Showing papers on "Constraint programming published in 2007"

Nonlinear Programming Theory and Algorithms

Abstract: (2007). Nonlinear Programming Theory and Algorithms. Technometrics: Vol. 49, No. 1, pp. 105-105.

MiniZinc: towards a standard CP modelling language

Abstract: There is no standard modelling language for constraint programming (CP) problems. Most solvers have their own modelling language. This makes it difficult for modellers to experiment with different solvers for a problem. In this paper we present MiniZinc, a simple but expressive CP modelling language which is suitable for modelling problems for a range of solvers and provides a reasonable compromise between many design possibilities. Equally importantly, we also propose a low-level solver-input language called FlatZinc, and a straightforward translation from MiniZinc to FlatZinc that preserves all solver-supported global constraints. This lets a solver writer support MiniZinc with a minimum of effort-- they only need to provide a simple FlatZinc front-end to their solver, and then combine it with an existing MiniZinc-to-FlatZinc translator. Such a front-end may then serve as a stepping stone towards a full MiniZinc implementation that is more tailored to the particular solver. A standard language for modelling CP problems will encourage experimentation with and comparisons between different solvers. Although MiniZinc is not perfect--no standard modelling language will be--we believe its simplicity, expressiveness, and ease of implementation make it a practical choice for a standard language.

Constraint Integer Programming

Abstract: This thesis introduces the novel paradigm of constraint integer programming (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method.

Integration of AI and OR techniques in constraint programming for combinatorial optimization problems : 4th International Conference, CPAIOR 2007, Brussels, Belgium, May 23-26, 2007 : proceedings

Abstract: Minimum Cardinality Matrix Decomposition into Consecutive-Ones Matrices: CP and IP Approaches.- Connections in Networks: Hardness of Feasibility Versus Optimality.- Modeling the Regular Constraint with Integer Programming.- Hybrid Local Search for Constrained Financial Portfolio Selection Problems.- The "Not-Too-Heavy Spanning Tree" Constraint.- Eliminating Redundant Clauses in SAT Instances.- Cost-Bounded Binary Decision Diagrams for 0-1 Programming.- YIELDS: A Yet Improved Limited Discrepancy Search for CSPs.- A Global Constraint for Total Weighted Completion Time.- Computing Tight Time Windows for RCPSPWET with the Primal-Dual Method.- Necessary Condition for Path Partitioning Constraints.- A Constraint Programming Approach to the Hospitals / Residents Problem.- Best-First AND/OR Search for 0/1 Integer Programming.- A Position-Based Propagator for the Open-Shop Problem.- Directional Interchangeability for Enhancing CSP Solving.- A Continuous Multi-resources cumulative Constraint with Positive-Negative Resource Consumption-Production.- Replenishment Planning for Stochastic Inventory Systems with Shortage Cost.- Preprocessing Expression-Based Constraint Satisfaction Problems for Stochastic Local Search.- The Deviation Constraint.- The Linear Programming Polytope of Binary Constraint Problems with Bounded Tree-Width.- On Boolean Functions Encodable as a Single Linear Pseudo-Boolean Constraint.- Solving a Stochastic Queueing Control Problem with Constraint Programming.- Constrained Clustering Via Concavity Cuts.- Bender's Cuts Guided Large Neighborhood Search for the Traveling Umpire Problem.- A Large Neighborhood Search Heuristic for Graph Coloring.- Generalizations of the Global Cardinality Constraint for Hierarchical Resources.- A Column Generation Based Destructive Lower Bound for Resource Constrained Project Scheduling Problems.

Constraint Logic Programming using Eclipse

Abstract: Providing an introduction to constraint programming, as well as a systematic introduction to the Eclipse system, this text shows how to write constraint programs that solve complex problems, and illustrates the power, versatility and utility of Eclipse.

Planning and Scheduling by Logic-Based Benders Decomposition

Abstract: We combine mixed-integer linear programming (MILP) and constraint programming (CP) to solve an important class of planning and scheduling problems. Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logic-based Benders decomposition. Tasks assigned to a facility may run in parallel subject to resource constraints (cumulative scheduling). We solve problems in which the objective is to minimize cost, makespan, or total tardiness. We obtain significant computational speedups, of several orders of magnitude for the first two objectives, relative to the state of the art in both MILP and CP. We also obtain better solutions and bounds for problems than cannot be solved to optimality.

