Edward A. Silver

The University of Calgary · Haskayne School of Business

In this article, we investigate the (R, S) periodic review, order-up-to level inventory control system with stochastic demand and variable leadtimes. Variable leadtimes can lead to order crossover, in which some orders arrive out of sequence. Most theoretical studies of order-up-to inventory systems under variable leadtimes assume that crossovers d...

In this paper we consider an inventory management problem motivated by a specific practical context that concerns the commonly used periodic review, reorder point, order-up-to-level control system. In particular, the reorder point and order-up-to level have to be selected so as to satisfy two management-specified constraints. First, from a marketin...

In this paper, we consider a periodic review order-up-to-level (or base stock) inventory control system under normally distributed demand. For such circumstances, an expression for the exact fill rate (fraction of demand satisfied without backordering) has been available in the literature, but has not been widely known, let alone used by practition...

In this paper, we address an important practical situation, namely where the usual replenishment lead time (when the supplier's production facility is operating) is a random variable and the supplier shuts down for an interval of known duration (for maintenance, vacation, etc.) each year. The demand rate is constant and any demand when out of stock...

Models for the economic design of control charts have, in the past, implicitly assumed that the rate at which the process goes out of control is known or can be easily estimated. It is suggested that there are difficulties with this assumption; in particular, the most obvious estimators generally perform badly. Four estimators for this rate are dev...

In this paper we consider a periodic review, reorder point, order-up-to-level system, a type commonly used in practice. Motivated by a specific practical context, we present a novel approach to determining the reorder point and order-up-to-level (for a given review interval) so as to target desired values of (i) customer fill rate and (ii) average...

This paper is concerned with movement from a current operating point so as to reach a 2D efficient frontier. After a discussion of different criteria for deciding on which point on the frontier to target, we focus, as an illustration, on a particular inventory management context and the use of the criterion of minimum distance from the current poin...

Through literature sources, personal trial and error experiences, and observations of students and other colleagues, the author has developed a collection of ideas regarding enhancing research productivity. These are laid out in this article, including a number of examples and relevant quotes from the literature. The paper should be of particular r...

This paper presents an overview of inventory management. This includes a categorization, by a number of dimensions, of inventory problems and associated models. Relevant literature references are provided within the dimensions. The paper points out the continuing gap between theory and practice, followed by a number of suggested research topics to...

In this paper, we revisit the well-known joint replenishment problem. There is a family of items with a major fixed cost associated with any replenishment of the family and a minor, (item-dependent) setup cost for each item that is included in the replenishment. In contrast with optimization methods and sophisticated (iterative) heuristics that hav...

In this paper we consider a single item, discrete time, lot sizing situation where demand is random and its parameters (e.g., mean and standard deviation) can change with time. For the appealing criterion of minimizing expected total relevant costs per unit time until the moment of the next replenishment we develop two heuristic ways of selecting a...

The periodic review inventory system is not only pervasive, but also has an extensive literature dealing with various aspects, from its theoretical underpinnings through to its performance. However, the selection of the best review period and its behaviour with respect to basic inventory parameters such as demand and supply variability appears to b...

This paper first presents a tutorial overview of inventory management. This includes a categorization, by a number of dimensions, of inventory problems and associated models. Relevant Canadian contributions to fundamental and applied theory are listed within the dimensions. Then substantial evidence is provided of widespread Canadian applications o...

We explain deviations away from the usual estimate of the expected number of false alarms per unit time in the use of a Statistical Process Control (SPC) chart. Using applied probability we develop an exact analytic expression for the expected number of false alarms per unit time. The derivation involves determining the expected value of a modulus...

The paper addresses the problem of determining replenishment timing and quantities in a multi-product environment where the aggregate level of inventory at any time is limited by the available warehouse capacity. Demands are random following a compound-Poisson process and replenishment lead times are assumed to be negligible. We formulate the probl...

Decision makers often face scheduling problems in which processing times are not known with certainty. Non-regular performance measures, in which both earliness and tardiness are penalized, are also becoming more common in both manufacturing and service operations. We model a managerial environment with task processing times (which include sequence...

