Constraint Programming Methods Research Articles

CP and MIP Methods for Ship Scheduling with Time-Varying Draft

Existing ship scheduling approaches either ignore constraints on ship draft (distance between the waterline and the keel), or model these in very simple ways, such as a constant draft limit that does not change with time. However, in most ports the draft restriction changes over time due to variation in environmental conditions. More accurate consideration of draft constraints would allow more cargo to be scheduled for transport on the same set of ships. We present constraint programming (CP) and mixed integer programming (MIP) models for the problem of scheduling ships at a port with time-varying draft constraints so as to optimise cargo throughput at the port. We also investigate the effect of several variations to the CP model, including a model containing sequence variables, and a model with ordered inputs. Our model allows us to solve realistic instances of the problem to optimality in a very short time, and produces better schedules than both scheduling with constant draft, and manual scheduling approaches used in practice at ports.

Improvements of constraint programming and hybrid methods for scheduling of tests on vehicle prototypes

In the automotive industry, a manufacturer must perform several hundreds of tests on prototypes of a vehicle before starting its mass production. Tests must be allocated to suitable prototypes and ordered to satisfy temporal constraints and various kinds of test dependencies. The manufacturer aims to minimize the number of prototypes required. We present improvements of constraint programming (CP) and hybrid approaches to effectively solve random instances from an existing benchmark. CP mostly achieves better solutions than the previous heuristic technique and genetic algorithm. We also provide customized search schemes to enhance the performance of general search algorithms. The hybrid approach applies mixed integer linear programming (MILP) to solve the planning part and CP to find the complete schedule. We consider several logical principles such that the MILP model can accurately estimate the prototype demand, while its size particularly for large instances does not exceed memory capacity. Moreover, the robustness is alleviated when we allow CP to partially change the allocation obtained from the MILP model. The hybrid method can contribute to optimal solutions in some instances.

Sports scheduling: Problems and applications

AbstractSports scheduling problems mainly consist in determining the date and the venue in which each game of a tournament will be played. Integer programming, constraint programming, metaheuristics, and hybrid methods have been successfully applied to the solution of different variants of this problem. This paper provides an introductory review of fundamental problems in sports scheduling and their formulations, followed by a survey of applications of optimization methods to scheduling problems in professional leagues of different sport disciplines such as football, baseball, basketball, cricket, and hockey. A case study illustrates a real‐life application of integer programming to the schedule of the yearly Brazilian football tournament.

Optimal methods for resource allocation and scheduling: a cross-disciplinary survey

Classical scheduling formulations typically assume static resource requirements and focus on deciding when to start the problem activities, so as to optimize some performance metric. In many practical cases, however, the decision maker has the ability to choose the resource assignment as well as the starting times: this is a far-from-trivial task, with deep implications on the quality of the final schedule. Joint resource assignment and scheduling problems are incredibly challenging from a computational perspective. They have been subject of active research in Constraint Programming (CP) and in Operations Research (OR) for a few decades, with quite difference techniques. Both the approaches report individual successes, but they overall perform equally well or (from a different perspective) equally poorly. In particular, despite the well known effectiveness of global constraints for scheduling, comparable results for joint filtering of assignment and scheduling variables have not yet been achieved. Recently, hybrid approaches have been applied to this class of problems: most of them work by splitting the overall problem into an assignment and a scheduling subparts; those are solved in an iterative and interactive fashion with a mix of CP and OR techniques, often reporting impressive speed-ups compared to pure CP and OR methods. Motivated by the success of hybrid algorithms on resource assignment and scheduling, we provide a cross-disciplinary survey on such problems, including CP, OR and hybrid approaches. The main effort is to identify key modeling and solution techniques: they may then be applied in the construction of new hybrid algorithms, or they may provide ideas for novel filtering methods (possibly based on decomposition, or on alternative representations of the domain store). In detail, we take a constraint-based perspective and, according to the equation CP = model + propagation + search, we give an overview of state-of-art models, propagation/bounding techniques and search strategies.

