This paper discusses the use of dynamic programming in plant piping for the purpose of determining optimal welding locations where welding costs are minimized. Pipes are constructed and assembled by welding elbows, T-joints, and material pipes together. Partial pipes, called spools, which are small enough to be transported, are welded at the factory; these parts are then assembled and welded at the site to form an entire pipe system. The conditions we have considered here include the following: the unit costs for factory welding and for site welding; the size of the spools, restricted by the transportation mode; some parts of the pipes where welding is not possible and other parts where welding is necessary; and the lengths of material pipes. The functions in which the welding costs are minimized are the following: [numerical formula] [numerical formula] Here, x, y, and t represent positions on the pipe; the center line of the pipe is partitioned into meshes, and the boundaries of these meshes are labeled with numbers beginning at 0, the initial point, and ending at m, the terminal point. Y(x) is the set of points y such that the part between y and x can form one spool which can be transported. W(y,x) is the set of points t between y and x which can be the factory-welding spot closest to y. F(x) gives the minimum value for the sum of the site welding cost and the factory welding cost from the initial point to x, whereas G(y,x) gives the minimum value for the factory welding cost within the spool from y to x. C1 and C2 are the unit costs for site welding and factory welding, respectively. In order to demonstrate the applicability of the present method, it is applied to several examples and its results are compared with those by the conventional, manual method. It indicates that the new method reduces the cost by 10% and that the welding locations are determined with only about 20% of the work processes required by manual calculation.

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In this paper we show three methods for solving optimization problems of expected value of multiplicative functions with negative values; multi-stage stochastic decision tree, Markov bidecision process and invariant imbedding approach.

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This paper investigates the search problem for a moving target when a search path is given in advance. The searcher's strategy is represented by a randomized look strategy, that is, the probability with which he looks in his current position at each discrete time. The searcher knows the probabilistic law about which one of some options the target selects as his path. If the detection of the target occurs by the looking, the searcher gains some value but must expend some cost for the looking. The searcher wants to determine his optimal look strategy in order to maximize the expected reward which is defined as the expected value minus the expected cost. He can randomize his look strategy by the probability that he looks in his current position. We prove the NP-completeness of this problem and propose a dynamic programming method to give an optimal solution which becomes the bang-bang control in the result. We derive some characteristics of the optimal look strategy and analyze them by some numerical examples.

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Recently, Paskov reported that the use of a certain pseudo-random number generator, ran1(), given in Numerical Recipes in C, First Edition makes Monte Carlo simulations for pricing financial derivatives converge to wrong values. In this paper, we trace Paskov's experiment, investigate the characteristics and the generation algorithm of the pseudo-random number generator in question, and explain why the wrong convergences occur. We then present a method for avoiding such wrong convergences. A variance reduction procedure is applied, together with a method for obtaining more precise values, and its correctness is examined. We also investigate whether statistical tests for pseudo-random numbers can detect the cause of wrong convergences.

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By developing an algorithm, Herings, Talman and Yang [6] recently proved the following interesting and deep theorem: A correspondence ζ from the n-dimensional unit cube U^n to the n-dimensional Euclidean space R^n has a continuum of zero points containing the origin and the vector of all-ones if the correspondence satisfies certain conditions. In this note we give an alternative proof of the theorem in the case where the correspondence ζ: U^n → R^n is single-valued.

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The characteristic polynomial of a positive reciprocal matrix has some noteworthy properties. They are deeply related to the notion of consistency of a pairwise comparison matrix of AHP. Based on the results, we propose a method for estimating a missing entry of an incomplete pairwise comparison matrix.

