Ravi Anupindi

Coordination and Flexibility in Supply Contracts with Options

We investigate the role of options (contingent claims) in a buyer-supplier system. Specifically using a two-period model with correlated demand, we illustrate how options provide flexibility to a buyer to respond to market changes in the second period. We also study the implications of such arrangements between a buyer and a supplier for coordination of the channel. We show that, in general, channel coordination can be achieved only if we allow the exercise price to be piecewise linear. We develop sufficient conditions on the cost parameters such that linear prices coordinate the channel. We derive the appropriate prices for channel coordination which, however, violate the individual rationality constraint for the supplier. Contrary to popular belief (based on simpler models) we show that credit for returns offered by the supplier does not always coordinate the channel and alleviate the individual rationality constraint. Credit for returns are useful only on a subset of the feasibility region under which channel coordination is achievable with linear prices. Finally, we demonstrate (numerically) the benefits of options in improving channel performance and evaluate the magnitude of loss due to lack of coordination.

Single-Period Multiproduct Inventory Models with Substitution

We study a single period multiproduct inventory problem with substitution and proportional costs and revenues. We consider N products and N demand classes with full downward substitution, i.e., excess demand for class i can be satisfied using product j for i ≥ j. We first discuss a two-stage profit maximization formulation for the multiproduct substitution problem. We show that a greedy allocation policy is optimal. We use this to write the expected profits and its first partials explicitly. This in turn enables us to prove additional properties of the profit function and several interesting properties of the optimal solution. In a limited computational study using two products, we illustrate the benefits of solving for the optimal quantities when substitution is considered at the ordering stage over similar computations without considering substitution while ordering. Specifically, we show that the benefits are higher with high demand variability, low substitution cost, low profit margins (or low price to cost ratio), high salvage values, and similarity of products in terms of prices and costs.

Analysis of supply contracts with total minimum commitment

In this paper we analyze a supply contract for a single product that specifies that the cumulative orders placed by a buyer, over a finite horizon, be at least as large as a (contracted upon) given quantity. We assume that the demand for the product is uncertain, and the buyer places orders periodically. We derive the optimal purchase policy for the buyer for a given total minimum quantity commitment and a discounted price. We show that the policy is characterized by the order-up-to levels of the corresponding finite horizon and a single-period standard newsboy problem with no commitment but with discounted price. We show that this policy can be computed easily. We can use this to evaluate any discount schedule characterized by a set of (price discount, minimum commitment) pairs.

Supply Contracts with Quantity Commitments and Stochastic Demand

Supply chain management deals with the management of material, information, and financial flows in a network consisting of vendors, manufacturers, distributors and customers. Managing flows in this network is a major challenge due to the complexity (in space and time) of the network, the proliferation of products (often with short life cycles) that flow through this network, and the presence of multiple decision makers who each own and operate a piece of this network and optimize a private objective function. Supply chain management clearly involves a variety of issues including product/process design, production, third party logistics and outsourcing, supplier contracting, incentives and performance measures, multi-location inventory coordination, etc.

Capacity Investment Under Postponement Strategies, Market Competition, and Demand Uncertainty

We consider duopoly models where firms make decisions on capacity, production, and price under demand uncertainty. Capacity and price decisions are made, respectively, ex ante and ex post demand realizations. The interplay between the timings of demand realization and production decision endows firms with different capabilities. Flexible firms can postpone production decisions until the actual demand curve is observed, but inflexible firms cannot. Under general demand structures and cost functions, we characterize the equilibrium for a symmetric duopoly and establish the strategic equivalence of price and quantity competitions when firms are flexible. We investigate the stochastic order properties of capacity and profit and show that they both increase for a flexible firm when the market is more volatile. We find that flexibility allows a firm to increase investment in capacity and earn a higher profit while benefiting customers by keeping the price in a narrower range; strategic equivalence implies that these properties are robust to market conjectures. We also show that flexibility plays an important role in mitigating the destructive effect of competition when the demand shock is additive; the destructive effect is nonexistent for firms facing multiplicative demand shock. When flexibility decision is endogenous, a firms strategic flexibility choice depends on the cost of technology as well as the nature of demand shock. In particular, faced with a multiplicative demand shock, firms always choose to be inflexible, whereas all the possible equilibria are observed under additive demand shocks.

