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

순서의존적 작업준비가 있는 생산계획 문제에 대한 정수 최적화 및 근사 동적 계획법 기반 해법

학위논문(박사) -- 서울대학교대학원 : 공과대학 산업공학과, 2022. 8. 이경식.Lot-sizing and scheduling problem, an integration of the two important decision making problems in the production planning phase of a supply chain, determines both the production amounts and sequences of multiple items within a given planning horizon to meet the time-varying demand with minimum cost. Along with the growing importance of coordinated decision making in the supply chain, this integrated problem has attracted increasing attention from both industrial and academic communities. However, despite vibrant research over the recent decades, the problem is still hard to be solved due to its inherent theoretical complexity as well as the evolving complexity of the real-world industrial environments and the corresponding manufacturing processes. Furthermore, when the setup activity occurs in a sequence-dependent manner, it is known that the problem becomes considerably more difficult. This dissertation aims to propose integer optimization and approximate dynamic programming approaches for solving the lot-sizing and scheduling problem with sequence-dependent setups. Firstly, to enhance the knowledge of the structure of the problem which is strongly NP-hard, we consider a single-period substructure of the problem. By analyzing the polyhedron defined by the substructure, we derive new families of facet-defining inequalities which are separable in polynomial time via solving maximum flow problems. Through the computational experiments, these inequalities are demonstrated to provide much tighter lower bounds than the existing ones. Then, using these results, we provide new integer optimization models which can incorporate various extensions of the lot-sizing and scheduling problem such as setup crossover and carryover naturally. The proposed models provide tighter linear programming relaxation bounds than standard models. This leads to the development of an efficient linear programming-based heuristic algorithm which provides a primal feasible solution quickly. Finally, we devise an approximate dynamic programming algorithm. The proposed algorithm incorporates the value function approximation approach which makes use of both the tight lower bound obtained from the linear programming relaxation and the upper bound acquired from the linear programming-based heuristic algorithm. The results of computational experiments indicate that the proposed algorithm has advantages over the existing approaches.공급망의 생산 계획 단계에서의 주요한 두 가지 단기 의사결정 문제인 Lot-sizing 문제와 Scheduling 문제가 통합된 문제인 Lot-sizing and scheduling problem (LSP)은 계획대상기간 동안 주어진 복수의 제품에 대한 수요를 최소의 비용으로 만족시키기 위한 단위 기간 별 최적의 생산량 및 생산 순서를 결정한다. 공급망 내의 다양한 요소에 대한 통합적 의사 결정의 중요성이 커짐에 따라 LSP에 대한 관심 역시 산업계와 학계 모두에서 지속적으로 증가하였다. 그러나 최근 수십 년에 걸친 활발한 연구에도 불구하고, 문제 자체가 내포하는 이론적 복잡성 및 실제 산업 환경과 제조 공정의 고도화/복잡화 등으로 인해 LSP를 해결하는 것은 여전히 어려운 문제로 남아있다. 특히 순서의존적 작업준비가 있는 경우 문제가 더욱 어려워진다는 것이 잘 알려져 있다. 본 논문에서는 순서의존적 작업준비가 있는 LSP를 해결하기 위한 정수 최적화 및 근사 동적 계획법 기반의 해법을 제안한다. 먼저, 이론적으로 강성 NP-hard에 속한다는 사실이 잘 알려진 LSP의 근본 구조에 대한 이해를 높이기 위하여 단일 기간만을 고려하는 부분구조에 대해 다룬다. 