Operations Research current issue

In This Issue

2009-06-27

No abstract available.

OR FORUM-Rocket Science Retailing: The 2006 Philip McCord Morse Lecture

Fisher, M. — 2009-06-27

Retailing is a huge industry. In the United States, retail business represents about 40% of the economy and is the largest employer. Retail supply chain management is still more art than science, but this is changing rapidly as retailers begin to apply analytic models to the huge volume of data they are collecting on consumer purchases and preferences. This industry-wide movement resembles the transformation of Wall Street that occurred in the 1970s when physicists and other "rocket scientists" applied their analytic skills to investment decisions.

The Consortium for Operational Excellence in Retailing (COER) (codirected by Ananth Raman, Harvard Business School, and myself) is a group of academics working with about 50 leading retailers to assess their progress towards rocket science retailing and to accelerate that progress through selected research projects.

After some brief comments on the current state of industry practice in retail supply chain management, this paper will describe examples of COER research in four areas: assortment planning, pricing, inventory optimization, and store execution.

Dynamic Hedging Under Jump Diffusion with Transaction Costs

Kennedy, J. S., Forsyth, P. A., Vetzal, K. R. — 2009-06-27

If the price of an asset follows a jump diffusion process, the market is in general incomplete. In this case, hedging a contingent claim written on the asset is not a trivial matter, and other instruments besides the underlying must be used to hedge in order to provide adequate protection against jump risk. We devise a dynamic hedging strategy that uses a hedge portfolio consisting of the underlying asset and liquidly traded options, where transaction costs are assumed present due to a relative bid-ask spread. At each rebalance time, the hedge weights are chosen to simultaneously (i) eliminate the instantaneous diffusion risk by imposing delta neutrality, and (ii) minimize an objective that is a linear combination of a jump risk and transaction cost penalty function. Because reducing the jump risk is a competing goal vis-à-vis controlling for transaction cost, the respective components in the objective must be appropriately weighted. Hedging simulations of this procedure are carried out, and our results indicate that the proposed dynamic hedging strategy provides sufficient protection against the diffusion and jump risk while not incurring large transaction costs.

Portfolio Selection with Robust Estimation

DeMiguel, V., Nogales, F. J. — 2009-06-27

Mean-variance portfolios constructed using the sample mean and covariance matrix of asset returns perform poorly out of sample due to estimation error. Moreover, it is commonly accepted that estimation error in the sample mean is much larger than in the sample covariance matrix. For this reason, researchers have recently focused on the minimum-variance portfolio, which relies solely on estimates of the covariance matrix, and thus usually performs better out of sample. However, even the minimum-variance portfolios are quite sensitive to estimation error and have unstable weights that fluctuate substantially over time. In this paper, we propose a class of portfolios that have better stability properties than the traditional minimum-variance portfolios. The proposed portfolios are constructed using certain robust estimators and can be computed by solving a single nonlinear program, where robust estimation and portfolio optimization are performed in a single step. We show analytically that the resulting portfolio weights are less sensitive to changes in the asset-return distribution than those of the traditional portfolios. Moreover, our numerical results on simulated and empirical data confirm that the proposed portfolios are more stable than the traditional minimum-variance portfolios, while preserving (or slightly improving) their relatively good out-of-sample performance.

Why Defeating Insurgencies Is Hard: The Effect of Intelligence in Counterinsurgency Operations--A Best-Case Scenario

Kress, M., Szechtman, R. — 2009-06-27

In insurgency situations, the government-organized force is confronted by a small guerrilla group that is dispersed in the general population with no or a very small signature. Effective counterinsurgency operations require good intelligence. Absent intelligence, not only might the insurgents escape unharmed and continue their violent actions, but collateral damage caused to the general population from poor targeting may generate adverse response against the government and create popular support for the insurgents, which may result in higher recruitment to the insurgency. We model the dynamic relations among intelligence, collateral casualties in the population, attrition, recruitment to the insurgency, and reinforcement to the government force. Even under best-case assumptions, we show that the government cannot totally eradicate the insurgency by force. The best it can do is contain it at a certain fixed level.

Incremental Network Optimization: Theory and Algorithms

Seref, O., Ahuja, R. K., Orlin, J. B. — 2009-06-27

In an incremental optimization problem, we are given a feasible solution x⁰ of an optimization problem P, and we want to make an incremental change in x⁰ that will result in the greatest improvement in the objective function. In this paper, we study the incremental optimization versions of six well-known network problems. We present a strongly polynomial algorithm for the incremental minimum spanning tree problem. We show that the incremental minimum cost flow problem and the incremental maximum flow problem can be solved in polynomial time using Lagrangian relaxation. We consider two versions of the incremental minimum shortest path problem, where increments are measured via arc inclusions and arc exclusions. We present a strongly polynomial time solution for the arc inclusion version and show that the arc exclusion version is NP-complete. We show that the incremental minimum cut problem is NP-complete and that the incremental minimum assignment problem reduces to the minimum exact matching problem, for which a randomized polynomial time algorithm is known.