Large-Scale Evacuation Using Subway and Bus Transit: Approach and Application in City of Toronto

Public transportation systems play a significant role in emergency evacuation. Therefore, this paper is geared towards harnessing subway and bus transit to alleviate congestion pressure during evacuation of busy urban areas. Routing and scheduling of transit vehicles and subway operation is envisioned as a new variant of the well-established vehicle routing problem. The model presented in this paper combines multiple variants of the traditional vehicle routing problem while reflecting on the operational characteristics during emergency evacuation, to include (1) multiple depots to better distribute the transit fleet, (2) time constraints to account for the evacuation time window, and (3) constraints for pick-up and delivery locations of evacuees. The evacuation problem is hereafter defined as a multi-depot time-constrained pick-up delivery vehicle route problem. A framework, using constraint programming and local search methods, was developed to model and solve the problem. An optimal spatio-temporal evacuation model was performed first to optimize evacuation of background vehicular traffic, generating transit travel cost (i.e., link travel times) as an input to the evacuation problem. The methodology was applied to evacuate the entire city of Toronto. The results show that the Toronto Transit Commission fleet is capable of evacuating the transit-dependent population (1.34 million) within 2 h on average. The four subway lines of the city of Toronto carry approximately 0.62 million people and can evacuate these people in less than 3 h on average. Toronto Transit Commission shuttle buses (1,320 vehicles) can evacuate the remainder of the transit-dependent population (0.72 million) in approximately 1.5 h on average.

A New Graph-Theoretical Model for the Guillotine-Cutting Problem

We consider the problem of determining whether a given set of rectangular items can be cut from a larger rectangle using so-called guillotine cuts only. We introduce a new class of arc-colored directed graphs called guillotine graphs and show that each guillotine graph can be associated with a specific class of pattern solutions that we call a guillotine-cutting class. The properties of guillotine graphs are examined, and some effective algorithms for dealing with guillotine graphs are proposed. As an application, we then describe a constraint programming method based on guillotine graphs, and we propose effective filtering techniques that use the graph model properties in order to reduce the search space efficiently. Computational experiments are reported on benchmarks from the literature: our algorithm outperforms previous methods when solving the most difficult instances exactly.

Solving Vehicle Routing Problems Using Constraint Programming and Lagrangean Relaxation in a Metaheuristics Framework

This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology provides a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem.

Preface to the special issue

The series “International Symposium on Artificial Intelligence and Mathematics” is held every two years in Fort Lauderdale. While the general scope of the series is broad, and even includes the interaction of AI with Operations Research, each meeting typically includes a special focus on one or two particular topics. In the 2010 meeting, about half the papers were in the track Boolean and pseudo-Boolean functions which was organized by Endre Boros and Yves Crama, another four were in the track Constraint programming and hybrid methods organized by John Hooker, and the remaining papers covered a wide range of topics that tie AI to mathematics. This special issue of AMAI contains four papers from the third track and one from the first track, chosen by the organizers to be expanded and polished into journal versions by their excellence in written and oral presentation. Two papers are on Bayesian updating, one on the multi-armed bandit problem, one on concept analysis in data mining, and one on minimizing clausal data bases. In what follows, we try to set the context for each of these papers. A Bayesian network is a probabilistic directed acyclic graph for modeling a set of random variables and their conditional dependencies, for example, between a set of diseases and symptoms. Bayesian networks may be used to generate explanations for given evidence, such as finding the most likely instantiation of a set of target variables given partial evidence on the remaining variables. Most Relevant Explanation (MRE) is the problem of generating an explanation that emphasizes the role of the most relevant target variables: if e is a set of observed variable values and if y1, y2, ... are all possible instantiations of all possible subsets of target variables, then MRE = maxi Pr(e|yi)/Pr(e|¬yi). In some respects, MRE estimates may be better than other Bayesian estimates. However, in the paper “Most Relevant Explanation: Computational complexity and approximation methods”, Yuan, Lim, and Littman show that MRE is NP-hard. The authors then propose and evaluate

