they happen? Risks and opportunities measure the likelihood that the alternatives will happen and make positive and respectively negative contributions. Each one is a hierarchy (or a network) by itself. The overall priorities of the

alternatives with respect to each of these are then combined by forming the ratios: Benefits x Opportunities Costs x Risks to obtain their final overall priorities for a decision. 6 Weighting The Components In the ANP one often needs to prioritize the influence of the components themselves on each other component to which the elements belong. This influence is assessed through paired comparisons with respect to a control criterion. The priority of each component is used to weight the priorities of all the elements in that component. The reason for doing this is to enable us to perform feedback multiplication of priorities by other priorities in a cycle, an infinite number of times. The process would not converge unless the resulting matrix of priorities is column stochastic (each of its columns adds to one). To see that one must compare clusters in real life, we note that if a person is introduced as the president it makes much difference, for example, whether he or she is the President of the United States or the president of a local labor group. 7 Functional - Structural Criteria Independence - Dependence 1--Criteria completely independent from alternatives - Absolute Measurement, Intensities and Standards. 2--Criteria quasi dependent on alternatives - Relative Measurement: Rescale the weight of a criterion by the number of alternatives and their measurement (normalization). 3--Criteria completely dependent on alternatives - Feedback network - the Supermatrix. 8 Why ANP? The power of the Analytic Network Process (ANP) lies in its use of ratio scales to capture all kinds of interactions and make accurate predictions, and, even further, to make better decisions. So far, it has proven itself to be a success when expert knowledge is used with it to predict sports outcomes, economic turns, business, social and political decision outcomes. The ANP is a mathematical theory that makes it possible for one to deal systematically with all kinds of dependence and feedback. The reason for its success is the way it elicits judgments and uses measurement to derive ratio scales. Priorities as ratio scales are a fundamental kind of number amenable to performing the basic arithmetic operations of adding within the same scale and multiplying different scales meaningfully as required by the ANP. 9 Mutual Influence Among Several Elements In order to distinguish among the influence of several homogeneous elements that is exerted on a single element, the number of influencing elements cannot be more than a few. The reason is that the element that is influenced must be able to distinguish between the various influences and respond to them in relatives terms. If their number is large, the relative influence of each would be a small part of the total. On the other hand, if the number of elements is small, the relative influence of each one on any other single element would be large and distinguishable. A small change in the influence of any of these elements would not alter the receiving elements estimation of its overall influence. When the number of elements is large, they need to be put in different clusters. Unidirectional Influence A single powerful element may influence numerous other elements that do not influence it in return or influence each other. If many elements influence a single element without feedback, their number can be arbitrarily large. 10 The Questions to Answer About the Dominance of Influence Two kinds of questions encountered in the ANP: 1. Given a criterion, which element has greater influence (is more dominant) with respect to that criterion? Use one of the following two questions throughout an exercise. 2. Given a criterion and given an element X in any cluster, which element in the same cluster or a different cluster has greater influence on X with respect to that criterion? 2. Given a criterion and given an element X in any cluster, which element in the same or in a different cluster is influenced more by X with respect to that criterion. 11 Example of Control Hierarchy Optimum Function of A System

Environmental Economic Social Influence is too general a concept and must be specified in terms of particular criteria. It is analyzed according to each criterion and then synthesized by weighting with these priorities of the control criteria belonging to a hierarchy or to a system. 12 The Supermatrix Take a control criterion. The priorities of the elements derived from paired comparisons with respect to that control criterion are arranged both vertically and horizontally according to components. The elements in each component are listed for that component in a matrix known as the Supermatrix. Each vector taken from a paired comparison matrix is part of the column of the supermatrix representing the impact with respect to the control criterion of the elements of that component on a single element of the same or another component listed at the top. The Weighted Supermatrix All the clusters are pairwise compared according to their influence on a given cluster X with respect to the control criterion. This yields a vector of priorities of the impact of all the clusters on a given criterion. Each component of a vector is used to weight all the elements in the block of column priorities of the supermatrix corresponding to the impact of the elements of that cluster on X. The process is repeated for all the clusters resulting in a weighted supermatrix. In each block of the supermatrix, a column is either a normalized eigenvector with possibly some zero entries, or all of its elements are equal to zero. In either case it is weighted by the priority of the corresponding cluster on the left. If it is zero, that column of the supermatrix must be normalized after weighting by the clusters weights. This operation is equivalent to assigning a zero value to the cluster on the left when weighting a column of a block with zero entries and then re-normalizing the weights of the remaining clusters. 13 The Limiting Supermatrix The weighted supermatrix is now column stochastic from which one then derives the limiting supermatrix. There are four major cases to consider in deriving the limiting supermatrix depending on the simplicity or multiplicity of the principle eigenvalue and on the reducibility and irreducibility of the matrix. How to Read Off the Answer The desired priorities of the criteria and alternatives with respect to the corresponding control criterion can be read off the supermatrix as given or they may be structurally adjusted according to the number of elements in each cluster and appropriately re-weighted. How to Combine Benefits, Costs, Opportunities, Risks One must first combine the supermatrices for the benefits, then for the costs, then for the opportunities and then for the risks by using the weights of the control criteria for each. One then takes the ratio benefits x opportunities / costs x risks for the alternatives and selects the alternative with the largest ratio. 14 Networks and the Supermatrix c1 W= c1 e11 e12 c2 e1n 211 e22 cN e2n N12 eN2 e11e12 c2 e1n1 e21e22

cN e2n2 eN1eN2 eNnN W11 W12 W1N W21 W22 W2N WN1 WN2 WNN eNuN 15 where (j1) Wi1 Wij = (j1) Wi2 (j1) Wini (j2) Wi1 (j2) Wi2 (j2) Wini (jnj) Wi1 (jnj) Wi2 (jnj) Wini 16 Supermatrix of a Hierarchy 0 W= 0 0 0 0 W21 0 0 0 0

0 0 0 0 0 0 0 0 W32 0 0 0 0 0 Wn-1, n-2 0 0 0 Wn, n-1 I 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Wk= Wn,n-1 Wn-1,n-2 W32 W21 Wn,n-1 Wn-1,n-2 W32 ... Wn,n-1 Wn-1,n-2 Wn,n-1 I for k>n-1 18 The School Hierarchy as Supermatrix Goal Learning Friends School life Vocational training College preparation Music classes Alternative A Alternative B Alternative C Goal 0 0.32 0.14 0.03 0.13 0.24 0.14

0 0 0 Learning 0 0 0 0 0 0 0 0.16 0.59 0.25 Friends 0 0 0 0 0 0 0 0.33 0.33 0.34 School life Vocational trainingCollege preparation Music classes 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.45 0.77 0.25 0.69 0.09 0.06 0.5 0.09 0.46 0.17 0.25 0.22 A 0 0 0 0 0 0 0 1 0 0 B 0 0 0

0 0 0 0 0 1 0 C 0 0 0 0 0 0 0 0 0 1 Limiting Supermatrix & Hierarchic Composition Goal Learning Friends School life Vocational training College preparation Music classes Alternative A Alternative B Alternative C Goal 0 0 0 0 0 0 0 0.3676 0.3781 0.2543 Learning 0 0 0 0 0 0 0 0.16 0.59 0.25 Friends 0 0 0 0 0 0 0 0.33 0.33 0.34 School life Vocational trainingCollege preparation Music classes 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0.45 0.77 0.25 0.69 0.09 0.06 0.5 0.09 0.46 0.17 0.25 0.22 A 0 0 0 0 0 0 0 1 0 0 B 0 0 0 0 0 0 0 0 1 0 C 0 0 0 0 0 0 0 0 0 1 19 Criteria Independent from Alternatives When the criteria do not depend on the alternatives, the latter are kept out of the supermatrix and are evaluated in the usual hierarchic way by the distributive or ideal modes to make possible rank preservation or reversal as desired. The priorities of the criteria in terms of which the alternatives are evaluated hierarchically are taken from the limiting supermatrix. Here again benefit, cost, opportunity, and risk evaluation can be made to determine the ranks of the alternatives. 20 Structural Adjust After & Before the Final Results After computing the limiting results, if it is desired to group together elements from two or more clusters to determine their relative influence, the priorities of each cluster may be multiplied by the relative number of elements in that cluster to the total number in the set of clusters and then the entire set is normalized. One may think to do such structural adjustment in the weighting process of the original supermatrix. There may be occasions where that is what should be done.

