在連續生產的化工廠中，設備可能會隨著時間而老化。所以為了維持化學工廠之連續操作不間斷，會設置適當的備援系統，維持製程運作順利。雖然在先前有相關研究，但目前沒有研究考慮設備會隨著時間增加而老化，亦即故障速率隨時間增加的情境。為了使數學規劃模型預測的結果更符合實際，又由於威布分佈具有設備之故障速率隨時間增加而增加的特性，故有必要以威布分佈來建立一套完整且可靠的數學規劃模型未幫助備援系統的設計及保養。本研究的目的為建立通用的數學模型，並藉由最小化生命週期期望支出來為各製程訂定合適的備援機制。在本研究中我們透過二階多項式來近威布分佈之不可靠度，並利用基因演算法來執行最佳化的計算，且最後會以案例測試，來驗證本研究提出之數學模型的正確性與可行性。從最適化結果中可以得到備援系統最佳設計規格，其中包含 (1) 關鍵元件個數，(2) 各個量測通道的邏輯閘，(3) 各個量測通道需要的線上與備用感測器數量，(4) 備用切換裝置的數量，(5) 線上切換裝置的檢測週期，(6) 暖備件的檢測週期，以及(7) 冷備件的數量。

In a continuously operated chemical plant, process units usually age over time. In these situations, it is necessary to incorporate standby mechanisms so as to maintain uninterrupted production throughout the operation horizon. Even though a few related studies have been reported in literature, e.g., Chan and Chang (2020), Chan and Chang (2021), Tu and Chang (2022), a comprehensive analysis of standby components with time-dependent failure rates has not been carried out. A generalized mathematical programming model formulated on the basis of Weibull distribution has been developed in this research to automatically synthesize the optimal designs and maintenance policies of the standby mechanisms for any given processes by minimizing the total expected lifecycle expenditure. To facilitate efficient calculations, the unreliability of the Weibull distribution has been approximated with the second-order polynomial and a Matlab code has also been developed to carry out the optimization runs via genetic algorithm. The feasibility and effectiveness of the proposed model and solution procedure are demonstrated with case studies concerning the pump systems in a typical chemical plant.
From the optimization results, one can obtain the optimum design of the standby mechanisms, which include: (1) the number of protection layers, (2) the corresponding voting-gate logic in each measurement channel, (3) the numbers of both online and spare sensors in each measurement channel, (4) the number of spares for switch, (5) the inspection interval of online switch, (6) the inspection intervals for warm standbys, and (7) the number of cold standbys.

Amari, S. V., Xing, Shrestha, Akers, and Trivedi. (2010). "Performability analysis of multistate computing systems using multivalued decision diagrams." IEEE Transactions on Computers 59(10): 1419-1433.

Badia, F. G.,Breeade, and Campos. (2001). "Optimization of inspection intervals based on cost." Journal of Applied Probability 38(4): 872-881.

Chan, S.-Z. and C.-T. Chang (2021). "Optimization of Multilayer Standby Mechanisms in Continuous Processes under Varying Loads." Chemical Engineering Research and Design 166: 86-96.

Chan, S.-Z., Liu, Luo, and Chang. (2020). "Optimization of Multilayer Standby Mechanisms in Continuous Chemical Processes." Industrial & Engineering Chemistry Research 59(5): 2049-2059.

Duarte, J. A. C., and Trigo. (2006). "Optimization of the preventive maintenance plan of a series components system." International Journal of Pressure Vessels and Piping 83(4): 244-248.

Haupt, R. L. and S. Ellen Haupt (2004). "Practical genetic algorithms."

Hellmich, M. and H.-P. Berg (2015). "Markov analysis of redundant standby safety systems under periodic surveillance testing." Reliability Engineering & System Safety 133: 48-58.

Jia, H., Ding, Peng, and Song. (2017). "Reliability evaluation for demand-based warm standby systems considering degradation process." IEEE Transactions on Reliability 66(3): 795-805.

Jia, H., Liu, Li, Ding, Liu, and Peng. (2020). "Reliability evaluation of power systems with multi-state warm standby and multi-state performance sharing mechanism." Reliability Engineering & System Safety 204: 107139.

Jia, H. P., Peng, Yang, Wu, Liu, and Li. (2022). "Reliability evaluation of demand-based warm standby systems with capacity storage." Reliability Engineering & System Safety 218: 12.
 
