Stephen C. Graves

A Review of Production Scheduling

Production scheduling can be defined as the allocation of available production resources over time to best satisfy some set of criteria. Typically, the scheduling problem involves a set of tasks to be performed, and the criteria may involve both tradeoffs between early and late completion of a task, and between holding inventory for the task and frequent production changeovers. The intent of this paper is to present a broad classification for various scheduling problems, to review important theoretical developments for these problem classes, and to contrast the currently available theory with the practice of production scheduling. This paper will highlight problem areas for which there is both a significant discrepancy between practice and theory, and for which the practice corresponds closely to the theory.

A Single-Item Inventory Model for a Nonstationary Demand Process

In this paper, we consider an adaptive base-stock policy for a single-item inventory system, where the demand process is nonstationary. In particular, the demand process is an integrated moving average process of order (0, 1, 1), for which an exponential-weighted moving average provides the optimal forecast. For the assumed control policy we characterize the inventory random variable and use this to find the safety stock requirements for the system. From this characterization, we see that the required inventory, both in absolute terms and as it depends on the replenishment lead-time, behaves much differently for this case of nonstationary demand compared with stationary demand. We then show how the single-item model extends to a multi-stage, or supply-chain context; in particular we see that the demand process for the upstream stage is not only nonstationary but also more variable than that for the downstream stage. We also show that for this model there is no value from letting the upstream stages see the exogenous demand. The paper concludes with some observations about the practical implications of this work.

Optimizing Strategic Safety Stock Placement in Supply Chains

Manufacturing managers face increasing pressure to reduce inventories across the supply chain. However, in complex supply chains, it is not always obvious where to hold safety stock to minimize inventory costs and provide a high level of service to the final customer. In this paper we develop a framework for modeling strategic safety stock in a supply chain that is subject to demand or forecast uncertainty. Key assumptions are that we can model the supply chain as a network, that each stage in the supply chain operates with a periodic-review base-stock policy, that demand is bounded, and that there is a guaranteed service time between every stage and its customers. We develop an optimization algorithm for the placement of strategic safety stock for supply chains that can be modeled as spanning trees. Our assumptions allow us to capture the stochastic nature of the problem and formulate it as a deterministic optimization. As a partial validation of the model, we describe its successful application by product flow teams at Eastman Kodak. We discuss how these flow teams have used the model to reduce finished goods inventory, target cycle time reduction efforts, and determine component inventories. We conclude with a list of needs to enhance the utility of the model.

Process Flexibility in Supply Chains

Process flexibility, whereby a production facility can produce multiple products, is a critical design consideration in multiproduct supply chains facing uncertain demand. The challenge is to determine a cost-effective flexibility configuration that is able to meet the demand with high likelihood. In this paper, we present a framework for analyzing the benefits from flexibility in multistage supply chains. We find two phenomena, stage-spanning bottlenecks and floating bottlenecks, neither of which are present in single-stage supply chains, which reduce the effectiveness of a flexibility configuration. We develop a flexibility measureg and show that increasing this measure results in greater protection from these supply-chain inefficiencies. We also identify flexibility guidelines that perform very well for multistage supply chains. These guidelines employ and adapt the single-stage chaining strategy of Jordan and Graves (1995) to multistage supply chains.

Technology portfolio management: optimizing interdependent projects over multiple time periods

In order to maintain competitiveness, companies need to continually invest in technology projects. However, resource limitations require an organization to strategically allocate resources to a subset of possible projects. A variety of tools and methods can be used to select the optimal set of technology projects. However, these methods are only applicable when projects are independent and are evaluated in a common funding cycle. When projects are interdependent, the complexity of optimizing even a moderate number of projects over a small number of objectives and constraints can become overwhelming. This paper presents a model developed for the Boeing Company, Seattle, WA, USA, to optimize a portfolio of product development improvement projects. Using a dependency matrix, which quantifies the interdependencies between projects, a nonlinear, integer program model was developed to optimize project selection. The model also balances risk, overall objectives and the cost and benefit of the entire portfolio. Once the optimum strategy is identified, the model enables the team to quickly quantify and evaluate small changes to the portfolio.

Optimizing the Supply Chain Configuration for New Products

We address how to configure the supply chain for a new product for which the design has already been decided. The central question is to determine what suppliers, parts, processes, and transportation modes to select at each stage in the supply chain. There might be multiple options to supply a raw material, to manufacture or assemble the product, and to transport the product to the customer. Each of these options is differentiated by its lead time and direct cost added. Given these various choices along the supply chain, the configuration problem is to select the options that minimize the total supply chain cost. We develop a dynamic program with two state variables to solve the supply chain configuration problem for supply chains that are modeled as spanning trees. We illustrate the problem and its solution with an industrial example. We use the example to show the benefit from optimization relative to heuristics and to form hypotheses concerning the structure of optimal supply chain configurations. We conduct a computational experiment to test these hypotheses.