21 The Management of a Water Reservoir Here we are faced with the decision to choose one of the possibilities of maintaining the water level in a dam at: Low (L), Medium (M) or High (H) depending on the relative importance of Flood Control (F), Recreation (R) and the generation of Hydroelectric Power (E) respectively for the three levels. The first set of three matrices gives the prioritization of the alternatives with respect to the criteria and the second set, those of the criteria in terms of the alternatives. 22 A Feedback System with Two Components Flood Control Low Level Recreation Intermediate Level HydroElectric Power High Level 23 1) Which level is best for flood control? Flood Control Low Low 1 Medium 1/5 High 1/7 Med 5 1 1/4 High Eigenvector 7 .722 4 .205 1 .073 Consistency Ratio = .107 3) Which level is best for power generation? Power Generation Low Low 1 Medium 5 High 9 Med 1/5 1 5 High Eigenvector 1/9 .058 1/5 .207 1 .735 2) Which level is best for recreation? Recreation

Low Low 1 Medium 7 High 5 Med 1/7 1 1/3 High Eigenvector 1/5 .072 3 .649 1 .279 Consistency Ratio = .056 Consistency Ratio = .101 24 Flood Control Recreation Hydro-Electric Power 2) At Intermediate Level, which attribute is satisfied best? Flood Control Recreation Hydro-Electric Power Low Level Dam F R E 1 3 5 1/3 1 3 1/5 1/3 1 Eigenvector .637 .258 .105 Consistency Ratio = .033 Flood Control Recreation Hydro-Electric Power F 1 5 9 Intermediate Level Dam F R E 1 1/3 1 3 1 3 1 1/3 1 1) At Low

Level, which attribute is satisfied best? Eigenvector .200 .600 .200 Consistency Ratio = .000 High Level Dam R E 1/5 1/9 1 1/4 4 1 Eigenvector .060 .231 .709 3) At High Level, which attribute is satisfied best? Consistency Ratio = .061 25 The six eigenvectors were then introduced as columns of the following stochastic supermatrix. F F R E L M H 0 0 0 .722 .205 .073 R E L M H 0 0 0 .072 .649 .279 0 0 0 .058 .207 .735 .637 .258 .105 0 0 0 .200 .600 .200

0 0 0 .060 .231 .709 0 0 0 One must ensure that all columns sum to unity exactly. 26 The final priorities for both, the height of the dam and for the criteria were obtained from the limiting power of the supermatrix. The components were not weighted here because the matrix is already column stochastic and would give the same limiting result for the ratios even if multiplied by the weighting constants. Its powers stabilize after a few iterations. We have F R E L M H F R E L M H 0 0 0 .223 .372 .405 0 0 0 .223 .372 .405 0 0 0 .223 .372 .405 .241 .374 .385 0 0 0 .241 .374 .385 0 0 0 .241 .374 .385 0 0 0 27 The columns of each block of this matrix are

identical, so that in the top right block we can read off the overall priority of each of the three criteria from any column, and read off the overall priorities of the three alternatives from any column of the bottom left block. It is clear from this analysis that for the kind of judgments provided, there is preference for a high dam with priority .405 for hydroelectric power generation with priority .385. 28 Choosing a Car: Foreign or Domestic? Cost A E J A E J 1 1/5 1/3 5 1 3 3 1/3 1 Eigenvector .637 .105 .258 American C R D C 1 1/3 1/4 R D 3 1 1 4 1 1 Consistency Ratio = .033 Repair Cost A E J A E 1 1/5 1/2 J 5 1 3 Eigenvector 2

1/3 1 .582 .109 .309 A E J A E J 1 5 3 1/5 1 1/3 1/3 3 1 Eigenvector .105 .637 .258 Consistency Ratio = .033 .634 .192 .174 Consistency Ratio = .008 European C R D C R D 1 1 2 1 1 2 1/2 1/2 1 Consistency Ratio = .003 Durability Eigenvector Eigenvector .250 .250 .500 Consistency Ratio = .008 Japanese C R D C R D

1 1/2 1 2 1 2 1 1/2 1 Eigenvector .400 .200 .400 Consistency Ratio = .000 29 Original Supermatrix C R D A E J C 0 0 0 .637 .105 .258 R 0 0 0 .582 .109 .309 D 0 0 0 .105 .637 .258 A .634 .192 .174 0 0 0 E .250 .250 .500 0 0 0 J .400 .200 .400 0 0 0 Limiting Supermatrix C R D A E J

C 0 0 0 .452 .279 .269 R 0 0 0 .452 .279 .269 D 0 0 0 .452 .279 .269 A .464 .210 .326 0 0 0 E .464 .210 .326 0 0 0 J .464 .210 .326 0 0 0 Choose an American car. Cost is the dominant criterion. 30 Date and Strength of Recovery of U.S. Economy Primary Factors Subfactors Adjustment Period Required for Turnaround Conventional adjustment Economic Restructuring Consumption (C) Exports (X) Investment (I) Fiscal Policy (F) Monetary Policy (M) Confidence (K) Financial Sector (FS) Defense Posture (DP) Global Competition (GC) 3 months 6 months 12 months 24 months The U.S. Holarchy of Factors for Forecasting Turnaround in Economic Stagnation

31 Table 1: Matrices for subfactor importance relative to primary factors influencing the Timing of Recovery Panel A: Which subfactor has the greater potential to influence Conventional Adjustment and how strongly? Consumption (C) Exports (E) Investment (I) Confidence (K) Fiscal Policy (F) Monetary Policy (M) C E I K F M Vector Weights 1 1/7 1/5 5 2 5 7 1 5 5 5 7 5 1/5 1 5 3 5 1/5 1/5 1/5 1 1/5 1 1/2 1/5 1/3 5 1 5 1/5 1/7 1/5 1 1/5 1 0.118 0.029 0.058 0.334 0.118 0.343 Panel B: Which subfactor has the greater potential to influence Economic Restructuring and how strongly? Financial Sector

(FS) Defense Posture (DS) Global Competition (GC) FS DP GC Vector Weights 1 3 3 0.584 1/3 1 3 0.281 1/3 1/3 1 0.135 32 Table 2: Matrices for relative influence of subfactors on periods of adjustment (months) (Conventional Adjustment) For each panel below, which time period is more likely to indicate a turnaround if the relevant factor is the sole driving force? Panel A: Relative importance of targeted time periods for consumption to drive a turnaround 3 6 12 24 Vec. Wts. 3 months 6 months 12 months 24 months 1 5 7 7 1/5 1 5 5 1/7 1/5 1 3 1/7 1/5 1/3 1 .043 .113 .310 .534 Panel C: Relative importance of targeted time periods for investment to drive a turnaround 3 6 12

24 Vec. Wts. 3 months 6 months 12 months 24 months 1 1 5 5 1 1 5 5 1/5 1/5 1 3 1/5 1/5 1/3 1 .078 .078 .305 .538 Panel E: Relative importance of targeted time periods for monetary policy to drive a turnaround 3 3 months 6 months 12 months 24 months 1 1/5 1/7 1/7 6 12 24 5 1 1/5 1/7 7 5 1 5 7 7 1/5 1 Vec. Wts. .605 .262 .042 .091 Panel B: Relative importance of targeted time periods for exports to drive a turnaround 3 3 months 6 months 12 months 24 months 1 1 5 5 6

1 1 5 5 12 24 1/5 1/5 1 1 1/5 1/5 1 1 Vec. Wts. .083 .083 .417 .417 Panel D: Relative importance of targeted time periods for fiscal policy to drive a turnaround 3 6 12 24 Vec. Wts. 3 months 6 months 12 months 24 months 1 1 3 5 1 1 5 5 1/3 1/5 1 1 1/5 1/5 1 1 .099 .087 .382 .432 Panel F: Expected time for a change of confidence indicators of consumer and investor activity to support a turnaround in the economy 3 6 12 24 Vec. Wts. 3 months 6 months 12 months 24 months 1 1/3 1/5 1/5 3 1 1/5 1/5 5

5 1 1/5 5 5 5 1 .517 .305 .124 .054 33 Table 3: Matrices for relative influence of subfactors on periods of adjustment (months) (Economic Restructuring) For each panel below, which time period is more likely to indicate a turnaround if the relevant factor is the sole driving force? Panel A: Financial system restructuring time 3 6 12 24 Vec. Wts. 3 months 6 months 12 months 24 months 1 3 5 7 1/3 1 5 7 1/5 1/5 1 5 1/7 1/7 1/5 1 Panel B: Defense readjustment time 3 6 12 24 .049 .085 .236 .630 3 months 6 months 12 months 24 months 1 3 5 7 1/3 1 5 7 1/5 1/5 1 5 1/7 1/7 1/5 1

Vec. Wts. .049 .085 .236 .630 Panel C: Global competition adjustment time 3 6 12 24 Vec. Wts. 3 months 6 months 12 months 24 months 1 1 5 5 1 1 5 5 1/5 1/5 1 3 1/5 1/5 1/3 1 .078 .078 .305 .538 Table 4: Most likely factor to dominate during a specified time period Which factor is more likely to produce a turnaround during the specified time period? Panel A: 3 Months CA R CA 1 1/5 R 5 1 Vec. Wts. .833 .167 Panel B: 6 Months CA CA 1 R 1/5 R 5 1 Vec. Wts. .833 .167 Conventional Adjustment Restructuring Panel C: 1 Year CA CA 1 R 1 R 1 1

