Traveling Salesman Problem Formulation Research Articles

Ant Colony Optimization and Beam-Ant Colony Optimization on Traveling Salesman Problem with Traffic Congestion

The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem that has various implications in a variety of industries. Even the purest formulation of TSP has applications on from logistics routes to microchip manufacturing, unexpectedly, it can be used on DNA sequencing with slight modification as a sub-problem. In this paper, two versions of TSP were studied, a classical TSP and the TSP containing traffic congestion data. Two state-of-the-art solution methods were used, Ant Colony Optimization (ACO) and Beam-ACO. These algorithms were hybridized with 2-Opt local search and their performances compared on the same benchmark instances. The experimental results show the efficiency of Beam-ACO compared to ACO.

Route design in sheepdog system–traveling salesman problem formulation and evolutionary computation solution–

ABSTRACT In this study, we consider the guidance control problem of the sheepdog system, which involves the guidance of a flock using the characteristics of the sheepdog and sheep. Sheepdog systems require strategies to guide sheep agents to a goal area using small numbers of sheepdog agents, and various methods have been proposed. Previous studies have proposed a guidance control law to reliably guide sheep herds; however, the movement distance required by the sheepdog for guidance has not been considered. Therefore, in this study, we propose a novel guidance algorithm in which a supposedly efficient route for guiding a flock of sheep is designed via the traveling salesman problem and evolutionary computation. Numerical simulations were performed to confirm whether sheep flocks could be guided and controlled using the obtained guidance routes. We revealed that the proposed method reduces both the guidance failure rate and the guidance distance.

Machine learning-enhanced aircraft landing scheduling under uncertainties

Aircraft delays lead to safety concerns and financial losses, which can propagate for several hours during extreme scenarios. Developing an efficient landing scheduling method is one of the effective approaches to reducing flight delays and safety concerns. Existing scheduling practices are mostly done by air traffic controllers (ATC) with heuristic rules. This paper proposes a novel machine learning-enhanced methodology for aircraft landing scheduling. Data-driven machine learning (ML) models are proposed to enhance automation and safety. ML enhancement is adopted for both prediction and optimization. First, the flight arrival delay scenarios are analyzed to identify the delay-related factors, where strong multimodal distributions and arrival flight time duration clusters are observed. A multi-stage conditional ML predictor is proposed for improved prediction performance of separation time conditioned on flight events. Next, we propose incorporating the ML predictions as safety constraints of the time-constrained traveling salesman problem formulation. The scheduling problem is then solved with mixed-integer linear programming (MILP). Additionally, uncertainties between successive flights from historical flight recordings and model predictions are included to ensure reliability. We demonstrate the real-world applicability of our method using the flight track and event data from the Sherlock database of the Atlanta Air Route Traffic Control Center (ARTCC ZTL). The case studies provide evidence that the proposed method is capable of reducing the total landing time by an average of 17.2% across three case studies, when compared to the First-Come-First-Served (FCFS) rule. Unlike the deterministic heuristic FCFS rule, the proposed methodology also considers the uncertainties between aircraft and ensures confidence in the scheduling. Finally, several concluding remarks and future research directions are given. The code used can be retrieved from Link.

Route Optimization in Precision Agriculture Settings: A Multi-Steiner TSP Formulation

In this work, we propose a route planning strategy for heterogeneous mobile robots in Precision Agriculture (PA) settings. Given a set of agricultural tasks to be performed at specific locations, we formulate a multi-Steiner Traveling Salesman Problem (TSP) to define the optimal assignment of these tasks to the robots as well as the respective optimal paths to be followed. The optimality criterion aims to minimize the total time required to execute all the tasks, as well as the cumulative execution times of the robots. Costs for travelling from one location to another, for maneuvering and for executing the task as well as limited energy capacity of the robots are considered. In addition, we propose a sub-optimal formulation to mitigate the computational complexity by leveraging the fact that generally in PA settings only a few locations require agricultural tasks in a certain period of interest compared to all possible locations in the field. A formal analysis of the optimality gap between the optimal and the sub-optimal formulations is provided. The effectiveness of the approach is validated in a simulated orchard where three heterogeneous aerial vehicles perform inspection tasks. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper aims at providing an efficient solution to PA needs by deploying a team of robots able to perform agricultural tasks at given locations in large-scale orchards. In particular, a novel general optimization problem is proposed that, given a set of mobile and possibly heterogeneous robots and a set of agricultural tasks to carry out, defines the assignment of these tasks to the robots as well as the routes to follow, while minimizing the total and the cumulative execution times of the robots. Existing approaches for route optimization in PA generally involves complete coverage of the field by one or multiple robots and do not account for maneuvering costs with general layouts of the field. We consider costs for travelling from one location to another, for executing the task and for maneuvering without any restriction on the layout of the plants as well as we take into account the limited energy capacity of the robots. We also provide a sub-optimal formulation which reduces the computational burden by relaxing the optimization of the maneuvering costs at the locations where agricultural tasks are carried out and formally derive the optimality gap. The proposed approach is flexible and can be easily adapted to any PA setting involving multiple mobile robots that are required to accomplish given tasks in an area of interest. We validate its effectiveness in a realistic simulated setup composed of three heterogeneous aerial vehicles performing inspection tasks. In future research, we aim to design algorithms to solve the proposed optimization problems in an efficient manner as well as to validate the formulations on real-world robotic platforms.

