Yehuda Bassok

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.

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.

Random Yield and Random Demand in a Production System with Downward Substitution

in this paper, we present and solve a single-period, multiproduct, downward substitution model. Our model has one raw material as the production input and produces N different products as outputs. The demands and yields for the products are random. We determine the optimal production input and allocation of the N products to satisfy demands. The problem is modeled as a two-stage stochastic program, which we show can be decomposed into a parameterized network flow problem. We present and compare three different solution methods: a stochastic linear program, a decomposition resulting in a series of network flow subproblems, and a decomposition where the same network flow subproblems are solved by a new greedy algorithm.

Rendezvous Search on the Line

We present two new results for the asymmetric rendezvous problem on the line. We first show that it is never optimal for one player to be stationary during the entire search period in the two-player rendezvous. Then we consider the meeting time of n-players in the worst case and show that it has an asymptotic behavior of n = 2 + O(log n).

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.

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.

Design of component-supply contract with commitment revision flexibility

In this paper, we study a type of supply contract that is frequently used in the electronics industry. A common feature of these supply contracts is that at the beginning of the contract, the buyer makes purchasing commitments to the supplier for each period. The buyer may have some flexibility to purchase quantities that actually deviate from the original commitments. Moreover, as time passes and more information about the actual demand is collected, the buyer may update the previous commitments, in a way that is described. We develop a heuristic that is easy to implement and that determines nearly optimal commitments and purchasing quantities. We show that in many cases of practical interest, the heuristic results in solutions that are close to the optimal.

An inventory model for delayed customization: A hybrid approach

We present a variant of the delayed customization problem in which the company considers using both regular production and postponement. We determine the optimal inventory levels for the generic product (to be used for postponement) and the regular products. Even when the use of the generic product is significantly more expensive than other products, maintaining an inventory of the generic product significantly increases profits. We present an algorithm to optimally solve the single period and heuristically solve the infinite horizon problems for any number of products, and present computational results.

Rendezvous Search on the Interval and the Circle

Two people are placed randomly and independently on a street of unit length. They attempt to find each other in the shortest possible expected time. We solve this problem, assuming each searcher knows where he or she is on the street, for monotonic density functions for the initial placement (this includes the uniform pdf as a special case). This gives an example of a rendezvous search problem where there is no advantage in being allowed to use asymmetric strategies. We also solve some corresponding problems for the circle when asymmetric strategies are permitted: One of these shows that it can sometimes be optimal for one player to wait for the other to find him.