Vec. Wts. .500 .500 CA R Panel D: 2 Years CA CA 1 R 5 R 1/5 1 Vec. Wts. .167 .833 34 Table 5: The Completed Supermatrix Conven. Adjust Economic. Restru. Conven. Economic. Consum. Adjust Restruc. Exports Invest. Confid. Fiscal Policy Monet. Policy Financ. Sector Defense Posture Global Compet. 3 mo. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.500 0.167 0.833 0.833 0.500 0.833 0.167 0.167 +-------------------------- ------+ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.118 0.0 Exports 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.029 0.0 Invest. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.058 0.0 Confid. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.334 0.0 Fiscal 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.118 0.0 Policy Monetary 0.343 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Policy ------+ +----+ Financ. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.584 0.0 Sector Defense 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.281 0.0 Posture Global 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.135 0.0 Compet. +----+ +------------------------------------------------------------------+ 3 months 0.0 0.0 0.043 0.083 0.078 0.517 0.099 0.605 0.049 0.049 0.089 0.0 6 months 0.0 0.0 0.113 0.083 0.078 0.305 0.086 0.262

0.085 0.085 0.089 0.0 1 year 0.0 0.0 0.310 0.417 0.305 0.124 0.383 0.042 0.236 0.236 0.209 0.0 0.0 0.534 0.417 0.539 0.054 0.432 0.091 0.630 0.630 0.613 0.0 2 years 0.0 Consum. 6 mo. 1 yr. 2 years 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 35 Table 6: The Limiting Supermatrix Conven. Economic. Consum. Adjust Restruc. Exports Invest. Confid. Fiscal Policy Monet. Policy Financ. Sector Defense Posture Global Compet. 6 mo. 1 yr. 2 years 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Conven. Adjust Economic Restru. 0.0

Consum. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Exports 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Invest. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Confid. 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 Fiscal Policy Monetary Policy Financ. Sector Defense Posture Global Compet. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.484 0.484 0.484 0.484 0.484 0.484 0.484 0.484 0.484 0.0 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 3 mo. 0.224

0.224 0.0 6 months 0.151 0.151 0.0 1 year 0.201 0.201 0.0 0.424 0.424 0.0 2 years 3 months 0.057 0.057 0.057 0.057 0.014 0.014 0.014 0.014 0.028 0.028 0.028 0.028 0.162 0.162 0.162 0.162 0.057 0.057 0.057 0.057 0.166 0.166 0.166 0.166 0.301 0.301 0.301 0.301 0.145 0.145 0.145 0.145 0.070 0.070 0.070 0.070 36 Synthesis/Results When the judgments were made, the AHP framework was used to perform a synthesis which produced the following results. First a meaningful turnaround in the economy would likely require an additional ten to eleven months, occurring during the fourth quarter of 1992. This forecast is derived from weights generated in the first column of the limiting matrix in Table 6, coupled with the mid-points of the alternate time periods (so as to provide unbiased estimates: .224 x 1.5 + .151 x 4.5 + .201 x 9 + .424 x 18 = 10.45 months from late December 1991/early January 1992 37 The Strength of Recovery Primary Factors Conventional Adjustment Subfactors

Consumption (C) Exports (X) Investment (I) Fiscal Policy (F) Monetary Policy (M) Confidence (K) Very Strong (5.5-6.5% GNP) Strong (4.5-5.5% GNP) Economic Restructuring Financial Sector (FS) Defense Posture (DP) Global Competition (GC) Moderate (3-4.5% GNP) Weak (2-3% GNP) 38 Table 7: Matrices for Primary and Subfactors for Strength of Recovery Panel A: Which primary factor will be more influential in determining the Strength of Recovery? Vector CA R Weights Conventional Adjustment (CA) Restructuring (R) 1 5 1/5 1 .167 .833 Panel B: Which subfactor is more important in influencing Conventional Adjustment? Consumption (C) Exports (E) Investment (I) Confidence (K) Fiscal Policy (F) Monetary Policy (M) C E I K F M Vector Weights 1 1/7 1/3 1 1/7 1/3 7 1 5 5

1 7 3 1/5 1 3 3 5 1 1/5 1/3 1 1/7 1/3 7 1 1/3 7 1 7 3 1/5 1/5 3 1/7 1 0.317 0.037 0.099 0.305 0.035 0.207 Panel C: Which subfactor is more important in influencing Economic Restructuring? Financial Sector (FS) Defense Posture (DS) Global Competition (GC) FS DP GC Vector Weights 1 1/5 1/3 0.105 5 1 3 0.637 3 1/3 1 0.258 CI = 0.037 39

Table 8: Matrices for relative influence of subfactors on Strength of Recovery (Conventional Adjustment) For each panel below, which intensity is more likely to obtain if the designated factor drives the recovery? Panel A: Relative likelihood of the strength of recovery if consumption drives the expansion V S M W Vec. Wts. Very Strong (V) 1 Strong (S) 1 1/5 Moderate (M) 1/7 Weak (W) 1 1 1/5 1/7 5 5 1 1/3 7 7 3 1 .423 .423 .104 .051 CI = 0.028 Panel B: Relative likelihood of the strength of recovery if exports drives the expansion V Very Strong (V) 1 Strong (S) 1 3 Moderate (M) 5 Weak (W) S M 1 1 3 5 1/3 1/3 1 3 W 1/5 1/5 1/3 1 Vec. Wts. .095 .095 .249 .560 CI = 0.016 Panel C: Relative likelihood of the strength of recovery if investment drives the expansion V S M W Vec. Wts.

Panel D: Relative likelihood of the strength of recovery if confidence drives the expansion V S M W Vec. Wts. Very Strong (V) 1 Strong (S) 1 3 Moderate (M) 1/2 Weak (W) Very Strong (V) 1 Strong (S) 1 1/3 Moderate (M) 1/5 Weak (W) 1 1 3 1/2 1/3 1/3 1 1/6 2 2 6 1 .182 .182 .545 .091 CI = 0.0 1 1 1/3 1/5 3 3 1/3 1/7 5 5 7 1 .376 .376 .193 .054 CI = 0.101 Panel E: Relative likelihood of the strength of recovery if fiscal policy drive the expansion V S M W Vec. Wts. Panel F: Relative likelihood of the strength of recovery if monetary policy drives the expansion V S M W Vec. Wts. Very Strong (V) Strong (S)

Very Strong (V) 1 Strong (S) 1 5 Moderate (M) 3 Weak (W) Moderate (M) Weak (W) 1 1 5 1 1 1 5 1 1/5 1/5 1 1/5 CI = 0.0 1 1 5 1 .125 .125 .625 .125 1 1 5 3 1/5 1/5 1 1/7 1/3 1/3 7 1 .084 .084 .649 .183 CI = 0.101 40 Table 9: Matrices for relative influence of subfactors on Strength of Recovery (Restructuring) For each panel below, which intensity is more likely to obtain if the designated factor drives the recovery? Panel A: Relative likelihood of the strength of recovery if financial sector drives the expansion V S M W Vec. Wts. Very Strong (V) 1 Strong (S) 1 3 Moderate (M) 5 Weak (W) 1 1 3 5 1/3 1/3

1 1/3 1/5 1/5 1/3 1 Panel B: Relative likelihood of the strength of recovery if defense posture drives the expansion V S M W Vec. Wts. .095 .095 .249 .560 Very Strong (V) 1 Strong (S) 3 5 Moderate (M) 7 Weak (W) CI = 0.016 1/3 1 3 5 1/5 1/3 1 3 1/7 1/5 1/3 1 .055 .118 .262 .565 CI = 0.044 Panel C: Relative likelihood of the strength of recovery if global competition drives the expansion V S M W Vec. Wts. Very Strong (V) 1 Strong (S) 1 3 Moderate (M) 5 Weak (W) 1 1 3 5 1/3 1/3 1 1 1/5 1/5 1 1 .101 .101 .348

.449 CI = 0.012 Table 10: Overall Results for Strength of Recovery % GNP Growth Very Strong (5.56.5) Strong (4.5-.5) Moderate (3-4.5) Weak (2-3) 0.108 0.141 0.290 0.461 % GNP Recovery Rate* 3.6 *% GNP Recovery rate calculated using the relative strength of conventional adjustment and restructuring in Table 5 Panel A each used to multiply midpoints of % GNP Growth and then summed. 41 Hamburger Model Estimating Market Share of Wendys, Burger King and McDonalds with respect to the single economic control criterion 42 How to Pose the Question to Make Paired Comparisons One must answer questions of the following kind: given McDonalds (in the Alternatives cluster) is its economic strength derived more from Creativity or from Frequency (both in the Advertising cluster)? Conversely, given Creativity in the Advertising cluster who is more dominant, McDonalds or Burger King? Then, again, by comparing the dominance impact of the clusters of Advertising and Quality of Food on the economic success of McDonald by weighting and normalizing we can relate the relative effect of elements in these different clusters. 43 Hamburger Model Supermatrix Other O t h e r Q Ad C o m p Quality Advertising Competition Local: Menu Cleanli ness Speed Service Location Price