UMLtoCSP: a tool for the formal verification of UML/OCL models using constraint programming

Abstract: We present UMLtoCSP, a tool for the formal verification of UML/OCL models. Given a UML class diagram annotated with OCL constraints, UMLtoCSP is able to automatically check several correctness properties, such as the strong and weak satisfiability of the model or the lack of redundant constraints. The tool uses Constraint Logic Programming as the underlying formalism and the constraint solver ECLiPSe as the verification engine.

Feasibility and Infeasibility in Optimization:: Algorithms and Computational Methods

Abstract: Constrained optimization models are core tools in business, science, government, and the military with applications including airline scheduling, control of petroleum refining operations, investment decisions, and many others. Constrained optimization models have grown immensely in scale and complexity in recent years as inexpensive computing power has become widely available. Models now frequently have many complicated interacting constraints, giving rise to a host of issues related to feasibility and infeasibility. For example, it is sometimes difficult to find any feasible point at all for a large model, or even to accurately determine if one exists, e.g. for nonlinear models. If the model is feasible, how quickly can a solution be found? If the model is infeasible, how can the cause be isolated and diagnosed? Can a repair to restore feasibility be carried out automatically? Researchers have developed numerous algorithms and computational methods in recent years to address such issues, with a number of surprising spin-off applications in fields such as artificial intelligence and computational biology. Over the same time period, related approaches and techniques relating to feasibility and infeasibility of constrained problems have arisen in the constraint programming community. Feasibility and Infeasibility in Optimization is a timely expository book that summarizes the state of the art in both classical and recent algorithms related to feasibility and infeasibility in optimization, with a focus on practical methods. All model forms are covered, including linear, nonlinear, and mixed-integer programs. Connections to related work in constraint programming are shown. Part I of the book addresses algorithms for seeking feasibility quickly, including new methods for the difficult cases of nonlinear and mixed-integer programs. Part II provides algorithms for analyzing infeasibility by isolating minimal infeasible (or maximum feasible) subsets of constraints, or by finding the best repair for the infeasibility. Infeasibility analysis algorithms have arisen primarily over the last two decades, and the book covers these in depth and detail. Part III describes applications in numerous areas outside of direct infeasibility analysis such as finding decision trees for data classification, analyzing protein folding, radiation treatment planning, automated test assembly, etc. A main goal of the book is to impart an understanding of the methods so that practitioners can make immediate use of existing algorithms and software, and so that researchers can extend the state of the art and find new applications. The book is of interest to researchers, students, and practitioners across the applied sciences who are working on optimization problems.

A constraint programming model for real-time train scheduling at junctions

Abstract: In this paper, we present a constraint programming model for the routing and scheduling of trains running through a junction. The model uses input data from relevant time events of train runs calculated by a simulator. The model can be integrated into a decision support system used by operators who make decisions to change train routes or orders to avoid conflicts and delays. The model has been applied to a set of problem instances. This set has been defined from a real case study of traffic on the Pierrefitte-Gonesse node, North of Paris. Preliminary results show that the solution identified by the model yields a significant improvement in performance within an acceptable computation time.

Using Decomposition Techniques and Constraint Programming for Solving the Two-Dimensional Bin-Packing Problem

Abstract: The two-dimensional bin-packing problem is the problem of orthogonally packing a given set of rectangles into a minimum number of two-dimensional rectangular bins. The problem is NP-hard and very difficult to solve in practice as no good mixed integer programming (MIP) formulation has been found for the packing problem. We propose an algorithm based on the well-known Dantzig-Wolfe decomposition where the master problem deals with the production constraints on the rectangles while the subproblem deals with the packing of rectangles into a single bin. The latter problem is solved as a constraint-satisfaction problem (CSP), which makes it possible to formulate a number of additional constraints that may be difficult to formulate as MIP models. This includes guillotine-cutting requirements, relative positions, fixed positions and irregular bins. The CSP approach uses forward propagation to prune inferior arrangements of rectangles. Unsuccessful attempts to pack rectangles into a bin are brought back to the master model as valid inequalities. Hence, CSP is used not only to solve the pricing problem but also to generate valid inequalities in a branch-and-cut system. Using delayed column-generation, we obtain lower bounds of very good quality in reasonable time. In all instances considered, we obtain similar or better bounds than previously published. Several instances with up to n = 100 rectangles are solved to optimality through the developed branch-and-price-and-cut algorithm.

