Technique for order preference by similarity to ideal solution (TOPSIS) is a well-known multi attribute decision making (MADM) method and it has been widely used in materials selection. However, the main drawback of the traditional TOPSIS is that it has a rank reversal phenomenon. To overcome this drawback, we propose an improved TOPSIS without rank reversal based on linear max-min normalization with absolute maximum and minimum values by modifying normalization formula and ideal solutions. Moreover, to study the impacts of changing attribute weights on relative closeness values of alternatives, we propose a sensitivity analysis method to attribute weights on the relative closeness values of the alternatives. We applied the proposed method to select best absorbent layer material for thin film solar cells (TFSCs). As a result, copper indium gallium diselinide was selected as the best one and the next cadmium telluride from among five materials. When the alternative is added to or removed from the set of original alternatives, the elements of the normalized decision-matrix, PIS, NIS and the relative closeness values don’t change at all, they are always coincide with the corresponding elements of the original ones. The relative closeness values are absolute values irrelevant to the composition of the alternatives in the improved TOPSIS, while the relative closeness values are relative values relevant to the composition of the alternatives in the traditional TOPSIS. Therefore, the proposed TOPSIS overcomes the rank reversal phenomenon, perfectly. It could be actively applied to practical problems for materials selection.

Analytic hierarchy process (AHP) is a well-known attribute weighting method in multiattribute decision-making. Its major requirement is to satisfy the consistency of pairwise matrix (PM). To solve this problem, we first propose a new consistency improvement method of PM based on consistency ratio (CR) decreasing rate. In this method, we calculate the CR decreasing rates of all the PMs reconstituted by replacing all elements of the PM with the lower and upper neighbouring 9-point scales and find the element with maximum CR decreasing rate, and then modify it to its lower or upper neighbouring scale. Second, we develop third-order approximate polynomial for random consistency index using least square method. It enables to determine the RI value according to the number of attributes without a numerical table. Third, we propose the final PM determining method and final attribute weighting method considered decision makers’ levels based on the CR values of the individual PMs in case several decision makers perform their own pairwise comparisons. We test the performances of the proposed and some previous consistency improvement methods with two numerical examples. The results demonstrate that the proposed method improves the consistency of PM better and faster with smaller amount of modification than that of the previous methods, while it modify the elements of the PM to 9-point scales, necessarily. We apply the proposed method to hip joint prosthesis material selection. The proposed methods may be widely used in practical applications of AHP.

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

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