A note on minimizing the expected makespan in flowshops subject to breakdowns

Consider a two machine flow shop and n jobs. The processing time of job j on machine i is equal to the random variable X_ij One of the two machines is subject to breakdown and repair. The objective is to find the schedule that minimizes the expected makespan. Two results are shown. First, ifP(X_2j ≧ X_1j) = 1 for all j and the random variables X₁₁, X₁₂,…, X_1n are likelihood ratio ordered, then the SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process; if P(X_1j≧X_2j) = 1 and X₂₁, X₂₂,….,X_2n are likelihood ratio ordered, then the LEPT sequence minimizes the expected makespan when machine 1 is subject to an arbitrary breakdown process. A generalization is presented for flow shops with m machines. Second, consider the case where X_1j and X_2j are i.i.d. exponentially distributed with rate λ_j. The SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process and the LEPT sequence is optimal when machine 1 is subject to an arbitrary breakdown process. © 1995 John Wiley & Sons, Inc.     

On scheduling a single machine to minimize a piecewise linear objective function: A compact MIP formulation

We study the scheduling situation in which a set of jobs subjected to release dates and deadlines are to be performed on a single machine. The objective is to minimize a piecewise linear objective function ∑_jF_j where F_j(C_j) corresponds to the cost of the completion of job j at time C_j. This class of function is very large and thus interesting both from a theoretical and practical point of view: It can be used to model total (weighted) completion time, total (weighted) tardiness, earliness and tardiness, etc. We introduce a new Mixed Integer Program (MIP) based on time interval decomposition. Our MIP is closely related to the well‐known time‐indexed MIP formulation but uses much less variables and constraints. Experiments on academic benchmarks as well as on real‐life industrial problems show that our generic MIP formulation is efficient. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Scheduling deteriorating jobs to minimize makespan

We consider a single-machine problem of scheduling n independent jobs to minimize makespan, in which the processing time of job J_j grows by w_j with each time unit its start is delayed beyond a given common critical date d. This processing time is p_j if J_j starts by d. We show that this problem is NP-hard, give a pseudopolynomial algorithm that runs in time and O(nd) space, and develop a branch-and-bound algorithm that solves instances with up to 100 jobs in a reasonable amount of time. We also introduce the case of bounded deterioration, where the processing time of a job grows no further if the job starts after a common maximum deterioration date D > d. For this case, we give two pseudopolynomial time algorithms: one runs in O(n²d(D − d) time and O(nd(D − d)) space, the other runs in p_j)²) time and p_j) space. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 511–523, 1998     

Characterizations of the exponential distribution by higher-order gap

Suppose X₁,X₂, ?,X_n is a random sample of size n from a continuous distribution function F(x) and let X_1,n, ≦ X_2,n ≦ ? ≦ X_n,n be the corresponding order statistics. We define the jth-order gap g_i,j as g_i,j = X_i+j,n ? X_i,n, 1 ≦ i < n, 1 ≦ j ≦ n ? i. In this article characterizations of the exponential distribution are given by considering the distributional properties of g_k,n-k, 1 ≦ k ≦ n.     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.     

Selecting review periods for a coordinated multi‐item inventory model with staggered deliveries

Consider an N‐item, periodic review, infinite‐horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. For 1 ≤ j ≤ N, orders for item j can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting). Every T_j periods, one reviews the current stock level of item j and decides on deliveries for each of the next T_j periods, thus incurring an item‐by‐item fixed cost k_j. There is also a joint fixed cost whenever any item is reviewed. The problem is to find review periods T₁, T₂, …, T_N and an ordering policy satisfying the average cost criterion. The current article builds on earlier results for the single‐item case. We prove an optimal policy exists, give conditions where it has a simple form, and develop a branch and bound algorithm for its computation. We also provide two heuristic policies with O(N) computational requirements. Computational experiments indicate that the branch and bound algorithm can handle normal demand problems with N ≤ 10 and that both heuristics do well for a wide variety of problems with N ranging from 2 to 200; moreover, the performance of our heuristics seems insensitive to N. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:430–449, 2001     

Parallel machine scheduling with eligibility constraints: A composite dispatching rule to minimize total weighted tardiness

