New tighter polynomial length formulations for the asymmetric traveling salesman problem with and without precedence constraints

We propose a new formulation for the asymmetric traveling salesman problem, with and without precedence relationships, which employs a polynomial number of subtour elimination constraints that imply an exponential subset of certain relaxed Dantzig-Fulkerson-Johnson subtour constraints. Promising computational results are presented, particularly in the presence of precedence constraints.     

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.     

Arc routing problems with time-dependent service costs

This paper studies an arc routing problem with capacity constraints and time-dependent service costs. This problem is motivated by winter gritting applications where the “timing” of each intervention is crucial. The exact problem-solving approach reported here first transforms the arc routing problem into an equivalent node routing problem. Then, a column generation scheme is used to solve the latter. The master problem is a classical set covering problem, while the subproblems are time-dependent shortest path problems with resource constraints. These subproblems are solved using an extension of a previously developed algorithm. Computational results are reported on problems derived from a set of classical instances of the vehicle routing problem with time windows.     

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).     

Exact algorithms for the chance-constrained vehicle routing problem

We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for the VRP with stochastic demands require independent demands. We first study an edge-based formulation for the CCVRP, in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers. We then investigate the use of a branch-and-cut-and-price (BCP) algorithm. While BCP algorithms have been considered the state of the art in solving the deterministic VRP, few attempts have been made to extend this framework to the VRP with stochastic demands. In contrast to the deterministic VRP, we find that the pricing problem for the CCVRP problem is strongly \(\mathcal {NP}\)-hard, even when the routes being priced are allowed to have cycles. We therefore propose a further relaxation of the routes that enables pricing via dynamic programming. We also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust CCVRP problem. Numerical results indicate that the proposed methods can solve instances of CCVRP having up to 55 vertices.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

A Semidefinite Programming Heuristic for Quadratic Programming Problems with Complementarity Constraints

The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic programming problem is solved for each of these feasible solutions and the best resulting solution provides an estimate for the optimal solution to the quadratic program with complementarity constraints. Computational testing of such an approach is described for a problem arising in portfolio optimization.Research supported in part by the National Science Foundations VIGRE Program (Grant DMS-9983646).Research partially supported by NSF Grant number CCR-9901822.     

Un modèle ES–BGK pour des mélanges de gaz

In this note, we derive an ES–BGK model for gas mixtures that is able to give the correct Prandtl number obtained from the true Boltzmann equation. The derivation principle is based on the resolution of an entropy minimisation problem under moments constraints. The set of constraints is constructed by introducing a relaxation of some non-conserved moments. The non-conserved moments are dissipated according to some relaxation rates. Finally the BGK model that is obtained satisfies the classical properties of the Boltzmann operator for inert gas mixtures.     

Saving-based algorithms for vehicle routing problem with simultaneous pickup and delivery

The vehicle routing problem (VRP) with simultaneous pickup and delivery (VRPSPD) is an extension of the classical capacitated VRP (CVRP). In this paper, we present the saving heuristic and the parallel saving heuristic for VRPSPD. Checking the feasibility of a route in VRPSPD is difficult because of the fluctuating load on the route. In the saving heuristic, a new route is created by merging the two existing routes. We use a cumulative net-pickup approach for checking the feasibility when two existing routes are merged. The numerical results show that the performance of the proposed heuristics is qualitatively better than the existing insertion-based heuristics.     

Relaxation in the optimal control problem for the Goursat-Darboux system

We consider the optimal control problem for a system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls satisfying mixed nonconvex constraints. For this problem we prove an analog of the classical Bogolyubov relaxation theorem.     

Semidefinite relaxations of dynamical programs under discrete constraints

In this work we propose an exact semidefinite relaxation for non-linear, non-convex dynamical programs under discrete constraints in the state variables and the control variables. We outline some theoretical features of the method and workout the solutions of a benchmark problem in cybernetics and the classical inventory problem under discrete constraints.     

Relaxation Through Homogenization for Optimal Design Problems with Gradient Constraints

The problem of the relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of either a finite or infinite number of constraints. The relaxed formulation is used to identify minimizing sequences of configurations of phases.     

