On the optimal detection of an underwater intruder in a channel using unmanned underwater vehicles

Given a number of patrollers that are required to detect an intruder in a channel, the channel patrol problem consists of determining the periodic trajectories that the patrollers must trace out so as to maximized the probability of detection of the intruder. We formulate this problem as an optimal control problem. We assume that the patrollers' sensors are imperfect and that their motions are subject to turn‐rate constraints, and that the intruder travels straight down a channel with constant speed. Using discretization of time and space, we approximate the optimal control problem with a large‐scale nonlinear programming problem which we solve to obtain an approximately stationary solution and a corresponding optimized trajectory for each patroller. In numerical tests for one, two, and three underwater patrollers, an underwater intruder, different trajectory constraints, several intruder speeds and other specific parameter choices, we obtain new insight—not easily obtained using simply geometric calculations—into efficient patrol trajectory design under certain conditions for multiple patrollers in a narrow channel where interaction between the patrollers is unavoidable due to their limited turn rate.© 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

地地导弹突击目标火力分配模型

火力分配及分配的优化是地地导弹作战行动中需要解决的重要问题。通过具体分析,建立了地地导弹突击目标火力分配的数学模型,并主动运用动态规划的思想分析了地地导弹突击目标火力分配的选优过程,提出了对模型的优化解法。进而,对解法进行了计算机语言的实现。经过程序计算,快速、准确地得出了所给实例的优化分配方案。     

基于小波配点法的空间机械臂关节轨迹规划

针对平面型空间机械臂关节轨迹优化问题,提出一种基于小波配点法的数值算法。该算法在区间样条小波函数及其导数算法的基础上,对状态变量与控制变量在小波配点上作离散化处理,然后将原最优控制问题转化为以小波系数为优化参数的非线性规划问题,利用非线性规划算法求解该问题得到原问题的解。算法可充分利用小波具有非线性逼近的优点,提高计算精度和效率。对典型问题进行数值仿真,结果表明该算法对空间机械臂关节轨迹优化问题是有效的。     

A squared-variable transformation approach to nonlinear programming optimality conditions

We show that the well-known necessary and sufficient conditions for a relative maximum of a nonlinear differentiable objective function with nonnegative variables constrained by nonlinear differentiable inequalities may be derived using the classical theory of equality constrained optimization problems with unrestricted variables. To do this we transform the original inequality-constrained problem to an equivalent equality-constrained problem by means of a well-known squared-variable transformation. Our major result is to show that second order conditions must be used to obtain the Kuhn-Tucker conditions by this approach. Our nonlinear programming results are motivated by the development of some well-known linear programming results by this approach.     

Joint pricing,assortment, and inventory decisions for a retailer's product line

Consider a set of product variants that are differentiated by some secondary attributes such as flavor, color, or size. The retailer's problem is to jointly determine the set of variants to include in her product line (“assortment”), together with their prices and inventory levels, so as to maximize her expected profit. We model the consumer choice process using a multinomial logit choice model and consider a newsvendor type inventory setting. We derive the structure of the optimal assortment for some important special cases, including the case of horizontally differentiated items, and propose a dominance relationship for the general case that simplifies the search for an optimal assortment. We also discuss structural properties of the optimal prices. Finally, motivated by our analytical results, we propose a heuristic solution procedure, which is shown to be quite effective through a numerical study. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Dynamic pricing and replenishment: Optimality,bounds, and asymptotics

In many applications, managers face the problem of replenishing and selling products during a finite time horizon. We investigate the problem of making dynamic and joint decisions on product replenishment and selling in order to improve profit. We consider a backlog scenario in which penalty cost (resulting from fulfillment delay) and accommodation cost (resulting from shortage at the end of the selling horizon) are incurred. Based on continuous‐time and discrete‐state dynamic programming, we study the optimal joint decisions and characterize their structural properties. We establish an upper bound for the optimal expected profit and develop a fluid policy by resorting to the deterministic version of the problem (ie, the fluid problem). The fluid policy is shown to be asymptotically optimal for the original stochastic problem when the problem size is sufficiently large. The static nature of the fluid policy and its lack of flexibility in matching supply with demand motivate us to develop a “target‐inventory” heuristic, which is shown, numerically, to be a significant improvement over the fluid policy. Scenarios with discrete feasible sets and lost‐sales are also discussed in this article.     

