Showing papers in "Operations Research in 1961"

A Proof for the Queuing Formula: L = λW

Abstract: In a queuing process, let 1/λ be the mean time between the arrivals of two consecutive units, L be the mean number of units in the system, and W be the mean time spent by a unit in the system. It is shown that, if the three means are finite and the corresponding stochastic processes strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = λW.

A Linear Programming Approach to the Cutting-Stock Problem

Abstract: The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approximate solution is being sought by linear programming. In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the problem. The technique enables one to compute always with a matrix which has no more columns than it has rows.

Nonlinear Follow-the-Leader Models of Traffic Flow

Abstract: A variety of nonlinear follow-the-leader models of traffic flow are discussed in the light of available observational and experimental data. Emphasis is placed on steady-state flow equations. Some trends regarding the advantages of certain follow-the-leader functionals over others are established. However, it is found from extensive correlation studies that more data are needed before one can establish the unequivocal superiority of one particular model. A discussion is given of some ideas concerning the possible reasons for the existence of a bimodal flow versus concentration curve especially for multilane highways.

Nonlinear Effects in the Dynamics of Car Following

Abstract: It is assumed that the velocity of a car at time t is some nonlinear function of the spacial headway at time t-Δ, so the equations of motion for a sequence of cars consists of a set of differential-difference equations. There is a special family of velocity-headway relations that agrees well with experimental data for steady flow, and that also gives differential equations which for Δ = 0 can be solved explicitly. Some exact solutions of these equations show that a small amplitude disturbance propagates through a series of cars in the manner described by linear theories except that the dependence of the wave velocity on the car velocity causes an accleration wave to spread as it propagates and a deceleration wave to form a stable shock. These conclusions are then shown to hold for quite general types of velocity-headway relations, and to yield a theory that in certain limiting cases gives all the results of the linear car-following theories and in other cases all the features of the nonlinear continuum theories, plus a detailed picture of the shock structure.

Critical-Path Planning and Scheduling: Mathematical Basis

Abstract: This paper is concerned with establishing the mathematical basis of the Critical-Path Method---a new tool for planning, scheduling, and coordinating complex engineering-type projects. The essential ingredient of the technique is a mathematical model that incorporates sequence information, durations, and costs for each component of the project. It is a special parametric linear program that, via the primal-dual algorithm, may be solved efficiently by network flow methods. Analysis of the solutions of the model enables operating personnel to answer questions concerning labor needs, budget requirements, procurement and design limitations, the effects of delays, and communication difficulties.

Parallel Sequencing and Assembly Line Problems

Abstract: This paper deals with a new sequencing problem in which n jobs with ordering restrictions have to be done by men of equal ability. Assume every man can do any of the n jobs. The two questions considered in this paper are 1 How to arrange a schedule that requires the minimum number of men so that all jobs are completed within a prescribed time T, and 2 if m men are available, arrange a schedule that completes all jobs at the earliest time.

The Greatest of a Finite Set of Random Variables

Abstract: The variables ξ1,..., ξn have a joint normal distribution. We are concerned with the calculation or approximation of maxξ1,..., ξn. Current analyses and tables handle the case in which the ξi are independently distributed with common expected values and common variances. This paper presents formulas and tables for the most general case with n = 2. When n > 2, the problem becomes cumbersome. This paper presents formulas and tables that permit approximations to the moments in case n > 2. The moments are approximated by iteration of a three-parameter computation or, alternatively, through successive use of a three-parameter table, which is given. Recent applications of the theory are described.

Car-Following and Steady-State Theory for Noncongested Traffic

Abstract: Car-following and continuum theories have been shown by other investigators to describe a number of aspects of the complex phenomenon of vehicular traffic flow in both quantitative and qualitative terms. It is the purpose of this paper to suggest a variation in these theories in an effort to make them more accurate for traffic densities less than optimum. The theory as modified for noncongested traffic may help to describe quantitatively the sudden change of state occurring in a traffic stream going from a relatively free-flowing condition to a crawling stop-and-go condition and back again.

