Showing papers in "Operations Research in 1964"

Scheduling of Vehicles from a Central Depot to a Number of Delivery Points

Abstract: The optimum routing of a fleet of trucks of varying capacities from a central depot to a number of delivery points may require a selection from a very large number of possible routes, if the number of delivery points is also large. This paper, after considering certain theoretical aspects of the problem, develops an iterative procedure that enables the rapid selection of an optimum or near-optimum route. It has been programmed for a digital computer but is also suitable for hand computation.

Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph

Abstract: The concepts of the "center" and the "median vertex" of a graph are generalized to the "absolute center" and the "absolute median" of a weighted graph a graph with weights attached to its vertices as well as to its branches. These results are used to find the optimum location of a "switching center" in a communication network and to locate the best place to build a "police station" in a highway system. It is shown that the optimum location of a switching center is always at a vertex of the communication network while the best location for the police station is not necessarily at an intersection. Procedures for finding these locations are given.

Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem

Abstract: We consider a machine with a single real variable x that describes its state. Jobs J1, ', JN are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state Bi. This means that Ji can be started only when x = Ai and, at the completion of the job, x = Bi. There is a cost, which may represent time or money, etc., for changing the machine state x so that the next job may start. The problem is to find the minimal cost sequence for the N jobs. This problem is a special case of the traveling salesman problem. We give a solution requiring only 0N2 simple steps. A solution is also provided for the bottleneck form of this traveling salesman problem under special cost assumptions. This solution permits a characterization of those directed graphs of a special class which possess Hamiltonian circuits.

On an Integer Program for a Delivery Problem

Abstract: An actual truck delivery problem of general applicability is stated as an integer program. The successful computational performance of Gomory's "cutting plane" algorithm for a set of nine particular problems is discussed. Finally, the structure of the general problem is shown to be a natural generalization of covering problems of graph theory.

Removing Arcs from a Network

Abstract: This paper proposes an algorithm for making a sensitivity analysis on a flow network, specifically, the problem of removing n arcs in a network such that the reduction in the maximum flow between an origin and destination node is maximized. Such an algorithm might be helpful in determining how sensitive a transportation system is to having its roads closed for repairs.

An Analytical Study of the PERT Assumptions

Abstract: This paper presents the results of a mathematical analysis of the standard assumptions used in PERT calculations. The objectives of this analysis were four-fold: 1 to pull together the mathematical aspects of the PERT model, 2 to suggest relevant analytic techniques, 3 to obtain an indication of the magnitude and direction of errors introduced by the assumptions, and 4 to suggest possible modifications and improvements in the model. The first part of the paper analyzes those assumptions relevant to the individual activities. Three possible sources of error are considered here: 1 the beta distribution assumption, 2 the standard deviation assumption and the approximation formula for the mean, and 3 the imprecise time estimates. The second part of the paper deals with the PERT network as a whole and analyzes the calculations underlying the project mean, standard deviation, and probability statements. The concept of relative criticalness is explored for the PERT stochastic model. Techniques for network reduction are outlined.

Optimum Control of a System of Oversaturated Intersections

Abstract: The problem of optimizing the control of two oversaturated traffic intersections is solved by using the semi-graphical methods employed in a previous paper for an isolated intersection. As in the case of a single intersection the optimum control involves values of the control variables that lie along edges of the control region, which in this case is defined by the permissible ranges of the green phase splits. An analytical formulation of the method using Pontryagin's control theory is also given.

On a Generalized M/G/1 Queuing Process in Which the First Customer of Each Busy Period Receives Exceptional Service

Abstract: The following generalization of the M/G/1 queue is considered. If a customer arrives when the server is busy, his service time has a distribution function, Gbx; while if he arrives when the server is idle, his service time has a different distribution function, Gex. Results are obtained that characterize the transient and asymptotic distributions of the queue size, waiting time, and waiting-plus-service time. These results are then applied to the special case of a queue with a single service time distribution function, but with additional independent delay times that have one distribution function for the customers arriving when the server is idle and another for the customers arriving when the server is busy.

