Heuristic approaches for the family traveling salesman problem

In this article, we address the family traveling salesman problem (FTSP), an NP‐hard problem that may be seen as a generalization of the traveling salesman problem. In the FTSP, the set of cities is partitioned into several families and one wants to find the minimum cost route that visits a given number of cities in each family. We propose two metaheuristics, a population‐based method and a local search method, and a hybrid algorithm, which combines a branch‐and‐cut algorithm with a local search procedure. All the proposed methods improve the best known upper bounds from the literature. The local search method and the hybrid algorithm improve the best known upper bounds for all the benchmark instances with unknown optimal value, while the population‐based method improves the best known upper bounds for all instances, except two. Furthermore, we developed an instance generator to create additional test instances with different visit patterns. These newly generated instances were considered in the computational experiment and, thus, we provide optimal values or upper bounds for each instance. Additionally, we created a website dedicated to the FTSP, where this information is made available to the scientific community ( http://familytsp.rd.ciencias.ulisboa.pt/ ).     

Heuristic solutions to the long-term unit commitment problem with cogeneration plants

We consider a long-term version of the unit commitment problem that spans over one year divided into hourly time intervals. It includes constraints on electricity and heating production as well as on biomass consumption. The problem is of interest for scenario analysis in long-term strategic planning. We model the problem as a large mixed integer programming problem. Two solutions to this problem are of interest but computationally intractable: the optimal solution and the solution derived by market simulation. To achieve good and fast approximations to these two solutions, we design heuristic algorithms, including mixed integer programming heuristics, construction heuristics and local search procedures. Two setups are the best: a relax and fix mixed integer programming approach with an objective function reformulation and a combination of a dispatching heuristic with stochastic local search. The work is developed in the context of the Danish electricity market and the computational analysis is carried out on real-life data.     

The pickup and delivery traveling salesman problem with first-in-first-out loading

This paper addresses a variation of the traveling salesman problem with pickup and delivery in which loading and unloading operations have to be executed in a first-in-first-out fashion. It provides an integer programming formulation of the problem. It also describes five operators for improving a feasible solution, and two heuristics that utilize these operators: a probabilistic tabu search algorithm, and an iterated local search algorithm. The heuristics are evaluated on data adapted from TSPLIB instances.     

An iterative time‐bucket refinement algorithm for a high‐resolution resource‐constrained project scheduling problem

We consider a resource‐constrained project scheduling problem originating in particle therapy for cancer treatment, in which the scheduling has to be done in high resolution. Traditional mixed integer linear programming techniques such as time‐indexed formulations or discrete‐event formulations are known to have severe limitations in such cases, that is, growing too fast or having weak linear programming relaxations. We suggest a relaxation based on partitioning time into so‐called time‐buckets. This relaxation is iteratively solved and serves as basis for deriving feasible solutions using heuristics. Based on these primal and dual solutions and bounds, the time‐buckets are successively refined. Combining these parts, we obtain an algorithm that provides good approximate solutions soon and eventually converges to an optimal solution. Diverse strategies for performing the time‐bucket refinement are investigated. The approach shows excellent performance in comparison to the traditional formulations and a metaheuristic.     

A note on a single-machine lot scheduling problem with indivisible orders

In this paper, a lot scheduling problem on a single machine with indivisible orders is studied. The objective is to minimize the total completion time of all orders. We show that the problem is NP-hard in the strong sense. Then, a binary integer programming approach and four simple heuristics are proposed to solve the problem. The binary integer programming approach with running time limit is considered as one heuristic method. As compared to a lower bound, the average performances of the heuristic method are really good and better than those of the four simple heuristics.     

Searching for optimal integer solutions to set partitioning problems using column generation

In this paper, we describe a new approach to increase the possibility of finding integer feasible columns to a set partitioning problem (SPP) directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP‐relaxation as quickly as possible without any concern for the integer properties of the columns formed. In our approach, we aim to generate columns forming an optimal integer solution while simultaneously solving the LP‐relaxation. Using this approach, we can improve the possibility of finding integer solutions by heuristics at each node in the branch‐and‐bound search. In addition, we improve the possibility of finding high‐quality integer solutions in cases where only the columns in the root node are used to solve the problem. The basis of our approach is a subgradient technique applied to a Lagrangian dual formulation of the SPP extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations and how the multiplier values are computed. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate optimal integer columns in a large set of well‐known test problems as compared to both standard and stabilized column generation, and simultaneously keep the number of columns smaller than standard column generation. This is also supported by tests on a case study with work‐shift generation.     

