Cyclic best first search: Using contours to guide branch‐and‐bound algorithms

The cyclic best‐first search (CBFS) strategy is a recent search strategy that has been successfully applied to branch‐and‐bound algorithms in a number of different settings. CBFS is a modification of best‐first search (BFS) that places search tree subproblems into contours which are collections of subproblems grouped in some way, and repeatedly cycles through all non‐empty contours, selecting one subproblem to explore from each. In this article, the theoretical properties of CBFS are analyzed for the first time. CBFS is proved to be a generalization of all other search strategies by using a contour definition that explores the same sequence of subproblems as any other search strategy. Further, a bound is proved between the number of subproblems explored by BFS and the number of children generated by CBFS, given a fixed branching strategy and set of pruning rules. Finally, a discussion of heuristic contour‐labeling functions is provided, and proof‐of‐concept computational results for mixed‐integer programming problems from the MIPLIB 2010 database are shown. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 64–82, 2017     

Capacitated lot‐sizing and scheduling with parallel machines,back‐orders,and setup carry‐over

We address the capacitated lot‐sizing and scheduling problem with setup times, setup carry‐over, back‐orders, and parallel machines as it appears in a semiconductor assembly facility. The problem can be formulated as an extension of the capacitated lot‐sizing problem with linked lot‐sizes (CLSPL). We present a mixed integer (MIP) formulation of the problem and a new solution procedure. The solution procedure is based on a novel “aggregate model,” which uses integer instead of binary variables. The model is embedded in a period‐by‐period heuristic and is solved to optimality or near‐optimality in each iteration using standard procedures (CPLEX). A subsequent scheduling routine loads and sequences the products on the parallel machines. Six variants of the heuristic are presented and tested in an extensive computational study. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Strategic capacity decision‐making in a stochastic manufacturing environment using real‐time approximate dynamic programming

In this study, we illustrate a real‐time approximate dynamic programming (RTADP) method for solving multistage capacity decision problems in a stochastic manufacturing environment, by using an exemplary three‐stage manufacturing system with recycle. The system is a moderate size queuing network, which experiences stochastic variations in demand and product yield. The dynamic capacity decision problem is formulated as a Markov decision process (MDP). The proposed RTADP method starts with a set of heuristics and learns a superior quality solution by interacting with the stochastic system via simulation. The curse‐of‐dimensionality associated with DP methods is alleviated by the adoption of several notions including “evolving set of relevant states,” for which the value function table is built and updated, “adaptive action set” for keeping track of attractive action candidates, and “nonparametric k nearest neighbor averager” for value function approximation. The performance of the learned solution is evaluated against (1) an “ideal” solution derived using a mixed integer programming (MIP) formulation, which assumes full knowledge of future realized values of the stochastic variables (2) a myopic heuristic solution, and (3) a sample path based rolling horizon MIP solution. The policy learned through the RTADP method turned out to be superior to polices of 2 and 3. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Expansion planning for waste‐to‐energy systems using waste forecast prediction sets

We consider an expansion planning problem for Waste‐to‐Energy (WtE) systems facing uncertainty in future waste supplies. The WtE expansion plans are regarded as strategic, long term decisions, while the waste distribution and treatment are medium to short term operational decisions which can adapt to the actual waste collected. We propose a prediction set uncertainty model which integrates a set of waste generation forecasts and is constructed based on user‐specified levels of forecasting errors. Next, we use the prediction sets for WtE expansion scenario analysis. More specifically, for a given WtE expansion plan, the guaranteed net present value (NPV) is evaluated by computing an extreme value forecast trajectory of future waste generation from the prediction set that minimizes the maximum NPV of the WtE project. This problem is essentially a multiple stage min‐max dynamic optimization problem. By exploiting the structure of the WtE problem, we show this is equivalent to a simpler min‐max optimization problem, which can be further transformed into a single mixed‐integer linear program. Furthermore, we extend the model to optimize the guaranteed NPV by searching over the set of all feasible expansion scenarios, and show that this can be solved by an exact cutting plane approach. We also propose a heuristic based on a constant proportion distribution rule for the WtE expansion optimization model, which reduces the problem into a moderate size mixed‐integer program. Finally, our computational studies demonstrate that our proposed expansion model solutions are very stable and competitive in performance compared to scenario tree approaches. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 47–70, 2016     

Optimizing usage and maintenance decisions for k‐out‐of‐n systems of moving assets