The Created Response Surface Technique for Optimizing Nonlinear, Restrained Systems

Abstract: A different approach to the general problem of optimization of nonlinear, restrained systems is presented. This approach, called the Created Response Surface Technique, is based, essentially, on steepest ascents up a succession of created response surfaces within the solution space. The most important characteristic of the approach is that it automatically avoids restraint boundary violations during the optimization. This article discusses application to fully developed steady-state mathematical models. By the use of appropriate experimental designs, the method could be used for the characterization and optimization of mathematically incomplete, nonlinear, restrained systems. In addition, it seems possible that the Created Response Surface concept could be applied to attain dynamic process control in certain nonlinear, restrained control situations. However, the mathematical implications of the new approach must be studied further.

The Fundamental Theorem of Exponential Smoothing

Abstract: Exponential smoothing is a formalization of the familiar learning process, which is a practical basis for statistical forecasting. Higher orders of smoothing are defined by the operator Snt(x) = αSn−1t(x) + (1 − α) Snt−1(x), where S0t(x) = xt, 0 < α < 1. If one assumes that the time series of observations {xt} is of the form xt = nt + ∑i=Ni=0 aiti where nt is a sample from some error population, then least squares estimates of the coefficients a, can be obtained from linear combinations of the operators S, S2, …, SN+1. Explicit forms of the forecasting equations are given for N = 0, 1, and 2. This result makes it practical to use higher order polynomials as forecasting models, since the smoothing computations are very simple, and only a minimum of historical statistics need be retained in the file from one forecast to the next.

Reliability Analysis of a One-Unit System

Abstract: This paper considers an electronic system that upon failure is repaired Only two states---the “on” state and the “off” state---are distinguished. The time-to-failure and the time-to-repair are random variables with general distribution functions. The purpose of the paper is to make available, in single unified treatment, mathematical information that is relevant for a reliability analysis of a one-unit system. The following questions are answered a What is the probability that the system will be on at any given time t? b What is the probability that it will be on t hours or more during a given time interval 0, T? c What is the expected fractional amount of time that it will be on during 0, T? d What is the probability distribution of the number of failures during a given time interval? e What is the expected number of failures during a given time interval?. In particular, it is noted that “down” time is approximately normally distributed for large time intervals. The special cases of exponential failure with exponential and constant repair are given as examples.

Optimum Distribution of Effort: An Extension of the Koopman Basic Theory

Abstract: The fundamental problem of search theory is to allocate a given amount of search effort in such a way as to maximize the over-all probability of discovering an object located in a given space. The problem has already been solved under the assumption that the probability of detecting the object is a negative exponential function of the search effort density, this assumption seems to be fairly realistic in many military applications, but has definite drawbacks in others. In this paper a method is provided for solving the problem in the general case, where no assumption is made concerning the form of the detection probability function. A theorem is derived that gives a general relation governing the optimal solution.

Preemptive Resume Priority Queue

Abstract: The queue length probability generating function for a preemptive resume priority queue characterized by Poisson arrivals and general service time distributions has been obtained by using the “supplemetary vanable method”. The preempted item follows the “resume” rule so that upon re-entry the service on the nonpriority unit is started at the point where it was interrupted when preemption occurred. Apart from the steady-state solution, the Laplace transform of the time dependent probability generating function and the length of busy periods has also been obtained.

Comments on the Highway-Crossing Problem

Abstract: A discussion is given of the gap acceptance problem confronting a driver at a stop sign who is attempting to cross a single lane of traffic. This problem is discussed in the light of a gap acceptance probability that is taken in the form αt = 0 when t T, rather than a step function as has been considered in the past. Some experimental information is discussed concerning the values of the parameters λ and T in such situations. This information coupled with theoretical considerations allows one to estimate quantities such as mean delay time, probability of zero delay, and highway transparency. Comments are made concerning the influence of car performance on road crossing and on entering highways.

Driver Response to the Amber Phase of Traffic Signals

Abstract: Observations of motorist response to the amber phase of traffic signals obtained at five intersections, representing three speed zones, are presented. The data from these observations give an estimate of the probability of stopping for vehicles as a function of their distance from the intersection at the onset of the amber phase of the traffic signal. The results lend no support to a popular hypothesis, i.e., that drivers tend to “take advantage” of a long amber phase by treating it as an extension of the green. The results of the study are compared with other investigations pertaining to amber phase lengths and implications of this work for the design of amber phases is discussed.

