Open-shop scheduling problems with dominated machines

This article deals with special cases of open-shop scheduling where n jobs have to be processed by m, m ?3, machines to minimize the schedule length. The main result obtained is an O(n) algorithm for the three-machine problem with a dominated machine.     

Discrete repairable reliability systems

Consider a reliability system consisting of n components. The failures and the repair completions of the components can occur only at positive integer-valued times k ϵ N₊₊ ϵ (1, 2, …). At any time k ϵ N₊₊ each component can be in one of two states: up (i.e., working) or down (i.e., failed and in repair). The system state is also either up or down and it depends on the states of the components through a coherent structure function τ. In this article we formulate mathematically the above model and we derive some of its properties. In particular, we identify conditions under which the first failure times of two such systems can be stochastically ordered. A variety of special cases is used in order to illustrate the applications of the derived properties of the model. Some instances in which the times of first failure have the NBU (new better than used) property are pointed out. © 1993 John Wiley & Sons, Inc.     

Polaroids: A new tool in non-convex and in integer programming

This paper presents a generalization, called polaroid, of the concept of polar sets A list of properties satisfied by polaroids is established indicating that the new concept nay be fruitfully used in an area of non-convex (called here polar) programming as well as in integer programming, by means of polaroid cuts; this class of new cuts contains the ones defined by Tuy for concave programming (a special case of polar programming) and by Balas integer programming; it furthermore provides for new degrees of freedom in the construction of algorithms in the above-mentioned areas of mathematical programming.     

Future supply uncertainty in EOQ models

This article deals with an inventory problem where the supply is available only during an interval of (random) length X. The unavailability of supply lasts for a random duration Y. Using concepts from renewal theory, we construct an objective function (average cost/time) in terms of the order-quantity decision variable Q. We develop the individual cost components as order, holding, and shortage costs after introducing two important random variables. Due to the complexity of the objective function when X and Y are general random variables, we discuss two special cases and provide numerical examples with sensitivity analysis on the cost and noncost parameters. The article concludes with a discussion of the comparison of the current model with random yield and random lead-time models. Suggestions for further research are also provided.     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Testing for positive quadrant dependence in ordinal contingency table

Two new randomization tests are introduced for ordinal contingency tables for testing independence against strictly positive quadrant dependence, i.e., P(X > x,Y > y) ≥ P(X > x)P(Y > y) for all x,y with strict inequality for some x and y. For a number of cases, simulation is used to compare the estimated power of these tests versus those standard tests based on Kendall's T, Spearman's p, Pearson's X², the usual likelihood ratio test, and a test based upon the log-odds ratio. In these cases, subsets of the alternative region are identified where each of the testing statistics is superior. The new tests are found to be more powerful than the standard tests over a broad range of the alternative regions for these cases.     

Hybrid minimal repair and age replacement maintenance policies,with nonnegligible repair times

A system undergoes minimal repair during [0, T] with a failure replacement on first failure during [T, a], or a planned replacement if the system is still functioning at elapsed time a. Repairs and replacements are not necessarily instantaneous. An expression is obtained for the asymptotic expected cost rate, and sufficient conditions are obtained for the optimum T* > 0. Several special cases are considered. A numerical investigation for a Weibull distributed time to first failure compares this elapsed-time policy with replacement on failure only, and also a policy based on system operating time or age. It is found that in many cases the elapsed-time-based policy is only marginally worse than one based on system age, and may therefore be preferred in view of its administrative convenience. © 1994 John Wiley & Sons, Inc.     

The N-seasons S-servers loss system

We consider a class of loss systems with exponential service times and a Poisson arrival process with a rate that varies periodically among N levels called seasons. For two special cases, we derive transient and steady-state solutions and provide simple proofs that losses are minimized when the arrival rates for all seasons are equal. In the general case, we describe a straightforward procedure to derive the steady-state probabilities. We also prove that when S=1, the server is generally busier during the high arrival rate seasons.     