CC-Pi: a constraint-based language for specifying service level agreements

Abstract: Service Level Agreements are a key issue in Service Oriented Computing. SLA contracts specify client requirements and service guarantees, with emphasis on Quality of Service (cost, performance, availability, etc.). In this work we propose a simple model of contracts for QoS and SLAs that also allows to study mechanisms for resource allocation and for joining different SLA requirements. Our language combines two basic programming paradigms: name-passing calculi and concurrent constraint programming (cc programming). Specifically, we extend cc programming by adding synchronous communication and by providing a treatment of names in terms of restriction and structural axioms closer to nominal calculi than to variables with existential quantification. In the resulting framework, SLA requirements are constraints that can be generated either by a single party or by the synchronisation of two agents. Moreover, restricting the scope of names allows for local stores of constraints, which may become global as a consequence of synchronisations. Our approach relies on a system of named constraints that equip classical constraints with a suitable algebraic structure providing a richer mechanism of constraint combination. We give reduction-preserving translations of both cc programming and the calculus of explicit fusions.

Municipal Solid Waste Management Under Uncertainty: A Mixed Interval Parameter Fuzzy-Stochastic Robust Programming Approach

Abstract: A mixed interval parameter fuzzy-stochastic robust programming (MIFSRP) model is developed and applied to the planning of solid waste management systems under uncertainty The MIFSRP can explicitly address system uncertainties with multiple presentations It can be used as an extension of the existing interval-parameter fuzzy robust programming, interval-parameter linear programming, and chance constraint programming methods In this MIFSRP model, the hybrid uncertainties can be directly communicated into the optimization process and resulting solution through representing the uncertain parameters as interval numbers and fuzzy membership functions with random characteristics Highly uncertain information arising from simultaneous appearance of fuzziness and randomness for the lower and upper bounds of interval parameters can be effectively addressed through integrating chance constraint programming, interval linear programming, and fuzzy robust programming methods into a general optimization framework Th

Workforce optimization: identification and assignment of professional workers using constraint programming

Abstract: Matching highly skilled people to available positions is a high-stakes task that requires careful consideration by experienced resource managers. A wrong decision may result in significant loss of value due to understaffing, underqualification or overqualification of assigned personnel, and high turnover of poorly matched workers. While the importance of quality matching is clear, dealing with pools of hundreds of jobs and resources in a dynamic market generates a significant amount of pressure to make decisions rapidly. We present a novel solution designed to bridge the gap between the need for high-quality matches and the need for timeliness. By applying constraint programming, a subfield of artificial intelligence, we are able to deal successfully with the complex constraints encountered in the field and reach near-optimal assignments that take into account all resources and positions in the pool. The considerations include constraints on job role, skill level, geographical location, language, potential retraining, and many more. Constraints are applied at both the individual and team levels. This paper introduces the technology and then describes its use by IBM Global Services, where large numbers of service and consulting employees are considered when forming teams assigned to customer projects.

From precedence constraint posting to partial order schedules: A CSP approach to Robust Scheduling

Abstract: Constraint-based approaches to scheduling have typically formulated the problem as one of finding a consistent assignment of start times for each goal activity. In contrast, we are pursuing an approach that operates with a problem formulation more akin to least-commitment frameworks, where the objective is to post sufficient additional precedence constraints between pairs of activities contending for the same resources to ensure feasibility with respect to time and resource constraints. One noteworthy characteristic of this Precedence Constraint Posting (PCP) approach, is that solutions generated in this way generally encapsulate a set of feasible schedules (i.e., a solution contains the sets of activity start times that remain consistent with posted sequencing constraints). Such solutions can offer advantages when there is temporal uncertainty associated with executing activities. In this paper, we consider the problem of generating temporally flexible schedules that possess good robustness properties. We first introduce the concept of a Partial Order Schedule (POS), a type of temporally flexible schedule in which each embedded temporal solution is also guaranteed to be resource feasible, as a target class of solutions that exploit flexibility in a robust way. We then present and analyze two PCP-based methods for generating POSs. The first method uses a pure least commitment approach, where the set of all possible time-feasible schedules is successively winnowed into a smaller resource-feasible set. The second method alternatively utilizes a focused analysis of one possible solution, and first generates a single, resource-feasible, fixed-times schedule. This point solution is then transformed into a POS in a second post-processing phase. Somewhat surprisingly, this second method is found to be a quite effective means of generating robust schedules.

