Bayesian selling problem with partial information

We introduce an optimal stopping problem for selling an asset when the fixed but unknown distribution of successive offers is from one of n possible distributions. The initial probabilities as to which is the true distribution are given and updated in a Bayesian manner as the successive offers are observed. After receiving an offer, the seller has to decide whether to accept the offer or continue to observe the next offer. Each time an offer is observed a fixed cost is incurred. We consider both the cases where recalling a past offer is allowed and where it is not allowed. For each case, a dynamic programming model and some heuristic policies are presented. Using simulation, the performances of the heuristic methods are evaluated and upper bounds on the optimal expected return are obtained. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Optimal location of a facility relative to area demands

This paper deals with the Weber single-facility location problem where the demands are not only points but may be areas as well. It provides an iterative procedure for solving the problem with l^p distances when p > 1 (a method of obtaining the exact solution when p = 1 and distances are thus rectangular already exists). The special case where the weight densities in the areas are uniform and the areas are rectangles or circles results in a modified iterative process that is computationally much faster. This method can be extended to the simultaneous location of several facilities.     

An M/M/1 queue with a general bulk service rule

A model of an M/M/1, bulk queue with service rates dependent on the batch size is developed. The operational policy is to commence service when at least L customers are available with a maximum batch size of K. Arriving customers are not allowed to join in-process service. The solution procedure utilizes the matrix geometric methodology and reduces to obtaining the inverse of a square matrix of dimension K + 1 - L. For the case where the service rates are not batch size dependent, the limiting probabilities can be written in closed form. A numerical example illustrates the variability of the system cost as a function of the minimum batch service size L.     

Dynamic lot‐sizing model with production time windows

We consider a dynamic lot‐sizing model with production time windows where each of n demands has earliest and latest production due dates and it must be satisfied during the given time window. For the case of nonspeculative cost structure, an O(nlogn) time procedure is developed and it is shown to run in O(n) when demands come in the order of latest production due dates. When the cost structure is somewhat general fixed plus linear that allows speculative motive, an optimal procedure with O(T⁴) is proposed where T is the length of a planning horizon. Finally, for the most general concave production cost structure, an optimal procedure with O(T⁵) is designed. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

An economic two-stage screening procedure with a prescribed outgoing quality in logistic and normal models

An economic two-stage screening procedure based on a dichotomous performance variable T and a continuous screening variable X is proposed. X is measured first to decide whether an item should be accepted, rejected, or additional observations should be taken. If no terminal decision is reached, T is then observed to classify the undecided items. Two models are considered; (i) the logistic model, where P(T = 1|X = x) is assumed to be a logistic function of x, and (ii) the normal model, where X given T is assumed to be normally distributed. A simple economic model based on inspection and misclassification costs is constructed. Optimal cutoff values on the screening variable are obtained by minimizing the expected cost subject to the constraint that the average outgoing quality attains a pre-specified level. Solutions are provided for both known-parameter and unknown-parameter cases. © 1995 John Wiley & Sons, Inc.     

Approximation schemes for ordered vector packing problems

In this paper we deal with the d‐dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors a_i, a_j, either a_i is componentwise not smaller than a_j or a_j is componentwise not smaller than a_i. An asymptotic polynomial‐time approximation scheme is constructed for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge. We also extend the result to instances with constant Dilworth number, i.e., instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Quality control chart design under Jidoka

We consider design of control charts in the presence of machine stoppages that are exogenously imposed (as under jidoka practices). Each stoppage creates an opportunity for inspection/repair at reduced cost. We first model a single machine facing opportunities arriving according to a Poisson process, develop the expressions for its operating characteristics and construct the optimization problem for economic design of a control chart. We, then, consider the multiple machine setting where individual machine stoppages may create inspection/repair opportunities for other machines. We develop exact expressions for the cases when all machines are either opportunity‐takers or not. On the basis of an approximation for the all‐taker case, we then propose an approximate model for the mixed case. In a numerical study, we examine the opportunity taking behavior of machines in both single and multiple machine settings and the impact of such practices on the design of an X – Q C chart. Our findings indicate that incorporating inspection/repair opportunities into QC chart design may provide considerable cost savings. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

On queues with dependent interarrival and service times

AnM/G/1 queueing system is studied in which the service time required by a customer is dependent on the interarrival time between his arrival and that of his predecessor Assuming the two variables are “associated,” we prove that the expected delay in this system is less than or equal to than of a conventional M/G/1 queue This conclusion has been verified via simulation by Mitchell and Paulson [9] for a special class of dependent M/M/1 queue. Their model is a special case of the one we consider here. We also study another modified GI/G/1 queue. where the arrival process and/or the service process are individually “associated”.     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.     