Reputa tion Take Out Portion Taste Nutri tion Freq uency Promo tion Creativ ity Wendys Burger King McDonalds Menu Item Cleanliness Speed Service Location Price Reputation Take-Out Portion Taste Nutrition Frequency Promotion Creativity Wendy's Burger King McDonalds 0.0000 0.6370 0.1940 0.0000 0.0530 0.1170 0.0000 0.0000 0.2290 0.6960 0.0750 0.7500 0.1710 0.0780 0.3110 0.1960 0.4930 0.0000 0.0000 0.7500 0.0780 0.1710 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5000 0.2500 0.2500 0.0000 0.0000 0.0000

0.1880 0.0000 0.0000 0.0810 0.7310 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0990 0.3640 0.5370 0.0000 0.5190 0.2850 0.0000 0.0980 0.0000 0.0980 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5280 0.1400 0.3330 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0950 0.2500 0.6550 0.0000 0.0000 0.0000 0.0000 0.5000 0.0000 0.0000 0.5000 0.8330 0.0000 0.1670 0.1670 0.8330 0.0000 0.0950 0.2500 0.6550 0.1930 0.2390 0.0830 0.0450 0.2640 0.0620 0.0570 0.0570 0.2800 0.6270 0.0940 0.5500 0.3680 0.0820 0.1010 0.2260 0.6740

0.0000 0.0000 0.2900 0.0550 0.6550 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1960 0.3110 0.4940 0.0000 0.0000 0.0000 0.0000 0.0000 0.8570 0.0000 0.1430 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2760 0.1280 0.5950 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6050 0.1050 0.2910 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5940 0.1570 0.2490 0.3110 0.0000 0.0000 0.0000 0.1960 0.0000 0.4930 0.0000 0.0000 0.0000 0.0000 0.0000 0.5000

0.5000 0.0880 0.1950 0.7170 0.1670 0.0000 0.0000 0.0000 0.0000 0.8330 0.0000 0.0000 0.0000 0.0000 0.0000 0.6670 0.0000 0.3330 0.0880 0.1950 0.7170 0.1350 0.0000 0.0000 0.0000 0.7100 0.0000 0.1550 0.0000 0.0000 0.0000 0.0000 0.8750 0.1250 0.0000 0.1170 0.2680 0.6140 0.1570 0.2760 0.0640 0.0650 0.1420 0.0300 0.2070 0.0590 0.0940 0.2800 0.6270 0.6490 0.0720 0.2790 0.0000 0.2500 0.7500 0.0510 0.1100 0.1400 0.1430 0.2240 0.2390 0.0420 0.0510 0.6490 0.0720 0.2790 0.7090 0.1130 0.1790 0.1670 0.0000 0.8330 0.1590 0.3330 0.0480 0.0240 0.1070 0.0330 0.2230 0.0740 0.5280

0.1400 0.3320 0.6610 0.1310 0.2080 0.2000 0.8000 0.0000 Cluster Priorities Matrix Cluster: Other Quality Advertising Competition Other 0.198 0.066 0.607 0.129 Quality 0.500 0.000 0.000 0.500 Advertising 0.131 0.000 0.622 0.247 Competition 0.187 0.066 0.533 0.215 44 Weighted Supermatrix Weighted: Menu Cleanli ness Speed Service Location Price Reputa tion Take Out Portion Taste Nutri tion Freq uency Promo tion Creativ ity Wendys Burger King McDonalds

Menu Item Cleanliness Speed Service Location Price Reputation Take-Out Portion Taste Nutrition Frequency Promotion Creativity Wendy's Burger King McDonald s 0.0000 0.1262 0.0384 0.0000 0.0105 0.0232 0.0000 0.0000 0.0151 0.0460 0.0050 0.4554 0.1038 0.0474 0.0401 0.0253 0.0636 0.0000 0.0000 0.4544 0.0473 0.1036 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1974 0.0987 0.0987 0.0000 0.0000 0.0000 0.1138 0.0000 0.0000 0.0490 0.4426 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0391 0.1436 0.2118 0.0000 0.3141 0.1725 0.0000 0.0593 0.0000 0.0593 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.2082 0.0552 0.1313 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0950 0.2500 0.6550 0.0000 0.0000 0.0000 0.0000 0.0990 0.0000 0.0000 0.0990 0.0550 0.0000 0.0110 0.1014 0.5056 0.0000 0.0123 0.0323 0.0845 0.0382 0.0473 0.0164 0.0089 0.0523 0.0123 0.0113 0.0113 0.0185 0.0414 0.0062 0.3338 0.2233 0.0498 0.0130 0.0291 0.0869 0.0000 0.0000 0.1755 0.0333 0.3964 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0773 0.1226 0.1948 0.0000 0.0000 0.0000 0.0000 0.0000 0.4287 0.0000 0.0715 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.1381 0.0640 0.2976 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6044 0.1049 0.2907 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5940 0.1570 0.2490 0.0407 0.0000 0.0000 0.0000 0.0257 0.0000 0.0646 0.0000 0.0000 0.0000 0.0000 0.0000 0.3110 0.3110 0.0217 0.0482 0.1771 0.0219 0.0000 0.0000 0.0000 0.0000 0.1091 0.0000 0.0000 0.0000 0.0000 0.0000 0.4149 0.0000 0.2071 0.0217 0.0482 0.1771 0.0177 0.0000 0.0000 0.0000 0.0930

0.0000 0.0203 0.0000 0.0000 0.0000 0.0000 0.5444 0.0778 0.0000 0.0289 0.0662 0.1517 0.0293 0.0516 0.0120 0.0121 0.0265 0.0056 0.0387 0.0110 0.0062 0.0185 0.0413 0.3455 0.0383 0.1485 0.0000 0.0537 0.1611 0.0095 0.0205 0.0261 0.0267 0.0418 0.0446 0.0078 0.0095 0.0428 0.0047 0.0184 0.3773 0.0601 0.0953 0.0359 0.0000 0.1788 0.0297 0.0622 0.0090 0.0045 0.0200 0.0062 0.0417 0.0138 0.0348 0.0092 0.0219 0.3519 0.0697 0.1107 0.0429 0.1718 0.0000 Synthesized: Global Menu Cleanli ness Speed Service Location Price Reputa tion

Take Out Portion Taste Nutri tion Freq uency Promo tion Creativ ity Wendys Burger King McDonalds Menu Item Cleanliness Speed Service Location Price Reputation Take-Out Portion Taste Nutrition Frequency Promotion Creativity Wendy's Burger King McDonalds 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244

0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203

0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784

0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518 0.1279 0.1388 0.0435 0.0784 0.1579 0.0234 0.0203 0.0185 0.0072 0.0397 0.0244 0.0296 0.0152 0.0114 0.0049 0.0073 0.2518

0.1279 0.1388 0.0435 0.0784 0.1579 Limiting Supermatrix { Relative local weights: Wendys= 0.156, Burger King =0.281, and McDonalds=0.566 45 Validation The same problem worked as a simple and a complex hierarchy and as a feedback network. 46 Hamburger Model Synthesized Local: Other Quality Synthesized Local Contd: Menu Item Cleanliness Speed Service Location Price Reputation Take-Out Portion Taste Nutrition Simple Hierarchy (Three Level) Wendys Burger King McDonalds 0.3055 0.2305 0.4640 0.132 0.115 0.104 0.040 0.224 0.138 0.167 0.086 0.494 0.214 0.316 Advertising Competition Complex Hierarchy (Several Levels) 0.1884 0.2689 0.5427 Frequency Promotion Creativity Wendys Burger King McDonalds Feedback Network 0.156 0.281 0.566 0.485 0.246

0.267 0.156 0.281 0.566 Actual Market Share 0.1320 0.2857 0.5823 47 48 Strategic Planning for the Future of the University of Pittsburgh Medical Center Using the Analytic Network Process(ANP) 49 Evaluate Strategies for the University Health Network to Compete in a Managed Care Environment Benefits Control Model Costs Control Model B e n e f it s S o c ia l Social Benefits Network Submodel Risks Control Model C o s ts E c o n o m ic P o lit ic a l S o c ia l Economic Benefits Network Submodel Political Benefits Network Submodel Social Costs Network Submodel E c o n o m ic Economic Costs Network Submodel R is k s P o lit ic a l Political Costs Network Submodel S o c ia l E c o n o m ic Social Risks Network Submodel

