Kurt M. Bretthauer

The nonlinear knapsack problem - algorithms and applications

Abstract We present a survey of algorithms and applications for the nonlinear knapsack problem (or, the nonlinear resource allocation problem). In its most general form, the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint, bounds on the variables, and, in some cases, a set of specially structured constraints such as generalized upper bounds (GUBs). This problem is encountered either directly, or as a subproblem, in a variety of applications, including production planning, financial modeling, stratified sampling, and capacity planning in manufacturing, health care, and computer networks. By taking advantage of the special structure of the problem, efficient solution methods can be developed. Problem classes addressed here include continuous and integer problems, convex and nonconvex problems, separable and nonseparable problems, and problems with additional specially structured constraints.

The Nonlinear Resource Allocation Problem

In this paper we study the nonlinear resource allocation problem, defined as the minimization of a convex function over one convex constraint and bounded integer variables. This problem is encountered in a variety of applications, including capacity planning in manufacturing and computer networks, production planning, capital budgeting, and stratified sampling. Despite its importance to these and other applications, the nonlinear resource allocation problem has received little attention in the literature. Therefore, we develop a branch-and-bound algorithm to solve this class of problems. First we present a general framework for solving the continuous-variable problem. Then we use this framework as the basis for our branch-and-bound method. We also develop reoptimization procedures and a heuristic that significantly improve the performance of the branch-and-bound algorithm. In addition, we show how the algorithm can be modified to solve nonconvex problems so that a concave objective function can be handled. The general algorithm is specialized for the applications mentioned above and computational results are reported for problems with up to 200 integer variables. A computational comparison with a 0, 1 linearization approach is also provided.

Online/In-Store Integration and Customer Retention

Reducing the risks believed to be associated with product availability can be critical to increasing consumer retention rates. This study considers the role that perceptions of channel integration have on such beliefs and their impact on purchasing decisions. Surveys distributed to purchasers of specific goods both online and in-store provide data used in the analysis of these effects. The findings suggest that firms simultaneously managing both online and in-store channels should not only reassess the repercussions of availability failures but also consider efforts that encourage the transparency of channel integration.

A pegging algorithm for the nonlinear resource allocation problem

In this paper we present a new algorithm for solving the nonlinear resource allocation problem. The nonlinear resource allocation problem is defined as the minimization of a convex function over a single convex constraint and bounded integer variables. We first present a pegging algorithm for solving the continuous variable problem, and then incorporate the pegging method in a branch and bound algorithm for solving the integer variable problem. We compare the computational performance of the pegging branch and bound algorithm with three other methods: a multiplier search branch and bound algorithm, dynamic programming, and a 0,1 linearization method. The computational results demonstrate that the pegging branch and bound algorithm advances the state of the art in methods for solving the nonlinear resource allocation problem.

The value of virtual pooling in dual sales channel supply chains

Recently the most significant growth in online retailing has been attributed to traditional offline retailers extending their brands online. Unfortunately, there is little research addressing the value of better information in retail/e-tail organizations. To fill this gap, this paper examines how investing in the continuous monitoring of online demands and inventory positions can provide economic benefit for companies that handle both in-store and online sales. Specifically, we develop and evaluate two dynamic assignment policies that incorporate real time information to specify which of a firms e-fulfillment locations will handle each of its Internet sales. Computational results indicate that investing in dynamic assignment capability can reduce system cost (holding, backorder, and transportation) by as much as 8.2% over the optimal static policy. The percentage of sales occurring online plays a critical role in determining the magnitude of the benefit.

