Efficient distributions of arms‐control inspection effort

A rule that constrains decision‐makers is enforced by an inspector who is supplied with a fixed level of inspection resources—inspection personnel, equipment, or time. How should the inspector distribute its inspection resources over several independent inspectees? What minimum level of resources is required to deter all violations? Optimal enforcement problems occur in many contexts; the motivating application for this study is the role of the International Atomic Energy Agency in support of the Treaty on the Non‐Proliferation of Nuclear Weapons. Using game‐theoretic models, the resource level adequate for deterrence is characterized in a two‐inspectee problem with inspections that are imperfect in the sense that violations can be missed. Detection functions, or probabilities of detecting a violation, are assumed to be increasing in inspection resources, permitting optimal allocations over inspectees to be described both in general and in special cases. When detection functions are convex, inspection effort should be concentrated on one inspectee chosen at random, but when they are concave it should be spread deterministicly over the inspectees. Our analysis provides guidance for the design of arms‐control verification operations, and implies that a priori constraints on the distribution of inspection effort can result in significant inefficiencies. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017     

An inspection game: Taking account of fulfillment probabilities of players' aims

This paper deals with an inspection game of customs and a smuggler. The customs can take two options of assigning a patrol or not. The smuggler has two strategies of shipping its cargo of contraband or not. Two players have several opportunities to take actions during a limited number of days. When both players do, there are some possibilities that the customs captures the smuggler and, simultaneously, the smuggler possibly makes a success of the smuggling. If the smuggler is captured or there remain no days for playing the game, the game ends. In this paper, we formulate the problem into a multi‐stage two‐person zero‐sum stochastic game and investigate some characteristics of the equilibrium solution, some of which are given in a closed form in a special case. There have been some studies so far on the inspection game. However, some consider the case that the smuggler has only one opportunity of smuggling or the perfect‐capture case that the customs can certainly arrest the smuggler on patrol, and others think of a recursive game without the probabilities of fulfilling the players' purposes. In this paper, we consider the inspection game taking account of the fulfillment probabilities of the players' aims. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

A price leadership method for solving the inspector's non-constant-sum game

An inspector's game is a non-constant-sum two-person game in which one player has promised to perform a certain duty and the other player is allowed to inspect and verify occasionally that the duty has indeed been performed. A solution to a variant of such a game is given in this paper, based on the assumption that the inspector can announce his mixed strategy in advance, if he so wishes, whereas the other player, who has already given his promise, cannot threaten by explicitly saying that he will not keep his word.     

Cost allocation rules for elastic single‐attribute situations

Many cooperative games, especially ones stemming from resource pooling in queueing or inventory systems, are based on situations in which each player is associated with a single attribute (a real number representing, say, a demand) and in which the cost to optimally serve any sum of attributes is described by an elastic function (which means that the per‐demand cost is non‐increasing in the total demand served). For this class of situations, we introduce and analyze several cost allocation rules: the proportional rule, the serial cost sharing rule, the benefit‐proportional rule, and various Shapley‐esque rules. We study their appeal with regard to fairness criteria such as coalitional rationality, benefit ordering, and relaxations thereof. After showing the impossibility of combining coalitional rationality and benefit ordering, we show for each of the cost allocation rules which fairness criteria it satisfies. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 271–286, 2017     

Revisiting inspection game and inspector leadership through reaction networks

The inspection game is a two-player noncooperative game that models a situation where an inspector verifies whether the inspectee complies with the rules (on the assumption that the inspectee has the tendency to violate at least one of the rules). The usual approach in the analysis of this game seeks to find an optimal strategic inspection scheme for each of the two players yielding favorable payoffs. Recently, there have been some developments in the study of such games that use a mathematical structure known as reaction network involving a set of molecular species and the existing reactions among these species. In this paper, we use a reaction network to analyze the inspection game giving an alternative way of modeling the social situation. The molecular species play the role of the players' decision moves and their resulting gain or loss, while the reactions are the encounters of the decisions of the players which, as expected, yield payoffs. We reexamine the dynamics of the inspection game through the lens of reaction network theory and consider various situations that call for more detailed analyses such as equal or unequal reaction rates and inspection leadership. Conditions concerning reaction rates, initial population of decision species, benefits, and costs are determined in order to identify strategies that yield better payoffs both for the inspector and inspectee. These results illustrate practical insights rooted from the formulated simple game models.     

