Showing papers in "Operations Research in 1978"
TL;DR: General conditions under which a collection of optimization problems, with the objective function and the constraint set depending on a parameter, has optimal solutions that are an isotone function of the parameter are given.
Abstract: This paper gives general conditions under which a collection of optimization problems, with the objective function and the constraint set depending on a parameter, has optimal solutions that are an isotone function of the parameter. Relating to this, we present a theory that explores and elaborates on the problem of minimizing a submodular function on a lattice.
1,393 citations
TL;DR: Inner approximation algorithms have had two major roles in the mathematical programming literature: their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle and recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs.
Abstract: Inner approximation algorithms have had two major roles in the mathematical programming literature. Their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle. However, recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs. Avriel and Williams' Avriel, M., A. C. Williams. 1970. Complementary geometric programming. SIAM J. Appl. Math.19 125-141. complementary geometric programming algorithm, Duffin and Peterson's Duffin, R. J., E. L. Peterson. 1972. Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J.22 531-550. reversed geometric programming algorithms, Reklaitis and Wilde's Reklaitis, G. V., D. J. Wilde. 1974. Geometric programming via a primal auxiliary problem. AIIE Trans.6 308-317. primal reversed geometric programming algorithm, and Bitran and Novaes' Bitran, G. R., A. G. Novaes. 1973. Linear programming with a fractional objective function. Opns. Res.21 22-29. linear fractional programming algorithm are all examples of this class of inner approximation algorithms. A sequence of approximating convex programs are solved in each of these algorithms. Rosen's Rosen, J. B. 1966. Iterative solution of nonlinear optimal control problems. SIAM J. Control4 223-244. inner approximation algorithm is a special case of the general inner approximation algorithm presented in this note.
957 citations
TL;DR: This approach has obtained and verified optimal solutions to all the Kuehn-Hamburger location problems in well under 0.1 seconds each on an IBM 360/91 computer, with no branching required.
Abstract: We develop and test a method for the uncapacitated facility location problem that is based on a linear programming dual formation. A simple ascent and adjustment procedure frequently produces optimal dual solutions, which in turn often correspond directly to optimal integer primal solutions. If not, a branch-and-bound procedure completes the solution process. This approach has obtained and verified optimal solutions to all the Kuehn-Hamburger location problems in well under 0.1 seconds each on an IBM 360/91 computer, with no branching required. Computational tests on problems with as many as 100 potential facility locations provide evidence that this approach is superior to several other methods.
914 citations
TL;DR: The paper develops easily implemented approximations to stationary policies based on finitely transient policies and shows that the concave hull of an approximation can be included in the well-known Howard policy improvement algorithm with subsequent convergence.
Abstract: This paper treats the discounted cost, optimal control problem for Markov processes with incomplete state information. The optimization approach for these partially observable Markov processes is a generalization of the well-known policy iteration technique for finding optimal stationary policies for completely observable Markov processes. The state space for the problem is the space of state occupancy probability distributions the unit simplex. The development of the algorithm introduces several new ideas, including the class of finitely transient policies, which are shown to possess piecewise linear cost functions. The paper develops easily implemented approximations to stationary policies based on these finitely transient policies and shows that the concave hull of an approximation can be included in the well-known Howard policy improvement algorithm with subsequent convergence. The paper closes with a detailed example illustrating the application of the algorithm to the two-state partially observable Markov process.
620 citations
TL;DR: This work studies the problem of scheduling periodic-time-critical tasks on multiprocessor computing systems and considers two heuristic algorithms that are easy to implement and yield a number of processors that is reasonably close to the minimum number.
Abstract: We study the problem of scheduling periodic-time-critical tasks on multiprocessor computing systems. A periodic-time-critical task consists of an infinite number of requests, each of which has a prescribed deadline. The scheduling problem is to specify an order in which the requests of a set of tasks are to be executed and the processor to be used, with the goal of meeting all the deadlines with a minimum number of processors. Since the problem of determining the minimum number of processors is difficult, we consider two heuristic algorithms. These are easy to implement and yield a number of processors that is reasonably close to the minimum number. We also analyze the worst-case behavior of these heuristics.
616 citations
TL;DR: The use of precedence constraints between jobs that have to be respected in every feasible schedule is illustrated by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
Abstract: Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
589 citations
TL;DR: It is shown that finding minimum finish time preemptive and non-preemptive schedules for flow shops and job shops is NP-complete.
Abstract: We show that finding minimum finish time preemptive and non-preemptive schedules for flow shops and job shops is NP-complete. Bounds on the performance of various heuristics to generate reasonably good schedules are also obtained.
