A heuristic approach for big bucket multi-level production planning problems

Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper, we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other heuristics, it generates high quality lower bounds using strong formulations, and its simple scheme allows it to be easily implemented in the Xpress-Mosel modeling language. Extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic, particularly for challenging problems

Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems.Mixed Integer Programming;Formulations;Symmetry;Lotsizing

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming 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 New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated Lot Sizing Problem (CLSP),as already proposed by Manne in 1958, has an important structural deficiency. Imposingintegrality constraints on the variables in the full blown master will not necessarily give theoptimal IP solution as only production plans which satisfy the Wagner-Whitin condition canbe selected. It is well known that the optimal solution to a capacitated lot sizing problem willnot necessarily have this Wagner-Whitin property. The columns of the traditionaldecomposition model include both the integer set up and continuous production quantitydecisions. Choosing a specific set up schedule implies also taking the associated Wagner-Whitin production quantities. We propose the correct Dantzig-Wolfe decompositionreformulation separating the set up and production decisions. This formulation gives the samelower bound as Manne's reformulation and allows for branch-and-price. We use theCapacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computationalexperiments are presented on data sets available from the literature. Column generation isspeeded up by a combination of simplex and subgradient optimization for finding the dualprices. The results show that branch-and-price is computationally tractable and competitivewith other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulationcan be generalized to other mixed integer programming problems, whereas in the literature,branch-and-price algorithms are almost exclusively developed for pure integer programmingproblems

A priori reformulations for joint rolling-horizon scheduling of materials processing and lot-sizing problem

In many production processes, a key material is prepared and then transformed into different final products. The lot sizing decisions concern not only the production of final products, but also that of material preparation in order to take account of their sequence-dependent setup costs and times. The amount of research in recent years indicates the relevance of this problem in various industrial settings. In this paper, facility location reformulation and strengthening constraints are newly applied to a previous lot-sizing model in order to improve solution quality and computing time. Three alternative metaheuristics are used to fix the setup variables, resulting in much improved performance over previous research, especially regarding the use of the metaheuristics for larger instances