We study supply contracts for deterministic demand but in an environment of uncertain prices. We develop valuation methodologies for different types of supply contracts. A "time-inflexible contract" requires the firm to specify not only how many units it will purchase, but also the timing of the purchase. A "time-flexible contract" allows the firm to specify the purchase amount over a given period of time without specifying the exact time of purchase. Other than time flexibility, the suppliers may offer "quantity flexibility" to the firm as well, i.e., purchase quantities could be within a prespecified quantity window. Finally, "risk-sharing" features can be incorporated in the contract in terms of the purchase price that the firm eventually pays to a supplier. Within a prespecified price window the firm pays the realized price, but outside of it the firm shares, in an agreed way, added costs or benefits. Given the structure of a supply contract, we study the firm's decision when to purchase and how many units in each purchase such that the expected net present value of the purchase cost plus inventory holding cost is minimized. We discuss optimal purchasing strategies for both time-flexible and time-inflexible contracts with risk-sharing features. Other interesting results include the analysis of two-supplier sourcing environments and the exploitation of quantity flexibility in such contracts. Our discussion illustrates how time flexibility, quantity flexibility, supplier selection, and risk sharing, when carefully exercised can effectively reduce the sourcing cost in environments of price uncertainty.supply contracts, global operations, flexibility, binomial lattice

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Abstract. We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed.Key words. fully polynomial time approximation schemes, stochastic dynamic programming, K-approximation AMS subject classifications. 68Q25, 68W25, 90B05, 90B06, 90C15, 90C39, 90C40, 90C56, 90C59

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In this paper, we consider a spare parts inventory problem faced by a manufacturer of electronic machines with expensive parts that are located at various customer locations. The parts fail infrequently according to a Poisson process. To serve customers when a failure occurs, the manufacturer operates a central warehouse and many field depots that stock spare parts. The central warehouse acts as a repair facility and replenishes stock at the field depots. There is a centralized decision maker who manages the inventory in both the central warehouse and the field depots. We develop a continuous review, base stock policy for this two-echelon, multi-item spare parts inventory system. We formulate a model to minimize the system-wide inventory cost subject to a response time constraint at each field depot. We present an efficient heuristic algorithm and study its computational effectiveness.

This paper studies two important variants of the dynamic economic lot-sizing problem that are applicable to a wide range of real-world situations. In the first model, production in each time period is restricted to a multiple of a constant batch size, where backlogging is allowed and all cost parameters are time varying. Several properties of the optimal solution are discussed. Based on these properties, an efficient dynamic programming algorithm is developed. The efficiency of the dynamic program is further improved through the use of Monge matrices. Using the results developed for the first model, an O(n3log n) algorithm is developed to solve the second model, which has a general form of product acquisition cost structure, including a fixed charge for each acquisition, a variable unit production cost, and a freight cost with a truckload discount. This algorithm can also be used to solve a more general problem with concave cost functions.

Most scheduling literature considers a "one-job-on-one-processor" pattern, which assumes that a processor processes exactly one job at a time. In this paper we consider a new scheduling problem with a "multiple-job-on-one-processor" pattern, where several jobs can be processed by a single processor simultaneously, provided that the total size of the jobs being processed does not exceed the capacity of the processor at any point in time. This problem is motivated by the operation of berth allocation, which is to allocate vessels (jobs) to a berth (processor), where the vessels, if small in dimension, may share the berth with some other vessels for loading/un!oading the goods. We consider the problem to minimize the makespan of the schedule. The well-known First-Fit Decreasing heuristic is generalized and applied to several variations of the problem, and the worst-case behavior of the generalized heuristics is studied. Worst-case error bounds are obtained for those models. Computational experiments are conducted to test the heuristics. The results suggest that the heuristics are effective in producing near-optimal solutions.