Bicriteria product design optimization: An efficient solution procedure using AND/OR trees

Competitive imperatives are causing manufacturing firms to consider multiple criteria when designing products. However, current methods to deal with multiple criteria in product design are ad hoc in nature. In this paper we present a systematic procedure to efficiently solve bicriteria product design optimization problems. We first present a modeling framework, the AND/OR tree, which permits a simplified representation of product design optimization problems. We then show how product design optimization problems on AND/OR trees can be framed as network design problems on a special graph—a directed series‐parallel graph. We develop an enumerative solution algorithm for the bicriteria problem that requires as a subroutine the solution of the parametric shortest path problem. Although this parametric problem is hard on general graphs, we show that it is polynomially solvable on the series‐parallel graph. As a result we develop an efficient solution algorithm for the product design optimization problem that does not require the use of complex and expensive linear/integer programming solvers. As a byproduct of the solution algorithm, sensitivity analysis for product design optimization is also efficiently performed under this framework. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 574–592, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10031     

Interdiction models for delaying adversarial attacks against critical information technology infrastructure

Information technology (IT) infrastructure relies on a globalized supply chain that is vulnerable to numerous risks from adversarial attacks. It is important to protect IT infrastructure from these dynamic, persistent risks by delaying adversarial exploits. In this paper, we propose max‐min interdiction models for critical infrastructure protection that prioritizes cost‐effective security mitigations to maximally delay adversarial attacks. We consider attacks originating from multiple adversaries, each of which aims to find a “critical path” through the attack surface to complete the corresponding attack as soon as possible. Decision‐makers can deploy mitigations to delay attack exploits, however, mitigation effectiveness is sometimes uncertain. We propose a stochastic model variant to address this uncertainty by incorporating random delay times. The proposed models can be reformulated as a nested max‐max problem using dualization. We propose a Lagrangian heuristic approach that decomposes the max‐max problem into a number of smaller subproblems, and updates upper and lower bounds to the original problem via subgradient optimization. We evaluate the perfect information solution value as an alternative method for updating the upper bound. Computational results demonstrate that the Lagrangian heuristic identifies near‐optimal solutions efficiently, which outperforms a general purpose mixed‐integer programming solver on medium and large instances.     

基于边界交叉法的高超声速滑翔飞行器多目标轨迹优化研究

针对高超声速滑翔飞行器复杂约束条件下多目标轨迹设计问题，基于边界交叉法和伪谱法提出了其多目标轨迹优化方法。首先，分析了高超声速滑翔飞行器复杂约束轨迹优化问题的特点，提出了多目标轨迹优化问题。然后，采用边界交叉法和伪谱法将多目标轨迹优化问题转化为一组单目标优化子问题，利用非线性规划算法分别求解。在优化过程中，将已求解子问题的解作为下一个子问题的初始值。利用上述方法求解了最大横程和最小峰值热流轨迹优化问题，仿真结果表明：本文方法能够有效搜索到优化轨迹的Pareto前沿，可以为高超声速滑翔飞行器轨迹设计提供参考。     

Benders decomposition with alternative multiple cuts for a multi‐product closed‐loop supply chain network design model

In this article, we consider a multi‐product closed‐loop supply chain network design problem where we locate collection centers and remanufacturing facilities while coordinating the forward and reverse flows in the network so as to minimize the processing, transportation, and fixed location costs. The problem of interest is motivated by the practice of an original equipment manufacturer in the automotive industry that provides service parts for vehicle maintenance and repair. We provide an effective problem formulation that is amenable to efficient Benders reformulation and an exact solution approach. More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch‐and‐cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch‐and‐cut approach. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Modeling and solution for the ship stowage planning problem of coils in the steel industry

