Showing papers in "Operations Research in 1956"

Shock Waves on the Highway

Abstract: A simple theory of traffic flow is developed by replacing individual vehicles with a continuous “fluid” density and applying an empirical relation between speed and density. Characteristic features of the resulting theory are a simple “graph-shearing” process for following the development of traffic waves in time and the frequent appearance of shock waves. The effect of a traffic signal on traffic streams is studied and found to exhibit a threshold effect wherein the disturbances are minor for light traffic but suddenly build to large values when a critical density is exceeded.

The Output of a Queuing System

Abstract: For a queuing system with Poisson input, a single waiting line without defections, and identically distributed independent negative exponential service times, the equilibrium distribution of the number of service completions in an arbitrary time interval is shown to be the same as the input distribution, for any number of servers. This result has applications in problems of tandem queuing. The essence of the proof is the demonstration of the independence of an interdeparture interval and the state of the system at the end of the interval.

The Traveling-Salesman Problem

Abstract: The traveling-salesman problem is that of finding a permutation P = (1 i2 i3 … in) of the integers from 1 through n that minimizes the quantity \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a_{1i_2} + a_{i_2i_3} + a_{i_3i_4} + \cdots + a_{i_n1},$$ \end{document} where the aαβ are a given set of real numbers. More accurately, since there are only (n − 1)′ possibilities to consider, the problem is to find an efficient method for choosing a minimizing permutation. This problem was posed, in 1934, by Hassler Whitney in a seminar talk at Princeton University. There are as yet no acceptable computational methods, and surprisingly few mathematical results relative to the problem.

A Competitive-Bidding Strategy

Abstract: A method is presented that determines optimum bids in a competitive-bidding situation where each competitor submits one closed bid. The number of bidders may be large or may be unknown. This method makes use of the previous “bidding patterns” of all possible opposition bidders and in the case where the bidding is on contracts, the estimated probability distribution of the cost of fulfilling the contract. The case where a number of bids are to be submitted simultaneously is also discussed.

Sequential Arrays of Waiting Lines

Abstract: The problem considered in this paper is that in which a sequence of service operations must be performed on the units to be serviced. Poisson arrivals and exponential service times are assumed. Four particular cases of service facilities in series are treated, involving infinite storage space between stages, no storage space between stages, finite storage space between stages, and the case of the unpaced belt-production line. A comparison is given for two stages, and the maximum possible utilization is obtained for all cases discussed.

Production Scheduling by the Transportation Method of Linear Programming

Abstract: With fluctuating sales, a manufacturer must have fluctuating production, or fluctuating inventory, or both. Penalties are associated with either type of fluctuation. Several papers place this problem into a conventional linear-programming framework. This paper suggests that the same problem may be placed into a transportation-method framework and, further, that many transportation problems may be extended to include multiple time periods where this is meaningful. A generalized scheduling problem is placed here into the standard form of the transportation table.

The Theory of Search. II. Target Detection

Abstract: “Kinematic bases,” the first paper of this series, discussed the geometric and kinematic factors involved in search---the positions, motions, and contacts of observers and targets. Probability was introduced only in assuming specific relative positions for the observer and target. The present paper discusses the uncertainties inherent in the act of detection under various specific conditions of contact. In the course of the discussion a body of methods for applying probability to problems of detection is developed. It must be emphasized, however, that these methods are conditioned by the particular situation in the case of visual detection because the different elementary acts of looking or “glimpses” are essentially independent trials. The reason for the distinction follows.

The Theory of Search. I. Kinematic Bases

Abstract: In the conventional search situation, there are three general features. I The kinematic bases, involving the positions, geometrical configurations, and motions in the searchers and targets, with particular reference to the statistics of their contacts and the probabilities of their reaching various specified relative positions. II The probabilistic behavior of the instrument of detection (eye, radar, sonar, etc) when making a given passage relative to the target. III The over-all result—the probability of contact under general stated conditions, along with the possibility of optimizing the results by improving the methods of directing the search. The present paper is devoted to the first feature, and is intended to provide the kinematic bases of the theory. It studies fairly simple cases of relative motion and draws inspiration from the kinetic theory of matter. The second paper, entitled Target Detection, will deal with the second feature. It is based on the methods of survival probabilities familiar in ...

