This study considered the problems of congestion situation and edge selection simultaneously with regard to the Vehicle Routing Problem with Time Windows (VRPTW). This problem is called the Time Dependent Alternative Vehicle Routing Problem (TDAVRP). After investigating the properties of the TDAVRP, the problem with a multigraph was formulated into a Mixed Integer Programming model. Due to its NP nature, after using Solomon’s benchmark problem with 5 customer nodes to verify the model by ILOG CPLEX, we have developed an algorithm based on Particle Swarm Optimization (PSO) to speed up the solution procedure. According to the proper encoding and decoding methods, a two-stage PSO with local improvement was proposed. By using different sets of Solomon’s benchmark problems and continuous travel time functions, the accuracy and efficiency of the two-stage PSO were evaluated. The computational results have shown that the proposed algorithm is capable of deriving optimal or near optimal solutions in a short period of time when the size of the problem is small; and when the problem size increases that ILOG CPLEX cannot solve, our algorithm can give a satisfactory solution within a reasonable time. In addition, Sensitivity Analysis was used to evaluate the performances of the results when adjusting some parameters or data. Finally, the existence of the alternate edges has been particularly discussed respective to the TDVRP, which has shown the importance with the clustered customers.

本研究在具時窗限制的車輛途程問題的規劃中同時考量交通擁塞和道路選擇的問題，並稱此問題為「依時性可選擇道路之車輛途程問題」(TDAVRP)。經過比較相關的依時性車輛途程問題之文獻及總結TDAVRP的問題特性後，在多重圖的結構下提出一個混合整數規劃的模型，經由索羅門五位顧客點的測試例題以ILOG CPLEX軟體驗證模型的可行性。由於TDAVRP為非指數時間可解的複雜問題，故進一步參考粒子群演算法(PSO)的原理發展了演算法。藉由適當的編碼與解碼和區域改進，研究中的兩階段粒子群演算法在各階段處理了不同但相關聯的議題，結合兩部分的結果可得到一組完整的路徑解。在ILOG CPLEX軟體可解的小問題下，使用索羅門的測試例題和文獻中的連續型時間函數來驗證PSO的精確度和效率，比較結果顯示所提出的演算法可在短時間內找到最佳解或近似解。此外敏感度分析也用來評估參數或資料改變對結果的影響。當問題的顧客分佈為群聚型態時，替代路徑的存在較其他問題為重要，最後TDAVRP問題的時窗違反情況和成本都比依時性車輛途程問題(TDVRP)還低。

[1] Ai, T. J., & Kachitvichyanukul, V. (2009). A particle swarm optimization for the vehicle routing problem with simultaneous pickup and delivery. Computers and Operations Research, 36(5), 1693-1702.
[2] Augerat, P., Belenguer, J. M., Benavent, E., Corber, A., & Naddef, D. (1998). Separating capacity constraints in the CVRP using tabu search. European Journal of Operational Research, 106, 546–557.
[3] Diestel, R. (2006). Graph theory (3rd ed.). Germany：Springer.
[4] Donati, A.V., Montemanni, R., Casagrande, N., Rizzoli A.E., & Gambardella L.M. (2008). Time dependent vehicle routing problem with a multi ant colony system. European Journal of Operational Research, 185(3), 1174-1191.
[5] Fleischmann, B., Gietz, M., & Gnutzmann, S. (2004). Time-varying travel times in vehicle routing. Transportation Science, 38(2), 160–173.
[6] Ichoua, S., Gendreau, M., & Potvin J. Y. (2003). Vehicle Dispatching With Time-Dependent Travel times. European Journal of Operational Research, 144(2), 379-396.
[7] Jabali, O., Woensel, T. V., De Kok, A.G., Lecluyse, C., & Peremans, H. (2009). Time-dependent vehicle routing subject to time delay perturbations. IIE Transactions, 41, 1049-1066.
[8] Jiang, W., Zhang, Y., & Xie, J. (2009). A Particle Swarm Optimization Algorithm with Crossover for Vehicle Routing Problem with Time Windows. Proceedings of the 2009 IEEE Symposium on Computational Intelligence in Scheduling (IEEE CISched 2009), 103‐106.
[9] Kachitvichyanukul, V., 2010. Intensive Course Plan：Adaptive Particle Swarm Optimization. The 6th International Conference on Intelligent Manufacturing & Logistics Systems, IML 2010.
[10] Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proceedings of IEEE international conference on neural networks, 4, 1942-1948.
[11] Malandraki, C., & Daskin, M.S. (1992). Time dependent vehicle routing problems: Formulations, properties and heuristic algorithms. Transportation Science, 26 (3), 185–200.
[12] Miao, Z., Yang F., Fu K., & Xu D. (2010). Transshipment service through crossdocks with both soft and hard time windows. Annals of Operations Research.
[13] Rao, G. S. (2002). Discrete Mathematical Structures, Bangalore, India：New Age International.
[14] Solomon, M. M. (1987). Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research, 35(2), 254–265.
[15] Tang, W. H., & Wu, Q. H. (2011). Condition Monitoring and Assessment of Power Transformers Using Computational Intelligence, Springer.
[16] Wang, F.L., Deng, H., & Lei, J. (2010). Artificial Intelligence and Computational Intelligence: International Conference, Aici 2010, Sanya, China, October 23-24, 2010, Proceedings, PartⅡ, Springer.
[17] Wu,Y., Ye, C. M., & Ma, H.M. (2007). Parallel particle swarm optimization algorithm for vehicle routing problems with time windows[J]. Computer Engineering and Applications, 43(14), 223-226.
[18] Hu, X. (2010, August). Particle Swarm Optimization [Web blog message]. Retrieved from http://www.swarmintelligence.org/
[19] Wikipedia. (2011, June 5). Particle Swarm Optimization [Online encyclopedia]. Retrieved from http://en.wikipedia.org/wiki/Particle_swarm_optimization
[20] TOP transportation optimization portal. (2010, October 21). Solomon Benchmark [Web blog message]. Retrieved from http://www.sintef.no/Projectweb/TOP/Problems/VRPTW/Solomon-benchmark/
[21] International Energy Agency (IEA) website. (2009) Transport, Energy and CO2: Moving towards Sustainability - executive summary. [IEA report] Retrieved from
http://www.iea.org/Textbase/npsum/transport2009SUM.pdf