Data structures for generalised arc consistency for extensional constraints

Abstract: Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised arc consistency on extensional constraints. The first, the Next-Difference list, is novel and has been developed with this application in mind. The second, the trie, is well known but its use in this context is novel. Empirical analyses demonstrate the efficiency of the resulting approaches, both in GAC-schema, and in the watched-literal table constraint in Minion.

FSTTCS 2007 : foundations of software technology and theoretical computer science : 27th International Conference, New Delhi, India, December 12-14, 2007 : proceedings

Abstract: Invited Papers.- The Multicore Revolution.- Streaming Algorithms for Selection and Approximate Sorting.- Adventures in Bidirectional Programming.- Program Analysis Using Weighted Pushdown Systems.- The Complexity of Zero Knowledge.- Contributed Papers.- The Priority k-Median Problem.- "Rent-or-Buy" Scheduling and Cost Coloring Problems.- Order Scheduling Models: Hardness and Algorithms.- On Simulatability Soundness and Mapping Soundness of Symbolic Cryptography.- Key Substitution in the Symbolic Analysis of Cryptographic Protocols.- Symbolic Bisimulation for the Applied Pi Calculus.- Non-mitotic Sets.- Reductions to Graph Isomorphism.- Strong Reductions and Isomorphism of Complete Sets.- Probabilistic and Topological Semantics for Timed Automata.- A Theory for Game Theories.- An Incremental Bisimulation Algorithm.- Logspace Algorithms for Computing Shortest and Longest Paths in Series-Parallel Graphs.- Communication Lower Bounds Via the Chromatic Number.- The Deduction Theorem for Strong Propositional Proof Systems.- Satisfiability of Algebraic Circuits over Sets of Natural Numbers.- Post Embedding Problem Is Not Primitive Recursive, with Applications to Channel Systems.- Synthesis of Safe Message-Passing Systems.- Automata and Logics for Timed Message Sequence Charts.- Propositional Dynamic Logic for Message-Passing Systems.- Better Algorithms and Bounds for Directed Maximum Leaf Problems.- Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs.- Covering Graphs with Few Complete Bipartite Subgraphs.- Safely Composing Security Protocols.- Computationally Sound Typing for Non-interference: The Case of Deterministic Encryption.- Bounding Messages for Free in Security Protocols.- Triangulations of Line Segment Sets in the Plane.- Reconstructing Convex Polygons and Polyhedra from Edge and Face Counts in Orthogonal Projections.- Finding a Rectilinear Shortest Path in R 2 Using Corridor Based Staircase Structures.- Compressed Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space.- Stochastic Muller Games are PSPACE-Complete.- Solving Parity Games in Big Steps.- Efficient and Expressive Tree Filters.- Markov Decision Processes with Multiple Long-Run Average Objectives.- A Formal Investigation of Diff3.- Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem.- Undirected Graphs of Entanglement 2.- Acceleration in Convex Data-Flow Analysis.- Model Checking Almost All Paths Can Be Less Expensive Than Checking All Paths.- Closures and Modules Within Linear Logic Concurrent Constraint Programming.

