A new formulation and approach for the black and white traveling salesman problem

This paper proposes a new formulation and a column generation approach for the black and white traveling salesman problem. This problem is an extension of the traveling salesman problem in which the vertex set is divided into black vertices and white vertices. The number of white vertices visited and the length of the path between two consecutive black vertices are constrained. The objective of this problem is to find the shortest Hamiltonian cycle that covers all vertices satisfying the cardinality and the length constraints. We present a new formulation for the undirected version of this problem, which is amenable to the Dantzig–Wolfe decomposition. The decomposed problem which is defined on a multigraph becomes the traveling salesman problem with an extra constraint set in which the variable set is the feasible paths between pairs of black vertices. In this paper, a column generation algorithm is designed to solve the linear programming relaxation of this problem. The resulting pricing subproblem is an elementary shortest path problem with resource constraints, and we employ acceleration strategies to solve this subproblem effectively. The linear programming relaxation bound is strengthened by a cutting plane procedure, and then column generation is embedded within a branch-and-bound algorithm to compute optimal integer solutions. The proposed algorithm is used to solve randomly generated instances with up to 80 vertices.     

An integer programming-based local search for the covering salesman problem

We consider a generalized version of the well known Traveling Salesman Problem called Covering Salesman problem. In this problem, we are given a set of vertices while each vertex i can cover a subset of vertices within its predetermined covering distance r_i. The goal is to construct a minimum length Hamiltonian cycle over a subset of vertices in which those vertices not visited on the tour has to be within the covering distance of at least one vertex visited on the tour. The paper proposes an Integer Linear Programming based heuristic method which takes advantage of Integer Linear Programming techniques and heuristic search to improve the quality of the solutions. Extensive computational tests on the standard benchmark instances and on a new set of large sized datasets show the effectiveness of the proposed approach.     

基于基因库求解旅行商问题的遗传算法

针对传统遗传算法(genetic algorithm, GA)求解旅行商问题(traveling salesman problem, TSP)存在寻优效率低、实验结果缺乏一致性等问题,提出了一种基于基因库的遗传算法(genetic algorithm based on genes pool, GPGA)。GPGA从种群中搜索减小哈密顿圈长度的边,并当做优良基因构成基因库。父代哈密顿圈在基因库引导下产生更优的子代哈密顿圈,基因库也随着种群的不断进化而同步更新,引导种群个体逐步向最优解靠近。算例结果表明在同样条件下,GPGA比传统遗传算法和几种改进遗传算法的性能更优。     

无向哈密顿图的一个充分必要条件及计算公式

哈密顿图的判定问题是一个NP完全问题,是图论理论中尚未解决的主要问题之一。1968年,Grinberg证明了一个必要条件,提高了判定非哈密顿可平面图的效率,由此产生了很多3-正则3-连通非哈密顿可平面图的研究成果。根据无向哈密顿图的特征,提出了基本圈的分解、合并、单条公共边连通,原子圈等概念。任何一个简单连通无向图G是哈密顿图,当且仅当,哈密顿圈要么其本身就是一个包含所有顶点的原子圈;要么总是可以分解成若干个原子圈,这些原子圈按照某种次序以单条公共边连通。根据这个充分必要条件,推导出了一个必要条件计算公式。它不仅能处理平面图,也能处理非平面图;甚至能处理某些Grinberg条件不能处理的平面图。此外,对一些实际案例的测试结果验证了充分必要条件和计算公式的有效性。     

A column generation heuristic for a dynamic generalized assignment problem

This paper studies the dynamic generalized assignment problem (DGAP) which extends the well-known generalized assignment problem by considering a discretized time horizon and by associating a starting time and a finishing time with each task. Additional constraints related to warehouse and yard management applications are also considered. Three linear integer programming formulations of the problem are introduced. The strongest one models the problem as an origin–destination integer multi-commodity flow problem with side constraints. This model can be solved quickly for instances of small to moderate size. However, because of its computer memory requirements, it becomes impractical for larger instances. Hence, a column generation algorithm is used to compute lower bounds by solving the linear program (LP) relaxation of the problem. This column generation algorithm is also embedded in a heuristic aimed at finding feasible integer solutions. Computational experiments on large-scale instances show the effectiveness of the proposed approach.     

