在本研究中，我們研究

In this research, we research the scheduling problem with n jobs that can be divided into batches and m parallel machines under availability constraint. Due to the eligibility constraint, each jobs has its own recipe, not all m machines can process job’s recipe. And the jobs have the waiting time before the processing, they can process together when their arrival time smaller than the batch’s waiting time. The objective of our scheduling problem is to minimize the total completion time.
In order to find the best solution to this problem, this research proposes a branch and bound algorithm. In the algorithm of this study, first, we resort a sequence according to the job ‘s arrival time. The available batch positions of the machines are branched, and we determine whether schedule the job to process on the batch of the machine, and then how to plan the remaining job entering the same or different batches.

Contents
摘要 i
Abstract ii
Contents iii
List of Figures v
List of Tables vi
Chapter 1 Introduction 1
1.1 Research background and motivation 1
1.2 Problem definition 2
1.3 Research objective 4
1.4 Research methodology 4
1.5 Research framework 5
Chapter 2 Literature Review 7
2.1 Parallel machine scheduling with batch processing 7
2.2 Scheduling problem with eligibility constraints 8
2.3 Parallel machine with release time constraint 8
Chapter 3 Methodology 10
3.1 Notation 10
3.2 Branching scheme 11
3.3 Propositions 16
3.4 Bounding scheme 20
3.5 Branch and bound Algorithm 21
Chapter 4 Analysis 25
4.1 Instance Generation 25
4.2Validation of the branch and bound algorithm 26
4.3 Performance of the branch and bound algorithm 27
Chapter 5 Conclusion 31
5.1 Research Contribution 31
5.2 Research Limitation 32
5.3 Future Research 32
Reference 33
Appendix 35
List of Figures
Figure 1. Demonstration of machine environment. 3
Figure 2. Research framework. 6
Figure 3. Searching tree of branching scheme 15
Figure 4. If Si,d<Si,d', we don’t branch out node d' 17
Figure 5. SPT sequencing of batches 18
Figure 6. Proof of Proposition 2 – original schedule 19
Figure 7. Proof of Proposition 2 – after pairwise interchange schedule 19
Figure 8. Flow chart of branch and bound algorithm 24
Figure 9. Percentage of nodes eliminated by branch and bound algorithm 28
Figure A. 1 Execution of the branch and bound on the numerical example 36
Figure A. 2 The beginning of the tree is root node n1 37
Figure A. 3 The branching out child node n2 from root node n1 38
Figure A. 4 The branching out child node n3 from node n2 40
Figure A. 5 The branching out child node n4 from node n3 42
Figure A. 6 Backtracking from node n4 and right branching node n5 44
List of Tables
Table 1. Job data 25
Table 2. Machine data 25
Table 3. Validation results of branch and bound algorithm 33
Table 4. Instance that can solve by our branch and bond algorithm 36

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