Decomposing global grammar constraints

Abstract: A wide range of constraints can be specified using automata or formal languages. The GRAMMAR constraint restricts the values taken by a sequence of variables to be a string from a given context-free language. Based on an AND/OR decomposition, we show that this constraint can be converted into clauses in conjunctive normal form without hindering propagation. Using this decomposition, we can propagate the GRAMMAR constraint in O(n3) time. The decomposition also provides an efficient incremental propagator. Down a branch of the search tree of length k, we can enforce GAC k times in the same O(n3) time. On specialized languages, running time can be even better. For example, propagation of the decomposition requires just O(n|δ|) time for regular languages where |δ| is the size of the transition table of the automaton recognizing the regular language. Experiments on a shift scheduling problem with a constraint solver and a state of the art SAT solver show that we can solve problems using this decomposition that defeat existing constraint solvers.

Solving two-stage stochastic programming problems with level decomposition

Abstract: We propose a new variant of the two-stage recourse model. It can be used e.g., in managing resources in whose supply random interruptions may occur. Oil and natural gas are examples for such resources. Constraints in the resulting stochastic programming problems can be regarded as generalizations of integrated chance constraints. For the solution of such problems, we propose a new decomposition method that integrates a bundle-type convex programming method with the classic distribution approximation schemes. Feasibility and optimality issues are taken into consideration simultaneously, since we use a convex programming method suited for constrained optimization. This approach can also be applied to traditional two-stage problems whose recourse functions can be extended to the whole space in a computationally efficient way. Network recourse problems are an example for such problems. We report encouraging test results with the new method.

Algorithmic aspects of stable matching problems

Abstract: The Stable Marriage problem (SM), the Hospitals/Residents problem (HR) and the Stable Roommates problem (SR) are three classical stable matching problems that were first studied by Gale and Shapley in 1962. These problems have widespread practical application in centralised automated matching schemes, which assign applicants to posts based on preference lists and capacity constraints in both the UK and internationally. Within such schemes it is often the case that an agent's preference list may be incomplete, and agents may also be allowed to express indifference in the form of ties. In the presence of ties, three stability criteria can be defined, namely weak stability, strong stability and super-stability. In this thesis we consider stable matching problems from an algorithmic point of view. Some of the problems that we consider are derived from new stable matching models, whilst others are obtained from existing stable matching models involving ties and incomplete lists, with additional natural restrictions on the problem instance. Furthermore, we also explore the use of constraint programming with both SM and HR. We first study a new variant of the Student-Project Allocation problem in which each student ranks a set of acceptable projects in preference order and similarly each lecturer ranks his available projects in preference order. In this context, two stability definitions can be identified, namely weak stability and strong stability. We show that the problem of finding a maximum weakly stable matching is NP-hard. However, we describe two 2-approximation algorithms for this problem. Regarding strong stability, we describe a polynomial-time algorithm for finding such a matching or reporting that none exists. Next we investigate SM with ties and incomplete lists (SMTI), and HR with ties (HRT), where the length of each agent's list is subject to an upper bound. We present both polynomial-time algorithms and NP-hardness results for a range of problems that are derived from imposing upper bounds on the length of the lists on one or both sides. We also consider HRT, and SR with ties and incomplete lists (SRTI), where the preference lists of one or both sets of agents (as applicable) are derived from one or two master lists in which agents are ranked. For super-stability, in the case of each of HRT and SRTI with a master list, we describe a linear-time algorithm that simplifies the algorithm used in the general case. In the case of strong stability, for each of HRT and SRTI with a master list, we describe an algorithm that is faster than that for the general case. We also show that, given an instance I of SRTI with a master list, the problem of finding a weakly stable matching is polynomial-time solvable. However, we show that given such an I, the problem of finding a maximum weakly stable matching is NP-hard. Other new stable matching models that we study are the variants of SMTI and SRTI with symmetric preferences. In this context we consider two models that are derived from alternative ways of interpreting the rank of an agent in the presence of ties. For both models we show that deciding if a complete weakly stable matching exists is NP-complete. Then for one of the models we show that each of the problem of finding a minimum regret and an egalitarian weakly stable matching is NP-hard and that the problem of determining if a (man,woman) pair belongs to a weakly stable matching is NP-complete. We then describe algorithms for each of the problems of finding a super-stable matching and a strongly stable matching, or reporting that none exists, given instances of SRTI and HRT with symmetric preferences (regardless of how the ranks are interpreted). Finally, we use constraint programming techniques to model instances of SM and HR. We describe two encodings of SM in terms of a constraint satisfaction problem. The first model for SM is then extended to the case of HR. This encoding for HR is then extended to create a model for HRT under weak stability. Using this encoding we can obtain, with the aid of search, all the weakly stable matchings, given an instance of HRT.

