The Stochastic Economic Lot Scheduling Problem: A Survey
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paper_title: Lot Sizes, Lead Times and In-Process Inventories

paper_content:

In many manufacturing problems, the total lead time taken to manufacture a product is an important consideration. Long lead times impose costs due to higher work-in-process inventory, increased uncertainty about requirements, larger safety stocks and poorer performance to due dates. Traditional lot sizing models ignore lead time related costs, although there are systematic relationships between lot sizes and lead times. This paper examines these relationships and their implications for lot sizing and work-in-process inventories, for batch manufacturing shops with queues.
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paper_title: Inventory Management and Production Planning and Scheduling

paper_content:

THE CONTEXT AND IMPORTANCE OF INVENTORY MANAGEMENT AND PRODUCTION PLANNING AND SCHEDULING. The Importance of Inventory Management and Production Planning and Scheduling. Strategic Issues. Frameworks for Inventory Management and Production Planning and Scheduling. Forecasting. TRADITIONAL REPLENISHMENT SYSTEMS FOR MANAGING INDIVIDUAL--ITEM INVENTORIES. Order Quantities When Demand is Approximately Level. Lot Sizing for Individual Items with Time--Varying Demand. Individual Items with Probabilistic Demand. SPECIAL CLASSES OF ITEMS. Managing the Most Important (Class A) Inventories. Managing Routine (Class C) Inventories. Style Goods and Perishable Items. THE COMPLEXITIES OF MULTIPLE ITEMS AND MULTIPLE LOCATIONS. Coorinated Replenishments at a Single Stocking Point. Supply Chain Management and Multiechelon Inventories. PRODUCTION PLANNING AND SCHEDULING. An Overall Framework for Production Planning and Scheduling. Medium--Range Aggregate Production Planning. Material Requirements Planning and its Extensions. Just--in--Time and Optimized Production Technology. Short--Range Production Scheduling. Summary. Appendices. Indexes.
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paper_title: A review of the stochastic lot scheduling problem

paper_content:

Abstract This paper surveys the current research literature on the stochastic lot scheduling problem which deals with scheduling production of multiple products with random demand on a single facility with limited production capacity and significant change-overs between products. The deterministic version of this problem has received significant coverage in the literature; however, the stochastic problem has been addressed only recently. Furthermore, a range of distinctly different analytical methods have been applied to this problem. This paper provides a unifying framework for discussing these approaches and offers some explanation and clarification of the different analytical methods for this problem. After discussing some of the modeling and managerial implications of this problem, a detailed review of both continuous and discrete time control strategies is given, and areas for further research are outlined.
---
paper_title: Multi-item, single-machine scheduling problem with stochastic demands: a comparison of heuristics

paper_content:

This paper discusses the results of simulation tests for multi-item, single-machine production systems facing stochastic, time-varying demands. Six different heuristics are compared with the objective of minimizing the sum of changeover and inventory costs. Tests are performed to evaluate the effect of the stationarity of the demand process and of the level of forecast errors on the performance of the heuristics tested.
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paper_title: Foundations of Inventory Management

paper_content:

Systems and models one item with a constant demand rate time-varying demands several products and locations stochastic demand - one item with constant leadtimes stochastic leadtimes - the structure of the supply system several items with stochastic demands time-varying, stochastic demand - policy optimization. Appendices: optimization and convexity dynamical systems probability and stochastic processes notational conventions.
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paper_title: Managing Stochastic Multi-product Systems: Model, Measures, and Analysis

paper_content:

We consider a model for managing a single stage that produces multiple items. The production rates are finite, and there are switchover times. The interarrival times and quantities of demands for the items are random, and demand may occur for a set of items. We consider order focussed measures: cost based on response times; service levels based on quoted lead times and Type-1 service. We operate the stage in the following manner: (1) there is a cyclic schedule that determines the sequence of items and the number of times a particular item is produced in a cycle; (2) given a cyclic schedule, production of each item follows a modified base-stock policy or a (s, S) policy. We present a simulation based procedure to obtain good values for the base stock levels or S (for any fixed S-s) for each of the above performance measures. Numerical results indicate that good solutions can be obtained with modest computational effort. We also report on a real world implementation of this model.
---
paper_title: Demand Management and Production Control in Process Industries

paper_content:

In a changing and more dynamic market of the 1980s, available mathematical programming algorithms on which production planning in process industries was based did not provide the required responsiveness. Although it is suggested in the literature that a variable cycle times policy will enable the system to react to short‐term demand fluctuations, proposes the use of a fixed cycle times policy. Presents simulation results which show a considerable improvement in service level at a high level of utilization. Presents a three‐tiered hierarchical model which is based on this fixed cycle times policy, which gives insight into the influence that the demand manager has upon the results in the production department. On the other hand, it also reflects the commercial interests. Suggests that for both departments which are this strongly intertwined, a common reference is necessary to improve the general results of the business.
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paper_title: The Impact of Adding a Make-To-Order Item to a Make-To-Stock Production System

paper_content:

Stochastic Economic Lot Scheduling Problems (ELSPs) involve settings where several items need to be produced in a common facility with limited capacity, under significant uncertainty regarding demands, unit production times, setup times, or combinations thereof. We consider systems where some products are made-to-stock while another product line is made-to-order. We present a rich and effective class of strategies for which a variety of cost and performance measures can be evaluated and optimized efficiently by analytical methods. These include inventory level and waiting-time distributions, as well as average setup, holding, and backlogging costs. We also characterize how strategy choices are affected by the system parameters. The availability of efficient analytical evaluation and optimization methods permits us to address the impact of product line diversification or standardization on the performance of the manufacturing system, in particular the logistical implications of adding low-volume specialized models to a given make-to-stock product line.
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paper_title: Combining make to order and make to stock

paper_content:

In inventory control and production planning one is tempted to use one of two strategies: produce all demand to stock or produce all demand to order. The disadvantages are well-known. In the ‘make everything to order’ case (MTO) the response times may become quite long if the load is high, in the ‘make everything to stock’ case (MTS) one gets an enormous inventory if the number of different products is large.
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paper_title: A review of scheduling research involving setup considerations

paper_content:

