A random-key genetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem is a variation of the well-known traveling salesman problem in which the set of nodes is divided into clusters; the objective is to find a minimum-cost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.     

Solving the Probabilistic TSP with Ant Colony Optimization

In this paper, we describe new ways to apply Ant Colony Optimization (ACO) to the Probabilistic Traveling Salesperson Problem (PTSP). PTSP is a stochastic extension of the well known Traveling Salesperson Problem (TSP), where each customer will require a visit only with a certain probability. The goal is to find an a priori tour visiting all customers with minimum expected length, customers not requiring a visit simply being skipped in the tour.We show that ACO works well even when only an approximative evaluation function is used, which speeds up the algorithm, leaving more time for the actual construction. As we demonstrate, this idea can also be applied successfully to other state-of-the-art heuristics. Furthermore, we present new heuristic guidance schemes for ACO, better adapted to the PTSP than what has been used previously. We show that these modifications lead to significant improvements over the standard ACO algorithm, and that the resulting ACO is at least competitive to other state-of-the-art heuristics.     

An integrated approach to shift-starting time selection and tour-schedule construction

A common practice in personnel tour scheduling is to limit the number of permissible daily shift starting times. Accordingly, an important subproblem that arises in tour-scheduling problems is the selection of those starting times that are conducive to efficient satisfaction of labour requirements. A previously published heuristic for this problem is sequential in nature, whereby a subset of starting times is selected in the first stage and tours are constructed using the chosen starting times in the second stage. We propose an integrative procedure that overcomes many of the limitations inherent in the sequential approach. Our procedure, which uses tabu search for starting time selection and a cutting plane method for tour construction, was compared to integer programming and the previously published heuristic in two experiments using test conditions from the airport staffing literature. Although all methods performed well when the labour requirement patterns were consistent across days of the week, the new procedure yielded the best performance when such patterns were inconsistent.     

The time-dependent capacitated profitable tour problem with time windows and precedence constraints

We introduce the time-dependent capacitated profitable tour problem with time windows and precedence constraints. This problem concerns determining a tour and its departure time at the depot that maximizes the collected profit minus the total travel cost (measured by total travel time). To deal with road congestion, travel times are considered to be time-dependent. We develop a tailored labeling algorithm to find the optimal tour. Furthermore, we introduce dominance criteria to discard unpromising labels. Our computational results demonstrate that the algorithm is capable of solving instances with up to 150 locations (75 pickup and delivery requests) to optimality. Additionally, we present a restricted dynamic programing heuristic to improve the computation time. This heuristic does not guarantee optimality, but is able to find the optimal solution for 32 instances out of the 34 instances.     

Heuristics for the Stochastic Eulerian Tour Problem

The Stochastic Eulerian Tour Problem (SETP) seeks the Eulerian tour of minimum expected length on an undirected Eulerian graph, when demand on the arcs that have to be serviced is probabilistic. The SETP is NP-hard and in this paper, we develop three constructive heuristics for this problem. The first two are greedy tour construction heuristics while the third is a sub-tour concatenation heuristic. Our experimental results show that for grid networks, the sub-tour concatenation heuristic performs well when the probability of service of each edge is greater than 0.1. For Euclidean networks, as the number of edges increases, the second heuristic performs the best among the three. Also, the expected length of our overall best solution is lower than the expected length of a random tour by up to 10% on average for grid networks and up to 2% for Euclidean networks.     

Match twice and stitch: a new TSP tour construction heuristic

We present a new symmetric traveling salesman problem tour construction heuristic. Two sequential matchings yield a set of cycles over the given point set; these are then stitched to form a tour. Our method outperforms all previous tour construction methods, but is dominated by several tour improvement heuristics.     

On the tour partitioning heuristic for the unit demand capacitated vehicle routing problem

The tour partitioning heuristic for the vehicle routing problem assumes an unlimited supply of vehicles. If the number of vehicles is fixed, this heuristic may produce infeasible solutions. We modify the heuristic to guarantee feasibility in this situation and we analyze the worst-case performance of the modified heuristic.     

