Control of entry to a nonstationary queuing system

In this article the control of entry of customers to a queuing system with s servers is considered. It is assumed that the arrivals form a nonstationary Poisson process with a periodic rate. The service times are assumed to be exponentially distributed with a parameter independent of time. The cost structure considered is the same as that of Naor. It is demonstrated numerically that, like the stationary cases, the average expected benefit of customers per unit of time is a unimodal function of the critical point. And, also, the social critical point is smaller than the individual critical point. These suggest the use of a search technique for finding the social critical point. The results show successful application of the discrete version of the Fibonacci search.     

Optimal admission pricing and service rate control of an M[x]/M/s queue with reneging

In this article we consider the optimal control of an M^[X]/M/s queue, s ≧ 1. In addition to Poisson bulk arrivals we incorporate a reneging function. Subject to control are an admission price p and the service rate μ. Thus, through p, balking response is induced. When i customers are present a cost h(i,μ,p) per unit time is incurred, discounted continuously. Formulated as a continuous time Markov decision process, conditions are given under which the optimal admission price and optimal service rate are each nondecreasing functions of i. In Section 4 we indicate how the infinite state space may be truncated to a finite state space for computational purposes.     

Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low-Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma-distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non-increasing in p. We prove that the optimal admission fee p^* is a non-decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low-Lippman model to include a state-dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.     

A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders

In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make‐to‐stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand‐based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time‐dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

An (s,S) model for inventory with exponential lifetimes and renewal demands

Inventory control of products with finite lifetimes is important in many modern business organizations. It has been an important and difficult research subject. Here, we study the (s, S) continuous review model for items with an exponential random lifetime and a general renewal demand process through a Markov process. We derive a fundamental rate conservation theorem and show that all the other system performance measures can be obtained easily through the expected reorder cycle length. This leads to a simple expression for the total expected long run cost rate function in terms of the expected reorder cycle length. Subsequently, we derive formulas for computing the expected cycle lengths for the general renewal demand as well as for a large class of demands characterized by the phase type interdemand time distribution. We show analytically when the cost as a function of the reorder level is monotone, concave, or convex. We also show analytically that, depending on the behavior of the expected reorder cycle, the cost as a function of the order‐up level is either monotone increasing or unimodal. These analytical properties enable us to understand the problem and make the subsequent numerical optimization much easier. Numerical studies confirm and illustrate some of the analytical properties. The results also demonstrate the impact of various parameters on the optimal policy and the cost. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 39–56, 1999     

Optimal service rates of a service facility with perishable inventory items

In this paper we optimally control service rates for an inventory system of service facilities with perishable products. We consider a finite capacity system where arrivals are Poisson‐distributed, lifetime of items have exponential distribution, and replenishment is instantaneous. We determine the service rates to be employed at each instant of time so that the long‐run expected cost rate is minimized for fixed maximum inventory level and capacity. The problem is modelled as a semi‐Markov decision problem. We establish the existence of a stationary optimal policy and we solve it by employing linear programming. Several numerical examples which provide insight to the behavior of the system are presented. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 464–482, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10021     

Lot sizes under continuous demand: The backorder case

We study a deterministic lot-size problem, in which the demand rate is a (piecewise) continuous function of time and shortages are backordered. The problem is to find the order points and order quantities to minimize the total costs over a finite planning horizon. We show that the optimal order points have an interleaving property, and when the orders are optimally placed, the objective function is convex in the number of orders. By exploiting these properties, an algorithm is developed which solves the problem efficiently. For problems with increasing (decreasing) demand rates and decreasing (increasing) cost rates, monotonicity properties of the optimal order quantities and order intervals are derived.     

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     

A continuous review inventory model with compound poisson demand process and stochastic lead time

Using Markov renewal theory, we derive analytic expressions for the expected average cost associated with (s, S) policies for a continuous review inventory model with a compound Poisson demand process and stochastic lead time, under the (restrictive) assumption that only one order can be outstanding.     

A robust policy for serial agile production systems

One of the major problems in modeling production systems is how to treat the job arrival process. Restrictive assumptions such as Markovian arrivals do not represent real world systems, especially if the arrival process is generated by job departures from upstream workstations. Under these circumstances, cost‐effective policies that are robust with respect to the nature of the arrival process become of interest. In this paper, we focus on minimizing the expected total holding and setup costs in a two‐stage produce‐to‐order production system operated by a cross‐trained worker. We will show that if setup times are insignificant in comparison with processing times, then near‐optimal policies can be generated with very robust performances with respect to the arrival process. We also present conditions under which these near‐optimal policies can be obtained by using only the arrival and service rates. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A (Q,R) inventory model with lost sales and erlang-distributed lead times

The exact expression is derived for the average stationary cost of a (Q,R) inventory system with lost sales, unit Poisson demands, Erlang-distributed lead times, fixed order cost, fixed cost per unit lost sale, linear holding cost per unit time, and a maximum of one order outstanding. Explicit expressions for the state probabilities and a fast method of calculating them are obtained for the case of Q greater than R. Exponential lead times are analyzed as a special case. A simple cyclic coordinate search procedure is used to locate the minimum cost policy. Examples of the effect of lead time variability on costs are given.     

Managing waiting times of backordered demands in single‐stage (Q,r) inventory systems

We present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028     

TTT transforms and age replacements with discounted costs

The ordinary age replacement problem consists of finding an optimal age at which a unit, needed in a continuous production process, should be replaced in order to minimize the average long-run cost per unit time. Bergman introduced a graphical procedure based on the total-time-on-test (TTT) concept for the analysis of the age replacement problem. In this article, that idea is generalized to the situation of discounted costs. We also study a more general age replacement problem in which we have a form of imperfect repair.     

