Technical note: Worst‐case performance of power‐of‐two policies for serial inventory systems with incremental quantity discounts

This study presents power‐of‐two policies for a serial inventory system with constant demand rate and incremental quantity discounts at the most upstream stage. It is shown that an optimal solution is nested and follows a zero‐inventory ordering policy. To prove the effectiveness of power‐of‐two policies, a lower bound on the optimal cost is obtained. A policy that has a cost within 6% of the lower bound is developed for a fixed base planning period. For a variable base planning period, a 98% effective policy is provided. An extension is included for a system with price dependent holding costs. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Stage life testing

In progressive censoring, items are removed at certain times during the life test. Commonly, it is assumed that the removed items are used for further testing. In order to take into account information about these additional testing in inferential procedures, we propose a two‐step model of stage life testing with one fixed stage‐change time which incorporates information about both the removed items (further tested under different conditions) and those remaining in the current life test. We show that some marginal distributions in our model correspond either to progressive censoring with a fixed censoring time or to a simple‐step stress model. Furthermore, assuming a cumulative exposure model, we establish exact inferential results for the distribution parameters when the lifetimes are exponentially distributed. An extension to Weibull distributed lifetimes is also discussed.     

Dynamic coordination of production planning and sales admission control in the presence of a spot market

We consider the decision‐making problem of dynamically scheduling the production of a single make‐to stock (MTS) product in connection with the product's concurrent sales in a spot market and a long‐term supply channel. The spot market is run by a business to business (B2B) online exchange, whereas the long‐term channel is established by a structured contract. The product's price in the spot market is exogenous, evolves as a continuous time Markov chain, and affects demand, which arrives sequentially as a Markov‐modulated Poisson process (MMPP). The manufacturer is obliged to fulfill demand in the long‐term channel, but is able to rein in sales in the spot market. This is a significant strategic decision for a manufacturer in entering a favorable contract. The profitability of the contract must be evaluated by optimal performance. The current problem, therefore, arises as a prerequisite to exploring contracting strategies. We reveal that the optimal strategy of coordinating production and sales is structured by the spot price dependent on the base stock and sell‐down thresholds. Moreover, we can exploit the structural properties of the optimal strategy to conceive an efficient algorithm. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

The reformulation‐linearization technique (RLT) is a methodology for constructing tight linear programming relaxations of mixed discrete problems. A key construct is the multiplication of “product factors” of the discrete variables with problem constraints to form polynomial restrictions, which are subsequently linearized. For special problem forms, the structure of these linearized constraints tends to suggest that certain classes may be more beneficial than others. We examine the usefulness of subsets of constraints for a family of 0–1 quadratic multidimensional knapsack programs and perform extensive computational tests on a classical special case known as the 0–1 quadratic knapsack problem. We consider RLT forms both with and without these inequalities, and their comparisons with linearizations derived from published methods. Interestingly, the computational results depend in part upon the commercial software used. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

A sequential auction‐bargaining procurement model

One of the most important decisions that a firm faces in managing its supply chain is a procurement decision: selecting suitable suppliers among many potential competing sellers and reducing the purchase cost. While both auctions and bargaining have been extensively studied in the literature, the research that combines auctions and bargaining is limited. In this article, we consider a combined auction‐bargaining model in a setting where a single buyer procures an indivisible good from one of many competing sellers. The procurement model that we analyze is a sequential model consisting of the auction phase followed by the bargaining phase. In the auction phase, the sellers submit bids, and the seller with the lowest bid is selected as the winning bidder. In the bargaining phase, the buyer audits the cost of the winning seller and then negotiates with him to determine the final price. For this auction‐bargaining model, we find a symmetric equilibrium bidding strategy for the sellers in a closed form, which is simple to understand and closely related to the classical results in the auction and bargaining literature. We also show that the auction‐bargaining model generates at least as much profit to the buyer as the standard auction or sequential bargaining model. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Path optimization for the resource‐constrained searcher

We formulate and solve a discrete‐time path‐optimization problem where a single searcher, operating in a discretized three‐dimensional airspace, looks for a moving target in a finite set of cells. The searcher is constrained by maximum limits on the consumption of one or more resources such as time, fuel, and risk along any path. We develop a specialized branch‐and‐bound algorithm for this problem that uses several network reduction procedures as well as a new bounding technique based on Lagrangian relaxation and network expansion. The resulting algorithm outperforms a state‐of‐the‐art algorithm for solving time‐constrained problems and also is the first algorithm to solve multi‐constrained problems. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Newsvendor bounds and heuristics for serial supply chains with regular and expedited shipping

We study an infinite‐horizon, N‐stage, serial production/inventory system with two transportation modes between stages: regular shipping and expedited shipping. The optimal inventory policy for this system is a top–down echelon base‐stock policy, which can be computed through minimizing 2N nested convex functions recursively (Lawson and Porteus, Oper Res 48 (2000), 878–893). In this article, we first present some structural properties and comparative statics for the parameters of the optimal inventory policies, we then derive simple, newsvendor‐type lower and upper bounds for the optimal control parameters. These results are used to develop near optimal heuristic solutions for the echelon base‐stock policies. Numerical studies show that the heuristic performs well. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

On the availability of R out of N repairable systems

An R out of N repairable system consisting of N components and operates if at least R components are functioning. Repairable means that failed components are repaired, and upon repair completion they are as good as new. We derive formulas for the expected up‐time, expected down‐time, and the availability of the system, using Markov renewal processes. We assume that either the repair times of the components are generally distributed and the components' lifetimes are exponential or vice versa. The analysis is done for systems with either cold or warm stand‐by. Numerical examples are given for several life time and repair time distributions. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 483–498, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10025     

Two‐stage component test plans for testing the reliability of a series system

A system reliability is often evaluated by individual tests of components that constitute the system. These component test plans have advantages over complete system based tests in terms of time and cost. In this paper, we consider the series system with n components, where the lifetime of the i‐th component follows exponential distribution with parameter λ_i. Assuming test costs for the components are different, we develop an efficient algorithm to design a two‐stage component test plan that satisfies the usual probability requirements on the system reliability and in addition minimizes the maximum expected cost. For the case of prior information in the form of upper bounds on λ_i's, we use the genetic algorithm to solve the associated optimization problems which are otherwise difficult to solve using mathematical programming techniques. The two‐stage component test plans are cost effective compared to single‐stage plans developed by Rajgopal and Mazumdar. We demonstrate through several numerical examples that our approach has the potential to reduce the overall testing costs significantly. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 95–116, 2002; DOI 10.1002/nav.1051