A single server queue with arrival rate dependent on server breakdowns

This paper considers a single server queueing system that alternates stochastically between two states: operational and failed. When operational, the system functions as an M/E_k/1 queue. When the system is failed, no service takes place but customers continue to arrive according to a Poisson process; however, the arrival rate is different from that when the system is operational. The durations of the operating and failed periods are exponential with mean 1/cβ and Erlang with mean 1/cβ, respectively. Generating functions are used to derive the steady-state quantities L and W, both of which, when viewed as functions of c, decrease at a rate inversely proportional to c². The paper includes an analysis of several special and extreme cases and an application to a production-storage system.     

Primal dual algorithms for the vehicle refueling problem

Mehrez, Stern, and Ronen have defined a vehicle refueling problem in which a fleet of vehicles travels on a round-trip, self-contained mission from a common origin, with the objective of maximizing the operational range of the fleet. They have defined a “pure refueling chain” strategy for transferring fuel between vehicles in the fleet, and have solved the problem in the special cases when all vehicles have the same fuel capacity or consumption rate. In this article we present algorithms for the general case, where vehicles have different capacities and consumption rates. Our approach is based on a new primal dual formulation of the problem. The exact algorithm was effective to find the optimal solution for a fleet size n ⩽13. For larger fleets, we present an approximation version of it, which very quickly found a solution within 1% of the maximum possible range for arbitrarily large (up to n = 200) fleets. We also show that a small number of the best vehicles can always reach almost the same range as a large fleet. © 1992 John Wiley & Sons, Inc.     

A heuristic routine for solving large loading problems

The loading problem involves the optimal allocation of n objects, each having a specified weight and value, to m boxes, each of specified capacity. While special cases of these problems can be solved with relative ease, the general problem having variable item weights and box sizes can become very difficult to solve. This paper presents a heuristic procedure for solving large loading problems of the more general type. The procedure uses a surrogate procedure for reducing the original problem to a simpler knapsack problem, the solution of which is then employed in searching for feasible solutions to the original problem. The procedure is easy to apply, and is capable of identifying optimal solutions if they are found.     

A vertex-closing approach to the p-center problem

This article uses a vertex-closing approach to investigate the p-center problem. The optimal set of vertices to close are found in imbedded subgraphs of the original graph. Properties of these subgraphs are presented and then used to characterize the optimal solution, to establish a priori upper and lower bounds, to establish refined lower bounds, and to verify the optimality of solutions. These subgraphs form the foundation of two polynomial algorithms of complexity O(|E| log |E|) and O(|E|²). The algorithms are proven to converge to an optimum for special cases, and computational evidence is provided which suggests that they produce very good solutions more generally. Both algorithms perform very well on problems where p is large relative to the number of vertices n, specifically, when p/n ≥ 0.30. One of the algorithms is especially efficient for solving a sequence of problems on the same graph.     

Mathematical aspects of the 3 × n job-shop sequencing problem

The paper discusses mathematical properties of the well-known Bellman-Johnson 3 × n sequencing problem. Optimal rules for some special cases are developed. For the case min B_i ≥ maxA_j we find an optimal sequence of the 2 × n problem for machines B and C and move one item to the front of the sequence to minimize (7); when min B_i ≥ max C_j we solve a 2 × n problem for machines A and B and move one item to the end of the optimal sequence so as to minimize (9). There is also given a sufficient optimality condition for a solution obtained by Johnson's approximate method. This explains why this method so often produces an optimal solution.     

On maximization of the integral of a bell-shaped function over a symmetric set

Given a target T in Euclidean n-space Rⁿ and a point bomb whose point of impact in Rⁿ is governed by a probability distribution about the aim point a, what choice of a maximizes the probability of a hit v_a(T)? Of course, only in special cases is an exact solution of this problem obtainable. This paper treats targets T which are symmetric about the origin o and demonstrates conditions on the extent of T and the impact density f, a density with respect to Lebesgue measure, sufficient for v_a(T) to be monotone in the distance from a to o and maximized at a = o. The results are applied to various tactical situations.     

