System of automated timetabling

Handbook of Constraint Programming

Constraint Programming typically uses the technique of depth-first branch and bound as the method of solving optimization problems. Although this method can give the optimal solution, for large problems, the time needed to find the optimal can be prohibitive. This paper introduces a method for using local search techniques within a Constraint Programming framework, and applies this technique to vehicle routing problems. We introduce a Constraint Programming model for vehicle routing, and a system for integrating Constraint Programming and local search techniques. We then describe how the method can be accelerated by handling core constraints using fast local checks, while other more complex constraints are left to the constraint propagation system. We have coupled our local search method with a meta-heuristic to avoid the search being trapped in local minima. Several meta-heuristics are investigated ranging from a simple Tabu Search method to Guided Local Search. An empirical study over benchmark problems shows the relative merits of these techniques. Investigations indicate that the specific long-term memory technique used by Guided Local Search can be used as a diversification method for Tabu Search, resulting in significant benefit. Several new best solutions on the Solomon problems are found in relatively few iterations of our algorithm.

Many real-world problems in Artificial Intelligence (AI) as well as in other areas of computer science and engineering can be efficiently modeled and solved using constraint programming techniques. In many real-world scenarios the problem is partially known, imprecise and dynamic such that some effects of actions are undesired and/or several unforeseen incidences or changes can occur. Whereas expressivity, efficiency, and optimality have been the typical goals in the area, there are several issues regarding robustness that have a clear relevance in dynamic Constraint Satisfaction Problems (CSPs). However, there is still no clear and common definition of robustness-related concepts in CSPs. In this paper, we propose two clearly differentiated definitions for robustness and stability in CSP solutions. We also introduce the concepts of recoverability and reliability, which arise in temporal CSPs. All these definitions are based on related well-known concepts, which are addressed in engineering and other related areas.

The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

Constraint programming techniques have been widely used in many different types of applications. However for NP-hard problems, such as scheduling, resources allocation, etc, basic constraint programming techniques may not be enough solve efficiently. This paper describes a design and implementation of a simplified nurse rostering system using constraint programming and automatic implied constraint generation by meta-level reasoning. The nurse rostering system requires generating a weekly timetable by assigning work shifts to nurse. Although the problem set is simplified, the search is difficult because it involves more than hundred constraints with a search space of about $3.74 \times 10^{50}$. Using only traditional constraint programming techniques, even in addition with popular heuristics, no timetable can be generated in reasonable time. To improve the search, we propose to use automatic implied constraint generation by meta-level reasoning. Several solvable and non-solvable problem instances were tested. With our approach, these instances can be solved or identified as nonsolvable within one second.

Held and Karp have proposed, in the early 1970s, a relaxation for the Traveling Salesman Problem (TSP) as well as a branch-and-bound procedure that can solve small to modest-size instances to optimality [4, 5]. It has been shown that the Held-Karp relaxation produces very tight bounds in practice, and this relaxation is therefore applied in TSP solvers such as Concorde [1]. In this short paper we show that the Held-Karp approach can benefit from well-known techniques in Constraint Programming (CP) such as domain filtering and constraint propagation. Namely, we show that filtering algorithms developed for the weighted spanning tree constraint [3, 8] can be adapted to the context of the Held and Karp procedure. In addition to the adaptation of existing algorithms, we introduce a special-purpose filtering algorithm based on the underlying mechanisms used in Prim’s algorithm [7]. Finally, we explored two different branching schemes to close the integrality gap. Our initial experimental results indicate that the addition of the CP techniques to the Held-Karp method can be very effective.The paper is organized as follows: section 2 describes the Held-Karp approach while section 3 gives some insights on the Constraint Programming techniques and branching scheme used. In section 4 we demonstrate, through preliminary experiments, the impact of using CP in combination with Held and Karp based branch-and-bound on small to modest-size instances from the TSPlib.KeywordsSpan TreeMinimum Span TreeTravel Salesman ProblemTravel Salesman ProblemConstraint ProgrammingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Constraint Satisfaction Problems (CSP) provide a modelling framework for many computer aided decision making problems. Many of these problems are associated to an optimization criterion. Solving a CSP consists in finding an assignment of values to the variables that satisfies the constraints and optimizes a given objective function (in case of an optimization problem). In this paper, we extend our framework for genetic algorithms (GA) as suggested by the reviewers of our previous ICLP paper [5]. Our purpose is not to solve efficiently the Balanced Academic Curriculum Problem (BACP) [2] but to combine a genetic algorithm with constraint programming techniques and to propose a general modelling framework to precisely design such hybrid resolution process and highlight their characteristics and properties.

