Showing papers in "Naval Research Logistics Quarterly in 1961"
TL;DR: In this paper, a fixed-cost transportation problem is formulated as an integer program, and an approximate method of solution is proposed to solve it using integer programming techniques, and examples are given to demonstrate the approximation technique.
Abstract: This paper formulates a fixed-cost transportation problem as an integer program, describes some of its special properties, and suggests an approximate method of solution. Examples are given to demonstrate the approximation technique.
297 citations
TL;DR: In this paper, the Lagrange multiplier method is used to solve the problem of nonlinear constrained maximization with nonlinear maxirnands or constraint functions or both, and sufficient conditions for the validity of the multiplier method are discussed.
Abstract: Many problems arising in logistics and in the application of mathematics to industrial planning are in the form of constrained maximizations with nonlinear maxirnands or constraint functions or both. Thus a depot facing random demands for several items may wish to place orders for each in such a way as to maximize the expected number of demands which are fulfilled; the total of orders placed is limited by a budget constraint. In this case, the maximand is certainly nonlinear. The constraint would also be nonlinear if, for example, the marginal cost of storage of the goods were increasing. Practical methods for solving such problems in nonlinear programming almost invariably depends on some use of Lagrange multipliers, either by direct solution of the resulting system of equations or by a gradient method of successive approximations (see [5], Part II). This article discusses a part of the sufficient conditions for the validity of the multiplier method.
144 citations
TL;DR: In this article, an inventory model is constructed in which there is a fixed lag time of one period for delivery of orders, but there is also defined an emergency situation with respect to initial stock at any particular inventory point.
Abstract: Abstract : An inventory model is constructed in which there is a fixed lag time of one period for delivery of orders, but in which there is also defined an emergency situation with respect to initial stock at any particular inventory point. When such an emergency situation obtains, an additional order of a specified fixed size is taken with immediate delivery. While the particular structure of emergency is laid down, there is a free parameter left to be chosen for each period to render the definition of emergency fully specific, and this parameter is determined as part of the optimization. There is no attempt to achieve the fullest generality in every respect. The principal object is to investigate the indicated emergency character of the model. It is anticipated that the n-period cases, with n > 1, will yield their essential properties by the device of inductive argument, and that from these, by suitable limit considerations, the stationary case may be studied. This first investigation is directed to the case of the single-period problem, so formulated as to present the features which may be expected to arise in the general n period case.
139 citations
45 citations
44 citations
TL;DR: In this paper, a simple ordering and disposal policy is proposed for the multiple installation model with the assumption that the decision to be made at the beginning of each period is always one of the following: ordering, disposal, no ordering and no disposal.
Abstract: In this paper we analyze the inventory model in which the disposal of surplus items is taken into consideration as well as the purchase of stock in anticipation of future demand. Our main concern is to formulate the optimal ordering and disposal policy in a simple form, and it is assumed that the decision to be made at the beginning of each period is always one of the following: (1) ordering, (2) disposal, (3) no ordering and no disposal. The multiple installation model is treated by the method of implied penalty described by Clark and Scarf. The simple optimal policies are explicitly determined under various sets of assumptions on the cost structure of the inventory model.
37 citations
TL;DR: In this article, a sequence of approximations to the optimal policy for (a) is considered; in the present paper, we consider the problem of minimizing the maximum distance traveled in reaching shore, minimizing the statistical expectation of the distance travelled, or maximizing the probability of reaching shore within a given distance travelled.
Abstract: A problem posed by Bellman and considered by Isbell is as follows: Suppose one is a mile from a straight shore with no means whatsoever of ascertaining its direction. What is the optimum path to follow so as to (a) minimize the maximum distance travelled in reaching shore, (b) minimize the statistical expectation of the distance travelled, or (c) maximize the probability of reaching shore within a given distance travelled? Isbell found the solution to (a); in the present paper a sequence of approximations to the optimal policy for (b) is considered.
30 citations
TL;DR: In this article, the search policy is considered to be optimal when the statistical expectation of the total cost of the search is minimized, and an optimal search strategy is found in the case where the pi* s are all equal, and approximately optimal strategies are presented when pi is proportional to i. The latter case has application to defense situations where complete searches occur at successive intervals of time.