Synchronizing Traffic Signals for Maximal Bandwidth

Abstract: Traffic signals can be synchronized so that a car, starting at one end of a street and traveling at preassigned speeds, can go to the other end without stopping for a red light. The portion of a signal cycle for which this is possible is called the bandwidth for that direction. Ordinarily the bandwidth in each direction is single, i.e., is not split into two or more intervals within a cycle. We solve two problems for this case: 1 Given an arbitrary number of signals along a street, a common signal period, the green and red times for each signal, and specified vehicle speeds in each direction between adjacent signals, synchronize the signals to produce bandwidths that are equal in each direction and as large as possible. 2 Adjust the synchronization to increase one bandwidth to some specified, feasible value and maintain the other as large as is then possible. The method of calculation has been programmed for a digital computer and results have been used to synchronize signals on a street in Cleveland.

Some Properties of Schedules for Large Projects with Limited Resources

Abstract: The Critical Path Method, PERT, and related techniques that make use of arrow diagrams for scheduling large projects implicitly assume that unlimited resources are available for assignment to project activities. When resources are limited, however, the usual concepts of "critical path" and "job slack" basic to these methods lose their normal meaning. Jobs may be delayed by the unavailability of resources as well as by technological orderings. A scheme for analyzing project schedules in the limited resource case is developed in this paper, with the intent of preserving for the project scheduler the operational utility of the slack concept. A new procedure for calculating slack values leads to the identification of a "critical sequence" of jobs in a certain class of project schedules-a notion analogous to that of a critical path in the unlimited resource case. Some properties of project schedules when resources are limited are explored and suggestions are made for utilizing the revised concept of slack in scheduling large projects.

Composition Principles for Synthesis of Optimal Multistage Processes

Abstract: Any multistage process may be constructed by a sequence of serial and/or parallel compositions. With certain restrictions on the form of the return composition function, optimal processes may be constructed in an efficient manner from optimal subprocesses. A number of simple sufficient conditions on the return functions are given and several approximation procedures are outlined.

Critical Path Analyses Via Chance Constrained and Stochastic Programming

Abstract: Chance-constrained programming methods are applied to examine some statistical properties of PERT networks. Using duality, the PERT method is shown to be equivalent to use of the crudest linear decision rule and the confidence or lack thereof in meeting constraints is explicitly presented. The distribution of completion times = Tintner's stochastic programming follows easily and may often be multimodal, contrasting with erroneous central limit theorem usages in the literature. Possible extensions and developments of PERT using more adequate chance-constrained models and techniques are suggested and will be presented elsewhere.

Development of M-Stage Decision Rule for Scheduling N Jobs Through M Machines

Abstract: This paper describes an algorithm that will yield an optimum sequence for n jobs requiring processing through M machines when no passing is allowed. Theoretical development utilizing combinatorial analysis precedes the algorithm and an example problem.

Multiple Choice Programming A Procedure for Linear Programming with Zero-One Variables

Abstract: A new procedure for mixed integer programming is presented. It is applicable where the integer variables must be either zero or one, and where the integer variables are divided into sets such that the variables in each set sum to unity. Numerous practical applications fit this formulation. A justification for the algorithm is given, but proof of convergence is not known. The procedure has solved all test problems so far, and this computational experience is discussed along with some applications.