Mathematical Programs with Vanishing Constraints: Optimality Conditions,Sensitivity, and a Relaxation Method

We consider a class of optimization problems with switch-off/switch-on constraints, which is a relatively new problem model. The specificity of this model is that it contains constraints that are being imposed (switched on) at some points of the feasible region, while being disregarded (switched off) at other points. This seems to be a potentially useful modeling paradigm, that has been shown to be helpful, for example, in optimal topology design. The fact that some constraints “vanish” from the problem at certain points, gave rise to the name of mathematical programs with vanishing constraints (MPVC). It turns out that such problems are usually degenerate at a solution, but are structurally different from the related class of mathematical programs with complementarity constraints (MPCC). In this paper, we first discuss some known first- and second-order necessary optimality conditions for MPVC, giving new very short and direct justifications. We then derive some new special second-order sufficient optimality conditions for these problems and show that, quite remarkably, these conditions are actually equivalent to the classical/standard second-order sufficient conditions in optimization. We also provide a sensitivity analysis for MPVC. Finally, a relaxation method is proposed. For this method, we analyze constraints regularity and boundedness of the Lagrange multipliers in the relaxed subproblems, derive a sufficient condition for local uniqueness of solutions of subproblems, and give convergence estimates. Research of the first author was supported by the Russian Foundation for Basic Research Grants 07-01-00270, 07-01-00416 and 07-01-90102-Mong, and by RF President’s Grant NS-9344.2006.1 for the support of leading scientific schools. The second author was supported in part by CNPq Grants 301508/2005-4, 490200/2005-2 and 550317/2005-8, by PRONEX-Optimization, and by FAPERJ.     

A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.     

A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows

We propose lifted versions of the Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows (TW). The constraints are valid for problems such as the travelling salesman problem with TW, the vehicle routing problem with TW, the generalized travelling salesman problem with TW, and the general vehicle routing problem with TW. They are corrected versions of the constraints proposed by Desrochers and Laporte (1991).     

A single machine scheduling problem with availability constraints and sequence-dependent setup costs

We study a single machine scheduling problem with availability constraints and sequence-dependent setup costs, with the aim of minimizing the makespan. To the authors’ knowledge, this problem has not been treated as such in the operations research literature. We derive in this paper a mixed integer programming model to deal with such scheduling problem. Computational tests showed that commercial solvers are capable of solving only small instances of the problem. Therefore, we propose two ways for reducing the execution time, namely a valid inequality that strengthen the linear relaxation and an efficient heuristic procedure that provides a starting feasible solution to the solver. A substantial gain is achieved both in terms of the linear programming relaxation bound and in terms of the time to obtain an integer optimum when we use the enhanced model in conjunction with providing to the solver the solution obtained by the proposed heuristic.     

A library of local search heuristics for the vehicle routing problem

The vehicle routing problem (VRP) is a difficult and well-studied combinatorial optimization problem. Real-world instances of the VRP can contain hundreds and even thousands of customer locations and can involve many complicating constraints, necessitating the use of heuristic methods. We present a software library of local search heuristics that allows one to quickly generate solutions to VRP instances. The code has a logical, object-oriented design and uses efficient data structures to store and modify solutions. The core of the library is the implementation of seven local search operators that share a similar interface and are designed to be extended to handle additional options with minimal code change. The code is well-documented, straightforward to compile, and is freely available online. The code contains several applications that can be used to generate solutions to the capacitated VRP. Computational results indicate that these applications are able to generate solutions that are within about one percent of the best-known solution on benchmark problems.     

Unrelated machine scheduling with time-window and machine downtime constraints: An application to a naval battle-group problem

This paper deals with an unrelated machine scheduling problem of minimizing the total weighted flow time, subject to time-window job availability and machine downtime side constraints. We present a zero-one integer programming formulation of this problem. The linear programming relaxation of this formulation affords a tight lower bound and often generates an integer optimal solution for the problem. By exploiting the special structures inherent in the formulation, we develop some classes of strong valid inequalities that can be used to tighten the initial formulation, as well as to provide cutting planes in the context of a branch-and-cut procedure. A major computational bottleneck is the solution of the underlying linear programming relaxation because of the extremely high degree of degeneracy inherent in the formulation. In order to overcome this difficulty, we employ a Lagrangian dual formulation to generate lower and upper bounds and to drive the branch-and-bound algorithm. As a practical instance of the unrelated machine scheduling problem, we describe a combinatorial naval defense problem. This problem seeks to schedule a set of illuminators (passive homing devices) in order to strike a given set of targets using surface-to-air missiles in a naval battle-group engagement scenario. We present computational results for this problem using suitable realistic data.     

Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem

Robust design optimization (RDO) problems can generally be formulated by incorporating uncertainty into the corresponding deterministic problems. In this context, a careful formulation of deterministic equality constraints into the robust domain is necessary to avoid infeasible designs under uncertain conditions. The challenge of formulating equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is itself a complex task. A judicious formulation of equality constraints adds to this complexity because additional tradeoffs are introduced between constraint satisfaction under uncertainty and multiobjective performance. Equality constraints under uncertainty in multiobjective problems can therefore pose a complicated decision making problem. In this paper, we provide a new problem formulation that can be used as an effective multiobjective decision making tool, with emphasis on equality constraints. We present two numerical examples to illustrate our theoretical developments.