Tabu search for solving the black-and-white travelling salesman problem

The black-and-white travelling salesman problem (BWTSP) is an extension to the well-known TSP by partitioning the set of vertices into black and white vertices, and imposing cardinality and length constraints between two consecutive black vertices in a Hamiltonian tour. BWTSP has various applications in aircraft routing, telecommunication network design and logistics. In this paper, we develop several tabu search (TS) heuristics for solving the BWTSP. Our TS is built upon a new efficient neighbourhood structure, which exploits both the permutation and knapsack features of BWTSP. We also embed our TS as a heuristic procedure to improve the upper bound in a mixed-integer linear programming method. Extensive computational experiment on both benchmark and randomly generated instances shows effectiveness and efficiency of our algorithms. Our algorithms are able to obtain optimal and near optimal solutions to small instances in seconds, and find feasible solutions to large instances that have not been solved by the existing methods in the literature.     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

The travelling salesman problem: new solvable cases and linkages with the development of approximation algorithms

We identify new solvable cases of the travelling salesman problem (TSP) by an indirect analysis that has useful consequences. First, we develop new procedures for the TSP that require only linear time to execute and yield TSP tours that are better than an exponential number of alternative tours. We then identify special subgraphs, easily generated, so that our method yields these outcomes for every instance of these subgraphs. Finally, when the associated costs satisfy prescribed conditions, we show the solutions produced by these algorithms are optimal and thus we have new solvable cases of the TSP. Besides possible practical applications to problems that may exhibit these cost conditions, our algorithms may also be applied as subroutines within more complex metaheuristics. Our methods extend in a natural way to bottleneck TSP formulations, and their underlying results raise new theoretical questions about the analysis of heuristics for hard combinatorial problems.     

Two-level evolutionary approach to the survivable mesh-based transport network topological design

The complete topology design problem of survivable mesh-based transport networks is to address simultaneously design of network topology, working path routing, and spare capacity allocation based on span-restoration. Each constituent problem in the complete design problem could be formulated as an Integer Programming (IP) and is proved to be NP\mathcal{NP} -hard. Due to a large amount of decision variables and constraints involved in the IP formulation, to solve the problem directly by exact algorithms (e.g. branch-and-bound) would be impractical if not impossible. In this paper, we present a two-level evolutionary approach to address the complete topology design problem. In the low-level, two parameterized greedy heuristics are developed to jointly construct feasible solutions (i.e., closed graph topologies satisfying all the mesh-based network survivable constraints) of the complete problem. Unlike existing “zoom-in”-based heuristics in which subsets of the constraints are considered, the proposed heuristics take all constraints into account. An estimation of distribution algorithm works on the top of the heuristics to tune the control parameters. As a result, optimal solution to the considered problem is more likely to be constructed from the heuristics with the optimal control parameters. The proposed algorithm is evaluated experimentally in comparison with the latest heuristics based on the IP software CPLEX, and the “zoom-in”-based approach on 28 test networks problems. The experimental results demonstrate that the proposed algorithm is more effective in finding high-quality topologies than the IP-based heuristic algorithm in 21 out of 28 test instances with much less computational costs, and performs significantly better than the “zoom-in”-based approach in 19 instances with the same computational costs.     

Construction heuristics for the asymmetric TSP

Non-Euclidean traveling salesman problem (TSP) construction heuristics, and especially asymmetric TSP construction heuristics, have been neglected in the literature by comparison with the extensive efforts devoted to studying Euclidean TSP construction heuristics. This state of affairs is at odds with the fact that asymmetric models are relevant to a wider range of applications, and indeed are uniformly more general that symmetric models. Moreover, common construction approaches for the Euclidean TSP have been shown to produce poor quality solutions for non-Euclidean instances. Motivation for remedying this gap in the study of construction approaches is increased by the fact that such methods are a great deal faster than other TSP heuristics, which can be important for real time problems requiring continuously updated response. The purpose of this paper is to describe two new construction heuristics for the asymmetric TSP and a third heuristic based on combining the other two. Extensive computational experiments are performed for several different families of TSP instances, disclosing that our combined heuristic clearly outperforms well-known TSP construction methods and proves significantly more robust in obtaining (relatively) high quality solutions over a wide range of problems.     

