Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

A multiperiod optimization model to schedule large-scale petroleum development projects

This dissertation solves an optimization problem in the area of scheduling large-scale petroleum development projects under several resources constraints. The dissertation focuses on the application of a metaheuristic search Genetic Algorithm (GA) in solving the problem. The GA is a global search method inspired by natural evolution. The method is widely applied to solve complex and sizable problems that are difficult to solve using exact optimization methods. A classical resource allocation problem in operations research known under Knapsack Problems (KP) is considered for the formulation of the problem. Motivation of the present work was initiated by certain petroleum development scheduling problem in which large-scale investment projects are to be selected subject to a number of resources constraints in several periods. The constraints may occur from limitations in various resources such as capital budgets, operating budgets, and drilling rigs. The model also accounts for a number of assumptions and business rules encountered in the application that motivated this work. The model uses an economic performance objective to maximize the sum of Net Present Value (NPV) of selected projects over a planning horizon subject to constraints involving discrete time dependent variables. Computational experiments of 30 projects illustrate the performance of the model. The application example is only illustrative of the model and does not reveal real data. A Greedy algorithm was first utilized to construct an initial estimate of the objective function. GA was implemented to improve the solution and investigate resources constraints and their effect on the assets value. The timing and order of investment decisions under constraints have the prominent effect on the economic performance of the assets. The application of an integrated optimization model provides means to maximize the financial value of the assets, efficiently allocate limited resources and to analyze more scheduling alternatives in less time

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

Local cuts and two-period convex hull closures for big-bucket lot-sizing problems

Despite the significant attention they have drawn, big bucket lot-sizing problems remain notoriously difficult to solve. Previous work of Akartunali and Miller (2012) presented results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multi-period submodels. We therefore consider the simplest such submodel, a multi-item, two-period capacitated relaxation. We propose a methodology that can approximate the convex hulls of all such possible relaxations by generating violated valid inequalities. To generate such inequalities, we separate two-period projections of fractional LP solutions from the convex hulls of the two-period closure we study. The convex hull representation of the two-period closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation, and therefore can be used efficiently in a regular branch-and-bound procedure. We present computational results that illustrate how these two-period models could be effective in solving complicated problems

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed

Mixed policies for recovery and disposal of multiple-type consumer products

New European government policies aim at the closure of material flows as part of integrated chain management (ICM). One of the main implementation instruments is extended producer responsibility, which makes original equipment manufacturers (OEMs) formally responsible for take-back, recovery, and reuse of discarded products. One of the key problems for OEMs is to determine a recovery strategy, i.e., determine to what extent return products must be disassembled and which recovery and disposal (RD) options should be applied. On a tactical management level, this involves anticipation of problems such as meeting legislation, limited volumes of secondary end markets, bad quality of return products, and facility investments in recycling infrastructure. In this paper, a model is presented that can be used to determine a recovery strategy for multiple-type consumer products. The objective function incorporates technical, ecological, and commercial decision criteria and optimization occurs using a two-level optimization procedure. First, a set of potential product recovery and disposal (PRD) strategies is generated for each separate product type. Secondly, optimal PRD strategies are assigned to the products within a coherent multiproduct or product group policy. The aim is to find an optimal balance between maximizing net profit and meeting constraints like recovery targets, limited market volumes, and processing capacities. A TV case is worked out to illustrate the working of the model. Also, the managerial use of the model is discussed in view of establishing an economically and ecologically sound base for achieving ICM

A Robust Optimization of Capacity Allocation Policies in the Third-Party Warehouse

We study the capacity allocation policies of a third-party warehouse center, which supplies several different level services on different prices with fixed capacity, on revenue management perspective. For the single period situation, we use three different robust methods, absolute robust, deviation robust, and relative robust method, to maximize the whole revenue. Then we give some numerical examples to verify the practical applicability. For the multiperiod situation, as the demand is uncertain, we propose a stochastic model for the multiperiod revenue management problem of the warehouse. A novel robust optimization technique is applied in this model to maximize the whole revenue. Then we give some numerical examples to verify the practical applicability of our method