Economic Risks Network Submodel P o lit ic a l Political Risks Network Submodel 50 List of Clusters and Elements (Not all the Clusters appear in all 9 of the sub-models.) Cluster Names Clients Cluster Elements Businesses- businesses that offer employees health care plans Consumers- individuals who purchase their own health coverage Insurers- companies who sell health insurance Competition Competitors- other hospitals in Pittsburgh that compete w/ UPMC Convenience Time- expended by customer scheduling, traveling, and actual waiting room Safety- safety of location Internal Stakeholders Physicians- working for UPMC Administrators- planners, managers, decision makers of UPMC Alliances- outside organizations, involved: isurers, hospitals, physician networks Staff- non-physician, non-administrative personel Public Relations Public Relations- UPMCs public image: TV, Newspaper, Radio Quality Specialty quality non-general health services, Diversity- range of health services offered by UPMC Care- quality of general health services Research- quality of research at UPMC *Strategies Improve and Measure Outcomes- measure effectiveness to improve service Capitalization- negotiated insurance contracts with fixed payments Develop a Primary Network- increase number of primary care physicians Internal Cost Reduction- cut facilities, employees, and high cost procedures Teach Primary Care- shift focus from curative care to preventive care Variety of Services Internal Medicine and Surgery- Curative specialty services and hospitalization Cancer Treatment- cancer treatment cure Outpatient Care- preventive care and short term medical treatments 51 *Strategies apperar in every sub-moadel as alternatives of choice Clusters and Elements in the Economic Benefits Sub-model 52 Benefits & Costs Predicting the Superbowl Winner 96 & 97 53 Pre-start (early December 1995) 1996 Playoffs Wild Card Games Team Benefits Costs B/C Miami vs. Buffalo 0.701 0.745 0.612 0.590 1.145

1.263 Indianapolis vs. San Diego 0.687 0.660 0.622 0.650 1.105 1.015 Detroit vs. Philadelphia 0.625 0.695 0.636 0.580 0.983 1.198 Atlanta vs. Green Bay 0.590 0.785 0.612 0.515 0.964 1.524 All predictions correct except for two games below. Second Round Pittsburgh vs. Buffalo 0.740 0.704 0.581 0.605 1.274 1.164 Indianapolis vs. Kansas City 0.695 0.750 0.590 0.575 1.178 1.304 Kansas kicker missed 3 field goals & ruined them. No way to know his ailments that day. Green Bay vs. San Francisco 0.755 0.751 0.590 0.585 1.280 1.284 Was too close to determine the winner. Green Bay won. Philadelphia vs. Dallas

0.732 0.759 0.641 0.576 1.142 1.318 Divisional Playoffs Dallas vs. Green Bay 0.742 0.756 0.540 0.561 1.370 1.350 Pittsburgh vs. Indianapolis 0.699 0.741 0.555 0.598 1.260 1.240 The Super Bowl Dallas vs. Pittsburgh 0.761 0.748 0.728 0.735 1.045 1.018 54 Pre-start (early December 1996) The first predictions were wrong on three games which then required revision. PlayoffPredictions Pre-Start Predicted Outcomes AFC 1997 Playoffs Team Benefits Costs B/C Winner Las Vegas Wild Cards Indianapolis Pittsburgh 0.588 0.592 0.489 0.477 1.2 1.24

Pittsburgh Jacksonville Buffalo 0.601 0.594 0.501 0.487 1.2 1.22 Buffalo Wrong prediction. Pittsburgh N ew England 0.609 0.516 0.479 0.419 1.27 1.23 Pittsburgh Wrong prediction. Buffalo Denver 0.551 0.62 0.488 0.447 1.13 1.39 Denver Wrong prediction. Pittsburgh Denver 0.633 0.686 0.523 0.5318 1.21 1.29 Denver Philadelphia San Francisco 0.557 0.621 0.467 0.444 1.19 1.4 San Francisco Minnesota Dallas 0.545 0.571 0.488

0.476 1.12 1.2 Dallas San Francisco Green Bay 0.585 0.685 0.5 0.46 1.17 1.49 Green Bay Dallas Carolina 0.522 0.51 0.494 0.448 1.06 1.14 Carolina Carolina Green Bay 0.511 0.643 0.498 0.521 1.03 1.23 Green Bay 0.618 0.556 0.457 0.476 1.35 1.17 Green Bay Conference Finals N FC Wild Cards Conference Finals Super Bowl Green Bay Denver 55 Post-start (before Conference Finals) PlayoffPredictions A gains error in one game. Predicted Outcomes AFC Team Benefits Costs

B/C Winner Actual ConferenceFinals Jacksonville Denver 0.545 0.612 0.488 0.447 1.12 Denver 1.37 Jacksonville New England 0.576 0.645 0.515 0.519 1.12 N ew Eng 1.24 NE 0.627 0.653 0.554 0.506 1.13 1.29 Green Bay Jax Super Bowl New England Green Bay 56 The Benefits The Costs 57 Benefits Supermatrix Local Weights Offense QB Ability Running Emotions Play Above Ability Coaching Emotional State Outside Home Field Road Ahead Teams Dallas Green Bay Offensive Global 0.0297 0.3140 Local 0.0864 0.9136 0.0037 0.0235 0.0923 0.0433 0.3670

0.1227 0.0039 0.0309 0.1962 0.7724 0.1055 0.8945 0.9693 0.0308 QB Ability Emotions Running Play Above 1.0000 Outside Coaching 0.8000 0.2000 Emotions 1.0000 Home Field .8000 .2000 Road Ahead Teams Dallas 1.0000 Green Bay 0.8000 0.2000 1.0000 0.7500 0.2500 1.0000 1.0000 1.0000 1.0000 0.7500 0.2500 1.0000 1.0000 1.0000 0.2000 0.8000 1.0000 CLUSTER WEIGHTS Offense Offense 0.0000 Emotions 0.2176 Outside 0.0914 Team 0.6910 Emotions 0.2449 0.0000 0.0902 0.6648 Outside 0.6442 0.0852 0.0000 0.2706

Teams 0.7172 0.1947 0.0881 0.0000 58 Weighted Supermatrix Cluster Weights Offense QB Ability Running Emotions Play Above Ability Coaching Emotional State Outside Home Field Road Ahead Teams Dallas Green Bay Offensive Global 0.0297 0.3140 Local 0.0864 0.9136 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 0.0309 0.1962 0.7724 0.1055 0.8945 0.9693 0.0308 QB Ability Emotions Running Play Above 0.7308 Coaching 0.1959 0.0490 Outside Emotions 0.7308 Home Field 0.5154 0.1288 Teams Road Ahead Dallas Green Bay 0.7125 0.1781 0.6442 0.0852 0.0639 0.0213 1.0000

1.0000 0.2692 0.0902 0.4986 0.1662 0.6885 0.3115 0.0538 0.2153 0.2706 0.0219 0.0875 0.2706 Limiting Benefits Supermatrix Offensive Offense Emotions Outside Teams QB Ability Running Play Above Ability Coaching Emotional State Home Field Road Ahead Dallas Green Bay Emotions Global 0.0297 0.3140 Local 0.0297 0.3140 QB Ability 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 Running 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433

0.3670 0.1227 0.0039 Play Above 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 Outside Teams Coaching 0.0297 0.3140 Emotions 0.0297 0.3140 Home Field 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 Road Ahead 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 Dallas 0.0297 0.3140 Green Bay 0.0297 0.3140 0.0037 0.0235 0.0923 0.0433 0.3670 0.1227 0.0039 59 0.0037 0.0235

0.0923 0.0433 0.3670 0.1227 0.0039 Costs Supermatrix Local Weights Offense History Outside Teams Road Ahead Immature Players Not Full Strength Cinderella Play Bey Ability Past Failures Mental State Weather Dallas Green Bay Offensive Global 0.1529 0.0000 0.2261 0.0011 0.2002 0.0738 0.0121 0.1683 0.1653 0.0000 Local 0.4034 0.0000 0.5966 0.0041 0.7278 0.2683 0.0673 0.9332 1.0002 0.0000 Emotions Immature Not Full Road Ahead Players Strength Outside Cinderella 1.0000 Play beyond Past Mental Ability Failures State 0.8000 0.8000 0.2000 1.0000 0.2000 1.0000 0.8333 0.1667 0.8571 0.1429 1.0000 Teams Weather 1.0000 0.7500 0.2500

1.0000 1.0000 1.0000 Dallas Green Bay 0.7500 0.2000 0.2500 0.8000 0.7500 0.2500 1.0000 0.8333 1.000 0.1667 1.0000 CLUSTER WEIGHTS Offense Offense 0.0000 Emotions 0.0877 Outside 0.1392 Team 0.7731 Emotions 0.3614 0.0000 0.0650 0.5736 Outside 0.6267 0.0936 0.0000 0.2797 Teams 0.7172 0.1947 0.0881 0.0000 60 Weighted Supermatrix Cluster Weighted Offense History Outside Teams Road Ahead Immature Players Not Full Strength Cinderella Play Bey Ability Past Failures Mental State Weather Dallas Green Bay Offensive Global 0.1529 0.0000 0.2261 0.0011 0.2002 0.0738 0.0121 0.1683 0.1653 0.0000 Local 0.4034 0.0000 0.5966 0.0041