A Flow-Based Formulation of the Travelling Salesman Problem with Penalties on Nodes

The travelling salesman problem (TSP) is one of combinatorial optimization problems of huge importance to practical applications. However, the TSP in its “pure” form may lack some essential issues for a decision maker—e.g., time-dependent travelling conditions. Among those shortcomings, there is also a lack of possibility of not visiting some nodes in the network—e.g., thanks to the existence of some more cost-efficient means of transportation. In this article, an extension of the TSP in which some nodes can be skipped at the cost of penalties for skipping those nodes is presented under a new name and in a new mathematical formulation. Such an extension can be applied as a model for transportation cost reduction due to the possibility of outsourcing deliveries to some nodes in a TSP route. An integer linear programming formulation of such a problem based on the Gavish–Graves-flow-based TSP formulation is introduced. This formulation makes it possible to solve the considered problem by using any integer linear programming optimization software. Numerical examples and opportunities for further research are presented.

Energy Efficient Path Planning for 3D Aerial Inspections

The use of Unmanned Aerial Vehicles (UAVs) in different inspection tasks is increasing. This technology reduces inspection costs and collects high quality data of distinct structures, including areas that are not easily accessible by human operators. However, the reduced energy available on the UAVs limits their flight endurance. To increase the autonomy of a single flight, it is important to optimize the path to be performed by the UAV, in terms of energy loss. Therefore, this work presents a novel formulation of the Travelling Salesman Problem (TSP) and a path planning algorithm that uses a UAV energy model to solve this optimization problem. The novel TSP formulation is defined as Asymmetric Travelling Salesman Problem with Precedence Loss (ATSP-PL), where the cost of moving the UAV depends on the previous position. The energy model relates each UAV movement with its energy consumption, while the path planning algorithm is focused on minimizing the energy loss of the UAV, ensuring that the structure is fully covered. The developed algorithm was tested in both simulated and real scenarios. The simulated experiments were performed with realistic models of wind turbines and a UAV, whereas the real experiments were performed with a real UAV and an illumination tower. The inspection paths generated presented improvements over 24% and 8%, when compared with other methods, for the simulated and real experiments, respectively, optimizing the energy consumption of the UAV.

A new integer programming formulation of the graphical traveling salesman problem

In the Traveling Salesman Problem (TSP), a salesman wants to visit a set of cities and return home. There is a cost $$c_{ij}$$ of traveling from city i to city j, which is the same in either direction for the Symmetric TSP. The objective is to visit each city exactly once, minimizing total travel costs. In the Graphical TSP, a city may be visited more than once, which may be necessary on a sparse graph. We present a new integer programming formulation for the Graphical TSP requiring only two classes of polynomial-sized constraints while addressing an open question proposed by Denis Naddef. We generalize one of these classes, and present promising preliminary computational results.

Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method

There are many sub-tour elimination constraint (SEC) formulations for the traveling salesman problem (TSP). Among the different methods found in articles, usually three apply more than others. This study examines the Danzig–Fulkerson–Johnson (DFJ), Miller–Tucker–Zemlin (MTZ), and Gavish–Graves (GG) formulations to select the best asymmetric traveling salesman problem (ATSP) formulation. The study introduces five criteria as the number of constraints, number of variables, type of variables, time of solving, and differences between the optimum and the relaxed value for comparing these constraints. The reason for selecting these criteria is that they have the most significant impact on the mathematical problem-solving complexity. A new and well-known multiple-criteria decision making (MCDM) method, the simultaneous evaluation of the criteria and alternatives (SECA) method was applied to analyze these criteria. To use the SECA method for ranking the alternatives and extracting information about the criteria from constraints needs computational computing. In this research, we use CPLEX 12.8 software to compute the criteria value and LINGO 11 software to solve the SECA method. Finally, we conclude that the Gavish–Graves (GG) formulation is the best. The new web-based software was used for testing the results.