Practical minimization of pairwise-covering test configurations using constraint programming

. The industrial and commercial worlds are increasingly competitive, requiring companies to be more productiveand more responsive to market changes (e.g. globalisation and privatisation). As a consequence, there is a strong need for solutions to large scale optimization problems, in domains such as production scheduling, transport, finance and network management. This means that more experts in constraint programming and optimization technology are required to develop adequate software. Given the computational complexity of Large Scale Combinatorial Optimization problems, a key question is how to help/guide in the tackling of LSCO problems in industry. Optimization technology is certainly reaching a level of maturity. Having emerged in the 50s within the Operational Research community, it has evolved and comprises new paradigms such as constraint programming and stochastic search techniques. There is a practical need, i.e. efficiency, scalability and tractability, to integrate techniques from the different paradigms. This adds complexity to the design of LSCO models and solutions. Various forms of guidance are available in the literature in terms of 1) case studies that map powerful algorithms to problem instances, and 2) visualization and programming tools that ease the modelling and solving of LSCOs. However, there is little guidance to address the process of building applications for new LSCO problems (independently of any language). This article gives an overview of the CHIC-2 methodology which aims at filling a gap in this direction. In particular, we describe some management issues specific to LSCOs such as risk management and team structures, and focus on the technical development guidance for scoping, designing and implementing LSCO appl...

Over the last years, interest on hybrid metaheuristics has risen considerably in the field of optimization. The best results found for many real-life or classical optimization problems are obtained by hybrid algorithms. Combinations of algorithms such as metaheuristics, mathematical programming, constraint programming and machine learning techniques have provided very powerful search algorithms. Four different types of combinations are considered in this chapter:

In this article we are particularly interested in optimizing the assignment of containers of different sizes to wagons on a train while respecting the business constraints. We present a model of the problem based on Constraint Programming (CP) techniques.

The ability to solve various constraints is a principal factor of automatic constraint solvers. Most object-oriented languages treat a character string as a primitive data type which is manipulated by string library functions. Most constraint solvers have limitations on their input constraints, such as strong restrictions on the expressiveness of constraints or lack of the ability to solve hybrid constraints. These limitations hinder applying automated constraint solvers on program analysis techniques for programs containing strings and string manipulation functions. We propose an approach to automatically solve program constraints involving strings and string manipulation functions. Based on the character array model, we design a constraint language which contains primitive operations to precisely describe the constraints of commonly used string manipulation functions. The translated string constraints together with numeric constraints are then solved by a two-phase test generation procedure: firstly, a partial solution is obtained to satisfy the arithmetic constraints of the position variables, and the solution is utilized to simplify the string constraints into pure character array constraints; secondly, the pure array constraints are solved by an off-the-shelf array-specific theory based constraint solver. We integrate the approach into an automated testing tool to support the generation of string test cases, and then perform experiments. The results of the experiments prove that the integration of the proposed approach promotes the testing coverage of the existing testing tool, and the integrated tool has an advantage of handling specific string manipulation functions compared with an existing string solver.

Different approaches in the hybridization of constraint programming and local search techniques have been recently proposed in the literature. In this paper we investigate two of them, namely the employment of local search to improve a solution found by constraint programming and the exploitation of a constraint model to perform the exploration of the local neighborhood. We apply the two approaches to a real-world personnel rostering problem arising at the department of neurology of the Udine University hospital and we report on computational studies on both real-world and randomly generated structured instances. The results highlight the benefits of the hybridization approach w.r.t. their component algorithms.

This paper introduces a high level language that allows for the specification and manipulation of solver independent models and allows for easily generating complex solvers in the Comet language. As Constraint Programming (CP) techniques have increased in complexity, it has become more difficult and time consuming to implement models that take advantage of state-of-the-art modeling techniques and search heuristics. This is particularly problematic for problems that have not been well studied as it is often unclear a priori which modeling technologies and search strategies will be effective. This work builds on previous solver independent languages by introducing a more general framework based on abstract models and model operators. Model operators represent complex model transformations that can be applied in various combinations to yield a wide array of concrete solvers, including hybrid solvers. Furthermore, Local Search (LS) is fully supported allowing for sequential and parallel bounds-passing hybrids that have not been possible in previous solver independent languages. Large Neighborhood Search (LNS) and column generation based models are also demonstrated.

We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLib we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.