Strategic Inventories in Vertical Contracts

Classical reasons for carrying inventory include fixed (nonlinear) production or procurement costs, lead times, nonstationary or uncertain supply/demand, and capacity constraints. The last decade has seen active research in supply chain coordination focusing on the role of incentive contracts to achieve first-best levels of inventory. An extensive literature in industrial organization that studies incentives for vertical controls largely ignores the effect of inventories. Does the ability to carry inventory influence the problem of vertical control? Conversely, can inventories arise purely due to incentive effects? This paper explicitly considers both incentives and inventories, and their interplay, in a dynamic model of an upstream firm (supplier) and a downstream firm (buyer) who can carry inventories. In our model, none of the classical reasons for carrying inventory exists. However, as we prove, the buyers optimal strategy in equilibrium is to carry inventories, and the supplier is unable to prevent this. These inventories arise out of purely strategic considerations not yet identified in the literature, and have a significant impact on the equilibrium solution as well as supplier, buyer, and channel profits. We prove that strategic inventories play a pivotal role under arbitrary contractual structures, general (arbitrary) demand functions and general (finite or infinite) horizon lengths. As one example, two-part tariff contracts do not lead to optimal channel performance, nor can the supplier extract away all of the channel profits, in our dynamic model. Our results imply that firms can and must carry inventories strategically, and that optimal vertical contracts must take the possibility of inventories into account.

Control of batch processing systems in semiconductor wafer fabrication facilities

Loading policies for a batch processing machine, i.e. a machine that can process more than one job at a time, when the arrival times of jobs to the machine are uncertain, are described. The motivation for the study is the structure of process flows and the predominance of batch processing systems in a semiconductor wafer fabrication facility. A two stage serial-batch system with the serial stage (e.g. photolithography) feeding the batch (e.g. furnace) is considered. Machines in the serial stage process one job at a time; further, these machines are subject to failure. A control limit policy for loading the batch machine is assumed, i.e. load if the queue length >or=Q, else wait until the number of jobs in queue is at least Q. The basic tradeoffs considered are delay (waiting too long) vs. capacity utilization (loading early with very few jobs). An average cost analysis is done and optimized to compute the critical number Q. In an extension to the basic model, the effect of due dates on the critical number is analyzed. Comparison with simulation results is very encouraging. >

Approximations for multiproduct contracts with stochastic demands and business volume discounts: Single supplier case

In this paper we analyze contracts for multiple products when the supplier offers business volume discounts. The contract takes the form of price discounts for total minimum dollar volume commitments over the horizon with flexibility to adjust the total purchases upwards by a fixed percentage about this minimum commitment. The optimal policy could be complex since it involves the solution of a constrained multi-period multi-product dynamic program. We suggest approximations that give us an upper bound. To develop this upper bound, we present the optimal policy for a similar contract for a single product problem. We then develop a lower bound for the multiproduct problem and show, numerically, that the gap is small. Given the approximations standard single period resource allocation algorithms can be used to solve the problem. The resulting solution methodology is fast and permits evaluation of various what-if scenarios.

Integrated Optimization of Procurement, Processing, and Trade of Commodities

We consider the integrated optimization problem of procurement, processing, and trade of commodities in a multiperiod setting. Motivated by the operations of a prominent commodity processing firm, we model a firm that procures an input commodity and has processing capacity to convert the input into a processed commodity. The processed commodity is sold using forward contracts, while the input itself can be traded at the end of the horizon. We solve this problem optimally and derive closed-form expressions for the marginal value of input and output inventory. We find that the optimal procurement and processing decisions are governed by price-dependent inventory thresholds. We use commodity markets data for the soybean complex to conduct numerical studies and find that approximating the joint price processes of multiple output commodities using a single, composite output product and using the approximate price process to determine procurement and processing decisions is near optimal. Compared to a myopic spread-option-based heuristic, the optimization-based dynamic programming policy provides significant benefits under conditions of tight processing capacities and high price volatilities. Finally, we propose an approximation procedure to compute heuristic policies and an upper bound to compare the heuristic against, when commodity prices follow multifactor processes.

Managing Stochastic Multiproduct Systems: Model, Measures, and Analysis

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.