We consider an integrated usage and maintenance optimization problem for a k‐out‐of‐n system pertaining to a moving asset. The k‐out‐of‐n systems are commonly utilized in practice to increase availability, where n denotes the total number of parallel and identical units and k the number of units required to be active for a functional system. Moving assets such as aircraft, ships, and submarines are subject to different operating modes. Operating modes can dictate not only the number of system units that are needed to be active, but also where the moving asset physically is, and under which environmental conditions it operates. We use the intrinsic age concept to model the degradation process. The intrinsic age is analogous to an intrinsic clock which ticks on a different pace in different operating modes. In our problem setting, the number of active units, degradation rates of active and standby units, maintenance costs, and type of economic dependencies are functions of operating modes. In each operating mode, the decision maker should decide on the set of units to activate (usage decision) and the set of units to maintain (maintenance decision). Since the degradation rate differs for active and standby units, the units to be maintained depend on the units that have been activated, and vice versa. In order to minimize maintenance costs, usage and maintenance decisions should be jointly optimized. We formulate this problem as a Markov decision process and provide some structural properties of the optimal policy. Moreover, we assess the performance of usage policies that are commonly implemented for maritime systems. We show that the cost increase resulting from these policies is up to 27% for realistic settings. Our numerical experiments demonstrate the cases in which joint usage and maintenance optimization is more valuable. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 418–434, 2017     

Single‐commodity stochastic network design under demand and topological uncertainties with insufficient data

Stochastic network design is fundamental to transportation and logistic problems in practice, yet faces new modeling and computational challenges resulted from heterogeneous sources of uncertainties and their unknown distributions given limited data. In this article, we design arcs in a network to optimize the cost of single‐commodity flows under random demand and arc disruptions. We minimize the network design cost plus cost associated with network performance under uncertainty evaluated by two schemes. The first scheme restricts demand and arc capacities in budgeted uncertainty sets and minimizes the worst‐case cost of supply generation and network flows for any possible realizations. The second scheme generates a finite set of samples from statistical information (e.g., moments) of data and minimizes the expected cost of supplies and flows, for which we bound the worst‐case cost using budgeted uncertainty sets. We develop cutting‐plane algorithms for solving the mixed‐integer nonlinear programming reformulations of the problem under the two schemes. We compare the computational efficacy of different approaches and analyze the results by testing diverse instances of random and real‐world networks. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 154–173, 2017     

A branch‐and‐bound‐based solution approach for dynamic rerouting of airborne platforms

This article is a sequel to a recent article that appeared in this journal, “An extensible modeling framework for dynamic reassignment and rerouting in cooperative airborne operations” [ 17 ], in which an integer programming formulation to the problem of rescheduling in‐flight assets due to changes in battlespace conditions was presented. The purpose of this article is to present an improved branch‐and‐bound procedure to solve the dynamic resource management problem in a timely fashion, as in‐flight assets must be quickly re‐tasked to respond to the changing environment. To facilitate the rapid generation of attractive updated mission plans, this procedure uses a technique for reducing the solution space, supports branching on multiple decision variables simultaneously, incorporates additional valid cuts to strengthen the minimal network constraints of the original mathematical model, and includes improved objective function bounds. An extensive numerical analysis indicates that the proposed approach significantly outperforms traditional branch‐and‐bound methodologies and is capable of providing improved feasible solutions in a limited time. Although inspired by the dynamic resource management problem in particular, this approach promises to be an effective tool for solving other general types of vehicle routing problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Route optimization for multiple searchers

We consider a discrete time‐and‐space route‐optimization problem across a finite time horizon in which multiple searchers seek to detect one or more probabilistically moving targets. This article formulates a novel convex mixed‐integer nonlinear program for this problem that generalizes earlier models to situations with multiple targets, searcher deconfliction, and target‐ and location‐dependent search effectiveness. We present two solution approaches, one based on the cutting‐plane method and the other on linearization. These approaches result in the first practical exact algorithms for solving this important problem, which arises broadly in military, rescue, law enforcement, and border patrol operations. The cutting‐plane approach solves many realistically sized problem instances in a few minutes, while existing branch‐and‐bound algorithms fail. A specialized cut improves solution time by 50[percnt] in difficult problem instances. The approach based on linearization, which is applicable in important special cases, may further reduce solution time with one or two orders of magnitude. The solution time for the cutting‐plane approach tends to remain constant as the number of searchers grows. In part, then, we overcome the difficulty that earlier solution methods have with many searchers. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Managing components for assemble‐to‐order products with lead‐time‐dependent pricing: The full‐shipment model

We study a component inventory planning problem in an assemble‐to‐order environment faced by many contract manufacturers in which both quick delivery and efficient management of component inventory are crucial for the manufacturers to achieve profitability in a highly competitive market. Extending a recent study in a similar problem setting by the same authors, we analyze an optimization model for determining the optimal component stocking decision for a contract manufacturer facing an uncertain future demand, where product price depends on the delivery times. In contrast to our earlier work, this paper considers the situation where the contract manufacturer needs to deliver the full order quantity in one single shipment. This delivery requirement is appropriate for many industries, such as the garment and toy industries, where the economies of scale in transportation is essential. We develop efficient solution procedures for solving this optimization problem. We use our model results to illustrate how the different model parameters affect the optimal solution. We also compare the results under this full‐shipment model with those from our earlier work that allows for multiple partial shipments. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A multi‐stage stochastic programming approach for network capacity expansion with multiple sources of capacity