단일 기간 부분구조에 의해 정의되는 다면체에 대한 이론적 분석을 통해 새로운 유효 부등식 군을 도출하고 해당 유효 부등식들이 극대면(facet)을 정의할 조건에 대해 밝힌다. 또한, 도출된 유효 부등식들이 다항시간 내에 분리 가능함을 보이고, 최대흐름문제를 활용한 다항시간 분리 알고리듬을 고안한다. 실험 결과를 통해 제안한 유효 부등식들이 모형의 선형계획 하한강도를 높이는 데 큰 영향을 줌을 확인한다. 또한 해당 부등식들이 모두 추가된 모형과 이론적으로 동일한 하한을 제공하는 확장 수리모형(extended formulation)을 도출한다. 이를 활용하여, 실제 산업에서 발생하는 LSP에서 종종 고려하는 주요한 추가 요소들을 다룰 수 있는 새로운 수리 모형을 제안하며 해당 모형이 기존의 모형에 비해 더욱 강한 선형계획 하한을 제공함을 보인다. 이 모형을 바탕으로 빠른 시간 내에 가능해를 찾을 수 있는 선형계획 기반 휴리스틱 알고리듬을 개발한다. 마지막으로 해당 문제에 대한 근사 동적 계획법 알고리듬을 제안한다. 제안하는 알고리듬은 가치함수 근사 기법을 활용하며 특정 상태의 가치를 근사하기 위해 해당 상태에서의 근사함수의 상한 및 하한을 활용한다. 이 때, 좋은 상한 및 하한값을 구하기 위해 제안된 모형의 선형계획 완화문제와 선형계획 기반 휴리스틱 알고리듬을 사용한다. 실험 결과를 통해 제안한 알고리듬이 기존의 방법들과 비교하여 우수한 성능을 보임을 확인한다.Abstract i Contents iii List of Tables vii List of Figures ix Chapter 1 Introduction 1 1.1 Backgrounds 1 1.2 Integrated Lot-sizing and Scheduling Problem 6 1.3 Literature Review 9 1.3.1 Optimization Models for LSP 9 1.3.2 Recent Works on LSP 14 1.4 Research Objectives and Contributions 16 1.5 Outline of the Dissertation 19 Chapter 2 Polyhedral Study on Single-period Substructure of Lot-sizing and Scheduling Problem with Sequence-dependent Setups 21 2.1 Introduction 22 2.2 Literature Review 27 2.3 Single-period Substructure 30 2.3.1 Assumptions 31 2.3.2 Basic Polyhedral Properties 32 2.4 New Valid Inequalities 37 2.4.1 S-STAR Inequality 37 2.4.2 Separation of S-STAR Inequality 42 2.4.3 U-STAR Inequality 47 2.4.4 Separation of U-STAR Inequality 49 2.4.5 General Representation of the Inequalities 52 2.5 Extended Formulations 55 2.5.1 Single-commodity Flow Formulations 55 2.5.2 Multi-commodity Flow Formulations 58 2.5.3 Time-ow Formulations 59 2.6 Computational Experiments 63 2.6.1 Experiment Settings 63 2.6.2 Experiment Results on Single-period Instances 65 2.6.3 Experiment Results on Multi-period Instances 69 2.7 Summary 73 Chapter 3 New Optimization Models for Lot-sizing and Scheduling Problem with Sequence-dependent Setups, Crossover, and Carryover 75 3.1 Introduction 76 3.2 Literature Review 78 3.3 Integer Optimization Models 80 3.3.1 Standard Model (ST) 82 3.3.2 Time-ow Model (TF) 84 3.3.3 Comparison of (ST) and (TF) 89 3.3.4 Facility Location Reformulation 101 3.4 LP-based Naive Fixing Heuristic Algorithm 104 3.5 Computational Experiments 108 3.5.1 Test Instances 108 3.5.2 LP Bound 109 3.5.3 Computational Performance with MIP Solver 111 3.5.4 Performance of LPNF Algorithm 113 3.6 Summary 115 Chapter 4 Approximate Dynamic Programming Algorithm for Lot-sizing and Scheduling Problem with Sequence-dependent Setups 117 4.1 Introduction 118 4.1.1 Markov Decision Process 118 4.1.2 Approximate Dynamic Programming Algorithms 121 4.2 Markov Decision Process Reformulation 124 4.3 Approximate Dynamic Programming Algorithm 127 4.4 Computational Experiments 131 4.4.1 Comparison with (TF-FL) Model 131 4.4.2 Comparison with Big Bucket Model 134 4.5 Summary 138 Chapter 5 Conclusion 139 5.1 Summary and Contributions 139 5.2 Future Research Directions 141 Bibliography 145 Appendix A Pattern-based Formulation in Chapter 2 159 Appendix B Detailed Test Results in Chapter 2 163 Appendix C Detailed Test Results in Chapter 3 169 국문초록 173박

