Saif Benjaafar

The carbon-constrained EOQ

Abstract In this paper, we provide analytical support for the notion that it may be possible, via operational adjustments alone, to significantly reduce emissions without significantly increasing cost. Using the EOQ model, we provide a condition under which it is possible to reduce emissions by modifying order quantities. We also provide conditions under which the relative reduction in emissions is greater than the relative increase in cost and discuss factors that affect the difference in the magnitude of emission reduction and cost increase. We discuss the applicability of the results to systems under a variety of environmental regulations, including strict carbon caps, carbon tax, cap-and-offset, and cap-and-price.

Production and Inventory Control of a Single Product Assemble-to-Order System with Multiple Customer Classes

We consider the optimal production and inventory control of an assemble-to-order system with m components, one end-product, and n customer classes. A control policy specifies when to produce each component and, whenever an order is placed, whether or not to satisfy it from on-hand inventory. We formulate the problem as a Markov decision process and characterize the structure of an optimal policy. We show that a base-stock production policy is optimal, but the base-stock level for each component is dynamic and depends on the inventory level of all other components (more specifically, it is nondecreasing). We show that the optimal inventory allocation for each component is a rationing policy with different rationing levels for different demand classes. The rationing levels for each component are dynamic and also nondecreasing in the inventory level of all other components. We compare the performance of the optimal policy to heuristic policies, including the commonly used base-stock policy with fixed base-stock levels, and find them to perform surprisingly well, especially for systems with lost sales.

Outsourcing via Service Competition

We consider a single buyer who wishes to outsource a fixed demand for a manufactured good or service at a fixed price to a set of potential suppliers. We examine the value of competition as a mechanism for the buyer to elicit service quality from the suppliers. We compare two approaches the buyer could use to orchestrate this competition: (1) a supplier-allocation (SA) approach, which allocates a proportion of demand to each supplier with the proportion allocated to a supplier increasing in the quality of service the supplier promises to offer, and (2) a supplier-selection (SS) approach, which allocates all demand to one supplier with the probability that a particular supplier is selected increasing in the quality of service to which the supplier commits. In both cases, suppliers incur a cost whenever they receive a positive portion of demand, with this cost increasing in the quality of service they offer and the demand they receive. The analysis reveals that (a) a buyer could indeed orchestrate a competition among potential suppliers to promote service quality, (b) under identical allocation functions, the existence of a demand-independent service cost gives a distinct advantage to SS-type competitions, in terms of higher service quality for the buyer and higher expected profit for the supplier, (c) the relative advantage of SS versus SA depends on the magnitude of demand-independent versus demand-dependent service costs, (d) in the presence of a demand-independent service cost, a buyer should limit the number of competing suppliers under SA competition but impose no such limits under SS competition, and (e) a buyer can induce suppliers to provide higher service levels by selecting an appropriate allocation function. We illustrate the impact of these results through three example applications.

Using Imperfect Advance Demand Information in Production-Inventory Systems with Multiple Customer Classes

We consider a make-to-stock supplier that operates a production facility with limited capacity. The supplier receives orders from customers belonging to several demand classes. Some of the customer classes share advance demand information with the supplier by announcing their orders ahead of their due date. However, this advance demand information is not perfect because the customer may decide to order prior to or later than the expected due date or may decide to cancel the order altogether. Customer classes vary in their demand rates, expected due dates, cancellation probabilities, and shortage costs. The supplier must decide when to produce and, whenever an order becomes due, whether or not to satisfy it from on-hand inventory. Hence, the supplier is faced with a joint production-control and inventory-allocation problem. We formulate the problem as a Markov decision process and characterize the structure of the optimal policy. We show that the optimal production policy is a state-dependent base-stock policy with a base-stock level that is nondecreasing in the number of announced orders. We show that the optimal inventory-allocation policy is a state-dependent multilevel rationing policy, with the rationing level for each class nondecreasing in the number of announced orders (regardless of whether the class provides advance information). From numerical results, we obtain several insights into the value of advance demand information for both supplier and customers.

