Local search with constraint propagation and conflict-based heuristics

Search algorithms for solving csp (Constraint Satisfaction Problems) usually fall into one of two main families: local search algorithms and systematic algorithms. Both families have their advantages. Designing hybrid approaches seems promising since those advantages may be combined into a single approach. In this paper, we present a new hybrid technique. It performs a local search over partial assignments instead of complete assignments, and uses filtering techniques and conflict-based techniques to efficiently guide the search. This new technique benefits from both classical approaches: a priori pruning of the search space from filtering-based search and possible repair of early mistakes from local search. We focus on a specific version of this technique: tabu decision-repair. Experiments done on open-shop scheduling problems show that our approach competes well with the best highly specialized algorithms.     

Explaining the <Literal>cumulative</Literal> propagator

The global cumulative constraint was proposed for modelling cumulative resources in scheduling problems for finite domain (FD) propagation. Since that time a great deal of research has investigated new stronger and faster filtering techniques for cumulative, but still most of these techniques only pay off in limited cases or are not scalable. Recently, the “lazy clause generation” hybrid solving approach has been devised which allows a finite domain propagation engine possible to take advantage of advanced SAT technology, by “lazily” creating a SAT model of an FD problem as computation progresses. This allows the solver to make use of SAT explanation and autonomous search capabilities. In this article we show how, once we use lazy clause generation, modelling the cumulative constraint by decomposition creates a highly competitive version of cumulative. Using decomposition into component parts automatically makes the propagator incremental and able to explain itself. We then show how, using the insights from the behaviour of the decomposition, we can create global cumulative constraints that explain their propagation. We compare these approaches to explaining the cumulative constraint on resource constrained project scheduling problems. All our methods are able to close a substantial number of open problems from the well-established PSPlib benchmark library of resource-constrained project scheduling problems.     

Cost-based filtering algorithms for a Capacitated Lot Sizing Problem and the Constrained Arborescence Problem

Constraint Programming (CP) is a paradigm derived from artificial intelligence, operational research, and algorithmics that can be used to solve combinatorial problems. CP solves problems by interleaving search (assign a value to an unassigned variable) and propagation. Constraint propagation aims at removing/filtering inconsistent values from the domains of the variables in order to reduce the search space of the problem. In this thesis, we develop filtering algorithms for two complex combinatorial optimization problems: a Capacitated Lot Sizing Problem (CLSP) and the Constrained Arborescence Problem (CAP). Each of these problems has many variants and practical applications.The CLSP is the problem of finding an optimal production plan for single or multiple items while satisfying demands of clients and respecting resource restrictions. The CLSP finds important applications in production planning. In this thesis, we introduce a CLSP in CP. In many lot sizing and scheduling problems, in particular when the planning horizon is discrete and finite, there are stocking costs to be minimized. These costs depend on the time spent between the production of an order and its delivery. We focus on developing specialized filtering algorithms to handle the stocking cost part of a class of the CLSP. We propose the global optimization constraint StockingCost when the perperiod stocking cost is the same for all orders; and its generalized version, the IDStockingCost constraint (ID stands for Item Dependent).In this thesis, we also deal with a well-known problem in graph theory: the Minimum Weight Arborescence (MWA) problem. Consider a weighted directed graph in which we distinguish one vertex r as the root. An MWA rooted at r is a directed spanning tree rooted at r with minimum total weight. We focus on the CAP that requires one to find an arborescence that satisfies some side constraints (for example resource, degree, or diameter constraints) and that has minimum weight. The CAP has many real life applications in telecommunication networks, computer networks, transportation problems, scheduling problems, etc. After sensitivity analysis of the MWA, we introduce the CAP in CP. We propose a dedicated global optimization constraint to handle any known variant of the CAP in CP: the MinArborescence constraint. All the proposed filtering algorithms are analyzed theoretically and evaluated experimentally. The different experimental evaluations of these propagators against the state-of-the-art propagators show their respective efficiencies.     