Generating constrained length personalized bicycle tours

In the context of recreational routing, the problem of finding a route which starts and ends in the same location (while achieving a length between specified upper and lower boundaries) is a common task, especially for tourists or cyclists who want to exercise. The topic of finding a tour between a specified starting and ending location while minimizing one or multiple criteria is well covered in literature. In contrast to this, the route planning task in which a pleasant tour with length between a maximum and a minimum boundary needs to be found is relatively underexplored. In this paper, we provide a formal definition of this problem, taking into account the existing literature on which route attributes influence cyclists in their route choice. We show that the resulting problem is NP-hard and devise a branch-and-bound algorithm that is able to provide a bound on the quality of the best solution in pseudo-polynomial time. Furthermore, we also create an efficient heuristic to tackle the problem and we compare the quality of the solutions that are generated by the heuristic with the bounds provided by the branch-and-bound algorithm. Also, we thoroughly discuss the complexity and running time of the heuristic.     

Algorithms for a stochastic selective travelling salesperson problem

In this paper, the selective travelling salesperson problem with stochastic service times, travel times, and travel costs (SSTSP) is addressed. In the SSTSP, service times, travel times and travel costs are known a priori only probabilistically. A non-negative value of reward for providing service is associated with each customer and there is a pre-specified limit on the duration of the solution tour. It is assumed that not all potential customers can be visited within this tour duration limit, even under the best circumstances. And, thus, a subset of customers must be selected. The objective of the SSTSP is to design an a priori tour that visits each chosen customer once such that the total profit (total reward collected by servicing customers minus travel costs) is maximized and the probability that the total actual tour duration exceeds a given threshold is no larger than a chosen probability value. We formulate the SSTSP as a chance-constrained stochastic program and propose both exact and heuristic approaches for solving it. Computational experiments indicate that the exact algorithm is able to solve small- and moderate-size problems to optimality and the heuristic can provide near-optimal solutions in significantly reduced computing time.     

Validating vehicle routing zone construction using Monte Carlo simulation

The primary purpose of this paper is to validate a clustering procedure used to construct contiguous vehicle routing zones (VRZs) in metropolitan regions. Given a set of customers with random demand for pickups and deliveries over the day, the goal of the design problem is to cluster the customers into zones that can be serviced by a single vehicle. Monte Carlo simulation is used to determine the feasibility of the zones with respect to package count and tour time. For each replication, a separate probabilistic traveling salesman problem (TSP) is solved for each zone. For the case where deliveries must precede pickups, a heuristic approach to the TSP is developed and evaluated, also using Monte Carlo simulation. In the testing, performance is measured by overall travel costs and the probability of constraint violations. Gaps in tour length, tour time and tour cost are the measure used when comparing exact and heuristic TSP solutions.     

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.     

An ensemble of discrete differential evolution algorithms for solving the generalized traveling salesman problem

In this paper, an ensemble of discrete differential evolution algorithms with parallel populations is presented. In a single populated discrete differential evolution (DDE) algorithm, the destruction and construction (DC) procedure is employed to generate the mutant population whereas the trial population is obtained through a crossover operator. The performance of the DDE algorithm is substantially affected by the parameters of DC procedure as well as the choice of crossover operator. In order to enable the DDE algorithm to make use of different parameter values and crossover operators simultaneously, we propose an ensemble of DDE (eDDE) algorithms where each parameter set and crossover operator is assigned to one of the parallel populations. Each parallel parent population does not only compete with offspring population generated by its own population but also the offspring populations generated by all other parallel populations which use different parameter settings and crossover operators. As an application area, the well-known generalized traveling salesman problem (GTSP) is chosen, where the set of nodes is divided into clusters so that the objective is to find a tour with minimum cost passing through exactly one node from each cluster. The experimental results show that none of the single populated variants was effective in solving all the GTSP instances whereas the eDDE performed substantially better than the single populated variants on a set of problem instances. Furthermore, through the experimental analysis of results, the performance of the eDDE algorithm is also compared against the best performing algorithms from the literature. Ultimately, all of the best known averaged solutions for larger instances are further improved by the eDDE algorithm.     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

New TSP Construction Heuristics and Their Relationships to the 2-Opt

Correction heuristics for the traveling salesman problem (TSP), with the 2-Opt applied as a postprocess, are studied with respect to their tour lengths and computation times. This study analyzes the 2-Opt dependency, which indicates how the performance of the 2-Opt depends on the initial tours built by the construction heuristics. In accordance with the analysis, we devise a new construction heuristic, the recursive-selection with long-edge preference (RSL) method, which runs faster than the multiple-fragment method and produces a comparable tour when they are combined with the 2-Opt.     