Approximate Solutions to the Three-Machine Scheduling Problem

Abstract: This paper summarizes experimental results from applying several computational methods for solving the classic three-machine scheduling model. The basic mathematical problem is to select an optimal permutation of n items, where the objective function employed is the total amount of processing time elapsing for the completion of all n items on three machines. The methods tested are integer linear programming, ordinary linear programming with answers rounded to integers, a heuristic algorithm, and random sampling. Throughout the experimentation n = 6. A dual integer programming code is tested on six trial problems. Consideration is given to studying the phenomenon of input-form sensitivity. The results are not encouraging and confirm previous experimental evidence as to the strong effect of varying the input form. However, the addition of a powerful bound on the objective function has a significant beneficial effect on the convergence of the dual algorithm. A sampling study of the statistical characteristics of this model is made with 100 sets of data randomly generated. Frequency distributions of minimum and mean total processing time are tabulated and analyzed. On this same set of problems, a linear programming approach is examined for effectiveness in producing nearly optimal schedules. Such an approach tends to give answers that are better than would be obtained by a single permutation drawn at random, but the LP rounded solutions average about an 11 per cent increase over the optimal processing time. A heuristic algorithm based on Johnson's method is tested on 20 of the 100 cases and gives excellent results. Finally attention is turned to the method of randomly sampling the permutations, possibly applying a simple heuristic gradient method to improve each sampled schedule. A feature of this method is that elementary probability theory can be applied to yield corresponding statements about the accuracy guaranteed by the approach.

A Theory of the Diffusion of Traffic Platoons

Abstract: A theoretical investigation is made of the diffusion of a traffic platoon as it moves down a road. The basic assumption of the mathematical model is a kinematical one-that the speeds of the cars in the platoon are distributed normally. The parameters of this distribution are related to a diffusion constant that measures the spreading of the platoon. For certain assumed initial conditions, analytical and numerical solutions of the model are presented and applied to the problem of the coordination of two successive traffic lights.

Queues for a Vehicle-Actuated Traffic Light

Abstract: A vehicle-actuated traffic light controls two intersecting traffic streams keeping the light green for lane i, i = 1 or 2 until any existing queue has been discharged and a headway of duration at least βi, is detected in the subsequent arrivals. The object of this paper is to investigate how one should choose the βi so as to minimize the average delay per vehicle at the intersection. A model is considered in which the arrival headways are exponentially distributed random variables, departure headways have any specified distribution, and there is a random lost time for each switch of the traffic light. Formulas are derived for the moments of the cycle times, the average wait per vehicle, and the optimal βi.

Algorithm for Traffic Control

Abstract: A linear control algorithm is proposed for the computer operation of a signalized traffic intersection. A theoretical analysis of the algorithm under the assumption of constant vehicular arrival and departure rates shows that the control is stable under unsaturated conditions, and with proper choice of control constants usually gives a rapid convergence to a limit cycle that minimizes the average delay.

Inequalities for Stochastic Nonlinear Programming Problems

Abstract: Many actual situations can be represented in a realistic manner by the two-stage stochastic nonlinear programming problem MinxEminy[φx + ψy] subject to gx + hy ≧ b, where b is a random vector with a known distribution, and E denotes expectation taken with respect to the distribution of b. Madansky has obtained upper and lower bounds on the optimum solution to this two-stage problem for the completely linear case. In the present paper these results are extended, under appropriate convexity, concavity, and continuity conditions, to the two-stage nonlinear problem. In many cases of practical interest the calculation of these bounds will require only slightly more effort than two solutions of a deterministic problem of the same size, that is, a problem with a known constant value for the vector b. A small nonlinear numerical example illustrates the calculation of these bounds. For this example the bounds closely bracket the optimum solution to the two-stage problem.

The Use of Cumulative Sum Cusum Techniques for the Control of Routine Forecasts of Product Demand

Abstract: The cumulative sum technique has proved its value in process control and this paper discusses its use when applied to sales forecasting systems. The two main applications of cusum are i to detect when the pattern of customer demand changes, and ii to detect when a forecasting system is no longer adequate. The former use generally applies when demand is reasonably steady and the latter when changes are more frequent. The question of re-estimating the parameters used in the forecasting system is considered and, in particular, the problem of differentiating between a change in trend and a change in the mean level of demand is studied. Finally, it is shown how, the cusum control scheme can be simplified so that its operation is suitable for a computer.

Bargaining Behavior: A Comparison Between Mature Industrial Personnel and College Students

Abstract: The study reported here was conducted for two purposes. The primary purpose was to provide an experimental test of certain theoretical models of bargaining behavior. A second purpose, emphasized in this paper, was to study the bargaining behavior of experienced personnel in business as contrasted with the bargaining behavior of college students. The design of the study was dictated by the requirements of the theoretical model; the interest in comparing businessmen with students was subordinate, and did not in itself influence the fundamental features of the design. Thus, while the test of the hypothesis is formal and has some rigor, the comparison of the two subject populations is informal, and is suggestive and provocative, perhaps, rather than definitive. In general the groups were notably similar in regard to their bidding patterns and the equilibrium solutions they reached.

