ABSTRACT A truck platoon is a set of trucks that drive behind one another at short headways to save fuel, reduce emissions, and improve traffic throughput. Despite the potential benefits of platooning, road operators have raised concerns about the impact of platoons on surrounding traffic. The use of dedicated platoon lanes helps prevent potentially dangerous situations when interacting with regular traffic. Furthermore, dedicated platooning lanes can help increase platooning benefits and prolong the life of infrastructure. Such types of infrastructure design decisions typically have to be made when the demand that the infrastructure needs to support is uncertain. One way to represent uncertain demand in optimization models is by means of possible realizations or scenarios. Using many scenarios is likely to produce better solutions, but it also creates computational challenges. We propose an approach that blends solutions, each obtained using an optimization model with relatively few scenarios. We use this approach in the context of locating dedicated truck platoon lanes. Computational experiments show that blending designs obtained with a few scenarios is as effective, but much more efficient, as creating a design using many scenarios.

Finding a shortest path in a network is a fundamental optimization problem. We focus on settings in which the travel time on an arc in the network depends on the time at which traversal of the arc begins. In such settings, reaching the destination as early as possible is not the only objective of interest. Minimizing the duration of the path, that is, the difference between the arrival time at the destination and the departure from the origin, and minimizing the travel time along the path from origin to destination, are also of interest. We introduce dynamic discretization discovery algorithms to efficiently solve such time-dependent shortest path problems with piecewise linear arc travel time functions. The algorithms operate on partially time-expanded networks in which arc costs represent lower bounds on the arc travel time over the subsequent time interval. A shortest path in this partially time-expanded network yields a lower bound on the value of an optimal path. Upper bounds are easily obtained as by-products of the lower bound calculations. The algorithms iteratively refine the discretization by exploiting breakpoints of the arc travel time functions. In addition to time discretization refinement, the algorithms permit time intervals to be eliminated, improving lower and upper bounds, until, in a finite number of iterations, optimality is proved. Computational experiments show that only a small fraction of breakpoints must be explored and that the fraction decreases as the length of the time horizon and the size of the network increases, making the algorithms highly efficient and scalable. Summary of Contribution: New data collection techniques have increased the availability and fidelity of time-dependent travel time information, making the time-dependent variant of the classic shortest path problem an extremely relevant problem in the field of operations research. This paper provides novel algorithms for the time-dependent shortest path problem with both the minimum duration and minimum travel time objectives, which aims to address the computational challenges faced by existing algorithms. A computational study shows that our new algorithm is indeed significantly more efficient than existing approaches.

Abstract The service network design problem is commonly used to represent the tactical decisions encountered by a consolidation carrier operating a hub‐and‐spoke network: what transportation services to operate between hubs and how to route commodities from their origin to their destination through the network. In most settings, the capacity at hubs is not a limiting factor and can safely be ignored. However, in the context of city logistics networks, where space is limited and expensive, hub capacities typically have to be taken into account. The presence of hub capacity (and time) constraints implies that, contrary to traditional service network design problems, the existence of a feasible solution is no longer guaranteed. We present an exact dynamic discretization discovery algorithm for a variant of the service network design problem in which the number of vehicles that can be loaded and unloaded simultaneously at a hub is restricted. Novel techniques are introduced in the algorithm to handle the hub capacity constraints. A computational study using instances derived from real‐world data shows the potential of dynamic discretization discovery for this class of problems: integer program sizes are reduced by a factor of up to one thousand and small to mid size instances can be (optimally) solved in an acceptable amount of time.