Abstract: Consider a model in which there are N neighboring cells in one of which there is an object that it is required to find. The a priori probabilities of the object being in cells 1, …, N are p1, …, pN respectively, and the costs of examination of these cells are tl, …, tN respectively; the search policy is considered to be optimal when the statistical expectation of the total cost of the search is minimized. For the case in which the ti†s are constant throughout the search, an optimal policy solution has previously been found by Bellman and by Smith. In the present paper it is assumed that the costs comprise a travel cost dependent upon the distance from the last cell examined, in addition to a fixed examination cost: initially, assuming that the searcher is next to cell 1, ti = i + t, where t is constant; and from then onwards, assuming that the jth cell has just been examined, ti = | i - j | + t. An optimal search strategy is found in the case where the pi† s are all equal, and approximately optimal strategies in the case where pi is proportional to i. The latter case has application to defense situations where complete searches occur at successive intervals of time, and hence the enemy objects are thinned out the nearer they come to the defense base.
22 citations
TL;DR: In this article, the capacity of a network as a function of the capacities of its individual arcs was investigated, and it was found that each pair of arcs either consistently help each other or consistently hinder each other.
Abstract: This article covers an investigation of the capacity of a network as a function of the capacities of its individual arcs. The case of two variables was investigated in detail, and it was found that each pair of arcs either consistently help each other or consistently hinder each other. Interaction types were computed for a number of special cases, such as arcs in parallel or arcs in series.
18 citations
TL;DR: A FORTRAN II transportation code, using Kuhn's Hungarian Method, was reported upon at the RAND Symposium on Mathematical Programming in March 1959, and several problems were run at the IBM Research Center to obtain data regarding computing times and frequencies of various internal loops.
Abstract: A FORTRAN II transportation code, using Kuhn's Hungarian Method, was reported upon at the RAND Symposium on Mathematical Programming in March 1959. The algorithm was based upon a proof of the Konig-Egervary theorem, presented by the present author at the 1958 Symposium on Combinatorial Problems sponsored by the American Mathematical Society. The code was entirely rewritten (in FORTRAN II), during the 1959 IBM Summer Institute on Combinatorial Problems, and several problems were run at the IBM Research Center to obtain data regarding computing times and frequencies of various internal loops. A CLOCK Subroutine yields readings, in hundredths of minutes, for each time through selected portions of the computing run. The version reported upon at RAND solved a 29 × 116 pseudorandom transportation problem in 8.01 min, as compared with 3.17 min using the fastest competing code (SHARE 464) then available. The present version solved this same problem in 2.87 min, and another 29 × 116 problem in 1.89 min. This paper presents the algorithm and reports upon computing experience.
17 citations
TL;DR: In this paper, it was shown that the problem of finding a circle of known radius, of known distance from a starting point in its plane, is a limiting case of the problem posed as the radius of the circle tends to infinity.
Abstract: The problem has been considered by Isbell of determining the path that minimizes the maximum distance along the path to a line in the same plane whose distance from the starting point of the search is known, while its direction is unknown. We consider in this article the analogous problem for the search for a circle of known radius, of known distance from a starting point in its plane. It is further shown that Isbell's solution is a limiting case of the problem posed as the radius of the circle tends to infinity. The problem appears to be of some practical significance, since it is equivalent to that of searching for an object a given distance away which will be spotted when we get sufficiently close—that is, within a specific radius.
TL;DR: It is concluded that good taxi redistribution policies markedly decrease customer waiting times, except when the level of taxi utilization is too high.
Abstract: In dispatching a taxi fleet to meet random customer demand, two types of strategies must be defined: (a) the dispatching strategy, which sets the rules for choosing a taxi to serve a given call, and (b) the idle time strategy, which determines how the idle taxis are to be distributed most advantageously to meet future demand. The effect of dispatching and idle-time strategies on taxi utilization and on passenger service is demonstrated under several conditions of operation. First, the simples taxi model is examined: random demand between two points serviced by one taxi. Next, we consider the effect of different strategies on the operation of a taxi fleet when demand is uniformly distributed in a two-dimensional region. In the last section, the main properties of these simplified models are exhibited in an intermediate case by a Monte Carlo simulation of taxi service between a number of discrete points. The purpose of the paper is to provide simple and practical methods to determine the payoff in customer waiting time and/or taxi savings resulting from more sophisticated dispatching and idle-time strategies. We conclude that good taxi redistribution policies markedly decrease customer waiting times, except when the level of taxi utilization is too high. Our results apply most directly to private transportation fleets, rather than to automobile taxi companies where cruising may be important.