This paper considers a Markov process in which there are rewards associated with state transitions. Two or more actions are available to the decision maker, and the transition probabilitities and the rewards are influenced by the actions selected. A key assumption is that control cannot be a function of the current state of the process. In other wo...

Executives and academics alike are expressing increasing interest in supplier trade credit. We consider a form of credit known as ‘date-terms’ where the credit period extends to a specified date in the month following the invoice. We extend published research to consider both demand and supply uncertainty, employing a gamma distribution to model de...

The goal of this paper is to propose a tabu search heuristic for the car sequencing problem (CSP) used for the ROADEF 2005 international Challenge. This NP-hard problem was proposed by the automobile manufacturer Renault. The rst objective of the car industry is to assign a production day to each customer-ordered car and the second one consists of...

Most inventory modelling has assumed stochastic demands and constant lead times. However, here we consider a problem for which the opposite situation holds; namely, there is a known constant demand rate, but lead times are random variables. Moreover, the probability distributions of the lead times change in a seasonal fashion. Also, shortages of ra...

An inventory system that consists of a depot (central warehouse) and retailers (regional warehouses) is considered. The system is replenished regularly on a fixed cycle by an outside supplier. Most of the stock is direct shipped to the retailer locations but some stock is sent to the central warehouse. At the beginning of any one of the periods dur...

We analyze a multi-product inventory replenishment problem with Poisson demands where the aggregate inventory level at any time is restricted by a common budget or space limitation. We present a semi-Markov decision problem formulation, several heuristics for finding the replenishment quantities (given the current inventories) and a comparison from...

We consider an organization that has more than one outlet (retailer or regional warehouse) where stock is held to meet customer demands. In such a situation, transshipments can be used to improve customer service and/or reduce the aggregate stock required. Unfortunately, the determination of an optimal transshipment policy (i.e. when to transship a...

This paper was motivated by a practical situation in the fast food industry. We consider the choice of the order quantities for a group of end items, each facing random demand in a period of interest. There is the additional option of having a limited number of units of unfinished stock that can be customized to produce any of the end items once de...

A process monitored with an chart is considered. It may go out of control due to the occurrence of one of several independent assignable causes. After the process has gone out of control and the assignable cause has been determined, the process undergoes improvement that results in a reduction of the rate due to that cause. A Bayesian estimator of...

In this article a challenge is provided to the operations management community to broaden our perspective from the traditional, functional one to the cross-functional (interdisciplinary) view of process management. At the same time, the challenge is to de-emphasize mathematical optimization and instead seek improvements through better representatio...

This note presents a variation of the well-known Silver–Meal heuristic to deal with lot sizing under a combination of a known, but time-varying, demand pattern along with an incremental quantity discount structure. The heuristic is shown to perform very well on a set of experiments presented in a recent paper in this journal. Additional experiments...

Besides those analysts who are already familiar with a number of heuristic methods, this paper should also be of interest to those analysts and managers who, although not yet aware of specific heuristic approaches, are quite comfortable with the use of mathematical modelling as an aid to decision making. It is concerned with obtaining usable soluti...

This paper examines a process that is monitored by an (X) over bar chart. It is assumed that the process may go out of control due to the occurrence of any of several independent assignable causes. The time until each specific assignable cause occurs is exponential, but the distributional parameters of the various causes are unknown and are not nec...

We extend past research on the economic lot scheduling problem to address some of the limitations of the earlier work. In particular we develop mixed integer programming formulations with the assumption of a production precedence sequence. When evaluated over a variety of problems from the literature, it is clear that finding a schedule that minimi...

Utilising emergency lateral transshipments between retailers to meet customer demand can be an effective means for companies to improve service levels and/or reduce costs. This paper provides a decision rule for how much (if any) to source and from which retailer(s), taking into account the financial impact of the transshipment decision on future s...

This paper considers a periodic-review, two-echelon inventory system with one central warehouse and several retailers facing stochastic demand. The retailers replenish their stock from the warehouse, which in turn places orders at an outside supplier with infinite capacity. Transportation times and costs are constant. No ordering costs are consider...

