Marginal optimization method to improve the inconsistent comparison matrix in the analytic hierarchy process

To improve the inconsistency in the analytic hierarchy process(AHP), a new method based on marginal optimization theory is proposed. During the improving process, this paper regards the reduction of consistency ratio(CR) as benefit, and the maximum modification compared to the original pairwise comparison matrix(PCM) as cost, then the improvement of consistency is transformed to a benefit/cost analysis problem. According to the maximal marginal effect principle, the elements of PCM are modified by a fixed increment(or decrement) step by step till the consistency ratio becomes acceptable, which can ensure minimum adjustment to the original PCM so that the decision makers’ judgment is preserved as much as possible. The correctness of the proposed method is proved mathematically by theorem. Firstly, the marginal benefit/cost ratio is calculated for each single element of the PCM when it has been modified by a fixed increment(or decrement).Then, modification to the element with the maximum marginal benefit/cost ratio is accepted. Next, the marginal benefit/cost ratio is calculated again upon the revised matrix, and followed by choosing the modification to the element with the maximum marginal benefit/cost ratio. The process of calculating marginal effect and choosing the best modified element is repeated for each revised matrix till acceptable consistency is reached, i.e., CR<0.1. Finally,illustrative examples show the proposed method is more effective and better in preserving the original comparison information than existing methods.

Optimal margin in nephnm-sparing surgery for renal cell carcinoma 4 cm or less in diameter.

Objective To investigate the optimal margin in nephron-sparing surgery (NSS) for renal cell carcinoma (RCC) 4 cm or less in diameter. Methods Eighty-two kidneys with RCC 4 cm or less in diameter resected by radical nephrectomy were prospectively studied. The kidney samples were sectioned at 3 mm interval and examined for multicentricity. On each layer of tissue sectioned, parenchyma margin of 15 mm beyond pseudocapsule was continuously sectioned and examined for completeness of pseudocapsule and extra-pseudocapsule cancer lesion. The farthest distance between extra-pseudocapsule lesion and primary tumor was measured. PCNA expression was detected in 41 patients by using standard SP immunohistochemistry technique. Results The diameter of 82 primary tumors was 3. 4 ± 0. 8 cm (range 1.5-4.0 cm).Of these,31.7% (26/82) were found without intact pseudocapsules and 17.1% (14/82) with positive cancer lesions beyond pseudocapsule. The average distance between extra-pseudocapsule cancer lesion and primary tumor