Transit vessel scheduling

A transit vessel traffic scheduling algorithm has been developed to limit the negative effects on cargo volume throughput in two‐way waterways where separation distances between transiting vessels must be maintained and passage restrictions may hold. It runs in time that is polynomial in the number of ships involved in the computation and finds schedules which increase the utilization of waterways. Three examples illustrate its use. The first example is situated in the Sunda Strait where the algorithm is used to enhance the safety of merchant shipping against a terrorist threat. It illustrates important features of the algorithm and demonstrates how it can be used with cross traffic. The second example is situated in the Strait of Istanbul and offers a comparison between the developed algorithm and the transit vessel scheduling algorithm of Ulusçu et al., J Navig 62 (2009), 59–77. This was done using a plausible model of the Strait of Istanbul. The third and last example shows how the algorithm can be used to schedule transit vessel traffic in two‐way waterways with junctions. This feature is especially useful in congested waters with a high risk of collisions like the Inland Sea of Japan. An extreme test case proves that the developed algorithm is a practical algorithm ready for such use. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 225–248, 2017     

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     

Instant transportation solutions

This paper presents direct noniterative methods for solving such known problems as shoploading and aggregate scheduling. There is given a simple optimal rule for the shop-loading problem which is quite surprising. The crucial point in solving this problem is what not to assign rather than what to assign. The development of those methods was possible since the discussed problems can be converted into a special transportation model where the given cost coefficients c_ij assume a form cij = ui + uj.     

Multiechelon repairable-item provisioning in a time-varying environment using the randomization technique

Multiechelon repairable-item provisioning systems are considered under a time-varying environment. Such conditions could arise, for example, in a military context where a shift from peacetime operation to wartime operation takes place; or, in a civilian setting where a public transit system decides to increase its hours of operation or frequency of service. Exact Markovian models, incorporating a finite population of repairable components and limited repair capacity (nonample service), are treated, with transient solutions obtained using the randomization technique. The exact models are compared with the approximate Dyna-METRIC model which assumes an infinite population of components and ample repair capacity.     

Batch scheduling on a two‐machine jobshop with machine‐dependent setup times

The problem of minimum makespan on an m machine jobshop with unit execution time (UET) jobs (m ≥ 3) is known to be strongly NP‐hard even with no setup times. We focus in this article on the two‐machine case. We assume UET jobs and consider batching with batch availability and machine‐dependent setup times. We introduce an efficient \begin{align*}(O(\sqrt{n}))\end{align*} algorithm, where n is the number of jobs. We then introduce a heuristic for the multimachine case and demonstrate its efficiency for two interesting instances. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Stability in a general oligopoly model

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the “regular” extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models. We identify a fully best response operator which is monotonically increasing so that the market converges to a Nash equilibrium, when firms dynamically adjust their prices, as best responses to their competitors' prices, at least when starting in one of two price regions. Moreover, geometrically fast convergence to a common equilibrium can be guaranteed for an arbitrary starting point, under an additional condition for the price sensitivity matrix.     

Maximizing the ratio of two convex functions over a convex set

The purpose of this article is to present an algorithm for globally maximizing the ratio of two convex functions f and g over a convex set X. To our knowledge, this is the first algorithm to be proposed for globally solving this problem. The algorithm uses a branch and bound search to guarantee that a global optimal solution is found. While it does not require the functions f and g to be differentiable, it does require that subgradients of g can be calculated efficiently. The main computational effort of the algorithm involves solving a sequence of subproblems that can be solved by convex programming methods. When X is polyhedral, these subproblems can be solved by linear programming procedures. Because of these properties, the algorithm offers a potentially attractive means for globally maximizing ratios of convex functions over convex sets. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

A chance-constrained distribution problem

The transportation model with supplies (S_i) and demands (D_i) treated as bounded variables developed by Charnes and Klingman is extended to the case where the S_i and D_i are independently and uniformly distributed random variables. Chance constraints which require that demand at the jth destination will be satisfied with probability at least β_i and that stockout at the ith origin will occur with probability less than α_i are imposed. Conversion of the chance constraints to their linear equivalents results in a transportation problem with one more row and column than the original with some of the new arcs capacitated. The chance-constrained formulation is extended to the transshipment problem.     

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

We consider the problem of scheduling N jobs on M parallel machines so as to minimize the maximum earliness or tardiness cost incurred for each of the jobs. Earliness and tardiness costs are given by general (but job-independent) functions of the amount of time a job is completed prior to or after a common due date. We show that in problems with a nonrestrictive due date, the problem decomposes into two parts. Each of the M longest jobs is assigned to a different machine, and all other jobs are assigned to the machines so as to minimize their makespan. With these assignments, the individual scheduling problems for each of the machines are simple to solve. We demonstrate that several simple heuristics of low complexity, based on this characterization, are asymptotically optimal under mild probabilistic conditions. We develop attractive worst-case bounds for them. We also develop a simple closed-form lower bound for the minimum cost value. The bound is asymptotically accurate under the same probabilistic conditions. In the case where the due date is restrictive, the problem is more complex only in the sense that the set of initial jobs on the machines is not easily characterized. However, we extend our heuristics and lower bounds to this general case as well. Numerical studies exhibit that these heuristics perform excellently even for small- or moderate-size problems both in the restrictive and nonrestrictive due-date case. © 1997 John Wiley & Sons, Inc.