Combining moving averages has been suggested as a simple and practical means to improve sales forecasting. Here we present a natural extension whereby combinations of all possible moving averages up to a given number of periods are employed. We evaluate the method’s performance relative to other methods, such as simple moving averages and exponenti...

We develop a method for finding the economic sampling plan that minimizes expected total costs, subject to a constraint on labour time. We explicitly recognize the labour time required for sampling, investigation, extra processing effort per unit due to operating the process out of control, and the time to put the process back into control. These t...

The economic lot-scheduling problem (ELSP) is an important production sched-uling problem that has been intensively studied over 40 years. Numerous heuristic algorithms have been developed since the problem is NP-hard. Dobson's heuristic has been regarded as the best in its performance. The present paper provides a hybrid genetic algorithm based on...

With over 30 years of academic experience in both engineering and management faculties, involving trial and error experimentation in teaching as well as reading relevant literature and observing other instructors in action, the author has accumulated a number of ideas, regarding the preparation and delivery of a university course, that should be of...

This paper deals with the situation of a number of end items, each facing uncertain demand in a single period of interest. Besides being able to purchase units of the end items there is also available a stock of units that can be converted into end items but at unit costs that depend on the specific end item. Efficient solution procedures are prese...

We address a problem of setting reorder points (expressed as time supplies) of a population of items, subject to a restricted set of possible time supplies as well as a budget on the total amount of safety stocks, both important practical constraints. We provide a branch-and-bound algorithm for obtaining the optimal solution. In addition, a simple...

In this paper we consider a group of end items facing level demand patterns and opportunities for regular purchasing. In addition, there arc a number of units that can be converted into any one of the end items, but at different unit costs of conversion. A constrained optimization model is developed for the problem of how best to allocate the conve...

This paper deals with a multi-item newsvendor problem subject to a budget constraint on the total value of the replenishment quantities. Fixed costs for non-zero replenishments have been explicitly considered. Dynamic programming procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated...

Consider an organization whose capability to produce an item and whose customer demand are both stochastic. In such a context “take-or-pay” contracts can be attractive. Under such a contract the organization agrees to purchase from a supplier a fixed quantity per period over a specified number of periods. Simulation is too slow an analysis approach...

We address a problem of setting reorder intervals (time supplies) of a population of items, subject to a restricted set of possible intervals as well as a limit on the total number of replenishments per unit time, both important practical constraints. We provide a dynamic programming formulation for obtaining the optimal solution. In addition, a si...

In this paper we consider an important decision faced in many large-scale construction projects. Specifically, there is a surplus stock of an item at the end of the construction phase. Moreover, the item is to be used (as a spare part or for routine replacement) in the on-going operations of the facility. There is an opportunity to dispose of some...

Abstract In this paper we consider the use of an,- chart for controlling a single-stage process. X The three controllable variables, as usual, are the interval between consecutive samples, the sample size, and the distance away from the in-control process mean at which the control limits are placed. Costs considered are those of sampling, false ala...

Periodic review systems are commonly employed by distributors and retailers to replenish their inventories (for example, to co-ordinate in-bound transportation). It is also often the case that vendors specify minimum purchase quantities for physical (for example, packaging) or strategic reasons. When inventory systems recommend order quantities bel...

This paper is devoted to the study of the effects of payment terms on the replenishment strategy. We use a discounting approach with Poisson demand, a linear inventory carrying cost and a very short replenishment lead time. The case of repeated identical payments at the end of a specified period of t days, as well as the case of a one time longer p...

In the field of combinatorial optimization, it may be possible to more accurately represent reality through stochastic models rather than deterministic ones. When randomness is present in a problem, algorithm designers face new difficulties which complicate their task significantly. Finding a proper mathematical formulation and a fast evaluation of...

In this paper we consider the situation where we know the sum of n independent observations from the same probability distribution. We investigate how to empirically determine the marginal probability distributions of the different order statistics conditional upon knowing the sum. This research was motivated by explorations in process improvement...

A common approach used in quality or process improvement is cause-and-effect analysis. It involves identifying a (large) number of possible causes of an undesirable effect or problem, and then choosing the sequence of investigation of the possible causes. The philosophy of continuous improvement dictates that any associated modelling should use rel...

