Revisiting dynamic programming for precedence-constrained traveling salesman problem and its time-dependent generalization

The precedence constrained traveling salesman problem (TSP-PC), or the sequential ordering problem (SOP), consists of finding an optimal TSP tour that will also satisfy the namesake precedence constraints, typically specified as a partial order or a directed acyclic graph. Its dynamic programming (DP) solution was proposed as early as 1979, however, by late 1990s, it mostly fell out of use in plain TSP-PC. Revisiting this method, we are able to close one of the long-standing TSPLIB SOP problem instances, ry48p.3.sop, and provide improved bounds on its time complexity. Harnessing the “omnivorous” nature of DP, we prove the validity of DP optimality principle for TSP-PC with both (i) abstract cost aggregation function, which may be the arithmetic + operation as in “ordinary” TSP or $\max$ as in Bottleneck TSP, or any other left-associative nondecreasing in the first argument operation and (ii) travel cost functions depending on the set of pending tasks (“sequence dependence”). Using the latter generalization, we close several TD-SOP (time-dependent TSP-PC) instances based on TSPLIB SOP as proposed by Kinable et al., including rbg253a.sop. Through the restricted DP heuristic, which was originally formulated for time-dependent TSP by Malandraki and Dial, we improve the state-of-the-art upper bounds for all yet unsolved TSPLIB-based TD-SOP instances, including those with more than 100 cities. We also improve worst-case complexity estimates for DP in TSP-PC.     

A self-organising model for the travelling salesman problem

This work describes a new algorithm, based on a self-organising neural network approach, to solve the Travelling Salesman Problem (TSP). Firstly, various features of the available adaptive neural network algorithms for TSP are reviewed and a new algorithm is proposed. In order to investigate the performance of the algorithms, a comprehensive empirical study has been provided. The simulations, which are conducted on a series of standard data, evaluate the overall performance of this approach by comparing the results with the best known or the optimal solutions of the problems. The proposed algorithm shows significant advances in both the quality of the solution and computational effort for most of the experimental data. The deviation from the optimal solution of this algorithm was, in the worst case, around 2%. This fact indicates that the self-organising neural network may be regarded as a promising heuristic approach for optimisation problems.     

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.     

Uncertain multiobjective traveling salesman problem

Traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century, yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it. A novel TSP variation, called uncertain multiobjective TSP (UMTSP) with uncertain variables on the arc, is proposed in this paper on the basis of uncertainty theory, and a new solution approach named uncertain approach is applied to obtain Pareto efficient route in UMTSP. Considering the uncertain and combinatorial nature of UMTSP, a new ABC algorithm inserted with reverse operator, crossover operator and mutation operator is designed to this problem, which outperforms other algorithms through the performance comparison on three benchmark TSPs. Finally, a new benchmark UMTSP case study is presented to illustrate the construction and solution of UMTSP, which shows that the optimal route in deterministic TSP can be a poor route in UMTSP.     

Heuristics for the flow line problem with setup costs

This paper presents two new heuristics for the flowshop scheduling problem with sequence-dependent setup times (SDSTs) and makespan minimization objective. The first is an extension of a procedure that has been very successful for the general flowshop scheduling problem. The other is a greedy randomized adaptive search procedure (GRASP) which is a technique that has achieved good results on a variety of combinatorial optimization problems. Both heuristics are compared to a previously proposed algorithm based on the traveling salesman problem (TSP). In addition, local search procedures are developed and adapted to each of the heuristics. A two-phase lower bounding scheme is presented as well. The first phase finds a lower bound based on the assignment relaxation for the asymmetric TSP. In phase two, attempts are made to improve the bound by inserting idle time. All procedures are compared for two different classes of randomly generated instances. In the first case where setup times are an order of magnitude smaller than the processing times, the new approaches prove superior to the TSP-based heuristic; for the case where both processing and setup times are identically distributed, the TSP-based heuristic outperforms the proposed procedures.     

Multi-depot Multiple TSP: a polyhedral study and computational results

We study the Multi-Depot Multiple Traveling Salesman Problem (MDMTSP), which is a variant of the very well-known Traveling Salesman Problem (TSP). In the MDMTSP an unlimited number of salesmen have to visit a set of customers using routes that can be based on a subset of available depots. The MDMTSP is an NP-hard problem because it includes the TSP as a particular case when the distances satisfy the triangular inequality. The problem has some real applications and is closely related to other important multi-depot routing problems, like the Multi-Depot Vehicle Routing Problem and the Location Routing Problem. We present an integer linear formulation for the MDMTSP and strengthen it with the introduction of several families of valid inequalities. Certain facet-inducing inequalities for the TSP polyhedron can be used to derive facet-inducing inequalities for the MDMTSP. Furthermore, several inequalities that are specific to the MDMTSP are also studied and proved to be facet-inducing. The partial knowledge of the polyhedron has been used to implement a Branch-and-Cut algorithm in which the new inequalities have been shown to be very effective. Computational results show that instances involving up to 255 customers and 25 possible depots can be solved optimally using the proposed methodology.     

