Probabilistic Traveling Salesman Problem Research Articles

A combined ant colony optimization with Levy flight mechanism for the probabilistic traveling salesman problem with deadlines

In this paper, we are interested in the Probabilistic Traveling Salesman Problem with Deadlines (PTSPD) where clients must be contacted, in addition to their random availability before a set deadline. The main objective is to find an optimal route that covers a random subset of visitors in the same order as they appear on the tour, attempting to keep the path as short as possible. This problem is regarded as being ♯P-hard. Ant Colony Optimization (ACO) has been frequently employed to resolve this challenging optimization problem. However, we suggest an enhanced ACO employing the Levy flight algorithm in this study. This allows some ants to take longer jumps based on the Levy distribution, helping them escape from local optima situations. Our computational experiments using standard benchmark datasets demonstrate that the proposed algorithm is more efficient and accurate than traditional ACO.

The Traveling Salesman Problem with Stochastic and Correlated Customers

It is well-known that the cost of parcel delivery can be reduced by designing routes that take into account the uncertainty surrounding customers’ presences. Thus far, routing problems with stochastic customer presences have relied on the assumption that all customer presences are independent from each other. However, the notion that demographic factors retain predictive power for parcel-delivery efficiency suggests that shared characteristics can be exploited to map dependencies between customer presences. This paper introduces the correlated probabilistic traveling salesman problem (CPTSP). The CPTSP generalizes the traveling salesman problem with stochastic customer presences, also known as the probabilistic traveling salesman problem (PTSP), to account for potential correlations between customer presences. I propose a generic and flexible model formulation for the CPTSP using copulas that maintains computational and mathematical tractability in high-dimensional settings. I also present several adaptations of existing exact and heuristic frameworks to solve the CPTSP effectively. Computational experiments on real-world parcel-delivery data reveal that correlations between stochastic customer presences do not always affect route decisions, but could have a considerable impact on route cost estimates. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0005 .

Investigation of ant colony optimization with Levy flight technique for a class of stochastic combinatorial optimization problem

The demand for efficient solutions to optimization problems with uncertain and stochastic data is increasing. Probabilistic traveling salesman problem (PTSP) is a class of Stochastic Combinatorial Optimization Problems (SCOPs) involving partially unknown information about problem data with a known probability distribution. It consists to minimize the expected length of the tour where each customer requires a visit only with a given probability, at which customers who do not need a tour are just ignored without further optimization. Since the PTSP is NP-hard, the usage of metaheuristic methods is necessary to solve the problem. In this paper, we present the Ant Colony Optimization (ACO) algorithm combined with the Levy Flight mechanism (LFACO), which is based on Levy distribution to balance searching space and speed global optimization. Experimental results on a large number of instances show that the proposed Levy ACO algorithm on the probabilistic traveling salesman problem allows to obtain better results compared with the classical ACO algorithm.

Multiple probabilistic traveling salesman problem in the coordination of drug transportation-In the context of sustainability goals and Industry 4.0.

Improving the effectiveness of route planning, especially in road transport deliveries is a challenge we need to face in the context of advancing climate change and the sustainable development goals. The main aim of the paper is to demonstrate the above average and utilitarian significance of the multiple probabilistic traveling salesman problem (MPTSP) in the coordination and modeling of sustainable product transportation, which is a novelty at the theoretical, conceptual, methodological and empirical level. We propose a new, hybrid algorithm of solving MPTSP instances (it connects harmony search, k-means and 2-opt), which can be successfully used in economic practice for coordination and modeling of Industry 4.0. The effectiveness of proposed approach is tested using a case study of drugs distribution services and datasets obtained from the transportation enterprise located in Poland. The study focuses on the issue of planning routes, with particular emphasis on the changing demand of customers. It should be stressed that this work may be of interest to researchers but also to management practitioners. The value added of this research lies in the innovative modeling the coordination of sustainable drug transportation as an instance of MPTSP and proposing an effective method to solve it. The main research results confirm that proposed method contributes to overall sustainability of studied supply chain.

