1,785 research outputs found

    A computational analysis of lower bounds for big bucket production planning problems

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    In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research

    A Computational Analysis of Lower Bounds for Big Bucket Production Planning Problems

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    In this paper, we analyze a variety of approaches to obtain lower bounds for multilevel production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provides valuable insights on why these problems are hard to solve. We conclude with computational results from widely used test sets and discussion of future research

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

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    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

    On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

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    Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

    Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

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    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

    ์ˆœ์„œ์˜์กด์  ์ž‘์—…์ค€๋น„๊ฐ€ ์žˆ๋Š” ์ƒ์‚ฐ๊ณ„ํš ๋ฌธ์ œ์— ๋Œ€ํ•œ ์ •์ˆ˜ ์ตœ์ ํ™” ๋ฐ ๊ทผ์‚ฌ ๋™์  ๊ณ„ํš๋ฒ• ๊ธฐ๋ฐ˜ ํ•ด๋ฒ•

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    ํ•™์œ„๋…ผ๋ฌธ(๋ฐ•์‚ฌ) -- ์„œ์šธ๋Œ€ํ•™๊ต๋Œ€ํ•™์› : ๊ณต๊ณผ๋Œ€ํ•™ ์‚ฐ์—…๊ณตํ•™๊ณผ, 2022. 8. ์ด๊ฒฝ์‹.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.๊ณต๊ธ‰๋ง์˜ ์ƒ์‚ฐ ๊ณ„ํš ๋‹จ๊ณ„์—์„œ์˜ ์ฃผ์š”ํ•œ ๋‘ ๊ฐ€์ง€ ๋‹จ๊ธฐ ์˜์‚ฌ๊ฒฐ์ • ๋ฌธ์ œ์ธ Lot-sizing ๋ฌธ์ œ์™€ Scheduling ๋ฌธ์ œ๊ฐ€ ํ†ตํ•ฉ๋œ ๋ฌธ์ œ์ธ Lot-sizing and scheduling problem (LSP)์€ ๊ณ„ํš๋Œ€์ƒ๊ธฐ๊ฐ„ ๋™์•ˆ ์ฃผ์–ด์ง„ ๋ณต์ˆ˜์˜ ์ œํ’ˆ์— ๋Œ€ํ•œ ์ˆ˜์š”๋ฅผ ์ตœ์†Œ์˜ ๋น„์šฉ์œผ๋กœ ๋งŒ์กฑ์‹œํ‚ค๊ธฐ ์œ„ํ•œ ๋‹จ์œ„ ๊ธฐ๊ฐ„ ๋ณ„ ์ตœ์ ์˜ ์ƒ์‚ฐ๋Ÿ‰ ๋ฐ ์ƒ์‚ฐ ์ˆœ์„œ๋ฅผ ๊ฒฐ์ •ํ•œ๋‹ค. ๊ณต๊ธ‰๋ง ๋‚ด์˜ ๋‹ค์–‘ํ•œ ์š”์†Œ์— ๋Œ€ํ•œ ํ†ตํ•ฉ์  ์˜์‚ฌ ๊ฒฐ์ •์˜ ์ค‘์š”์„ฑ์ด ์ปค์ง์— ๋”ฐ๋ผ LSP์— ๋Œ€ํ•œ ๊ด€์‹ฌ ์—ญ์‹œ ์‚ฐ์—…๊ณ„์™€ ํ•™๊ณ„ ๋ชจ๋‘์—์„œ ์ง€์†์ ์œผ๋กœ ์ฆ๊ฐ€ํ•˜์˜€๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ตœ๊ทผ ์ˆ˜์‹ญ ๋…„์— ๊ฑธ์นœ ํ™œ๋ฐœํ•œ ์—ฐ๊ตฌ์—๋„ ๋ถˆ๊ตฌํ•˜๊ณ , ๋ฌธ์ œ ์ž์ฒด๊ฐ€ ๋‚ดํฌํ•˜๋Š” ์ด๋ก ์  ๋ณต์žก์„ฑ ๋ฐ ์‹ค์ œ ์‚ฐ์—… ํ™˜๊ฒฝ๊ณผ ์ œ์กฐ ๊ณต์ •์˜ ๊ณ ๋„ํ™”/๋ณต์žกํ™” ๋“ฑ์œผ๋กœ ์ธํ•ด LSP๋ฅผ ํ•ด๊ฒฐํ•˜๋Š” ๊ฒƒ์€ ์—ฌ์ „ํžˆ ์–ด๋ ค์šด ๋ฌธ์ œ๋กœ ๋‚จ์•„์žˆ๋‹ค. ํŠนํžˆ ์ˆœ์„œ์˜์กด์  ์ž‘์—…์ค€๋น„๊ฐ€ ์žˆ๋Š” ๊ฒฝ์šฐ ๋ฌธ์ œ๊ฐ€ ๋”์šฑ ์–ด๋ ค์›Œ์ง„๋‹ค๋Š” ๊ฒƒ์ด ์ž˜ ์•Œ๋ ค์ ธ ์žˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์ˆœ์„œ์˜์กด์  ์ž‘์—…์ค€๋น„๊ฐ€ ์žˆ๋Š” LSP๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•œ ์ •์ˆ˜ ์ตœ์ ํ™” ๋ฐ ๊ทผ์‚ฌ ๋™์  ๊ณ„ํš๋ฒ• ๊ธฐ๋ฐ˜์˜ ํ•ด๋ฒ•์„ ์ œ์•ˆํ•œ๋‹ค. ๋จผ์ €, ์ด๋ก ์ ์œผ๋กœ ๊ฐ•์„ฑ NP-hard์— ์†ํ•œ๋‹ค๋Š” ์‚ฌ์‹ค์ด ์ž˜ ์•Œ๋ ค์ง„ LSP์˜ ๊ทผ๋ณธ ๊ตฌ์กฐ์— ๋Œ€ํ•œ ์ดํ•ด๋ฅผ ๋†’์ด๊ธฐ ์œ„ํ•˜์—ฌ ๋‹จ์ผ ๊ธฐ๊ฐ„๋งŒ์„ ๊ณ ๋ คํ•˜๋Š” ๋ถ€๋ถ„๊ตฌ์กฐ์— ๋Œ€ํ•ด ๋‹ค๋ฃฌ๋‹ค. ๋‹จ์ผ ๊ธฐ๊ฐ„ ๋ถ€๋ถ„๊ตฌ์กฐ์— ์˜ํ•ด ์ •์˜๋˜๋Š” ๋‹ค๋ฉด์ฒด์— ๋Œ€ํ•œ ์ด๋ก ์  ๋ถ„์„์„ ํ†ตํ•ด ์ƒˆ๋กœ์šด ์œ ํšจ ๋ถ€๋“ฑ์‹ ๊ตฐ์„ ๋„์ถœํ•˜๊ณ  ํ•ด๋‹น ์œ ํšจ ๋ถ€๋“ฑ์‹๋“ค์ด ๊ทน๋Œ€๋ฉด(facet)์„ ์ •์˜ํ•  ์กฐ๊ฑด์— ๋Œ€ํ•ด ๋ฐํžŒ๋‹ค. ๋˜ํ•œ, ๋„์ถœ๋œ ์œ ํšจ ๋ถ€๋“ฑ์‹๋“ค์ด ๋‹คํ•ญ์‹œ๊ฐ„ ๋‚ด์— ๋ถ„๋ฆฌ ๊ฐ€๋Šฅํ•จ์„ ๋ณด์ด๊ณ , ์ตœ๋Œ€ํ๋ฆ„๋ฌธ์ œ๋ฅผ ํ™œ์šฉํ•œ ๋‹คํ•ญ์‹œ๊ฐ„ ๋ถ„๋ฆฌ ์•Œ๊ณ ๋ฆฌ๋“ฌ์„ ๊ณ ์•ˆํ•œ๋‹ค. ์‹คํ—˜ ๊ฒฐ๊ณผ๋ฅผ ํ†ตํ•ด ์ œ์•ˆํ•œ ์œ ํšจ ๋ถ€๋“ฑ์‹๋“ค์ด ๋ชจํ˜•์˜ ์„ ํ˜•๊ณ„ํš ํ•˜ํ•œ๊ฐ•๋„๋ฅผ ๋†’์ด๋Š” ๋ฐ ํฐ ์˜ํ–ฅ์„ ์คŒ์„ ํ™•์ธํ•œ๋‹ค. ๋˜ํ•œ ํ•ด๋‹น ๋ถ€๋“ฑ์‹๋“ค์ด ๋ชจ๋‘ ์ถ”๊ฐ€๋œ ๋ชจํ˜•๊ณผ ์ด๋ก ์ ์œผ๋กœ ๋™์ผํ•œ ํ•˜ํ•œ์„ ์ œ๊ณตํ•˜๋Š” ํ™•์žฅ ์ˆ˜๋ฆฌ๋ชจํ˜•(extended formulation)์„ ๋„์ถœํ•œ๋‹ค. ์ด๋ฅผ ํ™œ์šฉํ•˜์—ฌ, ์‹ค์ œ ์‚ฐ์—…์—์„œ ๋ฐœ์ƒํ•˜๋Š” LSP์—์„œ ์ข…์ข… ๊ณ ๋ คํ•˜๋Š” ์ฃผ์š”ํ•œ ์ถ”๊ฐ€ ์š”์†Œ๋“ค์„ ๋‹ค๋ฃฐ ์ˆ˜ ์žˆ๋Š” ์ƒˆ๋กœ์šด ์ˆ˜๋ฆฌ ๋ชจํ˜•์„ ์ œ์•ˆํ•˜๋ฉฐ ํ•ด๋‹น ๋ชจํ˜•์ด ๊ธฐ์กด์˜ ๋ชจํ˜•์— ๋น„ํ•ด ๋”์šฑ ๊ฐ•ํ•œ ์„ ํ˜•๊ณ„ํš ํ•˜ํ•œ์„ ์ œ๊ณตํ•จ์„ ๋ณด์ธ๋‹ค. ์ด ๋ชจํ˜•์„ ๋ฐ”ํƒ•์œผ๋กœ ๋น ๋ฅธ ์‹œ๊ฐ„ ๋‚ด์— ๊ฐ€๋Šฅํ•ด๋ฅผ ์ฐพ์„ ์ˆ˜ ์žˆ๋Š” ์„ ํ˜•๊ณ„ํš ๊ธฐ๋ฐ˜ ํœด๋ฆฌ์Šคํ‹ฑ ์•Œ๊ณ ๋ฆฌ๋“ฌ์„ ๊ฐœ๋ฐœํ•œ๋‹ค. ๋งˆ์ง€๋ง‰์œผ๋กœ ํ•ด๋‹น ๋ฌธ์ œ์— ๋Œ€ํ•œ ๊ทผ์‚ฌ ๋™์  ๊ณ„ํš๋ฒ• ์•Œ๊ณ ๋ฆฌ๋“ฌ์„ ์ œ์•ˆํ•œ๋‹ค. ์ œ์•ˆํ•˜๋Š” ์•Œ๊ณ ๋ฆฌ๋“ฌ์€ ๊ฐ€์น˜ํ•จ์ˆ˜ ๊ทผ์‚ฌ ๊ธฐ๋ฒ•์„ ํ™œ์šฉํ•˜๋ฉฐ ํŠน์ • ์ƒํƒœ์˜ ๊ฐ€์น˜๋ฅผ ๊ทผ์‚ฌํ•˜๊ธฐ ์œ„ํ•ด ํ•ด๋‹น ์ƒํƒœ์—์„œ์˜ ๊ทผ์‚ฌํ•จ์ˆ˜์˜ ์ƒํ•œ ๋ฐ ํ•˜ํ•œ์„ ํ™œ์šฉํ•œ๋‹ค. ์ด ๋•Œ, ์ข‹์€ ์ƒํ•œ ๋ฐ ํ•˜ํ•œ๊ฐ’์„ ๊ตฌํ•˜๊ธฐ ์œ„ํ•ด ์ œ์•ˆ๋œ ๋ชจํ˜•์˜ ์„ ํ˜•๊ณ„ํš ์™„ํ™”๋ฌธ์ œ์™€ ์„ ํ˜•๊ณ„ํš ๊ธฐ๋ฐ˜ ํœด๋ฆฌ์Šคํ‹ฑ ์•Œ๊ณ ๋ฆฌ๋“ฌ์„ ์‚ฌ์šฉํ•œ๋‹ค. ์‹คํ—˜ ๊ฒฐ๊ณผ๋ฅผ ํ†ตํ•ด ์ œ์•ˆํ•œ ์•Œ๊ณ ๋ฆฌ๋“ฌ์ด ๊ธฐ์กด์˜ ๋ฐฉ๋ฒ•๋“ค๊ณผ ๋น„๊ตํ•˜์—ฌ ์šฐ์ˆ˜ํ•œ ์„ฑ๋Šฅ์„ ๋ณด์ž„์„ ํ™•์ธํ•œ๋‹ค.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 ๊ตญ๋ฌธ์ดˆ๋ก 173๋ฐ•

