近幾十年來，網路可靠度廣泛應用於許多現實世界的系統。在實際應用中，許多日常生活的網路會受到角度的限制，例如:自動搬運車和軍用機器人。因此，一個新的二元狀態網路及多態流量網路（BN/MFN）的模型稱為二維二元狀態角度限制網路（2DBAN/2DMFAN），其中每個弧具有故障的概率以及每個節點有特定的轉彎角度限制，用來處理角度限制網路這種現實生活的問題。使用餘弦法和歐式距離公式去搜尋角度最小路徑集合（AMPS），這是一個簡單的路徑滿足所有角度的限制，從源節點到2DBAN/2DMFAN匯聚節點的所有最小路徑所成的集合。一個2DBAN/2DMFAN的可靠度可以使用所有MPS計算。最後，通過實例來說明如何進行2DBAN/2DMFAN的可靠度計算。

In current decades, we used the network reliability widely into several realistic situations. In practical application, the transmitting utility may be restricted by specific swerve angle i.e., automated guided vehicle, the oil transmission system and military robot. Hence, a novel binary-state network (BN)/multi-state flow network (MFN) model called the 2D binary-state angle network (2DBAN)/2D multi-state flow angle network (2DMFAN), in which each arc has a probability of different capacity as well as each node has a specific swerve angle restriction, is offered to deal with those realistic problems. An efficient algorithm constructed with the cosine law and Euclidean distance formula is introduced in searching for all minimal path set at capacity level d (d-MPs) which is a simple path satisfied all angle restrictions and capacity level d to the sink node from the source node in 2DBAN/2DMFAN. We can calculate the reliability of a 2DBAN/2DMFAN by all MPs (d-MPs) satisfy the swerve angle restriction. Finally, we use some examples to illustrate how the reliability of a 2DBAN and 2DMFAN are evaluated using the proposed algorithm.

ABSTRACT I
中文摘要 II
致謝 III
CHAPTER 1 Introduction 1
1.1 Background and Motivation 1
1.2 Problem Statement and Objectives 2
1.3 Research Framework 3
CHAPTER 2 Literature Review 4
2.1 Network Reliability 4
2.2 Binary-state network & Multi-state Flow network 5
CHAPTER 3 Research Methodology 6
3.1 Acronyms 6
3.2 Notations 6
3.3 Nomenclature 8
3.4 Assumptions 9
3.5 Theorem and Lemma 9
CHAPTER 4 Algorithms 14
4.1 An Algorithm for Binary-state Angle Network 14
4.2 An Algorithm for Multi-state Flow Angle Network 16
CHAPTER 5 Examples 18
5.1 An Example for Binary-state Angle Network 18
5.2 An Example for Multi-state Flow Angle Network 25
CHAPTER 6 Conclusion 36
6.1 Conclusion 36
6.2 Future Work 36
References 38

[1] Samad, M.A., An efficient algorithm for simultaneously deducing minimal paths as well as cuts of a communication network. Microelectronics Reliability, 1987. 27(3): p. 437-441.
[2] Aven, T., Availability evaluation of oil/gas production and transportation systems. Reliability Engineering, 1987. 18(1): p. 35-44.
[3] Lee, S.H., Reliability Evaluation of a Flow Network. Reliability, IEEE Transactions on, 1980. R-29(1): p. 24-26.
[4] Shier, D.R., Network reliability and algebraic structures. 1991: Clarendon Press. 144.
[5] Yeh, W.-C., Evaluating the reliability of a novel deterioration-effect multistate flow network. Information Sciences, 2013. 243: p. 75-85.
[6] Yeh, W.-C., A Novel Node-based Sequential Implicit Enumeration Method for finding all d-MPs in a Multistate Flow Network. Information Sciences, 2015. 297: p. 283-292..
[7] Abraham, J.A., An Improved Algorithm for Network Reliability. Reliability, IEEE Transactions on, 1979. R-28(1): p. 58-61.
[8] Aven, T., Reliability Evaluation of Multistate Systems with Multistate Components. Reliability, IEEE Transactions on, 1985. R-34(5): p. 473-479.
[9] Yeh, W.-C., An Improved Sum-of-Disjoint-Products Technique for Symbolic Multi-state Flow Network Reliability. Reliability Engineering & System Safety,
2007. 92(2): p. 260-268.
[10] Heidtmann, K., Statistical comparison of two sum-of-disjoint-product algorithms for reliability and safety evaluation. In: SAFECOMP. 2002. p. 70-81.
[11] DOHMEN, K., Inclusion-exclusion and network reliability. Journal of Combinatorics, 1998, 5: 537-544.
[12] Yeh, W.-C., An Evaluation of the Multistate-Node Networks Reliability Using the Traditional Binary-State Networks Reliability Algorithm, Reliability Engineering & System Safety, 2003. 81(1): p. 1-7.
[13] Colbourn, C.J., The Combinatorics of Network Reliability. 1987: Oxford University Press, Inc. 176.
[14] Levitin, G., Reliability of multi-state systems with common bus performance sharing. IIE Transactions, 2011. 43(7): p. 518-524.
[15] Janan, X., On Multistate System Analysis. Reliability, IEEE Transactions on, 1985. R-34(4): p. 329-337.
[16] Yeh, W.-C, A revised layered-network algorithm to search for all d-minpaths of a limited-flow acyclic network. Reliability, IEEE Transactions on, 1998. 47(4): p.436-442.
[17] Yeh, W.-C., A simple algorithm to search for all d-MPs with unreliable nodes. Reliability Engineering & System Safety, 2001. 73(1): p. 49-54.
[18] Yeh, W. -C., A Simple Method to Verify All d-Minimal Path Candidates of a Limited-Flow Network and its Reliability. The International Journal of Advanced Manufacturing Technology, 2002. 20(1): p. 77-81.
[19] Yeh, W.-C., A novel method for the network reliability in terms of capacitated minimal-paths without knowing minimal-paths in advance. Journal of the Operational Research Society, 2005. 56: p. 1235-1240.
[20] Yeh, W.-C., Search for MC in modified networks. Computers & Operations Research, 2001. 28(2): p. 177-184.
[21] Yeh, W.-C., A simple approach to search for all d-MCs of a limited-flow network. Reliability Engineering & System Safety, 2001. 71(1): p. 15-19.
[22] Yeh, W.-C., Search for all d-Mincuts of a limited-flow network. Computers & Operations Research, 2002. 29(13): p. 1843-1858.
[23] Yeh, W.-C., Search for all MCs in networks with unreliable nodes and arcs. Reliability Engineering & System Safety, 2003. 79(1): p. 95-101.