Considering the strategic importance of business process improvement, it is imperative that educators, students, and practitioners be familiar with this topic. Thus, the intention of this tutorial is to provide a guiding framework for carrying out improvements of business processes. We present numerous illustrative examples, taken from our personal...

This article presents a simple result involving a symmetric integration of the cumulative distribution function of an arbitrary symmetric distribution. Some background is provided regarding an inventory control policy that motivated the development of the aforementioned result. A more general relation, related to order statistics, is also presented...

This paper is concerned with a multi-period inventory problem where the demand and the supplier capacity of a given key resource are discrete random variables with known probability distribution. The procurement manager has to answer two types of questions: how much supplier quantity to reserve for each period (where all of these reservations have...

Spencer and Cox (1995), in a recent thought-provoking article, reported on a major study that led to a very effective definition of repetitive manufacturing as well as to a significant revision of the product-process matrix originally presented by Hayes and Wheelwright (1979). The study included a detailed field investigation (and associated case s...

Stochastic combinatorial problems are inherently difficult to solve. In this paper we explore the use of a relatively small subset of all of the possible random scenarios, thus drastically reducing the size of such problems. Two different methods of selecting the sample scenarios are investigated. Encouraging results are found for two variations of...

An extension of a preemptive open-shop scheduling problem is introduced. All processing times are integral and in each period i there is a cost ci, for each task which is processed in that period. Finding a schedule with minimum total cost is shown to be NP-hard; some solvable cases are discussed; bounds on the cost of an optimum schedule are compu...

In this article we consider the single period procurement strategy for an item with uncertainty in its demand and uncertainty in the capacity of the supplier. Dedicated capacity can be ensured by paying a premium charge to the supplier. The other decision variable is the replenishment quantity to request. It turns out to be very easy to select the...

In a consulting study for a major North American distributor it was observed that the materials manager had implemented a simple plausible method of establishing the reorder point of a stock keeping unit. Specifically, the demands in the last n periods (months) were observed. Then the current value of the reorder point was set as the maximum of the...

We consider a piece of equipment that is subject to breakdown. If breakdowns are very costly, it may be attractive to undertake preventive maintenance on a regular basis. We model the time until breakdown as a discrete random variable whose probability distribution is not exactly known (due to limited historical data). An optimal Bayesian strategy...

An extension of a preemptive open-shop scheduling problem is introduced. All processing times are integral and in each period i there is a cost c, for each task which is processed in that period. Finding a schedule with minimum total cost is shown to be NP-hard; some solvable cases are discussed; bounds on the cost of an optimum schedule are comput...

The situation of a family of items sharing a piece of production equipment is considered. Known level demand rates are assumed. The items are scheduled in a cyclic fashion where the production cycles through all of them every T units of time. There are shelf life limits for each of the items. I deal specifically with the situation where the cost-mi...

Practitioners of maintenance scheduling sometimes use a simple analytic formula, derived based on inspection scheduling, as a heuristic to determine the length of the preventive maintenance period. The sensitivity of this heuristic solution is analyzed and the cost penalties are calculated compared to the exact solution that utilizes the lifetime d...

In this paper we consider the context of a major project whose execution takes several periods. Specifically we are interested in the procurement policy for a resource (material, commodity or engineered item) that is used throughout the project. Two types of uncertainty are considered, namely (i) uncertainty in the requirements, period by period, a...

This paper presents a stochastic dynamic programming model for determining the optimal ordering policy for a perishable or potentially obsolete product so as to satisfy known time-varying demand over a specified planning horizon. We have considered random life time perishability where, at the end of each discrete period, the total remaining invento...

This paper introduces an alternative approach for determining lot sizes for purchased components in MRP environments where discounts are available from vendors. This new method is a variation of the least period cost procedure, which was previously found to be a poor performer relative to the least unit cost, McLaren's order moment, and the Chung e...

In this paper we consider the situation where the typical assumptions underlying the EOQ model are appropriate for an item, in particular a reasonably level demand rate exists. However, opportunities for special replenishments at a reduced unit cost occur at random. The following type of 3-parameter policy is considered. When a non-special (regular...

