Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon. A closely related problem is the single-period newsvendor model. It is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement (ordering) cost is fixed plus linear. The optimal policy for the single-item lot-sizing model is also well known under similar assumptions. In this paper we show that the classical (single-period) newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle. We provide a fully polynomial time approximation scheme for the nonlinear single-item stochastic lot-sizing problem, when demand distribution is given by an oracle, procurement costs are provided as nondecreasing oracles, holding/backlogging/disposal costs are linear, and lead time is positive. Similar results exist for the nonlinear newsvendor problem. These approximation schemes are designed by extending the technique of K-approximation sets and functions.National Science Foundation (U.S.) (Contract CMMI-0758069)United States. Office of Naval Research (Grant N000141110056

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

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

A Computationally Efficient FPTAS for Convex Stochastic Dynamic Programs

We propose a computationally efficient fully polynomial-time approximation scheme (FPTAS) to compute an approximation with arbitrary precision of the value function of convex stochastic dynamic programs, using the technique of K-approximation sets and functions introduced by Halman et al. [Math. Oper. Res., 34, (2009), pp. 674-685]. This paper deals with the convex case only, and it has the following contributions. First, we improve on the worst-case running time given by Halman et al. Second, we design and implement an FPTAS with excellent computational performance and show that it is faster than an exact algorithm even for small problem instances and small approximation factors, becoming orders of magnitude faster as the problem size increases. Third, we show that with careful algorithm design, the errors introduced by floating point computations can be bounded, so that we can provide a guarantee on the approximation factor over an exact infinite-precision solution. We provide an extensive computational evaluation based on randomly generated problem instances coming from applications in supply chain management and finance. The running time of the FPTAS is both theoretically and experimentally linear in the size of the uncertainty set

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Fully polynomial-time approximation schemes for time–cost tradeoff problems in series–parallel project networks

2009-2010 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Solving Practical Dynamic Pricing Problems with Limited Demand Information

Dynamic pricing problems have received considerable attention in the operations management literature in the last two decades. Most of the work has focused on structural results and managerial insights using stylized models without considering business rules and issues commonly encountered in practice. While these models do provide general, high-level guidelines for managers in practice, they may not be able to generate satisfactory solutions to practical problems in which business norms and constraints have to be incorporated. In addition, most of the existing models assume full knowledge about the underlying demand distribution. However, demand information can be very limited for many products in practice, particularly, for products with short life-cycles (e.g., fashion products). In this dissertation, we focus on dynamic pricing models that involve selling a fixed amount of initial inventory over a fixed time horizon without inventory replenishment. This class of dynamic pricing models have a wide application in a variety of industries. Within this class, we study two specific dynamic pricing problems with commonly-encountered business rules and issues where there is limited demand information. Our objective is to develop satisfactory solution approaches for solving practically sized problems and derive managerial insights. This dissertation consists of three parts. We first present a survey of existing pricing models that involve one or multiple sellers selling one or multiple products, each with a given initial inventory, over a fixed time horizon without inventory replenishment. This particular class of dynamic pricing problems have received substantial attention in the operations management literature in recent years. We classify existing models into several different classes, present a detailed review on the problems in each class, and identify possible directions for future research. We then study a markdown pricing problem that involves a single product and multiple stores. Joint inventory allocation and pricing decisions have to be made over time subject to a set of business rules. We discretize the demand distribution and employ a scenario tree to model demand correlation across time periods and among the stores. The problem is formulated as a MIP and a Lagrangian relaxation approach is proposed to solve it. Extensive numerical experiments demonstrate that the solution approach is capable of generating close-to-optimal solutions in a short computational time. Finally, we study a general dynamic pricing problem for a single store that involves two substitutable products. We consider both the price-driven substitution and inventory-driven substitution of the two products, and investigate their impacts on the optimal pricing decisions. We assume that little demand information is known and propose a robust optimization model to formulate the problem. We develop a dynamic programming solution approach. Due to the complexity of the DP formulation, a fully polynomial time approximation scheme is developed that guarantees a proven near optimal solution in a manageable computational time for practically sized problems. A variety of managerial insights are discussed