The majority of scheduling research assumes setup as negligible or part of the processing time. While this assumption simplifies the analysis and/or reflects certain applications, it adversely affects the solution quality for many applications which require explicit treatment of setup. Such applications, coupled with the emergence of production concepts like time-based competition and group technology, have motivated increasing interest to include setup considerations in scheduling problems. This paper provides a comprehensive review of the literature on scheduling problems involving setup times (costs). It classifies scheduling problems into batch and non-batch, sequence-independent and sequence-dependent setup, and categorizes the literature according to the shop environments of single machine, parallel machines, flowshops, and job shops. The suggested classification scheme organizes the scheduling literature involving setup considerations, summarizes the current research results for different problem types, and finally provides guidelines for future research.
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paper_title: A hierarchical approach for capacity coordination in multiple products single-machine production systems with stationary stochastic demands

paper_content:

We introduce a two-level hierarchical planning and scheduling approach for multi-item, single-machine production systems facing stochastic demand. The hierarchical approach extends the existing heuristics to handle situations where demand levels are high compared to the available production capacity. At the top level of the hierarchy the approach deploys a heuristic procedure for coordinating capacity in the medium term. The heuristic produces target production cycle times and target service levels for each of the products. It iteratively allocates capacity to individual products for maximizing the expected profit while meeting the individual product service level targets set by the management. The bottom level of the hierarchy focuses on the short term and aims to realize the target service level while keeping the production cycles stable. At this level, a fixed-sequence order up-to operational scheduling policy which permits a limited amount of flexibility for reacting to short-term demand variations is implemented. Demand which cannot be met from inventory gets lost. We also present and discuss the results of extensive simulation tests under a wide range of environments that serve to demonstrate the superiority of the approach, especially when the demand level is high.
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paper_title: The Economic Lot Scheduling Problem (ELSP): Review and Extensions

paper_content:

The ELSP is a time-honored problem that "has been around" since 1915. It is the problem of accommodating cyclical production patterns when several products are made on a single facility. Recent contributions to its resolution resulted in either analytical approaches to a restricted problem, or heuristic approaches to the entire problem. This paper reviews critically the various contributions to the problem, and extends the analysis in the following four directions: (a) An improved analytical approach, (b) A test for feasibility, (c) A systematic means for escape from infeasibility, and (d) A procedure for the determination of a basic period for a given set of multipliers to achieve a feasible schedule.
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paper_title: A review of the stochastic lot scheduling problem

paper_content:

Abstract This paper surveys the current research literature on the stochastic lot scheduling problem which deals with scheduling production of multiple products with random demand on a single facility with limited production capacity and significant change-overs between products. The deterministic version of this problem has received significant coverage in the literature; however, the stochastic problem has been addressed only recently. Furthermore, a range of distinctly different analytical methods have been applied to this problem. This paper provides a unifying framework for discussing these approaches and offers some explanation and clarification of the different analytical methods for this problem. After discussing some of the modeling and managerial implications of this problem, a detailed review of both continuous and discrete time control strategies is given, and areas for further research are outlined.
---
paper_title: Multi-item, single-machine scheduling problem with stochastic demands: a comparison of heuristics

paper_content:

This paper discusses the results of simulation tests for multi-item, single-machine production systems facing stochastic, time-varying demands. Six different heuristics are compared with the objective of minimizing the sum of changeover and inventory costs. Tests are performed to evaluate the effect of the stationarity of the demand process and of the level of forecast errors on the performance of the heuristics tested.
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paper_title: Scheduling Rules For The Multiple Product Single Machine System With Stochastic Demand

paper_content:

AbstractThe scheduling of multiple products with stochastic demand on a single machine system was modelled. Simulation of six scheduling rules under varying cost and system parameters was performed. It was shown that several of the commonly proposed rules resulted in very costly schedules. It was also shown that the best scheduling rule for a particular system depends on the number of products, distribution of demand among those products, and the proportion of capacity utilization, as well as on the relative costs for carrying inventory, setup, and shortages.
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paper_title: The Multi-Product Production Cycling Problem

paper_content:

Abstract The multi-product production cycling problem is concerned with the determination of a production/inventory policy for a single capacitated production facility which is dedicated to producing a family of products. This paper studies this problem assuming stochastic demand. The one-product problem may be analyzed as a Markov decision problem and solved as such. For the multi-product problem, a heuristic decision rule is proposed based on the analysis of the one-product problem, and a new notion: the composite product. This heuristic is then tested by simulation on a series of problems, and is shown to be the most effective of the heuristics considered for the set of test problems
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paper_title: Scheduling the Production of Several Items with Random Demands in a Single Facility

paper_content:

Consider the problem of scheduling the production of several items in a single facility that can produce only one item at a time. This problem occurs since it is often economic to produce several items in a single facility. The objective is to reduce the long run average holding, backorder and setup costs. ::: ::: We assume that demands are random with constant expected rates. We allow backorders and charge holding and backlogging costs at linear time weighted rates. Items are produced at continuous constant rates. Setup times and setup costs are item dependent constants. These parameters, however, are independent of the order of setups. ::: ::: A real-time scheduling tool is developed in three steps. First, with demands replaced by their expectations, we compute an optimal or near-optimal target cyclic schedule. Next, we study the problem of scheduling the facility after a single disruption perturbs the inventories. The goal is to recover the target cyclic schedule at minimal excess over the average cost of the cyclic schedule. We formulate this as a control problem and obtain a linear recovery policy that is optimal for a large configuration of disruptions. Finally, we select safety stocks to minimize the long run average cost of following the target schedule with the recovery policy. We show that optimal safety stocks are unique and have the property that in the long run the proportion of time that an item is in stock is the ratio of backorder to holding plus backorder cost. We present an example that integrates the cyclic schedule, the control policy and the safety stocks.
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paper_title: Determining Production Schedules Under Base-Stock Policies in Single Facility Multi-Item Production Systems

paper_content:

In this paper we address periodic base-stock policies for stochastic economic lot scheduling problems. These represent manufacturing settings in which multiple items compete for the availability of a common capacity source, in the presence of setup times and/or costs, incurred when switching between items, and in the presence of uncertainty regarding demand patterns, production, and setup times. Under periodic base-stock policies, items are produced according to a given periodic item-sequence. This paper derives effective heuristics for the design of a periodic item-sequence minimizing system-wide costs. This sequence is constructed based on desirable production frequencies for the items, obtained as the solution of lower bound mathematical programs. An extensive numerical study gauges the quality of the proposed heuristics.
---
paper_title: The strategic use of capacity slack in the economic lot scheduling problem with random demand

paper_content:

Growing interest in designing systems with capacity slack as one form of flexibility raises many questions about its use and its usefulness. In the framework of the economic lot scheduling problem with stochastic demand, we develop on optimization-based model that considers capacity slack, safety stock, and overtime explicitly, and has the objective of minimizing the expected cost per unit time of inventory, overtime, and, where applicable, setup costs. The solution is a continuous-time production plan that consists of a time-dependent inventory trajectory for each of the parts, including the placement of planned idle time in the schedule. We consider schedule stability to be desirable because of potential effects on upstream and downstream operations in multistage production settings. Thus, the plan also has certain characteristics that contribute to achieving stability. Our results on the relative merits of capacity slack and safety stock indicate that capacity slack in the form of planned idle time is not a cost-effective hedge against demand uncertainty in this context. Thus, it is essential that management carefully identify and evaluate other reasons for including idle time in a plan, and use the idle time effectively. For managers who face situations not fully represented by our model, this paper provides analytic results that will guide them in the placement of planned idle time and the choice of safety stock levels.
---
paper_title: Managing Stochastic Multi-product Systems: Model, Measures, and Analysis

paper_content:

We consider a model for managing a single stage that produces multiple items. The production rates are finite, and there are switchover times. The interarrival times and quantities of demands for the items are random, and demand may occur for a set of items. We consider order focussed measures: cost based on response times; service levels based on quoted lead times and Type-1 service. We operate the stage in the following manner: (1) there is a cyclic schedule that determines the sequence of items and the number of times a particular item is produced in a cycle; (2) given a cyclic schedule, production of each item follows a modified base-stock policy or a (s, S) policy. We present a simulation based procedure to obtain good values for the base stock levels or S (for any fixed S-s) for each of the above performance measures. Numerical results indicate that good solutions can be obtained with modest computational effort. We also report on a real world implementation of this model.
---
paper_title: Heavy Traffic Analysis of Dynamic Cyclic Policies: A Unified Treatment of the Single Machine Scheduling Problem

paper_content:

This paper examines how setups, due dates, and the mix of standardized and customized products affect the scheduling of a single machine operating in a dynamic and stochastic environment. We restrict ourselves to the class of dynamic cyclic policies, where the machine busy/idle policy and lot-sizing decisions are controlled in a dynamic fashion, but different products must be produced in a fixed sequence. As in earlier work, we conjecture that an averaging principle holds for this queueing system in the heavy traffic limit, and optimize over the class of dynamic cyclic policies. The results allow for a detailed discussion of the interactions between the due-date, setup, and product mix facets of the problem.
---
paper_title: A review of the stochastic lot scheduling problem

paper_content:

Abstract This paper surveys the current research literature on the stochastic lot scheduling problem which deals with scheduling production of multiple products with random demand on a single facility with limited production capacity and significant change-overs between products. The deterministic version of this problem has received significant coverage in the literature; however, the stochastic problem has been addressed only recently. Furthermore, a range of distinctly different analytical methods have been applied to this problem. This paper provides a unifying framework for discussing these approaches and offers some explanation and clarification of the different analytical methods for this problem. After discussing some of the modeling and managerial implications of this problem, a detailed review of both continuous and discrete time control strategies is given, and areas for further research are outlined.
---
paper_title: Multi-item, single-machine scheduling problem with stochastic demands: a comparison of heuristics

paper_content:

This paper discusses the results of simulation tests for multi-item, single-machine production systems facing stochastic, time-varying demands. Six different heuristics are compared with the objective of minimizing the sum of changeover and inventory costs. Tests are performed to evaluate the effect of the stationarity of the demand process and of the level of forecast errors on the performance of the heuristics tested.
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paper_title: The stochastic economic lot scheduling problem: heavy traffic analysis of dynamic cyclic policies

paper_content:

We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processesN customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class has its own general service time distribution, renewal demand process, and holding and backordering cost rates. In the first problem, a setup cost is incurred when the server switches class, and the setup cost is replaced by a setup time in the second problem. In both problems we employ a long-run average cost criterion and restrict ourselves to a class of dynamic cyclic policies, where idle periods and lot sizes are state-dependent, but theN classes must be served in a fixed sequence. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these scheduling problems by diffusion control problems. Our analysis of the approximating setup cost problem yields a closed-form dynamic lot-sizing policy and a computational procedure for an idling threshold. We derive structural results and an algorithmic procedure for the setup time problem. A computational study compares the proposed policy and several alternative policies to the numerically computed optimal policy.
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paper_title: When is a base stock policy optimal in recovering disrupted cyclic schedules

paper_content:

The extended economic lot scheduling problem (EELSP) is concerned with scheduling the production of a set of items in a single facility to minimize the long-run average holding, backlogging, and setup costs. Given an efficient cyclic production schedule for the EELSP, called the target schedule, we consider the problem of how to schedule production after a single schedule disruption. We propose a base stock policy, characterized by a base stock vector, that prescribes producing an item until its inventory level reaches the peak inventory of the target schedule corresponding to the item's position in the production sequence. We show that the base stock policy is always successful in recovering the target schedule. Moreover, the base stock policy recovers the target schedule at minimal excess over average cost whenever the backorder costs are proportional to the processing times. This condition holds, for example, when the value of the items is proportional to their processing times, and a common inventory carrying cost and a common service level is used for all the items. Alternatively, the proportionality condition holds if the inventory manager is willing to select the service levels from a certain set that is large enough to guarantee any minimal level of service, and then uses the imputed values for the backorder costs. When the proportionality condition holds we provide a closed-form expression for the total relevant excess over average cost of recovering the target schedule. We assess the performance of the base stock policy when the proportionality condition does not hold through a numerical study, and suggest some heuristic uses of the base stock policy. © 1994 John Wiley & Sons, Inc.
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paper_title: The strategic use of capacity slack in the economic lot scheduling problem with random demand

paper_content:

Growing interest in designing systems with capacity slack as one form of flexibility raises many questions about its use and its usefulness. In the framework of the economic lot scheduling problem with stochastic demand, we develop on optimization-based model that considers capacity slack, safety stock, and overtime explicitly, and has the objective of minimizing the expected cost per unit time of inventory, overtime, and, where applicable, setup costs. The solution is a continuous-time production plan that consists of a time-dependent inventory trajectory for each of the parts, including the placement of planned idle time in the schedule. We consider schedule stability to be desirable because of potential effects on upstream and downstream operations in multistage production settings. Thus, the plan also has certain characteristics that contribute to achieving stability. Our results on the relative merits of capacity slack and safety stock indicate that capacity slack in the form of planned idle time is not a cost-effective hedge against demand uncertainty in this context. Thus, it is essential that management carefully identify and evaluate other reasons for including idle time in a plan, and use the idle time effectively. For managers who face situations not fully represented by our model, this paper provides analytic results that will guide them in the placement of planned idle time and the choice of safety stock levels.
---
paper_title: Managing Stochastic Multi-product Systems: Model, Measures, and Analysis

paper_content:

We consider a model for managing a single stage that produces multiple items. The production rates are finite, and there are switchover times. The interarrival times and quantities of demands for the items are random, and demand may occur for a set of items. We consider order focussed measures: cost based on response times; service levels based on quoted lead times and Type-1 service. We operate the stage in the following manner: (1) there is a cyclic schedule that determines the sequence of items and the number of times a particular item is produced in a cycle; (2) given a cyclic schedule, production of each item follows a modified base-stock policy or a (s, S) policy. We present a simulation based procedure to obtain good values for the base stock levels or S (for any fixed S-s) for each of the above performance measures. Numerical results indicate that good solutions can be obtained with modest computational effort. We also report on a real world implementation of this model.
---
paper_title: Demand Management and Production Control in Process Industries

paper_content:

In a changing and more dynamic market of the 1980s, available mathematical programming algorithms on which production planning in process industries was based did not provide the required responsiveness. Although it is suggested in the literature that a variable cycle times policy will enable the system to react to short‐term demand fluctuations, proposes the use of a fixed cycle times policy. Presents simulation results which show a considerable improvement in service level at a high level of utilization. Presents a three‐tiered hierarchical model which is based on this fixed cycle times policy, which gives insight into the influence that the demand manager has upon the results in the production department. On the other hand, it also reflects the commercial interests. Suggests that for both departments which are this strongly intertwined, a common reference is necessary to improve the general results of the business.
---
paper_title: The stochastic dynamic product cycling problem

paper_content:

Abstract This paper studies the stochastic version of the dynamic, discrete period product cycling problem, in which several products are to be sequenced one at a time, on a single machine, to meet time-varying stochastic demands. The demand distribution for each product in each period is taken to be known, and there are variable costs incurred for holding inventory between periods, for backorders, and for production. At each changeover, fixed costs are incurred and production time is lost. All these costs can be dynamic. The problem is formulated as a multiperiod, dynamic program. A ‘restricted’ Lagrangean method is used to develop three alternative decompositions. One of these is used to investigate bounds and heuristic solutions. Preliminary numerical results on small problems are presented.
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paper_title: Optimization-based planning for the stochastic lot-scheduling problem

paper_content:

We describe a finite-horizon stochastic optimization model for the stochastic lot-scheduling problem and procedures for finding near-optimal solutions. Several different products are produced by a single-stage process with significant changeover times and costs, and the demand for these products is random. The deterministic version of this problem, the economic lot-scheduling problem, is the subject of a great deal of research. However, the problem with random demand for the products is commonly found in practice but is not as well researched. The models developed in this paper address the problem of dynamically planning the timing and size of production runs in this kind of production environment. We also report some computational results that indicate the quality of the resulting production schedules.
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paper_title: A hierarchical approach for capacity coordination in multiple products single-machine production systems with stationary stochastic demands

paper_content:

We introduce a two-level hierarchical planning and scheduling approach for multi-item, single-machine production systems facing stochastic demand. The hierarchical approach extends the existing heuristics to handle situations where demand levels are high compared to the available production capacity. At the top level of the hierarchy the approach deploys a heuristic procedure for coordinating capacity in the medium term. The heuristic produces target production cycle times and target service levels for each of the products. It iteratively allocates capacity to individual products for maximizing the expected profit while meeting the individual product service level targets set by the management. The bottom level of the hierarchy focuses on the short term and aims to realize the target service level while keeping the production cycles stable. At this level, a fixed-sequence order up-to operational scheduling policy which permits a limited amount of flexibility for reacting to short-term demand variations is implemented. Demand which cannot be met from inventory gets lost. We also present and discuss the results of extensive simulation tests under a wide range of environments that serve to demonstrate the superiority of the approach, especially when the demand level is high.
---
paper_title: Multiproduct production/inventory control under random demands

paper_content:

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the "norm-based" approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time. >
---
paper_title: The Multi-Product Production Cycling Problem

paper_content:

Abstract The multi-product production cycling problem is concerned with the determination of a production/inventory policy for a single capacitated production facility which is dedicated to producing a family of products. This paper studies this problem assuming stochastic demand. The one-product problem may be analyzed as a Markov decision problem and solved as such. For the multi-product problem, a heuristic decision rule is proposed based on the analysis of the one-product problem, and a new notion: the composite product. This heuristic is then tested by simulation on a series of problems, and is shown to be the most effective of the heuristics considered for the set of test problems
---
paper_title: Scheduling the Production of Several Items with Random Demands in a Single Facility

paper_content:

Consider the problem of scheduling the production of several items in a single facility that can produce only one item at a time. This problem occurs since it is often economic to produce several items in a single facility. The objective is to reduce the long run average holding, backorder and setup costs. ::: ::: We assume that demands are random with constant expected rates. We allow backorders and charge holding and backlogging costs at linear time weighted rates. Items are produced at continuous constant rates. Setup times and setup costs are item dependent constants. These parameters, however, are independent of the order of setups. ::: ::: A real-time scheduling tool is developed in three steps. First, with demands replaced by their expectations, we compute an optimal or near-optimal target cyclic schedule. Next, we study the problem of scheduling the facility after a single disruption perturbs the inventories. The goal is to recover the target cyclic schedule at minimal excess over the average cost of the cyclic schedule. We formulate this as a control problem and obtain a linear recovery policy that is optimal for a large configuration of disruptions. Finally, we select safety stocks to minimize the long run average cost of following the target schedule with the recovery policy. We show that optimal safety stocks are unique and have the property that in the long run the proportion of time that an item is in stock is the ratio of backorder to holding plus backorder cost. We present an example that integrates the cyclic schedule, the control policy and the safety stocks.
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paper_title: Note: An Improvement to the Dynamic Cycle Lengths Heuristic for Scheduling the Multi-Item, Single-Machine

paper_content:

A heuristic scheduling policy for multi-item, single-machine scheduling systems facing stochastic, time-varying demands was previously introduced by Leachman and Gascon. In this technical note we provide an improvement to the calculations of the policy. Simulation tests of the revised version of the policy indicate that the improvements maintain or enhance customer service while reducing total inventory and setup costs on the order of 1-7%.
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paper_title: Heavy Traffic Analysis of Dynamic Cyclic Policies: A Unified Treatment of the Single Machine Scheduling Problem

paper_content:

This paper examines how setups, due dates, and the mix of standardized and customized products affect the scheduling of a single machine operating in a dynamic and stochastic environment. We restrict ourselves to the class of dynamic cyclic policies, where the machine busy/idle policy and lot-sizing decisions are controlled in a dynamic fashion, but different products must be produced in a fixed sequence. As in earlier work, we conjecture that an averaging principle holds for this queueing system in the heavy traffic limit, and optimize over the class of dynamic cyclic policies. The results allow for a detailed discussion of the interactions between the due-date, setup, and product mix facets of the problem.
---
paper_title: SINGLE-STAGE, MULTI-PRODUCT PRODUCTION/INVENTORY SYSTEMS WITH BACKORDERS

paper_content:

A production/inventory system consisting of a single-stage facility producing multiple product types and a warehouse is considered. The demand process is assumed to be Poisson and the excess demand is backordered. Products have a priority structure and the processor is shared according to a switching rule. (R, r) continuous-review inventory control policies are used to start and stop production of each product. Production of a product continues until its target level is reached. Then, a switch-over takes place to the highest priority product among the ones that need the processor's attention. A set-up takes place every time a switch-over occurs. Both processing and set-up times are assumed to have arbitrary distributions. The number of switch-overs among the high priority products is restricted in case low priority products need attention. The objective is to obtain performance measures such as the average inventory and backorder levels as well as die probability of having a backorder upon arrival for eac...
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paper_title: A review of the stochastic lot scheduling problem

paper_content:

Abstract This paper surveys the current research literature on the stochastic lot scheduling problem which deals with scheduling production of multiple products with random demand on a single facility with limited production capacity and significant change-overs between products. The deterministic version of this problem has received significant coverage in the literature; however, the stochastic problem has been addressed only recently. Furthermore, a range of distinctly different analytical methods have been applied to this problem. This paper provides a unifying framework for discussing these approaches and offers some explanation and clarification of the different analytical methods for this problem. After discussing some of the modeling and managerial implications of this problem, a detailed review of both continuous and discrete time control strategies is given, and areas for further research are outlined.
---
paper_title: Determination of an order-up-to policy in the stochastic economic lot scheduling model

paper_content:

We consider a production–inventory problem with compound renewal item demand. The model consists of stockpoints, one for each item, controlled according to (R,S)-policies and one machine which replenishes them. The replenishment orders are produced with a fixed rate on the machine with significant setup times and costs, which are stochastic and sequence dependent. The time between the release and the production of the replenishment order is called the waiting time. We develop analytical approximations for the first two moments of this waiting time, the order-up-to levels and the average physical inventory levels for all stockpoints, given the target fill rates. These analytical approximations allows for a quick evaluation of the waiting time which is important when optimization of the system is considered.
---
paper_title: Multi-item, single-machine scheduling problem with stochastic demands: a comparison of heuristics

paper_content:

This paper discusses the results of simulation tests for multi-item, single-machine production systems facing stochastic, time-varying demands. Six different heuristics are compared with the objective of minimizing the sum of changeover and inventory costs. Tests are performed to evaluate the effect of the stationarity of the demand process and of the level of forecast errors on the performance of the heuristics tested.
---
paper_title: Determining Production Schedules Under Base-Stock Policies in Single Facility Multi-Item Production Systems

paper_content:

In this paper we address periodic base-stock policies for stochastic economic lot scheduling problems. These represent manufacturing settings in which multiple items compete for the availability of a common capacity source, in the presence of setup times and/or costs, incurred when switching between items, and in the presence of uncertainty regarding demand patterns, production, and setup times. Under periodic base-stock policies, items are produced according to a given periodic item-sequence. This paper derives effective heuristics for the design of a periodic item-sequence minimizing system-wide costs. This sequence is constructed based on desirable production frequencies for the items, obtained as the solution of lower bound mathematical programs. An extensive numerical study gauges the quality of the proposed heuristics.
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paper_title: The Stochastic Economic Lot Scheduling Problem: Cyclical Base-Stock Policies with Idle Times

paper_content:

In this paper we discuss stochastic Economic Lot Scheduling Problems ELSP, i.e., settings where several items need to be produced in a common facility with limited capacity, under significant uncertainty regarding demands, production times, setup times, or combinations thereof. We propose a class of production/inventory strategies for stochastic ELSPs and describe how a strategy which minimizes holding, backlogging, and setup costs within this class can be effectively determined and evaluated. The proposed class of strategies is simple but rich and effective: when the facility is assigned to a given item, production continues until either a specific target inventory level is reached or a specific production batch has been completed; the different items are produced in a given sequence or rotation cycle, possibly with idle times inserted between the completion of an item's production batch and the setup for the next item. An optimal strategy within the class can be determined, and all relevant performance measures can be evaluated in just a few CPU seconds, using a 486-based PC. We also derive a number of easily computable lower bounds for the optimal cost value and establish a comparison with deterministic ELSPs.
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paper_title: Analysis of Multi-Product Kanban Systems with State-Dependent Setups and Lost Sales

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We propose a decomposition-based approximation method that generates fairly accurate estimates for steady-state performance measures of a kanban-controlled production system. The manufacturing facility of this system can process items of several different products. Setup and processing times are assumed to be exponentially distributed. Customers arrive according to mutually independent Poisson processes. A customer whose demand cannot be met from stock leaves the system and satisfies his demand elsewhere (lost sales). The manufacturing facility processes items of a product until a target inventory level given by the number of kanbans has been reached. Then the manufacturing facility is set up for the next product according to a fixed setup sequence if the next product's inventory level is below target. Otherwise, this product is skipped (cyclic-exhaustive processing with state-dependent setups). The manufacturing facility idles when the inventory levels of all products are at their target levels. Copyright Kluwer Academic Publishers 2004
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paper_title: The stochastic economic lot scheduling problem: heavy traffic analysis of dynamic cyclic policies