Stability Analysis for the Modification Method Under the a Priori Strategy of the PTSP

We propose in this paper a new formulation for the stability of the Traveling Salesman Problem (TSP) compared with its probabilistic version, the Probabilistic Traveling Salesman Problem (PTSP). It is a real extension of the TSP, where the number of customers to be served each time is a random variable. That is only a subset of customers will need its services, moreover this subset varies from day to day. From the literature, several methods of resolution of the TSP have been proposed. In order to use these methods as they are for the PTSP came the idea of the study of stability. It is interested in finding cases where the solution for the TSP is also for the PTSP. First we survey and comment a number of easy TSPs. We also present via the notion of “Master tour” the stable problems in the TSP-PTSP context. An exact Branch and Bound is used for recognizing the TSPs that are not stable. Finally, we propose a new modification method, -different from the usual method of PTSP-, called the method of Taxi Driver, it takes the structure of the tour into consideration.

PTSP Solution Strategy for Motion Trajectory of UAV in Ubiquitous Sensor Network

Abstract Nowadays, thanks to technological progress, the Unmanned Aerial Vehicles (UAVs) have appeared and allow to realize practical applications at lower costs with higher quality. These are widely used in the ubiquitous sensor networks USN. To improve their performance, it is interesting to plan their complex missions in reasonable solution times. In this paper we propose a modeling of the UAV motion in USN as being a Probabilistic Traveling Salesman Problem (PTSP), where the number of sensors to be served each time is a random variable. This method consists in optimizing the movement trajectory of the UAV in order to reduce the time for delivery of information to the users. This modeling makes it possible to recover the essential information of the system despite the disturbance of a subset of its elementary components, while ensuring the resolution of the addressed situation according to the a priori strategy of the PTSP. It ensures the existence of a solution adaptable to a multitude of TSPs in real time. We proposed numerical results based on exact Branch and Bound algorithm that provides an optimal solution through each set of sensors which occur with certain probabilities.

Probabilistic Traveling Salesman Problem and Harmony Search Algorithms in Pharmacy Supply Optimization

This paper demonstrates the utilitarian significance of the Probabilistic Traveling Salesman Problem (PTSP) in planning travel routes by companies which provide distribution services for pharmacies, with a particular consideration of variable customer demand. The optimization problem was solved using the Harmony Search (HS) algorithm, thus verifying its utility based on one real instance of PTSP (representing the problem of pharmacy supply reliability) and three tasks from the public TSPLIB library (adjusted to PTSP). As a result of the conducted research, significant utility of the hybrid approach was identified, assuming the combination of HS with popular 2‑opt method, which enabled achievement of good results within acceptable period (in practical applications).

An Adaptive Bumble Bees Mating Optimization algorithm

The finding of the suitable parameters of an evolutionary algorithm, as the Bumble Bees Mating Optimization (BBMO) algorithm, is one of the most challenging tasks that a researcher has to deal with. One of the most common used ways to solve the problem is the trial and error procedure. In the recent few years, a number of adaptive versions of every evolutionary and nature inspired algorithm have been presented in order to avoid the use of a predefined set of parameters for all instances of the studied problem. In this paper, an adaptive version of the BBMO algorithm is proposed, where initially random values are given to each one of the parameters and, then, these parameters are adapted during the optimization process. The proposed Adaptive BBMO algorithm is used for the solution of the Multicast Routing Problem (MRP). As we would like to prove that the proposed algorithm is suitable for solving different kinds of combinatorial optimization problems we test the algorithm, also, in the Probabilistic Traveling Salesman Problem (PTSP) and in the Hierarchical Permutation Flowshop Scheduling Problem (HPFSP). Finally, the algorithm is tested in four classic benchmark functions for global optimization problems (Rosenbrock, Sphere, Rastrigin and Griewank) in order to prove the generality of the procedure. A number of benchmark instances for all problems are tested using the proposed algorithm in order to prove its effectiveness.