Bounds for Maximin Latin Hypercube Designs

van Dam, E. R., Rennen, G., Husslage, B. — 2009-06-27

Latin hypercube designs (LHDs) play an important role when approximating computer simulation models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maximin LHDs can be quite time consuming when the number of dimensions and design points increase. In these cases, we can use heuristical maximin LHDs. In this paper, we construct bounds for the separation distance of certain classes of maximin LHDs. These bounds are useful for assessing the quality of heuristical maximin LHDs. Until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs, i.e., for maximin designs without a Latin hypercube structure. The separation distance of maximin LHDs also satisfies these "unrestricted" bounds. By using some of the special properties of LHDs, we are able to find new and tighter bounds for maximin LHDs. Within the different methods used to determine the upper bounds, a variety of combinatorial optimization techniques are employed. Mixed-integer programming, the traveling salesman problem, and the graph-covering problem are among the formulations used to obtain the bounds. Besides these bounds, also a construction method is described for generating LHDs that meet Baer's bound for the distance measure for certain values of n.

Large-Scale, Less-than-Truckload Service Network Design

Jarrah, A. I., Johnson, E., Neubert, L. C. — 2009-06-27

We present a novel formulation for the service network design problem in the context of large-scale, less-than-truckload (LTL) freight operations. The formulation captures the basic network design constraints; the load-planning requirement that all freight at a location, irrespective of the freight's origin, loads to the same next terminal; and other important LTL-specific requirements. Our modeling scheme fragments the underlying massive network design model with up to 1.3 million 0–1 variables and 1.3 million rows into a separate and efficient integer programming (IP) problem for each destination terminal along with a coordinating master network design problem. We produce high-quality solutions in very reasonable CPU times (~2 hours) using slope scaling and load-planning tree generation with corresponding potential annual savings of $20–25 million dollars for the target company for which the research was conducted.

Optimal Policies for Inventory Systems with Separate Delivery-Request and Order-Quantity Decisions

Li, Q., Wu, X., Cheung, K. L. — 2009-06-27

Motivated by logistics practices, we consider a retailer that replenishes its inventory by making a delivery request without specifying a quantity, then deciding the quantity when the delivery vehicle arrives after one period. A fixed cost is incurred whenever a delivery request is made, regardless of the quantity ordered later. The new feature of this research relative to previous work is the separation of the delivery request and the quantity decision, or the postponement of ordering until one-period demand information is observed. Due to such separation, both the state space and the action space must be augmented in the model. We show that the optimal policy for delivery requests is of a threshold type: A delivery request is made if and only if the inventory on hand is below a threshold. The optimal decision on ordering is more complex, and there might be multiple order-up-to levels. Our numerical studies show, nonetheless, that the cost of an ordering policy that considers (at most) two order-up-to levels is close to the minimal when the planning horizon is not too short. We also identify conditions under which a base-stock policy is optimal for ordering. To understand the effects of ordering postponement, we compare our model with the traditional model in which the two decisions must be made at the same time. We show that postponement leads not only to a lower cost, but also a higher threshold for making delivery requests.

Using Lagrangian Relaxation to Compute Capacity-Dependent Bid Prices in Network Revenue Management

Topaloglu, H. — 2009-06-27

We propose a new method to compute bid prices in network revenue management problems. The novel aspect of our method is that it explicitly considers the temporal dynamics of the arrivals of the itinerary requests and generates bid prices that depend on the remaining leg capacities. Our method is based on relaxing certain constraints that link the decisions for different flight legs by associating Lagrange multipliers with them. In this case, the network revenue management problem decomposes by the flight legs, and we can concentrate on one flight leg at a time. When compared with the so-called deterministic linear program, we show that our method provides a tighter upper bound on the optimal objective value of the network revenue management problem. Computational experiments indicate that the bid prices obtained by our method perform significantly better than the ones obtained by standard benchmark methods.

An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and Integer Constraints

Bonami, P., Lejeune, M. A. — 2009-06-27

In this paper, we study extensions of the classical Markowitz mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint, which imposes that the expected return of the constructed portfolio must exceed a prescribed return threshold with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the deterministic equivalents are second-order cone programs and give closed-form formulations. Second, we account for real-world trading constraints (such as the need to diversify the investments in a number of industrial sectors, the nonprofitability of holding small positions, and the constraint of buying stocks by lots) modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a nonlinear branch-and-bound algorithm that features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and is called portfolio risk branching. The two branching rules are implemented and tested using the open-source Bonmin framework. The comparison of the computational results obtained with state-of-the-art MINLP solvers (MINLP_BB and CPLEX) and with our approach shows the effectiveness of the latter, which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time. The practicality of the approach is illustrated through its use for the construction of four fund-of-funds now available on the major trading markets.

Dynamic Capacity Management with Substitution

Shumsky, R. A., Zhang, F. — 2009-06-27

We examine a multiperiod capacity allocation model with upgrading. There are multiple product types, corresponding to multiple classes of demand, and the firm purchases capacity of each product before the first period. Within each period, after demand arrives, products are allocated to customers. Customers who arrive to find that their product has been depleted can be upgraded by at most one level. We show that the optimal allocation policy is a simple two-step algorithm: First, use any available capacity to satisfy same-class demand, and then upgrade customers until capacity reaches a protection limit, so that in the second step the higher-level capacity is rationed. We show that these results hold both when all capacity is salvaged at the end of the last demand period as well as when capacity can be replenished (in the latter case, an order-up-to policy is optimal for replenishment). Although finding the optimal protection limits is computationally intensive, we describe bounds for the optimal protection limits that take little effort to compute and can be used to effectively solve large problems. Using these heuristics, we examine numerically the relative value of strictly optimal capacity and dynamic rationing, the value of perfect demand information, and the impact of demand and economic parameters on the value of optimal substitution.

Staffing to Maximize Profit for Call Centers with Alternate Service-Level Agreements

Baron, O., Milner, J. — 2009-06-27

To ensure quality from outsourced call centers, firms sign service-level agreements (SLAs). These define service measures such as what constitutes an acceptable delay or an acceptable abandonment rate. They may also dictate penalties for failing to meet agreed-upon targets. We introduce a period-based SLA that measures performance over a short duration such as a rush hour. We compare it to alternate SLAs that measure service by individual and over a long horizon. To measure the service levels for these SLAs, we develop several approximations. We approximate the probability an acceptable delay is met by generalizing the heavy-traffic quality and efficiency driven regime. We also provide a new approximation for the abandonment rate. Further, we prove a central limit theorem for the probability of meeting a service level measured by the percentage of customers acceptably served during a period. We demonstrate how an outsourced call center operating in an environment with uncertain demand and abandonment can determine its staffing policy to maximize the expected profit for these SLAs. Numerical experiments demonstrate a high degree of accuracy for the approximations and the resulting staffing levels. We indicate several salient features of the behavior of the period-based SLA.

Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables

Li, H.-L., Lu, H.-C. — 2009-06-27

Many optimization problems are formulated as generalized geometric programming (GGP) containing signomial terms f(X)·g(Y), where X and Y are continuous and discrete free-sign vectors, respectively. By effectively convexifying f(X) and linearizing g(Y), this study globally solves a GGP with a lower number of binary variables than are used in current GGP methods. Numerical experiments demonstrate the computational efficiency of the proposed method.

On a Data-Driven Method for Staffing Large Call Centers

Bassamboo, A., Zeevi, A. — 2009-06-27

We consider a call center model with multiple customer classes and multiple server pools. Calls arrive randomly over time, and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner. The objective is to minimize the sum of personnel costs and expected abandonment penalties by selecting an appropriate staffing level for each server pool. We propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class; in this sense the method is said to be data-driven. The efficacy of the proposed method is illustrated via numerical examples. An asymptotic analysis establishes that the prescribed staffing levels achieve near-optimal performance and characterizes the magnitude of the optimality gap.

Discrete Hit-and-Run for Sampling Points from Arbitrary Distributions Over Subsets of Integer Hyperrectangles

2009-06-27

We consider the problem of sampling a point from an arbitrary distribution over an arbitrary subset S of an integer hyperrectangle. Neither the distribution nor the support set S are assumed to be available as explicit mathematical equations, but may only be defined through oracles and, in particular, computer programs. This problem commonly occurs in black-box discrete optimization as well as counting and estimation problems. The generality of this setting and high dimensionality of S precludes the application of conventional random variable generation methods. As a result, we turn to Markov chain Monte Carlo (MCMC) sampling, where we execute an ergodic Markov chain that converges to so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to . Unfortunately, classical Markov chains, such as the nearest-neighbor random walk or the coordinate direction random walk, fail to converge to because they can get trapped in isolated regions of the support set. To surmount this difficulty, we propose discrete hit-and-run (DHR), a Markov chain motivated by the hit-and-run algorithm known to be the most efficient method for sampling from log-concave distributions over convex bodies in Rⁿ. We prove that the limiting distribution of DHR is as desired, thus enabling us to sample approximately from by delivering the last iterate of a sufficiently large number of iterations of DHR. In addition to this asymptotic analysis, we investigate finite-time behavior of DHR and present a variety of examples where DHR exhibits polynomial performance.

