Knowledge that Transforms

The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is also significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In this paper, we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery. The objective includes a fixed cost of opening ORs and a variable cost of overtime relative to a fixed length-of-day. We describe two types of models. The first is a two-stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage. The second is its robust counterpart, in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations. We describe the mathematical models, bounds on the optimal solution, and solution methodologies, including an easy-to-implement heuristic. Numerical experiments based on real data from a large health-care provider are used to contrast the results for the two models and illustrate the potential for impact in practice. Based on our numerical experimentation, we find that a fast and easy-to-implement heuristic works fairly well, on average, across many instances. We also find that the robust method performs approximately as well as the heuristic, is much faster than solving the stochastic recourse model, and has the benefit of limiting the worst-case outcome of the recourse problem.

In many applications involving make-to-order or time-sensitive (e.g., perishable, seasonal) products, finished orders are often delivered to customers immediately or shortly after the production. Consequently, there is little or no finished product inventory in the supply chain such that production and outbound distribution are very intimately linked and must be scheduled jointly to achieve a desired on-time delivery performance at minimum total cost. Research on integrated scheduling models of production and outbound distribution is relatively recent but is growing very rapidly. In this paper, we provide a survey of such existing models. We present a unified model representation scheme, classify existing models into several different classes, and for each class of the models give an overview of the optimality properties, computational tractability, and solution algorithms for the various problems studied in the literature. We clarify the tractability of some open problems left in the literature and some new problems by providing intractability proofs or polynomial-time exact algorithms. We also identify several problem areas and issues for future research.

We study the modeling of nonconvex piecewise-linear functions as mixed-integer programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise-linear functions with special attention paid to multivariate nonseparable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study the extension of these formulations to lower semicontinuous piecewise-linear functions.

We extend the basic theory of kriging, as applied to the design and analysis of deterministic computer experiments, to the stochastic simulation setting. Our goal is to provide flexible, interpolation-based metamodels of simulation output performance measures as functions of the controllable design or decision variables, or uncontrollable environmental variables. To accomplish this, we characterize both the intrinsic uncertainty inherent in a stochastic simulation and the extrinsic uncertainty about the unknown response surface. We use tractable examples to demonstrate why it is critical to characterize both types of uncertainty, derive general results for experiment design and analysis, and present a numerical example that illustrates the stochastic kriging method.

We propose a continuous-time stochastic model for the dynamics of a limit order book. The model strikes a balance between three desirable features: it can be estimated easily from data, it captures key empirical properties of order book dynamics, and its analytical tractability allows for fast computation of various quantities of interest without resorting to simulation. We describe a simple parameter estimation procedure based on high-frequency observations of the order book and illustrate the results on data from the Tokyo Stock Exchange. Using simple matrix computations and Laplace transform methods, we are able to efficiently compute probabilities of various events, conditional on the state of the order book: an increase in the midprice, execution of an order at the bid before the ask quote moves, and execution of both a buy and a sell order at the best quotes before the price moves. Using high-frequency data, we show that our model can effectively capture the short-term dynamics of a limit order book. We also evaluate the performance of a simple trading strategy based on our results.

Reliable facility location models consider unexpected failures with site-dependent probabilities, as well as possible customer reassignment. This paper proposes a compact mixed integer program (MIP) formulation and a continuum approximation (CA) model to study the reliable uncapacitated fixed charge location problem (RUFL), which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios. The MIP determines the optimal facility locations as well as the optimal customer assignments and is solved using a custom-designed Lagrangian relaxation (LR) algorithm. The CA model predicts the total system cost without details about facility locations and customer assignments, and it provides a fast heuristic to find near-optimum solutions. Our computational results show that the LR algorithm is efficient for mid-sized RUFL problems and that the CA solutions are close to optimal in most of the test instances. For large-scale problems, the CA method is a good alternative to the LR algorithm that avoids prohibitively long running times.

Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.

Data envelopment analysis (DEA) is known as a nonparametric mathematical programming approach to productive efficiency analysis. In this paper, we show that DEA can be alternatively interpreted as nonparametric least-squares regression subject to shape constraints on the frontier and sign constraints on residuals. This reinterpretation reveals the classic parametric programming model by Aigner and Chu [Aigner, D., S. Chu. 1968. On estimating the industry production function. Amer. Econom. Rev. 58 826–839] as a constrained special case of DEA. Applying these insights, we develop a nonparametric variant of the corrected ordinary least-squares (COLS) method. We show that this new method, referred to as corrected concave nonparametric least squares (C2NLS), is consistent and asymptotically unbiased. The linkages established in this paper contribute to further integration of the econometric and axiomatic approaches to efficiency analysis.

Working in collaboration with Spain-based retailer Zara, we address the problem of distributing, over time, a limited amount of inventory across all the stores in a fast-fashion retail network. Challenges specific to that environment include very short product life cycles, and store policies whereby an article is removed from display whenever one of its key sizes stocks out. To solve this problem, we first formulate and analyze a stochastic model predicting the sales of an article in a single store during a replenishment period as a function of demand forecasts, the inventory of each size initially available, and the store inventory management policy just stated. We then formulate a mixed-integer program embedding a piecewise-linear approximation of the first model applied to every store in the network, allowing us to compute store shipment quantities maximizing overall predicted sales, subject to inventory availability and other constraints. We report the implementation of this optimization model by Zara to support its inventory distribution process, and the ensuing controlled pilot experiment performed to assess the model's impact relative to the prior procedure used to determine weekly shipment quantities. The results of that experiment suggest that the new allocation process increases sales by 3% to 4%, which is equivalent to $275 M in additional revenues for 2007, reduces transshipments, and increases the proportion of time that Zara's products spend on display within their life cycle. Zara is currently using this process for all of its products worldwide.