We study a parallel machine scheduling problem, where a job j can only be processed on a specific subset of machines M_j, and the M_j subsets of the n jobs are nested. We develop a two‐phase heuristic for minimizing the total weighted tardiness subject to the machine eligibility constraints. In the first phase, we compute the factors and statistics that characterize a problem instance. In the second phase, we propose a new composite dispatching rule, the Apparent Tardiness Cost with Flexibility considerations (ATCF) rule, which is governed by several scaling parameters of which the values are determined by the factors obtained in the first phase. The ATCF rule is a generalization of the well‐known ATC rule which is very widely used in practice. We further discuss how to improve the dispatching rule using some simple but powerful properties without requiring additional computation time, and the improvement is quite satisfactory. We apply the Sequential Uniform Design Method to design our experiments and conduct an extensive computational study, and we perform tests on the performance of the ATCF rule using a real data set from a large hospital in China. We further compare its performance with that of the classical ATC rule. We also compare the schedules improved by the ATCF rule with what we believe are Near Optimal schedules generated by a general search procedure. The computational results show that especially with a low due date tightness, the ATCF rule performs significantly better than the well‐known ATC rule generating much improved schedules that are close to the Near Optimal schedules. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 249–267, 2017     

A note on scheduling parallel machines subject to breakdown and repair

In this paper we consider n jobs and a number of machines in parallel. The machines are identical and subject to breakdown and repair. The number may therefore vary over time and is at time t equal to m(t). Preemptions are allowed. We consider three objectives, namely, the total completion time, ∑ C_j, the makespan C_max, and the maximum lateness L_max. We study the conditions on m(t) under which various rules minimize the objective functions under consideration. We analyze cases when the jobs have deadlines to meet and when the jobs are subject to precedence constraints. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Approximation schemes for ordered vector packing problems

In this paper we deal with the d‐dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors a_i, a_j, either a_i is componentwise not smaller than a_j or a_j is componentwise not smaller than a_i. An asymptotic polynomial‐time approximation scheme is constructed for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge. We also extend the result to instances with constant Dilworth number, i.e., instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

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.     

Readiness and the optimal redeployment of resources

This paper considers the problem of the optimal redeployment of a resource among different geographical locations. Initially, it is assumed that at each location i, i = 1,…, n, the level of availability of the resource is given by a₁ ≧ 0. At time t > 0, requirements R_f(t) ≧ 0 are imposed on each location which, in general, will differ from the a₁. The resource can be transported from any one location to any other in magnitudes which will depend on t and the distance between these locations. It is assumed that ΣR_j > Σa_t The objective function consideis, in addition to transportation costs incurred by reallocation, the degree to which the resource availabilities after redeployment differ from the requirements. We shall associate the unavailabilities at the locations with the unreadiness of the system and discuss the optimal redeployment in terms of the minimization of the following functional forms: \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj) + } $\end{document} transportation costs, Max \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {Max}\limits_j \,[kj(Rj - yj)] + $\end{document} transportation costs, and \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj)^2 + } $\end{document} transportation costs. The variables y_j represent the final amount of the resource available at location j. No benefits are assumed to accrue at any location if y_j > R_j. A numerical three location example is given and solved for the linear objective.     

Openshop scheduling under linear resources constraints

Consider n jobs (J₁, …, J_n), m working stations (M₁, …, M_m) and λ linear resources (R₁, …, R_λ). Job J_i consists of m operations (O_i1, …, O_im). Operation O_ij requires P_k(i, j) units of resource R_k to be realized in an M_j. The availability of resource R_k and the ability of the working station M_h to consume resource R_k, vary over time. An operation involving more than one resource consumes them in constant proportions equal to those in which they are required. The order in which operations are realized is immaterial. We seek an allocation of the resources such that the schedule length is minimized. In this paper, polynomial algorithms are developed for several problems, while NP-hardness is demonstrated for several others. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 51–66, 1998     

A note on a stochastic production-maximizing transportation problem

A stochastic production-maximizing problem with transportation constraints is considered where the production rates, R_ij, of man i — job j combinations are random variables rather than constants. It is shown that for the family of Weibull distributions (of which the Exponential is a special case) with scale parameters λ_ij and shape parameter β, the plan that maximizes the expected rate of the entire line is obtained by solving a deterministic fixed charge transportation problem with no linear costs and with “set-up” cost matrix ‖λ_ij‖.     

An inventory problem with obsolescence

A stochastic single product convex cost inventory problem is considered in which there is a probability, π_j, that the product will become obsolete in the future period j. In an interesting paper, Barankin and Denny essentially formulate the model, but do not describe some of its interesting and relevant ramifications. This paper is written not only to bring out some of these ramifications, but also to describe some computational results using this model. The computational results show that if obsolescence is a distinct possibility in the near future, it is quite important that the probabilities of obsolescence be incorporated into the model before computing the optimal policies.     

A note on optimal switching between two activities

Let f₁ and f₂ map [0, T] into the real numbers. A system is following either f₁ or f₂ and earning the associated reward ∫ f₁ or ∫ f₂, respectively. It is possible at any time to switch from f_i to f_j by paying a switching cost b > 0. We determine a switching policy which maximizes the total reward. Conditions which guarantee a planning horizon are established.     