416 citations
TL;DR: Information theory is used to derive three complementary tests that help analysts select a "best" disaggregate model and extends the information test to examine the relationships among successively more powerful null hypotheses.
Abstract: Disaggregate demand models predict the choice behavior of individual consumers. But while such models predict choice probabilities 0
311 citations
TL;DR: The optimization method described here combines finding shortest paths in an associated multipartite network with subgradient optimization and some branch-and-bound enumeration and gives computational results for the weighted tardiness problem.
Abstract: The time-dependent traveling salesman problem may be stated as a scheduling problem in which n jobs have to be processed at minimum cost on a single machine. The set-up cost associated with each job depends not only on the job that precedes it, but also on its position time in the sequence. The optimization method described here combines finding shortest paths in an associated multipartite network with subgradient optimization and some branch-and-bound enumeration. Minimizing the tardiness costs in one-machine scheduling in which the cost is a non-decreasing function of the completion time of each job is then attacked by this method. A branch-and-bound algorithm is designed for this problem. It uses a related time-dependent traveling salesman problem to compute the required lower bounds. We give computational results for the weighted tardiness problem.
298 citations
TL;DR: These two results permit a very compact computer implementation of a dynamic programming algorithm for solving one-machine sequencing problems with precedence constraints and this algorithm appears to be much more efficient than previous ones for certain one- machine sequencing problems.
Abstract: Consider a set of tasks that are partially ordered by precedence constraints. A subset of tasks is called feasible if, for every task in the subset, all predecessors are also in the subset. The major results are 1 a method for enumerating all feasible subsets and 2 a method for assigning to each feasible subset an easily computed label that can be used as a physical address for storing information about the subset. These two results permit a very compact computer implementation of a dynamic programming algorithm for solving one-machine sequencing problems with precedence constraints. This algorithm appears to be much more efficient than previous ones for certain one-machine sequencing problems.
244 citations
TL;DR: This exposition presents a state-of-the-art survey of results and algorithms for linear multicommodity network flow problems.
Abstract: This exposition presents a state-of-the-art survey of results and algorithms for linear multicommodity network flow problems.
TL;DR: A classification scheme for lower bounds that generates most previously known bounds and leads to a number of promising new ones as well as a report on computational experience that indicates the superiority of one of the new bounds.
Abstract: Branch-and-bound methods are commonly used to find a permutation schedule that minimizes maximum completion time in an m-machine flow-shop. In this paper we describe a classification scheme for lower bounds that generates most previously known bounds and leads to a number of promising new ones as well. After a discussion of dominance relations within this scheme and of the implementation of each bound, we report on computational experience that indicates the superiority of one of the new bounds.
TL;DR: This paper proves that the optimum solution structure for an n-period repairable inventory problem is completely defined by three period dependent values: I¸n, the repair-up-to-level; I¹n + I¾n,the scrap-down-to -level.
Abstract: This paper proves that the optimum solution structure for an n-period repairable inventory problem is completely defined by three period dependent values: I¸n, the repair-up-to-level; I´n, the purchase-up-to-level; and I¸n + I¾n, the scrap-down-to-level. The specific problem examined includes fixed periodic reviews, instantaneous delivery of purchased and repaired units, backlogging of unsatisfied demand, random demand for serviceables, random return of repairables with any relationship being permitted between demand and returns and a convex differentiable cost function. The basic solution methodology is a backward dynamic programming technique in two dimensions with the Kuhn-Tucker saddle point theorems applied in each stage.
TL;DR: It is proved that there always exists an optimal policy for the inventory model and that this policy is of a simple form.
Abstract: We formulate a continuous-time, infinite-horizon, discounted-cost cash management model with both fixed and proportional transactions costs and with linear holding and penalty costs. We model the cumulative demand for cash by a Wiener process with drift and use the optimal control technique of "impulse control" to find sufficient conditions under which an optimal policy exists. We show that these conditions are always met. Therefore, we prove that there always exists an optimal policy for the cash management problem and that this policy is of a simple form. When the proportional transactions cost of decreasing the cash balance is sufficiently high, it is never optimal to decrease the cash balance. Then the cash management model degenerates to the inventory model. We prove that there always exists an optimal policy for the inventory model and that this policy is of a simple form. Under special cases of the cash management model we obtain analytic expressions for the parameters of the optimal policy.
TL;DR: This paper presents an introduction to this approach to scheduling by describing its application to a well-known multiprocessor scheduling model and illustrating the variety of algorithms and results that are possible.