Minimum common string partition: on solving large‐scale problem instances

Minimum common string partition is an NP‐hard combinatorial optimization problem from the bioinformatics field. The current state‐of‐the‐art algorithm is a hybrid technique known as construct, merge, solve, and adapt (CMSA). This algorithm combines two main algorithmic components: generating solutions in a probabilistic way and solving reduced subinstances obtained from the tackled problem instances, if possible, to optimality. However, the CMSA algorithm was not intended for application to very large problem instances. Therefore, in this paper we present a technique that makes CMSA, and other available algorithms for this problem, applicable to problem instances that are about one order of magnitude larger than the largest problem instances considered so far. Moreover, a reduced variable neighborhood search (RVNS) for solving the tackled problem, based on integer programming, is introduced. The experimental results show that the modified CMSA algorithm is very strong for problem instances based on rather small alphabets. With growing alphabet size, it turns out that RVNS has a growing advantage over CMSA.     

Two-level particle swarm optimization for the multi-modal team orienteering problem with time windows

This study presents a new variant of the team orienteering problem with time windows (TOPTW), called the multi-modal team orienteering problem with time windows (MM-TOPTW). The problem is motivated by the development of a tourist trip design application when there are several transportation modes available for tourists to choose during their trip. We develop a mixed integer programming model for MM-TOPTW based on the standard TOPTW model with additional considerations of transportation mode choices, including transportation cost and transportation time. Because MM-TOPTW is NP-hard, we design a two-level particle swarm optimization with multiple social learning terms (2L-GLNPSO) to solve the problem. To demonstrate the applicability and effectiveness of the proposed model and algorithm, we employ the proposed 2L-GLNPSO to solve 56 MM-TOPTW instances that are generated based on VRPTW benchmark instances. The computational results demonstrate that the proposed 2L-GLNPSO can obtain optimal solutions to small and medium-scale instances. For large-scale instances, 2L-GLNPSO is capable of producing high-quality solutions. Moreover, we test the proposed algorithm on standard TOPTW benchmark instances and obtains competitive results with the state-of-art algorithms.     

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.     

Ensemble of multi-objective metaheuristic algorithms for multi-objective unconstrained binary quadratic programming problem

Metaheuristics have been widely utilized for solving NP-hard optimization problems. However, these algorithms usually perform differently from one problem to another, i.e., one may be effective on a problem but performs badly on another problem. Therefore, it is difficult to choose the best algorithm in advance for a given problem. In contrast to selecting the best algorithm for a problem, selection hyper-heuristics aim at performing well on a set of problems (instances). This paper proposes a selection hyper-heuristic based algorithm for multi-objective optimization problems. In the proposed algorithm, multiple metaheuristics exhibiting different search behaviors are managed and controlled as low-level metaheuristics in an algorithm pool, and the most appropriate metaheuristic is selected by means of a performance indicator at each search stage. To assess the performance of the proposed algorithm, an implementation of the algorithm containing four metaheuristics is proposed and tested for solving multi-objective unconstrained binary quadratic programming problem. Experimental results on 50 benchmark instances show that the proposed algorithm can provide better overall performance than single metaheuristics, which demonstrates the effectiveness of the proposed algorithm.     

An efficient hybrid algorithm for the Traveling Car Renter Problem

This paper tackles a Traveling Salesman Problem variant called Traveling Car Renter Problem, where one car renter desires to travel among cities using a rented vehicle. Basically, the car renter has two options when he/she arrives in a city: to return the vehicle and rent another one or to keep the same car until the next city. Every time a car is delivered in a city, a return fee must be paid. Travel cost between any pair of cities also depends on the chosen car. The objective is to establish a Hamiltonian cycle minimizing the travel costs and returning fees. An evolutionary algorithm (EA) and a hybrid method called Adaptive Local Search Procedure (ALSP) are proposed for this problem. Both were compared to the best known algorithm in literature and obtained better results for non-Euclidean instances. Such algorithms compose an efficient model for a better exploration of the problem solutions space. From the expert system point-of-view, we propose a novel inference engine with minimized results error.     