On the conditional residual lifetime of coherent systems under double regularly checking

In this paper, we consider a coherent system with n independent and identically distributed components under the condition that the system is monitored at time instances t₁ and t₂ (t₁ < t₂). First, various mixture representations for reliability function of the conditional residual lifetime of the coherent system are derived under different scenarios at times t₁ and t₂ (t₁ < t₂). Several stochastic comparisons between two systems are also made based on the proposed conditional random variables. Then, we consider the conditional residual lifetime of the functioning components of the system given that j components have failed at time t₁ and the system has failed at time t₂. Some stochastic comparisons on the proposed conditional residual lifetimes are investigated. Several illustrative graphs and examples are also provided.     

The discrete search problem and the construction of optimal allocations

Suppose one object is hidden in the k-th of n boxes with probability p(k). The boxes are to be searched sequentially. Associated with the j-th search of box k is a cost c(j,k) and a conditional probability q(j,k) that the first j - 1 searches of box k are unsuccessful while the j-th search is successful given that the object is hidden in box k. The problem is to maximize the probability that we find the object if we are not allowed to offer more than L for the search. We prove the existence of an optimal allocation of the search effort L and state an algorithm for the construction of an optimal allocation. Finally, we discuss some problems concerning the complexity of our problem.     

On labeling the vertices of products of complete graphs with distance constraints

Variations of Hale's channel assignment problem, the L(j, k)‐labeling problem and the radio labeling problem require the assignment of integers to the vertices of a graph G subject to various distance constraints. The λ_j,k‐number of G and the radio number of G are respectively the minimum span among all L(j, k)‐labelings, and the minimum span plus 1 of all radio labelings of G (defined in the Introduction). In this paper, we establish the λ_j,k‐number of ∏ K for pairwise relatively prime integers t₁ < t₂ < … < t_q, t₁ ≥ 2. We also show the existence of an infinite class of graphs G with radio number |V(G)| for any diameter d(G). © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Weighted vertex packing problem for specially structured geometric graphs

Consider a set of vertices V = {1, 2,…, n} placed on a two-dimensional Euclidean plane R² with each vertex attached a nonnegative weight w: V → R. For a given constant d>0, the geometric graph G = (V, E) is defined to have edge set E = {(i, j): d_ij ⩽ d} with d_ij being the Euclidean distance between vertices i and j. The geometric vertex packing (GVP) problem, which is often called the independent set problem, is defined as selecting the set of pairwise nonadjacent vertices with maximum total weight. We limit our attention to the special case that no vertex is within a distance βd of any other vertices where 0 ⩽ β < 1. A special value of β (= 1/2) is referred to frequently because of its correspondence to a manufacturing problem in circuit board testing. In this article we show that the weighted vertex packing problem for the specially structured geometric graph (SGVP) defined with the above restriction is NP-complete even for the case that all vertex weights are unity and for any β. Polynomial procedures have been designed for generating cuts to obtain tight LP upper bounds for the SGVP. Two heuristics with bounded worst-case performance are applied to the LP solution to produce a feasible solution and a lower bound. We then use a branch-and-bound procedure to solve the problem to optimality. Computational results on large-scale SGVP problems will be discussed. © 1995 John Wiley & Sons, Inc.     

Single batch machine scheduling with deliveries

We consider the problem of scheduling a set of n jobs on a single batch machine, where several jobs can be processed simultaneously. Each job j has a processing time p_j and a size s_j. All jobs are available for processing at time 0. The batch machine has a capacity D. Several jobs can be batched together and processed simultaneously, provided that the total size of the jobs in the batch does not exceed D. The processing time of a batch is the largest processing time among all jobs in the batch. There is a single vehicle available for delivery of the finished products to the customer, and the vehicle has capacity K. We assume that K = rD, where and r is an integer. The travel time of the vehicle is T; that is, T is the time from the manufacturer to the customer. Our goal is to find a schedule of the jobs and a delivery plan so that the service span is minimized, where the service span is the time that the last job is delivered to the customer. We show that if the jobs have identical sizes, then we can find a schedule and delivery plan in time such that the service span is minimum. If the jobs have identical processing times, then we can find a schedule and delivery plan in time such that the service span is asymptotically at most 11/9 times the optimal service span. When the jobs have arbitrary processing times and arbitrary sizes, then we can find a schedule and delivery plan in time such that the service span is asymptotically at most twice the optimal service span. We also derive upper bounds of the absolute worst‐case ratios in both cases. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 470–482, 2015