### Valid Inequalities Based on Demand Propagation for Chemical Production Scheduling MIP Models

"... Abstract. The planning of chemical production often involves the optimization of the size of the tasks to be performed subject to unit capacity constraints, as well as inventory constraints for intermediate materials. While several mixed-integer programming (MIP) models have been proposed that acco ..."

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Abstract. The planning of chemical production often involves the optimization of the size of the tasks to be performed subject to unit capacity constraints, as well as inventory constraints for intermediate materials. While several mixed-integer programming (MIP) models have been proposed that account for these features, the development of tightening methods for these formulations has received limited attention. In this paper, we develop a constraint propagation algorithm for the calculation of lower bounds on the number and size of tasks necessary to satisfy given demand. These bounds are then used to express three types of tightening constraints which greatly enhance the computational performance of the MIP scheduling model. Importantly, the proposed methods are applicable to a wide range of problem classes and time-indexed MIP models for chemical production scheduling.

### Cutting Plane Algorithms for Solving a Stochastic Edge-Partition Problem

, 2009

"... We consider the edge-partition problem, which is a graph theoretic problem arising in the design of Synchronous Optical Networks. The deterministic edge-partition problem considers an undirected graph with weighted edges, and simultaneously assigns nodes and edges to subgraphs such that each edge ap ..."

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We consider the edge-partition problem, which is a graph theoretic problem arising in the design of Synchronous Optical Networks. The deterministic edge-partition problem considers an undirected graph with weighted edges, and simultaneously assigns nodes and edges to subgraphs such that each edge appears in exactly one subgraph, and such that no edge is assigned to a subgraph unless both of its incident nodes are also assigned to that subgraph. Additionally, there are limitations on the number of nodes and on the sum of edge weights that can be assigned to each subgraph. In this paper, we consider a stochastic version of the edge-partition problem in which we assign nodes to subgraphs in a first stage, realize a set of edge weights from a finite set of alternatives, and then assign edges to subgraphs. We first prescribe a two-stage cutting plane approach with integer variables in both stages, and examine computational difficulties associated with the proposed cutting planes. As an alternative, we prescribe a hybrid integer programming/constraint programming algorithm capable of solving a suite of test instances within practical computational limits.

### A Heuristic Logic-Based Benders Method for the Home Health Care Problem

, 2012

"... We propose a heuristic adaptation of logic-based Benders decomposition to the home health care problem. The objective is design routes and schedules for health care workers who visit patient homes, so as to minimize cost while meeting all patient needs and work requirements. We solve the Benders mas ..."

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We propose a heuristic adaptation of logic-based Benders decomposition to the home health care problem. The objective is design routes and schedules for health care workers who visit patient homes, so as to minimize cost while meeting all patient needs and work requirements. We solve the Benders master problem by a greedy heuristic that is enhanced by the propagation facilities of constraint programming (CP). We solve the subproblems entirely by CP and generate logic-based Benders cuts that exploit problem structure. Many of the subproblem constraints are included in the master problem to compensate for the difficulty of designing strong Benders cuts, but they serve as guidance for the heuristic rather than constraints to be satisfied. We also experiment with local search in the master problem to hasten the discovery of a feasible solution. We report preliminary computational results for realistic problem instances. 1

### Optimization Methods in Logic

, 2003

"... Optimization can make at least two contributions to boolean logic. Its solution methods can address inference and satisfiability problems, and its style of analysis can reveal tractable classes of boolean problems that might otherwise have gone unnoticed. They key to linking optimization with logic ..."

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Optimization can make at least two contributions to boolean logic. Its solution methods can address inference and satisfiability problems, and its style of analysis can reveal tractable classes of boolean problems that might otherwise have gone unnoticed. They key to linking optimization with logic is to provide logical formulas a numerical interpretation or semantics. While syntax concerns the structure of logical expressions, semantics gives them meaning. Boolean semantics, for instance, focuses on truth functions that capture the meaning of logical propositions. To take an example, the function

### Project-Team RealOpt Reformulation based algorithms for Combinatorial Optimization

"... c t i v it y e p o r t 2009 Table of contents ..."

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### Team RealOpt Reformulation based algorithms for Combinatorial Optimization

"... c t i v it y e p o r t 2008 Table of contents ..."

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### Projection, Consistency, and George Boole

"... Abstract. Although best known for his work in symbolic logic, George Boole made seminal contributions in the logic of probabilities. He solved the probabilistic inference problem with a projection method, leading to the insight that inference (as well as optimization) is essentially a pro-jection pr ..."

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Abstract. Although best known for his work in symbolic logic, George Boole made seminal contributions in the logic of probabilities. He solved the probabilistic inference problem with a projection method, leading to the insight that inference (as well as optimization) is essentially a pro-jection problem. This unifying perspective has applications in constraint programming, because consistency maintenance is likewise a form of inference that can be conceived as projection. Viewing consistency in this light suggests a concept of J-consistency, which is achieved by projection onto a subset J of variables. We show how this projection problem can be solved for the satisfiability problem by logic-based Benders decom-position. We also solve it for among, sequence, regular, and all-different constraints. Maintaining J-consistency for global constraints can be more effective than maintaining traditional domain and bounds consistency when propagating through a richer structure than a domain store, such as a relaxed decision diagram. This paper is written in recognition of Boole’s 200th birthday. 1

### Robust Scheduling with Logic-Based Benders Decomposition

"... Abstract We study project scheduling at a large IT services delivery center in which there are unpredictable delays. We apply robust optimization to minimize tardiness while informing the customer of a reasonable worst-case completion time, based on empirically determined uncertainty sets. We introd ..."

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Abstract We study project scheduling at a large IT services delivery center in which there are unpredictable delays. We apply robust optimization to minimize tardiness while informing the customer of a reasonable worst-case completion time, based on empirically determined uncertainty sets. We introduce a new solution method based on logic-based Benders decomposition. We show that when the uncertainty set is polyhedral, the decomposition simplifies substantially, leading to a model of tractable size. Preliminary computational experience indicates that this approach is superior to a mixed integer programming model solved by state-of-the-art software. 1

### DETERMINISTIC AND STOCHASTIC MODELS FOR PRACTICAL SCHEDULING PROBLEMS

, 2012

"... This dissertation analyzes three scheduling problems motivated by real life situa-tions. In many manufacturing and service industries, scheduling is an important decision-making process. Unfortunately, scheduling problems are often computation-ally challenging to solve, and even modeling a schedulin ..."

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This dissertation analyzes three scheduling problems motivated by real life situa-tions. In many manufacturing and service industries, scheduling is an important decision-making process. Unfortunately, scheduling problems are often computation-ally challenging to solve, and even modeling a scheduling problem can be difficult. Chapter 2 considers single-facility non-preemptive scheduling problems with long time horizons having jobs with time windows (i.e., release times and due dates). I combine constraint programming (CP) and mixed integer linear programming (MILP) using a hybrid method: logic-based Benders decomposition. I first divide the long time horizon into segments to make the problem tractable. This gives rise to two versions of the single-facility scheduling problem: segmented and unsegmented. In the segmented problem, each job must be completed within one time segment. In the unsegmented problem, jobs can overlap two or more segments. I analyze different ob-jective functions, and introduce relevant Benders cuts. I find that for the segmented problem, logic-based Benders decomposition is always superior and should be used