The minmax multidimensional knapsack problem with application to a chance‐constrained problem

In this paper we present a new combinatorial problem, called minmax multidimensional knapsack problem (MKP), motivated by a military logistics problem. The logistics problem is a two‐period, two‐level, chance‐constrained problem with recourse. We show that the MKP is NP‐hard and develop a practically efficient combinatorial algorithm for solving it. We also show that under some reasonable assumptions regarding the operational setting of the logistics problem, the chance‐constrained optimization problem is decomposable into a series of MKPs that are solved separately. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A branch‐and‐price algorithm for a targeting problem

In this paper, we consider a new weapon‐target allocation problem with the objective of minimizing the overall firing cost. The problem is formulated as a nonlinear integer programming model, but it can be transformed into a linear integer programming model. We present a branch‐and‐price algorithm for the problem employing the disaggregated formulation, which has exponentially many columns denoting the feasible allocations of weapon systems to each target. A greedy‐style heuristic is used to get some initial columns to start the column generation. A branching strategy compatible with the pricing problem is also proposed. Computational results using randomly generated data show this approach is promising for the targeting problem. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Managing components for assemble‐to‐order products with lead‐time‐dependent pricing: The full‐shipment model

We study a component inventory planning problem in an assemble‐to‐order environment faced by many contract manufacturers in which both quick delivery and efficient management of component inventory are crucial for the manufacturers to achieve profitability in a highly competitive market. Extending a recent study in a similar problem setting by the same authors, we analyze an optimization model for determining the optimal component stocking decision for a contract manufacturer facing an uncertain future demand, where product price depends on the delivery times. In contrast to our earlier work, this paper considers the situation where the contract manufacturer needs to deliver the full order quantity in one single shipment. This delivery requirement is appropriate for many industries, such as the garment and toy industries, where the economies of scale in transportation is essential. We develop efficient solution procedures for solving this optimization problem. We use our model results to illustrate how the different model parameters affect the optimal solution. We also compare the results under this full‐shipment model with those from our earlier work that allows for multiple partial shipments. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders

In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make‐to‐stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand‐based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time‐dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Benders decomposition with alternative multiple cuts for a multi‐product closed‐loop supply chain network design model

In this article, we consider a multi‐product closed‐loop supply chain network design problem where we locate collection centers and remanufacturing facilities while coordinating the forward and reverse flows in the network so as to minimize the processing, transportation, and fixed location costs. The problem of interest is motivated by the practice of an original equipment manufacturer in the automotive industry that provides service parts for vehicle maintenance and repair. We provide an effective problem formulation that is amenable to efficient Benders reformulation and an exact solution approach. More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch‐and‐cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch‐and‐cut approach. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Simultaneous price,location, and capacity decisions on a line of time‐sensitive customers

Consider a monopolist who sells a single product to time‐sensitive customers located on a line segment. Customers send their orders to the nearest distribution facility, where the firm processes (customizes) these orders on a first‐come, first‐served basis before delivering them. We examine how the monopolist would locate its facilities, set their capacities, and price the product offered to maximize profits. We explicitly model customers' waiting costs due to both shipping lead times and queueing congestion delays and allow each customer to self‐select whether she orders or not, based on her reservation price. We first analyze the single‐facility problem and derive a number of interesting insights regarding the optimal solution. We show, for instance, that the optimal capacity relates to the square root of the customer volume and that the optimal price relates additively to the capacity and transportation delay costs. We also compare our solutions to a similar problem without congestion effects. We then utilize our single‐facility results to treat the multi‐facility problem. We characterize the optimal policy for serving a fixed interval of customers from multiple facilities when customers are uniformly distributed on a line. We also show how as the length of the customer interval increases, the optimal policy relates to the single‐facility problem of maximizing expected profit per unit distance. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Capacity allocation to sales agents in a decentralized logistics network

Many logistics systems operate in a decentralized way, while most optimization models assume a centralized planner. One example of a decentralized system is in some sea cargo companies: sales agents, who share ship capacity on a network, independently accept cargo from their location and contribute to the revenue of the system. The central headquarters does not directly control the agents' decisions but can influence them through system design and incentives. In this paper, we model the firm's problem to determine the best capacity allocation to the agents such that system revenue is maximized. In the special case of a single‐route, we formulate the problem as a mixed integer program incorporating the optimal agent behavior. For the NP‐hard multiple‐route case, we propose several heuristics for the problem. Computational experiments show that the decentralized system generally performs worse when network capacity is tight and that the heuristics perform reasonably well. We show that the decentralized system may perform arbitrarily worse than the centralized system when the number of locations goes to infinity, although the choice of sales incentive impacts the performance. We develop an upper bound for the decentralized system, where the bound gives insight on the performance of the heuristics in large systems. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Pricing bridges to cross a river

