Lower Bounds for Covering Problems

In this paper we present an (nlogn) lower bound proof for several covering problems including the optimal line bipartition problem, min-max covering by two axis parallel rectangles, discrete and continuous two-center problems, two-line center problem, etc. Our proofs are based on using the rotational reduce technique and well-known lower bound for the maximal gap problem.     

A Constraint Programming Framework for Local Search Methods

We propose in this paper a novel integration of local search algorithms within a constraint programming framework for combinatorial optimization problems, in an attempt to gain both the efficiency of local search methods and the flexibility of constraint programming while maintaining a clear separation between the constraints of the problem and the actual search procedure. Each neighborhood exploration is performed by branch-and-bound search, whose potential pruning capabilities open the door to more elaborate local moves, which could lead to even better approximate results. Two illustrations of this framework are provided, including computational results for the traveling salesman problem with time windows. These results indicate that it is one order of magnitude faster than the customary constraint programming approach to local search and that it is competitive with a specialized local search algorithm.     

Bounds and Tabu Search for a Cyclic Max-Min Scheduling Problem

A cyclic scheduling problem with applications to transport efficiency is considered. Given a set of regular polygons, whose vertices represent regularly occurring events and are lying on a common circle line, the objective is to maximize the distance between the closest vertices of different polygons on the circle line. Lower and upper bounds for the optimal solution of this NP-hard scheduling problem are presented. They are used to improve the quality of a procedure which is applied to solve this problem heuristically. It consists of a greedy starting algorithm and a Tabu Search algorithm. The numerical results show the efficiency of the procedure proposed.     

Heuristic Control of a Constraint-Based Algorithm for the Preemptive Job-Shop Scheduling Problem

In the recent years, constraint programming has been applied to a wide variety of academic and industrial non-preemptive scheduling problems, i.e., problems in which activities cannot be interrupted. In comparison, preemptive scheduling problems have received almost no attention from both the Operations Research and the Artificial Intelligence community. Motivated by the needs of a specific application, we engaged in a study of the applicability of constraint programming techniques to preemptive scheduling problems. This paper presents the algorithms we developed and the results we obtained on the preemptive variant of the famous job-shop scheduling problem. Ten heuristic search strategies, combined with two different constraint propagation techniques, are presented, and compared using two well-known series of job-shop scheduling instances from the literature. The best combination, which relies on limited discrepancy search and on edge-finding techniques, is shown to provide excellent solutions to the preemptive job-shop scheduling problem. A mean relative distance to the optimal solution of 0.32% is achieved in five minutes, on instances with 10 jobs and 10 machines (100 activities).     

On the Applicability of Lower Bounds for Solving Rectilinear Quadratic Assignment Problems in Parallel

The quadratic assignment problem (QAP) belongs to the hard core of NP-hard optimization problems. After almost forty years of research only relatively small instances can be solved to optimality. The reason is that the quality of the lower bounds available for exact methods is not sufficient. Recently, lower bounds based on decomposition were proposed for the so called rectilinear QAP that proved to be the strongest for a large class of problem instances. We investigate the strength of these bounds when applied not only at the root node of a search tree but as the bound function used in a Branch-and-Bound code solving large scale QAPs.     

The multiple TSP with time windows: vehicle bounds based on precedence graphs

Vehicle bounds for the multiple traveling salesman problem with time windows are found by covering two precedence graphs with the minimum number of paths. Instances with tight bounds are presented, as well as instances for which the bounds are loose. The similarity of these instances is discussed.     

调度问题中两类分离约束传播算法的比较及一种改进算法

约束传播算法是求解约束满足问题的一种重要方法。调度问题是一种特殊的约束满足问题。本介绍了调度问题中的Edge-Finding和Energy-Reasoning两种分离约束传播算法，并对它们进行了比较，中最后给出了一种结合Energy-Reasoning的Edge-Finding改进算法。     

An Enhanced TSP-Based Heuristic for Makespan Minimization in a Flow Shop with Setup Times

This paper presents an enhanced heuristic for minimizing the makespan of the flow shop scheduling problem with sequence-dependent setup times. The procedure transforms an instance of the problem into an instance of the traveling salesman problem by introducing a cost function that penalizes for both large setup times and bad fitness of schedule. This hybrid cost function is an improvement over earlier approaches that penalized for setup times only, ignoring the flow shop aspect of the problem. To establish good parameter values, each component of the heuristic was evaluated computationally over a wide range of problem instances. In the testing stage, an experimental comparison with a greedy randomized adaptive search procedure revealed the conditions and data attributes where the proposed procedure works best.     

带时间窗取送货问题的混合算法

为解决带时间窗的取送货问题,建立了集合划分模型,设计列生成算法与启发式规则相结合的CGA混合算法进行求解。首先,放松约束构建主问题及受限主问题,运用单纯形法与分支定界进行求解;其次,建立时空网络以构建子问题,基于修正的Dijkstra's算法,设计包含算法A、B1、B2的求解算法;最后,通过启发式算法解决节点重复覆盖问题。为验证算法有效性,进一步构建了OPT近似最优解算法;并基于CGA提出三种求解策略C1、C2、C3,做单因素方差分析,采用算例分析算法的性能。实验结果表明,对于客户点数量小于30的小规模算例,CGA与OPT所得结果相近,但CGA求解效率更显著;针对客户点数量为600的大规模算例,CGA至多在20分钟内求得结果,可见本文算法的精度和效率较高。而针对不同类型及规模的客户点的单因素方差分析结果显示,C1、C2、C3在“平均行驶距离成本”、“平均车辆数”、“平均求解时间”三个维度上差异性显著,经营者可根据实际需求进行策略选择。     