The economic effects of inspector error on attribute sampling plans

In this paper the effects of inspector error on a cost-based quality control system are investigated. The system examined is of a single sampling plan design involving several cost components. Both type I and type II inspector errors are considered. The model employs a process distribution, thus assuming that a stochastic process of some kind governs the quality of incoming lots. Optimal plan design is investigated under both error-free and error-prone inspection procedures and some comparisons are made.     

Probabilistic resource pooling games

We study a setting with a single type of resource and with several players, each associated with a single resource (of this type). Unavailability of these resources comes unexpectedly and with player‐specific costs. Players can cooperate by reallocating the available resources to the ones that need the resources most and let those who suffer the least absorb all the costs. We address the cost savings allocation problem with concepts of cooperative game theory. In particular, we formulate a probabilistic resource pooling game and study them on various properties. We show that these games are not necessarily convex, do have non‐empty cores, and are totally balanced. The latter two are shown via an interesting relationship with Böhm‐Bawerk horse market games. Next, we present an intuitive class of allocation rules for which the resulting allocations are core members and study an allocation rule within this class of allocation rules with an appealing fairness property. Finally, we show that our results can be applied to a spare parts pooling situation.     

New results on a Ruckle problem in discrete games of ambush

This article deals with a two‐person zero‐sum game in which player I chooses in integer interval [1, N] two integer intervals consisting of p and q points where p + q < N, and player II chooses an integer point in [1, N]. The payoff to player I equals 1 if the point chosen by player II is at least in one of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, Lee, Garnaev, and Zoroa, Zoroa and Fernández‐Sáez. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 98–106, 2001     

A recursive algorithm for a summed multinomial density function

An algorithm for calculating the probabilities of a summed multinomial density function which is recursive with n (the number of trials) is presented. Having application in inspector error models for auditing and quality control problems with Cartesian product structures, the algorithm is discussed in the context of computing optimal economic sampling plans. Computational experience with the algorithm is presented.     

Generalized orienteering problem with resource dependent rewards

We introduce a generalized orienteering problem (OP) where, as usual, a vehicle is routed from a prescribed start node, through a directed network, to a prescribed destination node, collecting rewards at each node visited, to maximize the total reward along the path. In our generalization, transit on arcs in the network and reward collection at nodes both consume a variable amount of the same limited resource. We exploit this resource trade‐off through a specialized branch‐and‐bound algorithm that relies on partial path relaxation problems that often yield tight bounds and lead to substantial pruning in the enumeration tree. We present the smuggler search problem (SSP) as an important real‐world application of our generalized OP. Numerical results show that our algorithm applied to the SSP outperforms standard mixed‐integer nonlinear programming solvers for moderate to large problem instances. We demonstrate model enhancements that allow practitioners to represent realistic search planning scenarios by accounting for multiple heterogeneous searchers and complex smuggler motion. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Using container inspection history to improve interdiction logistics for illicit nuclear materials

Strengthening the United States' ability to prevent adversaries from smuggling nuclear materials into the country is a vital and ongoing issue. The prospect of additional countries, such as Iran, obtaining the know‐how and equipment to produce these special nuclear materials in the near future underscores the need for efficient and effective inspection policies at ports and border crossings. In addition, the reduction of defense and homeland security budgets in recent years has made it increasingly important to accomplish the interdiction mission with fewer funds. Addressing these complications, in this article, we present a novel two‐port interdiction model. We propose using prior inspection data as a low‐cost way of increasing overall interdiction performance. We provide insights into two primary questions: first, how should a decision maker at a domestic port use detection data from the foreign port to improve the overall detection capability? Second, what are potential limitations to the usefulness of prior inspection data—is it possible that using prior data actually harms decision making at the domestic port? We find that a boundary curve policy (BCP) that takes into account both foreign and domestic inspection data can provide a significant improvement in detection probability. This BCP also proves to be surprisingly robust, even if adversaries are able to infiltrate shipments during transit. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 433‐448, 2013     

