Distributionally Robust Optimization of Two‐Stage Lot‐Sizing Problems

This paper studies two‐stage lot‐sizing problems with uncertain demand, where lost sales, backlogging and no backlogging are all considered. To handle the ambiguity in the probability distribution of demand, distributionally robust models are established only based on mean‐covariance information about the distribution. Based on shortest path reformulations of lot‐sizing problems, we prove that robust solutions can be obtained by solving mixed 0‐1 conic quadratic programs (CQPs) with mean‐risk objective functions. An exact parametric optimization method is proposed by further reformulating the mixed 0‐1 CQPs as single‐parameter quadratic shortest path problems. Rather than enumerating all potential values of the parameter, which may be the super‐polynomial in the number of decision variables, we propose a branch‐and‐bound‐based interval search method to find the optimal parameter value. Polynomial time algorithms for parametric subproblems with both uncorrelated and partially correlated demand distributions are proposed. Computational results show that the proposed models greatly reduce the system cost variation at the cost of a relative smaller increase in expected system cost, and the proposed parametric optimization method is much more efficient than the CPLEX solver.     

联合经济批量模型中的损失补贴与利益分配问题研究

针对单供应商和单采购商构成的供应链系统,分采购商占主导地位和供应商占主导地位两种情形,讨论了采购商的最优订货决策与供应商的最优生产决策,以及它们合作情形下的系统最优决策,给出了采购商和供应商各自独立决策与联合决策下,它们各自的成本函数以及供应链系统的总成本函数.最后,通过比较独立决策与联合决策下采购商和供应商的成本函数,分析了联合经济批量模型中的损失补贴与利益分配问题.     

Lot‐sizing Decisions Under Limited‐time Price Reduction*

We consider the optimal lot‐sizing policy for an inventoried item when the vendor offers a limited‐time price reduction. We use the discounted cash flow (DCF) approach in our analysis, thereby eliminating the sources of approximation found in most of the earlier studies that use an average annual cost approach. We first characterize the optimal lot‐sizing policies and their properties, then develop an algorithm for determining the optimal lot sizes. We analytically demonstrate that the lot sizes derived using an average annual cost approach for the different variants of the problem are, in general, larger than the DCF optimum. While DCF analysis is more rigorous and yields precise lot sizes, we recognize that the associated mathematical models and the solution procedure are rather complex. Since simple and easy‐to‐understand policies have a strong practical appeal to decision makers, we propose a DCF version of a simple and easy‐to‐implement heuristic called the “Early Purchase” (EP) strategy and discuss its performance. We supplement our analytical developments with a detailed computational analysis and discuss the implications of our findings for decision making.     

Stabilized‐Cycle Strategy for Capacitated Lot Sizing with Multiple Products: Fill‐Rate Constraints in Rolling Schedules

In practice, deterministic, multi‐period lot‐sizing models are implemented in rolling schedules since this allows the revision of decisions beyond the frozen horizon. Thus, rolling schedules are able to take realizations and updated forecasts of uncertain data (e.g., customer demands) into account. Furthermore, it is common to hold safety stocks to ensure given service levels (e.g., fill rate). As we will show, this approach, implemented in rolling schedules, often results in increased setup and holding costs while (over‐)accomplishing given fill rates. A well‐known alternative to deterministic planning models are stochastic, static, multi‐period planning models used in the static uncertainty strategy, which results in stable plans. However, these models have a lack of flexibility to react to the realization of uncertain data. As a result, actual costs may differ widely from planned costs, and downside deviations of actual fill rates from those given are very high. We propose a new strategy, namely the stabilized cycle. This combines and expands upon ideas from the literature for minimizing setup and holding costs in rolling schedules, while controlling actual product‐specific fill rates for a finite reporting period. A computational study with a multi‐item capacitated medium‐term production planning model has been executed in rolling schedules. On the one hand, it demonstrates that the stabilized‐cycle strategy yields a good compromise between costs and downside deviations. Furthermore, the stabilized‐cycle strategy weakly dominates the order‐based strategy for both constant and seasonal demands.     

Coordinated Capacitated Lot‐Sizing Problem with Dynamic Demand: A Lagrangian Heuristic

Coordinated replenishment problems are common in manufacturing and distribution when a family of items shares a common production line, supplier, or a mode of transportation. In these situations the coordination of shared, and often limited, resources across items is economically attractive. This paper describes a mixed‐integer programming formulation and Lagrangian relaxation solution procedure for the single‐family coordinated capacitated lot‐sizing problem with dynamic demand. The problem extends both the multi‐item capacitated dynamic demand lot‐sizing problem and the uncapacitated coordinated dynamic demand lot‐sizing problem. We provide the results of computational experiments investigating the mathematical properties of the formulation and the performance of the Lagrangian procedures. The results indicate the superiority of the dual‐based heuristic over linear programming‐based approaches to the problem. The quality of the Lagrangian heuristic solution improved in most instances with increases in problem size. Heuristic solutions averaged 2.52% above optimal. The procedures were applied to an industry test problem yielding a 22.5% reduction in total costs.     

Benefactors and Beneficiaries: The Effects of Giving and Receiving on Cost‐Coalitional Problems

Motivated by supply chain collaborations in practice, we introduce a class of cost‐coalitional problems, which are based on a priori information about the cost faced by each agent in each set that it could belong to. Our focus is on problems with decreasingly monotonic coalitional costs. In this class of problems, we study the effects of giving and receiving when there exist players whose participation in an alliance always contributes to the savings of all alliance members (we refer to these players as benefactors), and there also exist players whose cost decreases in such an alliance (we call them beneficiaries). We use linear and quadratic norm cost games to analyze the role played by benefactors and beneficiaries in achieving stability of different cooperating alliances. We consider different notions of stability (the core and the bargaining set) and provide conditions for stability of an all‐inclusive alliance of agents which leads to minimum value of total cost incurred by all agents.     