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

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    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

    A theoretical study of two-period relaxations for lot-sizing problems with big-bucket capacities

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    In this paper, we study two-period subproblems proposed by Akartunali et al. (2015) for lot-sizing problems with big-bucket capacities and nonzero setup times, complementing our previous work investigating the special case of zero setup times. In particular, we study the polyhedral structure of the mixed integer sets related to various two-period relaxations. We derive several families of valid inequalities and investigate their facet-defining conditions. We also discuss the separation problems associated with these valid inequalities

    Capacitated lot-sizing and scheduling with sequence-dependent, period-overlapping and non-triangular setups

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    In production planning, sequence dependent setup times and costs are often incurred for switchovers from one product to another. When setup times and costs do not respect the triangular inequality, a situation may occur where the optimal solution includes more than one batch of the same product in a single period - in other words, at least one sub tour exists in the production sequence of that period. By allowing setup crossovers, flexibility is increased and better solutions can be found. In tight capacity conditions, or whenever setup times are significant, setup crossovers are needed to assure feasibility. We present the first linear mixed-integer programming extension for the capacitated lot-sizing and scheduling problem incorporating all the necessary features of sequence sub tours and setup crossovers. This formulation is more efficient than other well known lot-sizing and scheduling models. ยฉ Springer Science+Business Media, LLC 2010
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