In networks, there are often more than one sources of capacity. The capacities can be permanently or temporarily owned by the decision maker. Depending on the nature of sources, we identify the permanent capacity, spot market capacity, and contract capacity. We use a scenario tree to model the uncertainty, and build a multi‐stage stochastic integer program that can incorporate multiple sources and multiple types of capacities in a general network. We propose two solution methodologies for the problem. Firstly, we design an asymptotically convergent approximation algorithm. Secondly, we design a cutting plane algorithm based on Benders decomposition to find tight bounds for the problem. The numerical experiments show superb performance of the proposed algorithms compared with commercial software. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 600–614, 2017     

Discrete equal‐capacity p‐median problem

This paper examines the discrete equal‐capacity p‐median problem that seeks to locate p new facilities (medians) on a network, each having a given uniform capacity, in order to minimize the sum of distribution costs while satisfying the demand on the network. Such problems arise, for example, in local access and transport area telecommunication network design problems where any number of a set of p facility units can be constructed at the specified candidate sites (hence, the net capacity is an integer multiple of a given unit capacity). We develop various valid inequalities, a separation routine for generating cutting planes that are specific members of such inequalities, as well as an enhanced reformulation that constructs a partial convex hull representation that subsumes an entire class of valid inequalities via its linear programming relaxation. We also propose suitable heuristic schemes for this problem, based on sequentially rounding the continuous relaxation solutions obtained for the various equivalent formulations of the problem. Extensive computational results are provided to demonstrate the effectiveness of the proposed valid inequalities, enhanced formulations, and heuristic schemes. The results indicate that the proposed schemes for tightening the underlying relaxations play a significant role in enhancing the performance of both exact and heuristic solution methods for this class of problems. © 2000 John & Sons, Inc. Naval Research Logistics 47: 166–183, 2000.     

Rank‐Cluster‐and‐Prune: An algorithm for generating clusters in complex set partitioning problems

Clustering problems are often difficult to solve due to nonlinear cost functions and complicating constraints. Set partitioning formulations can help overcome these challenges, but at the cost of a very large number of variables. Therefore, techniques such as delayed column generation must be used to solve these large integer programs. The underlying pricing problem can suffer from the same challenges (non‐linear cost, complicating constraints) as the original problem, however, making a mathematical programming approach intractable. Motivated by a real‐world problem in printed circuit board (PCB) manufacturing, we develop a search‐based algorithm (Rank‐Cluster‐and‐Prune) as an alternative, present computational results for the PCB problem to demonstrate the tractability of our approach, and identify a broader class of clustering problems for which this approach can be used. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Balancing a one‐way corridor capacity and safety‐oriented reliability: A stochastic optimization approach for metro train timetabling

In urban rail transit systems of large cities, the headway and following distance of successive trains have been compressed as much as possible to enhance the corridor capacity to satisfy extremely high passenger demand during peak hours. To prevent train collisions and ensure the safety of trains, a safe following distance of trains must be maintained. However, this requirement is subject to a series of complex factors, such as the uncertain train braking performance, train communication delay, and driver reaction time. In this paper, we propose a unified mathematical framework to analyze the safety‐oriented reliability of metro train timetables with different corridor capacities, that is, the train traffic density, and determine the most reliable train timetable for metro lines in an uncertain environment. By employing a space‐time network representation in the formulations, the reliability‐based train timetabling problem is formulated as a nonlinear stochastic programming model, in which we use 0‐1 variables to denote the time‐dependent velocity and position of all involved trains. Several reformulation techniques are developed to obtain an equivalent mixed integer programming model with quadratic constraints (MIQCP) that can be solved to optimality by some commercial solvers. To improve the computational efficiency of the MIQCP model, we develop a dual decomposition solution framework that decomposes the primal problem into several sets of subproblems by dualizing the coupling constraints across different samples. An exact dynamic programming combined with search space reduction strategies is also developed to solve the exact optimal solutions of these subproblems. Two sets of numerical experiments, which involve a relatively small‐scale case and a real‐world instance based on the operation data of the Beijing subway Changping Line are implemented to verify the effectiveness of the proposed approaches.     

