Under the situation of the furious competition among the leasing companies, the important problem for a leasing company's management is to determine a lease charge on each lease contract suitably from the management viewpoint. A lease charge is considered to be suitable when the lea,sing company can get an appropriate profit through the contract. But quite a lot of leasing companies adopt a full payout method for the lease charge determination. It may not be a desirable method in the above mentioned situation because the comparative high lease charge makes its competitive power down. So, it is important for a leasing company to determine suitable lease charges which satisfy both the leasing company's need and its client requirements. This paper proposes a lease charge determination method for a leasing company based on the risk analysis. The method which we propose in this paper has two eminent characteristics. One of them is that the risk can be treated probabilistically. The other is that the concept of two profit levels (the "necessary profit level" as minimum necessary profit and the "sufficient profit level" as maximum obtainable profit) is introduced as the lease charge determination criterion. The advantages of this method are as follows: (1) the lease charge which satisfies given risk conditions can be determined by using the disposal price distribution of the lease object, (2) the suitable lease charge can be given in the form of "a certain range" which makes the lease charge negotiation with its client flexible. In this paper, firstly, we explain t,he lease charge determination method, secondly, we give the lease charge determination procedures, thirdly, we discuss some characteristics of this method and finally, we show numerical examples.

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The Laguerre transform developed by Keilson, Nunn and Sumita (1979,1981, 1981) provides an algorithmic basis for mechanizing various continuum operations such as multiple convolutions, Integration, differentiation, and multiplication by polynomials and exponential functions. In this paper, we develop a numerical procedure for finding the Laguerre coefficients of a product of functions in terms of the Laguerre coefficients of individual functions. The procedure enables one to evaluate multiple convolutions and other continuum operations described above even when product form functions are involved, thereby further enhancing the power of the Laguerre transform.

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Given a set Ω of R^n and a function f : Ω→R^n the problem of finding a point x^2⋴Ω such that (x - x^s)^tf(x^s) >__-0 for any x⋴Ω is referred to as a stationary point problem and x^s is called a stationary point. For the problem with conical Ω and strongly copositive f we propose a system of equations whose solution set contains a path connecting a trivial starting point to a stationary point. We also develop an algorithm to trace the path when f is an affine function with. a copositive plus matrix. Starting with an appropriate point the algorithm provides a stationary point or shows that there exist no stationary points.

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This paper shows a property that a value of production rate which maximizes profit per unit time lies between the highest production rate and the production rate which minimizes production cost per unit product. A similar property holds among the unit production costs under these production rates. Moreover, some sufficient conditions for which the optimal value of decision variable which realizes the maximum profit rate lies between the optimal values realizing the other two criteria are derived. The last result is an extension of the theorem concerning the economic cutting speed for a machine tool, which says that the maximum-profit cutting speed lies between the minimum-production-cost cutting speed and the maximum-production-rate cutting speed.

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In this paper, a general fuzzy mathematical programming problem is formulated, considering the difference between the fuzziness in the preference of the decision maker and the fuzziness in the coefficients. While the fuzziness in the preference of the decision maker expresses the indefiniteness in nature or character as idea, feelings, etc., the fuzziness in the coefficients expresses uncertainty on the coefficients. Firstly, the fuzziness in the preference of decision maker is represented by fuzzy goals given on the deviations of right-hand sides from left-hand sides in the constraints as treated in flexible programming. On the other hand, the fuzziness in the coefficients is represented by possibility distributions, and the deviations of right-hand sides from left-hand sides are obtained as possibility distributions. At this juncture, two kinds of deviational possibility distributions can be considered. One is the possibility distribution obtained through all possible combinations of values restricted by two possibility distributions, and the other is the possibility distribution which is necessary for a possibility distribution to coincide with the other Possibility distribution. Using the possibility measures and the necessity measures derived from deviational possibility distributions, the fuzzy mathematical programming problem is formulated. Next, since the idea of goal programming is close to that of flexible programming, given a regret function as a preference model of the decision maker and a fuzzy goal on the regret function space, the fuzzy mathematical programming problem is formulated again in the similar manner of goal programming. A numerical example is given to illustrate their ideas. Lastly, the formulated problems based on the ideas of flexible programming and goal programming are compared. It is shown that the problem based on the idea of goal programming is a special case of the problem based on the idea of flexible programming, when the membership value of the decision set is regarded as the degree of satisfaction of the decision maker.