Mathematical modeling and heuristic approaches to flexible job shop scheduling problems

Scheduling for the flexible job shop is very important in both fields of production management and combinatorial optimization. However, it is quite difficult to achieve an optimal solution to this problem in medium and actual size problem with traditional optimization approaches owing to the high computational complexity. For solving the realistic case with more than two jobs, two types of approaches have been used: hierarchical approaches and integrated approaches. In hierarchical approaches assignment of operations to machines and the sequencing of operations on the resources or machines are treated separately, i.e., assignment and sequencing are considered independently, where in integrated approaches, assignment and sequencing are not differentiated. In this paper, a mathematical model and heuristic approaches for flexible job shop scheduling problems (FJSP) are considered. Mathematical model is used to achieve optimal solution for small size problems. Since FJSP is NP-hard problem, two heuristics approaches involve of integrated and hierarchical approaches are developed to solve the real size problems. Six different hybrid searching structures depending on used searching approach and heuristics are presented in this paper. Numerical experiments are used to evaluate the performance of the developed algorithms. It is concluded that, the hierarchical algorithms have better performance than integrated algorithms and the algorithm which use tabu search and simulated annealing heuristics for assignment and sequencing problems consecutively is more suitable than the other algorithms. Also the numerical experiments validate the quality of the proposed algorithms.     

Heuristics for hybrid flow shops with controllable processing times and assignable due dates

This paper considers a generalization of the permutation flow shop problem that combines the scheduling function with the planning stage. In this problem, each work center consists of parallel identical machines. Each job has a different release date and consists of ordered operations that have to be processed on machines from different machine centers in the same order. In addition, the processing times of the operations on some machines may vary between a minimum and a maximum value depending on the use of a continuously divisible resource. We consider a nonregular optimization criterion based on due dates which are not a priori given but can be fixed by a decision-maker. A due date assignment cost is included into the objective function. For this type of problems, we generalize well-known approaches for the heuristic solution of classical problems and propose constructive algorithms based on job insertion techniques and iterative algorithms based on local search. For the latter, we deal with the design of appropriate neighborhoods to find better quality solution. Computational results for problems with up to 20 jobs and 10 machine centers are given.Scope and purposeTraditional research to solve multi-stage scheduling problems has focused on regular measures of performance based on a single criterion and assumes that several decisions related to due dates and processing times have already been made. However, in many industrial scheduling practices, managers develop schedules based on multicriteria and have to decide the due dates and processing times as part of the scheduling activities. Further, in practical scheduling situations, there are multiple machines at each stage and the objective function often reflects the total cost of processing, earliness and tardiness. Such scheduling problems require significantly more effort in finding acceptable solutions and hence have not received much attention in the literature. For this reason, this paper considers one such hybrid flow shop scheduling problem involving nonregular measures of performance, controllable processing times, and assignable due dates. We combine and generalize the existing approaches for the classical flow shop problem to the problem under consideration. Computational experiments are used to evaluate the utility of the proposed algorithms for the generalized scheduling problems. Brah and Hunsucker (European Journal of Operational Research 1991;51:88–99) and Nowicki and Smutnicki (European Journal of Operational Research 1998;106:226–253) describe branch and bound and tabu search algorithms for the approach used in the development of heuristic algorithms can also be adapted to several other complex practical scheduling problems.     

A Cost-Regular Based Hybrid Column Generation Approach

Constraint Programming (CP) offers a rich modeling language of constraints embedding efficient algorithms to handle complex and heterogeneous combinatorial problems. To solve hard combinatorial optimization problems using CP alone or hybrid CP-ILP decomposition methods, costs also have to be taken into account within the propagation process. Optimization constraints, with their cost-based filtering algorithms, aim to apply inference based on optimality rather than feasibility. This paper introduces a new optimization constraint, cost-regular. Its filtering algorithm is based on the computation of shortest and longest paths in a layered directed graph. The support information is also used to guide the search for solutions. We believe this constraint to be particularly useful in modeling and solving Column Generation subproblems and evaluate its behaviour on complex Employee Timetabling Problems through a flexible CP-based column generation approach. Computational results on generated benchmark sets and on a complex real-world instance are given.A preliminary version of this paper appeared as [7]. This research was supported by the Mathematics of Information Technology and Complex Systems (MITACS) Internship program in association with Omega Optimisation Inc. (CA).     