We consider a ship stowage planning problem where steel coils with known destination ports are to be loaded onto a ship. The coils are to be stowed on the ship in rows. Due to their heavy weight and cylindrical shape, coils can be stowed in at most two levels. Different from stowage problems in previous studies, in this problem there are no fixed positions on the ship for the coils due to their different sizes. At a destination port, if a coil to be unloaded is not at a top position, those blocking it need to be shuffled. In addition, the stability of ship has to be maintained after unloading at each destination port. The objective for the stowage planning problem is to minimize a combination of ship instability throughout the entire voyage, the shuffles needed for unloading at the destination ports, and the dispersion of coils to be unloaded at the same destination port. We formulate the problem as a novel mixed integer linear programming model. Several valid inequalities are derived to help reducing solution time. A tabu search (TS) algorithm is developed for the problem with the initial solution generated using a construction heuristic. To evaluate the proposed TS algorithm, numerical experiments are carried out on problem instances of three different scales by comparing it with a model‐based decomposition heuristic, the classic TS algorithm, the particle swarm optimization algorithm, and the manual method used in practice. The results show that for small problems, the proposed algorithm can generate optimal solutions. For medium and large practical problems, the proposed algorithm outperforms other methods. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 564–581, 2015     

Analysis of algorithms for two‐stage flowshops with multi‐processor task flexibility

In this article we introduce a 2‐machine flowshop with processing flexibility. Two processing modes are available for each task: namely, processing by the designated processor, and processing simultaneously by both processors. The objective studied is makespan minimization. This production environment is encountered in repetitive manufacturing shops equipped with processors that have the flexibility to execute orders either individually or in coordination. In the latter case, the product designer exploits processing synergies between two processors so as to execute a particular task much faster than a dedicated processor. This type of flowshop environment is also encountered in labor‐intensive assembly lines where products moving downstream can be processed either in the designated assembly stations or by pulling together the work teams of adjacent stations. This scheduling problem requires determining the mode of operation of each task, and the subsequent scheduling that preserves the flowshop constraints. We show that the problem is ordinary NP‐complete and obtain an optimal solution using a dynamic programming algorithm with considerable computational requirements for medium and large problems. Then, we present a number of dynamic programming relaxations and analyze their worst‐case error performance. Finally, we present a polynomial time heuristic with worst‐case error performance comparable to that of the dynamic programming relaxations. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Optimal retail assortments for substitutable items purchased in sets

We consider the problem of optimizing assortments in a multi‐item retail inventory system. In addition to the usual holding and stockout costs, there is a fixed cost for including any item in the assortment. Customers' preferences for items include both probabilistic substitution patterns and the desire to purchase sets of complementary items. We develop a demand model to capture this behavior, and derive tractable approximations that allow us to formulate the optimization problem as a 0–1 mixed integer linear program. Numerical examples are solved to illustrate key insights into how both complementary and substitute items affect the optimal assortment and the expected profit. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 793–822, 2003.     

A two product dynamic economic lot size production model with either-or production constraints

This paper considers the production of two products with known demands over a finite set of periods. The production and inventory carrying costs for each product are assumed to be concave. We seek the minimum cost production schedule meeting all demands, without backlogging, assuming that at most one of the two products can be produced in any period. The optimization problem is first stated as a nonlinear programming problem, which allows the proof of a result permitting the search for the optimal policy to be restricted to those which produce a product only when its inventory level is zero. A dynamic programming formulation is given and the model is then formulated as a shortest route problem in a specially constructed network.     

航天测控调度问题的拉格朗日启发式算法

通过分析航天测控调度问题的测控需求,建立了航天测控调度0-1整数规划模型,运用拉格朗日松弛方法对模型中的任务约束和设备约束进行了松弛,运用次梯度优化算法求得了航天测控调度问题上界,同时得到了决策变量对应的拉格朗日权重,可以作为决策变量在最优解中是否被调度的启发式信息,对拉格朗日权重进行分析,提出了求解问题可行解的拉格朗日启发式算法。最后,通过对两个场景的试验分析验证了拉格朗日启发式算法所求可行解的优越性。     