A multi-space sampling heuristic for the vehicle routing problem with stochastic demands

The vehicle routing problem with stochastic demands consists in designing transportation routes of minimal expected cost to satisfy a set of customers with random demands of known probability distributions. This paper proposes a simple yet effective heuristic approach that uses randomized heuristics for the traveling salesman problem, a tour partitioning procedure, and a set partitioning formulation to sample the solution space and find high-quality solutions for the problem. Computational experiments on benchmark instances from the literature show that the proposed approach is competitive with the state-of-the-art algorithm for the problem in terms of both accuracy and efficiency. In experiments conducted on a set of 40 instances, the proposed approach unveiled four new best-known solutions (BKSs) and matched another 24. For the remaining 12 instances, the heuristic reported average gaps with respect to the BKS ranging from 0.69 to 0.15 % depending on its configuration.     

Capacitated Multiple Item Ordering with Incremental Quantity Discounts

This paper presents a heuristic algorithm for determining order quantities for multiple items given incremental quantity discounts and a single resourse constraint. The heuristic is based on Lagrangian relaxation. The performance of the heuristic is compared, for small problems, with a procedure that generates optimal solutions. Results from computational experiments are given that demonstrate the quality and computational efficiency of the heuristic algorithm.     

Customized simulated annealing based decision algorithms for combinatorial optimization in VLSI floorplanning problem

Modern very large scale integration technology is based on fixed-outline floorplan constraints, mostly with an objective of minimizing area and wirelength between the modules. The aim of this work is to minimize the unused area, that is, dead space in the floorplan, in addition to these objectives. In this work, a Simulated Annealing Algorithm (SAA) based heuristic, namely Simulated Spheroidizing Annealing Algorithm (SSAA) has been developed and improvements in the proposed heuristic algorithm is also suggested to improve its performance. Exploration capability of the proposed algorithm is due to the mechanism of reducing the uphill moves made during the initial stage of the algorithm, extended search at each temperature and the improved neighborhood search procedure. The proposed algorithm has been tested using two kinds of benchmarks: Microelectronics Center of North Carolina (MCNC) and Gigascale Systems Research Center (GSRC). The performance of the proposed algorithm is compared with that of other stochastic algorithms reported in the literature and is found to be efficient in producing floorplans with very minimal dead space. The proposed SSAA algorithm is also found more efficient for problems of larger sizes.     

A breadth-first search applied to the minimization of the open stacks

This paper presents a heuristic for the minimization of the open stacks problem (MOSP). The proposed heuristic is based on a simple breadth-first search in MOSP graphs and two new greedy rules to overcome errors. The performance of the proposed heuristic is compared with the best exact and heuristic methods available in the literature. The results show that in addition to the suggested heuristic having much shorter running times than the exact algorithm, the error gap between them is small for a substantial proportion of almost 4500 benchmark instances taken from the literature. The proposed heuristic also has a more robust behaviour than the best heuristic for the MOSP, although less accurate. The proposed heuristic therefore constitutes a viable and cost-effective alternative for solving or obtaining good upper bounds for the MOSP.     

Iterated greedy for the maximum diversity problem

The problem of choosing a subset of elements with maximum diversity from a given set is known as the maximum diversity problem. Many algorithms and methods have been proposed for this hard combinatorial problem, including several highly sophisticated procedures. By contrast, in this paper we present a simple iterated greedy metaheuristic that generates a sequence of solutions by iterating over a greedy construction heuristic using destruction and construction phases. Extensive computational experiments reveal that the proposed algorithm is highly effective as compared to the best-so-far metaheuristics for the problem under consideration.