0.7278 0.2683 0.0673 0.9332 1.0002 0.0000 Emotions Immature Not Full Road Ahead Players Strength Outside Cinderella 1.0000 Play beyond Past Mental Ability Failures State 0.2891 0.5014 0.0723 1.0000 0.1253 0.0936 0.8333 0.1667 1.0000 Teams Weather 0.6267 0.0702 0.0234 0.8571 0.1429 0.2797 Dallas Green Bay 0.5379 0.1434 0.1793 0.5738 0.1947 0.0734 0.0147 0.1460 0.0487 0.0881 0.2797 Limiting Costs Supermatrix Offensive Offense History Outside Teams Road Ahead Immature Players Not Full Strength Cinderella Play Bey Ability Past Failures Mental State Weather Dallas Green Bay Emotions Outside Teams Global Local 0.1529 0.1529

Immature Not Full Road Ahead Players Strength 0.1529 0.1529 0.1529 Cinderella 0.1529 Play beyond Ability 0.1529 Weather 0.1529 Dallas 0.1529 Green Bay 0.1529 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738

0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 Past Mental Failures State 0.1529 0.1529 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 0.2261 0.0011 0.2002 0.0738 0.0212 0.1683 0.1653 0.0000 61 Benefits Intensity Priorities Quarterback (0.030): Average (0.091) Running Game (0.314): Average (0.084) Good (0.281) Good (0.211)GB Play Above Potential (0.004): Average (0.075) Good (0.229)D Coaching Ability to Inspire (0.023): Not A lot (0.078) Emotional State (0.092): Apathy (0.082) Home Field Advantage (0.043): Neutral (0.105) The Road Ahead (0.367):

No Effect (0.082) Somewhat (0.205)D High Ability (0.691)GB,D High Ability (0.705)D High Play Level (0.696)GB Heroic (0.717)GB Mediocre (0.236) Excitement (0.682)GB,D Some Effect (0.258)GB Significant Effect (0.637)D Some Effect (0.236)D Very Confident (0.682)GB Dallas Effect on the Ultimate Outcome (0.123): Medium (0.280) Low Effect (0.094) Greatly Influenced (0.627)GB,D Green Bays Effect on the Ultimate Outcome (0.004): Medium (0.258) Not Much (0.105) Greatly Influenced (0.637)GB,D Costs Intensity Priorities The Road Ahead (0.153): Low Effect (0.085) Somewhat (0.271)GB,D High Effect (0.644) Not at Full Strength (0.226): Few Injuries (0.091) Some Injuries (0.218) Big Injury Problems (0.644) GB,D Playing Beyond Ability (0.200): Not a factor (0.094) GB,D May Falter (0.288) Venerable (0.627) Past Failures (0.074): Good History (0.082) GB Cant get it Gone (0.682) Mental State of Preparedness (0.012): Ready (0.122)GB Mixed Past (0.236)D May be Hurt (0.230)D Unready (0.648) Cinderalla Team (0.001): Good Team Not Cinderalla (0.082) GB,D Lucky (0.236) Its Midnight (0.682) Weather Sensitivity (0.168): Small

module to rate the players. Past Performance from the network model in the first stage was the first criterion added. Another two criteria: Technique and Conditioning were also included. Prediction: 7 of the top 8 players were correctly predicted to meet in the final rounds of the tournament with the final between Sampras and Chang. In reality, the final was a match between Moya and Sampras, with the top seed winning the outcome. As Moya was ranked 58th in the world prior to the start of the tournament, he was not even included in the model. 65 Sampras Chang Becker Agassi Ivanisevic Krajicek Muster Courier Kafelnikov Martin Washington Enqvist Ferreira Rios Costa mantilla 0.658 0.638 0.516 0.512 0.483 0.460 0.468 0.462 0.438 0.422 0.421 0.398 0.374 0.366 0.334 0.329 66 Hong Kong Competes with Singapore and somewhat less with Tokyo as Financial Center in Asia in the 21st Century Gang Hu (Tianjin), Chia-Shuan Huang (Taiwan), Hong Li (Beijing), Thomas Saaty (Pittsburgh), Torsten Schmidt (Germany), and Yu-Chan Wang (Taiwan). 67 The Purpose The purpose of this project is to study the potential impact imposed through the takeover of Hong Kong by China in 1997. The analysis focuses on the following questions: What set of criteria does an Asian location have to meet in order to be a Financial Center? Which city is the best candidate for the Financial Center in the Asia-Pacific region in 1996? What is the most likely policy of the Chinese government towards Hong Kong after 1997? What impact does the Chinese policy have towards Hong Kong as a Financial Center? Which city is the most likely candidate to be the Financial Center is the Asia-Pacific region in the year 2000? 68 The Approach These five questions are studied with the methodology and technique provided by a combination of the AHP and ANP. A dual-model approach was developed. The first model, the Financial Center model which is an ANP model, was used to examine the first two of the above questions. The second model, the Mainland China Policy model is an AHP model, used to focus on the third question above to generate a policy package most likely to be adopted by the Chinese government. Based on changed in the political, economic, and social environments incurred by the estimated policy package, the Financial Center model was re-evaluated. The fourth and fifth questions above are thus answered. The two models complement one another because: 1. The Financial Center model provides the relevant factors for a focused examination under the China model in order to find the relevant factors which may be changed by the Chinese government, and 2. The China model provides a package of feasible (for mainland China) and likely policies to be adopted by the Chinese government after 1997. Based on the package of policies, a second evaluation of the Financial Center model was made in order to estimate the future status of Hong Kong as a Financial Center.

69 Influencing Factors A).Economic-Benefits: 1. Geographic advantage. 2. Free flow of information. 3. Free flow of people. 4. Free flow of capital. 5. Internationalization. 6. Investment. 7. Educated workforce. 8. Convertible currencies. 9. Assistance from government. 11. Modern infrastructure. 12. Deregulated market. B).Political-Benefits: 1. Efficient government. 2. Independent legal system. 3. Assistance from government. 4. Free flow of people. 5. Free flow of information. C).Social-Benefits: 1. Free flow of people. 2. Free flow of information. 3. Educated workforce. 4. Open culture. 5. Internationalized language. 6. Availability of business professionals. D).Economic-Costs: 1. Labor cost. 2. Corruption. 3. Protection from government. 4. Operating cost. 5. Tax. E).Political-Costs: 1. Tax. 2. Corruption. 3. Protection from government. F).Social-Costs: 1. Environment. 2. Corruption. 3. Protection from government. G).Economic-Opportunities: 1. Investment. 2. Access to potential market. 3. Regional economic growth, membership of international organizations (GATT.WTO). H).Political- Opportunities: 1. Political credit. 2. Investment. 3. Membership of international organization (GATT,WTO). I).Social-Opportunities: 1. Social wealth. 2. Access to potential market. J).Political-Risks: 1. Political instability. 2. Instability of local government. 3. Political restriction. K).Economic-Risks: 1. Instability of local financial market. 2. Inflation. 3. Competition from local business. L).Social-Risks: 1. Industry resistance. 2. Public industry. 3. Instability of local society. 70 The Set of Four Control Hierarchies Benefits Control Model Opportunities Control Model * Risks Control Model Costs Control Model * 71

72 Costs Benefits 73 Risks Opportunities Economic Benefits Sub-Model 74 Results There are twelve supermatrices associated with the complete model. With each of these supermatrices is associated a cluster priority matrix, a weighted supermatrix, and a limiting supermatrix from which the priorities of the three contending centers are derived. These twelve sets of priorities are weighted by the priorities of the corresponding control criteria and summed to obtain the final ranking. 75 The output from the first Financial Center model We assume that all the situation will remain the same after 1997. In other words, the main land China government will adopt a set of feasible policies toward Hong Kong. Based on this assumption, we made the judgments. After synthesis, we got the results below: Opportunities Benefits Economic Political Social Hong Kong 0.4131 0.5034 0.4416 Singapore 0.2836 0.1503 0.2164 Tokyo 0.3033 0.3465 0.342 Political 0.3813 0.2935 0.3252 Social 0.4511 0.2227 0.3261 Economic Political 0.3086 0.4278 0.2186 0.4387 0.4728 0.1335 Social 0.4387