TL;DR: In this article, it was shown that under mild restrictions on G(x), a consistent estimate of C has a structure similar to that of estimates known to be optimal in certain important cases.
Abstract: If Y1 ≤ … ≤ Yn are ordered observations from a population with cumulative distribution function , probability density function , where B and C are unknown parameters with C > 0, and the function G(x) is known, it is shown that under mild restrictions on G(x), is a consistent estimate of C. In certain important cases, this estimate has a structure similar to that of estimates known to be optimal.
TL;DR: In this article, the distribution and expected value of the number of planned replacements, number of failures, and the total number of removals due to either planned replacement or failure replacement, corresponding to any specified length of time.
Abstract: Assume a component has a constant failure rate during the first τ unit of time following installation. After that, wearout sets in so that failure rate begins to increase sharply. Assume further that replacement is made at the time of component failure or at τ units of time following installation, whichever occurs first. Under these assumptions, we obtain explicit formulas for the distribution and expected value of the number of planned replacements, the number of failures, and the total number of removals due to either planned replacement or failure replacement, corresponding to any specified length of time. Examples are worked to show the application of these formulas. The results obtained in this article are of use in determining the number of spares to stock or the budget required to maintain the equipment.
TL;DR: A single echelon, multidepot supply system is studied for low demand items having a stationary Poisson probability distribution for demand, an extension of previous models in that redistribution costs and depot stockout costs are considered in the determination of the stockage objective for the system as a whole.
Abstract: Problems of inventory control for items with extremely low demand (say less than one unit per month average demand) have received relatively little attention in the literature. However, for military supply systems it is well known that an extremely large proportion of the total number of items fall into this category. The problem of designing a control system for these low demand items is therefore one of the most critical inventory management problems faced by the National Military Establishment.
In the following, a single echelon, multidepot supply system is studied for low demand items having a stationary Poisson probability distribution for demand. Instantaneous information concerning inventory levels is assumed to be available. Procurement lead time is assumed to be constant as well as the time required for either of two available modes of redistribution. Items are ordered one at a time and decision rules are developed for allocation of new procurement, redistribution of stocks among the depots, and for determining system and depot stockage objectives in order to minimize the expected costs resulting from system and depot stockouts, cost of redistributing stocks among the depots, and costs of transportation from the source. The model is an extension of previous models in that redistribution costs and depot stockout costs are considered in the determination of the stockage objective for the system as a whole. When more than a single unit is on order, the allocation of a unit ready to be delivered is determined by the solution of a dynamic programming problem. When only a single unit is on order (and this is the one ready to be delivered), the optimal allocation procedure is reduced to allocating the unit to the depot which has the greatest probability of using it in a time period T + (1/λ) where T is the procurement lead time and λ the system demand rate for the item.
TL;DR: It is concluded that this analytical expression for the downtime distribution of complex electronic equipments provides the basis for a quantitative and scientific approach to the measurement and prediction of maintainability.
Abstract: This article presents a method for predicting the downtime distribution for a new complex electronic equipment. A theoretical discussion is given which indicates that the downtime distribution for such an equipment can be represented by a lognormal function. Experimental data are shown which establish that the two-parameter lognormal distribution function adequately describes the downtimes of a diverse group of electronic equipments. It is concluded that this analytical expression for the downtime distribution of complex electronic equipments provides the basis for a quantitative and scientific approach to the measurement and prediction of maintainability. This approach takes account of equipment maintenance characteristics and maintenance personnel capabilities.
TL;DR: The principles and methods described in this article were developed as guides to the development and installation of a management system for programming, decision-making, and progress evaluation in weapon systems development, procurement, and production in the U. S. Air Force.
