Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.

In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.

Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs

The asymmetric travelling salesman problem with replenishment arcs (RATSP), arising from work related to aircraft routing, is a generalisation of the well-known ATSP. In this paper, we introduce a polynomial size mixed-integer linear programming (MILP) formulation for the RATSP, and improve an existing exponential size ILP formulation of Zhu [The aircraft rotation problem, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, 1994] by proposing two classes of stronger cuts. We present results that under certain conditions, these two classes of stronger cuts are facet-defining for the RATS polytope, and that ATSP facets can be lifted, to give RATSP facets. We implement our polyhedral findings and develop a Lagrangean relaxation (LR)-based branch-and-bound (BNB) algorithm for the RATSP, and compare this method with solving the polynomial size formulation using ILOG Cplex 9.0, using both randomly generated problems and aircraft routing problems. Finally we compare our methods with the existing method of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408–427]. It turns out that both of our methods are much faster than that of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408–427], and that the LR-based BNB method is more efficient for problems that resemble the aircraft rotation problems.

A novel Physarum-Based ant colony system for solving the real-world traveling salesman problem

The solutions to Traveling Salesman Problem can be widely applied in many real-world problems. Ant colony optimization algorithms can provide an approximate solution to a Traveling Salesman Problem. However, most ant colony optimization algorithms suffer premature convergence and low convergence rate. With these observations in mind, a novel ant colony system is proposed, which employs the unique feature of critical tubes reserved in the Physaurm-inspired mathematical model. A series of experiments are conducted, which are consolidated by two realworld Traveling Salesman Problems. The experimental results show that the proposed new ant colony system outperforms classical ant colony system, genetic algorithm, and particle swarm optimization algorithm in efficiency and robustness.

On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.

A modified Intelligent Water Drops algorithm and its application to optimization problems

The Intelligent Water Drop (IWD) algorithm is a recent stochastic swarm-based method that is useful for solving combinatorial and function optimization problems. In this paper, we investigate the effectiveness of the selection method in the solution construction phase of the IWD algorithm. Instead of the fitness proportionate selection method in the original IWD algorithm, two ranking-based selection methods, namely linear ranking and exponential ranking, are proposed. Both ranking-based selection methods aim to solve the identified limitations of the fitness proportionate selection method as well as to enable the IWD algorithm to escape from local optima and ensure its search diversity. To evaluate the usefulness of the proposed ranking-based selection methods, a series of experiments pertaining to three combinatorial optimization problems, i.e., rough set feature subset selection, multiple knapsack and travelling salesman problems, is conducted. The results demonstrate that the exponential ranking selection method is able to preserve the search diversity, therefore improving the performance of the IWD algorithm. © 2014 Elsevier Ltd. All rights reserved.

A universal optimization strategy for ant colony optimization algorithms based on the Physarum-inspired mathematical model

Ant colony optimization (ACO) algorithms often fall into the local optimal solution and have lower search efficiency for solving the travelling salesman problem (TSP). According to these shortcomings, this paper proposes a universal optimization strategy for updating the pheromone matrix in the ACO algorithms. The new optimization strategy takes advantages of the unique feature of critical paths reserved in the process of evolving adaptive networks of the Physarum-inspired mathematical model (PMM). The optimized algorithms, denoted as PMACO algorithms, can enhance the amount of pheromone in the critical paths and promote the exploitation of the optimal solution. Experimental results in synthetic and real networks show that the PMACO algorithms are more efficient and robust than the traditional ACO algorithms, which are adaptable to solve the TSP with single or multiple objectives. Meanwhile, we further analyse the influence of parameters on the performance of the PMACO algorithms. Based on these analyses, the best values of these parameters are worked out for the TSP.

An ensemble of intelligent water drop algorithms and its application to optimization problems

Crown Copyright © 2015 Published by Elsevier Inc. All rights reserved. The Intelligent Water Drop (IWD) algorithm is a recent stochastic swarm-based method that is useful for solving combinatorial and function optimization problems. In this paper, we propose an IWD ensemble known as the Master-River, Multiple-Creek IWD (MRMC-IWD) model, which serves as an extension of the modified IWD algorithm. The MRMC-IWD model aims to improve the exploration capability of the modified IWD algorithm. It comprises a master river which cooperates with multiple independent creeks to undertake optimization problems based on the divide-and-conquer strategy. A technique to decompose the original problem into a number of sub-problems is first devised. Each sub-problem is then assigned to a creek, while the overall solution is handled by the master river. To empower the exploitation capability, a hybrid MRMC-IWD model is introduced. It integrates the iterative improvement local search method with the MRMC-IWD model to allow a local search to be conducted, therefore enhancing the quality of solutions provided by the master river. To evaluate the effectiveness of the proposed models, a series of experiments pertaining to two combinatorial problems, i.e., the travelling salesman problem (TSP) and rough set feature subset selection (RSFS), are conducted. The results indicate that the MRMC-IWD model can satisfactorily solve optimization problems using the divide-and-conquer strategy. By incorporating a local search method, the resulting hybrid MRMC-IWD model not only is able to balance exploration and exploitation, but also to enable convergence towards the optimal solutions, by employing a local search method. In all seven selected TSPLIB problems, the hybrid MRMC-IWD model achieves good results, with an average deviation of 0.021% from the best known optimal tour lengths. Compared with other state-of-the-art methods, the hybrid MRMC-IWD model produces the best results (i.e. the shortest and uniform reducts of 20 runs) for all13 selected RSFS problems.

New cutting-planes for the time-and/or precedence-constrained ATSP and directed VRP

In this paper, we introduce five classes of new valid cutting planes for the precedence-constrained (PC) and/or time-window-constrained (TW) Asymmetric Travelling Salesman Problems (ATSPs) and directed Vehicle Routing Problems (VRPs). We show that all five classes of new inequalities are facet-defining for the directed VRP-TW, under reasonable conditions and the assumption that vehicles are identical. Similar proofs can be developed for the VRP-PC. As ATSP-TW and PC-ATSP can be formulated as directed identical-vehicle VRP-TW and PC-VRP, respectively, this provides a link to study the polyhedral combinatorics for the ATSP-TW and PC-ATSP. The first four classes of these new cutting planes are cycle-breaking inequalities that are lifted from the well-known D^-_k and D⁺_k inequalities (see Grötschel and Padberg in Polyhedral theory. The traveling salesman problem: a guided tour of combinatorial optimization, Wiley, New York, 1985). The last class of new cutting planes, the TW ₂ inequalities, are infeasible-path elimination inequalities. Separation of these constraints will also be discussed. We also present prelimanry numerical results to demonstrate the strengh of these new cutting planes.

Using autonomous mobile agents for efficient data collection in sensor networks

Sensor networks are emerging as the new frontier in sensing technology, however there are still issues that need to be addressed. Two such issues are data collection and energy conservation. We consider a mobile robot, or a mobile agent, traveling the network collecting information from the sensors themselves before their onboard memory storage buffers are full. A novel algorithm is presented that is an adaptation of a local search algorithm for a special case of the Asymmetric Traveling Salesman Problem with Time-windows (ATSPTW) for solving the dynamic scheduling problem of what nodes are to be visited so that the information collected is not lost. Our algorithms are given and compared to other work.