Scheduling projects with multi-skilled personnel by a hybrid MILP/CP benders decomposition algorithm

We study an assignment type resource-con- strained project scheduling problem with resources being multi-skilled personnel to minimize the total staffing costs. We develop a hybrid Benders decomposition (HBD) algorithm that combines the complimentary strengths of both mixed-integer linear programming (MILP) and constraint programming (CP) to solve this NP-hard optimization problem. An effective cut-generating scheme based on temporal analysis in project scheduling is devised for resolving resource conflicts. The computational study shows that our hybrid MILP/CP algorithm is both effective and efficient compared to the pure MILP or CP method alone.     

Temporal linear relaxation in IBM ILOG CP Optimizer

IBM ILOG CP Optimizer is a constraint solver that implements a model-and-run paradigm. For scheduling problems, CP Optimizer provides a relatively simple but very expressive modeling language based on the notion of interval variables. This paper presents the temporal linear relaxation (TLR) used to guide the automatic search when solving scheduling problems that involve temporal and resource allocation costs. We give the rationale of the TLR, describe its integration in the automatic search of CP Optimizer, and present the relaxation of most of the constraints and expressions of the model. An experimental study on a set of classical scheduling benchmarks shows that using the TLR is essential for problems with irregular temporal costs and generally helps for problems with resource allocation costs.     

基于推荐机制的网格资源匹配算法研究

针对网格计算环境下,参与计算用户和计算资源规模日益庞大,用户申请资源过程中所需的资源匹配过程逐步复杂化和大规模化,提出了一种基于推荐机制的网格资源匹配算法.以往的网格计算资源的匹配和调度算法需要在调度计算时遍历所有网格资源,而改进的基于SVD(奇异值分解)的协同过滤算法考虑了用户行为相关性和资源使用频度的相关性,通过用户对资源项的使用历史记录建立用户对资源的满意度评分体系,利用推荐机制给出用户推荐资源集以到达资源匹配的效果.从一个新的角度给出了解决大量资源匹配的方法.     

Optimization-Oriented Global Constraints

In this paper, we propose a way of exploiting Operations Research techniques within global constraints for cost-based domain filtering. In Constraint Programming, constraint propagation is aimed at removing from variable domains combinations of values which are proven infeasible. Pruning derives from feasibility reasoning. When coping with optimization problems, pruning can be performed also on the basis of costs, i.e., optimality reasoning. Cost-based filtering removes combination of values which are proven sub-optimal. For this purpose, we encapsulate in global constraints optimization components representing suitable relaxations of the constraint itself. These components embed efficient Operations Research algorithms computing the optimal solution of the relaxed problem and a gradient function representing the estimated cost of each variable-value assignment. We exploit these pieces of information for pruning and for guiding the search. We have applied these techniques to a couple of ILOG Solver global constraints (a constraint of difference and a path constraint) and tested the approach on a variety of combinatorial optimization problems such as Timetabling, Travelling Salesman Problems and Scheduling Problems with sequence dependent setup times. Comparisons with pure Constraint Programming approaches and related literature clearly show the benefits of the proposed approach.     

An effective hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problem

Flexible job-shop scheduling problem (FJSP) is an extension of the classical job-shop scheduling problem. Although the traditional optimization algorithms could obtain preferable results in solving the mono-objective FJSP. However, they are very difficult to solve multi-objective FJSP very well. In this paper, a particle swarm optimization (PSO) algorithm and a tabu search (TS) algorithm are combined to solve the multi-objective FJSP with several conflicting and incommensurable objectives. PSO which integrates local search and global search scheme possesses high search efficiency. And, TS is a meta-heuristic which is designed for finding a near optimal solution of combinatorial optimization problems. Through reasonably hybridizing the two optimization algorithms, an effective hybrid approach for the multi-objective FJSP has been proposed. The computational results have proved that the proposed hybrid algorithm is an efficient and effective approach to solve the multi-objective FJSP, especially for the problems on a large scale.     