The Development of Operations Research as a Science

Abstract: The development (rather than the history) of operations research as a science consists of the development of its methods, concepts, and techniques. Operations research is neither a method nor a technique, it is or is becoming a science and as such is defined by a combination of the phenomena it studies, its methods, and its techniques. Invited paper given at the Eighth National Meeting of the Society in Ottawa, Canada, on January 9, 1956.

Machine Repair as a Priority Waiting-Line Problem

Abstract: A paper by Cobham (Cobham, A. 1954. J. Opns. Res. Soc. Am. 2 70.) in this Journal, dealing with the assignment of priorities in waiting-line problems, is taken as the basis for treatment of a particular variety of machine-repair problem in which shortest jobs, rather than first arrivals, receive highest priority. Cobham's results for the single channel case are found to be easily applicable to this type of priority assignment. These results take a particularly simple form when the priority-labeling index is allowed to assume continuous, rather than discrete, values. They apply to any case in which a one-channel facility repairs random failures from a very large population of machines.

The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times

Abstract: The single-channel queuing equations considered in this paper are characterized by a Poisson-distributed arrival rate and a general class of holding-time distributions. General solutions of the equations are obtained for the case in which the traffic intensity i is a continuous function of time and possesses continuous derivatives of all orders. The following particular cases are considered in detail a i constant and the holding-time distribution Pearson type-III in this case the general solution is obtained in closed form in terms of a newly introduced function Inkz, many of the properties of which are derived in the Appendix, b i directly proportional to time and the holding time exponentially distributed.

Estimation of Reliability Growth in a Complex System with a Poisson-Type Failure

Abstract: Many modern complex systems, especially those involving numerous electronic components, are subject to failures with operating time that follow a Poisson-type distribution. In the course of development of such a system engineering changes are made, with the object of improving system performance or reliability. As a result, no two successive assemblies are alike in all components, although only a few changes may have been made across assemblies. It is nevertheless desirable to know at the earliest possible time in the development program 1 the level of reliability of the system, 2 whether reliability is increasing, and if so, how rapidly, and 3 the expected reliability at the end of the projected development program. In this paper, methods are developed for fitting typical reliability growth curves to experimental data by the method of maximum likelihood. Mathematical models are also developed for an idealized engineering development process to provide a basis for the selection of a suitable growth curve.

A Study of Sales Operations

Abstract: In a case study an experiment is executed to determine customer responses to various levels of effort. The experiment shows how much effort should be assigned to get customers, and how much to hold customers. The effort required, and the probabilities of success, determine the proper allocation of accounts to salesmen. By referring to the graph of customer size distribution, sizes of sales forces resulting from various allowed percentages for sales expense are derived. In addition, one method is presented for evaluating the expected profits from improved customer service.

Allocation of Sales Effort in the Lamp Division of the General Electric Company

Abstract: One of the authors of this paper was requested by the editor of Operations Research to prepare a more technical presentation of a case study he had published in 1955 Russell L. Ackoff, “The Allocation of Sales Effort,” Proceedings of the Conference on What Is Operations Research Accomplishing in Industry? Case Institute of Technology, April, 1955.. This request came shortly after the Conference on Operations-Research Education at The Johns Hopkins University in March of 1956. At this conference the need for detailed case histories, rather than highly rational reconstructions of OR projects, was recognized as serious in the field of education. Consequently, at the risk of boring professional analysts, but with the hope of assisting the novice, a detailed case history is presented here. It records the false starts, the blind alleys, and the tentativeness of the conclusions eventually reached. Not even those who worked on the project would do it over in the same way, knowing what they now know. The project raised more questions than it put to rest, but this, we believe, is the essence of scientific progress. Yet it should be pointed out, by way of anticipation, that even the tentative results were better than anything then available to management as a basis for decision making in the area involved. The results were used as recommended and the outcome that was forecast has occurred in the eighteen months during which they have been used.

Fallacies in Operations Research

Abstract: Address presented to the Fourth Annual Meeting of the Operations Research Society of America on May 10, 1956. Prepared comments to this address were made by Charles Hitch and Glen Camp. Hitch's comments are printed at the end of this paper.