0.2365 0.3248 Risks Costs Hong Kong Singapore Tokyo Hong Kong Singapore Tokyo Economic 0.4096 0.2874 0.303 Economic Political 0.2393 0.1922 0.2804 0.3625 0.4803 0.4453 Social 0.2519 0.1803 0.5678 Hong Kong Singapore Tokyo The overall result is listed below: Alternatives Hong Kong Singapore Tokyo Rank (B*O)/(C*R) 1 1.6498 2 0.906 3 0.5338 It is clear that Hong Kong has the highest priority, which means if mainland China government adopt all the policies described above toward Hong Kong, it will remain to be the financial center in the Asia-Pacific region. 76 Likely Policies Followed by China Affecting the Future of Hong Kong About 50 potential Chinese policies were identified and ranked in a hierarchy. The most likely policies were identified and the network sub-models were re-assessed given this information. The hierarchy and the policies are shown next. 77 Sample Hierarchy for Assessing Benefit Intensities 78 Policy Rating 79 The output from the Chinese policies model We picked 18 different factors (in the Financial Center model) which are highly dependent on the policies of the mainland China government. For each of the factors, we divided it into three situations(positive +, mutual 0, negative -), which denote the different Chinese policies toward it. And then, we put them into the China government model(absolute hierarchy model, including four sub-model: benefits, costs, opportunities and risks). After synthesis, we got the overall score for each policy. Based on the scores, we draw the optimal and most likely policies package(it is shown below). 1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Optimal and most likely policies free flow of information 0 free flow of people 0 educated workforce + convertible currency + deregulated market 0 assistance from government + inflation + independent legal system 0 political restrcitions 0 instability of local society + availability of business professionals + public insecurity + corruption + tax + protectionist barrier 0 investment + political credit + access to potential market + 80 The output from the second Financial Center model Based on the optimal and most likely policies package we got, we made another set of judgments for the Financial Center model. This is, with the assumptions we have made, an estimation of the location of the financial center in the Asian-Pacific region. The results are listed below: Benefits Hong Kong Singapore Tokyo Opportunities Economic Political 0.3814 0.4435 0.2992 0.2187 0.3194 0.3378 Social 0.4238 0.2306 0.3456 Costs Hong Kong Singapore Tokyo Hong Kong Singapore Tokyo Economic Political 0.4096 0.3813 0.2874 0.2935 0.303 0.3252 Social 0.4511 0.2227

0.3261 Economic Political 0.4278 0.4278 0.4387 0.4387 0.1335 0.1335 Social 0.4387 0.2365 0.3248 Risks Economic Political 0.2966 0.2814 0.2561 0.3161 0.4475 0.4025 The overall result is as below Social 0.3045 0.1657 0.5298 Alternatives Hong Kong Singapore Tokyo Hong Kong Singapore Tokyo Rank (B*O)/(C*R) 1 1.1822 2 1.1093 3 0.5949 We can see that Hong Kong can still maintain the financial center status after 1997, but the gap between Hong Kong and other cities is much smaller. Especially, Singapore becomes very competitive. 81 Original Economic Benefits Sub-Model Supermatrix (Truncated to save space) Economic Benefits Local: Sing Toky Hong assi free conv mode good dere Singapore 0

0 0 0.3196 0.1692 0.1396 0.1692 0.3333 0.2081 Tokyo 0 0 0 0.122 0.4434 0.5278 0.4434 0.3333 0.1311 Hong Kong 0 0 0 0.5584 0.3874 0.3325 0.3874 0.3333 0.6608 0.0538 0.0459 0.0501 0 0 0 0 0 0.044 0.0526 0.0369 0 0 0 0

0 convertible currency 0.0379 0.0796 0.0625 0 0 0 0 0 modern infrastructure 0.0843 0.189 0.0925 0 0 0 0 0 good auditing systems 0.105 0.0801 0.0619 0 0 0 0 0 deregulated market 0.0367 0.0296 0.1166 0 0 0 0 0 geographic advantages 0.1713 0.1599 0.1593 0 0

0 0 0 free flow of information 0.0168 0.0837 0.0594 0 0 0 0 0 free flow of capital 0.1389 0.0917 0.0928 0 0 0 0 0 educated workforce 0.0602 0.0996 0.0537 0 0 0 0 0 internationalized language 0.0961 0.0283 0.0555 0 0 0 0 0 investment from outside 0.1551 0.06 0.1588

0 0 0 0 0 assistance from government free flow of people 82 Weighted Economic Benefits Sub-Model Supermatrix (Truncated to save space) Economic Benefits Weighted: Sing Toky Hong assi free conv mode good dere Singapore 0 0 0 0.3196 0.1692 0.1396 0.1692 0.3333 0.2081 Tokyo 0 0 0 0.122 0.4434 0.5279 0.4434 0.3333 0.1311 Hong Kong 0 0

0 0.5584 0.3874 0.3325 0.3874 0.3333 0.6608 0.0538 0.0459 0.0501 0 0 0 0 0 0.044 0.0526 0.0369 0 0 0 0 0 convertible currency 0.0379 0.0796 0.0625 0 0 0 0 0 modern infrastructure 0.0843 0.189 0.0925 0 0 0 0 0 good auditing systems 0.105

0.0801 0.0619 0 0 0 0 0 deregulated market 0.0367 0.0296 0.1166 0 0 0 0 0 geographic advantages 0.1713 0.1599 0.1593 0 0 0 0 0 free flow of information 0.0168 0.0837 0.0594 0 0 0 0 0 free flow of capital 0.1389 0.0917 0.0928 0 0 0 0 0

educated workforce 0.0602 0.0996 0.0537 0 0 0 0 0 internationalized language 0.0961 0.0283 0.0555 0 0 0 0 0 investment from outside 0.1551 0.06 0.1588 0 0 0 0 0 assistance from government free flow of people 83 Limiting Economic Benefits Sub-Model Supermatrix (Truncated to save space) Economic Benefits Synthesized: Global Sing Toky Hong assi free conv mode good dere Singapore 0.2836

0.2836 0.2836 0.2836 0.2836 0.2836 0.2836 0.2836 0.2836 Tokyo 0.3033 0.3033 0.3033 0.3033 0.3033 0.3033 0.3033 0.3033 0.3033 Hong Kong 0.4131 0.4131 0.4131 0.4131 0.4131 0.4131 0.4131 0.4131 0.4131 assistance from government 0.0499 0.0499 0.0499 0.0499 0.0499 0.0499 0.0499 0.0499 0.0499 free flow of people 0.0437 0.0437 0.0437 0.0437

0.0437 0.0437 0.0437 0.0437 0.0437 convertible currency 0.0607 0.0607 0.0607 0.0607 0.0607 0.0607 0.0607 0.0607 0.0607 modern infrastructure 0.1194 0.1194 0.1194 0.1194 0.1194 0.1194 0.1194 0.1194 0.1194 good auditing systems 0.0796 0.0796 0.0796 0.0796 0.0796 0.0796 0.0796 0.0796 0.0796 deregulated market 0.0676 0.0676 0.0676 0.0676 0.0676 0.0676 0.0676

0.0676 0.0676 geographic advantages 0.1629 0.1629 0.1629 0.1629 0.1629 0.1629 0.1629 0.1629 0.1629 free flow of information 0.0547 0.0547 0.0547 0.0547 0.0547 0.0547 0.0547 0.0547 0.0547 free flow of capital 0.1055 0.1055 0.1055 0.1055 0.1055 0.1055 0.1055 0.1055 0.1055 educated workforce 0.0695 0.0695 0.0695 0.0695 0.0695 0.0695 0.0695 0.0695 0.0695 internationalized language

0.0588 0.0588 0.0588 0.0588 0.0588 0.0588 0.0588 0.0588 0.0588 investment from outside 0.1278 0.1278 0.1278 0.1278 0.1278 0.1278 0.1278 0.1278 0.1278 84 Normalized by Cluster - Results from Limiting Economic Benefits Sub-Model Supermatrix Economic Benefits Synthesized Local: Singapore 0.2836 Tokyo 0.3033 Hong Kong 0.4131 assistance from government 0.0499 free flow of people 0.0437 convertible currency 0.0607 modern infrastructure 0.1194 good auditing systems 0.0796 deregulated market 0.0675 geographic advantages 0.1629 free flow of information

0.0547 free flow of capital 0.1055 educated workforce 0.0695 internationalized language 0.0588 investment from outside 0.1278 85 The Result The first result from the Financial Center model: If the Chinese government is able to maintain the current status of Hong Kong, Hong Kong would still be the Financial Center is the Asia-Pacific region in 2000. The first result from the Mainland China Policy model: For interests of the mainland Chinese government, no negative policy should be adopted towards Hong Kong after 1997. A careful and sensitive approach towards the future Hong Kong policy is suggested by this result, which is also reinforced by the next result. The second result from the Financial Center model: Although Hong Kong may still be the best choice for a Financial Center, Singapore will become a strong competitor for the Center in 2000. 86 Conclusions 1) Based on the first output of our Financial Center model, we can see that if all conditions remain the same, in other words, if China adopts all the positive policies toward Hong Kong, in other words if the Chinese government is able to maintain or even improve the current status of Hong Kong, it is quite sure that Hong Kong will remain one of the important financial centers in the Asia-Pacific region. 2) Among the influencing factors of the financial center status, many of them are dependent directly on the governments policies. Therefore, Hong Kongs future as a financial center is highly dependent on the political attitude of the Chinese government. 3) Based on the result of our mainland China policy model, we found, among the 18 factors, the Chinese government should adopt positive policies on 12 of them, and mixed policies on 6 of them. In other words, for the interests of China itself (not Hong Kong), China should avoid implementing negative policies, as defined in this study towards Hong Kong as a financial center. 4) Based on the second output of the Financial Center model, Hong Kong will maintain its financial status after 1997. But at the same time, Singapore will become very competitive. Therefore, our conclusion is that if Chinese government adopts rational policies toward Hong Kong as estimated in this study, Hong Kong will remain the number one financial center of the Asian-Pacific region. But at the same time, the position of Hong Kong as a financial center will be weakened. If any negative policies are implemented, Singapore will become the number one financial center of this Asia-Pacific region followed by Hong Kong. 87 Feedback Measurement as the Limiting Power of the Supermatrix The eigenvectors of the paired comparison matrices are each part of a column of the supermatrix. The supermatrix may not be column stochastic. Its column blocks would be weighted by the priorities of the clusters to render the matrix stochastic. The supermatrix must now be raised to powers to capture all the interactions and feedback among its elements. What is desired is its limiting power limW k k The power of a matrix is function of that matrix. Entire functions (series expansion converges for all values) of a matrix can be represented by the formula: II(jI-W) Wk = n i=1 k i ji II(j-i) ji