Improving Humanitarian Operations through Technology‐Enabled Collaboration

Humanitarian supply chains involve many different entities, such as government, military, private, and non‐governmental organizations and individuals. Well‐coordinated interactions between entities can lead to synergies and improved humanitarian outcomes. Information technology (IT) tools can help facilitate collaboration, but cost and other barriers have limited their use. We document the use of an IT tool to improve last‐mile supply distribution and data management in one of many camps for internally displaced persons after the January 2010 earthquake in Haiti, and we describe other current uses of technology in camp management. Motivated by these examples and the interest among humanitarian organizations in expanding the use of such tools to facilitate coordination, we introduce a cooperative game theory model and explore insights about the conditions under which multi‐agency coordination is feasible and desirable. We also outline an agenda for future research in the area of technology‐enabled collaboration in the humanitarian sector.     

MRP Planned Orders in a Multiple‐Supplier Environment with Differing Lead Times

This study examines a deterministic material requirements planning (MRP) problem where lead times at subsequent ordering moments differ. Adequate replenishment methods that can cope with lead time differences are lacking because of the order crossover phenomenon, that is, replenishment orders are not received in the sequence they are ordered. This study specifies how to handle order crossovers and recalculate planned order releases after an update of gross requirements. The optimal (s, S) policy is based on dynamic programing. The state space is kept to a minimum due to three fundamental insights. The performance of the optimal solution approach is compared with two heuristics based on relaxations and a benchmark approach in which order crossovers are ignored. A numerical analysis reveals that average cost savings up to 25% are possible if the optimal policy is used instead of the benchmark approach. The contribution of this study is threefold: (1) it generalizes theory on MRP ordering, allowing for lead time differences and order crossovers; (2) it develops new fundamental insights and an optimal solution procedure, leading to substantial cost saving; and (3) it provides good‐performing heuristics for a general and realistic replenishment problem that can replace the current replenishment methods within MRP.     

Weakly Belief‐Free Equilibria in Repeated Games With Private Monitoring

Repeated games with imperfect private monitoring have a wide range of applications, but a complete characterization of all equilibria in this class of games has yet to be obtained. The existing literature has identified a relatively tractable subset of equilibria. The present paper introduces the notion of weakly belief‐free equilibria for repeated games with imperfect private monitoring. This is a tractable class which subsumes, as a special case, a major part of the existing literature (the belief‐free equilibria). It is shown that this class can outperform the equilibria identified by the previous work.     

On the Existence of Monotone Pure‐Strategy Equilibria in Bayesian Games

We generalize Athey's (2001) and McAdams' (2003) results on the existence of monotone pure‐strategy equilibria in Bayesian games. We allow action spaces to be compact locally complete metric semilattices and type spaces to be partially ordered probability spaces. Our proof is based on contractibility rather than convexity of best‐reply sets. Several examples illustrate the scope of the result, including new applications to multi‐unit auctions with risk‐averse bidders.     

Point‐of‐Dispensing Location and Capacity Optimization via a Decision Support System

Dispensing of mass prophylaxis can be critical to public health during emergency situations and involves complex decisions that must be made in a short period of time. This study presents a model and solution approach for optimizing point‐of‐dispensing (POD) location and capacity decisions. This approach is part of a decision support system designed to help officials prepare for and respond to public health emergencies. The model selects PODs from a candidate set and suggests how to staff each POD so that average travel and waiting times are minimized. A genetic algorithm (GA) quickly solves the problem based on travel and queuing approximations (QAs) and it has the ability to relax soft constraints when the dispensing goals cannot be met. We show that the proposed approach returns solutions comparable with other systems and it is able to evaluate alternative courses of action when the resources are not sufficient to meet the performance targets.     

A Comparison of the Bow‐Tie and STAMP Approaches to Reduce the Risk of Surgical Instrument Retention

Although relatively rare, surgical instrument retention inside a patient following central venous catheterization still presents a significant risk. The research presented here compared two approaches to help reduce retention risk: Bow‐Tie Analysis and Systems‐Theoretic Accident Model and Processes. Each method was undertaken separately and then the results of the two approaches were compared and combined. Both approaches produced beneficial results that added to existing domain knowledge, and a combination of the two methods was found to be beneficial. For example, the Bow‐Tie Analysis gave an overview of which activities keep controls working and who is responsible for each control, and the Systems‐Theoretic Accident Model and Processes revealed the safety constraints that were not enforced by the supervisor of the controlled process. Such two‐way feedback between both methods is potentially helpful for improving patient safety. Further methodology ideas to minimize surgical instrument retention risks are also described.     

A Comparison Between Probabilistic and Dempster‐Shafer Theory Approaches to Model Uncertainty Analysis in the Performance Assessment of Radioactive Waste Repositories

Model uncertainty is a primary source of uncertainty in the assessment of the performance of repositories for the disposal of nuclear wastes, due to the complexity of the system and the large spatial and temporal scales involved. This work considers multiple assumptions on the system behavior and corresponding alternative plausible modeling hypotheses. To characterize the uncertainty in the correctness of the different hypotheses, the opinions of different experts are treated probabilistically or, in alternative, by the belief and plausibility functions of the Dempster‐Shafer theory. A comparison is made with reference to a flow model for the evaluation of the hydraulic head distributions present at a radioactive waste repository site. Three experts are assumed available for the evaluation of the uncertainties associated with the hydrogeological properties of the repository and the groundwater flow mechanisms.