The no-wait flow shop with rejection

We consider the no-wait flow shop scheduling problem with rejection. We first show how to augment the travelling salesman problem formulation for the no-wait flow shop minimum makespan problem to incorporate the rejection option. We then focus on polynomially solvable cases by considering problems with ordered jobs. We present a third-order polynomial-time dynamic programming algorithm to minimise the sum of makespan and total rejection cost and faster quadratic algorithms for two special cases. We also present a third-order polynomial-time dynamic programming algorithm to minimise the sum of total completion time and total rejection cost with ordered jobs. We also exploit the duality between no-wait and no-idle flow shops to determine which operations to outsource in a no-idle flow shop with an outsourcing option.

A dual local search framework for combinatorial optimization problems with TSP application

In practice, solving realistically sized combinatorial optimization problems to optimality is often too time-consuming to be affordable; therefore, heuristics are typically implemented within most applications software. A specific category of heuristics has attracted considerable attention, namely local search methods. Most local search methods are primal in nature; that is, they start the search with a feasible solution and explore the feasible space for better feasible solutions. In this research, we propose a dual local search method and customize it to solve the traveling salesman problem (TSP); that is, a search method that starts with an infeasible solution, explores the dual space—each time reducing infeasibility, and lands in the primal space to deliver a feasible solution. The proposed design aims to replicate the designs of optimal solution methodologies in a heuristic way. To be more specific, we solve a combinatorial relaxation of a TSP formulation, design a neighborhood structure to repair such an infeasible starting solution, and improve components of intermediate dual solutions locally. Sample-based evidence along with statistically significant t-tests support the superiority of this dual design compared to its primal design counterpart.

A Combinatorial Approach for Developing Ring Communication Graphs for Vehicle Formations

In this paper, we study vehicle formations employing ring-structured communication strategies and propose a combinatorial approach for developing ring graphs for vehicle formations. In vehicle platoons, a ring graph is formed when each vehicle receives information from its predecessor, and the lead vehicle receives information from the last vehicle, thus forming a ring in its basic form. In such basic form, the communication distance between the first and the last vehicle increases with the platoon size, which creates implementation issues due to sensing range limitations. If one were to employ a communication protocol such as the token ring protocol, the delay in updating information and communication arises from the need for the token to travel across the entire graph. To overcome this limitation, alternative ring graphs which are formed by smaller communication distances between vehicles are proposed in this paper. For a given formation and a constraint on the maximum communication distance between any two vehicles, an algorithm to generate a ring graph is obtained by formulating the problem as an instance of the traveling salesman problem (TSP). In contrast to the vehicle platoons, generation of a ring communication graph is not straightforward for two- and three-dimensional formations; the TSP formulation allows this for both two- and three-dimensional formations with specific constraints. In addition, with ring communication structure, it is possible to devise simple ways to reconfigure the graph when vehicles are added/removed to/from the formation, which is discussed in the paper. Further, the experimental results using mobile robots for platooning and two-dimensional formations using ring graphs are shown and discussed.

Mixed Integer Linear Programming Based Approaches for Medium-Term Planning and Scheduling in Multiproduct Multistage Continuous Plants

This paper addresses the planning and scheduling problem of a multiproduct multistage continuous plant by three novel MILP-based (mixed integer linear programming) models. These models combine a TSP (traveling salesman problem) formulation with the main ideas of general precedence and unit-specific general precedence concepts to provide hybrid discrete/continuous time representations of the system. Also, an efficient solution approach involving rolling horizon and iterative-improvement algorithm is derived for solving medium-size instances of the problem. Results analyses for different model’s parameters demonstrate the benefits of the new formulations and the effectiveness of the solution approach presented in this work.

Traveling worker assembly line (re)balancing problem: Model, reduction techniques, and real case studies

The assembly line balancing problem arises from equally dividing the workload among all workstations. Several solution methods explore different variants of the problem, but no model includes all characteristics real assembly lines might contain. This paper presents a mixed integer linear programming model that solves the Traveling Worker Assembly Line Balancing Problem (TWALBP). In this problem, the tasks’ balancing along with the assignment of workers to one or more workstations is determined for a given layout. The assignment flexibility is solved with a traveling salesman problem formulation integrated in the balancing model. Adapted standard datasets and three real case scenarios are used as benchmark sets. These scenarios present particularities such as human and robotic workers, assignment restrictions, zoning constraints, automatic and common tasks. The model successfully determines the tasks’ assignments and the routing of every worker for a layout aware optimization of assembly lines. Better quality balancing solutions were achieved allowing workers to perform tasks at multiple stations, showing a trade-off between assignment flexibility and movement time.