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Recently many fuzzy mathematical programming methods were proposed as the decision making models under uncertainty. Fuzzy mathematical programming problems were formulated by replacing the coefficients of the conventional mathematical programming problems with the fuzzy number coefficients. In the formulations, it is not always easy to give a proper membership function to each fuzzy number coefficient. Thus we formulate a tractable problem called the interval programming problem by regarding the fuzziness of coefficients as intervals. In this paper, an interval 0-l programming problem with an interval objective function is formulated and analyzed to deal with a product-mix problem. First, two order relations between intervals are defined to maximize the interval objective function. The order relations are defined by the left and right limits of interval and by the center and width of interval, respectively. Furthermore the modified order relations to minimize the interval objective function are defined. Some properties of these order relations are investigated. Next, the interval 0-1 programming problem is formulated where a solution set is defined by the two order relations between intervals. A computational algorithm using the branch and bound method is proposed to obtain the solution set. Last, the interval 0-1 programming problem is applied to the product-mix problem where the demand of each product to be selected is estimated as an interval.

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This paper deals with the Hitchcock transportation problem with m supply points and n demand points. Assume that m <__- n and all the data are positive integers which are less than or equal to an integer M. We propose two polynomial time algorithms for solving the problems. The algorithms are based on the interior point algorithms for solving general linear programming problems. Using some features of the transportation problems, we decrease the computational complexities. We show that one of the algorithms requires at most O(m^3n^2 log nM + n^3) arithmetic operations and the other requires at most O(n^4 log nM) arithmetic operations.

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In usual scheduling problem, machine speed is considered to be constant. Here we assume that the machine speed is controllable for each job and consider the maximum lateness problem on the single machine. The objective is to determine an optimal schedule and optimal jobwise speed of the machine minimizing the total sum of costs associated with the maximum lateness and jobwise machine speed.

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This paper is concerned with the relationship between the ordering of' players and winning probability in two types of methods adopted in team sports. Suppose that there are two teams A and B each of which consists of more than one player. The team game between A and B is composed of a sequence of individual matches each of which is done between one player from A and one player from B, and the win or loss of team A is determined as a result of such individual matches. There are several known methods to organize such a team game. Among them, this paper deals with the following two methods. In both methods, an initial ordering of players in each team is determined in advance. The first method is called a tie (or elimination series) which is carried out as follows. Initially the first players in the orderings of teams A and B play an individual match against each other. The winning player successively plays a match against the second player of the other team. The player that once lost the match is eliminated from his team. In this manner, a player successively plays matches until he loses. The team that eliminated all players of the other team wins the game. This method is adopted in judo and kendo. The second method is the same as the first one except that when a player wins the match, he is place at the last position of the ordering of remaining players of his team, and that at each round the first two players in the current orderings of both teams play a next individual match. This method is called an exterminatory series and is adopted in soft tennis. This paper assumes that a nonnegative value is associated with each player that represents his strength, and that when player 1 with strength a plays an individual match against player 2 with strength b(a > 0 or b >0), the probability such that player 1 wins over player 2 is given by p(a, b). Under this assumption, we first give a necessary and sufficient condition under which the probability P_A such that team A wins the game over team B by both methods is independent of the initial orderings of both teams. We then show that under this condition, the probability P_A by the first method is equal to P_A by the second method. Thirdly, Bradley-Terry model that has been used in the literature (i.e., p(a, b) = a/(a + b)) satisfies this condition 1. Finally, we show that p(a, b) satisfies this condition if and only if p(a, b) = f(a)1(f(a) + f(b)), where f(a) is an arbitrary monotone increasing function with f(0) = 0.

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