哈密尔顿回路问题的DNA表面计算模型

首次提出用DNA表面计算模型来解决无向图哈密尔顿回路问题。该模型基于哈密尔顿回路问题的解空间,将问题解空间的DNA分子固定在固体载体上,对其进行荧光标记,然后通过相应的生化反应筛选出哈密尔顿回路问题的所有解。与已有的哈密尔顿路径问题的其它模型相比,新模型具有错误率低,编码简易,读取方便等更好的性能。     

A hybrid artificial bee colony for a nurse rostering problem

The nurse rostering problem (NRP) is a combinatorial optimization problem tackled by assigning a set of shifts to a set of nurses, each has specific skills and work contract, to a predefined rostering period according to a set constraints. The metaheuristics are the most successful methods for tackling this problem. This paper proposes a metaheuristic technique called a hybrid artificial bee colony (HABC) for NRP. In HABC, the process of the employed bee operator is replaced with the hill climbing optimizer (HCO) to empower its exploitation capability and the usage of HCO is controlled by hill climbing rate (HCR) parameter. The performance of the proposed HABC is evaluated using the standard dataset published in the first international nurse rostering competition 2010 (INRC2010). This dataset consists of 69 instances which reflect this problem in many real-world cases that are varied in size and complexity. The experimental results of studying the effect of HCO using different value of HCR show that the HCO has a great impact on the performance of HABC. In addition, a comparative evaluation of HABC is carried out against other eleven methods that worked on INRC2010 dataset. The comparative results show that the proposed algorithm achieved two new best results for two problem instances, 35 best published results out of 69 instances as achieved by other comparative methods, and comparable results in the remaining instances of INRC2010 dataset.     

A convex optimization approach for solving the single-vehicle cyclic inventory routing problem

This paper investigates the mathematical structure of the Single-Vehicle Cyclic Inventory Routing Problem (SV-CIRP). The SV-CIRP is an optimization problem consisting of finding a recurring distribution plan, from a single depot to a selected subset of retailers, that maximizes the collected rewards from the visited retailers while minimizing transportation and inventory costs. It appears as fundamental building block for all variants of the cyclic inventory routing problem (CIRP). One of the main complications in developing solution methods for the SV-CIRP using the current formulations is the non-convexity of the objective function. We demonstrate how the problem can be reformulated so that its continuous relaxation is a convex optimization problem. We further examine its mathematical properties and compare our findings with statements previously done in literature. Based of these findings we propose an algorithm that solves the SV-CIRP more effectively. We present experimental results on well-known benchmark instances, for which we are able to find optimal solutions for 22 out of 50 instances and obtained new best known solutions to 23 other instances.     

Load-dependent and precedence-based models for pickup and delivery problems

We address the one-to-one multi-commodity pickup and delivery traveling salesman problem (m-PDTSP) which is a generalization of the TSP and arises in several transportation and logistics applications. The objective is to find a minimum-cost directed Hamiltonian path which starts and ends at given depot nodes and such that the demand of each given commodity is transported from the associated source to its destination and the vehicle capacity is never exceeded. In contrast, the many-to-many one-commodity pickup and delivery traveling salesman problem (1-PDTSP) just considers a single commodity and each node can be a source or target for units of this commodity. We show that the m-PDTSP is equivalent to the 1-PDTSP with additional precedence constraints defined by the source–destination pairs for each commodity and explore several models based on this equivalence. In particular, we consider layered graph models for the capacity constraints and introduce new valid inequalities for the precedence relations. Especially for tightly capacitated instances with a large number of commodities our branch-and-cut algorithms outperform the existing approaches. For the uncapacitated m-PDTSP (which is known as the sequential ordering problem) we are able to solve to optimality several open instances from the TSPLIB and SOPLIB.     