Letter to the Editor---The Maximum Capacity Route Problem

Abstract: We consider the problem of finding maximum capacity routes between all pairs of nodes in a network.

Letter to the Editor---Comments on a Paper by Bowman

Abstract: In the paper, “Assembly-Line Balancing by Linear Programming,” that appeared in Operations Research8, 385--389 1960, E. H. Bowman presented two linear program formulations for the assembly-line balancing problem. These two formulations are referred to as the First Linear Program and the Second Linear Program, respectively. The following are comments about, and a reformulation of, the First Linear Program.

Production Smoothing and Inventory Control

Abstract: Let production of a product be continuous and subject to the following costs the direct cost of production is proportional to the rate of output, the cost of changing the rate is proportional to the size of the change, with (possibly) different coefficients for upward and downward shifts, the cost of storage is proportional to the stock level, and there is a proportional penalty for backlogs. No orders are lost demands for the product arise at random intervals and the number of units demanded is a random variable with known distribution. The problem is to find the optimal policy of adjusting the rate of production to stock. It is shown that there exists a zone of control depending on the stock level such that no adjustment is made as long as the rate of production falls inside this zone and the rate is brought to its nearest endpoint when it has fallen outside the zone. A discrete version of this problem has been programmed on the IBM 650 Results of some computations are presented.

Letter to the Editor—The kth Best Route Through a Network

Abstract: Determining the shortest route through a network is a problem for which many solutions have appeared. Not so well known, but of equal importance, is the generalized problem of determining the kth best route through a network. The purpose of this Letter is to present a formulation of the generalized problem, to reference several known solutions, and to describe a new simple method of solution which possesses some advantages.

Customer Behavior As a Markov Process

Abstract: Simple Markov processes appear to provide a fruitful conceptual basis for study of customer populations in the face of marketing activity. The purpose of this paper is to summarize characteristics and some implications and applications of the Markov model. Characteristic behavior of the model is compared with published data, and areas for further analytical and experimental effort are indicated.

The Probability Law of the Busy Period for Two Types of Queuing Processes

Abstract: A simple combinatorial method is given for the determination of the joint distribution of the length of the busy period and the number of customers served during this period for two types of single-Berver queuing processes. 1 Customers arrive at a counter according to a Poisson process and the service times have a general distribution, 2 customers arrive at a counter according to a recurrent process and the service times have an exponential distribution.

The Distribution of Urban Population and an Application to a Servicing Problem

Abstract: The growth of the urban component of a nation and the shift in population from rural to urban areas in the course of industrial development are processes that show both regularity and predictability. Consequently, it is possible to describe the concentration of population in terms of population density and area by comparatively simple expressions. In this paper expressions are examined for the distribution of people within cities, for the distribution of city sizes within the United States, and for the total population contained within boundaries described in various ways. These generalizations have many applications, both in industrial planning and in civil defense and other military considerations. The paper shows how it is possible to make approximate estimates of the total population serviced by operations of specified types as a function of the cost structure of the operation.

Letter to the Editor—A Doubled-Ended Queuing Problem of Kendall

Abstract: The distribution of the difference between two independent Poisson variables has been obtained by Irwin (Irwin, J. O. 1937. The frequency-distribution of the difference between two independent variates following the same Poisson distribution. J. Roy. Stat. Soc. 100(Part 3) 415–416.), when the means are equal, and by Skellam (Skellam, J. G. 1946. The frequency-distribution of the difference between two Poisson variates belonging to different populations. J. Roy. Stat. Soc. 109(Part 3) 296.), when the means are unequal. The purpose of this note is to point out that the Irwin-Skellam distribution still applies when α and β are time-dependent.

The Face of Bargaining

Abstract: The nature of a bargaining situation is discussed. Two general propositions about the conditions affecting the likelihood of a bargaining agreement are presented. The effects of the availability of threat and of permissive and compulsory communication upon interpersonal bargaining are investigated experimentally in a two-person bargaining game. The major findings are 1 Under conditions of bilateral i.e., available to both players threat, players have most difficulty in reaching agreements, unilateral available to one player threat produces somewhat less difficulty, however, only under conditions of no threat are subjects able to maximize their payoffs. 2 Under conditions of permissive communication, subjects' performance does not differ from that found when the opportunity to communicate is not present. 3 Under conditions of compulsory communication, an improvement in performance is noted in the unilateral threat condition only. These findings are discussed with respect to the balance of cooperative and competitive motivational forces present and the player's need to maintain “face” in interpersonal encounters.

On a Problem of Preemptive Priority Queuing

Abstract: A priority queuing system is considered possessing the following properties. There exist two populations of potential customers. One population of regular customers is infinitely large and its arrival characteristics are well-represented by a Poisson stream of constant intensity. The other population consists of a relatively small number of extraordinary customers. The arrival of each of these extraordinary customers is governed by an individual constant Poisson intensity. Service is given at a single station and regular customers on service are preempted by any arriving extraordinary customer. The expected value, the variance, etc., of the queue of ordinary customers is derived. Various relations given by previous authors are shown to represent special cases of the model discussed in this paper.

Some problems in the theory of intra-urban location

Abstract: A scientific understanding of urban-metropolitan structure and function is becoming increasingly important for metropolitan planning decisions and for governmental policy development, especially in the field of transportation. There is a strong coincidence between the needs of policy makers for adequate information about the implications of alternatives, and the interest of scientists in explaining the development of cities. Theories of urban form and intra-urban location are examined in some detail. It is concluded that existing theories of urban form are largely descriptive and taxonomic, while existing theories of locational behavior may be inadequate to deal with the inflexibilities and nonlinearities that arise in the urban framework. An adequate explanation and theory of metropolitan growth and structure depends on the accumulation of considerably more knowledge as to the behavior of locational decision makers, and the ways in which metropolitan policy impinges on their decisions. Some of the proble...

Double Queues and Impatient Customers with an Application to Inventory Theory

Abstract: This paper starts by observing that any queuing system can be viewed as two symmetric queues, one of customers in line for service, and one of idle clerks, awaiting customers. Usually at least one of the queues the idle clerks has a finite limit, but examples, such as passengers and taxis at a taxi stand, exist where both queues are unlimited. In such cases, if arrival and service rates are independent of queue length, at least one of the queues is unstable and grows indefinitely as time passes. In practice there has to be some factor that limits queue length, and a possible mechanism is to assume that both passengers and taxis become impatient and will leave if, after some fixed amount of time, they have not been “serviced.” The equations governing such a system are developed and steady-state solutions are found. It is shown that steady-state solutions always exist for both queues, the means and variances which are finite are given. In the last part of the paper an application to inventory theory is suggested. Here the backlog of orders corresponds to the queue of customers, and items in stock to the idle clerks. It is assumed that orders not filled in a certain time are cancelled and that items are perishable and cannot be sold if stocked too long.

Activity Subdivision and PERT Probability Statements

Abstract: The Program Evaluation and Review Technique PERT has been developed and implemented as a technique for planning and evaluating the progress of research and development programs. Probability theory is used in PERT to calculate a measure of the uncertainty in meeting scheduled dates. These computed probabilities can then be used by managers to aid their judgment in evaluating the validity of schedules for accomplishing program objectives. This paper investigates the effect of subdividing activities in PERT flow diagrams on computed probabilities for meeting scheduled dates. It is shown that the mechanics of activity subdivision can increase or decrease the computed probabilities, depending upon whether the scheduled date for an event is later or earlier in time than the expected date. Management personnel who are using PERT probabilities to aid their judgment in evaluating schedules should be aware of the potential biases inherent in the computation of probabilities.

Distribution of Gaps and Blocks in a Traffic Stream

Abstract: This paper studies some of the theoretical questions of large openings or gaps in a single stream of traffic. A gap in the traffic stream is defined as a headway between vehicles greater than or equal to some minimum size—say x. Several authors have studied the probability distribution of the wait that a randomly located observer must endure before he finds a gap. This paper, while briefly reviewing the solutions of this well-known problem, is primarily concerned with expressions for (i) the distribution of gap sizes, (ii) the distribution of spacings between vehicles and gaps, (iii) the mean and variance of intervehicle and intergap spacings, (iv) the stationary flow rates of gaps, and (v) the distribution of blocked and unblocked periods. It is assumed that the origin of measurements may be located (i) with the passing of a vehicle, (ii) at the beginning of a gap, or (iii) at random. It is also assumed that the distribution of intervehicle spacings are independently, but identically, distributed random ...