Operations Research current issue

In This Issue

2008-04-21

No abstract available.

The OR/MS Ecosystem: Strengths, Weaknesses, Opportunities, and Threats

Sodhi, M. S., Tang, C. S. — 2008-04-21

This paper is dedicated to Arthur Geoffrion, who serves as role model of a great researcher, educator, and practitioner.

We believe that research, teaching, and practice are becoming increasingly disengaged from one another in the OR/MS ecosystem. This ecosystem comprises researchers, educators, and practitioners in its core along with end users, universities, and funding agencies. Continuing disengagement will result in OR/MS occupying only niche areas and disappearing as a distinct field even though its tools would live on. To understand the reasons for this disengagement better and to engender discussion among academics and practitioners on how to counter it, we present the ecosystem's strengths, weaknesses, opportunities, and threats. Incorporated in this paper are insights from a cluster of sessions at the 2006 INFORMS meeting in Pittsburgh ("Where Do We Want to Go in OR/MS?") and from the literature.

OR PRACTICE--An Economic Equilibrium Model of the Market for Marine Transportation Services in Petroleum Products

Mudrageda, M., Murphy, F. H. — 2008-04-21

We describe a model of the market for petroleum tank vessels used for planning by Maritrans, Inc. This model is an enhanced version of an earlier model and more closely approximates the market for transportation services. Because of the better representation, we found that the market, which is defined around an index for transportation services, has the potential for multiple equilibria. We present how the model has been used in making major decisions at Maritrans and show how the index design leads to an anomaly where demand could increase with increasing prices, leading to the potential for multiple equilibria. We have not observed this phenomenon in the market. However, with the advent of forward markets for transportation services, known as freight-forward markets, if multiple equilibria do appear, it could become profitable for a player to move a market from one equilibrium to another.

Fast Pricing of Basket Default Swaps

Chen, Z., Glasserman, P. — 2008-04-21

A basket default swap is a derivative security tied to an underlying basket of corporate bonds or other assets subject to credit risk. The value of the contract depends on the joint distribution of the default times of the underlying assets. Valuing a basket default swap often entails Monte Carlo simulation of these default times. For baskets of high-quality credits and for swaps that require multiple defaults to trigger payment, pricing the swap is a rare-event simulation problem. The Joshi-Kainth algorithm is an innovative importance-sampling technique for this problem that forces a predetermined number of defaults to occur on each path. This paper analyzes, extends, and improves the Joshi-Kainth algorithm. We show that, in its original form, the algorithm can actually increase variance; we present an alternative that is guaranteed to reduce variance, even when defaults are not rare. Along the way, we provide a rigorous underpinning in a setting sufficiently general to include both the original method and the version proposed here.

Pricing Options in Jump-Diffusion Models: An Extrapolation Approach

Feng, L., Linetsky, V. — 2008-04-21

We propose a new computational method for the valuation of options in jump-diffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicit-explicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is first-order accurate in time. Second-order accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new high-order time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton's and Kou's jump-diffusion models, the diffusion-extended variance gamma model, as well as the two-dimensional Duffie-Pan-Singleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a one-year double-barrier option in Kou's jump-diffusion model, our scheme attains accuracy of 10^–5 in 72 time steps (in 0.05 seconds). In contrast, it takes the first-order IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the second-order IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive to financial engineering modelers.

Risk in Revenue Management and Dynamic Pricing

Levin, Y., McGill, J., Nediak, M. — 2008-04-21

We present a new model for optimal dynamic pricing of perishable services or products that incorporates a simple risk measure permitting control of the probability that total revenues fall below a minimum acceptable level. The formulation assumes that sales must occur within a finite time period, that there is a finite—possibly large—set of available prices, and that demand follows a price-dependent, nonhomogeneous Poisson process. This model is particularly appropriate for applications in which attainment of a revenue target is an important consideration for managers; for example, in event management, in seasonal clearance of high-value items, or for business subunits operating under performance targets. We formulate the model as a continuous-time optimal control problem, obtain optimality conditions, explore structural properties of the solution, and report numerical results on problems of realistic size.

A Linear Decision-Based Approximation Approach to Stochastic Programming

Chen, X., Sim, M., Sun, P., Zhang, J. — 2008-04-21

Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse.

Optimal Dynamic Trading Strategies with Risk Limits

Cuoco, D., He, H., Isaenko, S. — 2008-04-21

Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analysis of the optimal behavior of a trader subject to VaR limits has produced a negative view of VaR as a risk-control tool. In particular, VaR limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses. However, these conclusions are based on models that are either static or dynamically inconsistent. In this paper, we formulate a dynamically consistent model of optimal portfolio choice subject to VaR limits and show that the concerns expressed in earlier papers do not apply if, consistently with common practice, the VaR limit is reevaluated dynamically. In particular, we find that the optimal risk exposure of a trader subject to a VaR limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower. We also consider risk limits formulated in terms of tail conditional expectation (TCE), a coherent risk measure often advocated as an alternative to VaR, and show that in our dynamic setting it is always possible to transform a TCE limit into an equivalent VaR limit, and conversely.

Supply Function Equilibrium in a Constrained Transmission System

Wilson, R. — 2008-04-21

This paper characterizes a supply function equilibrium in an auction market constrained by limited capacities of links in a transportation network and limited input/output capacities of participants. The formulation is adapted to a wholesale spot market for electricity managed by the operator of the transmission system. The results are derived using the calculus of variations to obtain the Euler conditions and the transversality conditions that characterize a Nash equilibrium in an auction in which bids are as supply functions, and quantities and payments are based either on nodal prices or pay-as-bid.

Two-Stage Fleet Assignment Model Considering Stochastic Passenger Demands

Sherali, H. D., Zhu, X. — 2008-04-21

An airline's fleet typically contains multiple aircraft families, each having a specific cockpit design and crew requirement. Each aircraft family contains multiple aircraft types having different capacities. Given a flight schedule network, the fleet assignment model is concerned with assigning aircraft to flight legs to maximize profits with respect to captured itinerary-based demand. However, because of related yield management and crew-scheduling regulations, in particular, this decision needs to be made well in advance of departures when market demand is still highly uncertain, although subsequently at a later stage, reassignments of aircraft types within a given family can be made when demand forecasts improve, while preserving crew schedules. In this paper, we propose a two-stage stochastic mixed-integer programming approach in which the first stage makes only higher-level family-assignment decisions, while the second stage performs subsequent family-based type-level assignments according to forecasted market demand realizations. By considering demand uncertainty up-front at the initial fleeting stage, we inject additional flexibility in the process that offers more judicious opportunities for later revisions. We conduct a polyhedral analysis of the proposed model and develop suitable solution approaches. Results of some numerical experiments are presented to exhibit the efficacy of using the stochastic model as opposed to the traditional deterministic model that considers only expected demand, and to demonstrate the efficiency of the proposed algorithms as compared with solving the model using its deterministic equivalent.

Optimal Dynamic Production and Inventory Transshipment Policies for a Two-Location Make-to-Stock System

Zhao, H., Ryan, J. K., Deshpande, V. — 2008-04-21

Inventory sharing through transshipment has attracted a great deal of attention from researchers and practitioners due to its potential for increasing service levels while simultaneously decreasing stock levels. In this paper, we analyze the optimal production and transshipment policy for a two-location make-to-stock queueing system with exponential production and interarrival times. A key feature of our model is that we allow transshipments to be triggered by both demand arrivals and production completions. Thus, transshipment is used to achieve production flexibility through inventory reallocation, as well as to fill emergency demands. We also consider capacity issues in transshipment by modeling each location as a single-server, make-to-stock queueing system. In this setting, we prove that the optimal production policy for each location belongs to the "hedging point" family of policies, while the optimal demand filling policy belongs to the "state-dependent rationing" family of policies. We analyze the structural properties of the optimal policy and provide conditions under which the optimal policy can be simplified. Given the complex nature of the optimal policy, we develop three easy-to-implement heuristics that work very well for a large range of cost parameters.

Diversity Maximization Approach for Multiobjective Optimization

Masin, M., Bukchin, Y. — 2008-04-21

One of the most common approaches for multiobjective optimization is to generate the whole or partial efficient frontier and then decide about the preferred solution in a higher-level decision-making process. In this paper, a new method for generating the efficient frontier for multiobjective problems is developed, called the diversity maximization approach (DMA). This approach is capable of solving mixed-integer and combinatorial problems. The DMA finds Pareto optimal solutions by maximizing a proposed diversity measure and guarantees generating the complete set of efficient points. Given a subset of the efficient frontier, DMA finds the next Pareto optimal solution which, combined with the existing ones, yields the most diversified subset of efficient points. This solution is defined as the most diverse solution. In fact, it aims to maximize the distance between the new efficient point and the closest point in the given subset of the efficient frontier. The proposed approach can be applied to any problem that can be solved for the single-objective case. We can use the DMA by solving directly a modified version of the mixed-integer programming (MIP) formulation of the multiobjective problem. In this case, the Pareto optimal solutions are found sequentially in an iterative way. Consequently, as we terminate the procedure before completion, a partial efficient frontier is available. The diversity measure assures that in every stage of the procedure, the partial efficient frontier is well diversified. This partial efficient frontier can be perceived as a filtered set of the complete efficient frontier and can be used by the decision maker in case the complete efficient frontier contains too many points. An additional way of using DMA is by incorporating it in a problem oriented branch-and-bound algorithm. Detailed examples of both approaches are given.

An Optimization Algorithm for the Ordered Open-End Bin-Packing Problem

Ceselli, A., Righini, G. — 2008-04-21

The ordered open-end bin-packing problem is a variant of the bin-packing problem in which the items to be packed are sorted in a given order and the capacity of each bin can be exceeded by the last item packed into the bin. We present a branch-and-price algorithm for its exact optimization. The pricing subproblem is a special variant of the binary knapsack problem, in which the items are ordered and the last one does not consume capacity. We present a specialized optimization algorithm for this subproblem. The speed of the column generation algorithm is improved by subgradient optimization steps, allowing for multiple pricing and variable fixing. Computational results are presented on instances of different sizes and items with different correlations between their size and their position in the given order.

Evaluation and Optimization of Installation Base-Stock Policies in Supply Chains with Compound Poisson Demand

Zhao, Y. — 2008-04-21

In this paper, we establish an exact framework for a class of supply chains with at most one directed path between every two stages. External demands follow compound Poisson processes, the transit times are stochastic, sequential, and exogenous, and each stage controls its inventory by an installation base-stock policy under continuous review. Unsatisfied demand at each stage is fully backordered. This class of supply chains includes assembly, distribution, tree, and two-level general networks as special cases. We characterize the stockout delay for each unit of demand at each stage of the supply chain by developing an exact and unified approach that applies to various network topologies. We also present tractable approximations and decompositions that facilitate efficient evaluation and optimization (up to the approximations) of the base-stock policies in industry-size problems with a tree structure. We demonstrate the effectiveness of the solution by numerical studies.

Heavy-Traffic Optimality of a Stochastic Network Under Utility-Maximizing Resource Allocation

Ye, H.-Q., Yao, D. D. — 2008-04-21

We study a stochastic network that consists of a set of servers processing multiple classes of jobs. Each class of jobs requires a concurrent occupancy of several servers while being processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism. The allocation of the service capacities is a real-time control mechanism: in each network state, the resource allocation is the solution to an optimization problem that maximizes a general utility function. Whereas this resource allocation optimizes in a "greedy" fashion with respect to each state, we establish its asymptotic optimality in terms of (a) deriving the fluid and diffusion limits of the network under this allocation scheme, and (b) identifying a cost function that is minimized in the diffusion limit, along with a characterization of the so-called fixed-point state of the network.