On the Benefits of Pooling in Production-Inventory Systems

We study inventory pooling in systems with symmetric costs where supply lead times are endogenously generated by a finite-capacity production system. We investigate the sensitivity of the cost advantage of inventory pooling to various system parameters, including loading, service levels, demand and production time variability, and structure of the production system. The analysis reveals differences in how various parameters affect the cost reduction from pooling and suggests that these differences stem from the manner in which the parameters influence the induced correlation between lead-time demands of the demand streams. We compare these results with those obtained for pure inventory systems, where lead times are exogenous. We also compare inventory pooling with several forms of capacity pooling.

On the Effect of Product Variety in Production–Inventory Systems

In this paper, we examine the effect of product variety on inventory costs in a production–inventory system with finite capacity where products are made to stock and share the same manufacturing facility. The facility incurs a setup time whenever it switches from producing one product type to another. The production facility has a finite production rate and stochastic production times. In order to mitigate the effect of setups, products are produced in batches. In contrast to inventory systems with exogenous lead times, we show that inventory costs increase almost linearly in the number of products. More importantly, we show that the rate of increase is sensitive to system parameters including demand and process variability, demand and capacity levels, and setup times. The effect of these parameters can be counterintuitive. For example, we show that the relative increase in cost due to higher product variety is decreasing in demand and process variability. We also show that it is decreasing in expected production time. On the other hand, we find that the relative cost is increasing in expected setup time, setup time variability and aggregate demand rate. Furthermore, we show that the effect of product variety on optimal base stock levels is not monotonic. We use the model to draw several managerial insights regarding the value of variety-reducing strategies such as product consolidation and delayed differentiation.

Resequencing and feature assignment on an automated assembly line

We consider the problem of resequencing a prearranged set of jobs on a moving assembly line with the objective of minimizing changeover costs. A changeover cost is incurred whenever two consecutive jobs do not share the same feature. Features are assigned from a set of job-specific feasible features. Resequencing is limited by the availability of offline buffers. The problem is motivated by a vehicle resequencing and painting problem at a major U.S. automotive manufacturer. We develop a model for solving the joint resequencing and feature assignment problem and an efficient solution procedure for simultaneously determining optimal feature assignments and vehicle sequences. We show that our solution approach is amenable to implementation in environments where a solution must be obtained within tight time constraints. We also show that the effect of offline buffers is of the diminishing kind with most of the benefits achieved with very few buffers. This means that limited resequencing flexibility is generally sufficient. Furthermore, we show that the value of resequencing is sensitive to the feature density matrix, with resequencing having a significant impact on cost only when density is in the middle range.

Optimal Control of a Production-Inventory System with Both Backorders and Lost Sales

We consider the optimal control of a production inventory-system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state-dependent thresholds: a production base-stock level and two order-admission levels, one for each class. The production base-stock level determines when production takes place and how to allocate items that are produced. This base-stock level also determines when orders from the class with the lower shortage costs (class 2) are backordered and not fulfilled from inventory. The order-admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either non-increasing or non-decreasing) in the backorder level of class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.

Demand Allocation in Multiple-Product, Multiple-Facility, Make-to-Stock Systems

We consider the problem of allocating demand arising from multiple products to multiple production facilities with finite capacity and load-dependent lead times. Production facilities can choose to manufacture items either to stock or to order. Products vary in their demand rates, holding and backordering costs, and service-level requirements. We develop models and solution procedures to determine the optimal allocation of demand to facilities and the optimal inventory level for products at each facility. We consider two types of demand allocation, one in which we allow the demand for a product to be split among multiple facilities and the other in which demand from each product must be entirely satisfied by a single facility. We also consider two forms of inventory warehousing, one in which inventory locations are factory based and one in which they are centralized. For each case, we offer a solution procedure to obtain optimal demand allocations and optimal inventory base-stock levels. For systems with multiple customer classes, we also determine optimal inventory rationing levels for each class for each product. We use the models to characterize analytically several properties of the optimal solution. In particular, we highlight eight principles that relate the effects of cost, congestion, inventory pooling, multiple sourcing, customer segmentation, inventory rationing, and process and demand variability.

Optimal control of a production-inventory system with customer impatience

We consider the control of a production-inventory system with impatient customers. We show that the optimal policy can be described using two thresholds: a production base-stock level that determines when production takes place and an admission threshold that determines when orders should be accepted. We describe an algorithm for computing the performance of the system for any choice of base-stock level and admission threshold. In a numerical study, we compare the performance of the optimal policy against several other policies.