Abstract: Over the last several years, there has been considerable interest in the problems of effective programming, decision-making, and progress evaluation, in the development and production of weapon systems. The need for solving these problems was great when budget ceilings were less restrictive. At the present time when budgets are more stringent, the need is indeed urgent if defense is to be most adequate. The principles and methods described in this article were developed as guides to the development and installation of a management system for programming, decision-making, and progress evaluation in weapon systems development, procurement, and production in the U. S. Air Force. The principles, however, have applicability to programming, decision-making, and control in other governmental and in industrial and business organizations. Some of these principles and methods already have been applied in other situations; some are new. However, the amount that is new is relatively small; we are confident the system is feasible and economic within the existing state of the art, and that it would contribute to efficiency in the development and production of weapon systems.
TL;DR: An interval scale of quality of the intelligence reports was constructed with a view to providing some mathematical models that describe in a simple fashion the way in which the components of an intelligence report combine to determine the value of the report.
Abstract: A large-scale experiment was designed to obtain the intuitive evaluations of intelligence reports from a large group of experienced Army officers. An interval scale of quality of the intelligence reports was constructed with a view to providing some mathematical models that describe in a simple fashion the way in which the components of an intelligence report combine to determine the value of the report. The results of the study indicate (a) that officers judge such intelligence reports consistently, (b) that they use criteria in judging intelligence estimates different from those they use in judging courses of action, and (c) that the judgments depend to some extent on the combat experience of the officer. The different techniques of obtaining rating scales were found to produce essentially the same results.
TL;DR: A loading model is developed to estimate the relationship between cargo compartment dimensions and loading efficiency and suggests the relationshipBetween cargo compartment and payload capacity which would be optimum for this particular mission.
Abstract: An important mission of military cargo aircraft is the deployment of army equipment to areas of actual or potential warfare The large number of vehicles required to support army forces makes weight anti cube, the traditional measures of airlift capacity, insufficient for analysis of the amount of such cargo which can be carried by various aircraft This paper uses a representative list of army deployment cargo to investigate the capacity of alternative aircraft designs A loading model is developed to estimate the relationship between cargo compartment dimensions and loading efficiency The relationship between Compartment floor area and floor area of vehicles loaded is found to be remarkably stable over a large range of alternative compartment sizes A second model is used to minimize the distribution of total weight per aircraft when the aircraft are loaded in accordance with the criteria of the first model The resultant distributions, presented for various compartment sizes, suggest the relationship between cargo compartment and payload capacity which would be optimum for this particular mission
TL;DR: A recurrence relation for the direct computation of the waiting times of an array of c customers served by s parallel channels on a first come first served basis, where the time of arrival and the holding time of each customer are known, is worked out.
Abstract: A recurrence relation for the direct computation of the waiting times of an array of c customers served by s parallel channels on a first come first served basis, where the time of arrival and the holding time of each customer are known, is worked out. The number of operations required by this procedure is proportional to c.s, while the method commonly used, which is due to F. Pollaczek, requires a number of operations proportional to c2. In (c): this improvement is obtained by means of the adoption of sorting by insertion and by saving certain intermediate results.
TL;DR: In the present article, various possible types of budget constraints are discussed as well as their impact on certain types of operations research models.
Abstract: In application of operations research models in the area of logistics, important constraints are typically encountered in the form of limitations on the funds appropriated. A supply system must be operated within whatever budgeted funds are appropriated by Congress. In the present article, various possible types of budget constraints are discussed as well as their impact on certain types of operations research models. The possible misallocations that can result from the setting of specific budgets designated for particular purposes are discussed. The confusion existing in the literature concerning the type of constraints that are actually encountered versus those that are imposed upon the mathematical models is pointed out as well as the typical inconsistency of budget constraints and steady-state models. The difference between a steady-state budget and a transition-phase budget is clarified.
TL;DR: In this paper, a method by which economic retention quantity of a material having a known shelf-life and following a normal probability law of obsolescence and deterioration has been calculated.
Abstract: This paper gives a method by which economic retention quantity of a material having a known shelf-life and following a normal probability law of obsolescence and deterioration has been calculated The relevant equations are solved by trial and error method