Constraint Programming (CP) is a paradigm derived from artificial intelligence, operational research, and algorithmics that can be used to solve combinatorial problems. CP solves problems by interleaving search (assign a value to an unassigned variable) and propagation. Constraint propagation aims at removing/filtering inconsistent values from the domains of the variables in order to reduce the search space of the problem. In this thesis, we develop filtering algorithms for two complex combinatorial optimization problems: a Capacitated Lot Sizing Problem (CLSP) and the Constrained Arborescence Problem (CAP). Each of these problems has many variants and practical applications.The CLSP is the problem of finding an optimal production plan for single or multiple items while satisfying demands of clients and respecting resource restrictions. The CLSP finds important applications in production planning. In this thesis, we introduce a CLSP in CP. In many lot sizing and scheduling problems, in particular when the planning horizon is discrete and finite, there are stocking costs to be minimized. These costs depend on the time spent between the production of an order and its delivery. We focus on developing specialized filtering algorithms to handle the stocking cost part of a class of the CLSP. We propose the global optimization constraint StockingCost when the perperiod stocking cost is the same for all orders; and its generalized version, the IDStockingCost constraint (ID stands for Item Dependent).In this thesis, we also deal with a well-known problem in graph theory: the Minimum Weight Arborescence (MWA) problem. Consider a weighted directed graph in which we distinguish one vertex r as the root. An MWA rooted at r is a directed spanning tree rooted at r with minimum total weight. We focus on the CAP that requires one to find an arborescence that satisfies some side constraints (for example resource, degree, or diameter constraints) and that has minimum weight. The CAP has many real life applications in telecommunication networks, computer networks, transportation problems, scheduling problems, etc. After sensitivity analysis of the MWA, we introduce the CAP in CP. We propose a dedicated global optimization constraint to handle any known variant of the CAP in CP: the MinArborescence constraint. All the proposed filtering algorithms are analyzed theoretically and evaluated experimentally. The different experimental evaluations of these propagators against the state-of-the-art propagators show their respective efficiencies.

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容量批量问题和约束树状问题的基于成本的过滤算法

约束编程（CP）是源自人工智能，运筹学和算法的范式，可用于解决组合问题。CP通过交错搜索（将值分配给未分配的变量）和传播来解决问题。约束传播的目的是从变量的域中删除/过滤不一致的值，以减少问题的搜索空间。在本文中，我们针对两个复杂的组合优化问题开发了一种滤波算法：一个容量批量问题（CLSP）和约束树状问题（CAP）。这些问题中的每一个都有许多变体和实际应用。CLSP的问题是在满足客户需求并尊重资源限制的同时，为单个或多个项目找到最佳的生产计划。CLSP在生产计划中发现重要的应用。本文介绍了CP中的CLSP。在许多批量确定和计划问题中，尤其是当计划范围是离散且有限的时候，存在使库存成本最小化的问题。这些成本取决于订单生成和交付之间花费的时间。我们专注于开发专门的过滤算法以处理CLSP类的库存成本部分。当所有订单的周期库存成本相同时，我们提出全局优化约束StockingCost。在本文中，我们还处理了图论中一个众所周知的问题：最小权重树状结构（MWA）问题。考虑一个加权有向图，其中我们区分一个顶点r作为根。根为MWA [R是一个有向生成树上植根于[R总重量最小。我们专注于CAP，它要求人们找到满足某些侧面约束（例如资源，度数或直径约束）并且具有最小权重的树状结构。CAP在电信网络，计算机网络，运输问题，调度问题等方面有许多实际应用。在对MWA进行敏感性分析之后，我们将CAP引入CP中。我们提出了一个专用的全局优化约束来处理CP中CAP的任何已知变体：MinArborescence约束。对所有提出的滤波算法进行了理论分析和实验评估。这些繁殖器相对于最先进的繁殖器的不同实验评估表明了它们各自的效率。