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We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processesN customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class has its own general service time distribution, renewal demand process, and holding and backordering cost rates. In the first problem, a setup cost is incurred when the server switches class, and the setup cost is replaced by a setup time in the second problem. In both problems we employ a long-run average cost criterion and restrict ourselves to a class of dynamic cyclic policies, where idle periods and lot sizes are state-dependent, but theN classes must be served in a fixed sequence. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these scheduling problems by diffusion control problems. Our analysis of the approximating setup cost problem yields a closed-form dynamic lot-sizing policy and a computational procedure for an idling threshold. We derive structural results and an algorithmic procedure for the setup time problem. A computational study compares the proposed policy and several alternative policies to the numerically computed optimal policy.
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paper_title: Models for Design and Control of Stochastic, Multi-Item Batch Production Systems

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We propose an approach to modeling a production facility that makes many products in large, discrete batches, when demands and the production process are stochastic. This approach combines standard inventory and queueing submodels into classical optimization problems. The models developed in this paper to illustrate the approach represent the production facility by a single-server queueing system or by a network of queues; other systems can also be handled within the approach. The models optimize the approximate operating cost of a given facility over certain simple, plausible control policies; furthermore, the models are tractable enough to be used routinely for design studies.
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paper_title: When is a base stock policy optimal in recovering disrupted cyclic schedules

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The extended economic lot scheduling problem (EELSP) is concerned with scheduling the production of a set of items in a single facility to minimize the long-run average holding, backlogging, and setup costs. Given an efficient cyclic production schedule for the EELSP, called the target schedule, we consider the problem of how to schedule production after a single schedule disruption. We propose a base stock policy, characterized by a base stock vector, that prescribes producing an item until its inventory level reaches the peak inventory of the target schedule corresponding to the item's position in the production sequence. We show that the base stock policy is always successful in recovering the target schedule. Moreover, the base stock policy recovers the target schedule at minimal excess over average cost whenever the backorder costs are proportional to the processing times. This condition holds, for example, when the value of the items is proportional to their processing times, and a common inventory carrying cost and a common service level is used for all the items. Alternatively, the proportionality condition holds if the inventory manager is willing to select the service levels from a certain set that is large enough to guarantee any minimal level of service, and then uses the imputed values for the backorder costs. When the proportionality condition holds we provide a closed-form expression for the total relevant excess over average cost of recovering the target schedule. We assess the performance of the base stock policy when the proportionality condition does not hold through a numerical study, and suggest some heuristic uses of the base stock policy. © 1994 John Wiley & Sons, Inc.
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paper_title: The strategic use of capacity slack in the economic lot scheduling problem with random demand

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Growing interest in designing systems with capacity slack as one form of flexibility raises many questions about its use and its usefulness. In the framework of the economic lot scheduling problem with stochastic demand, we develop on optimization-based model that considers capacity slack, safety stock, and overtime explicitly, and has the objective of minimizing the expected cost per unit time of inventory, overtime, and, where applicable, setup costs. The solution is a continuous-time production plan that consists of a time-dependent inventory trajectory for each of the parts, including the placement of planned idle time in the schedule. We consider schedule stability to be desirable because of potential effects on upstream and downstream operations in multistage production settings. Thus, the plan also has certain characteristics that contribute to achieving stability. Our results on the relative merits of capacity slack and safety stock indicate that capacity slack in the form of planned idle time is not a cost-effective hedge against demand uncertainty in this context. Thus, it is essential that management carefully identify and evaluate other reasons for including idle time in a plan, and use the idle time effectively. For managers who face situations not fully represented by our model, this paper provides analytic results that will guide them in the placement of planned idle time and the choice of safety stock levels.
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paper_title: Managing Stochastic Multi-product Systems: Model, Measures, and Analysis

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We consider a model for managing a single stage that produces multiple items. The production rates are finite, and there are switchover times. The interarrival times and quantities of demands for the items are random, and demand may occur for a set of items. We consider order focussed measures: cost based on response times; service levels based on quoted lead times and Type-1 service. We operate the stage in the following manner: (1) there is a cyclic schedule that determines the sequence of items and the number of times a particular item is produced in a cycle; (2) given a cyclic schedule, production of each item follows a modified base-stock policy or a (s, S) policy. We present a simulation based procedure to obtain good values for the base stock levels or S (for any fixed S-s) for each of the above performance measures. Numerical results indicate that good solutions can be obtained with modest computational effort. We also report on a real world implementation of this model.
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paper_title: Demand Management and Production Control in Process Industries

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In a changing and more dynamic market of the 1980s, available mathematical programming algorithms on which production planning in process industries was based did not provide the required responsiveness. Although it is suggested in the literature that a variable cycle times policy will enable the system to react to short‐term demand fluctuations, proposes the use of a fixed cycle times policy. Presents simulation results which show a considerable improvement in service level at a high level of utilization. Presents a three‐tiered hierarchical model which is based on this fixed cycle times policy, which gives insight into the influence that the demand manager has upon the results in the production department. On the other hand, it also reflects the commercial interests. Suggests that for both departments which are this strongly intertwined, a common reference is necessary to improve the general results of the business.
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paper_title: Approximating queue size and waiting time distributions in general polling systems

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Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system withN stations, one finds the firstK probability density function values of the steady state queue size in any given station inO(max(N, K2) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings.
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paper_title: Pull-type manufacturing systems with multiple product types

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A multiple-product facility consisting of a manufacturing process and a finished product warehouse for several types of products is considered. An (R,r) continuous-review inventory control policy is applied to each type of product in the warehouse. The demand arrival process of each type is assumed to be Poisson. The processing times and the set-up times are arbitrarily distributed. We have developed an iterative procedure to approximately compute the average inventory level of each product in the warehouse under different priority schemes.
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paper_title: A decomposition method for multi-product kanban systems with setup times and lost sales

