The purpose of this paper is to find the robust facility location in consideration of facility opening and closing. For evaluating the robustness, we formulate a simple model of the regular facility locations (square lattice, triangular lattice, hexagonal lattice) and analyze the changes of travel distances to the nearest facility. We also question the accepted theory in the Central Place Theory that the triangular lattice is the optimal. First, we consider the case where some existing facilities are closed at random. This assumes the situation that some facilities suffer damage from disasters, or that the planner closes facilities without considering the spatial relationship. We theoretically derive the probability density function of kth-nearest-neighbor distance of the three regular locations and show that the triangular lattice is the optimal when 69% or more of facilities are survived. Then we apply the model to school locations in northern Tsukuba and confirm that the model is applicable to real problems. Second, we consider the case where some existing facilities are closed according to a plan. This assumes the situation that the planner controls facility closing in order not to decrease the efficiency of movement. In addition, the case where some new facilities are opened is considered. We derive the average travel distance and the standard deviation of travel distances as a function of the rate of closing (opening) and demonstrate that the hexagonal lattice is the optimal when the rate of closing (opening) is not less than 38% (27%). Comparing the above two cases, we show the effectiveness of planned closing. When closing 50% of facilities, the planner can reduce the average travel distance by 13% in the square lattice, 8% in the triangular lattice, and 15% in the hexagonal lattice.

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In this paper, we consider the optimal location problem for urban emergency vehicles. Examples of urban emergency vehicles are ambulances, police cars, etc. Emergency vehicles stand by the base until a service demand occurs. And if a service demand occurs, they move to give a service from its base to the demanded site. There are at least two important factors for performing effective services, i.e., 1) the location of bases for emergency vehicles until a service demand occurs, and 2) the number of the emergency vehicles stored there. We apply a continuous-time Markov chain to this problem, and propose a model for determining location of bases for emergency vehicles, and the number of the emergency vehicles deployed there. This model handles appropriately the mutual cooperation among emergency vehicles on congested situation, the assignment of the optimal emergency vehicles in each system state, and various service times. We apply this model to the emergency system of Seto city, Aichi, and evaluate its practicality.

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This paper deals with a production planning problem at Toyota Motor Corporation. Car dealerships order cars on the basis of weekly instructions given by Toyota Motor Corporation. A weekly production plan for each type of car is made according to the orders obtained from the car dealerships. If a certain type of car is produced on more than one assembly line, weekly production plans need to be made for those assembly lines. When making a weekly production plan for each assembly line, there are two objectives that we must achieve. One is to minimize the total cost of shipment from the assembly lines to car dealerships. The other is related to the production constraints on the assembly lines. Factories with the assembly lines and suppliers have already finished preparing for production based on monthly production plans. If the difference between the monthly and weekly production plans for each assembly line is minimal, cars and parts can be produced without loss. If the difference is much, however, then the weekly production plan becomes impossible without changing production preparations. Hence, the second objective is to minimize the differences between monthly and weekly production plans. To achieve two objectives, we firstly define the production constraint cost for the difference between two production plans and consider minimizing the sum of the shipment cost and the production constraint cost. As a result, this production planning problem can be formulated as a separable programming problem with integer constraints. From another point of view, this problem can be considered as a Hitchcock transportation problem with the piecewise linear objective function for the sum of some variables and integer constraints. The algorithm for this problem has not yet been developed. At first we show that we can convert the problem into the minimum cost flow problem, if the production constraints satisfy certain conditions. These conditions hold for many practical problems. And we also show that the practical problems that do not satisfy the conditions have an integer solution due to the structure of the production constraints. Moreover, a local optimal solution of the problem is a global optimal solution. Then we can apply the linear programming algorithms to the problem instead of separable programming and solve it efficiently.

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This paper presents exact and approximate solution methods for a production line with machine failures and finite intermediate buffers. Our ultimate objective is to develop a method which is better than 2stage decomposition method in accuracy and is reasonably short in computation times. We first clarify the structure of the balance equations of queueing models of the system and propose an exact solution method to exploit the structure of the transition rate matrix. We then seek an efficient approximate solution method to decompose the line into a set of three-machine and two-buffer blocks for evaluating the performance of the multistage production line. This approximate solution method leads to a simple and fast algorithm. Numerical experiments show that this approximate method is very accurate and efficient.

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A boom is a social phenomenon in which some commodity, fashion or the like is suddenly prevailed among people and is forgotten by most of them shortly after that. In this study, we introduce a mathematical model of booms and try to analyze such phenomena. This model is based on two assumptions. The first is that each of the consumers is in one of the four stages at a time : the stage in which he has not consumed the commodity yet, the stage in which he has begun to consume it after the start of the boom, the stage in which he stopped consuming and the stage of the regular consumer. Second, the increasing speed of the number of the consumers of each stage is assumed to depend only on populations of the former stages. A system of linear differential equations is formulated to describe the change in the numbers of the consumers in these four stages. The validity of the model is verified by fitting the solutions of the equations for some real data of booms such as "instant noodles boom", "clear liquor boom", "football boom" and so on. The model can explain some aspects of the mechanism of the boom. We develop some quantitative arguments about each boom, and the characteristics of regional consumers in Japan are also described by the estimated parameters in the model. Our model has so simple structure that we may be able to describe some more complicated phenomena by adding some elements to this model.

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