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Dimensioning an inbound call center using constraint programming

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Canon, Cyril Billaut, Jean-Charles Bouquard, Jean-Louis Vitalicom 643 avenue du grain d'or 41350 Vineuil France Université François-Rabelais de Tours Laboratoire d'Informatique 64 avenue Jean Portails 37200 Tours France

ISBN: (纸本)3540292381

One of the critical problems in the call center industries is the staffing problem, since they must face variable demands and because staff costs represent a major part of the costs of these industries. Prom a modeling point of view, a call center is generally modeled as a M/M/N system, also called the Erlang-C model. In [Koole and Mandelbaurn, 2001], the authors present a survey of the state-of-the-art about possible models of a call center. In this paper, the problem of dimensioning a call center is modeled as a deterministic scheduling problem, where resources represent the employees and jobs correspond to the calls that are received. To each resource are associated tools that correspond to the skills of employees, and a weigth corresponding to the cost of employment. To each job are associated a release date, a processing time, a deadline and a required tool. the objective is to schedule jobs so as to minimize the weighted number of resources needed to perform the jobs. For solving this problem, we propose exact methods based on constraint programming Models. three different models are presented. the second one is an improvement of the first one, both in terms of constraints definitions and instanciation methods. the last one introduces redundant constraints to speed up the resolution process. An approximated algorithm is also proposed, based on a priority rule and an assignment rule. We also compare our models to a Mixed Integer Linear Program, solved by CPLEX. Instances are randomly generated with around 45 resources and from 20 to 200 jobs. Feasible solutions are always obtained within two minutes by using these methods. CP models are solved using Eclipse 5.8-77 software with the time limit fixed to two minutes on a PC Pentium III, 1.4 GHz, 512 Mo. CP can solve the problem optimally with up to 80 jobs. For more than 100 jobs, the upper bound given by the heuristic algorithm is never improved within two minutes. © Springer-Verlag Berlin Heidelberg 2005.

关键词： constraint theory

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Uncertainty in soft constraint problems

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Pini, Maria Silvia Rossi, Francesca University of Padova Italy

ISBN: (纸本)3540292381

Preferences and uncertainty occur in many real-life problems. We are concerned with the coexistence of preference and uncertainty in the same problem. In particular, we consider uncertainty, defined by the theory of possibility [2], that is one non-probabilistic way of dealing with uncertainty, that, comes from lack of data or imprecise knowledge. We propose a method to integrate fuzzy preferences and uncertainty [3], which follows the approach in [2] and allows one to handle uncertainty within a fuzzy optimization engine, which at the same time observing separately the preference and the robustness of complete instantiations. To order the complete instantiations, we define three semantics, which correspond to the attitude to the risk. We generalize this method in [4] to deal also with other classes of preferences such as probabilistic or weighted constraints [1], and also probabilistic uncertainty. In particular, probabilistic constraints are very useful for modelling real-life scenarios where fuzzy constraints are not the ideal setting, because combining preferences multiplying them is better than taking their minimum value. Weighted constraints are useful to model problems where preferences are penalties (or costs) to be added, and where the best preferences are the smallest ones. We plan to develop a solver that can handle problems with several classes of soft constraints, together with uncertainty expressed via possibility or probability distributions. the solver will be able to generate solution orderings according the our semantics as well as others that we will define by following different optimistic or pessimistic approaches. We also plan to develop techniques for reasoning with uncertainty in problems with both positive and negative preferences. © Springer-Verlag Berlin Heidelberg 2005.

关键词： constraint theory

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Domain reduction for the circuit constraint

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Kaya, Latife Genc Hooker, John Carnegie Mellon University Pittsburgh PA 15213 United States