有向黑白旅行商问题

黑白旅行商问题是经典旅行商问题的推广,在基于SONET技术的光纤网络设计、航线调度等领域具有广泛的应用.已有研究工作集中在无向黑白旅行商问题上.文章研究该问题的更一般形式--有向黑白旅行商问题.首先,给出了有向黑白旅行商问题的混合整数线性规划公式.与目前无向黑白旅行商问题包含指数多个约束的规划公式相比,它仅包含多项式个约束.其次,给出了一种启发式算法.实验表明,该启发式算法能够有效地求解黑白旅行商问题的实例.由于无向黑白旅行商问题是有向黑白旅行商问题的特例,故文中的结论对于求解无向黑白旅行商问题同样有效.     

Hybrid approach for the two‐dimensional bin packing problem with two‐staged patterns

This paper presents a two‐phase heuristic approach for the two‐dimensional bin packing problem with two‐staged patterns and nonoriented items. A solution is generated in each phase and the better one is selected. Residual problems are solved by column generation in the first phase, where a partial admitting procedure is used to admit some of the patterns into the phase‐1 solution. The second solution is obtained from solving an integer linear programming problem over the set of all patterns generated in the first phase, where a time limit is used and subsequently the solution may not be optimal over the pattern set. The computational results indicate that the approach yields the best solution quality among the heuristics that use two‐staged or more complex patterns.     

Exact and heuristic algorithms for the just-in-time scheduling problem in a batch processing system

Batch processing systems are commonly used in many different environments such as chemical and semiconductor industries. In this research, a just-in-time scheduling problem in a batch processing system is investigated. Minimization of total earliness and tardiness of the jobs with respect to a common due date is considered as the objective function. First, the research problem is formulated as a mixed integer linear programming model. Then, to find the optimal schedule for a predetermined set of batches, a dynamic programming algorithm is proposed. Based on the proposed dynamic programming algorithm, several heuristics are also developed. A lower bounding method is presented, and then a branch and bound algorithm is proposed to solve the problem optimally. To demonstrate the performance of the proposed algorithms, several computational experiments are conducted.     

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.     

热轧实施计划中最优倒垛问题的整数规划模型及遗传算法

对钢铁企业板坯库中的最优倒垛问题建立了0和1整数规划模型.这一模型是一个 二次规划模型,且目标函数的系数与变量的取值相关联,属于NP-难问题,获得较大规模的最 优解是不可能或非常困难.为了求解此问题,本文构造了改进遗传算法:(1)提出了适合于最 优倒垛问题的遗传编码,运用此编码,不但能够产生可行的初始染色体,而且能够保证在交叉 和变异操作后的染色体仍然可行;(2)改进了遗传算法结构,在新的结构中,增加了一个培育 操作,改进了交叉操作.通过精选随机产生的问题例子的实验显示出,提出的算法的性能明显 好于原系统的启发式算法,最好的改进率达到7.04％.     

Single and parallel machine capacitated lotsizing and scheduling: New iterative MIP-based neighborhood search heuristics

We propose a general-purpose heuristic approach combining metaheuristics and mixed integer programming to find high quality solutions to the challenging single- and parallel-machine capacitated lotsizing and scheduling problem with sequence-dependent setup times and costs. Commercial solvers fail to solve even medium-sized instances of this NP-hard problem; therefore, heuristics are required to find competitive solutions. We develop construction, improvement and search heuristics all based on MIP formulations. We then compare the performance of these heuristics with those of two metaheuristics and other MIP-based heuristics that have been proposed in the literature, and to a state-of-the-art commercial solver. A comprehensive set of computational experiments shows the effectiveness and efficiency of the main approach, a stochastic MIP-based local search heuristic, in solving medium to large size problems. Our solution procedures are quite flexible and may easily be adapted to cope with model extensions or to address different optimization problems that arise in practice.     

Models for a traveling purchaser problem with additional side-constraints

The traveling purchaser problem (TPP) is the problem of determining a tour of a purchaser that needs to buy several items in different shops such that the total amount of travel and purchase costs is minimized. Motivated by an application in machine scheduling, we study a variant of the problem with additional constraints, namely, a limit on the maximum number of markets to be visited, a limit on the number of items bought per market and where only one copy per item needs to be bought. We present an integer linear programming (ILP) model which is adequate for obtaining optimal integer solutions for instances with up to 100 markets. We also present and test several variations of a Lagrangian relaxation combined with a subgradient optimization procedure. The relaxed problem can be solved by dynamic programming and can also be viewed as resulting from applying a state space relaxation technique to a dynamic programming formulation. The Lagrangian based method is combined with a heuristic that attempts to transform relaxed solutions into feasible solutions. Computational results for instances with up to 300 markets show that with the exception of a few cases, the reported differences between best upper bound and lower bound values on the optimal solutions are reasonably small.     