Waiting at the right location at the right time can be critically important in certain variants of time-dependent shortest path problems. We investigate the computational complexity of time-dependent shortest path problems in which there is either a penalty on waiting or a limit on the total time spent waiting at a given subset of the nodes. We show that some cases are nondeterministic polynomial-time hard, and others can be solved in polynomial time, depending on the choice of the subset of nodes, on whether waiting is penalized or constrained, and on the magnitude of the penalty/waiting limit parameter. Summary of Contributions: This paper addresses simple yet relevant extensions of a fundamental problem in Operations Research: the Shortest Path Problem (SPP). It considers time-dependent variants of SPP, which can account for changing traffic and/or weather conditions. The first variant that is tackled allows for waiting at certain nodes but at a cost. The second variant instead places a limit on the total waiting. Both variants have applications in transportation, e.g., when it is possible to wait at certain locations if the benefits outweigh the costs. The paper investigates these problems using complexity analysis and algorithm design, both tools from the field of computing. Different cases are considered depending on which of the nodes contribute to the waiting cost or waiting limit (all nodes, all nodes except the origin, a subset of nodes…). The computational complexity of all cases is determined, providing complexity proofs for the variants that are NP-Hard and polynomial time algorithms for the variants that are in P.

We explore the benefits of multivariable branching schemes for linear-programming-based branch-and-bound algorithms for the 0-1 knapsack problem—that is, the benefits of branching on sets of variables rather than on a single variable (the current default in integer-programming solvers). We present examples where multivariable branching has advantages over single-variable branching and partially characterize situations in which this happens. Chvátal shows that for a specific class of 0-1 knapsack instances, a linear-programming-based branch-and-bound algorithm (employing a single-variable branching scheme) must explore exponentially many nodes. We show that for this class of 0-1 knapsack instances, a linear-programming-based branch-and-bound algorithm employing an appropriately chosen multivariable branching scheme explores either three or seven nodes. Finally, we investigate the performance of various multivariable branching schemes for 0-1 knapsack instances computationally and demonstrate their potential; the multivariable branching schemes explored result in smaller search trees (some in search trees that are an order of magnitude smaller), and some also result in shorter solution times. Summary of Contribution: As a powerful modeling tool, mixed-integer programming (MIP) is ubiquitous in Operations Research and is usually solved via the branch-and-bound framework. However, solving MIPs is computationally challenging in general, where branching affects the performance of solvers dramatically. In this paper, we explore the benefits of branching on multiple variables, which can be viewed as a generalization of the standard single-variable branching. We analyze its theoretical behavior on a special instance introduced by Chvátal, which is proved to be hard for single-variable branching. We also partially characterize situations in which branching on multiple variables is superior to its single-variable counterpart. Lastly, we demonstrate its potential in reducing the overall computational time and possible memory usage for storing unexplored nodes through numerical experiments on 0-1 knapsack problems.

Despite recent interest in multiobjective integer programming, few algorithms exist for solving biobjective mixed integer programs. We present such an algorithm: the boxed line method. For one of its variants, we prove that the number of single-objective integer programs solved is bounded by a linear function of the number of nondominated line segments in the nondominated frontier. This is the first such complexity result. An extensive computational study demonstrates that the box line method is also efficient in practice and that it outperforms existing algorithms on a diverse set of instances.

Abstract Rapid urban growth, the increasing importance of e‐commerce and high consumer service expectations have given rise to new and innovative models for freight delivery within urban environments. Crowdsourced solutions—where drivers are not employed by a carrier but occasionally offer their services through on‐line platforms and are contracted as required by carriers—are receiving growing attention from industry. We consider a crowdsourced system where drivers express their availability to perform delivery tasks for a given period of time and the platform communicates a schedule with requests to serve. We investigate the potential benefits of introducing transfers to support driver activities. At transfer locations, drivers can drop off packages for pick up by other drivers at a later time. We frame the problem as a multidepot pickup and delivery problem with time windows and transfers, and propose an adaptive large neighborhood search algorithm that effectively identifies beneficial transfer opportunities and synchronizes driver operations. Computational experiments indicate that introducing transfer options can significantly reduce system‐wide travel distance as well as the number of drivers required to serve a given set of requests, especially when drivers have short availability and requests have high service requirements.

