Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a well known problem that is quite challenging to solve. It involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability. There are several proposed algorithms for the homogeneous version of the problem, where all customers have identical probability of being realized. From the literature, the most successful approaches involve local search procedures, with the most famous being the 2-p-opt and 1-shift procedures proposed by Bertsimas [D.J. Bertsimas, L. Howell, Further results on the probabilistic traveling salesman problem, European Journal of Operational Research 65 (1) (1993) 68–95]. Recently, however, evidence has emerged that indicates the equations offered for these procedures are not correct, and even when corrected, the translation to the heterogeneous version of the problem is not simple. In this paper we extend the analysis and correction to the heterogeneous case. We derive new expressions for computing the cost of 2-p-opt and 1-shift local search moves, and we show that the neighborhood of a solution may be explored in O(n²) time, the same as for the homogeneous case, instead of O(n³) as first reported in the literature.     

New integer linear programming formulation for the traveling salesman problem with time windows: minimizing tour duration with waiting times

The travelling salesman problem, being one of the most attractive and well-studied combinatorial optimization problems, has many variants, one of which is called ‘travelling salesman problem with Time Windows (TSPTW)’. In this problem, each city (nodes, customers) must be visited within a time window defined by the earliest and the latest time. In TSPTW, the traveller has to wait at a city if he/she arrives early; thus waiting times directly affect the duration of a tour. It would be useful to develop a new model solvable by any optimizer directly. In this paper, we propose a new integer linear programming formulation having O(n²) binary variables and O(n²) constraints, where (n) equals the number of nodes of the underlying graph. The objective function is stated to minimize the total travel time plus the total waiting time. A computational comparison is made on a suite of test problems with 20 and 40 nodes. The performances of the proposed and existing formulations are analysed with respect to linear programming relaxations and the CPU times. The new formulation considerably outperforms the existing one with respect to both the performance criteria. Adaptation of our formulation to the multi-traveller case and some additional restrictions for special situations are illustrated.     

The pyramidal capacitated vehicle routing problem

This paper introduces the pyramidal capacitated vehicle routing problem (PCVRP) as a restricted version of the capacitated vehicle routing problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the pyramidal traveling salesman problem (PTSP). A pyramidal route is defined as a route on which the vehicle first visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index.     

The multi-commodity one-to-one pickup-and-delivery traveling salesman problem

This paper concerns a generalization of the traveling salesman problem (TSP) called multi-commodity one-to-one pickup-and-delivery traveling salesman problem (m-PDTSP) in which cities correspond to customers providing or requiring known amounts of m different commodities, and the vehicle has a given upper-limit capacity. Each commodity has exactly one origin and one destination, and the vehicle must visit each customer exactly once. The problem can also be defined as the capacitated version of the classical TSP with precedence constraints. This paper presents two mixed integer linear programming models, and describes a decomposition technique for each model to find the optimal solution. Computational experiments on instances from the literature and randomly generated compare the techniques and show the effectiveness of our implementation.     

Formulations and Benders decomposition algorithms for multidepot salesmen problems with load balancing

This paper describes new models and exact solution algorithms for the fixed destination multidepot salesmen problem defined on a graph with n nodes where the number of nodes each salesman is to visit is restricted to be in a predefined range. Such problems arise when the time to visit a node takes considerably longer as compared to the time of travel between nodes, in which case the number of nodes visited in a salesman’s tour is the determinant of their ‘load’. The new models are novel multicommodity flow formulations with O(n²) binary variables, which is contrary to the existing formulations for the same (and similar) problems that typically include O(n³) binary variables. The paper also describes Benders decomposition algorithms based on the new formulations for solving the problem exactly. Results of the computational experiments on instances derived from TSPLIB show that some of the proposed algorithms perform remarkably well in cases where formulations solved by a state-of-the-art optimization code fail to yield optimal solutions within reasonable computation time.     

Heuristic for the Asymmetric Travelling Salesman Problem

The paper describes a heuristic algorithm for the asymmetric travelling salesman problem. The procedure is a generalization of Webb's Simple Loss 1 Method and is of the sequential type, i.e. in each step one link of the tour is fixed. The method proved to produce "high quality" near optimal solutions in a computing time, which only grows proportional to n^1.82.     

A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem

We present a variable neighborhood search approach for solving the one-commodity pickup-and-delivery travelling salesman problem. It is characterized by a set of customers such that each of the customers either supplies (pickup customers) or demands (delivery customers) a given amount of a single product, and by a vehicle, whose given capacity must not be exceeded, that starts at the depot and must visit each customer only once. The objective is to minimize the total length of the tour. Thus, the considered problem includes checking the existence of a feasible travelling salesman’s tour and designing the optimal travelling salesman’s tour, which are both NP-hard problems. We adapt a collection of neighborhood structures, k-opt, double-bridge and insertion operators mainly used for solving the classical travelling salesman problem. A binary indexed tree data structure is used, which enables efficient feasibility checking and updating of solutions in these neighborhoods. Our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best-known algorithms in terms of both the solution quality and computational efforts. Moreover, we improve the best-known solutions of all benchmark instances from the literature (with 200 to 500 customers). We are also able to solve instances with up to 1000 customers.     

On the core of routing games

A repairman makes a round-trip along a set of customers. He starts in his home location, visits each customer exactly once, and returns home. The cost of his trip has to be shared by the customers. A cooperative cost game, calledrouting game, is associated with this allocation problem, and anO(n ²) algorithm is given which computes a core element of a routing game if the core is non-empty. The non-emptiness of the core depends on the tour which is traversed by the repairman. Several procedures are given to construct tours which guarantee the non-emptiness of the core.     