Regulated Random Walks and the LCFS Backlog Probability: Analysis and Application

Baron, O. — 2008-04-21

Random walks have been used extensively within operations research models such as inventory systems and single-server queues to estimate performance measures. In this paper, we use sample-path analysis to express the steady-state probability of a one-sided regulated random walk to increase and be above a threshold, referred to as the last-come-first-serve (LCFS) backlog probability. We approximate the LCFS backlog probability under mild assumptions on the distribution of the random walk's steps and provide its exact expression when the steps are exponentially distributed, and a closed-form approximation when the steps are normally distributed. In our numerical experiments, the average relative gap between the approximated LCFS backlog probabilities and their simulated values is 5.13%. We further show that the LCFS backlog probability is an upper bound on the loss probability—the probability that a two-sided regulated random walk is at a boundary. This bound is tighter than the backlog probability—the probability that a random walk ever crosses a threshold—that also bounds the loss probability. In an inventory application, we demonstrate that using the LCFS backlog probability rather than the backlog probability reduces the inventory level required to satisfy a service-level constraint on the percentage of orders backlogged. In our examples, this reduction leads to cost savings of 31% on average.

Multivariate Bayesian Control Chart

Makis, V. — 2008-04-21

A multivariate Bayesian control chart for monitoring process mean under the assumption that the vector of process observations follows a multivariate normal distribution is considered. Traditional control charts such as Hotelling's T², EWMA, and CUSUM charts have been applied to control industrial processes characterized by several measurable variables. It is well known that these traditional, non-Bayesian process control techniques are not optimal, but very few results regarding the structure of the Bayesian control policy have been reported in the literature, all dealing with the univariate, finite-horizon case. In this paper, we formulate the multivariate Bayesian process control problem in the optimal stopping framework. The objective is to find a stopping rule under partial observations, minimizing the long-run expected average cost per unit time for a given sample size and sampling interval. Under standard operating and cost assumptions, it is proved that a control limit policy is optimal, and an algorithm is presented to find the optimal control limit and the minimum average cost.

Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

Jepsen, M., Petersen, B., Spoorendonk, S., Pisinger, D. — 2008-04-21

This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with time windows. The standard Dantzig-Wolfe decomposition of the arc flow formulation leads to a set-partitioning problem as the master problem and an elementary shortest-path problem with resource constraints as the pricing problem. We introduce the subset-row inequalities, which are Chvatal-Gomory rank-1 cuts based on a subset of the constraints in the master problem. Applying a subset-row inequality in the master problem increases the complexity of the label-setting algorithm used to solve the pricing problem because an additional resource is added for each inequality. We propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step-like structure of the objective function of the pricing problem. Computational experiments have been performed on the Solomon benchmarks where we were able to close several instances. The results show that applying subset-row inequalities in the master problem significantly improves the lower bound and, in many cases, makes it possible to prove optimality in the root node.

Visualizing and Constructing Cycles in the Simplex Method

Avis, D., Kaluzny, B., Titley-Peloquin, D. — 2008-04-21

Most examples of cycling in the simplex method are given without explanation of how they were constructed. An exception is Beale's example built around the geometry of the dual simplex method in the plane [Beale, E. 1955. Cycling in the dual simplex method. Naval Res. Logist. Quart. 2(4) 269–275]. Using this approach, we give a simple geometric explanation for a number of examples of cycling in the simplex method, including Hoffman's original example [Hoffman, A. 1953. Cycling in the Simplex Algorithm. National Bureau of Standards, Washington, D.C.]. This gives rise to a simple method for generating examples with cycles.

Technical Note--A Note on Parametric Analysis in Linear Assignment

Volgenant, A. — 2008-04-21

A classic application of the linear assignment problem is the assignment of people to jobs (or jobs to people). In this context, it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose; the length of the list is a parameter. A known list-generation procedure is based on an interior-point method followed by a parametric analysis. We describe a more efficient procedure, exploiting linear assignment theory and shortest-path computations. Further, we propose an alternative list-generation procedure, based on a special type of dual values for the linear assignment problem.

Contributors

2008-04-21

No abstract available.