We present the first criterion space search algorithm, the triangle splitting method, for finding all nondominated points of a biobjective mixed integer program. The algorithm is relatively easy to implement and converges quickly to the complete set of nondominated points. The algorithm maintains, at any point in time, a diverse set of nondominated points, and is thus ideally suited for fast approximation of the nondominated frontier. An extensive computational study demonstrates the efficacy of the triangle splitting method. Data, as supplemental material, are available at http://dx.doi.org/10.1287/ijoc.2015.0646 .

We present a new criterion space search algorithm, the balanced box method, for finding all nondominated points of a biobjective integer program. The method extends the box algorithm, is easy to implement, and converges quickly to the complete set of nondominated points. Because the method maintains, at any point in time, a diverse set of nondominated points, it is ideally suited for fast approximation of the efficient frontier. In addition, we present several enhancements of the well-known ε-constraint, augmented weighted Tchebycheff, and perpendicular search methods. An extensive computational study, using instances from different classes of combinatorial optimization problems, demonstrates the efficacy of the balanced box method.

We consider the variant of the shortest path problem in which a given set of paths is forbidden to occur as a subpath in an optimal path. We establish that the most-efficient algorithm for its solution, a dynamic programming algorithm, has polynomial time complexity; it had previously been conjectured that the algorithm has pseudo-polynomial time complexity. Furthermore, we show that this algorithm can be extended, without increasing its time complexity, to handle non elementary forbidden paths. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 63(3), 239–242 2014

We develop an exact algorithm for integer programs that uses restrictions of the problem to produce high-quality solutions quickly. Column generation is used both for generating these problem restrictions and for producing bounds on the value of the optimal solution. The performance of the algorithm is greatly enhanced by using structure, such as arises in network flow type applications, to help define the restrictions that are solved. In addition, local search around the current best solution is incorporated to enhance overall performance. The approach is parallelized and computational experiments on a classical problem in network design demonstrate its efficacy.

Consider a network 𝒩 =(N, A) and associate with each arc e ∈ A a fixed cost ce for using arc e, an interval [le, ue] (le, ue ∈ ℤ) specifying the range of allowable resource consumption quantities along arc e, and a per-unit cost [Formula: see text] for resource consumed along e. Furthermore, for each node n ∈ N, let Un ∈ ℤ be the maximum amount of resource consumption a path can accumulate before visiting node n. Given a source node ns ∈ N and sink node nt ∈ N, the fixed-charge shortest-path problem (FCSPP) seeks to find a minimum-cost-feasible path from ns to nt. When resource consumption is simply additive, the resource-constrained shortest-path problem (RCSPP) is a special case of FCSPP. We develop a new dynamic programming algorithm for FCSPP. The algorithm uses solutions from labeling and dominance techniques for standard RCSPPs on slightly modified problems, and it combines these solutions by exploiting the structure provided by certain classes of knapsack problems to efficiently construct an optimal solution to FCSPP. Computational experiments demonstrate that our algorithm is often several orders of magnitude faster than naive dynamic programming procedures.

We present a relaxation-based dynamic programming algorithm for solving resource-constrained shortest-path problems (RCSPPs) in the context of column generation for the dial-a-flight problem. The resulting network formulation and pricing problem require solving RCSPPs on extremely large time-expanded networks having a huge number of local resource constraints, i.e., constraints that apply to small subnetworks. The relaxation-based dynamic programming algorithm alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search space. Between consecutive searches, the relaxation is tightened using a set of critical resources and a set of critical arcs over which these resources are consumed. As a result, a relatively small state space is maintained, and many paths can be pruned while guaranteeing that an optimal path is ultimately found.

We design and implement an intensity-modulated radiation therapy plan generation technology that effectively and efficiently optimizes beam geometry as well as beam intensities. Our approach is based on an existing linear programming-based fluence map optimization model that approximates dose-volume requirements using conditional value-at-risk (C-VaR) constraints. We show how the parameters of the C-VaR constraints can be used to control various metrics of treatment plan quality. Next, we develop an automated search strategy for parameter tuning. Finally, beam angle selection is integrated with fluence map optimization. The beam angle selection scheme employs a bicriteria scoring of beam angle geometries and a selection mechanism to choose from among the set of nondominated geometries. The overall technology is automated and generates several high-quality treatment plans satisfying dose prescription requirements in a single invocation and without human guidance. The technology has been tested on various real-patient cases with uniform success.

