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2μ°¨μ 2λ¨κ³ λ°°λλ¬Έμ μ λν μ μκ³νλͺ¨ν λ° μ΅μ ν΄λ²
νμλ
Όλ¬Έ (μμ¬) -- μμΈλνκ΅ λνμ : 곡과λν μ°μ
곡νκ³Ό, 2021. 2. μ΄κ²½μ.In this thesis, we study integer programming models and exact algorithms for the two-dimensional two-staged knapsack problems, which maximizes the profit by cutting a single rectangular plate into smaller rectangular items by two-staged guillotine cuts. We first introduce various integer programming models, including the strip-packing model, the staged-pattern model, the level-packing model, and the arc-flow model for the problem. Then, a hierarchy of the strength of the upper bounds provided by the LP-relaxations of the models is established based on theoretical analysis. We also show that there exists a polynomial-size model that has not been proven yet as far as we know. Exact methods, including branch-and-price algorithms using the strip-packing model and the staged-pattern model, are also devised. Computational experiments on benchmark instances are conducted to examine the strength of upper bounds obtained by the LP-relaxations of the models and evaluate the performance of exact methods. The results show that the staged-pattern model gives a competitive theoretical and computational performance.λ³Έ λ
Όλ¬Έμ 2λ¨κ³ κΈΈλ‘ν΄ μ λ¨(two-staged guillotine cut)μ μ¬μ©νμ¬ μ΄μ€μ μ΅λννλ 2μ°¨μ 2λ¨κ³ λ°°λ λ¬Έμ (two-dimensional two-staged knapsack problem: μ΄ν 2TDK)μ λν μ μμ΅μ ν λͺ¨νκ³Ό μ΅μ ν΄λ²μ λ€λ£¬λ€. μ°μ , λ³Έ μ°κ΅¬μμλ μ€νΈλ¦½ν¨νΉλͺ¨ν, λ¨κ³ν¨ν΄λͺ¨ν, λ 벨ν¨νΉλͺ¨ν, κ·Έλ¦¬κ³ νΈ-νλ¦λͺ¨νκ³Ό κ°μ μ μμ΅μ ν λͺ¨νλ€μ μκ°νλ€. κ·Έ λ€, κ°κ°μ λͺ¨νμ μ νκ³νμνλ¬Έμ μ λν΄ μνκ°λλ₯Ό μ΄λ‘ μ μΌλ‘ λΆμνμ¬ μνκ°λ κ΄μ μμ λͺ¨νλ€ κ° μκ³λ₯Ό μ 립νλ€. λν, λ³Έ μ°κ΅¬μμλ 2TDKμ λ€νν¬κΈ°(polynomial-size) λͺ¨νμ μ‘΄μ¬μ±μ μ²μμΌλ‘ μ¦λͺ
νλ€. λ€μμΌλ‘ λ³Έ μ°κ΅¬λ 2TDKμ μ΅μ ν΄λ₯Ό ꡬνλ μκ³ λ¦¬μ¦μΌλ‘μ¨ ν¨ν΄κΈ°λ°λͺ¨νλ€μ λν λΆμ§νκ° μκ³ λ¦¬μ¦κ³Ό λ 벨ν¨νΉλͺ¨νμ κΈ°λ°μΌλ‘ ν λΆμ§μ λ¨ μκ³ λ¦¬μ¦μ μ μνλ€. λ¨κ³ν¨ν΄λͺ¨νμ΄ μ΄λ‘ μ μΌλ‘λ κ°μ₯ μ’μ μνκ°λλ₯Ό 보μ₯ν λΏλ§ μλλΌ, κ³μ° λΆμμ ν΅ν΄ λ¨κ³ν¨ν΄λͺ¨νμ κΈ°λ°μΌλ‘ ν λΆμ§νκ° μκ³ λ¦¬μ¦μ΄ μ νλ μκ° λ΄ μ’μ νμ§μ κ°λ₯ν΄λ₯Ό μ°Ύμμ νμΈνμλ€.Abstract i
Contents iv
List of Tables vi
List of Figures vii
Chapter 1 Introduction 1
1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2 Integer Programming Models for 2TDK 9
2.1 Pattern-based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Arc-flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Level Packing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 3 Theoretical Analysis of Integer Programming Models 20
3.1 Upper Bounds of AF and SM(1;1) . . . . . . . . . . . . . . . . . . 20
3.2 Upper Bounds of ML, PM(d), and SM(d; d) . . . . . . . . . . . . . . 21
3.3 Polynomial-size Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 4 Exact Methods 33
4.1 Branch-and-price Algorithm for the Strip Packing Model . . . . . . . 34
4.2 Branch-and-price Algorithm for the Staged-pattern Model . . . . . . 39
4.2.1 The Standard Scheme . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 The Height-aggregated Scheme . . . . . . . . . . . . . . . . . 40
4.3 Branch-and-cut Algorithm for the Modified Level Packing Model . . 44
Chapter 5 Computational Experiments 46
5.1 Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Upper Bounds Comparison . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 49
5.2.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 55
5.2.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Solving Instances to Optimality . . . . . . . . . . . . . . . . . . . . . 65
5.3.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 65
5.3.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 69
5.3.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 6 Conclusion 77
Bibliography 79
κ΅λ¬Έμ΄λ‘ 83Maste