Dual‐ascent and primal heuristics for production‐assembly‐distribution system design

This article proposes two dual‐ascent algorithms and uses each in combination with a primal drop heuristic embedded within a branch and bound framework to solve the uncapacitated production assembly distribution system (i.e., supply chain) design problem, which is formulated as a mixed integer program. Computational results indicate that one approach, which combines primal drop and dual‐ascent heuristics, can solve instances within reasonable time and prescribes solutions with gaps between the primal and dual solution values that are less than 0.15%, an efficacy suiting it for actual large‐scale applications. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

An exact branch‐and‐price algorithm for multitasking scheduling on unrelated parallel machines

We consider the multitasking scheduling problem on unrelated parallel machines to minimize the total weighted completion time. In this problem, each machine processes a set of jobs, while the processing of a selected job on a machine may be interrupted by other available jobs scheduled on the same machine but unfinished. To solve this problem, we propose an exact branch‐and‐price algorithm, where the master problem at each search node is solved by a novel column generation scheme, called in‐out column generation, to maintain the stability of the dual variables. We use a greedy heuristic to obtain a set of initial columns to start the in‐out column generation, and a hybrid strategy combining a genetic algorithm and an exact dynamic programming algorithm to solve the pricing subproblems approximately and exactly, respectively. Using randomly generated data, we conduct numerical studies to evaluate the performance of the proposed solution approach. We also examine the effects of multitasking on the scheduling outcomes, with which the decision maker can justify making investments to adopt or avoid multitasking.     

Cyclic scheduling in robotic flowshops with bounded work‐in‐process levels

This study addresses cyclic scheduling in robotic flowshops with bounded work‐in‐process (WIP) levels. The objective is to minimize the cycle time or, equivalently, to maximize the throughput, under the condition that the WIP level is bounded from above by a given integer number. We present several strongly polynomial algorithms for the 2‐cyclic robotic flowshop scheduling problems for various WIP levels. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 58: 1–16, 2011     

The one‐warehouse multiretailer problem with an order‐up‐to level inventory policy

We consider a two‐level system in which a warehouse manages the inventories of multiple retailers. Each retailer employs an order‐up‐to level inventory policy over T periods and faces an external demand which is dynamic and known. A retailer's inventory should be raised to its maximum limit when replenished. The problem is to jointly decide on replenishment times and quantities of warehouse and retailers so as to minimize the total costs in the system. Unlike the case in the single level lot‐sizing problem, we cannot assume that the initial inventory will be zero without loss of generality. We propose a strong mixed integer program formulation for the problem with zero and nonzero initial inventories at the warehouse. The strong formulation for the zero initial inventory case has only T binary variables and represents the convex hull of the feasible region of the problem when there is only one retailer. Computational results with a state‐of‐the art solver reveal that our formulations are very effective in solving large‐size instances to optimality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

求解整数规划问题的一个搜索方法

本文整数规划问题给出一种搜索方法,它类似于求解连续变量优化问题的迭代方法,从一个好的初始可行解出发,寻找一个搜索方向,沿着这个方向求出改进的可行解,然后又开始下一次迭代。此方法简单易行,可以求出问题的最优解或近似最优解,对于整数线性规划问题和整数非线性规划问题的求解都适用,并且容易推广到求解大规校整数线性规划问题。文中附有计算例子,说明方法是有效的。     

Approximating the nondominated frontiers of multi‐objective combinatorial optimization problems

Finding all nondominated vectors for multi‐objective combinatorial optimization (MOCO) problems is computationally very hard in general. We approximate the nondominated frontiers of MOCO problems by fitting smooth hypersurfaces. For a given problem, we fit the hypersurface using a single nondominated reference vector. We experiment with different types of MOCO problems and demonstrate that in all cases the fitted hypersurfaces approximate all nondominated vectors well. We discuss that such an approximation is useful to find the neighborhood of preferred regions of the nondominated vectors with very little computational effort. Further computational effort can then be spent in the identified region to find the actual nondominated vectors the decision maker will prefer. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Optimal FCFS allocation rules for periodic‐review assemble‐to‐order systems

In Assemble‐To‐Order (ATO) systems, situations may arise in which customer demand must be backlogged due to a shortage of some components, leaving available stock of other components unused. Such unused component stock is called remnant stock. Remnant stock is a consequence of both component ordering decisions and decisions regarding allocation of components to end‐product demand. In this article, we examine periodic‐review ATO systems under linear holding and backlogging costs with a component installation stock policy and a First‐Come‐First‐Served (FCFS) allocation policy. We show that the FCFS allocation policy decouples the problem of optimal component allocation over time into deterministic period‐by‐period optimal component allocation problems. We denote the optimal allocation of components to end‐product demand as multimatching. We solve the multi‐matching problem by an iterative algorithm. In addition, an approximation scheme for the joint replenishment and allocation optimization problem with both upper and lower bounds is proposed. Numerical experiments for base‐stock component replenishment policies show that under optimal base‐stock policies and optimal allocation, remnant stock holding costs must be taken into account. Finally, joint optimization incorporating optimal FCFS component allocation is valuable because it provides a benchmark against which heuristic methods can be compared. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 158–169, 2015