Huseyin Topaloglu

Approximate Dynamic Programming for Ambulance Redeployment

We present an approximate dynamic programming approach for making ambulance redeployment decisions in an emergency medical service system. The primary decision is where we should redeploy idle ambulances so as to maximize the number of calls reached within a delay threshold. We begin by formulating this problem as a dynamic program. To deal with the high-dimensional and uncountable state space in the dynamic program, we construct approximations to the value function that are parameterized by a small number of parameters. We tune the parameters using simulated cost trajectories of the system. Computational experiments demonstrate the performance of the approach on emergency medical service systems in two metropolitan areas. We report practically significant improvements in performance relative to benchmark static policies.

Dynamic-Programming Approximations for Stochastic Time-Staged Integer Multicommodity-Flow Problems

In this paper, we consider a stochastic and time-dependent version of the min-cost integer multicommodity-flow problem that arises in the dynamic resource allocation context. In this problem class, tasks arriving over time have to be covered by a set of indivisible and reusable resources of different types. The assignment of a resource to a task removes the task from the system, modifies the resource, and generates a profit. When serving a task, resources of different types can serve as substitutes of each other, possibly yielding different revenues. We propose an iterative, adaptive dynamic-programming-based methodology that makes use of linear or nonlinear approximations of the value function. Our numerical work shows that the proposed method provides high-quality solutions and is computationally attractive for large problems.

Learning Algorithms for Separable Approximations of Discrete Stochastic Optimization Problems

We propose the use of sequences of separable, piecewise linear approximations for solving nondifferentiable stochastic optimization problems. The approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself. We prove the convergence of several versions of such methods when the objective function is separable and has integer break points, and we illustrate their behavior on numerical examples. We then demonstrate the performance on nonseparable problems that arise in the context of two-stage stochastic programming problems, and demonstrate that these techniques provide near-optimal solutions with a very fast rate of convergence compared with other solution techniques.

Stochastic Programming in Transportation and Logistics

Freight transportation is characterized by highly dynamic information processes: customers call in orders over time to move freight; the movement of freight over long distances is subject to random delays; equipment failures require last minute changes; and decisions are not always executed in the field according to plan. The high-dimensionality of the decisions involved has made transportation a natural application for the techniques of mathematical programming, but the challenge of modeling dynamic information processes has limited their success. In this chapter, we explore the use of concepts from stochastic programming in the context of resource allocation problems that arise in freight transportation. Since transportation problems are often quite large, we focus on the degree to which some techniques exploit the natural structure of these problems. Experimental work in the context of these applications is quite limited, so we highlight the techniques that appear to be the most promising.

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

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.

Assortment Optimization Under Variants of the Nested Logit Model

We study a class of assortment optimization problems where customers choose among the offered products according to the nested logit model. There is a fixed revenue associated with each product. The objective is to find an assortment of products to offer so as to maximize the expected revenue per customer. We show that the problem is polynomially solvable when the nest dissimilarity parameters of the choice model are less than one and the customers always make a purchase within the selected nest. Relaxing either of these assumptions renders the problem NP-hard. To deal with the NP-hard cases, we develop parsimonious collections of candidate assortments with worst-case performance guarantees. We also formulate a convex program whose optimal objective value is an upper bound on the optimal expected revenue. Thus, we can compare the expected revenue provided by an assortment with the upper bound on the optimal expected revenue to get a feel for the optimality gap of the assortment. By using this approach, our computational experiments test the performance of the parsimonious collections of candidate assortments that we develop.

Appointment Scheduling Under Patient Preference and No-Show Behavior

Motivated by the rising popularity of electronic appointment booking systems, we develop appointment scheduling models that take into account the patient preferences regarding when they would like to be seen. The service provider dynamically decides which appointment days to make available for the patients. Patients arriving with appointment requests may choose one of the days offered to them or leave without an appointment. Patients with scheduled appointments may cancel or not show up for the service. The service provider collects a “revenue” from each patient who shows up and incurs a “service cost” that depends on the number of scheduled appointments. The objective is to maximize the expected net “profit” per day. We begin by developing a static model that does not consider the current state of the scheduled appointments. We give a characterization of the optimal policy under the static model and bound its optimality gap. Building on the static model, we develop a dynamic model that considers the current state of the scheduled appointments, and we propose a heuristic solution procedure. In our computational experiments, we test the performance of our models under the patient preferences estimated through a discrete choice experiment that we conduct in a large community health center. Our computational experiments reveal that the policies we propose perform well under a variety of conditions.

Robust Assortment Optimization in Revenue Management Under the Multinomial Logit Choice Model

We study robust formulations of assortment optimization problems under the multinomial logit choice model. The novel aspect of our formulations is that the true parameters of the logit model are assumed to be unknown, and we represent the set of likely parameter values by a compact uncertainty set. The objective is to find an assortment that maximizes the worst-case expected revenue over all parameter values in the uncertainty set. We consider both static and dynamic settings. The static setting ignores inventory consideration, whereas in the dynamic setting, there is a limited initial inventory that must be allocated over time. We give a complete characterization of the optimal policy in both settings, show that it can be computed efficiently, and derive operational insights. We also propose a family of uncertainty sets that enables the decision maker to control the trade-off between increasing the average revenue and protecting against the worst-case scenario. Numerical experiments show that our robust approach, combined with our proposed family of uncertainty sets, is especially beneficial when there is significant uncertainty in the parameter values. When compared to other methods, our robust approach yields over 10% improvement in the worst-case performance, but it can also maintain comparable average revenue if average revenue is the performance measure of interest.

Constrained Assortment Optimization for the Nested Logit Model

We study assortment optimization problems where customer choices are governed by the nested logit model and there are constraints on the set of products offered in each nest. Under the nested logit model, the products are organized in nests. Each product in each nest has a fixed revenue associated with it. The goal is to find a feasible set of products, i.e., a feasible assortment, to maximize the expected revenue per customer. We consider cardinality and space constraints on the offered assortment, which limit the number of products and the total space consumption of the products offered in each nest, respectively. We show that the optimal assortment under cardinality constraints can be obtained efficiently by solving a linear program. The assortment optimization problem under space constraints is NP-hard. We show how to obtain an assortment with a performance guarantee of 2 under space constraints. This assortment also provides a performance guarantee of 1/(1- (epsilon) ) when the space requirement of each product is at most a fraction (epsilon) of the space availability in each nest. Building on our results for constrained assortment optimization, we show that we can efficiently solve joint assortment optimization and pricing problems under the nested logit model, where we choose the assortment of products to offer to customers, as well as the prices of the offered products.Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2014.1931 . This paper was accepted by Dimitris Bertsimas, optimization.

The d-Level Nested Logit Model: Assortment and Price Optimization Problems

We consider assortment and price optimization problems under the d -level nested logit model. In the assortment optimization problem, the goal is to find the revenue-maximizing assortment of products to offer, when the prices of the products are fixed. Using a novel formulation of the d -level nested logit model as a tree of depth d , we provide an efficient algorithm to find the optimal assortment. For a d -level nested logit model with n products, the algorithm runs in O ( d n log n ) time. In the price optimization problem, the goal is to find the revenue-maximizing prices for the products, when the assortment of offered products is fixed. Although the expected revenue is not concave in the product prices, we develop an iterative algorithm that generates a sequence of prices converging to a stationary point. Numerical experiments show that our method converges faster than gradient-based methods, by many orders of magnitude. In addition to providing solutions for the assortment and price optimization problems, we give support for the d -level nested logit model by demonstrating that it is consistent with the random utility maximization principle and equivalent to the elimination by aspects model.