Crane scheduling with spatial constraints

In this work, we examine port crane scheduling with spatial and separation constraints. Although common to most port operations, these constraints have not been previously studied. We assume that cranes cannot cross, there is a minimum distance between cranes and jobs cannot be done simultaneously. The objective is to find a crane‐to‐job matching which maximizes throughput under these constraints. We provide dynamic programming algorithms, a probabilistic tabu search, and a squeaky wheel optimization heuristic for solution. Experiments show the heuristics perform well compared with optimal solutions obtained by CPLEX for small scale instances where a squeaky wheel optimization with local search approach gives good results within short times. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Using stochastic programming to solve an outpatient appointment scheduling problem with random service and arrival times

We study a stochastic outpatient appointment scheduling problem (SOASP) in which we need to design a schedule and an adaptive rescheduling (i.e., resequencing or declining) policy for a set of patients. Each patient has a known type and associated probability distributions of random service duration and random arrival time. Finding a provably optimal solution to this problem requires solving a multistage stochastic mixed‐integer program (MSMIP) with a schedule optimization problem solved at each stage, determining the optimal rescheduling policy over the various random service durations and arrival times. In recognition that this MSMIP is intractable, we first consider a two‐stage model (TSM) that relaxes the nonanticipativity constraints of MSMIP and so yields a lower bound. Second, we derive a set of valid inequalities to strengthen and improve the solvability of the TSM formulation. Third, we obtain an upper bound for the MSMIP by solving the TSM under the feasible (and easily implementable) appointment order (AO) policy, which requires that patients are served in the order of their scheduled appointments, independent of their actual arrival times. Fourth, we propose a Monte Carlo approach to evaluate the relative gap between the MSMIP upper and lower bounds. Finally, in a series of numerical experiments, we show that these two bounds are very close in a wide range of SOASP instances, demonstrating the near‐optimality of the AO policy. We also identify parameter settings that result in a large gap in between these two bounds. Accordingly, we propose an alternative policy based on neighbor‐swapping. We demonstrate that this alternative policy leads to a much tighter upper bound and significantly shrinks the gap.     

有限马氏链最优停止的线性规划解

离散时间的有限状态马尔可夫链最优停止的值函数存在的一个充分条件是所对应的线性规划有解,且其最优解等于值函数,本文证明这个条件还是必要的。     

Technical note – operational statistics: Properties and the risk‐averse case

Consider a repeated newsvendor problem for managing the inventory of perishable products. When the parameter of the demand distribution is unknown, it has been shown that the traditional separated estimation and optimization (SEO) approach could lead to suboptimality. To address this issue, an integrated approach called operational statistics (OS) was developed by Chu et al., Oper Res Lett 36 (2008) 110–116. In this note, we first study the properties of this approach and compare its performance with that of the traditional SEO approach. It is shown that OS is consistent and superior to SEO. The benefit of using OS is larger when the demand variability is higher. We then generalize OS to the risk‐averse case under the conditional value‐at‐risk (CVaR) criterion. To model risk from both demand sampling and future demand uncertainty, we introduce a new criterion, called the total CVaR, and find the optimal OS under this new criterion. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 206–214, 2015     

Shortest path interdiction problem with arc improvement recourse: A multiobjective approach

We consider the shortest path interdiction problem involving two agents, a leader and a follower, playing a Stackelberg game. The leader seeks to maximize the follower's minimum costs by interdicting certain arcs, thus increasing the travel time of those arcs. The follower may improve the network after the interdiction by lowering the costs of some arcs, subject to a cardinality budget restriction on arc improvements. The leader and the follower are both aware of all problem data, with the exception that the leader is unaware of the follower's improvement budget. The effectiveness of an interdiction action is given by the length of a shortest path after arc costs are adjusted by both the interdiction and improvement. We propose a multiobjective optimization model for this problem, with each objective corresponding to a different possible improvement budget value. We provide mathematical optimization techniques to generate a complete set of strategies that are Pareto‐optimal. Additionally, for the special case of series‐parallel graphs, we provide a dynamic‐programming algorithm for generating all Pareto‐optimal solutions.