Abstract: One approach to coping with the apparent difficulty of many schedule-optimization problems, such as occur in machine shops and computer processing, is to devise efficient algorithms that find schedules guaranteed to be "near-optimal." This paper presents an introduction to this approach by describing its application to a well-known multiprocessor scheduling model and illustrating the variety of algorithms and results that are possible. The paper concludes with a brief survey of what has been accomplished to date in the area of scheduling using this approach.
TL;DR: This paper solves the problem of vertex iterates and presents a general proof permitting a variable step length within certain bounds, used, in particular, to show the convergence of a modified gradient Newton-Raphson type of procedure.
Abstract: The location problem is to find a point M whose sum of weighted distances from m vertices in p-dimensional Euclidean space is a minimum. The best-known algorithm for solving the location problem is an iterative scheme devised by Weiszfeld in 1937. The procedure will not converge if some nonoptimal vertex is an iterate, however. This paper solves the problem of vertex iterates and presents a general proof permitting a variable step length within certain bounds. This property is used, in particular, to show the convergence of a modified gradient Newton-Raphson type of procedure.
TL;DR: For this problem, it is shown that there is a 0n2-time algorithm to find a schedule that minimizes the number of tardy jobs, under the assumption that ri.
Abstract: We consider a class of n-job one-machine scheduling problems with ready time ri, processing time pi, and due time di for each job i. Preemption is not allowed, and precedence constraints among jobs are not assumed. For this problem we show that there is a 0n2-time algorithm to find a schedule that minimizes the number of tardy jobs, under the assumption that ri
TL;DR: The problem of determining rates for a situation in which services are purchased in bulk but have to be paid for by a large number of small users is considered, formulated as a non-atomic game and solved by using the value of the game.
Abstract: We consider the problem of determining rates for a situation in which services are purchased in bulk but have to be paid for by a large number of small users. The desired rates must be “fair” and they must cover all costs. The problem is formulated as a non-atomic game and solved by using the value of the game. In addition to the general problem, we present a detailed actual case, together with computational methods and results.
TL;DR: A simple approximation for the queue length distribution of the M/G/m queue with finite waiting room is proposed and some numerical results indicate that the proposed approximation is rather accurate.
Abstract: We study the M/G/m queue and obtain difference-differential equations for the equilibrium joint distribution of the number of customers present and of the remaining holding times for services in progress. A study of these equations leads to the proposal of a simple approximation for the queue length distribution. Approximate waiting time and busy period distributions easily follow. Simple expressions for the means of the variables in question are obtained. Some numerical results on the mean waiting time indicate that the proposed approximation is rather accurate. The technique presented in this paper can also be applied to variants of the M/G/m queue. In particular, a short treatment of the queue with finite waiting room is included.
TL;DR: Computational results indicate that, when the curse of dimensionality can be dispelled, dynamic programming can be a useful procedure for large sequencing problems.
Abstract: We discuss the dynamic programming approach to finding an optimal sequence of a set of tasks when the tasks are related by precedence restrictions. We describe how to use this approach in problems where no explicit precedence relations exist. Computer implementation considerations played an important role in its development. Computational results indicate that, when the curse of dimensionality can be dispelled, dynamic programming can be a useful procedure for large sequencing problems.
TL;DR: An interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem.
Abstract: In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses e.g., timber production, hunting, grazing. The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10,000-acre sub-unit of a national forest.
TL;DR: This work formulate VSP as a pure integer programming problem and provides an exact algorithm that examines a sequence of feasibility capacitated transportation problems with job splitting elimination side constraints and offers an approximate solution procedure based on the entropy principle of informational smoothing.
Abstract: We treat the following problem: There are n jobs with given processing times and an interval for each job's starting time. Each job must be processed, without interruption, on any one of an unlimited set of identical machines. A machine may process any job, but no more than one job at any point in time. We want to find the starting time of each job such that the number of machines required to process all jobs is minimal. In addition, the assignment of jobs to each machine must be found. If every job has a fixed starting time the interval is a point, the problem is well-known as a special case of Dilworth's problem. We term it the fixed job schedule problem FSP. When the job starting times are variable, the problem is referred to as the variable job schedule problem VSP, for which no known exact solution procedure exists. We introduce the problems by reviewing previous solution methods to Dilworth's problem. We offer an approximate solution procedure for solving VSP based on the entropy principle of informational smoothing. We then formulate VSP as a pure integer programming problem and provide an exact algorithm. This algorithm examines a sequence of feasibility capacitated transportation problems with job splitting elimination side constraints. Our computational experience demonstrates the utility of the entropy approach.
TL;DR: This work presents a modification of the Fletcher-Reeves conjugate gradient algorithm, which indicates that this modification yields an improved algorithm.