An Exact Resolution for the Probabilistic Traveling Salesman Problem under the A Priori Strategy

The Probabilistic Traveling Salesman Problem (PTSP) is an extension of the classical Traveling Salesman Problem (TSP). The main difference is the stochastic presence of the customers, that is, the number of them to be visited each time is a random variable, where each customer associates with a given probability of occurrence. The resolution of problem consists in finding an a priori tour visiting all customers which minimizes the expected length over all possibilities. In this paper, we propose an exact algorithm for the solution of the PTSP. In this context, we derive new expressions for the lower bound and transitional, permanent evaluations as well as the mechanism of separation. The numerical results demonstrate the advantage of the proposed evaluations.

Estimation-based metaheuristics for the single vehicle routing problem with stochastic demands and customers

The vehicle routing problem with stochastic demands and customers (VRPSDC) requires finding the optimal route for a capacitated vehicle that delivers goods to a set of customers, where each customer has a fixed probability of requiring being visited and a stochastic demand. For large instances, the evaluation of the cost function is a primary bottleneck when searching for high quality solutions within a limited computation time. We tackle this issue by using an empirical estimation approach. Moreover, we adopt a recently developed state-of-the-art iterative improvement algorithm for the closely related probabilistic traveling salesman problem. We integrate these two components into several metaheuristics and we show that they outperform substantially the current best algorithm for this problem.

On the computational complexity of the Probabilistic Traveling Salesman Problem with Deadlines

The Probabilistic Traveling Salesman Problem with Deadlines (PTSPD) is a stochastic vehicle routing problem considering time dependencies in terms of deadlines. The evaluation of the PTSPD objective function is believed to be a computationally demanding task and all efforts to find a polynomial time approach have failed so far. On the other hand, no hardness results regarding the evaluation of the PTSPD objective function are known. We fill this gap and show that the evaluation of the PTSPD objective function is indeed a computationally demanding task: even for Euclidean instances this task is #P-hard. We then derive additional hardness results for different computational tasks related to the PTSPD. Among other results, we show that the decision variant and the optimization variant of the PTSPD are both #P-hard. Following this, we focus on polynomial time approximations of the PTSPD objective function. We prove the existence of an FPRAS under certain conditions and examine the approximation guarantees obtained by the existing approaches. Finally, we give a strong inapproximability result regarding the objective function of a slightly more general problem, the Dependent PTSPD.

The probabilistic traveling salesman problem with time windows

With time-definite services occupying a large part of the delivery business, the explicit consideration of time windows into a route design has the potential to reduce transportation costs and the penalty costs associated with late deliveries. In this paper, we incorporate time windows into a priori routes by introducing the probabilistic traveling salesman problem with time windows (PTSPTW). The PTSPTW is an extension of the well-known probabilistic traveling salesman problem, where in addition to stochastic customer presence, each customer has an associated time window during which deliveries must be made. We present a recourse model and a variable neighborhood search with variable neighborhood descent algorithm to solve problem instances. We also present computational experiments that demonstrate the value of incorporating stochasticity into the problem.

Heuristics for the probabilistic traveling salesman problem with deadlines based on quasi-parallel Monte Carlo sampling

The Probabilistic Traveling Salesman Problem with Deadlines (PTSPD) is a Stochastic Vehicle Routing Problem with a computationally demanding objective function. In this work we propose an approximation for that objective function based on Monte Carlo Sampling and using the novel approach of quasi-parallel evaluation of samples. We perform comprehensive computational studies that reveal the efficiency of this approximation. Additionally, we examine different Local Search Algorithms and present a Random Restart Local Search Algorithm for solving the PTSPD together with an extensive computational study on a large set of benchmark instances.

A metaheuristic framework for stochastic combinatorial optimization problems based on GPGPU with a case study on the probabilistic traveling salesman problem with deadlines

In this work we propose a general metaheuristic framework for solving stochastic combinatorial optimization problems based on general-purpose computing on graphics processing units (GPGPU). This framework is applied to the probabilistic traveling salesman problem with deadlines (PTSPD) as a case study. Computational studies reveal significant improvements over state-of-the-art methods for the PTSPD. Additionally, our results reveal the huge potential of the proposed framework and sampling-based methods for stochastic combinatorial optimization problems.