Kayedpour, Amiri, Rafizadeh, and Nia. (2017). "Multi-objective redundancy allocation problem for a system with repairable components considering instantaneous availability and strategy selection." Reliability Engineering & System Safety 160: 11-20.

Kuo, W. and M. J. Zuo (2003). Optimal reliability modeling: principles and applications, John Wiley & Sons.

Lai, C.-A., Chang, Ko, and Chen. (2003). "Optimal Sensor Placement and Maintenance
Strategies for Mass-Flow Networks." Industrial & Engineering Chemistry Research
42(19): 4366-4375.

Levitin, G., Xing, and Dai. (2015). "Non-Homogeneous 1-Out-of-${N} $ Warm Standby Systems With Random Replacement Times." IEEE Transactions on Reliability 64(2): 819-828.

Liang, K.-H. and C.-T. Chang (2008). "A simultaneous optimization approach to generate design specifications and maintenance policies for the multilayer protective systems in chemical processes." Industrial & Engineering Chemistry Research 47(15): 5543-5555.

Liao, Y.-C. and C.-T. Chang (2010). "Design and maintenance of multichannel protective systems." Industrial & Engineering Chemistry Research 49(22): 11421-11433.

Malhotra, R. and G. Taneja (2014). "Stochastic analysis of a two-unit cold standby system wherein both units may become operative depending upon the demand." Journal of Quality and Reliability Engineering 2014.

Malhotra, R. and G. Taneja (2015). "Comparative analysis of two single unit systems with production depending on demand." Ind Eng Lett 5(2): 43-48.

Malhotra, R. and G. Taneja (2015). "Comparative study between a single unit system and a two-unit cold standby system with varying demand." SpringerPlus 4: 705-705.

Naithani, Parashar, Bhatia, Taneja. (2017). "Probabilistic analysis of a 3-unit induced draft fan system with one warm standby with priority to repair of the unit in working state." International Journal of System Assurance Engineering and Management 8(2): 1383-1391.

Nakagawa, T. (1977). "A 2-Unit Repairable Redundant System with Switching Failure." IEEE Transactions on Reliability R-26(2): 128-130.
Nakagawa, T. and S. Osaki (1974). "Stochastic behaviour of a two-unit standby
redundant system." INFOR: Information Systems and Operational Research 12(1): 66-70.

Pan, J.-N. (1998). "Reliability prediction of imperfect switching systems subject to Weibull failures." Computers & industrial engineering 34(2): 481-492.

Raje, D.,Olaniya, Wakhare, Deshpande. (2000). "Availability assessment of a two-unit stand-by pumping system." Reliability Engineering & System Safety 68(3): 269-274.

Ruan, S.-J. and Y.-H. Lin (2021). "Reliability analysis and state transfer scheduling optimization of degrading load-sharing system equipped with warm standby components." Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability: 1748006X211001713.

Tannous, O., Xing, Peng. (2014). "Reliability of warm-standby systems subject to imperfect fault coverage." Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 228(6): 606-620.

Vaurio, J. (1999). "Availability and cost functions for periodically inspected preventively maintained units." Reliability Engineering & System Safety 63(2): 133-140.

Yun, W. Y. and J. H. Cha (2010). "Optimal design of a general warm standby system." Reliability Engineering & System Safety 95(8): 880-886.

 
Zhai, Q., Peng, Xing. (2013). "Binary decision diagram-based reliability evaluation of k-out-of-(n+ k) warm standby systems subject to fault-level coverage." Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 227(5): 540-548.

Zhai, Q.,Peng, Xing, Yang. (2015). "Reliability of demand-based warm standby systems subject to fault level coverage." Applied Stochastic Models in Business and Industry 31(3): 380-393.

Zhang, T., Xie, Horigome. (2006). "Availability and reliability of k-out-of-(M+ N): G warm standby systems." Reliability Engineering & System Safety 91(4): 381-387.

Zhong, C. and H. Jin (2014). "A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory." European Journal of Operational Research 232(2): 405-411.

Zhu, P., Lv, Guo, Si. (2018). "Optimal design of redundant structures by incorporating various costs." IEEE Transactions on Reliability 67(3): 1084-1095.