A Branch and Bound Algorithm for Integer Quadratic Knapsack Problems

We present a branch and bound algorithm for solving separable convex quadratic knapsack problems with lower and upper bounds on the integer variables. The algorithm solves a series of continuous quadratic knapsack problems via their Kuhn-Tucker conditions. We also discuss reoptimization procedures for efficiently solving the continuous subproblems. Computational results for problems with up to 300 integer variables are reported. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Inventory and distribution strategies for retail/e-tail organizations

In the retail sector many traditional bricks-and-mortar companies have added online sales channels to their supply chains. Unfortunately, even though the combined retailer/e-tailer is becoming a common business model, there is very limited research addressing retail/e-tail operations. To address this gap, this research considers where and how much inventory should be allocated and held at each site for a company that satisfies both in-store and online demand. Specifically, we determine how many and which of a firms capacitated locations should handle online sales to minimize total cost (holding, backorder, fixed operating, transportation, and handling costs). Our primary findings include the following: (i) when all costs are considered the percentage of sales occurring online plays a critical role in determining the number of sites providing e-fulfillment; (ii) when holding and backorder costs are the only consideration (i.e., the customer pays for shipping), the standard deviation of in-store demand controls where online inventory should be located, regardless of the percentage of demand occurring online; and (iii) an increase in unit shipping costs does not necessarily imply that adding online fulfillment locations will decrease total cost. Results from a computational study illustrate that the model provides good solutions even when demand is correlated or not normally distributed.

Service and cost benefits through clicks-and-mortar integration: Implications for the centralization/decentralization debate

Abstract Traditional “Brick-and-Mortar” operations face the challenge of adapting to a new set of competitive rules made necessary by consumers who want the option of ordering electronically via the Internet. To satisfy these customers, firms must develop strategies that integrate their standard retail in-store channel with this relatively new on-line channel. Therefore, this research is designed to provide insights into supply chain inventory management strategies relevant to “Clicks-and-Mortar” firms trying to satisfy both on-line and in-store sales. Specifically, this work considers the total cost implications of various inventory allocation strategies while maintaining target customer service levels. Analysis focuses on the development of models capable of handling new operating strategies made possible by electronic commerce. The implications of inventory risk pooling are considered in depth, revealing the existence of characteristics that determine whether completely centralized or decentralized policies are preferable.

Locating specialized service capacity in a multi-hospital network

Multi-hospital systems have become very common in todays healthcare environment. However, there has been limited published research examining the opportunities and challenges of pooling specialized services to a subset of hospitals in the network. Therefore, this paper considers how hospital networks with multiple locations can leverage pooling benefits when deciding where to position specialized services, such as magnetic resonance imaging (MRI), transplants, or neonatal intensive care. Specifically, we develop an optimization model to determine how many and which of a hospital networks hospitals should be set up to deliver a specialized service. Importantly, this model takes into account both financial considerations and patient service levels. Computational results illustrate the value of optimally pooling resources across a subset of hospitals in the network versus two alternate approaches: (1) delivering the service at all locations and requiring each site to handle its own demand, or (2) locating the service at one hospital that handles all network demand.

Strategies for Addressing the Nursing Shortage: Coordinated Decision Making and Workforce Flexibility

In this article, we present strategies to help combat the U.S. nursing shortage. Key considerations include providing an attractive work schedule and work environment—critical issues for retaining existing nurses and attracting new nurses to the profession—while at the same time using the set of available nurses as effectively as possible. Based on these ideas, we develop a model that takes advantage of coordinated decision making when managing a flexible workforce. The model coordinates scheduling, schedule adjustment, and agency nurse decisions across various nurse labor pools, each of differing flexibility levels, capabilities, and costs, allowing a much more desirable schedule to be constructed. Our primary findings regarding coordinated decision making and how it can be used to help address the nursing shortage include (i) labor costs can be reduced substantially because, without coordination, labor costs on average are 16.3% higher based on an actual hospital setting, leading to the availability of additional funds for retaining and attracting nurses, (ii) simultaneous to this reduction in costs, more attractive schedules can be provided to the nurses in terms of less overtime and fewer undesirable shifts, and (iii) the use of agency nurses can help avoid overtime for permanent staff with only a 0.7% increase in staffing costs. In addition, we estimate the cost of the shortage for a typical U.S. hospital from a labor cost perspective and show how that cost can be reduced when managers coordinate.