Assessing the value of demand sharing in supply chains

We consider the problem of assessing the value of demand sharing in a multistage supply chain in which the retailer observes stationary autoregressive moving average demand with Gaussian white noise (shocks). Similar to previous research, we assume each supply chain player constructs its best linear forecast of the leadtime demand and uses it to determine the order quantity via a periodic review myopic order‐up‐to policy. We demonstrate how a typical supply chain player can determine the extent of its available information in the presence of demand sharing by studying the properties of the moving average polynomials of adjacent supply chain players. The retailer's demand is driven by the random shocks appearing in the autoregressive moving average representation for its demand. Under the assumptions we will make in this article, to the retailer, knowing the shock information is equivalent to knowing the demand process (assuming that the model parameters are also known). Thus (in the event of sharing) the retailer's demand sequence and shock sequence would contain the same information to the retailer's supplier. We will show that, once we consider the dynamics of demand propagation further up the chain, it may be that a player's demand and shock sequences will contain different levels of information for an upstream player. Hence, we study how a player can determine its available information under demand sharing, and use this information to forecast leadtime demand. We characterize the value of demand sharing for a typical supply chain player. Furthermore, we show conditions under which (i) it is equivalent to no sharing, (ii) it is equivalent to full information shock sharing, and (iii) it is intermediate in value to the two previously described arrangements. Although it follows from existing literature that demand sharing is equivalent to full information shock sharing between a retailer and supplier, we demonstrate and characterize when this result does not generalize to upstream supply chain players. We then show that demand propagates through a supply chain where any player may share nothing, its demand, or its full information shocks (FIS) with an adjacent upstream player as quasi‐ARMA in—quasi‐ARMA out. We also provide a convenient form for the propagation of demand in a supply chain that will lend itself to future research applications. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 515–531, 2014     

Stochastic regret minimization for revenue management problems with nonstationary demands

We study an admission control model in revenue management with nonstationary and correlated demands over a finite discrete time horizon. The arrival probabilities are updated by current available information, that is, past customer arrivals and some other exogenous information. We develop a regret‐based framework, which measures the difference in revenue between a clairvoyant optimal policy that has access to all realizations of randomness a priori and a given feasible policy which does not have access to this future information. This regret minimization framework better spells out the trade‐offs of each accept/reject decision. We proceed using the lens of approximation algorithms to devise a conceptually simple regret‐parity policy. We show the proposed policy achieves 2‐approximation of the optimal policy in terms of total regret for a two‐class problem, and then extend our results to a multiclass problem with a fairness constraint. Our goal in this article is to make progress toward understanding the marriage between stochastic regret minimization and approximation algorithms in the realm of revenue management and dynamic resource allocation. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 433–448, 2016     

Multidirectional scheduling scheme in resource‐constrained project schedulingproblem

In 2000, Klein showed that bidirectional scheduling schemes (bidss) outperform single‐directional scheduling schemes (e.g., forward or backward schemes). In 2010, Yoosefzadeh, et al. [J Math Model Algor 9 (2010), 357–373] showed that depending on the nature of the problems and also the type of priority rules used, schedules produced by a so‐called tridirectional scheduling scheme (trdss) yields shorter makespans when compared to forward, backward, and even bidss. Since the justification technique is applied in many of the state‐of‐the‐art algorithms nowadays, we show that the tuned version of the trdss outperforms the double justification technique. Moreover, we investigate the circumstances under which the trdss is more probable to generate schedules with shorter makespans. To this end, we introduce a new measure of resource requirements and their distributions, namely total amount of overflows. Our analytical as well as empirical investigations show that when the new measure is increased, it is more probable to obtain schedules with shorter makespans using the trdss. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 44–55, 2014     