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We consider kanban controlled production systems with three or more different products processed on a single manufacturing facility. Customers for a product arrive according to a Poisson process. If a customer's demand cannot be met from stock, the customer leaves and satisfies his demand elsewhere (lost sales). Between the production of different products setup changes must be performed that take a significant time. Setup times and processing times are product-specific and follow exponential distributions. A production run continues until the target inventory level given by the number of kanbans for the product has been reached (exhaustive processing). Then the manufacturing facility is set up for the next product according to a fixed setup cycle. The exact model is mathematically intractable even for smaller systems. Therefore, we propose a decomposition-based approximation method for estimating steady-state performance measures
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paper_title: Combined make-to-order and make-to-stock in a food Production System

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The research into multi-product production/inventory control systems has mainly assumed one of the two strategies: Make-to-Order (MTO) or Make-to-Stock (MTS). In practice, however, many companies cater to an increasing variety of products with varying logistical demands (e.g. short due dates, specific products) and production characteristics (e.g. capacity usage, setup) to different market segments and so they are moving to more MTO-production. As a consequence they operate under a hybrid MTO-MTS strategy. Important issues arising out of such situations are, for example, which products should be manufactured to stock and which ones on order and, how to allocate capacity among various MTO-MTS products. This paper presents the state-of-the-art literature review of the combined MTO-MTS production situations. A variety of production management issues in the context of food processing companies, where combined MTO-MTS production is quite common, are discussed in details. The authors propose a comprehensive hierarchical planning framework that covers the important production management decisions to serve as a starting point for evaluation and further research on the planning system for MTO-MTS situations.
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paper_title: Customer waiting-time distributions under base-stock policies in single-facility multi-item production systems

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We derive efficient and highly accurate approximations for the customer waiting-time distributions experienced in stochastic economic lot scheduling systems (SELSPs) that are governed by general base-stock policies under a cyclic or more general periodic item sequence. SELSPs involve settings where several items need to be produced in a common facility with limited capacity, under significant uncertainty regarding demands, unit production times, setup times, or combinations thereof. Under a base-stock policy one continues production of a given item until a specific target level is reached; the different items are produced in a given periodic sequence, possibly with idle times inserted between the completion of an item's production batch and the setup of the next item. We also demonstrate how base-stock levels can be specified efficiently to guarantee any prescribed fill rate within a specific service window or distribution of service windows.
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paper_title: A hierarchical approach for capacity coordination in multiple products single-machine production systems with stationary stochastic demands

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We introduce a two-level hierarchical planning and scheduling approach for multi-item, single-machine production systems facing stochastic demand. The hierarchical approach extends the existing heuristics to handle situations where demand levels are high compared to the available production capacity. At the top level of the hierarchy the approach deploys a heuristic procedure for coordinating capacity in the medium term. The heuristic produces target production cycle times and target service levels for each of the products. It iteratively allocates capacity to individual products for maximizing the expected profit while meeting the individual product service level targets set by the management. The bottom level of the hierarchy focuses on the short term and aims to realize the target service level while keeping the production cycles stable. At this level, a fixed-sequence order up-to operational scheduling policy which permits a limited amount of flexibility for reacting to short-term demand variations is implemented. Demand which cannot be met from inventory gets lost. We also present and discuss the results of extensive simulation tests under a wide range of environments that serve to demonstrate the superiority of the approach, especially when the demand level is high.
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paper_title: A local search algorithm for the optimization of the stochastic economic lot scheduling problem

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Abstract This paper describes a model for the stochastic economic lot scheduling problem (SELSP) and a Local Search heuristic to find close to optimal solutions to this model. The SELSP considers multiple products, which have to be scheduled on a single facility with limited capacity and significant setup times and costs. The demand is modeled as a stationary compound renewal process. The objective is to find a schedule that minimizes the long-run average costs for setups and inventories while satisfying a given fill rate. We use a cyclic scheduling approach in which the individual cycle time of each product is a multiple of some basic period (fundamental cycle). For the deterministic version of the SELSP, efficient heuristics have been developed which guarantee the feasibility of the solution by adding an additional constraint to the problem. In our case this is not sufficient, because for the calculation of the average inventory levels and fill rates we need to develop a schedule with detailed timing of the lots. We present an efficient heuristic for this scheduling problem, which can also be used to check the feasibility of the solution. Thereby, the most time-consuming step (the calculation of average inventory levels and fill rates) is only performed for a limited set of candidates. The algorithm was tested on deterministic benchmark problems from literature and on a large set of stochastic instances. We report on the performance of the heuristic in both cases and try to identify the main factors influencing the objective.
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paper_title: Production to Order: Models and Rules for Production Planning