On the asymptotic behavior of subtour-patching heuristics in solving the TSP on permuted Monge matrices

We examine the performance of different subtour-patching heuristics for solving the strongly NP\mathcal{NP}-hard traveling salesman problem (TSP) on permuted Monge matrices. We prove that a well-known heuristic is asymptotically optimal for the TSP on product matrices and k-root cost matrices. We also show that the heuristic is provably asymptotically optimal for general permuted Monge matrices under some mild conditions. Our theoretical results are strongly supported by the findings of a large-scale experimental study on randomly generated numerical examples, which show that the heuristic is not only asymptotically optimal, but also finds optimal TSP tours with high probability that increases with the problem size. Thus the heuristic represents a practical tool to solve large instances of the problem.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

Multiobjective traveling salesperson problem on Halin graphs

In this paper, we study traveling salesperson (TSP) and bottleneck traveling salesperson (BTSP) problems on special graphs called Halin graphs. Although both problems are NP-Hard on general graphs, they are polynomially solvable on Halin graphs. We address the multiobjective versions of these problems. We show computational complexities of finding a single nondominated point as well as finding all nondominated points for different objective function combinations. We develop algorithms for the polynomially solvable combinations.     

A variable neighborhood search for the network design problem with relays

Given a set of commodities to be routed over a network, the network design problem with relays involves selecting a route for each commodity and determining the location of relays where the commodities must be reprocessed at certain distance intervals. We propose a hybrid approach based on variable neighborhood search. The variable neighborhood algorithm searches for the route for each commodity and the optimal relay locations for a given set of routes are determined by an implicit enumeration algorithm. We show that dynamic programming can be used to determine the optimal relay locations for a single commodity. Dynamic programming is embedded into the implicit enumeration algorithm to solve the relay location problem optimally for multiple commodities. The special structure of the problem is leveraged for computational efficiency. In the variable neighborhood search algorithm, the routes of the current solution are perturbed and reconstructed to generate neighbor solutions using random and greedy construction heuristics. Computational experiments on three sets of problems (80 instances) show that the variable neighborhood search algorithm with optimal relay allocations outperforms all existing algorithms in the literature.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’

Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.     

A benchmark library and a comparison of heuristic methods for the linear ordering problem

The linear ordering problem consists of finding an acyclic tournament in a complete weighted digraph of maximum weight. It is one of the classical NP-hard combinatorial optimization problems. This paper surveys a collection of heuristics and metaheuristic algorithms for finding near-optimal solutions and reports about extensive computational experiments with them. We also present the new benchmark library LOLIB which includes all instances previously used for this problem as well as new ones.     

An efficient genetic algorithm for single row facility layout

The single row facility layout is the NP-Hard problem of arranging facilities with given lengths on a line, so as to minimize the weighted sum of the distances between all pairs of facilities. Owing to its computational complexity, researchers have developed several heuristics to obtain good quality solutions. In this paper, we present a genetic algorithm called GENALGO to solve large single row facility layout problem instances. Our algorithm uses standard genetic operators and periodically improves the fitness of all individuals. Our computational experiments show that our genetic algorithm yields high quality solutions in spite of starting with an initial population that is randomly generated. Our algorithm improves the previously best known solutions for the 19 instances of 58 benchmark instances and is competitive for most of the remaining ones.     

A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem

We present a variable neighborhood search approach for solving the one-commodity pickup-and-delivery travelling salesman problem. It is characterized by a set of customers such that each of the customers either supplies (pickup customers) or demands (delivery customers) a given amount of a single product, and by a vehicle, whose given capacity must not be exceeded, that starts at the depot and must visit each customer only once. The objective is to minimize the total length of the tour. Thus, the considered problem includes checking the existence of a feasible travelling salesman’s tour and designing the optimal travelling salesman’s tour, which are both NP-hard problems. We adapt a collection of neighborhood structures, k-opt, double-bridge and insertion operators mainly used for solving the classical travelling salesman problem. A binary indexed tree data structure is used, which enables efficient feasibility checking and updating of solutions in these neighborhoods. Our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best-known algorithms in terms of both the solution quality and computational efforts. Moreover, we improve the best-known solutions of all benchmark instances from the literature (with 200 to 500 customers). We are also able to solve instances with up to 1000 customers.     

Solving the pallet loading problem

This paper presents new bounds, heuristics, and an exact algorithm for the Pallet Loading Problem (PLP). PLP maximizes the number of boxes placed on a rectangular pallet. All boxes have identical rectangular dimensions and, when placed, must be located completely within the pallet. Boxes may be rotated 90° so long as they are placed with edges parallel to the pallet’s edges. The set of all PLP instances with an area ratio (pallet area divided by box area) less than 101 boxes can be represented by 3,080,730 equivalent classes. Our G5-heuristic finds optimal solutions to 3,073,724 of these 3,080,730 classes and in the remaining 7006 classes only differs from the best known bound by one box. We develop three other heuristics that solve another 54 instances. Finally, we solve the 6952 remaining classes with our exact HVZ algorithm. Only a subset of these classes has been solved previously.     

