We formulate weighted, dynamic network range-adjusted measure (DN-RAM) and dynamic network slacks-based measure (DN-SBM), run robustness tests and compare results. To the best of our knowledge, the current paper is the first to compare two weighted dynamic network DEA models and it also represents the first attempt at formulating DN-RAM. We illustrate our models using simulated data on residential aged care. Insight gained by running DN-RAM in parallel with DN-SBM includes (a) identical benchmark groups, (b) a substantially wider range of efficiency estimates under DN-RAM, and (c) evidence of inefficient size bias. DN-RAM is also shown to have the additional desirable technical efficiency properties of translation-invariance and acceptance of free data. Managerial implications are also briefly discussed.

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A graph G with a perfect matching is called saturated if G+e has more perfect matchings than G for any edge e that is not in G. Lovasz gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which revealed canonical structures of general graphs with perfect matchings. Here, the cathedral theorem is derived in quite a natural way, providing more refined or generalized properties. Moreover, the new proof shows that it can be proved without using the Gallai-Edmonds structure theorem.

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Dynamic programming has been used to solve numerous complex problems in business and engineering. This study applies dynamic programming to a retail decision-making problem related to trade credit. A price, shelf-space, and time-dependent demand function is introduced to model the finite time horizon inventory. Trade credit was considered in the model because suppliers commonly provide retailers with credit periods. Consequently, the retailer is not required to pay for goods immediately upon receipt, and can instead earn interest on the retail price of the goods between the time the goods are sold and the end of the credit period. The objective of this paper is to determine the periodic retail price, shelf-space quantity, and ordering quantity that maximize total profit. The numerical examples explain the procedures of the solution approach and show that dynamic decision making is superior to fixed decision making regarding profit maximization.

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This paper presents an efficient algorithm that enumerates all the bottom-left stable positions for a given layout of rectangles. Its time complexity is O((n+K)log n), where n is the number of placed rectangles (i.e., input size) and K is the number of bottom-left stable positions (i.e., output size). This is the first non-trivial algorithm that works for layouts without bottom-left stability and with overlap. In addition to the theoretical analysis of the time complexity, we evaluate the computation time of the proposed algorithm via computational experiments and confirm that it is applicable to very large-scale instances having one million rectangles.

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