本論文的研究主題是工件(job)到達時間(release date)不同之下單一機器(single machine)上的排程問題(scheduling problem)。這是個關於單一機器處理一組工件的最佳化配置(optimal allocation)問題。基本上，我們假設機器能夠持續不斷地工作，但是同一時間只允許處理一個工件；而每一個工作在它的到達時間之後，隨時可被處理，且必須一次處理完成，不可中途間斷。排程的目標(objective)是使所有工件的完成時間總合為最小。這個排程問題模型記做 ，在理論上和實用上都很重要。文獻上已證實解這個問題的演算複雜度(complexity)為「Strongly NP-hard」，也就是說，不可能用多項式(polynomial)來表示求出最佳解(optimal solution)之演算次數與工件數目的關係。
　　本文將探討如何更快速且有效地解 問題，內容包括二部分：近似解演算法(approximation algorithm)和分支與界限法(branch-and-bound method)之最佳解演算法(optimality algorithm)。
　　在第一部份，首先導出一則局部最佳解(local optimality)的充分必要條件。所謂「局部最佳解」是指，在排序過程的每一步驟中，所有候選工件的最佳排序(optimal schedule)。這個等價條件提供一個優先權法則(prior rule)，用以構造三個演算複雜度為 的近似解演算法：LO-，A1LO- 和A2LO-法則。又，應用此條件規範一包含所有最佳解的「優勢子集合」(dominant subset)，它嚴格包含於(strictly contained)文獻上已知的優勢子集合。
　　在第二部分中，發展出三個分支與界限法的最佳解演算法。首先，應用ERD-優先法則推導出一個延遲時間(delay time)總和的上界(upper bound)，利用這個上界及其他條件，演譯出三個分支與界限流程中所需的優勢定理(dominance theorem)。利用近似解演算法LO-法則等作為分枝策略(branching schemes)，並且以其結果為上界限。其中，演算法BB_w使用橫向優先(width first)搜索路徑；演算法BB_d使用縱向優先(depth first)搜索路徑。另外，演算法BB_prior分為二階段，先篩選最優先工作(the prior jobs)，將原問題分解為獨立子問題，再對非最優先工作排序。
　　實驗模擬的結果顯示，經由LO-法則等近似演算法所得的最佳近似解，優於文獻上已知的相同演算複雜度之近似解演算法。同時，最佳解演算法BB_w，明顯優於其他文獻上之分支與界限法，而且能解含120個工件的問題，這是截至目前為止最好的結果。

In this dissertation, we study the single machine scheduling problem with an objective of minimizing the total completion time subject to release dates. The problem, denoted 1|rj ΣCj ,was known to be strongly NP-hard and both theoretically and practically important. The focus of the research in this dissertation is to develop the efficient algorithms for solving the 1|rj|ΣCj problem.
This thesis contains two parts.
In the first part, the theme concerns the approximation approach. We derive a necessary and sufficient condition for local optimality, which can be implemented as a priority rule and be used to construct three heuristic algorithms with running times of O(n log n). By ”local optimality”, we mean the optimality of all candidates whenever a job is selected in a schedule, without considering the other jobs preceding or following. This is the most broadly considered concepts of locally optimal rule. We also identify a dominant subset which is strictly contained in each of all known dominant subsets, where a dominant subset is a set of solutions containing all optimal schedules.
In the second part, we develop our optimality algorithms for the 1|rj |ΣCj problem. First, we present a lemma for estimating the sum of delay times of the rest jobs, if the starting time is delayed a period of time in a schedule. Then, using the lemma, partially, we proceed to develop a new partition property and three dominance theorems, that will be used and have improved the branch-and-bound algorithms for our optimization approach. By exploiting the insights gained from our heuristics as a branching scheme and by exploiting our heuristics as an upper bounding procedure, we propose three branch-and-bound algorithms. Our algorithms can optimally solve the problem up to 120 jobs, which is known to be the best till now.

Contents
1 Introduction 1
1.1 The Scheduling . . . . . . . . . . .. . . . . . . . . . . . . . 1
1.2 The 1jrj jPCj Problem . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Framework and Notation . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Approximation Approach . . . . . . . . . . . . . . . . . . . 5
1.2.3 Optimization Approach . . . . . . . . . . . . . . . . . . . . . 6
1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Articles concerning SMSP . . . . . . . . . . . . . . . . . . . 7
1.3.2 Approximation Algorithms for 1jrj jPCj in Literature . . . 8
1.3.3 Branch-and-Bound methods in Literature . . . . . . . . . . . 9
1.4 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . 11
2 Approximation Algorithms ．．．．．．．．．．．．．．．．．．．．．．．12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Notations and Terminologies . . . . . . . . . . . . . . . . . . 13
2.1.2 Heuristic Algorithms in Literatures . . . . . . . . . . . . . . 13
2.1.3 Contents of Chapter . . . . . . . . . . . . . . . . . . . . . . 15
2.2 An Equivalent Condition for Local Optimality . . . . . . . . . . 15
2.3 L-Active Schedule . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Heuristic Algorithms| LO, A1LO, and A2LO . . . . . . . . . . . 25
2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . 28
2.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . 29
3 Dominance Conditions ．．．．．．．．．．．．．．．．．．．．．．．．．33
3.1 The Proved Dominance Properties . . .. . . . . . . . . . . . . . 33
3.2 New Dominance Theorems . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 An Upper Bound for Total delay time . . . . . . . . . . . . . 33
3.2.2 Dominance Theorems . . . . . . . . . . . . . . . . . . . . . . 38
4 Optimal Algorithm : BB_w and BB_d ．．．．．．．．．．．．．．．47
4.1 Introduction . . . . .. . . . . . . . . . . . . . . . . . 47
4.2 Partitioning and Optimality . . . . . .. . . . . . . . . . . . . 48
4.3 Optimal Algorithm BB_w and BB_d. . . . . . . . . . . . . . . . . 50
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 60
5 Optimal Algorithm : BB_Prior ．．．．．．．．．．．．．．．．．．．67
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Problem System 1jrj ; priorjPCj . . . . . . . . . . . . . . . . . 68
5.3 Optimal Algorithm BB Prior . . . . . . . . . . . . . . . . . . . 69
6 Conclusion 74

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