Steady state waiting time in a multicenter job shop

This paper analyzes the waiting-time distribution of a specific job as it moves through a job-shop with multiple centers and exponential service times. The movement of the job through the shop is governed by a Markovian transition matrix and ends with the job's exit from the shop.     

Scheduling jobs that are subject to failure propagation

A single machine is available to process a collection of stochastic tasks. Processing is interrupted when the machine breaks down. We introduce a new model of breakdowns that more realistically incorporates the effects that job processing may have on the machine. This failure propagation model is equivalent to a Bayesian formulation in which learning about breakdown rates occurs as jobs evolve. Optimal scheduling policies are described in terms of Gittins indices and these indices are characterised in two special cases. For example, we obtain conditions which ensure that an optimal policy will only preempt a job's processing either at its completion or at a machine breakdown. We also bound the value lost by simplistic modelling which ignores the learning phenomenon. © 1996 John Wiley & Sons, Inc.     

Matrix-geometric solution of discrete time MAP/PH/1 priority queue

We use the matrix-geometric method to study the discrete time MAP/PH/1 priority queue with two types of jobs. Both preemptive and non-preemptive cases are considered. We show that the structure of the R matrix obtained by Miller for the Birth-Death system can be extended to our Quasi-Birth-Death case. For both preemptive and non-preemptive cases the distributions of the number of jobs of each type in the system are obtained and their waiting times are obtained for the non-preemptive. For the preemptive case we obtain the waiting time distribution for the high priority job and the distribution of the lower priority job's wait before it becomes the leading job of its priority class. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 23–50, 1998     

An approximate dynamic programing approach to the development of heuristics for the scheduling of impatient jobs in a clearing system

A single server is faced with a collection of jobs of varying duration and urgency. Each job has a random lifetime during which it is available for nonpreemptive service. Should a job's lifetime expire before its service begins then it is lost from the system unserved. The goal is to schedule the jobs for service to maximize the expected number served to completion. Two heuristics have been proposed in the literature. One (labeled π^S) operates a static priority among the job classes and works well in a “no premature job loss” limit, whereas the second (π^M) is a myopic heuristic which works well when lifetimes are short. Both can exhibit poor performance for problems at some distance from the regimes for which they were designed. We develop a robustly good heuristic by an approximative approach to the application of a policy improvement step to the asymptotically optimal heuristic π^S, in which we use a fluid model to obtain an approximation for the value function of π^S. The performance of the proposed heuristic is investigated in an extensive numerical study. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Due‐window assignment with unit processing‐time jobs

In due‐window assignment problems, jobs completed within a designated time interval are regarded as being on time, whereas early and tardy jobs are penalized. The objective is to determine the location and size of the due‐window, as well as the job schedule. We address a common due‐window assignment problem on parallel identical machines with unit processing time jobs. We show that the number of candidate values for the optimal due‐window starting time and for the optimal due‐window completion time are bounded by 2. We also prove that the starting time of the first job on each of the machines is either 0 or 1, thus introducing a fairly simple, constant‐time solution for the problem. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Production and transport logistics scheduling with two transport mode choices

This paper considers a new class of scheduling problems arising in logistics systems in which two different transportation modes are available at the stage of product delivery. The mode with the shorter transportation time charges a higher cost. Each job ordered by the customer is first processed in the manufacturing facility and then transported to the customer. There is a due date for each job to arrive to the customer. Our approach integrates the machine scheduling problem in the manufacturing stage with the transportation mode selection problem in the delivery stage to achieve the global maximum benefit. In addition to studying the NP‐hard special case in which no tardy job is allowed, we consider in detail the problem when minimizing the sum of the total transportation cost and the total weighted tardiness cost is the objective. We provide a branch and bound algorithm with two different lower bounds. The effectiveness of the two lower bounds is discussed and compared. We also provide a mathematical model that is solvable by CPLEX. Computational results show that our branch and bound algorithm is more efficient than CPLEX. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Opportunistic retooling of a flexible machine subject to failure

A set of jobs can be processed without interruption by a flexible machine only if the set of tools required by all jobs can be loaded in the tool magazine. However, in practice the total number of tools required by a job set would exceed the tool magazine capacity. In such situations, the job set has to be carefully partitioned at the start of the production run such that each partition can be processed without interruption. During the production run, if there are unscheduled machine downtimes due to machine failure, this provides an additional opportunity to optimally retool the magazine for a smaller job set consisting of just the unprocessed jobs. In this paper, we study job sequencing rules that allow us to minimize the total expected cost of machine down time due to machine failures and magazine retooling, assuming a dynamic re‐sequencing of the unprocessed jobs after each machine failure. Using these rules, we develop a branch‐and‐bound heuristic that allows us to solve problems of reasonable size. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 79–97, 2001     

Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost

In the last decade, there has been much progress in understanding scheduling problems in which selfish jobs aim to minimize their individual completion time. Most of this work has focused on makespan minimization as social objective. In contrast, we consider as social cost the total weighted completion time, that is, the sum of the agent costs, a standard definition of welfare in economics. In our setting, jobs are processed on restricted uniform parallel machines, where each machine has a speed and is only capable of processing a subset of jobs; a job's cost is its weighted completion time; and each machine sequences its jobs in weighted shortest processing time (WSPT) order. Whereas for the makespan social cost the price of anarchy is not bounded by a constant in most environments, we show that for our minsum social objective the price of anarchy is bounded above by a small constant, independent of the instance. Specifically, we show that the price of anarchy is exactly 2 for the class of unit jobs, unit speed instances where the finite processing time values define the edge set of a forest with the machines as nodes. For the general case of mixed job strategies and restricted uniform machines, we prove that the price of anarchy equals 4. From a classical machine scheduling perspective, our results establish the same constant performance guarantees for WSPT list scheduling. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Sojourn times in G/M/1 fork‐join networks

We consider a processing network in which jobs arrive at a fork‐node according to a renewal process. Each job requires the completion of m tasks, which are instantaneously assigned by the fork‐node to m task‐processing nodes that operate like G/M/1 queueing stations. The job is completed when all of its m tasks are finished. The sojourn time (or response time) of a job in this G/M/1 fork‐join network is the total time it takes to complete the m tasks. Our main result is a closed‐form approximation of the sojourn‐time distribution of a job that arrives in equilibrium. This is obtained by the use of bounds, properties of D/M/1 and M/M/1 fork‐join networks, and exploratory simulations. Statistical tests show that our approximation distributions are good fits for the sojourn‐time distributions obtained from simulations. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Single machine scheduling with a restricted rate‐modifying activity

In this paper we consider the problem of scheduling a set of jobs on a single machine on which a rate‐modifying activity may be performed. The rate‐modifying activity is an activity that changes the production rate of the machine. So the processing time of a job is a variable, which depends on whether it is scheduled before or after the rate‐modifying activity. We assume that the rate‐modifying activity can take place only at certain predetermined time points, which is a constrained case of a similar problem discussed in the literature. The decisions under consideration are whether and when to schedule the rate‐modifying activity, and how to sequence the jobs in order to minimize some objectives. We study the problems of minimizing makespan and total completion time. We first analyze the computational complexity of both problems for most of the possible versions. The analysis shows that the problems are NP‐hard even for some special cases. Furthermore, for the NP‐hard cases of the makespan problem, we present a pseudo‐polynomial time optimal algorithm and a fully polynomial time approximation scheme. For the total completion time problem, we provide a pseudo‐polynomial time optimal algorithm for the case with agreeable modifying rates. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Approximation algorithms for minimizing total weighted completion time of orders on identical machines in parallel

We consider the problem of scheduling orders on identical machines in parallel. Each order consists of one or more individual jobs. A job that belongs to an order can be processed by any one of the machines. Multiple machines can process the jobs of an order concurrently. No setup is required if a machine switches over from one job to another. Each order is released at time zero and has a positive weight. Preemptions are not allowed. The completion time of an order is the time at which all jobs of that order have been completed. The objective is to minimize the total weighted completion time of the orders. The problem is NP‐hard for any fixed number (≥2) of machines. Because of this, we focus our attention on two classes of heuristics, which we refer to as sequential two‐phase heuristics and dynamic two‐phase heuristics. We perform a worst case analysis as well as an empirical analysis of nine heuristics. Our analyses enable us to rank these heuristics according to their effectiveness, taking solution quality as well as running time into account. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Approximation of multiclass queueing networks with highly variable arrivals under deterministic routing

In this paper, we present the heavy‐traffic bottleneck phenomenon under multiclass deterministic routing and discuss how it can be addressed by decomposition approximation. Examples show that Bitran and Tirupati's method and Whitt's enhancements for deterministic routing may not properly account for this phenomenon. We propose refinements to these methods based on Whitt's variability functions. Results of numerical experiments on simple networks and semiconductor manufacturing process show significant improvement in the approximation of expected waiting time at bottleneck stations. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

On competitive analysis for polling systems

Polling systems have been widely studied, however most of these studies focus on polling systems with renewal processes for arrivals and random variables for service times. There is a need driven by practical applications to study polling systems with arbitrary arrivals (not restricted to time-varying or in batches) and revealed service time upon a job's arrival. To address that need, our work considers a polling system with generic setting and for the first time provides the worst-case analysis for online scheduling policies in this system. We provide conditions for the existence of constant competitive ratios, and competitive lower bounds for general scheduling policies in polling systems. Our work also bridges the queueing and scheduling communities by proving the competitive ratios for several well-studied policies in the queueing literature, such as cyclic policies with exhaustive, gated or l-limited service disciplines for polling systems.     