The purpose of this paper is to study, from a new perspective, the problem of allocating a given budget among product components, that are required for manufacturing the end items in an assemble-to-order environment. Since some of the components are common to a number of different end items, this constitutes a commonality inventory problem that in...

In this paper we consider a random variable that can take on N discrete values. The probability mass function is not assumed exactly known, rather the N probabilities have a prior N-dimensional Beta distribution. Data of two types are obtained: i). actual values of the random variable, ii). observations where all that is known is that the random va...

This paper is concerned with the relevance of mathematical modelling for organizations striving to achieve World Class Operations Management (WCOM) where the latter is viewed as a management philosophy that fosters rapid development and delivery, at reasonable cost, of high quality products and services. Continuous improvements are an integral part...

Most of the mathematical models of inventory problems, appearing in the literature, have had a rather narrow focus. Specifically, optimal solutions have been developed for mathematically stated problems, each typically subject to a number of so-called “givens”. As an example, the economic order quantity is the optimal replenishment size under a set...

When the operating time until failure of a piece of equipment is a random variable it often makes sense to undertake preventive maintenance at regular intervals. Mathematical models have been available for choosing the optimal preventive maintenance interval given that one assumes no uncertainty about the probability distribution of the operating t...

For low-usage (so-called C) items it makes sense to have rather low control costs, i.e. a simple control procedure. In this paper we develop a very simple graphical implementation aid for selecting either the reorder point in a continuous-review, reorder point, order quantity system or the order-up-to-level in a periodic-review, order-up-to-level s...

For low-usage (so-called C) items it makes sense to have rather low control costs, i.e. a simple control procedure. In this paper we develop a very simple graphical implementation aid for selecting either the reorder point in a continuous-review, reorder point, order quantity system or the order-up-to-level in a periodic-review, order-up-to-level s...

This paper addresses a so-called assemble-to-order environment. There are a number of end items each of which requires an assembly of several components. Some of the components are unique to specific end items while others are common to two or more final products. Components must, due to their long leadtimes, be ordered prior to knowing the end ite...

We consider a task being performed in an environment in which the work-rate (or efficiency of the processor varies with time in a deterministic manner. Given the probability density function (p.d.f.) of the total work-content of the task (i.e. processing time at 100% efficiency), we develop a closed-form expression for the p.d.f. of the task-proces...

In this article we consider an item for which a continuous review, reorder point, order quantity inventory control system is used. The amount of safety stock required depends upon, among other factors, the average value and variability of the length of the replenishment lead time. One way to reduce these quantities is to split orders among two or m...

In this paper we consider the policy of splitting each replenishment order among two or more vendors. This action reduces the expected time until the first replenishment arrives, as well as its variance. Thus using the same safety stock, a higher service level can be achieved or the safety stock can be reduced without decreasing the service level....

In this paper we consider a piece of manufacturing equipment dedicated to the production of a family of items. With known level demand rates for the individual items the policy used is to cycle through production of all of the items every T units of time. As usual, T is a decision variable, but we also permit the individual item production rates to...

In this paper we consider a piece of manufacturing equipment dedicated to the production of a family of items. With known level demand rates for the individual items the policy used is to cycle through production of all of the items every T units of time. The paper addresses the choice among two adjustment options when the cost-minimizing T value l...

A number of organizations sell products in containers that can be reused. The time from issue to return of an individual container is usually not known with certainty and there is a chance that the container is never returned (because of loss or irrepairable damage). Consequently, even under a level demand pattern new containers must be acquired fr...

This paper addresses a so-called assemble-to-order environment. There are a number of end items each of which requires an assembly of several components. Some of the components are unique to specific end items while others are common to two or more final products. Components must, due to the leadtimes, be ordered prior to knowing the end item deman...

In this paper we consider an item that is sold in a number of different prescribed sizes, an example being an item of clothing. We first deal with the situation where the total demand for a forthcoming period is known accurately but only historical or subjective estimates of the fractions of demand in each size are available. A procedure is develop...

This paper has addressed a number of issues and concerns related to materials management in large scale projects. Although the underlying interviews were conducted in the oil and gas industry in Canada it is likely that the concerns, suggestions and research topics identified are of interest in most other settings of major one-off projects.