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In a fairly recent paper Jonathan Barzilai (1998) showed that equivalent hierarchies can produce nonequivalent results. He used an example where two marketing strategies are compared on total annual revenue. The four executives of the company are each using their own model; the models differ in the way the five stores of the company are grouped together according to three territories. These four hierarchies are equivalent in the sense that they all are equivalent models or descriptions of the problem of choosing the best marketing strategy. Under the assumption that it does not matter from which store one extra dollar revenue originates and no other circumstantial reasons exist for making any difference in importance, all weights are made equal within each family of hierarchy elements. According to conventional AHP a weighted sum function is set up to compute the composite alternative priorities after the usual unity sum normalisations per node. Barzilai then shows that the weighted sum functions differ in the global weights of the stores thus producing different results, even rank reversals, despite the consistency of the judgements. He concludes that AHP's incorrect decomposition rule involving multiple normalisations must be the cause of the non-equivalent results and rank reversals that go with them, thereby invalidating AHP. 2. The improved structural weight adjustment procedure Three of the four marketing hierarchies are incomplete hierarchies where the nodes on a level are not connected to all nodes on its adjacent higher or lower level. In such hierarchies a structural imbalance exists if this incompleteness results in node families of unequal size on the same level, or if paths of unequal length exist from the hierarchy top to its bottom level. The former type of incompleteness occurs in Barzilai's example. This is particularly important in view of the unity sum normalisation. There, the average weight 1/n of the child of a parent-node depends on the size n of that node's family, and so does each individual child's weight. The global weights of the bottom level criteria therefore depend on the sizes of their (ancestor-)families. Conceptually equivalent hierarchies, but having different clustering of nodes can thus produce different global weights resulting in non-equivalent final priorities. A structural adjustment procedure adjusting the weights according to the hierarchical structure was proposed by Saaty in 1980 and implemented in the Expert Choice package, but presumably largely ignored thus far. It is good enough for our purpose. It is, however, not good enough for general situations with incomplete hierarchies of more than three levels or unequal path lengths from top to bottom. The adjustment is a very local procedure as it only considers two adjacent hierarchy levels at a time, and makes no distinction between alternatives and criteria. An improved procedure will be presented; it adjusts all weights at all levels above the bottom criterion level (i.e. the criterion leaf level), thus removing, as it were, the influence of the structure on that bottom criteria's global weights. The improved |
