Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations

A Markovian arrival process of order n, MAP(n), is typically described by two n × n transition rate matrices in terms of rate parameters. While it is straightforward and intuitive, the Markovian representation is redundant since the minimal number of parameters is n² for non‐redundant MAP(n). It is well known that the redundancy complicates exact moment fittings. In this article, we present a minimal and unique Laplace‐Stieltjes transform (LST) representations for MAP(n)s. Even though the LST coefficients vector itself is not a minimal representation, we show that the joint LST of stationary intervals can be represented with the minimum number of parameters. We also propose another minimal representation for MAP(3)s based on coefficients of the characteristic polynomial equations of the two transition rate matrices. An exact moment fitting procedure is presented for MAP(3)s based on two proposed minimal representations. We also discuss how MAP(3)/G/1 departure process can be approximated as a MAP(3). A simple tandem queueing network example is presented to show that the MAP(3) performs better than the MAP(2) in queueing approximations especially under moderate traffic intensities. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 549–561, 2016     

Parallel machine scheduling with batch deliveries to minimize total flow time and delivery cost

Motivated by the flow of products in the iron and steel industry, we study an identical and parallel machine scheduling problem with batch deliveries, where jobs finished on the parallel machines are delivered to customers in batches. Each delivery batch has a capacity and incurs a cost. The objective is to find a coordinated production and delivery schedule that minimizes the total flow time of jobs plus the total delivery cost. This problem is an extension of the problem considered by Hall and Potts, Ann Oper Res 135 (2005) 41–64, who studied a two‐machine problem with an unbounded number of transporters and unbounded delivery capacity. We first provide a dynamic programming algorithm to solve a special case with a given job assignment to the machines. A heuristic algorithm is then presented for the general problem, and its worst‐case performance ratio is analyzed. The computational results show that the heuristic algorithm can generate near‐optimal solutions. Finally, we offer a fully polynomial‐time approximation scheme for a fixed number of machines. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 492–502, 2016     

Convolutions and generalization of logconcavity: Implications and applications

Additive convolution of unimodal and α‐unimodal random variables are known as an old classic problem which has attracted the attention of many authors in theory and applied fields. Another type of convolution, called multiplicative convolution, is rather younger. In this article, we first focus on this newer concept and obtain several useful results in which the most important ones is that if is logconcave then so are and for some suitable increasing functions ?. This result contains and as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f(x) and and two comprehensive implications charts are provided. Then, these fundamental results are applied to aging properties, existence of moments and several kinds of ordered random variables. Multiplicative strong unimodality in the discrete case is also introduced and its properties are investigated. In the second part of the article, some refinements are made for additive convolutions. A remaining open problem is completed and a conjecture concerning convolution of discrete α‐unimodal distributions is settled. Then, we shall show that an existing result regarding convolution of symmetric discrete unimodal distributions is not correct and an easy alternative proof is presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 109–123, 2016     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Realignment in the National Football League: Did they do it right?

The National Football League (NFL) in the United States expanded to 32 teams in 2002 with the addition of a team in Houston. At that point, the league was realigned into eight divisions, each containing four teams. We describe a branch‐and‐cut algorithm for minimizing the sum of intradivisional travel distances. We consider first the case where any team can be assigned to any division. We also consider imposing restrictions, such as aligning the AFC (American Football Conference) and the NFC (National Football Conference) separately or maintaining traditional rivalries. We show that the alignment chosen by the NFL does not minimize the sum of intradivisional travel distances, but that it is close to optimal for an alignment that aligns the NFC and AFC separately and imposes some additional restrictions. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 683–701, 2003.     

Potential impact of recharging plug‐in hybrid electric vehicles on locational marginal prices

Plug‐in hybrid electric vehicles (PHEVs), upon market penetration, will create additional recharging loads to the electric power systems. This article considers different recharging scenarios and uses game theoretic models to study the potential impact of the recharging loads on locational marginal prices (wholesale electricity prices). Computational results from a Pennsylvania‐New Jersey‐Maryland Interconnection case study show that, under the existing recharging infrastructures, even a small magnitude of load increase caused by PHEV recharging could have a significant undesirable impact on locational marginal prices. The impact could be mitigated to a varying extent by the availability of possible future recharging infrastructures, including realtime pricing recharging meters, battery stations, or vehicle‐to‐grid technology.© 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Decomposing inventory routing problems with approximate value functions

We present a time decomposition for inventory routing problems. The methodology is based on valuing inventory with a concave piecewise linear function and then combining solutions to single‐period subproblems using dynamic programming techniques. Computational experiments show that the resulting value function accurately captures the inventory's value, and solving the multiperiod problem as a sequence of single‐period subproblems drastically decreases computational time without sacrificing solution quality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Scheduling a maintenance activity on parallel identical machines

We study a problem of scheduling a maintenance activity on parallel identical machines, under the assumption that all the machines must be maintained simultaneously. One example for this setting is a situation where the entire system must be stopped for maintenance because of a required electricity shut‐down. The objective is minimum flow‐time. The problem is shown to be NP‐hard, and moreover impossible to approximate unless P = NP. We introduce a pseudo‐polynomial dynamic programming algorithm, and show how to convert it into a bicriteria FPTAS for this problem. We also present an efficient heuristic and a lower bound. Our numerical tests indicate that the heuristic provides in most cases very close‐to‐optimal schedules. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Capacity expansion under a service‐level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service‐level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up‐and‐out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Optimal inventory and dynamic admission policies for a retailer of seasonal products with affiliate programs and drop‐shipping

This article investigates the optimal inventory and admission policies for a “Clicks‐and‐Bricks” retailer of seasonal products that, in addition to selling through its own physical and online stores, also sells through third‐party websites by means of affiliate programs. Through postings on partners' webpages, an affiliate program allows a retailer to attract customers who would otherwise be missed. However, this retailer needs to pay a commission for each sale that originates from the website operators participating in the program. The retailer may also refer online orders to other sources (such as distributors and manufacturers) for fulfillment through a drop‐shipping agreement and thus earns commissions. This would be an option when, for example, the inventories at the physical stores were running low. Therefore, during the selling horizon, the retailer needs to dynamically control the opening/closing of affiliate programs and decide on the fulfillment option for online orders. On the basis of a discrete‐time dynamic programming model, the optimal admission policy of the retailer is investigated in this paper, and the structural properties of the revenue function are characterized. Numerical examples are given to show the revenue impact of optimal admission control. The optimal initial stocking decisions at the physical stores are also studied. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009