Abstract: We present a modification of the Fletcher-Reeves conjugate gradient algorithm. Computational experience indicates that this modification yields an improved algorithm.
TL;DR: The basic structure of this vehicle routing and scheduling problem, formulate the street-sweeper routing problem, and explain the algorithm are presented and the computer implementation based on this algorithm is described.
Abstract: This paper discusses a computer-assisted method for routing and scheduling street sweepers in a municipality. We present the basic structure of this vehicle routing and scheduling problem, formulate the street-sweeper routing problem, and explain the algorithm. The computer implementation based on this algorithm is then described. Computational experience with the system in New York City and Washington, D.C., is presented and the obstacles to and successes with implementation are discussed.
TL;DR: An algorithm is presented that requires the solution of at most m-1 minimum cut problems on networks with at most n + 2 vertices and is easily extended to the same location problem on a tree graph.
Abstract: This paper is concerned with the problem of locating n new facilities in the plane when there are m facilities already located The objective is to minimize the weighted sum of rectilinear distances Necessary and sufficient conditions for optimality are established We show that the optimum locations of the new facilities are dependent on the relative orderings of old facilities along the two coordinate axes but not on the distances between them Based on these results an algorithm is presented that requires the solution of at most m-1 minimum cut problems on networks with at most n + 2 vertices All of these results are easily extended to the same location problem on a tree graph
TL;DR: This paper presents a preference order dynamic program for solving the traveling salesman problem with stochastic travel times, and introduces a branch-and-bound strategy in the solution procedure.
Abstract: Consider a traveling salesman problem with stochastic travel times. Our objective is to find a tour with maximum probability of completion by a specified time. This paper presents a preference order dynamic program for solving the problem. To facilitate computation, we introduce a branch-and-bound strategy in the solution procedure. Finally, we propose an implicit enumeration algorithm as an alternative approach.
TL;DR: It is shown how simulations of New York City's fire and police operations have been used to develop and validate simple analytic models that are now being used to analyze the deployment of resources in these two services.
Abstract: Simulation models are generally costly tools to use in systems analyses. Whenever applicable, a simple analytic model is preferable. However, in many cases the conditions assumed by solvable analytic models do not hold in the real world; hence an analyst would hesitate to use them. A simulation can be used to suggest an appropriate approximate model and to determine how good an approximation a given analytic model is. We show how simulations of New York City's fire and police operations have been used to develop and validate simple analytic models that are now being used to analyze the deployment of resources in these two services.
TL;DR: It appears that local search algorithms are ineffective when applied to symmetric traveling salesman problems, because there are many local optima with arbitrarily high cost.
Abstract: We construct instances of the symmetric traveling salesman problem with n = 8k cities that have the following property: There is exactly one optimal tour with cost n, and there are 2k-1k-1! tours that are next-best, have arbitrarily large cost, and cannot be improved by changing fewer than 3k edges. Thus, there are many local optima with arbitrarily high cost. It appears that local search algorithms are ineffective when applied to these problems. Even more catastrophic examples are available in the non-symmetric case.
TL;DR: In this paper, the authors present a new proof that for a work-conserving queue, the queuing discipline that always serves a job with the shortest remaining processing time minimizes the number of jobs in the system.
Abstract: We present a new proof of the fact that for a work-conserving queue, the queuing discipline that always serves a job with the shortest remaining processing time minimizes the number of jobs in the system. A key feature of the proof is a definition of work dominance, allowing comparison of two systems based on the remaining service times of jobs present. Work dominance is both necessary and sufficient for stochastic comparison of the number of jobs present under identical but arbitrary arrival processes.
TL;DR: A mathematical model of breast cancer is developed and used to evaluate the benefits of screening for breast cancer and determine the benefits under alternative assumptions about the woman screened, the number of screened women, the ages at which the screens are given, the reliability of the screening technique, and the rate of disease progression.
Abstract: A mathematical model of breast cancer is developed and used to evaluate the benefits of screening for breast cancer. This model consists of hypotheses concerning the age-specific incidence of the disease, the rate of disease progression, the tendency of the disease to be detected without benefit of scheduled screening examinations, and prognosis related to the extent of disease progression at treatment. We formulate these hypotheses quantitatively and estimate parameters by fitting the model statistically to published data on breast cancer. Model predictions are independently validated by comparison with data from breast cancer screening programs. On the basis of the model, we calculate the benefits of screening under alternative assumptions about the woman screened, the number of screens given, the ages at which the screens are given, the reliability of the screening technique, and the rate of disease progression. These calculations are then used to consider questions concerning the design of breast canc...