We consider a pricing problem in directed, uncapacitated networks. Tariffs must be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs in the network are assumed to be given. There are n clients, the followers, and after the tariffs have been determined, the clients route their demands independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by applications in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX‐hard. Moreover, we analyze the effect of uniform pricing, proving that it yields both an m approximation and a (1 + lnD)‐approximation. Here, D is upper bounded by the total demand of all clients. In addition, we consider the problem under the additional restriction that the operator must not reject any of the clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless P = NP, and we prove the existence of an n‐approximation algorithm. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Outbound supply chain network design with mode selection and lead time considerations

We present a large‐scale network design model for the outbound supply chain of an automotive company that considers transportation mode selection (road vs. rail) and explicitly models the relationship between lead times and the volume of flow through the nodes of the network. We formulate the problem as a nonlinear zero‐one integer program, reformulate it to obtain a linear integer model, and develop a Lagrangian heuristic for its solution that gives near‐optimal results in reasonable time. We also present scenario analyses that examine the behavior of the supply chain under different parameter settings and the performance of the solution procedures under different experimental conditions. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Dynamic learning,pricing, and ordering by a censored newsvendor

We address the problem of determining optimal ordering and pricing policies in a finite‐horizon newsvendor model with unobservable lost sales. The demand distribution is price‐dependent and involves unknown parameters. We consider both the cases of perishable and nonperishable inventory. A very general class of demand functions is studied in this paper. We derive the optimal ordering and pricing policies as unique functions of the stocking factor (which is a linear transformation of the safety factor). An important expression is obtained for the marginal expected value of information. As a consequence, we show when lost sales are unobservable, with perishable inventory the optimal stocking factor is always at least as large as the one given by the single‐period model; however, if inventory is nonperishable, this result holds only under a strong condition. This expression also helps to explain why the optimal stocking factor of a period may not increase with the length of the problem. We compare this behavior with that of a full information model. We further examine the implications of the results to the special cases when demand uncertainty is described by additive and multiplicative models. For the additive case, we show that if demand is censored, the optimal policy is to order more as well as charge higher retail prices when compared to the policies in the single‐period model and the full information model. We also compare the optimal and myopic policies for the additive and multiplicative models. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A m‐parallel crane scheduling problem with a non‐crossing constraint

In this paper, we study a m‐parallel machine scheduling problem with a non‐crossing constraint motivated by crane scheduling in ports. We decompose the problem to allow time allocations to be determined once crane assignments are known and construct a backtracking search scheme that manipulates domain reduction and pruning strategies. Simple approximation heuristics are developed, one of which guarantees solutions to be at most two times the optimum. For large‐scale problems, a simulated annealing heuristic that uses random neighborhood generation is provided. Computational experiments are conducted to test the algorithms. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.     

Production to order and off‐line inspection when the production process is partially observable

This study combines inspection and lot‐sizing decisions. The issue is whether to INSPECT another unit or PRODUCE a new lot. A unit produced is either conforming or defective. Demand need to be satisfied in full, by conforming units only. The production process may switch from a “good” state to a “bad” state, at constant rate. The proportion of conforming units in the good state is higher than in the bad state. The true state is unobservable and can only be inferred from the quality of units inspected. We thus update, after each inspection, the probability that the unit, next candidate for inspection, was produced while the production process was in the good state. That “good‐state‐probability” is the basis for our decision to INSPECT or PRODUCE. We prove that the optimal policy has a simple form: INSPECT only if the good‐state‐probability exceeds a control limit. We provide a methodology to calculate the optimal lot size and the expected costs associated with INSPECT and PRODUCE. Surprisingly, we find that the control limit, as a function of the demand (and other problem parameters) is not necessarily monotone. Also, counter to intuition, it is possible that the optimal action is PRODUCE, after revealing a conforming unit. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Collaboration in contingent capacities with information asymmetry

We study the optimal contracting problem between two firms collaborating on capacity investment with information asymmetry. Without a contract, system efficiency is lost due to the profit‐margin differentials among the firms, demand uncertainty, and information asymmetry. With information asymmetry, we demonstrate that the optimal capacity level is characterized by a newsvendor formula with an upward‐adjusted capacity investment cost, and no first‐best solution can be achieved. Our analysis shows that system efficiency can always be improved by the optimal contract and the improvement in system efficience is due to two factors. While the optimal contract may bring the system's capacity level closer to the first‐best capacity level, it prevents the higher‐margin firm from overinvesting and aligns the capacity‐investment decisions of the two firms. Our analysis of a special case demonstrates that, under some circumstances, both firms can benefit from the principal having better information about the agent's costs. © 2007 Wiley Periodicals, Inc. Naval Research Logistics 54:, 2007     

Flexibility and complexity in periodic distribution problems

In this paper, we explore trade‐offs between operational flexibility and operational complexity in periodic distribution problems. We consider the gains from operational flexibility in terms of vehicle routing costs and customer service benefits, as well as the costs of operational complexity in terms of modeling, solution methods, and implementation challenges for drivers and customers. The period vehicle routing problem (PVRP) is a variation of the classic vehicle routing problem in which delivery routes are constructed for a period of time; the PVRP with service choice (PVRP‐SC) extends the PVRP to allow service (visit) frequency to become a decision of the model. For the periodic distribution problems represented by PVRP and PVRP‐SC, we introduce operational flexibility levers and a set of quantitative measures to evaluate the trade‐offs between flexibility and complexity. We develop a Tabu Search heuristic to incorporate a range of operational flexibility options. We analyze the potential value and the increased operational complexity of the flexibility levers. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimal sample size allocation for tests with multiple levels of stress with extreme value regression