A new branching strategy for time constrained routing problems with application to backhauling

In this paper, we explore a new branching strategy for branch-and-bound approaches based on column generation for the vehicle routing problems with time windows. This strategy involves branching on resource variables (time or capacity) rather than on network flow variables. We also examine criteria for selecting network nodes for branching. To test the effectiveness of the branching strategy, we conduct computational experiments on time window constrained vehicle routing problems where backhauling is permitted only after all the shipments to clients have been made. The branching method proved very effective. In cases where time was the more binding constraint, time-based branching succeeded in decreasing the number of nodes explored by two thirds and the total computation time by more than half when compared to flow-based branching. The computational results also show that the overall algorithm was successful in optimally solving problems with up to 100 customers. It produced an average cost decrease of almost 7% when backhauling was permitted as compared to the cost involved when the client and the distributor routes were distinct.     

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.Supported by a grant from the Australia Research Council.     

求解带组换装时间单机调度问题的禁忌搜索算法

以包头某钢铁线材企业生产实际调度问题为背景,研究了一类带组换装时间的单机调度问题.由于该问题是NP难的,本文提出了一类适合该问题的禁忌搜索算法.此外,本文将问题性质引入了禁忌搜索算法以进一步提高算法寻优性能,降低算法运行时间.本文提出的算法在随机问题和实际问题上均进行了测试,实验结果表明,本文提出的算法能在不到10秒的时间内获得实际问题的一个近似最优解.     

Lower Bounds for the Maximum Modulus of $$zeta left( s right)$$ in Small Domains of the Critical Strip

Mathematical Notes -     

An exact algorithm for team orienteering problems

Optimising routing of vehicles constitutes a major logistic stake in many industrial contexts. We are interested here in the optimal resolution of special cases of vehicle routing problems, known as team orienteering problems. In these problems, vehicles are guided by a reward that can be collected from customers, while the length of routes is limited. The main difference with classical vehicle routing problems is that not all customers have to be visited. The solution method we propose here is based on a Branch & Price algorithm. It is, as far as we know, the first exact method proposed for such problems, except for a preliminary work from Gueguen (Methodes de résolution exacte pour problémes de tournées de véhicules. Thése de doctorat, école Centrale Paris 1999) and a work from Butt and Ryan (Comput Oper Res 26(4):427–441 1999). It permits to solve instances with up to 100 customers.      

Local search in routing problems with time windows

We develop local search algorithms for routing problems with time windows. The presented algorithms are based on thek-interchange concept. The presence of time windows introduces feasibility constraints, the checking of which normally requires O(N) time. Our method reduces this checking effort to O(1) time. We also consider the problem of finding initial solutions. A complexity result is given and an insertion heuristic is described.     

Heuristic algorithms for the operator-based relocation problem in one-way electric carsharing systems

This paper addresses an Electric Vehicle Relocation Problem (E-VReP), in one-way carsharing systems, based on operators who use folding bicycles to facilitate vehicle relocation. In order to calculate the economic sustainability of this relocation approach, a revenue associated with each relocation request satisfied and a cost associated with each operator used are introduced. The new optimization objective maximizes the total profit. To overcome the drawback of the high CPU time required by the Mixed Integer Linear Programming formulation of the E-VReP, two heuristic algorithms, based on the general properties of the feasible solutions, are designed. Their effectiveness is tested on two sets of realistic instances. In the first, all the requests have the same revenue, while, in the second, the revenue of each request has a variable component related to the user’s rent-time and a fixed part related to customer satisfaction. Finally, a sensitivity analysis is carried out on both the number of requests and the fixed revenue component.     

Three-stage approaches for optimizing some variations of the resource constrained shortest-path sub-problem in a column generation context

This paper presents a unified three-stage approach (TSA), comprising preprocessing, setup, and iterative solution stages, for solving several variations of the resource constrained shortest-path problem (RCSP). TSA is designed specially for column-generation applications in which sub-problems must be solved repetitively. The first two stages are implemented one time and only the third stage need be applied repetitively. In a companion paper, the authors proposed a TSA for solving RCSP on an acyclic graph with upper bound resource-limitation constraints. This paper shows that a TSA can be designed to solve each of several related problems: shortest-path with equality resource-limitation constraints; shortest-path with resource windows; resource-constrained, k-shortest path; and multiple-resource, multiple-choice knapsack. A numerical example demonstrates application of a TSA to design an international assembly system and its supply chain using branch and price with multiple-choice knapsack sub-problems. Computational results show that our TSA can solve this sub-problem effectively in such a column-generation environment.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.     

Algorithms for solving discrete optimal control problems with infinite time horizon and determining minimal mean cost cycles in a directed graph as decision support tool

Time-discrete systems with a finite set of states are considered. Discrete optimal control problems with infinite time horizon for such systems are formulated. We introduce a certain graph-theoretic structure to model the transitions of the dynamical system. Algorithms for finding the optimal stationary control parameters are presented. Furthermore, we determine the optimal mean cost cycles. This approach can be used as a decision support strategy within such a class of problems; especially so-called multilayered decision problems which occur within environmental emission trading procedures can be modelled by such an approach.