Kevin D. Glazebrook

Inventory models with lateral transshipments: A review

Lateral transshipments within an inventory system are stock movements between locations of the same echelon. These transshipments can be conducted periodically at predetermined points in time to proactively redistribute stock, or they can be used reactively as a method of meeting demand which cannot be satisfied from stock on hand. The elements of an inventory system considered, e.g. size, cost structures and service level definition, all influence the best method of transshipping. Models of many different systems have been considered. This paper provides a literature review which categorizes the research to date on lateral transshipments, so that these differences can be understood and gaps within the literature can be identified.

Whittle’s index policy for a multi-class queueing system with convex holding costs

Abstract. Multi-class service systems are of increasing importance in the practical modelling world but present a significant challenge for analysis. Most results to date concerning the optimal dynamic control of such systems have assumed holding cost rates to be linear in the number of customers present. In response to arguments that such an assumption is often inappropriate, we develop an index heuristic for a multi-class M/M/1 system with increasing convex holding cost rates. We use a prescription of Whittle to develop the required indices. A numerical study elucidates very strong performance of the index policy.

Stochastic scheduling with order constraints

A collection of jobs requires service. Each job has an associated cost function; normally costs accrue when a job is completed. The amount of service needed to complete a job is a random variable with a known probability distribution. Service may be allocated to jobs according to a sot of order constraints between the jobs. The problem is to find a strategy of allocating service which minimizes the total expected costs of all jobs in the system and which observes the order constraints.

The achievable region approach to the optimal control of stochastic systems

The achievable region approach seeks solutions to stochastic optimisation problems by: (i) characterising the space of all possible performances (the achievable region) of the system of interest, and (ii) optimising the overall system-wide performance objective over this space. This is radically different from conventional formulations based on dynamic programming. The approach is explained with reference to a simple two-class queueing system. Powerful new methodologies due to the authors and co-workers are deployed to analyse a general multiclass queueing system with parallel servers and then to develop an approach to optimal load distribution across a network of interconnected stations. Finally, the approach is used for the first time to analyse a class of intensity control problems.

On Single-Machine Scheduling with Precedence Relations and Linear or Discounted Costs

Some results of Sidney (Sidney, J. B. 1975. Decomposition algorithms for single machine sequencing with precedence relations and deferral costs. Opns. Res. 23 283–298.) concerning order-constrained deterministic scheduling with linear costs are extended to the more complex discounted costs case. Further, certain order-constrained stochastic scheduling problems are shown to have optimal strategies which are nonpreemptive. When this is so, results for the deterministic case may be utilized to obtain these optimal strategies. The models discussed are of interest in computer and job-shop scheduling and research planning.

An index heuristic for transshipment decisions in multi-location inventory systems based on a pairwise decomposition

In multi-location inventory systems, transshipments are often used to improve customer service and reduce cost. Determining optimal transshipment policies for such systems involves a complex optimisation problem that is only tractable for systems with few locations. Consequently simple heuristic transshipment policies are often applied in practice. This paper develops an approximate solution method which applies decomposition to reduce a Markov decision process model of a multi-location inventory system into a number of models involving only two locations. The value functions from the subproblems are used to estimate the fair charge for the inventory provided in a transshipment. This estimate of the fair charge is used as the decision criterion in a heuristic transshipment policy for the multi-location system. A numerical study shows that the proposed heuristic can deliver considerable cost savings compared to the simple heuristics often used in practice.

Index policies for a class of discounted restless bandits

The paper concerns a class of discounted restless bandit problems which possess an indexability property. Conservation laws yield an expression for the reward suboptimality of a general policy. These results are utilised to study the closeness to optimality of an index policy for a special class of simple and natural dual speed restless bandits for which indexability is guaranteed. The strong performance of the index policy is confirmed by a computational study.

Parallel Scheduling of Multiclass M/M/m Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance

We address the problem of scheduling a multiclass M/M/m queue with Bernoulli feedback on m parallel servers to minimize time-average linear holding costs. We analyze the performance of a heuristic priority-index rule, which extends Klimovs optimal solution to the single-server case: servers select preemptively customers with larger Klimov indices. We present closed-form suboptimality bounds approximate optimality for Klimovs rule, which imply that its suboptimality gap is uniformly bounded above with respect to i external arrival rates, as long as they stay within system capacity; and ii the number of servers. It follows that its relative suboptimality gap vanishes in a heavy-traffic limit, as external arrival rates approach system capacity heavy-traffic optimality. We obtain simpler expressions for the special no-feedback case, where the heuristic reduces to the classical cµ rule. Our analysis is based on comparing the expected cost of Klimovs rule to the value of a strong linear programming LP relaxation of the systems region of achievable performance of mean queue lengths. In order to obtain this relaxation, we derive and exploit a new set of work decomposition laws for the parallel-server system. We further report on the results of a computational study on the quality of the cµ rule for parallel scheduling.

Enhanced lateral transshipments in a multi-location inventory system

In managing an inventory network, two approaches to the pooling of stock have been proposed. Reactive transshipments respond to shortages at a location by moving inventory from elsewhere within the network, while proactive stock redistribution seeks to minimize the chance of future stockouts. This paper is the first to propose an enhanced reactive approach in which individual transshipments are viewed as an opportunity for proactive stock redistribution. We adopt a quasi-myopic approach to the development of a strongly performing enhanced reactive transshipment policy. In comparison to a purely reactive approach to transshipment, service levels are improved while a reduction in safety stock levels is achieved. The aggregate costs incurred in managing the system are significantly reduced, especially so for large networks. Moreover, an optimal policy is determined for small networks and it is shown that the enhanced reactive policy substantially closes the gap to optimality.

On the evaluation of suboptimal strategies for families of alternative bandit processes

Families of alternative bandit processes have been used as models for problems in a variety of areas. Optimal strategies for these decision processes are determined by dynamic allocation indices. These indices are here shown to play an important role in the evaluation of suboptimal strategies. BANDIT PROBLEM; DYNAMIC ALLOCATION INDEX; GITTINS INDEX; MARKOV DECISION PROCESS; SUBOPTIMAL STRATEGIES