A Dynamic Model for Requirements Planning with Application to Supply Chain Optimization

This paper develops a new model for studying requirements planning in multistage production-inventory systems. We first characterize how most industrial planning systems work, and we then develop a mathematical model to capture some of the key dynamics in the planning process. Our approach is to use a model for a single production stage as a building block for modeling a network of stages. We show how to analyze the single-stage model to determine the production smoothness and stability for a production stage and the inventory requirements. We also show how to optimize the tradeoff between production capacity and inventory for a single stage. We then can model the multistage supply chain using the single stage as a building block. We illustrate the multistage model with an industrial application, and we conclude with some thoughts on a research agenda.

A Tactical Planning Model for a Job Shop

We propose and develop a discrete-time, continuous-flow model for studying the operation of a job shop that sees a stationary input mix of job types. We are not concerned with issues of detailed scheduling, but rather hope to develop a tactical planning tool for a job-shop operation. With the model, we are able to characterize the operational behavior of each work center in the job shop for a given control policy. The control rule sets the production rate at a work center as a fixed proportion of its queue level in each time period, and is consistent with the assignment of a planned lead time to each work center. For these control rules, the model gives the steady-state distribution of the production levels at each work center, as well as the distribution of queue lengths. We show how to use the model not only to evaluate a choice of the controls but also to find a good specification that results in acceptable shop behavior. An example for a factory that produces grinding machines illustrates the use of the model.

Supply Chain Design: Safety Stock Placement and Supply Chain Configuration

Publisher Summary This chapter discusses two approaches to safety stock placement, which are termed as “stochastic-service model” and the “guaranteed-service model.” In the stochastic-service model, each stage in the supply chain maintains a safety stock sufficient to meet its service level target. In this setting, a stage that has one or more upstream-adjacent supply stages has to characterize its replenishment time taking into account the likelihood that these suppliers will meet a replenishment request from stock. In the guaranteed-service model, each stage provides a guaranteed service to its customer stages. In this setting, a supply stage sets a service time to its downstream customer and then holds sufficient inventory so that it can always satisfy the service-time commitment. A key assumption in this model is to assume that demand is bounded for the purpose of making the service-time guarantee. The chapter also discusses how the supply chain can be optimally configured. The notion of options are introduced for each stage in the supply chain, where the options differ in terms of lead-time and cost.

Scheduling of Re-Entrant Flow Shops

Abstract We propose and develop a scheduling system for a very special type of flow shop. This flow shop processes a variety of jobs that are identical from a processing point of view. All jobs have the same routing over the facilities of the shop and require the same amount of processing time at each facility. Individual jobs, though, may differ since they may have different tasks performed upon them at a particular facility. Examples of such shops are flexible machining systems and integrated circuit fabrication processes. In a flexible machining system, all jobs may have the same routing over the facilities, but the actual tasks performed may differ; for instance, a drilling operation may vary in the placement or size of the holes. Similarly, for integrated circuit manufacturing, although all jobs may follow the same routing, the jobs will be differentiated at the photolithographic operations. The photolitho-graphic process establishes patterns upon the silicon wafers where the patterns differ according to the mask that is used. The flow shop that we consider has another important feature, namely the job routing is such that a job may return one or more times to any facility. We say that when a job returns to a facility it reenters the flow at that facility, and consequently we call the shop a re-entrant flow shop. In integrated circuit manufacturing, a particular integrated circuit will return several times to the photolithographic process in order to place several layers of patterns on the wafer. Similarly, in a flexible machining system, a job may have to return to a particular station several times for additional metal-cutting operations. These re-entrant flow shops are usually operated and scheduled as general job shops, ignoring the inherent structure of the shop flow. Viewing such shops as job shops means using myopic scheduling rules to sequence jobs at each facility and usually requires large queues of work-in-process inventory in order to maintain high facility utilization, but at the expense of long throughput times. In this paper we develop a cyclic scheduling method that takes advantage of the flow character of the process. The cycle period is the inverse of the desired production rate (jobs per day). The cyclic schedule is predicated upon the requirement that during each cycle the shop should perform all of the tasks required to complete a job, although possibly on different jobs. In other words, during a cycle period we require each facility to do each task assigned to it exactly once. With this requirement, a cyclic schedule is just the sequencing and timing on each facility of all of the tasks that that facility must perform during each cycle period. This cyclic schedule is to be repeated by each facility each cycle period. The determination of the best cyclic schedule is a very difficult combinatorial optimization problem that we cannot solve optimally for actual operations. Rather, we present a computerized heuristic procedure that seems very effective at producing good schedules. We have found that the throughput time of these schedules is much less than that achievable with myopic sequencing rules as used in a job shop. We are attempting to implement the scheduling system at an integrated circuit fabrication facility.