ISBN: (纸本)3540292381

We present an incomplete filtering algorithm for the circuit constraint. the filter removes redundant values by eliminating non-Hamiltonian edges from the associated graph. We prove a necessary condition for an edge to be Hamiltonian, which provides the basis for eliminating edges of a smaller graph defined on a separator of the original graph. the circuit constraint, circuit(y1,,yn}, where yj ∈ {1,,n}, is true if and only if for each j ∈ {1,, n}, y j is the successor of j in some permutation of 1 n and yj ∈ Dj, where Dj is the domain of variable j. On a graph of vertices 1,,n, the circuit constraint can be thought as defining a directed Hamiltonian cycle. Nodes of the graph represent the variables. A directed edge (i, j) exists if and only if j is in the domain of variable i. Moreover, elimination of an edge (i, j) from the graph means elimination of the value j from the domain of variable i. With this representation, the problem of domain reduction for the circuit constraint reduces to identifying and eliminating non-Hamiltonian edges on a digraph. In this paper, we present a recursive algorithm that eliminates non-Hamiltonian edges from the graph. A much smaller but denser multi-graph is constructed from a vertex separator S of the original graph by adding certain labelled edges to the subgraph induced by the separator. A directed edge (v, w) with label C is added if C is a connected component separated by S and (v, ci) and (cj, w) are edges of G for some pair of vertices ci, cj in C. We prove that edges that appear in no Hamiltonian cycle containing at least one edge of each component label in the constructed graph are non-Hamiltonian in the original graph. the condition that the constructed graph contains such a Hamiltonian cycle is viewed as a constraint. Global cardinality constraint with vertex degree constraints is a relaxation of this constraint. then by applying a filtering algorithm for the global cardinality constraint together with in and out-vertex de

关键词： constraint theory

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Randomization for multi-agent constraint optimization

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Nguyen, Quang Huy Faltings, Boi V. Switzerland

ISBN: (纸本)3540292381

In a constraint optimization problem for multiple agents, the agents have conflicting preferences in the final solution and the goal is to find an optimal assignment that maximizes total utilities of all agents. Two major challenges when solving constraint optimization problems for multiple agents are the complexity of finding optimal solution and the incentive compatibility for participating agents. First, computing the optimal solution for large optimization problems are computationally infeasible and it can only be solved approximately by local search algorithms. Second, ensuring honest elicitation among self-interested agents is computationally expensive. It has been shown that the only known mechanism that guarantees truthfulness among agents requires computing optimal solutions, and sub-optimal solutions for such a mechanism will break the incentive compatibility ([2]). the long-term goal of our research is to solve these two challenges by using randomization in local search algorithms to find near-optimal solutions while ensuring incentive compatibility for bounded-rational agents. Our work is based on the observation that in real-world settings, the potential for manipulation is limited by uncertainty and risk. this uncertainty makes it difficult for a manipulator to predict the consequences of his manipulation and thus makes attempts at manipulating it uninteresting. In this paper we discuss a general randomization technique, called random subset optimization, for escaping from local minima in local search algorithms. In each local choice step, the local search procedure will randomly choose a part of the optimization function, and optimize for this part only. It turns out that this results in a more focussed optimization strategy and is shown to be especially effective in some hard optimization problems. We show that the uncertainty of our randomization algorithms can make the agents' manipulation hard, thus prevent bounded rational agents from manipulatin

关键词： Random processes

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Probabilistic Arc consistency

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Mehta, Deepak Van Dongen, M.R.C. Boole Centre for Research in Informatics Cork Constraint Computation Centre

ISBN: (纸本)3540292381

the two most popular algorithms for solving constraint Satisfaction Problems are Forward Checking (FC) [1] and Maintaining Arc Consistency (MAC) [2]. MAC maintains full arc consistency while FC maintains a limited form of arc consistency during search. there is no single champion algorithm: MAC is more efficient on sparse problems which are tightly constrained but FC has an increasing advantage as problems become dense and constraints loose. Ideally a good search algorithm should find the right balance - for any problem - between visiting fewer nodes in the search tree and reducing the work that is required to establish local consistency. In order to do so, we maintain probabilistic arc consistency during search. the idea is to assume that a support exists and skip the process of seeking a support if the probability of having some support for a value is at least equal to some, carefully chosen, stipulated threshold. Arc consistency involves revisions of domains, which require support checks to remove unsupported values. In many revisions, some or all values find some support. If we can predict the existence of a support then a considerable amount of work can be saved. In order to do so, we propose the notions of a Probabilistic Support Condition (PSC) and Probabilistic Revision Condition (PRC). If PSC holds then the probability of having some support for a value is at least equal to the threshold and the process of seeking a support is skipped. If PRC holds then for each value the probability of having some support is at least equal to the threshold and the corresponding revision is skipped. For hard dense problems constraint are generally loose, and on average each value has several supports, in which case the probability of having some support remains high for a while with respect to the number of values removed. this enables PSC and PRC to save checks. For hard sparse problems constraints are generally tight, and on average each value has only a few supports, in