Advanced planning and scheduling technology

This article presents research on the core technology in advanced planning and scheduling (APS) systems that are currently the state-of-art in supply chain planning and optimisation. Further research is needed to enable the opportunities given, especially the area of combined planning and scheduling. Mixed integer linear programming (MILP) methods have been successfully applied to solve diverse problems, such as planning and capital budgeting. Constraint programming (CP) methods have proved to be successful in solving highly constrained discrete optimisation and feasibility problems for scheduling and resource allocation and have been combined with MILP and titled Mixed linear logic programming (MLLP). In this article an integrated approach is presented that uses the complementary strengths of MILP and CP for solving problems that are otherwise intractable using either of these two methods alone. The presented optimisation approach can effectively handle the combined planning and scheduling problem within an APS system as part of the core optimisation engine.

A fuzzy-stochastic Constraint Programming Model for Hazmat Road Transportation Considering Terrorism Attacking

achieved on Hazmat transportation routing problems in order to minimize risk-related costs which is consequence of a natural accident. As the terrorist threat is growing, Hazmat shipment has became a new target of terrorist attacks, it is therefore necessary to consider the terrorist threat into the Hazmat routing model. In this work, we consider the following hazmat transportation scheduling problem in traffic engineering: a hazmat shipment has to be shipped over a road transportation network in order to transport a given amount of hazardous materials from a specific origin point to a specific destination point every day. We design a route to minimize the risks from the natural accident and the uncertainty of a terrorist threat, and develop a fuzzy-stochastic constraint programming method to deal with multiple uncertainties presented in terms of fuzzy sets, interval values and random variables which representing the probabilities and fatality rates of the terrorist attacking. Finally, it illustrates the methodology on a case study.

Growth Patterns in Incremental Self-Build Housing in Chile

Over the past three decades, a small community of eighty-four Chilean low-income families has built and improved their home incrementally, without any technical assistance, showing an impressive performance. A six square meters bathroom on a serviced plot of land with individual connection to potable water, sewerage, electricity and access roads, worked as a starting point back in 1974. However particular their rationale may seem, the individual history of their housing process reveals some general regularities in occurrence and duration of self-build activities, as well as size and allocation of the domestic spaces. A small random sample of fifteen households was selected to tell the story and explain the whys, hows, and whens of an ever-evolving housing process. Semi-structured interviews and building surveys were both combined to reconstruct the sequence of states of each housing process, with the awareness of the characteristic imprecision of oral information transfer. Alternative states were explored by constraint programming methods and spatial qualitative reasoning. Considering the hard constraints over the site morphology and services allocation, the results of the exploration stress how extraordinary lucid and intuitive the surveyed families are when making their design decisions. The article exposes a reconstructive case study on spontaneous growth patterns underlying an unassisted, incremental self-build housing dynamics.

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.

A Constraint Programming Approach for Solving a Queueing Control Problem

In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting time is minimized while the expected back room staffing is sufficient to perform all work. Three novel constraint programming models and several shaving procedures for these models are presented. Experimental results show that a model based on closed-form expressions together with a combination of shaving procedures is the most efficient. This model is able to find and prove optimal solutions for many problem instances within a reasonable run-time. Previously, the only available approach was a heuristic algorithm. Furthermore, a hybrid method combining the heuristic and the best constraint programming method is shown to perform as well as the heuristic in terms of solution quality over time, while achieving the same performance in terms of proving optimality as the pure constraint programming model. This is the first work of which we are aware that solves such queueing-based problems with constraint programming.

Models and methods for frequency assignment with cumulative interference constraints

AbstractIn this paper, a realistic modeling of interferences for frequency assignment in hertzian telecommunication networks is presented. In contrast with traditional interference models based only on binary interference constraints involving two frequencies, this new approach considers the case of cumulative disruptions that are modeled through a unique non‐binary constraint. To deal with these complex constraints, we propose extensions of classical integer linear programming formulations. On a set of realistic instances, we propose hybrid constraint programming and large neighborhood search solution methods to solve minimum interference and minimum span frequency assignment problems. We compare their performances with those of existing heuristics. Finally, we show how the end‐user benefits from using the cumulative model instead of the traditional one.