An inventory model with search for best ordering price

This paper presents a single-item inventory model with deterministic demand where the buyer is allowed to search for the most favorable price before deciding on the order quantity. In the beginning of each period, a sequential random sample can be taken from a known distribution and there is a fixed cost per search. The decision maker is faced with the task of deciding when to initiate and when to stop the search process, as well as determining the optimal order quantity once the search process is terminated. The objective is to minimize total expected costs while satisfying all demands on time. We demonstrate that a set of critical numbers determine the optimal stopping and ordering strategies. We present recursive expressions yielding the critical numbers, as well as the minimal expected cost from the beginning of every period to the end of the horizon.     

Optimal switchover times between two activities utilizing the same resource

The “gold‐mining” decision problem is concerned with the efficient utilization of a delicate mining equipment working in a number of different mines. Richard Bellman was the first to consider this type of a problem. The solution found by Bellman for the finite‐horizon, continuous‐time version of the problem with two mines is not overly realistic since he assumed that fractional parts of the same mining equipment could be used in different mines and this fraction could change instantaneously. In this paper, we provide some extensions to this model in order to produce more operational and realistic solutions. Our first model is concerned with developing an operational policy where the equipment may be switched from one mine to the other at most once during a finite horizon. In the next extension we incorporate a cost component in the objective function and assume that the horizon length is not fixed but it is the second decision variable. Structural properties of the optimal solutions are obtained using nonlinear programming. Each model and its solution is illustrated with a numerical example. The models developed here may have potential applications in other areas including production of items requiring the same machine or choosing a sequence of activities requiring the same resource. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 186–203, 2002; DOI 10.1002/nav.10008     

An application of yield management for Internet Service Providers

In this paper we study strategies for better utilizing the network capacity of Internet Service Providers (ISPs) when they are faced with stochastic and dynamic arrivals and departures of customers attempting to log‐on or log‐off, respectively. We propose a method in which, depending on the number of modems available, and the arrival and departure rates of different classes of customers, a decision is made whether to accept or reject a log‐on request. The problem is formulated as a continuous time Markov Decision Process for which optimal policies can be readily derived using techniques such as value iteration. This decision maximizes the discounted value to ISPs while improving service levels for higher class customers. The methodology is similar to yield management techniques successfully used in airlines, hotels, etc. However, there are sufficient differences, such as no predefined time horizon or reservations, that make this model interesting to pursue and challenging. This work was completed in collaboration with one of the largest ISPs in Connecticut. The problem is topical, and approaches such as those proposed here are sought by users. © 2001 John Wiley & Sons, Inc., Naval Research Logistics 48:348–362, 2001     

搜索路径给定时的搜索对策问题及其应用

搜索路径给定时的最优搜索方案问题,也可以理解为是关于搜索者和目标的二人对策问题,主要讨论了当搜索路径给定时的单个搜索者和单个目标的搜索对策问题。首先根据问题的特点,利用动态规划和迭代的方法,确定关于目标逃逸路径混合策略的最优分区,证明该分区是多面体凸集;针对目标不同逃逸路径的分区,求出搜索者的最大期望收益,再将问题转化为二人有限零和对策,计算出搜索者的支付矩阵,确定最优搜索策略。最后结合海军护航行动,对我舰载直升机搜索小型海盗船进行分析和计算,说明搜索路径给定时的最优搜索对策对于双方的资源分配和提高搜索效率具有一定的应用价值。     

Approximation methods for the analysis of a multicomponent,multiproduct assemble‐to‐order system

In this article, we consider the performance evaluation of a multicomponent, multiproduct assemble‐to‐order (ATO) system. Each component is managed independently using a base‐stock policy at a supply facility with limited production capacity and an infinite buffer. The arrivals of demands follow a multivariate Poisson process and unfilled demands are backlogged. Because exact analysis of the proposed system is not feasible, we propose two approximation methods which provide upper and lower bounds for various performance measures such as fill rate, average waiting time, and average number of backorders of the proposed system. Our computational experiments demonstrate the effectiveness of the two approximation methods under various system settings. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Transient solution of a finite-capacity M/Ga,b/1 queueing system

In this article we consider a single-server, bulk-service queueing system in which the waiting room is of finite capacity. Arrival process is Poisson and all the arrivals taking place when the waiting room is full are lost. The service times are generally distributed independent random variables and the distribution is depending on the batch size being served. Using renewal theory, we derive the time-dependent solution for the system-size probabilities at arbitrary time points. Also we give expressions for the distribution of virtual waiting time in the queue at any time t.     

Optimal state-dependent pricing policies for a class of stochastic multiunit service systems

This paper models a k-unit service system (e.g., a repair, maintenance, or rental facility) with Poisson arrivals, exponential service times, and no queue. If we denote the number of units that are busy as the state of the system, the state-dependent pricing model formalizes the intuitive notion that when most units are idle, the price (i.e., the service charge per unit time) should be low, and when most units are busy, the price should be higher than the average. A computationally efficient algorithm based on a nonlinear programming formulation of the problem is provided for determination of the optimal state-dependent prices. The procedure ultimately reduces to the search on a single variable in an interval to determine the unique intersection point of a concave increasing function and a linear decreasing function. The algorithm takes, on the average, only about 1/2 second per problem on the IBM 360/65 (FORTRAN G Compiler). A discrete optimal-control approach to the problem is shown to result in essentially the same procedure as the nonlinear-programming formulation. Several properties of the optimal state-dependent prices are given. Comparisons of the optimal values of the objective function for the state-dependent and state-independent pricing policies show that the former is on the average, only about 0.7% better than the latter, which may explain partly why state-dependent pricing is not prevalent in many service systems. Potential generalizations of the model are discussed.