关键词： Probabilistic logics

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2-Way vs. d-way branching for CSP

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Hwang, Joey Mitchell, David G. School of Computing Science Simon Fraser University Burnaby V5A 1S6 Canada

ISBN: (纸本)3540292381

Most CSP algorithms are based on refinements arid extensions of backtracking, and employ one of two simple "branching schemes": 2-way branching or d-way branching, for domain size d. the schemes are not equivalent, but little is known about their relative power. Here we compare them in terms of how efficiently they can refute an unsatisfiable instance with optimal branching choices, by studying two variants of the resolution proof system, denoted C-RES and NG-RES, which model the reasoning of CSP algorithms. the tree-like restrictions, tree-C-RES and tree-NG-RES, exactly capture the power of backtracking with 2-way branching and d-way branching, respectively. We give a family instances which require exponential sized search trees for backtracking with d-way branching, but have swc O(d2n) search trees for backtracking with 2-way branching. We also give a natural branching strategy with which backtracking with 2-way branching finds refutations of these instances in time O(d2n2). the unrestricted variants of C-RES and NG-RES can simulate the reasoning of algorithms which incorporate learning and k-consistency enforcement. We show exponential separations between C-RES and NG-RES, as well as between the tree-like and unrestricted versions of each system. All separations given are nearly optimal. © Springer-Verlag Berlin Heldelberg 2005.

关键词： constraint theory

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constraint-based inference: A bridge between constraint processing and probability inference

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Chang, Le Mackworth, Alan K. University of British Columbia 2366 Main Mall Vancouver BC V6T 1Z4 Canada

ISBN: (纸本)3540292381

constraint-Based Inference (CBI) [1] is an umbrella term for various superficially different problems including probabilistic inference, decision-making under uncertainty, constraint satisfaction, propositional satisfiability, decoding problems, and possibility inference. In this project we explicitly use the semiring concept to generalize various CBI problems into a single formal representation frame-work with a broader coverage of the problem space, based on the synthesis of existing generalized frameworks from both constraint processing and probability inference communities. Based on our generalized CBI framework, extensive comparative studies of exact and approximate inference approaches are commenced. First, we extend generalized arc consistency to probability inference based on a weaker condition [2]. All the existing arc consistency enforcing algorithms can be generalized and migrated to handle other concrete CBI problems that satisfy this condition. Second, based on our CBI framework we apply junction tree algorithms in probability inferences to solve soft CSPs [1]. We show that the message-passing schemes of junction tree algorithms can be modified to achieve better computational efficiency if the semiring of a CBI problem has additional properties. third, we study loopy message propagation in probability inference for general CBI problems. We claim in [1] that for CBI problems with a idempotent combination operator, the loopy message propagation is an exact inference approach. Our experimental results also show that the loopy message propagation yields high quality inference approximation for general CBI problems like Max CSPs. Finally, we discuss the possibilities of integrating stochastic approaches into our semiring-based CBI framework. We also discuss context-specific inference with backtracking as a promising inference approach for general CBI problems. In general, we are aiming at studying the most important common characteristics of various CBI prob

关键词： constraint theory

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From linear relaxations to global constraint propagation

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Quimper, Claude-Guy López-Ortiz, Alejandro School of Computer Science University of Waterloo Canada

ISBN: (纸本)3540292381

Recently, many algorithms have been designed to propagate global constraints. Unfortunately, some global constraints, such the AT-MOST-1 constraint and the EXTENDEDGCC are NP-Hard to propagate. Often, these constraints can easily be written as integer linear programs. Using linear relaxations and other techniques developed by the operation research community, we want to efficiently propagate such constraints. We model constraints as integer programs that we relax into linear programs. For each value v in a variable domain dom(x), we create a binary variable xv. the assignment xv= 1 indicates that x = v while xv= 0 indicates that x ≠ v. the binary variables are subject to the following linear program. Ax&leb 0&lexi&le1}P We find a solution to the relaxation using the interior point method. this method always converges to the interior of the solution polytope. Based on this observation, we conclude that if a variable xi subject to l &le xi &le u is assigned to one of its boundary value I or u, the variable is assigned to this value in any solution. therefore, if the interior point method assigns variable x *** 0, we conclude that v does not have a support in dom(x). We studied the consistency enforced by our propagator. We proved that if U is a totally unimodular matrix and that P has the following form, then GAC is enforced on the equations Ux &le b1. [U 0 B C] [x y] &le [b1 b2] 0 &le xi,yj &le 1} P Some constraints like the CARDINALITY-MATRIX constraint while written in the form Ax ≥ b have many totally unimodular sub-matrices of A. For each such sub-matrix, GAC is enforced on the corresponding variables. the resulting consistency is stronger than consistencies enforced by existing propagators. © Springer-Verlag Berlin Heidelberg 2005.