The Design of Military Supply Tables for Spare Parts

Abstract: This paper presents a technique for designing supply tables of spare parts which differs appreciably from those methods historically used in the military services. For a given size of table, the technique seeks that supply table which will minimize the expected number of shortages to be encountered during a given period of supply activity. A supply table is a pre-assembled group of spare parts to be used for a specified period without outside support. The proposed method makes use of probability distributions to represent future demand for spare parts, and considers the per-unit size of each part explicitly in designing the table. The computations give the quantity of each part to be included in the table, consistent with the specified size of the table. It also discusses how to compute follow-on supply tables. These are also pre-assembled, and are intended to support supply activity for a period following the use of the initial supply tables. The technique given for follow-on tables is very similar to th...

Inventory Management of Slow-Moving Parts

Abstract: We present in this paper a simple, fast, and accurate method for determining optimal stock levels for slow-moving parts which we worked out in March 1955. Independently of our work, Whitin and Youngs studied the same problem and published their results in a recent paper. “A Method for Calculating Optimal Inventory Levels and Delivery Time” in the Naval Research Logistics Quarterly, September 1955. While both studies are basically the same, our solution seems to be the more practical. Essential differences are the use of a second criterion not so complete as the “costs,” but more easily accepted by managers---the customer's satisfaction, and the graphical presentation, which allows the immediate determination of the optimal level. For practical reasons we assumed that the distribution of demand is Poisson, conceptually the distribution may be arbitrary. The method is restricted to slow-moving, non-seasonal parts with fairly long replenishment periods.

A two-phase method for the simplex tableau

Abstract: A version of a two-phase simplex technique is given for manually solving those linear-programming problems in which artificial vectors are introduced and subsequently driven out. The first phase of the method determines feasibility, provided it exists, the second phase, which follows, searches for optimality.

A Graphical Solution of the Transportation Problem

Abstract: This paper presents a simple graphical technique applicable to a wide variety of transportation problems involving a large number of production centers and market areas. The assumption is made that transportation costs are monotonically increasing as one moves out from a given production center, but the rate of increase need not be constant or uniform in all directions.

Statistics and Operations Research

Abstract: Usually when one thinks of applied mathematics, one thinks of applications in the physical sciences—physics or chemistry—or in the various branches of engineering. But there are other fields of applications, more closely related to the life sciences, where mathematics is also being applied. One small part of this new field of application is in what is coming to be called operations research. Paper presented to the meeting of the International Institute of Statistics, at Rio de Janeiro, Brazil, August, 1955.

Waiting-Line Theory as a Management Tool

Abstract: According to recent figures published by the U.S. Department of Commerce, approximately $28 billion were invested in new facilities and equipment by industries in this country during the year 1955. This large expenditure undoubtedly reflects the need for increased output in this country's rapidly expanding economy. But how have decisions been made in arriving at the amount and distribution of this additional productive effort? What proportion of this vast sum can be attributed to decisions based on intuition, judgment, and hunches? To what extent have expenditures been based upon sound scientific analysis? A great deal of inefficiency and waste of resources could certainly be avoided if management were to employ the many scientific techniques and tools that are currently available. In particular, waiting-line theory can be used to determine how many facilities one ought to have in a process or system for which output requirements can be specified. However, the theory has general applicability to a wide va...

The Queuiac: An Electromechanical Analog for the Simulation of Waiting-Line Problems

Abstract: Of interest to the Operations Research Office for several years has been the problem of improving the efficiency of Army communications systems. Involved in these studies, besides obvious problems of communications equipments, are problems of organization and human handling operations. To understand and offer solutions to these latter problems, electromechanical simulation equipment has been developed that displays the network of handling operations and provides estimates of congestion at any or all points in the system. In the analog each node consists of a display and register unit upon which columns of lights represent items in a waiting line. Distributions of arrival times of input items and servicing times are programmed on standard teletype tape fed to the analog by teletype transmitters. The register unit at each point of the network displays the queue, computes state occupancy times, and provides estimates of state probabilities. Continuous records of the system states are maintained by pen recorders so that individual delay times and empirical estimates of their distributions may be obtained. The register units may be connected in parallel or tandem, representing various sequential or parallel operations. Available as important queue disciplines are the handling of multiple inputs on a priority basis and the automatic adjustment of servicing rates according to transient loads.