汽配件颜色喷涂顺序问题的TSP转化与建模

汽配件颜色喷涂顺序问题通常以生产线上相邻汽配件颜色切换次数少为最优目标,以进一步降低生产成本.该类问题具有所有汽配件都必须喷涂一次且只喷涂一次的特点,为此提出了TSP转化与建模的方法.将待喷涂汽配件定义为TSP顶点,任意两个待喷涂汽配件的颜色切换定义为顶点的距离,仿照TSP问题构建0-1规划模型;类似于顶点距离,将某些...     

Integer programming models and linearizations for the traveling car renter problem

The traveling car renter problem (CaRS) is an extension of the classical traveling salesman problem (TSP) where different cars are available for use during the salesman’s tour. In this study we present three integer programming formulations for CaRS, of which two have quadratic objective functions and the other has quadratic constraints. The first model with a quadratic objective function is grounded on the TSP interpreted as a special case of the quadratic assignment problem in which the assignment variables refer to visitation orders. The second model with a quadratic objective function is based on the Gavish and Grave’s formulation for the TSP. The model with quadratic constraints is based on the Dantzig–Fulkerson–Johnson’s formulation for the TSP. The formulations are linearized and implemented in two solvers. An experiment with 50 instances is reported.     

A backbone based TSP heuristic for large instances

We introduce a reduction technique for large instances of the traveling salesman problem (TSP). This approach is based on the observation that tours with good quality are likely to share many edges. We exploit this observation by neglecting the less important tour space defined by the shared edges, and searching the important tour subspace in more depth. More precisely, by using a basic TSP heuristic, we obtain a set of starting tours. We call the set of edges which are contained in each of these starting tours as pseudo-backbone edges. Then we compute the maximal paths consisting only of pseudo-backbone edges, and transform the TSP instance to another one with smaller size by contracting each such path to a single edge. This reduced TSP instance can be investigated more intensively, and each tour of the reduced instance can be expanded to a tour of the original instance. Combining our reduction technique with the currently leading TSP heuristic of Helsgaun, we experimentally investigate 32 difficult VLSI instances from the well-known TSP homepage. In our experimental results we set world records for seven VLSI instances, i.e., find better tours than the best tours known so far (two of these world records have since been improved upon by Keld Helsgaun and Yuichi Nagata, respectively). For the remaining instances we find tours that are equally good or only slightly worse than the world record tours.     

Formulation and a two-phase matheuristic for the roaming salesman problem: Application to election logistics

In this paper we investigate a novel logistical problem. The goal is to determine daily tours for a traveling salesperson who collects rewards from activities in cities during a fixed campaign period. We refer to this problem as the Roaming Salesman Problem (RSP) motivated by real-world applications including election logistics, touristic trip planning and marketing campaigns. RSP can be characterized as a combination of the traditional Periodic TSP and the Prize-Collecting TSP with static arc costs and time-dependent node rewards. Commercial solvers are capable of solving small-size instances of the RSP to near optimality in a reasonable time. To tackle large-size instances we propose a two-phase matheuristic where the first phase deals with city selection while the second phase focuses on route generation. The latter capitalizes on an integer program to construct an optimal route among selected cities on a given day. The proposed matheuristic decomposes the RSP into as many subproblems as the number of campaign days. Computational results show that our approach provides near-optimal solutions in significantly shorter times compared to commercial solvers.     

Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well-known relaxations of the TSP are the subtour elimination problem and the 2-matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT ≤4/3, and that 2M/SUBT ≤10/9.In this paper, we exploit the structure of certain 1/2-integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2-matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solution found falls into our classes, the above two conjectures are true. Our proofs are constructive and could be implemented in polynomial time, and thus, for such cost functions, provide a 4/3 (or better) approximation algorithm for the TSP.     

A cutting plane procedure for the travelling salesman problem on road networks

A cutting plane algorithm for the exact solution of the symmetric travelling salesman problem (TSP) is proposed. The real tours on a usually incomplete road network, which are in general non-Hamiltonian, are characterized directly by an integer linear programming model. The algorithm generates special cutting planes for this model. Computational results for real road networks with up to 292 visiting places are reported, as well as for classical problems of the literature with up to 120 cities. Some of the latter problems have been solved for the first time with a pure cutting plane approach.     

