Comparing Dantzig-Wolfe decompositions and branch-and-price algorithms for the multi-item capacitated lot sizing problem

In this article, we consider the multi-item capacitated lot sizing problem with setup times. Starting from an original mixed integer programming model, we apply the standard Dantzig–Wolfe decomposition (DWD) in two different ways: defining the subproblems by items and defining the subproblems by periods.A third decomposition is developed in which the subproblems of both types are integrated in the same model. The linear relaxation of this last approach, which we denote as multiple DWD, provides lower bounds (equal to or) better than the bounds obtained by the other decompositions, which in turn, provide lower bounds (equal to or) better than the ones given by the original model. For solving the three decomposition models, we implemented three branch-and-price algorithms. We describe their main aspects and report on their computational results in instances from the literature

Lot-Sizing Problem for a Multi-Item Multi-level Capacitated Batch Production System with Setup Carryover, Emission Control and Backlogging using a Dynamic Program and Decomposition Heuristic

Wagner and Whitin (1958) develop an algorithm to solve the dynamic Economic Lot-Sizing Problem (ELSP), which is widely applied in inventory control, production planning, and capacity planning. The original algorithm runs in O(T^2) time, where T is the number of periods of the problem instance. Afterward few linear-time algorithms have been developed to solve the Wagner-Whitin (WW) lot-sizing problem; examples include the ELSP and equivalent Single Machine Batch-Sizing Problem (SMBSP). This dissertation revisits the algorithms for ELSPs and SMBSPs under WW cost structure, presents a new efficient linear-time algorithm, and compares the developed algorithm against comparable ones in the literature. The developed algorithm employs both lists and stacks data structure, which is completely a different approach than the rest of the algorithms for ELSPs and SMBSPs. Analysis of the developed algorithm shows that it executes fewer number of basic actions throughout the algorithm and hence it improves the CPU time by a maximum of 51.40% for ELSPs and 29.03% for SMBSPs. It can be concluded that the new algorithm is faster than existing algorithms for both ELSPs and SMBSPs. Lot-sizing decisions are crucial because these decisions help the manufacturer determine the quantity and time to produce an item with a minimum cost. The efficiency and productivity of a system is completely dependent upon the right choice of lot-sizes. Therefore, developing and improving solution procedures for lot-sizing problems is key. This dissertation addresses the classical Multi-Level Capacitated Lot-Sizing Problem (MLCLSP) and an extension of the MLCLSP with a Setup Carryover, Backlogging and Emission control. An item Dantzig Wolfe (DW) decomposition technique with an embedded Column Generation (CG) procedure is used to solve the problem. The original problem is decomposed into a master problem and a number of subproblems, which are solved using dynamic programming approach. Since the subproblems are solved independently, the solution of the subproblems often becomes infeasible for the master problem. A multi-step iterative Capacity Allocation (CA) heuristic is used to tackle this infeasibility. A Linear Programming (LP) based improvement procedure is used to refine the solutions obtained from the heuristic method. A comparative study of the proposed heuristic for the first problem (MLCLSP) is conducted and the results demonstrate that the proposed heuristic provide less optimality gap in comparison with that obtained in the literature. The Setup Carryover Assignment Problem (SCAP), which consists of determining the setup carryover plan of multiple items for a given lot-size over a finite planning horizon is modelled as a problem of finding Maximum Weighted Independent Set (MWIS) in a chain of cliques. The SCAP is formulated using a clique constraint and it is proved that the incidence matrix of the SCAP has totally unimodular structure and the LP relaxation of the proposed SCAP formulation always provides integer optimum solution. Moreover, an alternative proof that the relaxed ILP guarantees integer solution is presented in this dissertation. Thus, the SCAP and the special case of the MWIS in a chain of cliques are solvable in polynomial time