New Formulation for Traveling Salesman Problem with Separation Requirement and Cost

A unique integer linear programming formulation is proposed to define a Hamiltonian tour in which the cost between a node pair is determined by the number of nodes between each pair on the tour. In addition, a node separation requirement may be conditional on other node separations. Properties of this new traveling salesman problem formulation are discussed, along with an a priori solution to the probabilistic traveling salesman problem with equal node coverage probability as a linear programming alternative to the probabilistic programming method in the literature. A linear programming-based heuristic is proposed and numerically tested.

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. To test the methodology, a series of experiments were conducted using data provided by a leading shipping carrier for the Pittsburgh area. Currently, the region is divided into 73 VRZs, compared to 64 indicated by the clustering procedure. The simulation results showed that a redesign would yield approximately $334,360 in annual savings without any noticeable deterioration in service. In addition, when the heuristic TSP model was solved in place of the exact model, the average gap in tour duration increased by only 0.16 hours and 0.2 hours for the cases of 73 clusters and 64 clusters, respectively, indicating a small upward bias. However, runtimes decreased by almost 70%.

Different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem

The probabilistic traveling salesman problem (PTSP) is a topic of theoretical and practical importance in the study of stochastic network problems. It provides researchers with a modeling framework for exploring the stochastic effects in routing problems. This paper proposed three initial solution generators (NN1, NN2, RAN) under a genetic algorithm (GA) framework for solving the PTSP. A set of numerical experiments based on heterogeneous and homogeneous PTSP instances were conducted to test the effectiveness and efficiency of the proposed algorithms. The results from the heterogeneous PTSP show that the average E[@t] values obtained by the three generators under a GA framework are similar to those obtained by the ''Previous Best,'' but with an average computation time saving of 50.2%. As for the homogeneous PTSP instances, NN1 is a relatively better generator among the three examined, while RAN consistently performs worse than the other two generators in terms of average E[@t] values. Additionally, as compared to previously reported studies, no one single algorithm consistently outperformed the others across all homogeneous PTSP instances in terms of the best E[@t] values. The fact that no one initial solution generator consistently performs best in terms of the E[@t] value obtained across all instances in heterogeneous cases, and that the performance of each examined algorithm is dependent on the number of nodes (n) and probability (p) for homogeneous cases, suggest the possibility of context-dependent phenomenon. Finally, to obtain valid results, researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting PTSP experiments.

Estimation-based metaheuristics for the probabilistic traveling salesman problem

The probabilistic traveling salesman problem (PTSP) is a central problem in stochastic routing. Recently, we have shown that empirical estimation is a promising approach to devise highly effective local search algorithms for the PTSP. In this paper, we customize two metaheuristics, an iterated local search algorithm and a memetic algorithm, to solve the PTSP. This customization consists in adopting the estimation approach to evaluate the solution cost, exploiting a recently developed estimation-based local search algorithm, and tuning the metaheuristics parameters. We present an experimental study of the estimation-based metaheuristic algorithms on a number of instance classes. The results show that the proposed algorithms are highly effective and that they define a new state-of-the-art for the PTSP.

Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem. For this problem, recently an estimation-based local search algorithm using delta evaluation has been proposed. In this paper, we adopt two well-known variance reduction procedures in the estimation-based local search algorithm: the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in Monte Carlo evaluation; the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation. We investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one. Experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation-based local search.

Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem

The use of ant colony optimization for solving stochastic optimization problems has received a significant amount of attention in recent years. In this paper, we present a study of enhanced ant colony optimization algorithms for tackling a stochastic optimization problem, the probabilistic traveling salesman problem. In particular, we propose an empirical estimation approach to evaluate the cost of the solutions constructed by the ants. Moreover, we use a recent estimation-based iterative improvement algorithm as a local search. Experimental results on a large number of problem instances show that the proposed ant colony optimization algorithms outperform the current best algorithm tailored to solve the given problem, which also happened to be an ant colony optimization algorithm. As a consequence, we have obtained a new state-of-the-art ant colony optimization algorithm for the probabilistic traveling salesman problem.