Embedding Relaxations in Global Constraints for Solving TSP and TSPTW

Constraint Programming (CP) has been successfully applied to several combinatorial optimization problems. One of its advantages is the availability of complex global constraints performing efficient propagation and interacting with each other through shared variables. However, CP techniques have shown their limitations in dealing with optimization problems since the link between the objective function and problem decision variables is often quite loose and does not produce an effective propagation. We propose to integrate optimization components in global constraints, aimed at optimally solving a relaxation corresponding to the constraint itself. The optimal solution of the relaxation provides pieces of information which can be exploited in order to perform pruning on the basis of cost-based reasoning. In fact, we exploit reduction rules based on lower bound and reduced costs calculation to remove those branches which cannot improve the best solution found so far. The interest of integrating efficient well-known Operations Research (OR) algorithms into CP is mainly due to the smooth interaction between CP domain reduction and information provided by the relaxation acting on variable domains which can be seen as a communication channel among different techniques. We have applied this technique to symmetric and asymmetric Traveling Salesman Problem (TSP) instances both because the TSP is an interesting problem arising in many real-life applications, and because pure CP techniques lead to disappointing results for this problem. We have tested the proposed optimization constraints using ILOG solver. Computational results on benchmarks available from literature, and comparison with related approaches are described in the paper. The proposed method on pure TSPs improves the performances of CP solvers, but is still far from the OR state of the art techniques for solving the problem. However, due to the flexibility of the CP framework, we could easily use the same technique on TSP with Time Windows, a time constrained variant of the TSP. For this type of problem, we achieve results that are comparable with state of the art OR results.     

A Neurogenetic approach for the resource-constrained project scheduling problem

A variety of metaheuristic approaches have emerged in recent years for solving the resource-constrained project scheduling problem (RCPSP), a well-known NP-hard problem in scheduling. In this paper, we propose a Neurogenetic approach which is a hybrid of genetic algorithms (GA) and neural-network (NN) approaches. In this hybrid approach the search process relies on GA iterations for global search and on NN iterations for local search. The GA and NN search iterations are interleaved in a manner that allows NN to pick the best solution thus far from the GA pool and perform an intensification search in the solution's local neighborhood. Similarly, good solutions obtained by NN search are included in the GA population for further search using the GA iterations. Although both GA and NN approaches, independently give good solutions, we found that the hybrid approach gives better solutions than either approach independently for the same number of shared iterations. We demonstrate the effectiveness of this approach empirically on the standard benchmark problems of size J30, J60, J90 and J120 from PSPLIB.     

Local Search-based Hybrid Algorithms for Finding Golomb Rulers

The Golomb ruler problem is a very hard combinatorial optimization problem that has been tackled with many different approaches, such as constraint programming (CP), local search (LS), and evolutionary algorithms (EAs), among other techniques. This paper describes several local search-based hybrid algorithms to find optimal or near-optimal Golomb rulers. These algorithms are based on both stochastic methods and systematic techniques. More specifically, the algorithms combine ideas from greedy randomized adaptive search procedures (GRASP), scatter search (SS), tabu search (TS), clustering techniques, and constraint programming (CP). Each new algorithm is, in essence, born from the conclusions extracted after the observation of the previous one. With these algorithms we are capable of solving large rulers with a reasonable efficiency. In particular, we can now find optimal Golomb rulers for up to 16 marks. In addition, the paper also provides an empirical study of the fitness landscape of the problem with the aim of shedding some light about the question of what makes the Golomb ruler problem hard for certain classes of algorithm.     

一种高效的FDE并行传播算法

约束规划(constraint programming, CP)是表示和求解组合问题的经典范式之一.扩展约束(extensional constraint)或称表约束(table constraint)是约束规划中最为常见的约束类型.绝大多数约束规划问题都可以用表约束表达.在问题求解时,相容性算法用于缩减搜索空间.目前,最为高效的表约束相容性算法是简单表约缩减(simple table reduction, STR)算法簇,如Compact-Table (CT)和STRbit算法.它们在搜索过程中维持广义弧相容(generalized arc consistency, GAC).此外,完全成对相容性(full pairwise consistency, fPWC)是一种比GAC剪枝能力更强的相容性.最为高效的维持fPWC算法是PW-CT算法.多年来,人们提出了多种表约束相容性算法来提高剪枝能力和执行效率.因子分解编码(factor-decomposition encoding, FDE)通过对平凡问题重新编码.它一定程度地扩大了问题模型,使在新的问题上维持相对较弱的GAC等价于在原问题...     