Probabilistic analysis of an algorithm for the m-planar 3-index assignment problem on single-cycle permutations

Under study is them-planar 3-index assignment problem on single-cycle permutations, which is, in other words, the m peripatetic salesman problem (m-PSP) with different weight functions for each salesman. The problem is NP-hard for m ≥ 1. Some polynomial approximation algorithm is suggested for 1 <m < n/4 with time complexity O(nm ²). The performance ratios of the algorithm are established for the input data (elements of m× n × n matrix) which are assumed to be independent and identically distributed random variables on [a _n , b _n ], where 0 < a _n < b _n . If the distribution function is uniform or dominates (majorizes) the uniform distribution then some conditions on a _n , b _n , and m are obtained for the algorithm to be the asymptotically exact.     

A Lagrangian relaxation approach to the edge-weighted clique problem

The b-clique polytope CPⁿ_b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QPⁿ=CPⁿ_n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over CPⁿ_n. The problem of optimizing linear functions over CPⁿ_b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CPⁿ_b in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.     

An iterated local search heuristic for the capacitated prize-collecting travelling salesman problem

This paper considers a variant of the travelling salesman problem named the capacitated prize-collecting travelling salesman problem (CPCTSP), which is derived from the colour-coating production scheduling in a cold rolling mill. The objective of the CPCTSP is to minimize the travel cost and the penalties paid for unvisited customers in such a way that a sufficiently large prize is collected and the demand of the visited customers does not exceed the salesman's capacity. For this problem, we propose an iterated local search (ILS) heuristic adopting guided kick and enhanced dynasearch. The experimental results on randomly generated instances show that the proposed heuristic outperforms the improved tabu search algorithm using frequency-based memory, and the further experimental results on instances collected from real colour-coating production also show that the proposed ILS algorithm is more effective and efficient than the currently adopted manual scheduling method.     

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time.     

Finding efficient solutions for rectilinear distance location problems efficiently

Given n planar existing facility locations, a planar new facility location X is called efficient if there is no other location Y for which the rectilinear distance between Y and each existing facility is at least as small as between X and each existing facility, and strictly less for at least one existing facility. Rectilinear distances are typically used to measure travel distances between points via rectilinear aisles or street networks. We first present a simple arrow algorithm, based entirely on geometrical analysis, that constructs all efficient locations. We then present a row algorithm which is of order n(log n) that constructs all efficient locations, and establish that no alternative algorithm can be of a lower order.     

Worst-case analysis of two travelling salesman heuristics

In this paper we analyze the worst-case performance of some heuristics for the symmetric travelling salesman problem. We show that the worst-case ratios of tour length produced by the savings and greedy heuristics to that of a minimum tour are bounded by [log₂n]+1 and 0.5([log₂n]+1) respectively, where n is the number of cities.     

Algorithms for the universal and a priori TSP

We present two simple results for generalizations of the traveling salesman problem (TSP): for the universal TSP, we show that one can compute a tour that is universally optimal whenever the input is a tree metric. A (randomized) O(logn)-approximation algorithm for the a priori TSP follows as a corollary.     

The time constrained maximal covering salesman problem

We introduce the time constrained maximal covering salesman problem (TCMCSP) which is the generalization of the covering salesman and orienting problems. In this problem, we are given a set of vertices including a central depot, customer and facility vertices where each facility can supply the demand of some customers within its pre-determined coverage distance. Starting from the depot, the goal is to maximize the total number of covered customers by constructing a length constrained Hamiltonian cycle over a subset of facilities. We propose several mathematical programming models for the studied problem followed by a heuristic algorithm. The developed algorithm takes advantage of different procedures including swap, deletion, extraction-insertion and perturbation. Finally, an integer linear programming based improvement technique is designed to try to improve the quality of the solutions. Extensive computational experiments on a set of randomly generated instances indicate the effectiveness of the algorithm.     

The traveling salesman problem with few inner points

We propose two algorithms for the planar Euclidean traveling salesman problem. The first runs in O(k!kn) time and O(k) space, and the second runs in O(2^kk²n) time and O(2^kkn) space, where n denotes the number of input points and k denotes the number of points interior to the convex hull.     

A bilevel programming approach to the travelling salesman problem

We show that the travelling salesman problem is polynomially reducible to a bilevel toll optimization program. Based on natural bilevel programming techniques, we recover the lifted Miller-Tucker-Zemlin constraints. Next, we derive an O(n²) multi-commodity extension whose LP relaxation is comparable to the exponential formulation of Dantzig, Fulkerson and Johnson.     

An Integer Linear Programming Formulation and Branch-and-Cut Algorithm for the Capacitated m-Ring-Star Problem

We study the capacitated m-ring-star problem (CmRSP) that faces the design of minimum cost network structure that connects customers with m rings using a set of ring connections that share a distinguished node (depot), and optionally star connections that connect customers to ring nodes. Ring and star connections have some associated costs. Also, rings can include transit nodes, named Steiner nodes, to reduce the total network cost if possible. The number of customers in each ring-star (ringʼs customers and customer connected to it through star connections) have an upper bound (capacity).These kind of networks are appropriate in optical fiber urban environments. CmRSP is know to be NP-Hard. In this paper we propose an integer linear programming formulation and a branch-and-cut algorithm.     

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?ⁿ or ?ⁿ only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?ⁿ or ?ⁿ, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?