Optimal staffing in nonstationary service centers with constraints

This paper considers optimal staffing in service centers. We construct models for profit and cost centers using dynamic rate queues. To allow for practical optimal controls, we approximate the queueing process using a Gaussian random variable with equal mean and variance. We then appeal to the Pontryagin's maximum principle to derive a closed form square root staffing (SRS) rule for optimal staffing. Unlike most traditional SRS formulas, the main parameter in our formula is not the probability of delay but rather a cost‐to‐benefit ratio that depends on the shadow price. We show that the delay experienced by customers can be interpreted in terms of this ratio. Throughout the article, we provide theoretical support of our analysis and conduct extensive numerical experiments to reinforce our findings. To this end, various scenarios are considered to evaluate the change in the staffing levels as the cost‐to‐benefit ratio changes. We also assess the change in the service grade and the effects of a service‐level agreement constraint. Our analysis indicates that the variation in the ratio of customer abandonment over service rate particularly influences staffing levels and can lead to drastically different policies between profit and cost service centers. Our main contribution is the introduction of new analysis and managerial insights into the nonstationary optimal staffing of service centers, especially when the objective is to maximize profitability. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 615–630, 2017     

Decentralized admission control of a queueing system: A game‐theoretic model

Consider a distributed system where many gatekeepers share a single server. Customers arrive at each gatekeeper according to independent Poisson processes with different rates. Upon arrival of a new customer, the gatekeeper has to decide whether to admit the customer by sending it to the server, or to block it. Blocking costs nothing. The gatekeeper receives a reward after a customer completes the service, and incurs a cost if an admitted customer finds a busy server and therefore has to leave the system. Assuming an exponential service distribution, we formulate the problem as an n‐person non‐zero‐sum game in which each gatekeeper is interested in maximizing its own long‐run average reward. The key result is that each gatekeeper's optimal policy is that of a threshold type regardless what other gatekeepers do. We then derive Nash equilibria and discuss interesting insights. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 702–718, 2003.     

Revenue sharing contracts in a supply chain with uncontractible actions

We consider a supplier–customer relationship where the customer faces a typical Newsvendor problem of determining perishable capacity to meet uncertain demand. The customer outsources a critical, demand‐enhancing service to an outside supplier, who receives a fixed share of the revenue from the customer. Given such a linear sharing contract, the customer chooses capacity and the service supplier chooses service effort level before demand is realized. We consider the two cases when these decisions are made simultaneously (simultaneous game) or sequentially (sequential game). For each game, we analyze how the equilibrium solutions vary with the parameters of the problem. We show that in the equilibrium, it is possible that either the customer's capacity increases or the service supplier's effort level decreases when the supplier receives a larger share of the revenue. We also show that given the same sharing contract, the sequential game always induces a higher capacity and more effort. For the case of additive effort effect and uniform demand distribution, we consider the customer's problem of designing the optimal contract with or without a fixed payment in the contract, and obtain sensitivity results on how the optimal contract depends on the problem parameters. For the case of fixed payment, it is optimal to allocate more revenue to the supplier to induce more service effort when the profit margin is higher, the cost of effort is lower, effort is more effective in stimulating demand, the variability of demand is smaller or the supplier makes the first move in the sequential game. For the case of no fixed payment, however, it is optimal to allocate more revenue to the supplier when the variability of demand is larger or its mean is smaller. Numerical examples are analyzed to validate the sensitivity results for the case of normal demand distribution and to provide more managerial insights. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Service design at diagnostic service centers

We study a service design problem in diagnostic service centers, call centers that provide medical advice to patients over the phone about what the appropriate course of action is, based on the caller's symptoms. Due to the tension between increased diagnostic accuracy and the increase in waiting times more in‐depth service requires, managers face a difficult decision in determining the optimal service depth to guide the diagnostic process. The specific problem we consider models the situation when the capacity (staffing level) at the center is fixed, and when the callers have both congestion‐ and noncongestion‐related costs relating to their call. We develop a queueing model incorporating these features and find that the optimal service depth can take one of two different structures, depending on factors such as the nurses' skill level and the maximum potential demand. Sensitivity analyses of the two optimal structures show that they are quite different. In some situations, it may (or may not) be optimal for the manager to try to expand the demand at the center, and increasing skill level may (or may not) increase congestion. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Equilibrium customer strategies and social–profit maximization in the single‐server constant retrial queue

We consider the single‐server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward‐cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Workload distribution of discrete‐time parallel queues with two servers

We study discrete‐time, parallel queues with two identical servers. Customers arrive randomly at the system and join the queue with the shortest workload that is defined as the total service time required for the server to complete all the customers in the queue. The arrivals are assumed to follow a geometric distribution and the service times are assumed to have a general distribution. It is a no‐jockeying queue. The two‐dimensional state space is truncated into a banded array. The resulting modified queue is studied using the method of probability generating function (pgf) The workload distribution in steady state is obtained in form of pgf. A special case where the service time is a deterministic constant is further investigated. Numerical examples are illustrated. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 440–454, 2000     

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.     

Scheduling and due‐date quotation in a make‐to‐order supply chain

We consider a manufacturer, served by a single supplier, who has to quote due dates to arriving customers in a make‐to‐order production environment. The manufacturer is penalized for long lead times and for missing due dates. To meet due dates, the manufacturer has to obtain components from a supplier. We model this manufacturer and supplier as a two‐machine flow shop, consider several variations of this problem, and design effective due‐date quotation and scheduling algorithms for centralized and decentralized versions of the model. We perform extensive computational testing to assess the effectiveness of our algorithms and to compare the centralized and decentralized models to quantify the value of centralized control in a make‐to‐order supply chain. Since complete information exchange and centralized control is not always practical or cost‐effective, we explore the value of partial information exchange for this system. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Supply contracts in manufacturer‐retailer interactions with manufacturer‐quality and retailer effort‐induced demand

We consider a decentralized distribution channel where demand depends on the manufacturer‐chosen quality of the product and the selling effort chosen by the retailer. The cost of selling effort is private information for the retailer. We consider three different types of supply contracts in this article: price‐only contract where the manufacturer sets a wholesale price; fixed‐fee contract where manufacturer sells at marginal cost but charges a fixed (transfer) fee; and, general franchise contract where manufacturer sets a wholesale price and charges a fixed fee as well. The fixed‐fee and general franchise contracts are referred to as two‐part tariff contracts. For each contract type, we study different contract forms including individual, menu, and pooling contracts. In the analysis of the different types and forms of contracts, we show that the price only contract is dominated by the general franchise menu contract. However, the manufacturer may prefer to offer the fixed‐fee individual contract as compared to the general franchise contract when the retailer's reservation utility and degree of information asymmetry in costs are high. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Dynamic routing of prioritized warranty repairs

Recent years have seen a strong trend toward outsourcing warranty repair services to outside vendors. In this article we consider the problem of dynamically routing warranty repairs to service vendors when warranties have priority levels. Each time an item under warranty fails, it is sent to one of the vendors for repair. Items covered by higher priority warranty receive higher priority in repair service. The manufacturer pays a fixed fee per repair and incurs a linear holding cost while an item is undergoing or waiting for repair. The objective is to minimize the manufacturer's long‐run average cost. Because of the complexity of the problem, it is very unlikely that there exist tractable ways to find the optimal routing strategies. Therefore, we propose five heuristic routing procedures that are applicable to real‐life problems. We evaluate the heuristics using simulation. The simulation results show that the index‐based “generalized join the shortest queue” policy, which applies a single policy improvement step to an initial state‐independent policy, performs the best among all five heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal control policies in service systems with limited information on the downstream stage

Existing models in multistage service systems assume full information on the state of downstream stages. In this paper, we investigate how much the lack of such information impacts jobs' waiting time in a two‐stage system with two types of jobs at the first stage. The goal is to find the optimal control policy for the server at the first stage to switch between type‐1 and type‐2 jobs, while minimizing the long‐run average number of jobs in the system. We identify control policies and corresponding conditions under which having no or partial information, the system can still capture the most benefit of having full information.     

Dynamic capacity reservation and due date quoting in a make‐to‐order system

We consider a make‐to‐order manufacturer facing random demand from two classes of customers. We develop an integrated model for reserving capacity in anticipation of future order arrivals from high priority customers and setting due dates for incoming orders. Our research exhibits two distinct features: (1) we explicitly model the manufacturer's uncertainty about the customers' due date preferences for future orders; and (2) we utilize a service level measure for reserving capacity rather than estimating short and long term implications of due date quoting with a penalty cost function. We identify an interesting effect (“t‐pooling”) that arises when the (partial) knowledge of customer due date preferences is utilized in making capacity reservation and order allocation decisions. We characterize the relationship between the customer due date preferences and the required reservation quantities and show that not considering the t‐pooling effect (as done in traditional capacity and inventory rationing literature) leads to excessive capacity reservations. Numerical analyses are conducted to investigate the behavior and performance of our capacity reservation and due date quoting approach in a dynamic setting with multiple planning horizons and roll‐overs. One interesting and seemingly counterintuitive finding of our analyses is that under certain conditions reserving capacity for high priority customers not only improves high priority fulfillment, but also increases the overall system fill rate. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Computational analysis of MMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

This paper studies a queueing system with a Markov arrival process with marked arrivals and PH‐distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first‐come‐first‐served (FCFS) and last‐come‐first‐served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 399–421, 2000     

Dynamic evacuation routes for personnel on a naval ship

We propose a dynamic escape route system for emergency evacuation of a naval ship. The system employs signals that adapt to the causative contingency and the crew's physical distribution about the ship. A mixed‐integer nonlinear programming model, with underlying network structure, optimizes the evacuation process. The network's nodes represent compartments, closures (e.g., doors and hatches) and intersections, while arcs represent various types of passageways. The objective function integrates two potentially conflicting factors: average evacuation time and the watertight and airtight integrity of the ship after evacuation. A heuristic solves the model approximately using a sequence of mixed‐integer linear approximating problems. Using data for a Spanish frigate, with standard static routes specified by the ship's designers, computational tests show that the dynamic system can reduce average evacuation times, nearly 23%, and can improve a combined measure of ship integrity by up to 50%. In addition, plausible design changes to the frigate yield further, substantial improvements. Published 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Value management of diagnostic equipment with cancelation,no‐show,and emergency patients

Magnetic resonance imaging and other multifunctional diagnostic facilities, which are considered as scarce resources of hospitals, typically provide services to patients with different medical needs. This article examines the admission policies during the appointment management of such facilities. We consider two categories of patients: regular patients who are scheduled in advance through an appointment system and emergency patients with randomly generated demands during the workday that must be served as soon as possible. According to the actual medical needs of patients, regular patients are segmented into multiple classes with different cancelation rates, no‐show probabilities, unit value contributions, and average service times. Management makes admission decisions on whether or not to accept a service request from a regular patient during the booking horizon to improve the overall value that could be generated during the workday. The decisions should be made by considering the cancelation and no‐show behavior of booked patients as well as the emergency patients that would have to be served because any overtime service would lead to higher costs. We studied the optimal admission decision using a continuous‐time discrete‐state dynamic programming model. Identifying an optimal policy for this discrete model is analytically intractable and numerically inefficient because the state is multidimensional and infinite. We propose to study a deterministic counterpart of the problem (i.e., the fluid control problem) and to develop a time‐based fluid policy that is shown to be asymptotically optimal for large‐scale problems. Furthermore, we propose to adopt a mixed fluid policy that is developed based on the information obtained from the fluid control problem. Numerical experiments demonstrate that this improved policy works effectively for small‐scale problems. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 287–304, 2016     

Waiting time analysis for a queueing system with time‐limited service and exponential timer

We consider a single‐queue with exhaustive or gated time‐limited services and server vacations, in which the length of each service period at the queue is controlled by a timer, i.e., the server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then takes vacations. The customer whose service is interrupted due to the timer expiration may be attended according to nonpreemptive or preemptive service disciplines. For the M/G/1 exhaustive/gated time‐limited service queueing system with an exponential timer and four typical preemptive/nonpreemptive service disciplines, we derive the Laplace—Stieltjes transforms and the moment formulas for waiting times and sojourn times through a unified approach, and provide some new results for these time‐limited service disciplines. © John Wiley & Sons, Inc. Naval Research Logistics 48: 638–651, 2001.     

Analysis of the finite‐source multiclass priority queue with an unreliable server and setup time

In this article, we study a queueing system serving multiple classes of customers. Each class has a finite‐calling population. The customers are served according to the preemptive‐resume priority policy. We assume general distributions for the service times. For each priority class, we derive the steady‐state system size distributions at departure/arrival and arbitrary time epochs. We introduce the residual augmented process completion times conditioned on the number of customers in the system to obtain the system time distribution. We then extend the model by assuming that the server is subject to operation‐independent failures upon which a repair process with random duration starts immediately. We also demonstrate how setup times, which may be required before resuming interrupted service or picking up a new customer, can be incorporated in the model. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Should competing firms reveal their capacity?

In this article, we explore when firms have an incentive to hide (or reveal) their capacity information. We consider two firms that aim to maximize profits over time and face limited capacity. One or both of the firms have private information on their own capacity levels, and they update their beliefs about their rival's capacity based on their observation of the other firm's output. We focus on credible revelation mechanisms—a firm may signal its capacity through overproduction, compared to its myopic production levels. We characterize conditions when high‐capacity firms may have the incentive and capability to signal their capacity levels by overproduction. We show that prior beliefs about capacity play a crucial, and surprisingly complex, role on whether the firm would prefer to reveal its capacity or not. A surprising result is that, despite the fact that it may be best for the high‐capacity firm to overproduce to reveal its capacity when capacity information is private, it may end up with more profits than if all capacity information were public knowledge in the first place. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Estimating failure time distribution and its parameters based on intermediate data from a Wiener degradation model

Instead of measuring a Wiener degradation or performance process at predetermined time points to track degradation or performance of a product for estimating its lifetime, we propose to obtain the first‐passage times of the process over certain nonfailure thresholds. Based on only these intermediate data, we obtain the uniformly minimum variance unbiased estimator and uniformly most accurate confidence interval for the mean lifetime. For estimating the lifetime distribution function, we propose a modified maximum likelihood estimator and a new estimator and prove that, by increasing the sample size of the intermediate data, these estimators and the above‐mentioned estimator of the mean lifetime can achieve the same levels of accuracy as the estimators assuming one has failure times. Thus, our method of using only intermediate data is useful for highly reliable products when their failure times are difficult to obtain. Furthermore, we show that the proposed new estimator of the lifetime distribution function is more accurate than the standard and modified maximum likelihood estimators. We also obtain approximate confidence intervals for the lifetime distribution function and its percentiles. Finally, we use light‐emitting diodes as an example to illustrate our method and demonstrate how to validate the Wiener assumption during the testing. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Formulations for dynamic lot sizing with service levels

In this article, we study deterministic dynamic lot‐sizing problems with a service‐level constraint on the total number of periods in which backlogs can occur over a finite planning horizon. We give a natural mixed integer programming formulation for the single item problem (LS‐SL‐I) and study the structure of its solution. We show that an optimal solution to this problem can be found in \begin{align*}\mathcal O(n^2\kappa)\end{align*} time, where n is the planning horizon and \begin{align*}\kappa=\mathcal O(n)\end{align*} is the maximum number of periods in which demand can be backlogged. Using the proposed shortest path algorithms, we develop alternative tight extended formulations for LS‐SL‐I and one of its relaxations, which we refer to as uncapacitated lot sizing with setups for stocks and backlogs. {We show that this relaxation also appears as a substructure in a lot‐sizing problem which limits the total amount of a period's demand met from a later period, across all periods.} We report computational results that compare the natural and extended formulations on multi‐item service‐level constrained instances. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013