Algorithms for propagating resource constraints in AI planning and scheduling: Existing approaches and new results

This paper summarizes the main existing approaches to propagate resource constraints in Constraint-Based scheduling and identifies some of their limitations for using them in an integrated planning and scheduling framework. We then describe two new algorithms to propagate resource constraints on discrete resources and reservoirs. Unlike most of the classical work in scheduling, our algorithms focus on the precedence relations between activities rather than on their absolute position in time. They are efficient even when the set of activities is not completely defined and when the time window of activities is large. These features explain why our algorithms are particularly suited for integrated planning and scheduling approaches. All our algorithms are illustrated with examples. Encouraging preliminary results are reported on pure scheduling problems as well as some possible extensions of our framework.     

约束满足混合算法求解并行机Job-Shop调度问题

分析并行机Job-Shop调度问题的特点并建立其约束满足优化模型,结合约束满足与变邻域搜索技术设计了一个求解该问题的混合优化算法。该算法采用变量排序方法和值排序方法选择变量并赋值,利用回溯和约束传播消解资源冲突,生成初始可行调度,然后应用局部搜索技术增强收敛性,并通过结合问题特点设计的邻域结构的多样性提高求解质量。数据实验表明,提出的算法与其他两种算法相比,具有一定的可行性和有效性。     

A new hybrid parallel genetic algorithm for the job‐shop scheduling problem

The job‐shop scheduling problem (JSSP) is considered one of the most difficult NP‐hard problems. Numerous studies in the past have shown that as exact methods for the problem solution are intractable, even for small problem sizes, efficient heuristic algorithms must achieve a good balance between the well‐known themes of exploitation and exploration of the vast search space. In this paper, we propose a new hybrid parallel genetic algorithm with specialized crossover and mutation operators utilizing path‐relinking concepts from combinatorial optimization approaches and tabu search in particular. The new scheme relies also on the recently introduced concepts of solution backbones for the JSSP in order to intensify the search in promising regions. We compare the resulting algorithm with a number of state‐of‐the‐art approaches for the JSSP on a number of well‐known test‐beds; the results indicate that our proposed genetic algorithm compares fairly well with some of the best‐performing genetic algorithms for the problem.     

求解模糊Job Shop调度的遗传算法与蚁群算法融合研究

提出一种算法融合策略,解决单一算法求解模糊Job Shop调度问题存在的不足,提高这类问题的求解质量.算法融合策略中,采用遗传算法和蚁群算法进行并行搜索;根据模糊Job Shop调度问题解的特征,提出基于关键工序的邻域选择方法,并将基于这种邻域选择方法的禁忌搜索算法作为局部搜索算法,加强了遗传算法和蚁群算法的局部搜索能力.采用算法融合策略的混合优化算法对以13个难的benchmarks问题经模糊化得到实例进行求解,在较短的时间内,得到的平均满意度较并行遗传算法(PGA)提高5.24%、较TSAB算法提高8.40% .采用算法融合策略构造的混合算法具有较强的搜索能力,说明提出的混合搜索策略是有效的.     

Metaheuristic methods in hybrid flow shop scheduling problem

Memetic algorithms are hybrid evolutionary algorithms that combine global and local search by using an evolutionary algorithm to perform exploration while the local search method performs exploitation. This paper presents two hybrid heuristic algorithms that combine particle swarm optimization (PSO) with simulated annealing (SA) and tabu search (TS), respectively. The hybrid algorithms were applied on the hybrid flow shop scheduling problem. Experimental results reveal that these memetic techniques can effectively produce improved solutions over conventional methods with faster convergence.