On a class of branching problems in broadcasting and distribution

We introduce the following network optimization problem: given a directed graph with a cost function on the arcs, demands at the nodes, and a single source s, find the minimum cost connected subgraph from s such that its total demand is no less than lower bound D. We describe applications of this problem to disaster relief and media broadcasting, and show that it generalizes several well-known models including the knapsack problem, the partially ordered knapsack problem, the minimum branching problem, and certain scheduling problems. We prove that our problem is strongly NP-complete and give an integer programming formulation. We also provide five heuristic approaches, illustrate them with a numerical example, and provide a computational study on both small and large sized, randomly generated problems. The heuristics run efficiently on the tested problems and provide solutions that, on average, are fairly close to optimal.     

Min-degree constrained minimum spanning tree problem with fixed centrals and terminals: Complexity,properties and formulations

We consider a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. We prove that the optimization problem is NP-Hard even for complete graphs and the feasibility problem is NP-Complete even if there is an edge between each central and each terminal in the input graph. Actually, this complexity result still holds when the minimum degree of each central node is restricted to be a same value d ≥ 2. We derive necessary and sufficient conditions for feasibility. We present several integer linear programming formulations – based on known formulations for the minimum spanning tree problem – along with a theoretical comparison among the lower bounds provided by their linear relaxations. We propose three Lagrangian heuristics. Computational experiments compare the performances of the heuristics and the formulations.     

A heuristic for the vehicle routing problem with due times

This paper studies the vehicle routing problem with due times. The vehicles are supposed to visit customers within the due times, and a penalty cost is imposed in case the vehicle arrives past the due times. The objective is to minimize the weighted sum of the traveling time of vehicles and the tardiness of the service customers receive. A mixed integer programming formulation and a heuristic based on the tabu search for a practical use are suggested. Route-perturb and route-improvement method for the neighborhood generation is proposed. Performances are compared with other heuristics appeared in the literature using the bench-mark data set modified to be fit to the model. It is shown that the suggested heuristic gives a good solution in a short computation time.     

Two-stage hybrid flowshop scheduling with simultaneous processing machines

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(nlogn), where n is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.     

Exact and heuristic procedures for single machine scheduling with quadratic earliness and tardiness penalties

The present paper studies the single machine, no-idle-time scheduling problem with a weighted quadratic earliness and tardiness objective. We investigate the relationship between this problem and the assignment problem, and we derive two lower bounds and several heuristic procedures based on this relationship. Furthermore, the applicability of the time-indexed integer programming formulation is investigated. The results of a computational experiment on a set of randomly generated instances show (1) the high-quality results of the proposed heuristics, (2) the low optimality gap of one of the proposed lower bounds and (3) the applicability of the integer programming formulation to small and medium size cases of the problem.     

Genetic algorithm-based heuristics for an open shop scheduling problem with setup,processing, and removal times separated

This report proposes a solution to the open shop scheduling problem with the objective of minimizing total job tardiness in the system. Some practical processing restrictions, such as independent setup and dependent removal times, are taken into account as well. The addressed problem is first described as a 0–1 integer programming model, and is then solved optimally. Subsequently, some hybrid genetic-based heuristics are proposed to solve the problem in an acceptable computation time. To demonstrate the adaptability of these heuristics, some performance comparisons are made with solutions provided by running either a mathematical programming model or certain classic meta-heuristics such as genetic algorithm, simulated annealing, and tabu search in various manufacturing scenarios. The experimental results show that the hybrid genetic-based heuristics perform well, especially the DGA. However, these heuristics require some more additional computations but are still acceptable.     

Binary integer programming formulation and heuristics for differentiated coverage in heterogeneous sensor networks

Coverage is a fundamental task in sensor networks. We consider the minimum cost point coverage problem and formulate a binary integer linear programming model for effective sensor placement on a grid-structured sensor field when there are multiple types of sensors with varying sensing quality and price. The formulation is general and can be adapted to handle situations where sensing is perfect, imperfect or uncertain, and the coverage requirements are differentiated. Unfortunately, the new model suffers from the intractability of the binary integer programming formulations. We therefore suggest approximation algorithms and heuristics. Computational results indicate that the heuristic based on Lagrangean relaxation outperforms the others in terms of solution quality.