Laura Climent

Robustness and stability in constraint programming under dynamism and uncertainty

Many real life problems that can be solved by constraint programming, come from uncertain and dynamic environments. Because of the dynamism, the original problem may change over time, and thus the solution found for the original problem may become invalid. For this reason, dealing with such problems has become an important issue in the fields of constraint programming. In some cases, there is extant knowledge about the uncertain and dynamic environment. In other cases, this information is fragmentary or unknown. In this paper, we extend the concept of robustness and stability for Constraint Satisfaction Problems (CSPs) with ordered domains, where only limited assumptions need to be made as to possible changes. We present a search algorithm that searches for both robust and stable solutions for CSPs of this nature. It is well-known that meeting both criteria simultaneously is a desirable objective for constraint solving in uncertain and dynamic environments. We also present compelling evidence that our search algorithm outperforms other general-purpose algorithms for dynamic CSPs using random instances and benchmarks derived from real life problems.

Solving a hard cutting stock problem by machine learning and optimisation

We are working with a company on a hard industrial optimisation problem: a version of the well-known Cutting Stock Problem in which a paper mill must cut rolls of paper following certain cutting patterns to meet customer demands. In our problem each roll to be cut may have a different size, the cutting patterns are semi-automated so that we have only indirect control over them via a list of continuous parameters called a request, and there are multiple mills each able to use only one request. We solve the problem using a combination of machine learning and optimisation techniques. First we approximate the distribution of cutting patterns via Monte Carlo simulation. Secondly we cover the distribution by applying a k-medoids algorithm. Thirdly we use the results to build an ILP model which is then solved.

Robustness and stability in dynamic constraint satisfaction problems

Many real life problems come from uncertain and dynamic environments and therefore, the original problems, and consequently their associated Constraint Satisfaction Problem (CSP) models, may evolve over the time. In such situations, a solution that holds for the original problem can become invalid after changes occur. There exist two main approaches for dealing with these situations: reactive and proactive. Using reactive approaches entails re-solving the CSP after a solution is no longer a solution, which is time consuming. For this reason, such as mentioned in the Verfaillie and Jussien (2005) survey, a desirable objective is: “limit as much as possible the need for successive online problem solvings.”

Finding robust solutions for constraint satisfaction problems with discrete and ordered domains by coverings

Constraint programming is a paradigm wherein relations between variables are stated in the form of constraints. Many real life problems come from uncertain and dynamic environments, where the initial constraints and domains may change during its execution. Thus, the solution found for the problem may become invalid. The search for robust solutions for constraint satisfaction problems (CSPs) has become an important issue in the field of constraint programming. In some cases, there exists knowledge about the uncertain and dynamic environment. In other cases, this information is unknown or hard to obtain. In this paper, we consider CSPs with discrete and ordered domains where changes only involve restrictions or expansions of domains or constraints. To this end, we model CSPs as weighted CSPs (WCSPs) by assigning weights to each valid tuple of the problem constraints and domains. The weight of each valid tuple is based on its distance from the borders of the space of valid tuples in the corresponding constraint/domain. This distance is estimated by a new concept introduced in this paper: coverings. Thus, the best solution for the modeled WCSP can be considered as a most robust solution for the original CSP according to these assumptions.

Modeling Robustness in CSPs as Weighted CSPs

Many real life problems come from uncertain and dynamic environments, where the initial constraints and/or domains may undergo changes. Thus, a solution found for the problem may become invalid later. Hence, searching for robust solutions for Constraint Satisfaction Problems (CSPs) becomes an important goal. In some cases, no knowledge about the uncertain and dynamic environment exits or it is hard to obtain it. In this paper, we consider CSPs with discrete and ordered domains where only limited assumptions are made commensurate with the structure of these problems. In this context, we model a CSP as a weighted CSP (WCSP) by assigning weights to each valid constraint tuple based on its distance from the edge of the space of valid tuples. This distance is estimated by a new concept introduced in this paper: coverings. Thus, the best solution for the modeled WCSP can be considered as a robust solution for the original CSP according to our assumptions.

An Algorithm for Finding Robust and Stable Solutions for Constraint Satisfaction Problems with Discrete and Ordered Domains

Many real life problems come from uncertain and dynamic environments, which means that the original problem may change over time. Thus, the solution found for the original problem may become invalid. Dealing with such problems has become an important issue in the field of constraint programming. In some cases, there exists knowledge about the uncertain and dynamic environment. In other cases, this information is unknown or hard to obtain. In this paper, we extend the concept of robustness for Constraint Satisfaction Problems (CSPs) with discrete and ordered domains where the only assumptions made about changes are those inherent in the structure of these problems. We present a search algorithm that searches for both robust and stable solutions for such CSPs. Meeting both criteria simultaneously is a well-known desirable objective for constraint solving in uncertain and dynamic environments.

Robust solutions in changing constraint satisfaction problems

Constraint programming is a successful technology for solving combinatorial problems modeled as constraint satisfaction problems (CSPs). An important extension of constraint technology involves problems that undergo changes that may invalidate the current solution. These problems are called Dynamic Constraint Satisfaction Problems (DynCSP). Many works on dynamic problems sought methods for finding new solutions. In this paper, we focus our attention on studying the robustness of solutions in DynCSPs. Thus, most robust solutions will be able to absorb changes in the constraints of the problem. To this end, we label each constraint with two parameters that measure the degree of dynamism and the quantity of change. Thus, we randomly generate a set of more restricted CSPs by using these labels. The solutions that satisfy more random CSPs will have a higher probability of remain valid under constraints changes of the original CSP.

Uncertainty in dynamic constraint satisfaction problems

Many real life problems that can be modeled as Constraint Satisfaction Problems (CSPs) come from uncertain and dynamic environments, so the original solution found might become invalid after changes. Here, we study the problem of find- ing robust solutions for CSPs under the assumptions that domains are significantly ordered and that changes take the form of restrictions or relaxations at the borders of the solution space of the CSPs.

Robustness and Stability in Constraint Programming under Dynamism and Uncertainty

Because of the dynamism and uncertainty associated with many real life problems, these problems and their associated Constraint Satisfaction Problem (CSP) models may change over time; thus an earlier solution found for the latter may become invalid. Moreover, many approaches proposed in the literature cannot be applied when the required information about dynamism is unknown ([9], [4], [5], [11], [10], etc.). This fact has motivated us to consider dynamic situations where, in addition, only limited assumptions about changes can be made. Our analysis focuses on CSPs with ordered and discrete domains that model problems for which the order over the elements of the domain is significant. In these cases, a common type of change that problems may undergo is restrictive modifications over the bounds of the solution space. A discussion of these assumptions, their motivation and real life examples can be found in [3].

Bounding the Search Space of the Population Harvest Cutting Problem with Multiple Size Stock Selection

In this paper we deal with a variant of the Multiple Stock Size Cutting Stock Problem (MSSCSP) arising from population harvesting, in which some sets of large pieces of raw material (of different shapes) must be cut following certain patterns to meet customer demands of certain product types. The main extra difficulty of this variant of the MSSCSP lies in the fact that the available patterns are not known a priori. Instead, a given complex algorithm maps a vector of continuous variables called a values vector into a vector of total amounts of products, which we call a global products pattern. Modeling and solving this MSSCSP is not straightforward since the number of value vectors is infinite and the mapping algorithm consumes a significant amount of time, which precludes complete pattern enumeration. For this reason a representative sample of global products patterns must be selected. We propose an approach to bounding the search space of the values vector and an algorithm for performing an exhaustive sampling using such bounds. Our approach has been evaluated with real data provided by an industry partner.