A three‐stage procurement optimization problem under uncertainty

This article examines a problem faced by a firm procuring a material input or good from a set of suppliers. The cost to procure the material from any given supplier is concave in the amount ordered from the supplier, up to a supplier‐specific capacity limit. This NP‐hard problem is further complicated by the observation that capacities are often uncertain in practice, due for instance to production shortages at the suppliers, or competition from other firms. We accommodate this uncertainty in a worst‐case (robust) fashion by modeling an adversarial entity (which we call the “follower”) with a limited procurement budget. The follower reduces supplier capacity to maximize the minimum cost required for our firm to procure its required goods. To guard against uncertainty, the firm can “protect” any supplier at a cost (e.g., by signing a contract with the supplier that guarantees supply availability, or investing in machine upgrades that guarantee the supplier's ability to produce goods at a desired level), ensuring that the anticipated capacity of that supplier will indeed be available. The problem we consider is thus a three‐stage game in which the firm first chooses which suppliers' capacities to protect, the follower acts next to reduce capacity from unprotected suppliers, and the firm then satisfies its demand using the remaining capacity. We formulate a three‐stage mixed‐integer program that is well‐suited to decomposition techniques and develop an effective cutting‐plane algorithm for its solution. The corresponding algorithmic approach solves a sequence of scaled and relaxed problem instances, which enables solving problems having much larger data values when compared to standard techniques. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Robust capacity planning in semiconductor manufacturing

We present a stochastic programming approach to capacity planning under demand uncertainty in semiconductor manufacturing. Given multiple demand scenarios together with associated probabilities, our aim is to identify a set of tools that is a good compromise for all these scenarios. More precisely, we formulate a mixed‐integer program in which expected value of the unmet demand is minimized subject to capacity and budget constraints. This is a difficult two‐stage stochastic mixed‐integer program which cannot be solved to optimality in a reasonable amount of time. We instead propose a heuristic that can produce near‐optimal solutions. Our heuristic strengthens the linear programming relaxation of the formulation with cutting planes and performs limited enumeration. Analyses of the results in some real‐life situations are also presented. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Sequential and systematic sampling plans for the Markov‐dependent production process

Acceptance sampling plans are used to assess the quality of an ongoing production process, in addition to the lot acceptance. In this paper, we consider sampling inspection plans for monitoring the Markov‐dependent production process. We construct sequential plans that satisfy the usual probability requirements at acceptable quality level and rejectable quality level and, in addition, possess the minimum average sample number under semicurtailed inspection. As these plans result in large sample sizes, especially when the serial correlation is high, we suggest new plans called “systematic sampling plans.” The minimum average sample number systematic plans that satisfy the probability requirements are constructed. Our algorithm uses some simple recurrence relations to compute the required acceptance probabilities. The optimal systematic plans require much smaller sample sizes and acceptance numbers, compared to the sequential plans. However, they need larger production runs to make a decision. Tables for choosing appropriate sequential and systematic plans are provided. The problem of selecting the best systematic sampling plan is also addressed. The operating characteristic curves of some of the sequential and the systematic plans are compared, and are observed to be almost identical. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 451–467, 2001     

A bicriterion approach to common flow allowances due window assignment and scheduling with controllable processing times

We investigate a single‐machine scheduling problem for which both the job processing times and due windows are decision variables to be determined by the decision maker. The job processing times are controllable as a linear or convex function of the amount of a common continuously divisible resource allocated to the jobs, where the resource allocated to the jobs can be used in discrete or continuous quantities. We use the common flow allowances due window assignment method to assign due windows to the jobs. We consider two performance criteria: (i) the total weighted number of early and tardy jobs plus the weighted due window assignment cost, and (ii) the resource consumption cost. For each resource consumption function, the objective is to minimize the first criterion, while keeping the value of the second criterion no greater than a given limit. We analyze the computational complexity, devise pseudo‐polynomial dynamic programming solution algorithms, and provide fully polynomial‐time approximation schemes and an enhanced volume algorithm to find high‐quality solutions quickly for the considered problems. We conduct extensive numerical studies to assess the performance of the algorithms. The computational results show that the proposed algorithms are very efficient in finding optimal or near‐optimal solutions. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 41–63, 2017     

Technical note: Characterization of binomial semivalues through delegation games

Semivalues are allocation rules for cooperative games that assign to each player in a given game a weighted sum of his marginal contributions to all coalitions he belongs to, where the weighting coefficients depend only on the coalition size. Binomial semivalues are a special class of semivalues whose weighting coefficients are obtained by means of a unique parameter. In particular, the Banzhaf value is a binomial semivalue. In this article, we provide an axiomatic characterization for each binomial semivalue. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007