在現代的生產環境中，同時考慮訂單選擇(order selection)和生產排程(production scheduling)的決策是很重要的問題。在本研究的問題中，有一個已知的工件集合，各個工件有各自的屬性(attribute)包含可加工時間(ready date)、交期(due date)、權重(weight)和所需加工時間(processing time)。所有工件不允許延期完成或分段加工。因此，所有工件僅能被挑選加工或是拒絕，且問題的目標函式為最大化所挑選出來的工件權重總和。先前研究稱此問題為產出最大化問題(throughput maximization problem)。
針對此問題，本研究測試幾種不同求解方法的效率：混合整數規劃(mixed integer programming)、隱式窮舉法(implicit enumeration)、分支界線法(branch-and-bound method)、動態規劃法(Dynamic programming)。在這些方法中，混合整數規劃是相對簡單的方法，只要建立好MIP模型即可利用現有軟體加以求解。隱式窮舉法將利用4個優勢法則(dominance rules)來減少節點的個數。分支界線法將線性規劃放鬆(linear programming relaxation)加入隱式窮舉法中，企圖進一步縮短求解時間。動態規劃法為分支界線法的延伸，其重要的特性為加入記憶方案(memory scheme)，以紀錄各個子問題的解，避免計算相同的子問題。
從實驗結果顯示，動態規劃結合記憶方案相對於其餘方法，可以在較短的時間完成整個最佳化求解，在較大型的問題中，亦可以在限定時間內，提供一個較靠近最佳解的解。
關鍵字：生產排程、訂單選擇、產出最大化、分支界線法、動態規劃

In manufacturing environments, order selection and scheduling is a difficult, but important decision. In the problem investigated in this study, the attributes, including the ready date, the due date, the processing time, and the weight, of a number of given jobs are known. A selected job cannot be started before its ready date, or completed after its due date, or split into more than one processing period, or partially processed. The objective of the problem is to maximize the total weight of selected jobs. Such a problem is called throughput maximization problem in the previous literatures.
This study tests and compares several methods, including mixed integer programming (MIP), implicit enumeration method (IE), branch-and-bound method (B&B), and dynamic programming (DP). MIP is a traditional approach that requires only formulating the problem and generating MIP problem instances, which can be solved by standard software package, such as GUROBI (GUROBI, 2013). IE adopts the various dominance rules to reduce the number of nodes in the enumerative tree. B&B is extended from IE by adding the bounding features of linear programming relaxation. DP in essence is enhanced from IE by having a memory scheme to record the solution of previously solved sub-problems. Using such a memory scheme, DP can avoid solving the same sub-problems repeatedly.
The experiment result shows that DP with memory scheme requires much less computation time to solve a problem than the other methods. Further, according to the experiments, in a much larger problem, DP can more efficiently provide a good heuristic solution than the others one within a limited computation time.

Key words: production scheduling, order selection, throughput maximization, branch-and-bound, dynamic programming.

摘要 I
Abstract II
TABLE OF CONTENTS V
LIST OF FIGURES VII
LIST OF TABLES VIII
1. Introduction 1
2. Literature Review 7
3. Solution Methods 10
3.1. Heuristic Method 10
3.2. Mixed Integer Programming (MIP) 12
3.3. Implicit Enumeration (IE) 13
3.3.1. Partial Ordering 15
3.3.2. Dominance Rules 16
3.3.3. Depth-First Search 18
3.4. Branch-and-Bound (B&B) 20
3.4.1. The LP-Relaxation Formulation 20
3.4.2. Feasible Check of LP-Relaxation 22
3.4.3. Fathom Rules 23
3.5. Dynamic Programming (DP) 24
3.5.1. The Motivation of DP Approach 24
3.5.2. The Classification of Dominance Rules 25
3.5.3. DP Formulations 29
3.5.4. Reduction of State Variables 31
3.5.5. Saving to Memory Scheme 35
3.5.6. Retrieval from Memory Scheme 37
3.5.7. The Computation Procedure of DP Approach 38
3.5.8. A Numerical Example of the DP Approach 40
3.6. DP with LP-Relaxation 43
4. Computation Experiments 45
4.1. Experiment Parameters 45
4.2. Problem Generation Procedure 46
4.3. Experiment Setting 47
4.4. Result and Analysis 49
4.4.1. The Performance of Various Methods 49
4.4.2. The Progress of Solution Improvements 51
4.4.3. The Effectiveness of Dominance Rules, Fathom Rules, and LP-Relaxation 54
4.4.4. The Impacts of Various Control Factors on Computation Time 59
5. Conclusion and Future direction 63

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