if the eigenvalues are distinct, or if they are not then by: m W = k i=1 1 d mi-1 (mi-1)! d mi-1 One is the largest eigenvalue of a stochastic matrix. This follows from max max i ki (I-W)-1 n II (- i) i=1 n II (- i) i=mi+1 = i n a j=1 ij and the sum on the right is equal to the one for a column stochastic matrix. It is obvious that the moduli of the remaining eigenvalues of a stochastic matrix are less than or equal to one. One is a simple eigenvalue if the matrix is positive. It can be a multiple eigenvalue or there may be other eigenvalues whose moduli are equal to one if there is a sufficient number of zeros in the matrix so that it is reducible. When the supermatrix has some zero entries, it may be that some power of it is positive and hence the matrix remains positive for still larger powers and is called primitive. One is a simple eigenvalue of a primitive matrix. One may be a simple eigenvalue whether the matrix is primitive or not or a multiple eigenvalue yet there may not be other eigenvalues whose moduli is equal to one. The powers take on a certain form in the limit for each of these three cases. On the other hand if there are other such roots whose module is one, the powers of the supermatrix would cycle with a period of cyclicity and the limit is given by the same expression in all three cases namely when the supermatrix is imprimitive or when one is a simple or a multiple eigenvalue. 88 IrreducibleStochastic(=1isasim pleroot) W= N ootherrootsw ithm odulusequaltoone O therrootsw ithm odulusequal toone (prim itive) (im prim itivew ithcyclicityc). C aseA C aseA Prim itiveiftraceispositive. R aiseWtopow ers. 1 c 1 c ( I

W )( I W ) ( W ) A ll colum nsthesam eandanycolum ncanalsobe c 2 c obtainedasthesolutionoftheeigenvalue problemW w=w . 89 Reducible Stochastic W = No other roots with modulus equal to one Case B =1 Simple ( I W ) 1(1) Adjo int( I W ) normalized (1) (1) Case C =1 multiple n1 n1 ( 1) k k 0 n1! ( n1 k ) ( ) (I W ) k 1 | 1 ( n1 ) (n1 k ) ( ) Other roots with modulus equal to one (cyclic with cyclicity c). Case B 1 ( I W c )(I W ) 1 (W c ) c 2 c Case C 1 ( I W c )(I W ) 1 (W c ) c 2 c The desired outcome for Case C can often be obtained by introducing loops at all sinks and raising the matrix to limiting powers. 90 C o m p u ta tio n a lly , th e fo re g o in g c la s s ific a tio n m a y b e s im p lifie d a lo n g th e fo llo w in g lin e s . D e fin e 1. W e h av e: P rop er | i | < 1 1 = 1 s im p le ro o t = m ax Im p ro p er

| i | 1 (fo r s e v e ra l i) i > 1 A p rim itiv e s to c h a s tic m a trix is p ro p e r (1 ) F u lly R e g u la r R o o ts o f u n ity o f c y c lic ity c . (2 ) 1 (I W c T h e in d e x k = 1 in th e d ia g o n a l p rim itiv e b lo c k m a tric e s o f th e n o rm a l fo rm ( I W ) 1 ( 1 ) Adjo (1 ) c )( I W ) 1 ( W c ) c ) int( I W ) (1 ) N o rm a liz e th e c o lu m n s o f th e a d jo in t to g e t W . W h e n W is p rim itiv e o n e c a n s im p ly ra is e W to v e ry la rg e p o w e rs o n a p e rs o n a l c o m p u te r. (3 ) 1 = 1 m u ltip le ro o t o f m u ltip lic ity n 1 (4 ) I f a n d o n ly if m a tr ic e s A 1 , , A k in u p p e r p a r t o f d ia g o n a l o f n o rm a l fo rm a re p rim itiv e n n 1 1 k 0 ( 1) k n 1! (n1 k ) ( ) ( I W ) ( n 1 k )! ( n 1 ) ( ) k 1 | 1 (I W c c

)( I W ) 1 ( W 1 A s im p le p ra c tic a l ru le fo r o b ta in in g a lim itin g m a trix fo r a g iv e n n b y n n o n n e g a tiv e a n d s to c h a s tic s u p e r m a t r i x W i s f i r s t t o t e s t i t f o r i r r e d u c i b i l i t y w i t h t h e c o n d i t i o n ( I + W ) n -1 > 0 . I f i t i s i r r e d u c i b l e , t h e n m ax = 1 is s im p le a n d o n e o f th e tw o f o rm u la s a p p lie s . I t is th e n te s te d fo r c y c lic ity a n d th e a n s w e r is o b ta in e d u s in g th e a b o v e . W h e n a lte rn a tiv e s d o n o t fe e d b a c k in to th e c rite ria , it is b e s t n o t to in c lu d e th e m in th e s u p e rm a trix . T h e r e a s o n is th a t if th e s u p e rm a trix c y c le s , th e n th e a v e ra g e v a lu e w o u ld f irs t h a v e to b e c a lc u la te d . T h e a v e ra g e w e ig h ts o f th e c rite ria a re th e n u s e d to w e ig h t th e a lte rn a tiv e s in a s e p a ra te h ie ra rc h y . 91 ANP PROJECTS 1996 United States Presidential Election A Day at the Races: Predicting a Harness Race at the Meadows: An Application of the ANP A Prediction of Opportunities for Job Growth by U.S. Region Alternative Fuels for Automobiles An ANP Approach for Commodity Markets Demand/Supply Ratio Model Analysis of the Market for 32-Bit Operating Systems Bridge Management Decision Choosing the Best Location for Permanent Storage of High-Level Nuclear Waste Commodity Markets Convocation Center Corporate Market Value in the Computer Industry Corporate Restructuring at Chrysler Corporate Strategies for Competitors Crime and Punishment Disney America: Should Disney Build a Theme Park? Given \$10 Million, What Would be the Best Allocation to Each of the Proposed Programs that Contribute to Decreasing Gang Activity? Health Insurance Systems How to Implement Flex Time Justify the Existence of the Economic Black Market Lake Levels and Flow Releases Management Consulting Model Market Share Predictions for Aqueous Intra-Nasal Steroids Medical Center: Strategic Planning with the ANP Mergers and Acquisitions Mode of Transportation to School Modeling a Reservoir Operations for Managing of Ecological Interests Multi-Objective Decision Making Analysis with Engineering and Business Applications 92 ANP REFERENCES contd NBA Playoffs for 1991 Net Dollar Value for IBM, Apple, Intel and Microsoft Network Analysis of Illegal Drug Marketing in the United States Planning Strategies for Incubator Space using the ANP Predicting the Outcome of Legislative Debate over Superfund Reauthorization Predicting the Winner of the 1995-1996 NHL Stanley Cup Predicting the Winner of the 1996 Chase Championship with the ANP Prediction of 1997 Australian Tennis Open Prediction of the 1997 Wimbledon Tennis Championships Prediction of the CPU Market Prioritizing Flow Alternatives for Social Objectives Ranking Countries in Telecommunications as a Subset of Locating a Business Problem Stadium Placement and Optimal Funding Strategic Staffing Extra Care Providers Strategies for Improvement at the Joseph M. Katz Graduate School of Business Teenage Pregnancy Telecommunications Network Design and Performance The Decision to Market Nimbex (new drug) vs. Continuing to Market Tracruim (old drug) The Emerging Information Technologies of the Future: The Prize of Firms and Industries The Future of East Central Europe The Future of Major League Baseball in Pittsburgh: Strategic Planning with the ANP The Future of the University of Pittsburghs Medical Center The Middle East The Optimal MBA Program Structure The Teenage Smoking Problem Transportation to Work Understanding the Tiananmen Massacre in China What will be the worth? (Predicting Average Starting Salaries for MBA Graduates) Where to Invest in Capital Markets 93

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