LAND INEQUALITY AND CONFLICT IN LATIN AMERICA IN THE TWENTIETH CENTURY

Fiji's total debt stands at 65% of GDP. Domestic debt constitutes 55% of GDP. The goal of this paper is to investigate whether military expenditure has contributed to Fiji's exploding debt levels over the period 1970 to 2005. Our empirical analysis, conducted within a cointegration and vector error‐correction framework, suggests that, in the long‐run, military expenditure has had a statistically significant positive impact on both external debt and domestic debt, while income has had a statistically significant positive impact on domestic debt and a statistically significant negative impact on external debt. We explain the reasons behind this finding and draw some policy implications.     

Branch‐and‐price‐and‐cut for the manpower routing problem with synchronization constraints

In this article, we propose a branch‐and‐price‐and‐cut (BPC) algorithm to exactly solve the manpower routing problem with synchronization constraints (MRPSC). Compared with the classical vehicle routing problems (VRPs), the defining characteristic of the MRPSC is that multiple workers are required to work together and start at the same time to carry out a job, that is, the routes of the scheduling subjects are dependent. The incorporation of the synchronization constraints increases the difficulty of the MRPSC significantly and makes the existing VRP exact algorithm inapplicable. Although there are many types of valid inequalities for the VRP or its variants, so far we can only adapt the infeasible path elimination inequality and the weak clique inequality to handle the synchronization constraints in our BPC algorithm. The experimental results at the root node of the branch‐and‐bound tree show that the employed inequalities can effectively improve the lower bound of the problem. Compared with ILOG CPLEX, our BPC algorithm managed to find optimal solutions for more test instances within 1 hour. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 138–171, 2016     

Parallel machine scheduling with batch deliveries to minimize total flow time and delivery cost

Motivated by the flow of products in the iron and steel industry, we study an identical and parallel machine scheduling problem with batch deliveries, where jobs finished on the parallel machines are delivered to customers in batches. Each delivery batch has a capacity and incurs a cost. The objective is to find a coordinated production and delivery schedule that minimizes the total flow time of jobs plus the total delivery cost. This problem is an extension of the problem considered by Hall and Potts, Ann Oper Res 135 (2005) 41–64, who studied a two‐machine problem with an unbounded number of transporters and unbounded delivery capacity. We first provide a dynamic programming algorithm to solve a special case with a given job assignment to the machines. A heuristic algorithm is then presented for the general problem, and its worst‐case performance ratio is analyzed. The computational results show that the heuristic algorithm can generate near‐optimal solutions. Finally, we offer a fully polynomial‐time approximation scheme for a fixed number of machines. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 492–502, 2016     

Transshipment through crossdocks with inventory and time windows

A major challenge in making supply meet demand is to coordinate transshipments across the supply chain to reduce costs and increase service levels in the face of demand fluctuations, short lead times, warehouse limitations, and transportation and inventory costs. In particular, transshipment through crossdocks, where just‐in‐time objectives prevail, requires precise scheduling between suppliers, crossdocks, and customers. In this work, we study the transshipment problem with supplier and customer time windows where flow is constrained by transportation schedules and warehouse capacities. Transportation is provided by fixed or flexible schedules and lot‐sizing is dealt with through multiple shipments. We develop polynomial‐time algorithms or, otherwise, provide the complexity of the problems studied. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

An approximate planning model for distributed computing networks

We develop an approximate planning model for a distributed computing network in which a control system oversees the assignment of information flows and tasks to a pool of shared computers, and describe several optimization applications using the model. We assume that the computers are multithreaded, and have differing architectures leading to varying and inconsistent processing rates. The model is based on a discrete‐time, continuous flow model developed by Graves [Oper Res 34 (1986), 522–533] which provides the steady‐state moments of production and work‐in‐queue quantities. We make several extensions to Graves' model to represent distributed computing networks. First, we approximately model control rules that are nonlinear functions of the work‐in‐queue at multiple stations through a linearization approach. Second, we introduce an additional noise term on production and show its use in modeling the discretization of jobs. Third, we model groups of heterogeneous computers as aggregate, “virtual computing cells” that process multiple tasks simultaneously, using a judiciously selected control rule. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A note on combined job selection and sequencing problems

We investigate the solvability of two single‐machine scheduling problems when the objective is to identify among all job subsets with cardinality k,1≤k≤n, the one that has the minimum objective function value. For the single‐machine minimum maximum lateness problem, we conclude that the problem is solvable in O(n²) time using the proposed REMOVE algorithm. This algorithm can also be used as an alternative to Moore's algorithm to solve the minimum number of tardy jobs problem by actually solving the hierarchical problem in which the objective is to minimize the maximum lateness subject to the minimum number of tardy jobs. We then show that the REMOVE algorithm cannot be used to solve the general case of the single‐machine total‐weighted completion time problem; we derive sufficient conditions among the job parameters so that the total weighted completion time problem becomes solvable in O(n²) time. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 449–453, 2013