Pareto memetic algorithm for multiple objective optimization with an industrial application

Multiple objective combinatorial optimization problems are difficult to solve and often, exact algorithms are unable to produce optimal solutions. The development of multiple objective heuristics was inspired by the need to quickly produce acceptable solutions. In this paper, we present a new multiple objective Pareto memetic algorithm called PMS^MO. The PMS^MO algorithm incorporates an enhanced fine-grained fitness assignment, a double level archiving process and a local search procedure to improve performance. The performance of PMS^MO is benchmarked against state-of-the-art algorithms using 0–1 multi-dimensional multiple objective knapsack problem from the literature and an industrial scheduling problem from the aluminum industry.     

Exact algorithms for the traveling salesman problem with draft limits

This paper deals with the Traveling Salesman Problem (TSP) with Draft Limits (TSPDL), which is a variant of the well-known TSP in the context of maritime transportation. In this recently proposed problem, draft limits are imposed due to restrictions on the port infrastructures. Exact algorithms based on three mathematical formulations are proposed and their performance compared through extensive computational experiments. Optimal solutions are reported for open instances of benchmark problems available in the literature.     

An MILP for scheduling problems in an FMS with one vehicle

This paper concerns the mathematical formulation and optimal solutions for the Flexible Manufacturing Systems Scheduling Problem (FMSSP) with one vehicle. This linear formulation differs from the previously published ones as it takes into account the maximum number of jobs allowed in the system, limited input/output buffer capacities, empty vehicle trips and no-move-ahead trips simultaneously. Our objective is to propose optimal solutions for small and medium-sized instances and to examine a number of commonly used assumptions and heuristics. Computational experiments are carried out on instances adapted from Bilge and Ulusoy [Bilge, Ü., Ulusoy, G., 1995. A time window approach to simultaneous scheduling of machines and material handling system in an FMS. Operations Research 43, 1058–1070] and the following heuristics are evaluated: FIFO (First In First Out) rules for input/output buffer management; and FIFO, SPT (Shortest Processing Time), STT (Shortest Travel Time) and MOQS (Maximum Outgoing Queue Size) rules concerning the vehicle. The consequences of classical assumptions are also studied: ignoring empty trips, ignoring no-move-ahead constraints, and ignoring vehicle-disjunction constraints. The numerical experiments provide a set of optimal solutions and allow to evaluate the performances of heuristic search schemes.     

Multi-armed bandit-based hyper-heuristics for combinatorial optimization problems

There are significant research opportunities in the integration of Machine Learning (ML) methods and Combinatorial Optimization Problems (COPs). In this work, we focus on metaheuristics to solve COPs that have an important learning component. These algorithms must explore a solution space and learn from the information they obtain in order to find high-quality solutions. Among the metaheuristics, we study Hyper-Heuristics (HHs), algorithms that, given a number of low-level heuristics, iteratively select and apply heuristics to a solution. The HH we consider has a Markov model to produce sequences of low-level heuristics, which we combine with a Multi-Armed Bandit Problem (MAB)-based method to learn its parameters. This work proposes several improvements to the HH metaheuristic that yields a better learning for solving problem instances. Specifically, this is the first work in HHs to present Exponential Weights for Exploration and Exploitation (EXP3) as a learning method, an algorithm that is able to deal with adversarial settings. We also present a case study for the Vehicle Routing Problem with Time Windows (VRPTW), for which we include a list of low-level heuristics that have been proposed in the literature. We show that our algorithms can handle a large and diverse list of heuristics, illustrating that they can be easily configured to solve COPs of different nature. The computational results indicate that our algorithms are competitive methods for the VRPTW (2.16% gap on average with respect to the best known solutions), demonstrating the potential of these algorithms to solve COPs. Finally, we show how algorithms can even detect low-level heuristics that do not contribute to finding better solutions to the problem.     

Algorithms for common due-date assignment and sequencing on a single machine with sequence-dependent setup times

This paper focuses on the single machine sequencing and common due-date assignment problem for the objective of minimizing the sum of the penalties associated with earliness, tardiness and due-date assignment. Unlike the previous research articles on this class of scheduling problem, we consider sequence-dependent setup times that make the problem much more difficult. To solve the problem, a branch and bound algorithm, which incorporates the method to obtain lower and upper bounds as well as a dominance property to reduce the search space, is suggested that gives the optimal solutions for small-sized instances. Heuristic algorithms are suggested to obtain solutions for large-sized problems within a reasonable computation time. The performances of both the optimal and heuristic algorithms, in computational experiments on randomly generated test instances, are reported.