关键词： Linear systems

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Robust constraint solving using multiple heuristics

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Vidotto, Alfio Brown, Kenneth N. Beck, J. Christopher Cork Constraint Computation Centre Dept. of Computer Science UCC Cork Ireland Department of Mechanical and Industrial Engineering University of Toronto Canada

ISBN: (纸本)3540292381

Representing and solving problems in terms of constraints can be difficult to do effectively. A single problem can be modeled in many different ways, either in terms of representation or in terms of the solving process. Different approaches can outperform each other over different problem classes or even for different instances within the same class. It is possible that even the best combination of model and search on average is still too slow across a range of problems, taking orders of magnitude more time on some problems than combinations that are usually poorer. this fact complicates the use of constraints, and makes it very difficult for novice users to produce effective solutions. the modeling and solving process would be easier if we could develop robust algorithms, which perform acceptably across a range of problems. We present one method of developing a robust algorithm. We combine a single model and a single basic algorithm with a set of variable and value ordering heuristics, in a style similar to [1]. the aim is to exploit the variance among the orderings to get a more robust procedure, which may be slower on some problems, but avoids the significant deterioration on others. During the search, we allocate steadily increasing time slices to each ordering, restarting the search at each point. We demonstrate the performance of the multiple heuristic approach (MH) on a scheduling problem class [2] and on quasi groups with holes (QWH), showing that it is more robust and competitive than the standard recommended heuristic. We also compare to randomized restarts [3], the leading method for QWH and which uses a similar restart policy. We show that MH is poorer in run time and robustness on QWH, but better on the scheduling class. For the immediate future, we intend to investigate whether MH does perform better on insoluble problems (as indicated by the scheduling results). We will also tune the heuristics and time slices, and attempt to generate them automatical

关键词： Robustness (control systems)

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Scheduling with uncertain start dates

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11th international conference on principles and practice of constraint programming - CP 2005

作者： Wu, Christine Wei Brown, Kenneth N. Beck, J. Christopher Cork Constraint Computation Center Dept. of Computer Science UCC Ireland Dept. of Mechanical and Industrial Engineering University of Toronto Canada

ISBN: (纸本)3540292381

In manufacturing scheduling, jobs may have uncertain earliest start times, caused by supplier lead-time uncertainty. How should we build initial schedules to be robust to these uncertain release dates? We are attempting to adapt the online decision making methods from [1]: in particular the expectation and consensus methods, which combine sampling of future scenarios with deterministic optimization. First, we assume that we have a probability distribution over the release dates, and we use this to select samples, giving them a weight proportional to their probability, which we use when determining the best decision. For expectation, we consider all possible partial schedules of released jobs up to the earliest possible uncertain release date (t 0), for each sample extend each partial schedule, and return the one with the lowest expected makespan. For consensus, we find the optimal decision for each sample, and then choose the one with the highest sum of weights. We consider three possible types of initial decision: (i) partial schedules up to to;(ii) independent resource assignments for a single time step;and (in) a tuple of resource assignments for a single time step. the latter two require re-optimization at each time-step up to t0. We have implemented the above in Hog Scheduler 6.0, and tested them on modified JSP benchmarks [2]. the experiments show that our four adapted methods all provide good results (i.e. weighted relative error not being more than 2%). However, it is also possible to get close to the optimal without reasoning about the uncertain events in advance (e.g. using pure reactive methods). the problems appear to be too easy, and not sensitive to the uncertainties. Future work will investigate why this is the case, to determine the features that do make the problems sensitive to the uncertainty in the release dates. © Springer-Verlag Berlin Heidelberg 2005.

关键词： Scheduling

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