A Method of Computing Survival Probabilities of Several Targets Versus Several Weapons

Abstract: Infantry squad weapons are evaluated against the background of a rifle squad attacking a position defended by riflemen. The behavior of Attackers and Defenders is characterized by simultaneous action within each squad. The Defenders, acting independently of one another, each choose at random an Attacker as target and all Defenders fire simultaneously. The process is repeated in a sequence of volleys. As independent variables there are the original number, T, of Attackers, the number of Defenders, the single-shot hit probability, and the number of volleys fired. After each volley there is defined a set of probabilities of the survival of 0, 1,..., T Attackers. Taken as a sequence, these sets form a Markov chain, which is the basis for weapons comparison. All results of this paper are applications of the Markov theory. This technique is most useful for squads with a maximum of 15 men, some numerical examples are included. There are suggested ways in which tactical features can be reflected in the independent variables.

A Monte Carlo Model of Lanchester's Square Law

Abstract: F. W. Lanchester published, in 1916, the book entitled Aircraft in Warfare, the Dawn of the Fourth Arm (Lanchester, F. W. 1916. Aircraft in Warfare, the Dawn of the Fourth Arm. Constable, London.). In the course of the development of this work. Lanchester suggested equations, based upon probability analyses, that express various types of tactical situations. In the simplest form as proposed by Lanchester the theory, theorem, or law for a multiple-element engagement states “The over-all effectiveness of a given force equals the average effectiveness of the individual units multiplied by the square of the number of units engaged”. By multiple-element engagement is meant a situation in which each element of one force can fire on any or all elements of the other, as distinguished from a series of individual duels such as were experienced in early sword to shield warfare.

On Operations Analysis

Abstract: Among the staff elements of organization which have multiplied in recent years are those engaged in the analysis of operations. Their tasks are described variously as operations research, industrial engineering, operations analysis, operations evaluation, etc., terms which are not quite synonymous nor altogether acceptable to jealous practitioners in one or another of these fields. Here, however, they will mean the same thing staff engaged in the analysis of operations with the mission of improving performance. Their goal is research rather than service. This paper was presented as an invited address to the session on Operations Research and the Firm at the Fourth Annual Meeting of the Society, Washington, D.C., May 10, 1956.

Weapon-System Cost Analysis

Abstract: A method of estimating cost inputs for weapon-systems analyses is discussed. The major characteristics of the method are 1 Total cost concept, which provides for accumulation of “support” as well as “direct” costs to the tactical unit of capability e.g., wing or squadron in the case of the Air Force 2 Distinction between investment and annual operating costs 3 Incremental costing, which provides for taking into account “inheritance value” of existing assets when a new weapon system is phased in to replace an existing one 4 Time phasing of weapon systems and the costs related thereto 5 Emphasis on consistency in costing alternative weapon proposals, which, for purposes of systems analysis, is perhaps more important than striving for a high degree of accuracy. In addition to the application to formal weapon-systems analysis, use of the method as a tool in military planning and programming of future weapon systems is also discussed.

Approaches to operational problems in street and highway traffic-a review*

Abstract: Recently developed techniques for traffic operational changes involving traffic signal control equipment are discussed. Present practice is very largely based on intuition and qualitative analysis. Quantitative analysis is limited because a general theory of traffic flow has not yet been developed Studies aimed at the eventual formulation of a theory of traffic flow are being conducted at the Road Research Laboratory (British), Brown University, the University of Michigan, and the University of California. In combination with the flow theory the greatest promise appears to be in the simulation approach, some aspects of which are discussed.

A Brief History of the Development of Canadian Military Operational Research

Abstract: Invited paper presented to the session on Historical Development of Operations Research at the Ottawa Meeting of the Society on January 4, 1956.

Use of Transfer Functions for Company Planning

Abstract: In the aeronautical industry as in other manufacturing industries, one of the problems associated with planning an engineering program is to determine the amount and timing of engineering effort needed to support the sales forecast. In an effort to understand this problem better, a study was made of the relation of engineering effort to sales. It was found that this problem was analogous to that of a servomechanism system for which mathematical techniques have been developed and applied successfully. The study resulted in the development of a “transfer-function” formula which has proved to be a useful tool for measuring the relation between engineering effort and sales. The method employed to obtain the transfer function and the results obtained by its use are presented herein.