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

For many combinatorial optimization problems that are NP-hard, a number of special cases exist that can be solved in polynomial time. This paper addresses the issue of solving one such problem, the well-known m-median problem with mutual communication (MMMC), by exploiting polynomially solvable special cases of the problem. For MMMC, a dependency graph is defined that characterizes the structure of the interactions between decision variables. A Lagrangian decomposition scheme is proposed that partitions the problem into two or more subproblems, each having the same structure as the original problem, but with simpler dependency graphs. The dual problems are solved using subgradient or multiplier adjustment methods. An efficient method of adjusting the multiplier values is given. Computational results are reported that show the method to be quite effective. In addition, applications of the approach to other difficult location problems is discussed. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 791–815, 1998     

Introduction: Exploring the jungle of terminology

When twentieth-century authors wrote about ‘partisan warfare’, they usually meant an insurgency or asymmetric military operations conducted against a superior force by small bands of ideologically driven irregular fighters. By contrast, originally (i.e. before the French Revolution) ‘partisan’ in French, English, and German referred only to the leader of a detachment of special forces (party, partie, Parthey, détachement) which the major European powers used to conduct special operations alongside their regular forces. Such special operations were the classic definition of ‘small war’ (petite guerre) in the late seventeenth and in the eighteenth centuries. The Spanish word ‘la guerrilla’, meaning nothing other than ‘small war’, only acquired an association with rebellion with the Spanish War of Independence against Napoleon. Even after this, however, armies throughout the world have continued to employ special forces. In the late nineteenth century, their operations have still been referred to as prosecuting ‘la guerrilla’ or ‘small war’, which existed side by side with, and was often mixed with, ‘people's war’ or popular uprisings against hated regimes.     

Scheduling multiple products on parallel machines with setup costs

We consider a class of production scheduling models with m identical machines in parallel and k different product types. It takes a time p_i to produce one unit of product type i on any one of the machines. There is a demand stream for product type i consisting of n_i units with each unit having a given due date. Before a machine starts with the production of a batch of products of type i a setup cost c is incurred. We consider several different objective functions. Each one of the objective functions has three components, namely a total setup cost, a total earliness cost, and a total tardiness cost. In our class of problems we find a relatively large number of problems that can be solved either in polynomial time or in pseudo‐polynomial time. The polynomiality or pseudo‐polynomiality is achieved under certain special conditions that may be of practical interest; for example, a regularity pattern in the string of due dates combined with earliness and tardiness costs that are similar for different types of products. The class of models we consider includes as special cases discrete counterparts of a number of inventory models that have been considered in the literature before, e.g., Wagner and Whitin (Manage Sci 5 (1958), 89–96) and Zangwill (Oper Res 14 (1966), 486–507; Manage Sci 15 (1969), 506–527). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Single‐machine scheduling with deadlines to minimize the total weighted late work

We consider scheduling a set of jobs with deadlines to minimize the total weighted late work on a single machine, where the late work of a job is the amount of processing of the job that is scheduled after its due date and before its deadline. This is the first study on scheduling with the late work criterion under the deadline restriction. In this paper, we show that (i) the problem is unary NP‐hard even if all the jobs have a unit weight, (ii) the problem is binary NP‐hard and admits a pseudo‐polynomial‐time algorithm and a fully polynomial‐time approximation scheme if all the jobs have a common due date, and (iii) some special cases of the problem are polynomially solvable.     

Stock control in a many depot system

The chief problems considered are: (1) In a parallel set of warehouses, how should stocks be allocated? (2) In a system consisting of a central warehouse and several subsidiary warehouses, how much stock should be carried in each? The demands may have known, or unknown, distribution functions. For problem (1), the i-th stock n_i should usually be allocated in proportion to the i-th demand m_i; in special cases, a significant improvement is embodied in the formula (N = total allocable stock)

     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040