Air-traffic complexity resolution in multi-sector planning using constraint programming

Abstract: Using constraint programming, we effectively model and efficiently solve the problem of balancing and minimising the traffic complexities of an airspace of adjacent sectors. The traffic complexity of a sector is here defined in terms of the numbers of flights within it, near its border, and on non-level segments within it. The allowed forms of complexity resolution are the changing of the take-off times of not yet airborne flights, the changing of the remaining approach times into the chosen airspace of already airborne flights by slowing down and speeding up within the two layers of feeder sectors around that airspace, as well as the changing of the levels of passage over way-points in that airspace. Experiments with actual European flight profiles obtained from the Central Flow Management Unit (CFMU) show that these forms of complexity resolution can lead to significant complexity reductions and rebalancing.

Parallelizing constraint programs transparently

Abstract: The availability of commodity multi-core andmulti-processor machines and the inherent parallelism inconstraint programming search offer significant opportunities for constraint programming. They also present a fundamental challenge: how to exploit parallelism transparently to speed up constraint programs. This paper shows how to parallelize constraint programs transparently without changes to the code. The main technical idea consists of automatically lifting a sequential exploration strategy into its parallel counterpart, allowing workers to share and steal subproblems. Experimental results showthat the parallel implementationmay produces significant speedups on multi-core machines.

User-definable rule priorities for CHR

Abstract: This paper introduces CHRrp: Constraint Handling Rules with user-definable rule priorities. CHRrp offers flexible execution control which is lacking in CHR. A formal operational semantics for the extended language is given and is shown to be an instance of the theoretical operational semantics of CHR. It is discussed how the CHR rp semantics influences confluence results. A translation scheme for CHRrp programs with static rule priorities into (regular) CHR is presented. The translation is proven correct and bench-mark results are given. CHRrp is related to priority systems in other constraint programming and rule based languages.

Modeling the Regular Constraint with Integer Programming

Abstract: Many optimisation problems contain substructures involving constraints on sequences of decision variables. Such constraints can be very complex to express with mixed integer programming (MIP), while in constraint programming (CP), the global constraint regular easily represents this kind of substructure with deterministic finite automata (DFA). In this paper, we use DFAs and the associated layered graph structure built for the regular constraint consistency algorithm to develop a MIP version of the constraint. We present computational results on an employee timetabling problem, showing that this new modeling approach can significantly decrease computational times in comparison with a classical MIP formulation.

Minimum Cardinality Matrix Decomposition into Consecutive-Ones Matrices: CP and IP Approaches

Abstract: We consider the problem of decomposing an integer matrix into a positively weighted sum of binary matrices that have the consecutive-ones property. This problem is well-known and of practical relevance. It has an important application in cancer radiation therapy treatment planning: the sequencing of multileaf collimators to deliver a given radiation intensity matrix, representing (a component of) the treatment plan. Two criteria characterise the efficacy of a decomposition: the beam-on time(length of time the radiation source is switched on during the treatment), and the cardinality(the number of machine set-ups required to deliver the planned treatment). Minimising the former is known to be easy. However finding a decomposition of minimal cardinality is NP-hard. Progress so far has largely been restricted to heuristic algorithms, mostly using linear programming, integer programming and combinatorial enumerative methods as the solving technologies. We present a novel model, with corresponding constraint programming and integer programming formulations. We compare these computationally with previous formulations, and we show that constraint programming performs very well by comparison.

A constraint programming approach for the resource-constrained project scheduling problem

Abstract: A “pure” Constraint Programming approach for the Resource-Constrained Project Scheduling Problem (RCPSP) is presented. Our basic idea was to substitute the resource constraints by a set of “sub-constraints” generated as needed. Each of these sub-constraints corresponds to a set of tasks that cannot be executed together without violating one of the resource constraints. A filtering algorithm for these sub-constraints has been developed. When applied to the initial resource constraints together with known filtering algorithms, this new filtering algorithm provides very good numerical results.