Opher Baron

The equitable location problem on the plane

This paper considers the problem of locating M facilities on the unit square so as to minimize the maximal demand faced by each facility subject to closest assignments and coverage constraints. Focusing on uniform demand over the unit square, we develop upper and lower bounds on feasibility of the problem for a given number of facilities and coverage radius. Based on these bounds and numerical experiments we suggest a heuristic to solve the problem. Our computational results show that the heuristic is very efficient, as the average gap between its solutions and the lower bound is 4.34%.

Facility Location with Stochastic Demand and Constraints on Waiting Time

We analyze the problem of optimal location of a set of facilities in the presence of stochastic demand and congestion. Customers travel to the closest facility to obtain service; the problem is to determine the number, locations, and capacity of the facilities. Under rather general assumptions (spatially distributed continuous demand, general arrival and service processes, and nonlinear location and capacity costs) we show that the problem can be decomposed, and construct an efficient optimization algorithm. The analysis yields several insights, including the importance of equitable facility configurations (EFCs), the behavior of optimal and near-optimal capacities, and robust class of solutions that can be constructed for this problem.

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

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.

Continuous review inventory models for perishable items ordered in batches

This paper is an in-depth treatment of an inventory control problem with perishable items. We focus on two prototypes of perishability for items that have a common shelflife and that arrive in batches with zero lead time: (i) sudden deaths due to disasters (e.g., spoilage because of extreme weather conditions or a malfunction of the storage place) and (ii) outdating due to expirations (e.g., medicine or food items that have an expiry date). By using known mathematical tools we generalize the stochastic analysis of continuous review (s, S) policies to our problems. This is achieved by integrating with each inventory cycle stopping times that are independent of the inventory level. We introduce special cases of compound Poisson demand processes with negative jumps and consider demands (jumps) that are exponentially distributed or of a unit (i.e., Poisson) demand. For these special cases we derive a closed form expression of the total cost, including that of perishable items, given any order up to level. Since the stochastic analysis leads to tractable expressions only under specific assumptions, as an added benefit we use a fluid approximation of the inventory level to develop efficient heuristics that can be used in general settings. Numerical results comparing the solution of the heuristics with exact or simulated optimal solutions show that the approximation is accurate.

Pricing Time-Sensitive Services Based on Realized Performance

Services such as FedEx charge up-front fees but reimburse customers for delays. However, lead-time pricing studies ignore such delay refunds. This paper contributes to filling this gap. It studies revenue-maximizing tariffs that depend on realized lead times for a provider serving multiple time-sensitive customer types. We relax two key assumptions of the standard model in the lead-time pricing literature. First, customers may be risk averse RA with respect to payoff uncertainty, where payoff equals valuation, minus delay cost, minus payment. Second, tariffs may be arbitrary functions of realized lead times. The standard model assumes risk-neutral RN customers and restricts attention to flat rates. We report three main findings: 1 With RN customers, flat-rate pricing maximizes revenues but leaves customers exposed to payoff variability. 2 With RA customers, flat-rate pricing is suboptimal. If types are distinguishable, the optimal lead-time-dependent tariffs fully insure delay cost risk and yield the same revenue as under optimal flat rates for RN customers. With indistinguishable RA types, the differentiated first-best tariffs may be incentive-compatible even for uniform service, yielding higher revenues than with RN customers. 3 Under price and capacity optimization, lead-time-dependent pricing yields higher profits with less capacity compared to flat-rate pricing.

Now Playing: DVD Purchasing for a Multilocation Rental Firm

This paper studies the problem of purchasing and allocating copies of movies to multiple stores of a movie rental chain. A unique characteristic of this problem is the return process of rented movies. We formulate this problem for new movies as a newsvendor-like problem with multiple rental opportunities for each copy. We provide demand and return forecasts at the store-day level based on comparable movies. We estimate the parameters of various demand and return models using an iterative maximum-likelihood estimation and Bayesian estimation via Markov chain Monte Carlo simulation. Test results on data from a large movie rental firm reveal systematic underbuying of movies purchased through revenue-sharing contracts and overbuying of movies purchased through standard (nonrevenue-sharing) ones. For the movies considered, our model estimates an increase in the average profit per title for new movies by 15.5% and 2.5% for revenue sharing and standard titles, respectively. We discuss the implications of revenue sharing on the profitability of the rental firm.

M/M/c Queue with Two Priority Classes.

This paper provides the first exact analysis of a preemptive M/M/c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the two-dimensionally infinite Markov chain (MC), representing the two class state space, into a one-dimensionally infinite MC, from which the generating function (GF) of the number of low-priority jobs can be derived in closed form. (The high-priority jobs form a simple M/M/c system and are thus easy to solve.) We demonstrate our methodology for the c = 1, 2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c ≥ 2. Numerical examples demonstrate the accuracy of our algorithm and generate insights on (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compare...

Strategies for a Centralized Single Product Multiclass M/G/1 Make-to-Stock Queue

Make-to-stock queues are typically investigated in the M/M/1 settings. For centralized single-item systems with backlogs, the multilevel rationing (MR) policy is established as optimal and the strict priority (SP) policy is a practical compromise, balancing cost and ease of implementation. However, the optimal policy is unknown when service time is general, i.e., for M/G/1 queues. Dynamic programming, the tool commonly used to investigate the MR policy in make-to-stock queues, is less practical when service time is general. In this paper we focus on customer composition: the proportion of customers of each class to the total number of customers in the queue. We do so because the number of customers in M/G/1 queues is invariant for any nonidling and nonanticipating policy. To characterize customer composition, we consider a series of two-priority M/G/1 queues where the first service time in each busy period is different from standard service times, i.e., this first service time is exceptional. We characterize the required exceptional first service times and the exact solution of such queues. From our results, we derive the optimal cost and control for the MR and SP policies for M/G/1 make-to-stock queues.

Reducing the age of transfused red blood cells in hospitals: ordering and allocation policies.

Although recent randomized controlled trials have not found increased risk of morbidity/mortality with older red blood cells (RBCs), several large trials will be completed soon providing power to detect smaller risks if indeed they exist. Hence, there may still be a need for inventory management policies that could reduce the age of transfused RBCs without compromising availability or resulting in excessive outdates.

Exact Analysis of Capacitated Two-Echelon Inventory Systems with Priorities

We consider a two-echelon inventory system with a capacitated centralized production facility and several distribution centers (DCs). Both production and transportation times are stochastic with general distributions. Demand arrives at each DC according to an independent Poisson process and is backlogged if the DC is out of stock. We allow different holding and backlog costs at the different DCs. We assume that inventory at DCs is managed using the one-for-one replenishment policy. The main objective of this paper is to investigate the control of the multiechelon M/G/1 setting with general transportation times. To achieve this objective, we analyze several decentralized allocation policies including the first-come, first-served (FCFS), strict priority (SP), and multilevel rationing (MR) policies. For our analytic results, we assume no order crossing. We derive the cost function for a capacitated two-echelon inventory system with general transportation times under these policies. Our numerical examples sho...