We develop a solution approach for the fixed-charge network flow (FCNF) problem that produces provably high-quality solutions quickly. The solution approach combines mathematical programming algorithms with heuristic search techniques. To obtain high-quality solutions, it relies on neighborhood search with neighborhoods that involve solving carefully chosen integer programs derived from the arc-based formulation of FCNF. To obtain lower bounds, the linear programming relaxation of the path-based formulation of FCNF is used and strengthened with cuts discovered during the neighborhood search. The solution approach incorporates randomization to diversify the search and learning to intensify the search. Computational experiments demonstrate the efficacy of the proposed approach.

Abstract We propose a practical and flexible fixed routing system that preserves many of the benefits of traditional fixed routes but can be deployed in settings with medium to high variability and delivery time window constraints. The key ideas are the introduction of a new recourse strategy, in which customers are assigned to two planned routes, a primary and a backup, and recourse decisions can move customers to backup routes to regain feasibility or improve costs, and the use of sampling‐based techniques to handle the presence of delivery time windows during the construction of primary and backup routes. A computational study based on real‐life data demonstrates the efficacy of the proposed fixed routing system and the route construction techniques. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

Abstract We study a dynamic multiperiod routing problem where, at the beginning of each time period, a set of orders arrive that have to be fulfilled either that time period or the next. Thus, in each time period there are customers that have to be served and customers whose service may be postponed. Once it has been decided which customers to serve, an optimal route is constructed and executed. The objective of the problem is to minimize the total distance traveled during the planning horizon. Deciding which customers to serve in a time period is done on the basis of incomplete information, analyzing simultaneously customers in two consecutive periods. No knowledge is available about customers requiring service in future time periods. We introduce simple algorithms, ones which naturally arise in practice, and analyze these algorithms by studying their competitive ratio. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(4), 308–317 2007

Recently there has been much progress on the design of approximation algorithms for a variety of scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled. Many of these approximation algorithms have been inspired by polyhedral formulations of the scheduling problems and their use in computing optimal solutions to small instances. In this paper we demonstrate that the progress in the design and analysis of approximation algorithms for these problems also yields techniques with improved computational efficacy. Specifically, we give a comprehensive experimental study of a number of these approximation algorithms for 1|rj∑wjCj, the problem of scheduling jobs with release dates on one machine so as to minimize the average weighted completion time of the jobs scheduled. We study both the quality of lower bounds given for this problem by different linear-programming relaxations and combinatorial relaxations, and the quality of upper bounds delivered by a number of approximation algorithms based on them. The best algorithms, on almost all instances, come within a few percent of the optimal average weighted completion time. Furthermore, we show that this can usually be achieved with O(n log n) computation. In addition we observe that on most kinds of synthetic data used in experimental studies a simple greedy heuristic, used in successful combinatorial branch-and-bound algorithms for the problem, outperforms (on average) all of the LP-based heuristics. We identify, however, other classes of problems on which the LP-based heuristics are superior and report on experiments that give a qualitative sense of the range of dominance of each. We consider the impact of local improvement on the solutions as well. We also consider the performance of the algorithms for the average weighted flow-time criterion, which, although equivalent to average weighted completion time at optimality, is provably much harder to approximate. Nonetheless, we demonstrate that for most instances we consider that the algorithms give very good results for this criterion as well. Finally, we extend the techniques to a rather different and more complex problem that arises from an actual manufacturing application: resource-constrained project scheduling. In this setting as well, the techniques yield algorithms with improved performance; we give the best-known solutions for a set of instances provided by BASF AG, Germany.