On the refinement of bounds of heuristic algorithms for the traveling salesman problem

A number of heuristics for the traveling salesman problem (TSP) rely on the assumption that the triangle inequality (TI) is satisfied. When TI does not hold, the paper proposes a transformation such that for the transformed problem the TI holds. Consequently, the bounds obtained for heuristics are valid with appropriate modification. Moreover, for a TSP satisfying TI the same transformation strengthens such bounds. The transformation essentially maps the problem into one that is minimal with respect to the property that TI holds. For the symmetric TSP the transformation is particularly simple. For an application of the transformation in the asymmetric case we need the dual solution of an assignment problem.     

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.     

An improved upper bound for the TSP in cubic 3-edge-connected graphs

We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general.     

The travelling maintainer problem: integration of condition-based maintenance with the travelling salesman problem

The Travelling Salesman Problem (TSP) is one of the most studied problems in the literature due to its applicability to a large number of real cases. Most variants of the TSP consider total distance travelled. This paper presents a new generalised formulation of the TSP that aims to minimise the sum of functions of latencies to cities, rather than total distance travelled. Then, a new problem that uses a special function using the latency as input is presented, called the Travelling Maintainer Problem (TMP). The TMP integrates the output of prognostics in Condition-based Maintenance (CBM) with the TSP. CBM aims to minimise the failure and maintenance cost by identifying and predicting upcoming failures through the analysis of sensory information collected in real-time. Maintenance scheduling is performed using the predicted failure information obtained from the CBM. When the systems to be maintained are geographically distributed, maintenance scheduling requires integrated analysis of travel times and their effects on the failure progression in systems. This paper also presents Genetic Algorithm and Particle Swarm Optimisation-based solutions and their comparisons for the TMP on a case study.     

Performance characteristics of alternative genetic algorithmic approaches to the traveling salesman problem using path representation: An empirical study

The potential of Genetic Algorithmic (GA) approaches for solving order-based problems including the Traveling Salesman Problem (TSP) is recognized in a number of recent studies. By applying various GAs, these studies developed a set of unresolved GA design and configuration issues. The purpose of this study is to resolve the conflicting GA design and configuration issues by (1) concentrating on the classical TSP; and (2) developing, implementing, and testing a complete set of alternative GA configurations, 144 GAs are developed and evaluated by solvinh 5000 TSPs. A carefully designed statistical experimental plan accompanied by rigorous statistical analysis isolate the most promising configurations and identify their effect on solution time and quality. Although, the emphasis is on the TSP, the final results are applicable to other order-based problems that use sequence encoding.     

A new population seeding technique for permutation-coded Genetic Algorithm: Service transfer approach

Genetic Algorithm (GA) is a popular heuristic method for dealing complex problems with very large search space. Among various phases of GA, the initial phase of population seeding plays an important role in deciding the span of GA to achieve the best fit w.r.t. the time. In other words, the quality of individual solutions generated in the initial population phase plays a critical role in determining the quality of final optimal solution. The traditional GA with random population seeding technique is quite simple and of course efficient to some extent; however, the population may contain poor quality individuals which take long time to converge with optimal solution. On the other hand, the hybrid population seeding techniques which have the benefit of good quality individuals and fast convergence lacks in terms of randomness, individual diversity and ability to converge with global optimal solution. This motivates to design a population seeding technique with multifaceted features of randomness, individual diversity and good quality. In this paper, an efficient Ordered Distance Vector (ODV) based population seeding technique has been proposed for permutation-coded GA using an elitist service transfer approach. One of the famous combinatorial hard problems of Traveling Salesman Problem (TSP) is being chosen as the testbed and the experiments are performed on different sized benchmark TSP instances obtained from standard TSPLIB [54]. The experimental results advocate that the proposed technique outperforms the existing popular initialization methods in terms of convergence rate, error rate and convergence time.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

用遗传算法求解分组旅行推销员问题

在遗传算法能够有效解决TSP问题的基础上，根据遗传算法——通过搜索大规模，多样化的种群，在种群间交换个体所携带的遗传信息，保留种群中个体的优越遗传信息——的思想，设计了求解分组TSP问题的遗传算法。算法中染色体表示、评价函数的构造、杂交变异算子的设计经过实例计算的检验被证明较为可靠；算法运算速度快，容易获得有效解。