Uncertainty, Information Acquisition, and Technology Adoption

Ulu, C., Smith, J. E. — 2009-06-27

Consumers or firms contemplating purchasing a new product or adopting a new technology are often plagued by uncertainty: Will the benefits outweigh the costs? Should we buy now or wait and gather more information? In this paper, we study a dynamic programming model of this technology adoption problem. In each period, the consumer decides whether to adopt the technology, reject it, or wait and gather additional information by observing a signal about the technology's benefit. The technology's actual benefit may be constant or changing stochastically over time. The dynamic programming state variable is a probability distribution that describes the consumer's beliefs about the benefits of the technology. We allow general probability distributions on benefits and general signal processes and assume that the consumer updates her beliefs over time using Bayes' rule. We are interested in structural properties of this model. We show that improving the technology's benefit need not make the consumer better off and that first-order stochastic dominance improvements in the consumer's distribution on benefits need not increase the consumer's value function. Nevertheless, the model possesses a great deal of structure. For example, we obtain monotonic value functions and policies if we order distributions using likelihood-ratio dominance rather than first-order stochastic dominance. We also examine convexity properties and provide many comparative statics results.

Near-Optimal Dynamic Lead-Time Quotation and Scheduling Under Convex-Concave Customer Delay Costs

Ata, B., Olsen, T. L. — 2009-06-27

We consider a make-to-order system where customers are dynamically quoted lead times (and prices). Customers are homogenous but have general (nonlinear) disutility for delay. Because the firm is a monopolist, the pricing problem is trivial and the dynamic problem reduces to one of lead-time quotation and order sequencing. We also consider the (static) problem of up-front capacity installation. We use a large-capacity asymptotic regime to make the problem tractable. We provide recommended policies for convex, concave, and convex-concave lead-time cost functions and prove that these policies are asymptotically optimal. The policies are both highly intuitive and readily implementable. Moreover, they provide delay guarantees for all served customers. They are tested numerically; we find that significant benefits can accrue by using the prescribed dynamic policies instead of first-come-first-served type policies.

A Column Generation Algorithm for Choice-Based Network Revenue Management

Bront, J. J. M., Mendez-Diaz, I., Vulcano, G. — 2009-06-27

During the past few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges. One way to describe choice behavior is to assume that each customer belongs to a segment, which is characterized by a consideration set, i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature.

In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. (2004) [Gallego, G., G. Iyengar, R. Phillips, A. Dubey. 2004. Managing flexible products on a network. Technical Report CORC TR-2004-01, Department of Industrial Engineering and Operations Research, Columbia University, New York], and the follow-up dynamic programming decomposition heuristic of van Ryzin and Liu (2008) [van Ryzin, G. J., Q. Liu. 2008. On the choice-based linear programming model for network revenue management. Manufacturing Service Oper. Management 10(2) 288–310]. We focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-hard and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective and that the overall approach leads to high-quality, practical solutions.

Technical Note--A Multiperiod Model of Inventory Competition

Nagarajan, M., Rajagopalan, S. — 2009-06-27

This paper explores when it is important for firms to consider stockout-based substitution and competitor's inventory levels in making inventory decisions in the context of a duopoly model. To address this question, we consider a model where two newsvendors sell substitutable products in a market with aggregate market demand D. The two firms get a proportion p and (1 – p) of this demand, where p is random. We characterize the equilibrium inventory levels of the two firms in a single-period model and show the striking property that, under certain reasonable conditions on the cost parameters, the two firms ignore their competitor's inventory levels and potential substitution demand, i.e., their inventory decisions are decoupled. Furthermore, we show under slightly more restrictive conditions on the cost parameters that the single-period results can be extended to the case where D is random. Finally, we extend the decoupling property to a multiperiod periodic review scenario and show that the resulting Nash equilibrium can be characterized simply as the solution to a single-product dynamic newsvendor problem that ignores substitution demand.

Technical Note--A Note on "The Censored Newsvendor and the Optimal Acquisition of Information"

Bensoussan, A., Cakanyildirim, M., Sethi, S. P. — 2009-06-27

This paper revisits the finite-horizon model of a censored newsvendor by Ding et al. [Ding, X., M. L. Puterman, A. Bisi. 2002. The censored newsvendor and the optimal acquisition of information. Oper. Res. 50 517–527]. An important result claimed there without a proper proof is that the myopic order quantity is always less than or equal to the optimal order quantity. Lu et al. [Lu, X., J. S. Song, K. Zhu. 2008. Analysis of perishable inventory systems with censored demand data. Oper. Res. 56(4) 1034–1038.] supplied a correct proof of the result. We analyze the same model using the interesting concept of the unnormalized probability, which simplifies the dynamic programming equation considerably and facilitates the proof of the claim. Moreover, it produces the proof of the existence of an optimal solution for an infinite-horizon setting of the problem.

Contributors

2009-06-27

No abstract available.