Modelos e métodos para problemas de dimensionamento de lotes e escalonamento

Tese de doutoramento em Engenharia Industrial e de SistemasO trabalho que se apresenta nesta tese relaciona-se com o desenvolvimento de modelos e de métodos para a resolução de dois problemas de planeamento da produção de médio/curto prazo. A principal motivação consiste na exploração e comparação de diferentes abordagens, baseadas em programação inteira mista, em modelos/métodos de decomposição e em métodos heurísticos, para os problemas em estudo. O primeiro problema, é um problema clássico de dimensionamento de lotes, que está associado às decisões de planeamento da produção de médio-prazo. O problema consiste na determinação de um plano de produção para vários produtos finais ao longo de um determinado horizonte temporal, que minimize todos os custos envolvidos e respeite restrições de procura e de capacidade. Para este problema desenvolve-se um novo modelo exacto, que resulta da aplicação dos princípios da decomposição de Dantzig-Wolfe múltipla a uma formulação de programação inteira mista para o problema. Os princípios gerais de aplicação desta decomposição são também apresentados neste trabalho. A potencial mais valia deste modelo relaciona-se com a obtenção de limites inferiores de boa qualidade. O modelo que resulta da decomposição de Dantzig-Wolfe múltipla é comparado com dois modelos de decomposição alternativos, que se obtêm aplicando directamente os princípios da decomposição de Dantzig-Wolfe, e com o modelo de programação inteira mista, resolvido directamente através de um software de estado-da-arte. Para determinar a solução óptima inteira dos modelos de decomposição aplica-se o método de partição e geração de colunas (branchand- price). São apresentados resultados computacionais partindo de um conjunto de instâncias da literatura, para os vários modelos e métodos. O segundo problema estudado neste trabalho surge associado ao planeamento de curto-prazo e combina as decisões de dimensionamento de lotes, com as decisões de afectação e escalonamento desses lotes. Este estudo foi motivado por um problema real da indústria têxtil, no qual se pretende definir um plano de produção para uma secção de tricotagem, onde os principais componentes dos produtos finais são realizados num conjunto de máquinas paralelas idênticas. Para este problema propõe-se um novo modelo de programação inteira mista, que se resolve através de um software de estadoda- arte. Paralelamente, propõem-se vários métodos heurísticos. Duas das heurísticas propostas são: uma heurística de fluxos em rede e escalonamento e uma heurística de ordenação e escalonamento. Estas heurísticas visam a obtenção de soluções com alguma qualidade em pouco tempo. Propõem-se ainda quatro algoritmos de pesquisa local, que têm em consideração características específicas do problema e que tentam melhorar a qualidade das soluções das heurísticas anteriores. Atendendo ao desempenho dos algoritmos de pesquisa local, estes são combinados através de mudanças sistemáticas das vizinhanças, dando origem a duas meta-heurísticas: uma de descida em vizinhanças variáveis e outra de pesquisa em vizinhanças variáveis. Para avaliar as soluções do modelo de programação inteira mista e dos métodos heurísticos sugere-se uma função de avaliação inovadora, que minimiza os atrasos totais e os níveis em curso de fabrico entre duas etapas sucessivas do processo produtivo. É ainda sugerida uma nova função de avaliação nos métodos heurísticos, também baseada na minimização dos atrasos totais e na minimização dos níveis em curso de fabrico. A principal vantagem desta segunda medida de avaliação é contabilizar de um modo mais rigoroso os níveis em curso de fabrico. Para avaliar o desempenho e a qualidade das soluções do modelo de programação inteira mista e dos métodos heurísticos, desenvolveu-se um gerador de instâncias, que gera instâncias semelhantes às do problema real.This work is associated with the development of models and methods for two medium/short term production planning problems. Our main motivation is the exploration and comparison of different approaches, based on mixed integer programming, on decomposition models and methods and on heuristics, for those two problems. The first one is a classical lot sizing problem associated with the medium-term production planning decisions. The problem consists of finding a production plan for several final items over a given planning horizon that minimizes the overall costs involved, while respecting demand and capacity constraints. An exact model based on a multiple Dantzig-Wolfe decomposition is developed. The general principles of this decomposition are presented in this work too. The potential benefit of this decomposition is the achievement of good quality lower bounds, although our purpose is to obtain integer optimal solutions. The resulting model of multiple Dantzig-Wolfe decomposition is compared with two alternative decomposition models that are obtained when applying directly the Dantzig-Wolfe decomposition principles, and is also compared with an integer programming formulation solved by a state-of-art software. The integer optimal solutions of all the decomposition models are obtained through branch-and-price algorithms. We present computational results for a set of instances from the literature. The second problem studied in this work is a short-term production planning problem that integrates lot sizing, assignment and scheduling decisions. This study was motivated by a real problem from a textile industry. The aim is to define a production plan for a knitting section where the main components of the final items are processed on a set of identical parallel machines. A new mixed integer programming model is proposed for this problem, as well as several heuristics. Two of those heuristics are: a network flow and scheduling heuristic and an ordering and scheduling heuristic. The purpose of these heuristics is to find good quality solutions quickly. Four local search based algorithms that consider specific characteristics of the problem are developed too, in order to try to improve the solutions of the previous heuristics. Taking into account the performance of the four local search heuristics, we combine them through systematic changes of neighborhoods, testing two metaheuristics: variable neighborhood descent and variable neighborhood search. To evaluate the mixed integer programming model solutions and the solutions of all the heuristics, an innovative evaluation function that minimizes a weighted sum of total tardiness and work-in-process levels between two successive production processes is suggested. We study another new evaluation function for the heuristic methods, which is related to the previous one. The main advantage of the second evaluation function over the first one is that it calculates in a more precise way the levels of workin- process inventory. The performance and quality of solutions of all the above presented methods for the second problem are evaluated using a set of instances that are similar to the real ones. Those instances were generated by an instance generator developed by us.Fundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/38582/200