An ANTS heuristic for the frequency assignment problem

The problem considered in this paper consists in assigning frequencies to radio links between base stations and mobile transmitters in order to minimize the global interference over a given region. This problem is NP-hard and few results have been reported on techniques for solving it to optimality. We have applied to this problem an ANTS metaheuristic, that is, an approach following the ant colony optimization paradigm. Computational results, obtained on a number of standard problem instances, testify the effectiveness of the proposed approach.     

An efficient algorithm for the collapsing knapsack problem

The collapsing knapsack problem (CKP) is a type of nonlinear knapsack problem in which the knapsack size is a non-increasing function of the number of items included. This paper proposes an exact algorithm for CKP by partitioning CKP to some subproblems, then solving them with the improved expanding-core technique. The proposed algorithm solves the subproblems in the special processing order resulting in the reduction of computing time. Experimental results show that the proposed algorithm is an efficient approach for various random instances of size up to 1000.     

A hybrid GRASP/VND heuristic for the one-commodity pickup-and-delivery traveling salesman problem

We address in this paper the one-commodity pickup-and-delivery traveling salesman problem, which is characterized by a set of customers, each of them supplying (pickup customer) or demanding (delivery customer) a given amount of a single product. The objective is to design a minimum cost Hamiltonian route for a capacitated vehicle in order to transport the product from the pickup to the delivery customers. The vehicle starts the route from a depot, and its initial load also has to be determined. We propose a hybrid algorithm that combines the GRASP and VND metaheuristics. Our heuristic is compared with other approximate algorithms described in Hernández-Pérez and Salazar-González [Heuristics for the one-commodity pickup-and-delivery traveling salesman problem. Transportation Science 2004;38:245–55]. Computational experiments on benchmark instances reveal that our hybrid method yields better results than the previously proposed approaches.     

An efficient certifying algorithm for the Hamiltonian cycle problem on circular-arc graphs

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is an evidence that can be used to authenticate the correctness of the answer. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to determine whether or not a graph contains a Hamiltonian cycle. The best result for the Hamiltonian cycle problem on circular-arc graphs is an O(n²logn)-time algorithm, where n is the number of vertices of the input graph. In fact, the O(n²logn)-time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature. However, whether there exists an algorithm whose time complexity is better than O(n²logn) for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for two decades. In this paper, we present an O(Δn)-time certifying algorithm to solve this problem, where Δ represents the maximum degree of the input graph. The certificates provided by our algorithm can be authenticated in O(n) time.     

Simulating realistic set covering problems with known optimal solutions

This paper outlines a methodology to generate random Set Covering Problem (SCP) instances with known optimal solutions and correlated coefficients. Positive correlation between the objective function coefficients and the column sums of the SCP constraint matrix is known to affect the performance of SCP solution methods. Generating large SCP instances with known optimal solutions and realistic coefficient correlation provides a plethora of test problems with controllable problem characteristics, including correlation, and an ample opportunity to test the performance of SCP heuristics and algorithms without having to solve the problems to optimality. We describe the procedure for generating SCP instances and present the results of a computational demonstration conducted on SCP instances generated by our procedure. This computational demonstration shows that the heuristics’ relative errors increase as the correlation increases, that the likelihood of finding a non-optimal solution also increases with the level of correlation, and that the quality of the solutions found is affected by the number of constraints.     

A simple randomized algorithm for two-dimensional strip packing

Two-dimensional strip packing problem is to pack given rectangular pieces on a strip of stock sheet having fixed width and infinite height. Its aim is to minimize the height of the strip such that non-guillotinable and fix orientation constraints are meet. In this paper, an improved scoring rule is developed and the least waste priority strategy is introduced, and a randomized algorithm is presented for solving this problem. This algorithm is very simple and does not need to set any parameters. Computational results on a wide range of benchmark problem instances show that the proposed algorithm obtains a better or matching performance as compared to the most of the previously published meta-heuristics.