The stochastic inventory models analyzed in this paper explore the problem of lead time associated with setup cost reductions for the continuous review and periodic review inventory models. For these two models with a mixture of backorders and lost sales, we respectively assume that their mean and variance of the lead time demand and protection interval (i.e., lead time plus review period) demand are known, but their probability distributions are unknown. We develop a minimax distribution free procedure to find the optimal solution for each case.

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We have developed a simple approach to valuing risky corporate debt when corporations own securities issued by other corporations. We assume that corporate debt can be valued as an option on corporate business asset value, and derive payoff functions when there exist cross-holdings of stock or debt between two firms. Next we show that payoff functions with multiple cross-holdings can be solved by the contraction principle. The payoff functions which we derive provide a number of insights about the risk structure of company cross-holdings. First, the Modigliani-Miller theorem can obtain when there exist cross-holdings between firms. Second, by establishing cross-shareholdings each of stock holders distributes a part of its payoff values to the bond holder of the other's firm, so that both firms can decrease credit risks by cross-shareholdings. In the numerical examples, we show that the correlation in firms can be a critical condition for reducing credit risk by cross-holdings of stock using Monte Carlo simulation. Moreover we show we can calculate the default spread easily when complicated cross-holdings exist in many firms by a simulation that includes the contraction principle method. By using our method, we can reevaluate which cross-held shares are beneficial or disadvantageous for the cross-holding firms.

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This paper describes a heuristic algorithm for the two-dimensional free-form bin packing (2D-FBP) problem, which is also called the irregular cutting and packing, or nesting problem. Given a set of 2D free-form bins, which in practice may be plate materials, and a set of 2D free-form items, which in practice may be plate parts to be cut out of the materials, the 2D-FBP problem is to lay out items inside one or more bins in such a way that the number of bins used is minimized, and for each bin, the yield is maximized. The proposed algorithm handles the problem as a variant of the one-dimensional bin-packing problem; i.e., items and bins are approximated as sets of scanlines, and scanlines are packed. The details of the algorithm are given, and its application to a nesting problem in a shipbuilding company is reported. The proposed algorithm consists of the basic and the group placement algorithms. The basic placement algorithm is a variant of the first-fit decreasing algorithm which is simply extended from the one-dimensional case to the two-dimensional case by a novel scanline approximation. The group placement algorithm is an extension of the basic placement algorithm with recombination of input items. A numerical study with real instances shows that the basic placement algorithm has sufficient performance for most of the instances, however the group placement algorithm is required when items must be aligned in columns. The qualities of the resulting layouts are good enough for practical use, and the processing times required for both algorithms are much faster than those by manual nesting.

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Minoux considered the maximum balanced flow problem, i.e. the problem of finding a maximum flow in a two-terminal network N=(V, A) with source s and sink t satisfying the constraint that any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where V is vertex set and A is arc set. Several efficient algorithms, so far, have been proposed for this problem. As a natural generalization of this problem we focus on the problem of maximizing the total flow value of a generalized flow in a network N=(V, A) with gains γ(a)>0 (a∈A) satisfying any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where γ(a)f(a) units arrive at the vertex w for each arc-flow f(a) (a≡(v, w)∈A) entering vertex v in a generalized flow in AF. We call it the generalized maximum balanced flow problem and if γ(a)=1 for any a∈A then it is a maximum balanced flow problem. The authors believe that no algorithms have been shown for this generalized version. Our main results are to propose two polynomial algorithms for solving the generalized maximum balanced flow problem. The first algorithm runs in O(mM(n, m, B')log B) time, where B is the maximum absolute value among integral values used by an instance of the problem, and M(n, m, B') denotes the complexity of solving a generalized maximum flow problem in a network with n vertices, and m arcs, and a rational instance expressed with integers between 1 and B'. In the second algorithm we combine a parameterized technique of Megiddo with one of algorithms for the generalized maximum flow problem, and show that it runs in O({M(n, m, B')}^2)time.

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The internal rate of return (IRR) criterion is often used to evaluate profitability of investment projects. In this paper, we focus on a single-period project which consists of two types of cash flows; an investment at one period and a return at a succeeding period, and a financing at one period and a repayment at a succeeding period. We decompose the given investment project into a series of the singleperiod projects. From the viewpoint of the single-period project, we point out the applicability issue of the IRR criterion, namely the IRR criterion cannot be applied in which a project is composed of both investment type and financing type. Investigating the properties of a series of the single-period projects, we resolve the applicability issue of the IRR criterion and propose the decision procedure for profitability judgment toward any type of investment project based on the comparison between the IRR and the capital cost. We develop a new algorithm to obtain the value of the project investment rate (PIR) for the given project, which is a function of the capital cost, only using the standard IRR computing routine. This outcome is a theoretical breakthrough to widen the utilization of IRR in practical applications.

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Liang presented a fuzzy multiple criteria decision making (MCDM) method based on the concepts of ideal and anti-ideal points. Despite its merits, Liang method has the following limitations: (i) the objective criteria are converted into dimensionless indices and the subjective criteria are not converted, which may prevent compatibility for these criteria, (ii) the formulas for converting objective criteria are not reliable, and (iii) an unreliable ranking method, i.e. maximizing set and minimizing set, is applied to rank the fuzzy numbers. This paper applies the Hsu and Chen method and suggests a fuzzy number ranking method to propose an improved fuzzy MCDM model based on ideal and anti-ideal concepts to overcome the shortcomings of the Liang method. Numerical examples demonstrate the effectiveness and feasibility of the proposed ranking method and the improved model, respectively.

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This study discusses the mathematical structure of the dominant AHP and the concurrent convergence method which were originally developed by Kinoshita and Nakanishi. They introduced a new concept of a regulating alternative into an analyzing tool for a simple evaluation problem with a criterion set and an alternative set. Although the original idea of the dominant AHP and the concurrent convergence method is unique, the dominant AHP and the concurrent convergence method are not sufficiently analyzed in mathematical theory. This study shows that the dominant AHP consists of a pair of evaluation rules satisfying a certain property of overall evaluation vectors. This study also shows that the convergence of concurrent convergence method is guaranteed theoretically.

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We have launched the first artificial satellite in 1957. Since then, a new environmental problem has been brought about by ourselves. That is the space pollution. Everything we have lifted up can become space debris when we lose control of it. The space debris is accumulated around the outer atmosphere more than forty years and is propagating by collision and explosion of itself. As the debris has high kinetic energy, the space pollution affects the safety of our future space activities. For the sake of reducing the debris and preserving clear and safe space environment, we have been developing spacecraft of various types, such as for observation, for sweeping the debris up, or for shooting them down. Even if we may take any one of these responses to the debris, a concrete operating plan of spacecraft has not declared till now. In this paper, we propose one of the efficient operating plans of the spacecraft. If a spacecraft flies along the route constructed by our proposing procedure, it will be able to respond more efficient to individual drifting debris. When we construct the 3-dimensional standard route of the spacecraft, we extend the routing methods in 2-dimension that was applied to make standard routes of an aircraft in surveillance operation at sea. In these methods, the standard routes (piecewise linear) are constructed under the policy of maximizing the expected value of the detected ships locally within an operation. That objective value is calculated as integration, along a given route, of product value of the detecting probability from the plane and the density of the ships in the area. In this 3-dimensional extended case, we also use non-linear optimization technique (quasi-Newton method) to make the spacecraft routes. By numerical experiments, we can construct 3-dimensional standard routes that close to drifting debris.

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