AHP and ANP as Particular Cases of Markov Chains
Abstract: The models and methods of AHP/ANP analysis are considered. It is shown that AHP/ANP models are the particular cases of finite Markov chains, i.e. discrete random processes with Markov property. Applying models, methods and algorithms of Markov chains theory will stimulate the progress in researches on multicriteria decision making problems and its application in various spheres and particularly in investment planning and management. A new approach to sensitivity analysis in intervals of uncertainty of data is proposed.
Keywords: Analytic Hierarchy Process, Analytic Network Process, Markov Chains, Sensitivity Analysis
1. INTRODUCTION
AHP and ANP (Saaty T.L., 1980, Saaty T.L. and Vargas L. G. 1996, Saaty T. L. and Özdemir, M. S. 2005) are well known as multicriteria decision making methods that can help general decision operation by decomposing a complicated problem into a multilevel hierarchical structure of objective, criteria and alternatives. AHP and ANP are very effective when subjectivity exists and it is very suitable to solve problems where the decision criteria can be organized in logical schemes described by a direct graph, composed by a user. One can see three main parts in these techniques: 1) determination of relative priorities from discrete or continuous paired comparisons, 2) a logical schema (hierarchy or network) – directed graph (set of nodes connected by arcs) – for synthesizing the numerical weights or absolute priorities for each element (node of schema) taking in account all the data and 3) check and control on the consistency of the judgments for all elements of logical schema. A digraph of AHP is considered as a stratified schema which describes the logic of decision of the problem (its model), proposed by researcher. The choice of schema is constrained by definite rules: 1) a hierarchy must contain K+1 levels (sets of elements), k = 0,1,…, K, 2) only one node-top at 0-level (decision goal), 3) the actors, criteria, sub-criteria, … occupy levels k = 1,…, K-1, 4) the alternatives are situated on K-level, 5) each element of the schema, except for the top one, is subordinate to one or more other elements of precedent level and 6) each element of the schema must be connected by a path with at least one node-alternative. The loops and connections between the elements of one level are disallowed. Many decisions problems cannot be structured hierarchically because they involve the interaction and dependence of higher level elements in a hierarchy on lower level elements (Saaty T. L. and Özdemir, M. S. 2005). ANP is not constrained by strict rules of hierarchy. Its schema isn’t stratified, the arcs may connect the elements of one or different levels in unordered way. Here below we propose to consider AHP/ANP model as a particular case of finite-state Markov chains with discrete time (Kemeny J.G. and Snell J.L. 1960), determined by AHP/ANP digraph with the priorities on it’s arcs interpreted as the probabilities of transition between the Markov chain states (nodes of digraph). 2. AHP AS A NON-HOMOGENEOUS MARKOV CHAIN The transition probabilities (as AHP relative priorities of elements of next level) are depending only on the process state at moment t (number of level) and not on the way by which the process has attained this state. So Markov propriety is accomplished by the procedure of constructing the AHP model. The matrices At = ||aij(t)||, iSt , jSt+1 of AHP relative priorities (interpreted here as transition probabilities) are rectangular stochastic ntnt+1-matrices where nt is quantity of elements on t-level, t = 0, 1,…, K-1, K. In initial moment t = 0 the Markov process is in the state i = 0, S0 = {0}, with probability 0 = 1. In the next moment t = 1 the process occupy the states from S1 = {1,2,…,n1} with probabilities 1 = 0A0 = (0a01, 0a02,…, 0a0n1) = {a01, a02,…, a0n1}, etc. If t is nt-vector-row then the probability distribution of states on t+1-level is equal to The vector K determines the final probabilities (absolute priorities) of alternatives. It is easy to see identity of (1) to the main relationship of AHP.