Present State of the Lanchester Theory of Combat

Abstract: The mathematical models and methods of analysis of opposing forces, stemming from the original Lanchester equations are summarized in terms of two broad categories: a deterministic models, b stochastic models. The present limitations of previous methods are used to identify topics that require attention in the future.

Analysis of Experiments on Single-Lane Bus Flow

Abstract: The results of a series of experiments carried out to determine the characteristics of single-lane bus flow are reported. Using pairs of buses theoretical "car-following" models of single-lane traffic flow, which had been shown by previous work to be valid for automobiles, are examined for this type of heavy vehicle and found to provide a good representation of the detailed manner in which one bus follows another. Flow characteristics of buses were also determined by using platoons of up to ten buses. Both of these approaches, which can be termed microscopic and macroscopic, respectively, indicated that a single lane of buses form a stable stream of traffic and have an optimal flow of about 1,450 buses/hour at a speed of 33 miles/ hour for an expressway type facility.

A Statistical Evaluation of Recurrent Medical Examinations

Abstract: In this paper we consider the efficiency of different policies for scheduling medical examinations in a search for diseases like cancer, which start without overt symptoms. We treat both periodic and random examinations and allow for imperfect diagnosis depending on how long the disease has been present. The results of our theory are then checked against a rough data fit to the incidence curve of cervical cancer.

Letter to the Editor-A Modification of Fulkerson's PERT Algorithm

Abstract: Fulkerson has described a method for approximating expected earliest start times in a pert network in which activity durations are discrete random variables. This letter describes an extension of Fulkerson's algorithm to include the continuous case and derives a computationally feasible form for this algorithm.

Approximate Circular and Noncircular Offset Probabilities of Hitting

Abstract: For equal or unequal delivery errors and an offset point of aim, the chance that the burst point of a warhead occurs within a given distance of a selected point of the target is approximated by reference to weighted noncentral chi-square distributions Offset circular and noncircular probabilities of hitting for the two and three dimensional cases may thus be approximated with a single, straight-forward and rather simple technique by the use of an approximate central chi-square distribution with fractional number of degrees of freedom or a transformation to approximate normality Computations of probabilities of hitting are illustrated by examples The approximations recommended appear to be of sufficient accuracy for many weapon systems evaluation problems We do not claim originality for the various parts of theory involved, but rather our purpose is to provide the weapon systems analyst with a reasonable and useful analytical procedure

A Probability Problem Arising in Reliability and Traffic Studies

Abstract: A system is alternately operative and inoperative; operative times and inoperative times are identically and independently distributed according to arbitrary distributions. Demands for system service occur at exponentially distributed time intervals, and persist for exponentially distributed times thereafter. The system "disappointment time" is discussed, where the latter is the time until either the system fails while a demand is being satisfied, or a demand occurs during an inoperative period. Approximations to the expected disappointment time and to the distribution of disappointment time are suggested. Numerical examples are given.

Random Walks, Fire Damage Amount and Other Paretian Risk Phenomena

Abstract: Being one of the oldest branches of operations research, actuarial science has accumulated a substantial store of knowledge about the risks associated with living. The present paper will discuss one such question. Although it is relative to a specific problem of fire casualty, it illustrates more generally why the Paretian distribution of incomes and fortunes should constitute "a source of anxiety for the risk theory of insurance." Very similar mechanisms apply in many other problems.

Some Probability Distributions Associated with Commuter Travel in a Homogeneous Circular City

Abstract: A commuter living outside a circular city arrives at a point on the circumference and is equally likely to wish to travel to every point of the interior. In the present paper we give the probability distribution for trip length, and the probability distribution for traffic weight at each point, under the assumption of six different routing systems.