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A generalized M/G/1 queueing system is considered where the efficiency of the server varies as the number of customers served in a busy period increases due to server fatigue or service enforcement. More specifically, the k-th arriving customer within a busy period has the random service requirement V_k where the 0-th customer initiates the busy period and V_k (k = 0,1,2,・・・) are mutually independent but may have different distributions. This model includes an M/G/1 queueing model with delayed busy period as a special case where V_k are i.i.d. for k ≥ 1. Transform results are obtained for the system idle probability at time t, the busy period, and the number of customers at time t given that m customers have left the system at time t since the commencement of the current busy period. The virtual waiting time at time t is also analyzed. A special case that V_k are i.i.d. for k ≥ 2 is treated in detail, yielding simple and explicit solutions.

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We consider an M/GI/1/N queue with vacation time and exhaustive service discipline. We show that the embedded Markov chain formed by the customer departure epochs makes the analysis simple and it enables us to find an explicit solution for the server-vacation queue. We also find an explicit solution for the ordinary (non-vacation) queue.

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We analyze alternating traffic crossing a narrow one-lane bridge on a two-lane road. Once a car begins to cross the bridge in one direction, arriving cars from the other direction must wait, forming a queue, until all the arrivals in the first direction finish crossing the bridge. Such a situation can often be observed when road-maintenance work is carried out. Cars are assumed to arrive at the queues according to independent Poisson processes and to cross the bridge in a constant time. In addition, once cars join the queue, each car needs a starting delay, a constant, to start crossing the bridge. We model the situation where a signal controls the traffic so that the signal gives a priority to one direction at least for a fixed time. Under an assumption, the first two moments of a period during which the signal keeps giving a priority to one direction are obtained. Using a stochastic decomposition property the mean waiting times are obtained of cars to start crossing the bridge from each direction.

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We consider an airplane routing problem when it flies for surveillance operation at sea. The surveillance operation is done by the following procedure. First, an airplane surveys the given area along a given standard route (piecewise linear) by its own sensor (radar, visual apparatus etc.) and detect a ship. The range of the sensor is limited, so the plane cannot grasp the information of all ships within the mission area at once. Then the plane approaches the objective and discriminate what it is. The plane must go back to the base where it has left before its fuel is used up. We intend to give a standard route to accomplish the mission more efficiently. The precise information of the ships (position and speed) cannot get before takeoff, so we cannot utilize the combinational techniques, that is often used in vehicle routing problems. We formulate the problem as a non-linear optimization problem with a constraint for flight distance. The objective is to maximize the expected value of the detected ships within an operation, that 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. As the surveillance operation is carried out periodically, and the positions of the ships (such as fishing boats, ocean ships) is expectable to some degree, we assume that we can draw the density map of ship of the area. We also suppose that the detecting probability obeys the inverse cube law of the distance. The density of the ships is given as a sum of 2-dimensional normal density functions, centered at expected positions of ihdividual ships. We use Augmented Lagrangean Function Method to construct the algorithm of the problem. Numerical experiments show that the method proposed in the paper gives a good standard route in reasonable computation time.

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This paper investigates a non-linear integer programming problem with an exponential objective function and cost constraints, which is a general model and could be applied to the assignment problem or the search problem. As a representative example, we consider a following problem of assigning m types of missiles to n targets. The value of target i is V_i and the single shot kill probability (SSKP) of a j-type missile to target i is β_<ij> = 1 - exp(-α_<ij>) where α_<ij> &ge; 0. Denoting the number of j-type missiles assigned to target i by n_<ij>, the expected destroyed value is given by Σ^n_<i=1>V_i{1 - exp(-Σ_jα_<ij>n_<ij>)}. If a unit price of j-type missile is c_j and the total cost of missiles is limited by C, we have cost constraints Σ^m_<j=1>c_jΣ^n_<i=1>n_<ij>&le;C. To this problem which is NP-hard, we propose two methods, the dynamic programming method and the branch and bound method. An approximate algorithm and an estimation of the upper bound of the objective function are incorporated in the branch and bound method. Each of these methods has its characteristic merits for the computational time. We clarify their characteristics through the sensitivity analysis by some examples in this paper.

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