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

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...

A general heuristic for vehicle routing problems

We present a unified heuristic which is able to solve five different variants of the vehicle routing problem: the vehicle routing problem with time windows (VRPTW), the capacitated vehicle routing problem (CVRP), the multi-depot vehicle routing problem (MDVRP), the site-dependent vehicle routing problem (SDVRP) and the open vehicle routing problem (OVRP). All problem variants are transformed into a rich pickup and delivery model and solved using the adaptive large neighborhood search (ALNS) framework presented in Ropke and Pisinger [An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transportation Science, to appear]. The ALNS framework is an extension of the large neighborhood search framework by Shaw [Using constraint programming and local search methods to solve vehicle routing problems. In: CP-98, Fourth international conference on principles and practice of constraint programming, Lecture notes in computer science, vol. 1520, 1998. p. 417–31] with an adaptive layer. This layer adaptively chooses among a number of insertion and removal heuristics to intensify and diversify the search. The presented approach has a number of advantages: it provides solutions of very high quality, the algorithm is robust, and to some extent self-calibrating. Moreover, the unified model allows the dispatcher to mix various variants of VRP problems for individual customers or vehicles. As we believe that the ALNS framework can be applied to a large number of tightly constrained optimization problems, a general description of the framework is given, and it is discussed how the various components can be designed in a particular setting. The paper is concluded with a computational study, in which the five different variants of the vehicle routing problem are considered on standard benchmark tests from the literature. The outcome of the tests is promising as the algorithm is able to improve 183 best known solutions out of 486 benchmark tests. The heuristic has also shown promising results for a large class of vehicle routing problems with backhauls as demonstrated in Ropke and Pisinger [A unified heuristic for a large class of vehicle routing problems with backhauls. European Journal of Operational Research, 2004, to appear].

Logic-based Benders decomposition

Benders decomposition uses a strategy of ``learning from one's mistakes.'' The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an ``inference dual.'' Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out the details for propositional satisfiability and 0-1 programming problems. Computational tests are carried out for the latter, but the most promising contribution of logic-based Benders may be to provide a framework for combining optimization and constraint programming methods.

Two FMS scheduling methods based on Petri nets: A global and a local approach

We are interested in the Flexible Manufacturing System (FMS) scheduling problem. Different methods have been explored to solve this problem and to master its combinatorial complexity, which is NP-hard in the general case. In this paper we will give two different scheduling methods based on Petri nets. The first one tends to solve the general scheduling problem (acyclic schedule) using the Constraint Programming method to avoid exhaustive search. The second method is a dedicated cyclic scheduling method. The aim is not to compare the methods' performances (computation time, results' quality) because they do not solve exactly the same problem, but to compare their application domains in terms of parts number.

Decomposition techniques for multistage scheduling problems using mixed-integer and constraint programming methods

In this paper two strategies are presented to reduce the combinatorial complexity when solving single stage and multistage optimization scheduling problems that involve cost minimization and due dates. These problems can naturally be decomposed into assignment and sequencing subproblems. The proposed strategies rely on either combining mixed-integer programming (MILP) to model the assignment part and constraint programming (CP) for modeling the sequencing part, or else combining MILP models for both parts. The subproblems are solved sequentially by adding integer cuts to the first MILP to generate new assignments. Results are presented for both single and multistage systems.

A Declarative Modeling Framework that Integrates Solution Methods

Constraint programming offers modeling features and solution methods that are unavailable in mathematical programming but are often flexible and efficient for scheduling and other combinatorial problems. Yet mathematical programming is well suited to declarative modeling languages and is more efficient for some important problem classes. This raises this issue as to whether the two approaches can be combined in a declarative modeling framework. This paper proposes a general declarative modeling system in which the conditional structure of the constraints shows how to integrate any “checker” and any special-purpose “solver”. In particular this integrates constraint programming and optimization methods, because the checker can consist of constraint propagation methods, and the solver can be a linear or nonlinear programming routine.