We consider a very general online scheduling problem with an objective to minimize the maximum level of resource allocated. We find a simple characterization of an optimal deterministic online algorithm. We develop further results for the two, more specific problems of single resource scheduling and hierarchical line balancing. We determine how to compute optimal online algorithms for both problems using linear programming and integer programming, respectively. We show that randomized algorithms can outperform deterministic algorithms, but only if the amount of work done is a nonconcave function of resource allocation.

We describe a parallel, linear programming and implication-based heuristic for solving set partitioning problems on distributed memory computer architectures. Our implementation is carefully designed to exploit parallelism to greatest advantage in advanced techniques like preprocessing and probing, primal heuristics, and cut generation. A primaldual subproblem simplex method is used for solving the linear programming relaxation, which breaks the linear programming solution process into natural phases from which we can exploit information to find good solutions on the various processors. Implications from the probing operation are shared among the processors. Combining these techniques allows us to obtain solutions to large and difficult problems in a reasonable amount of computing time.

Time-indexed formulations for machine scheduling problems have received a great deal of attention; not only do the linear programming relaxations provide strong lower bounds, but they are good guides for approximation algorithms as well. Unfortunately, time-indexed formulations have one major disadvantage—their size. Even for relatively small instances the number of constraints and the number of variables can be large. In this paper, we discuss how Dantzig-Wolfe decomposition techniques can be applied to alleviate, at least partly, the difficulties associated with the size of time-indexed formulations. In addition, we show that the application of these techniques still allows the use of cut generation techniques.

This paper is about modeling and solving mixed integer programming (MIP) problems. In the last decade, the use of mixed integer programming models has increased dramatically. Fifteen years ago, mainframe computers were required to solve problems with a hundred integer variables. Now it is possible to solve problems with thousands of integer variables on a personal computer and obtain provably good approximate solutions to problems such as set partitioning with millions of binary variables. These advances have been made possible by developments in modeling, algorithms, software, and hardware. This paper focuses on effective modeling, preprocessing, and the methodologies of branch-and-cut and branch-and-price, which are the techniques that make it possible to treat problems with either a very large number of constraints or a very large number of variables. We show how these techniques are useful in important application areas such as network design and crew scheduling. Finally, we discuss the relatively new research areas of parallel integer programming and stochastic integer programming.

We investigate several complexity issues related to branch-and-cut algorithms for 0-1 integer programming based on lifted-cover inequalities (LCIs). We show that given a fractional point, determining a violated LCI over all minimal covers is NP-hard. The main result is that there exists a class of 0-1 knapsack instances for which any branch-and-cut algorithm based on LCIs has to evaluate an exponential number of nodes to prove optimality.

The branch-and-bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations has been used with great success for decades. Over the years, a variety of researchers have studied ways of making the basic algorithm more effective. Breakthroughs in the fields of computer hardware, computer software, and mathematics have led to increasing success at solving larger and larger MIP instances. The goal of this article is to survey many of the results regarding branch-and-bound search strategies and evaluate them again in light of the other advances that have taken place over the years. In addition, novel search strategies are presented and shown to often perform better than those currently used in practice.

We investigate the algorithmic and implementation issues related to the effective and efficient use of lifted cover inequalities and lifted GUB cover inequalities in a branch and cut algorithm for 0-1 integer programming. We have tried various strategies on several test problems and we identify the best ones for use in practice.

In the first part of the paper, we present a framework for describing basic techniques to improve the representation of a mixed integer programming problem. We elaborate on identification of infeasibility and redundancy, improvement of bounds and coefficients, and fixing of binary variables. In the second part of the paper, we discuss recent extensions to these basic techniques and elaborate on the investigation and possible uses of logical consequences. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

We investigate the implementation of edge-exchange improvement methods for the vehicle routing problem with time windows with minimization of route duration as the objective. The presence of time windows as well as the chosen objective cause verification of the feasibility and profitability of a single edge-exchange to require an amount of computing time that is linear in the number of vertices. We show how this effort can, on the average, be reduced to a constant. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.