Modeling supermarket re-layout from the owner’s perspective

It is well-known that if a customer follows a longer path while shopping then the expected value of his/her purchased amount is increased; therefore the sale amount of the supermarket can be increased. This study deals with a new problem: how to re-layout a supermarket the impulsive purchases of the average customer are maximized. Supermarket is a shop of limited size and is definitely smaller than the hypermarket. It is assumed that it is located in a living area and customers know its layout well. In many countries, there are plenty of shops like that. In a case study 27 clusters of customers are defined based on 13,300 real buying. To assume that actors behave in a rational way, is traditional in analysis of economic problems. Rationality means in that case that customers choose the shortest possible path according to their a priori purchase plan. Thus, traveling salesman problem (TSP) can be used to simulate the customer’s shopping path. Dantzig–Fulkerson–Johnson formulation of TSP is used to maximize the shortest traveled path of each customer type by rearranging the items of the supermarket as a max–min problem. The computational experiences on the case study show that the total distance is increased in the new layout proposed by the model.

Optimizing Movement Sequences for Step-and-Scan Lithography Equipment

The purpose of this work is to improve the throughput of step-and-scan lithography equipment to shorten the production time of a wafer. For this purpose, we propose a method for solving the MSOP (Movement Sequence Optimization Problem), which is the problem of computing the fastest schedule for visiting all shots on a wafer. It is well-known that the MSOP on step-and-repeat lithography equipment can be modeled as a traveling salesman problem. In contrast to step-and-repeat lithography equipment, a schedule for step-and-scan lithography equipment must also indicate the scanning direction of each shot, in addition to the sequence of the shots. For this reason, the traveling salesman problem formulation for step-and-repeat lithography equipment cannot be applied to solve the MSOP on step-and-scan lithography equipment directly. We overcame this difficulty by introducing auxiliary vertices to model the scanning directions in the traveling salesman problem formulation. By this method, we were able to compute exact optimal sequences considering the scanning directions of shots for several MSOP instances. Our numerical experiments demonstrated that our proposed method was capable of computing exact optimal solutions for real-world MSOP instances having up to 232 shots on a wafer. These optimal solutions gave a 0.25% to 4.66% improvement in productivity over solutions computed by previously known methods.

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.

Efficient reordering algorithm for color palette image using adaptive particle swarm technique

Palette re-ordering is a class of pre-processing methods aiming at finding a permutation of color palette such that the resulting image of indexes is more amenable for compression. The objective is to manipulate the palette index such that the adjacent symbols are assigned close indices in the symbol space, thus enhancing the compressibility of the image with many lossless compressors. Finding an exact reordered palette would certainly be exhaustive and computationally complex. To address this issue, a heuristic approach based on Traveling Salesman Problem formulation (NP hard) is sought. In this paper, the solution to this NP hard problem is presented by using an Adaptive Particle Swarm Optimization to achieve fast global convergence by maximizing the co-occurrences.

An optimisation-based seasonal sugarcane harvest scheduling decision support system for commercial growers in South Africa

An ongoing sugarcane decision support research project in South Africa is aimed at developing a decision support system capable of providing computerised support to those charged with the task of scheduling sugarcane harvesting operations in South Africa. In situations where the number of fields is large and when the conditions under which the crops are growing change frequently—for example as a result of climatic, biological or management-related events—computerised support is applicable. Commercial growers have provided data suitable for regression modelling of the parameters that govern the values and costs involved, and have participated in two consecutive preliminary system evaluation and development experiments conducted during the 2009 and 2010 harvesting seasons. The optimisation models underlying the decision support system are based on a time-dependent travelling salesman problem formulation and are solved approximately by means of a tabu search in a Microsoft Visual Basic for Applications (VBA) for Excel environment. The growers who participated in the evaluation experiments responded positively to the decision support system and stated that it may be useful to large-scale sugarcane producers as well as emerging growers. The authors’ findings are that the decision support system provides support in practical sugarcane harvest scheduling and that one series of regression fits is required for each agroclimatically and management-wise homogenous area.

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.

New formulations for the hop-constrained minimum spanning tree problem via Sherali and Driscoll's tightened 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, the new models based on the Miller–Tucker–Zemlin (MTZ) subtour elimination constraints are developed and computational results together with comparisons against MTZ-based, flow-based, and hop-indexed formulations are reported. The first model is obtained by adapting the MTZ-based Asymmetric Traveling Salesman Problem formulation of Sherali and Driscoll [18] and the other two models are obtained by combining topology-enforcing and MTZ-related constraints offered by Akgün and Tansel (submitted for publication) [20] for HMST with the first model appropriately. Computational studies show that the best LP bounds of the MTZ-based models in the literature are improved by the proposed models. The best solution times of the MTZ-based models are not improved for optimally solved instances. However, the results for the harder, large-size instances imply that the proposed models are likely to produce better solution times. The proposed models do not dominate the flow-based and hop-indexed formulations with respect to LP bounds. However, good feasible solutions can be obtained in a reasonable amount of time for problems for which even the LP relaxations of the flow-based and hop-indexed formulations can be solved in about 2 days.