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1. Introduction.- 1.1. Production to order.- 1.2. Description of the problem.- 1.3. The control rules.- 1.3.1. The control rules for the lead times.- 1.3.2. The control rules for the production planning.- 1.4. The performance of the control rules.- 1.5. Influences on the control rules.- 1.5.1. Influence of the market.- 1.5.2. Influence of the production facilities.- 1.6. The methods.- 1.7. Overview of the text.- 2. Preliminaries.- 2.1. The situation and the model.- 2.2. Costs as instruments for decision making.- 2.3. Control rules for production planning.- 2.4. Control rules for lead times and order acceptance.- 3. A Fixed Production Cycle.- 3.1. Introduction.- 3.2. Analysis of fixed production cycles.- 3.2.1. Extended gating cyclic service model.- 3.2.2. Normal gating cyclic service with strict capacity.- 3.2.3. Normal gating service with flexible capacity.- 4. Non Cyclic Methods for One Product Type.- 4.1. Introduction.- 4.2. Fixed lead times.- 4.2.1. Introduction.- 4.2.2. Description of the model.- 4.2.3. The optimal production policy.- 4.2.4. Heuristic procedures.- 4.2.4.1. Notation.- 4.2.4.2. The Silver-Meal approach.- 4.2.4.3. The Wagner-Whitin approach.- 4.2.4.4. The (x,T)-rule.- 4.2.5. Numerical results.- 4.2.5.1. Conclusions.- 4.2.6. Orders with different priorities.- 4.2.6.1. Introduction.- 4.2.6.2. Model.- 4.3. A fluctuating demand rate.- 4.3.1. Introduction.- 4.3.2. Model.- 4.3.3. Example of lateness-dependent demand.- 4.3.4. Numerical results.- 4.4. A decision rule for lead times.- 4.4.1. Introduction.- 4.4.2. The model.- 4.4.3. A cyclic approach.- 4.4.4. The dynamic programming rule.- 4.4.5. Simplified dynamic programming rule.- 4.4.6. Numerical results.- 5. Several Types of Products on One Machine.- 5.1. Introduction.- 5.1.1. Description of the model.- 5.1.2. Notation.- 5.2. No possibilities for working overtime.- 5.2.1 Introduction.- 5.2.2. Extended (x,T)-rule.- 5.2.2.1. Introduction to the extended (x,T)-rule.- 5.2.2.2. Formulation of the extended (x,T)-rule.- 5.2.3. The two-step rule.- 5.2.3.1. Introduction to the two-step rule.- 5.2.3.2. Formulation of the first part.- 5.2.3.3. Formulation of the second part.- 5.2.4. A semi-fixed cycle rule.- 5.2.4.1. Introduction.- 5.2.4.2. Determining a semi-fixed cycle.- 5.2.5. Numerical results.- 5.3. Situation in which extra capacity is available.- 5.3.1. Introduction.- 5.3.2. Extended overtime (x,T)-rule.- 5.3.2.1. Introduction to the extended overtime (x,T)-rule.- 5.3.2.2. Formulation of the extended overtime (x,T)-rule.- 5.3.3. The two-step overtime rule.- 5.3.4. The semi-fixed cycle rule.- 5.3.5. Numerical results.- 6. Analysis of Multi-Type Models.- 6.1. Introduction.- 6.2. The multi (x,T)-rule.- 6.2.1. Analysis of the multi (x,T)-rule.- 6.2.2. Numerical results.- 6.2.3. Other possibilities and conclusions.- 6.3. Continuous review model.- 6.3.1. Introduction.- 6.3.2. Production model.- 6.3.3. Decomposition model.- 6.3.4. Results for the normal extended service.- 6.3.5. Numerical results and comparison with fixed cycle.- 7. The (x,T)-Rule in Complex Situations.- 7.1. Description of the problem.- 7.2. The elements of the extended (x,T)-rule.- 7.2.1. The penalty points.- 7.2.2. The allocation of capacity.- 7.2.3. Planning on different machines.- 7.3. Determination of lead times.- 7.3.1. Acceptance of the lead times.- 7.4. Numerical results.- 8. Summary and Conclusions.- 8.1. Summary and conclusions.- References.
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paper_title: Multiproduct production/inventory control under random demands

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Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the "norm-based" approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time. >
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paper_title: Scheduling the Production of Several Items with Random Demands in a Single Facility

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Consider the problem of scheduling the production of several items in a single facility that can produce only one item at a time. This problem occurs since it is often economic to produce several items in a single facility. The objective is to reduce the long run average holding, backorder and setup costs. ::: ::: We assume that demands are random with constant expected rates. We allow backorders and charge holding and backlogging costs at linear time weighted rates. Items are produced at continuous constant rates. Setup times and setup costs are item dependent constants. These parameters, however, are independent of the order of setups. ::: ::: A real-time scheduling tool is developed in three steps. First, with demands replaced by their expectations, we compute an optimal or near-optimal target cyclic schedule. Next, we study the problem of scheduling the facility after a single disruption perturbs the inventories. The goal is to recover the target cyclic schedule at minimal excess over the average cost of the cyclic schedule. We formulate this as a control problem and obtain a linear recovery policy that is optimal for a large configuration of disruptions. Finally, we select safety stocks to minimize the long run average cost of following the target schedule with the recovery policy. We show that optimal safety stocks are unique and have the property that in the long run the proportion of time that an item is in stock is the ratio of backorder to holding plus backorder cost. We present an example that integrates the cyclic schedule, the control policy and the safety stocks.
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paper_title: Note: An Improvement to the Dynamic Cycle Lengths Heuristic for Scheduling the Multi-Item, Single-Machine

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A heuristic scheduling policy for multi-item, single-machine scheduling systems facing stochastic, time-varying demands was previously introduced by Leachman and Gascon. In this technical note we provide an improvement to the calculations of the policy. Simulation tests of the revised version of the policy indicate that the improvements maintain or enhance customer service while reducing total inventory and setup costs on the order of 1-7%.
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paper_title: Single‐stage, multi‐product production/inventory systems with lost sales

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A production/inventory system consisting of a single processor producing three product types and a warehouse is considered. For each product type, the demand process is assumed to be Poisson and the processing time is phase-type. Excess demand is lost. Products have a priority structure and the processor's attention is shared by all the products according to a switching rule. Production of a product continues until its target level is reached. Then, a switch-over takes place if another product needs the processor's attention. A set-up process takes place every time a switch-over occurs. An (R, r) continuous-review inventory control policy is used to start and stop the production of each product. The underlying Markov chain is studied and its steady-state distribution is obtained recursively. Through the recursive procedure, the steady-state balance equations to be dealt with are significantly reduced to a manageable set. The procedure is implemented on a supercomputer and examples are provided to show its efficiency and stability for a range of model parameters. We analyzed the joint behaviors of the inventory levels of the three products as their demand rates increase. Finally we introduced a cost minimizing objective function to guide design efforts. © 1995 John Wiley & Sons, Inc.
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Title: The Stochastic Economic Lot Scheduling Problem: A Survey

Section 1: Introduction
Description 1: This section introduces the significance of the bottleneck in production plants, classifies multi-product single-machine scheduling problems, and provides an overview of the stochastic economic lot scheduling problem (SELSP).

Section 2: Problem description
Description 2: The present section gives a detailed problem description of the SELSP and gives the relationship between the SELSP and its deterministic counterpart.

Section 3: Model
Description 3: This section describes a system with one single production capacity for multiple products and focuses on the main objective to minimize total expected costs while considering stochastic production and set-up times.

Section 4: Relation with the deterministic ELSP
Description 4: This section elaborates on the ELSP, comparing it with the SELSP, and highlights the analytical and heuristic approaches used in literature to resolve the deterministic problem.

Section 5: Classification of the literature
Description 5: Based on the elements of the production plan such as the production sequence and lot sizing policies, a classification of the strategies for the SELSP is made.

Section 6: Strategy classes for the SELSP
Description 6: In this section, different strategies for the SELSP based on the classifications introduced in Section 5 are discussed in detail.

Section 7: Discussion and future research
Description 7: This section summarizes the most important conclusions drawn from the literature review and identifies the gaps in the current research while suggesting areas for future research.