In this article, we discuss the optimal allocation problem in a multiple stress levels life‐testing experiment when an extreme value regression model is used for statistical analysis. We derive the maximum likelihood estimators, the Fisher information, and the asymptotic variance–covariance matrix of the maximum likelihood estimators. Three optimality criteria are defined and the optimal allocation of units for two‐ and k‐stress level situations are determined. We demonstrate the efficiency of the optimal allocation of units in a multiple stress levels life‐testing experiment by using real experimental situations discussed earlier by McCool and Nelson and Meeker. Monte Carlo simulations are used to show that the optimality results hold for small sample sizes as well. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Dynamic routing of prioritized warranty repairs

Recent years have seen a strong trend toward outsourcing warranty repair services to outside vendors. In this article we consider the problem of dynamically routing warranty repairs to service vendors when warranties have priority levels. Each time an item under warranty fails, it is sent to one of the vendors for repair. Items covered by higher priority warranty receive higher priority in repair service. The manufacturer pays a fixed fee per repair and incurs a linear holding cost while an item is undergoing or waiting for repair. The objective is to minimize the manufacturer's long‐run average cost. Because of the complexity of the problem, it is very unlikely that there exist tractable ways to find the optimal routing strategies. Therefore, we propose five heuristic routing procedures that are applicable to real‐life problems. We evaluate the heuristics using simulation. The simulation results show that the index‐based “generalized join the shortest queue” policy, which applies a single policy improvement step to an initial state‐independent policy, performs the best among all five heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Analysis of decentralized multi‐product pull systems with lost sales

We study a pull‐type, flexible, multi‐product, and multi‐stage production/inventory system with decentralized two‐card kanban control policies. Each stage involves a processor and two buffers with finite target levels. Production stages, arranged in series, can process several product types one at a time. Transportation of semi‐finished parts from one stage to another is performed in fixed lot sizes. The exact analysis is mathematically intractable even for smaller systems. We present a robust approximation algorithm to model two‐card kanban systems with batch transfers under arbitrary complexity. The algorithm uses phase‐type modeling to find effective processing times and busy period analysis to identify delays among product types in resource contention. Our algorithm reduces the effort required for estimating performance measures by a considerable margin and resolves the state–space explosion problem of analytical approaches. Using this analytical tool, we present new findings for a better understanding of some tactical and operational issues. We show that flow of material in small procurement sizes smoothes flow of information within the system, but also necessitates more frequent shipments between stages, raising the risk of late delivery. Balancing the risk of information delays vis‐à‐vis shipment delays is critical for the success of two‐card kanban systems. Although product variety causes time wasted in setup operations, it also facilitates relatively short production cycles enabling processors to switch from one product type to another more rapidly. The latter point is crucial especially in high‐demand environments. Increasing production line size prevents quick response to customer demand, but it may improve system performance if the vendor lead‐time is long or subject to high variation. Finally, variability in transportation and processing times causes the most damage if it arises at stages closer to the customer. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Machine scheduling with an availability constraint and job delivery coordination

In this paper we study the scheduling problem that considers both production and job delivery at the same time with machine availability considerations. Only one vehicle is available to deliver jobs in a fixed transportation time to a distribution center. The vehicle can load at most K jobs as a delivery batch in one shipment due to the vehicle capacity constraint. The objective is to minimize the arrival time of the last delivery batch to the distribution center. Since machines may not always be available over the production period in real life due to preventive maintenance, we incorporate machine availability into the models. Three scenarios of the problem are studied. For the problem in which the jobs are processed on a single machine and the jobs interrupted by the unavailable machine interval are resumable, we provide a polynomial algorithm to solve the problem optimally. For the problem in which the jobs are processed on a single machine and the interrupted jobs are nonresumable, we first show that the problem is NP‐hard. We then propose a heuristic with a worst‐case error bound of 1/2 and show that the bound is tight. For the problem in which the jobs are processed on either one of two parallel machines, where only one machine has an unavailable interval and the interrupted jobs are resumable, we propose a heuristic with a worst‐case error bound of 2/3. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Exact expected values of variance estimators for simulation

We formulate exact expressions for the expected values of selected estimators of the variance parameter (that is, the sum of covariances at all lags) of a steady‐state simulation output process. Given in terms of the autocovariance function of the process, these expressions are derived for variance estimators based on the simulation analysis methods of nonoverlapping batch means, overlapping batch means, and standardized time series. Comparing estimator performance in a first‐order autoregressive process and the M/M/1 queue‐waiting‐time process, we find that certain standardized time series estimators outperform their competitors as the sample size becomes large. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007