A decomposition algorithm for robust lot sizing problem with remanufacturing option

In this paper, we propose a decomposition procedure for constructing robust optimal production plans for reverse inventory systems. Our method is motivated by the need of overcoming the excessive computational time requirements, as well as the inaccuracies caused by imprecise representations of problem parameters. The method is based on a min-max formulation that avoids the excessive conservatism of the dualization technique employed by Wei et al. (2011). We perform a computational study using our decomposition framework on several classes of computer generated test instances and we report our experience. Bienstock and Özbay (2008) computed optimal base stock levels for the traditional lot sizing problem when the production cost is linear and we extend this work here by considering return inventories and setup costs for production. We use the approach of Bertsimas and Sim (2004) to model the uncertainties in the input

A theoretical and computational analysis of lot-sizing in remanufacturing with separate setups

Due to the stricter government regulations on end-of-life product treatment and the increasing public awareness towards environmental issues, remanufacturing has been a significantly growing industry over the last decades, offering many potential business opportunities. In this paper, we investigate a crucial problem apparent in this industry, the remanufacturing lot-sizing problem with separate setups. We first discuss two reformulations of this problem, and remark an important property with regards to their equivalence. Then, we present a theoretical investigation of a related subproblem, where our analysis indicates that a number of flow cover inequalities are strong for this subproblem under some general conditions. We then investigate the computational effectiveness of the alternative methods discussed for the original problem. Detailed numerical results are insightful for the practitioner, indicating that in particular when the return variability increases or when the remanufacturing setup costs decrease relevant to manufacturing setup costs, the flow covers can be very effective

Progressive selection method for the coupled lot-sizing and cutting-stock problem

The coupled lot-sizing and cutting-stock problem has been a challenging and significant problem for industry, and has therefore received sustained research attention. The quality of the solution is a major determinant of cost performance in related production and inventory management systems, and therefore there is intense pressure to develop effective practical solutions. In the literature, a number of heuristics have been proposed for solving the problem. However, the heuristics are limited in obtaining high solution qualities. This paper proposes a new progressive selection algorithm that hybridizes heuristic search and extended reformulation into a single framework. The method has the advantage of generating a strong bound using the extended reformulation, which can provide good guidelines on partitioning and sampling in the heuristic search procedure so as to ensure an efficient solution process. We also analyze per-item and per-period Dantzig-Wolfe decompositions of the problem and present theoretical comparisons. The master problem of the per-period Dantzig-Wolfe decomposition is often degenerate, which results in a tailing-off effect for column generation. We apply a hybridization of Lagrangian relaxation and stabilization techniques to improve the convergence. The discussion is followed by extensive computational tests, where we also perform detailed statistical analyses on various parameters. Comparisons with other methods indicate that our approach is computationally tractable and is able to obtain improved results

Large-Scale Optimisation in Operations Management: Algorithms and Applications

The main contributions of this dissertation are the design, development and application of optimisation methodology, models and algorithms for large-scale problems arising in Operations Management. The first chapter introduces constraint transformations and valid inequalities that enhance the performance of column generation and Lagrange relaxation. I establish theoretical connections with dual-space reduction techniques and develop a novel algorithm that combines Lagrange relaxation and column generation. This algorithm is embedded in a branch-and-price scheme, which combines large neighbourhood and local search to generate upper bounds. Computational experiments on capacitated lot sizing show significant improvements over existing methodologies. The second chapter introduces a Horizon-Decomposition approach that partitions the problem horizon in contiguous intervals. In this way, subproblems identical to the original problem but of smaller size are created. The size of the master problem and the subproblems are regulated via two scalar parameters, giving rise to a family of reformulations. I investigate the efficiency of alternative parameter configurations empirically. Computational experiments on capacitated lot sizing demonstrate superior performance against commercial solvers. Finally, extensions to generic mathematical programs are presented. The final chapter shows how large-scale optimisation methods can be applied to complex operational problems, and presents a modelling framework for scheduling the transhipment operations of the Noble Group, a global supply chain manager of energy products. I focus on coal operations, where coal is transported from mines to vessels using barges and floating cranes. Noble pay millions of dollars in penalties for delays, and for additional resources hired to minimize the impact of delays. A combination of column generation and dedicated heuristics reduces the cost of penalties and additional resources, and improves the efficiency of the operations. Noble currently use the developed framework, and report significant savings attributed to it

Local Cuts and Two-Period Convex Hull Closures for Big Bucket Lot-Sizing Problems.

Despite the significant attention that they have drawn over the years, big bucket lot-sizing problems remain notoriously difficult to solve. The authors have previously presented evidence that what make 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 model

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 literature contained results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multiperiod 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 linear programming solutions from the convex hulls of the two-period closure we study. The convex hull representation of the twoperiod closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation and can therefore 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

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes