現今自由行與商務旅客日益增加，且多數選擇搭乘大眾運輸至目的地後，於當地租車以提升機動性，面對擴增的租車需求，該如何有效地安排車輛，提升整體營運收益，對租車公司而言為相當實務且重要之議題。本篇研究在已知出租站所、車隊規模、訂單資訊、考量訂單獲利與車輛調度成本等條件下，探討出租車輛的訂單指派與調度策略，以最大化租車公司總收益。研究中首先針對此問題建構整數規劃模型，並發現其屬於困難的NP-complete問題，為了求解更符合實務應用的問題規模，本篇研究運用拉格朗日鬆弛(Lagrangian relaxation)之概念，鬆弛數學模型中的複雜限制式(complicating constraints)，將此問題轉為數個最短路徑問題(shortest path problem)，再結合次梯度演算法(subgradient heuristic)進行搜尋以逼近最佳解。透過數筆實驗結果印證，本篇研究建構之啟發式演算法能夠求解之問題規模，高於使用商用套裝求解軟體50%，顯示本研究架構更貼近實務之應用。

The adoption of an effective relocation strategy is critical to vehicle rental companies, as it can improve the profits of such companies. In this research, we investigate the rental car assignment and relocation problem. For given rental stations, exogenous customer reservations, customers’ rental preferences, published rental prices, known relocation costs and fixed fleet sizes, we attempt to determine an assignment and relocation strategy for the available rental cars of a company that maximizes the resulting profit. The problem is formulated as an integer programming problem. It is shown that the problem is an NP-complete optimization problem. Therefore, to solve the realistic size problem, we devise a Lagrangian relaxation-based solution approach that dualizes the bundle constraint and that results in multiple shortest path subproblems. A subgradient heuristic is then embedded to guide the search procedure so that the overall solution framework converges to a solution with a reasonable optimality gap. The proposed solution framework is empirically applied to realistic problems, and numerical results show that it can outperform a widely used optimization package and is suitable for practical use. Specifically, the proposed LR-based heuristic can solve problem size at least 50% greater than the optimization package within a reasonable computational time.

LIST OF FIGURES vii
LIST OF TABLES viii
1. INTRODUCTION 1
1.1 Research background and motivation 1
1.2 Research purpose and method 2
1.3 Research framework 3
2. LITERATURE REVIEW 5
2.1 Rental vehicle fleet sizing and deployment 5
2.2 Rental vehicle assignment and relocation 6
2.3 Related studies 7
2.4 Lagrangian relaxation approach 8
2.5 Summary 10
3. MATHEMATICAL FORMULATION 12
3.1 Problem statement 12
3.2 Notations 13
3.3 The integer programming formulation 15
3.4 Model reformulation 18
4. SOLUTION APPROACH 25
4.1 Problem decomposition 25
4.2 Subgradient heuristic 26
4.3 The overall solution framework 28
5. NUMERICAL EXPERIMENT 30
5.1 Validation of the proposed LR-solution approach 30
5.2 Sensitivity analysis 32
6. CONCLUSION 38
6.1 Concluding Remarks 38
6.2 Future Research 38

[1] Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows : theory, algorithms, and applications. Englewood Cliffs, N.J.: Prentice Hall.
[2] Angulo, E., Garcia-Rodenas, R., & Espinosa-Aranda, J. L. (2016). A Lagrangian relaxation approach for expansion of a highway network. Annals of Operations Research, 246(1-2), 101-126. doi:10.1007/s10479-014-1682-7
[3] Astorino, A., Gaudioso, M., & Miglionico, G. (2018). Lagrangian relaxation for the directional sensor coverage problem with continuous orientation. Omega-International Journal of Management Science, 75, 77-86. doi:10.1016/j.omega.2017.03.001
[4] Ernst, A. T., Gavriliouk, E. O., & Marquez, L. (2011). An efficient Lagrangean heuristic for rental vehicle scheduling. Computers & Operations Research, 38(1), 216-226. doi:10.1016/j.cor.2010.04.014
[5] Ernst, A. T., Horn, M., Kilby, P., & Krishnamoorthy, M. (2010). Dynamic scheduling of recreational rental vehicles with revenue management extensions. Journal of the Operational Research Society, 61(7), 1133-1143. doi:10.1057/jors.2009.78
[6] Garey, M. R., & Johnson, D. S. (1979). Computers and intractability : a guide to the theory of NP-completeness. San Francisco: W. H. Freeman.
[7] Gaudioso, M., Gorgone, E., Labbe, M., & Rodriguez-Chia, A. M. (2017). Lagrangian relaxation for SVM feature selection. Computers & Operations Research, 87, 137-145. doi:10.1016/j.cor.2017.06.001
[8] George, D. K., & Xia, C. H. (2011). Fleet-sizing and service availability for a vehicle rental system via closed queueing networks. European Journal of Operational Research, 211(1), 198-207. doi:10.1016/j.ejor.2010.12.015
[9] Hernandez-Leandro, N. A., Boyer, V., Salazar-Aguilar, M. A., & Rousseau, L. M. (2019). A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem. European Journal of Operational Research, 272(3), 859-867. doi:10.1016/j.ejor.2018.07.010
[10] Kulkarni, S., Patil, R., Krishnamoorthy, M., Ernst, A., & Ranade, A. (2018). A new two-stage heuristic for the recreational vehicle scheduling problem. Computers & Operations Research, 91, 59-78. doi:10.1016/j.cor.2017.11.004
[11] Li, D., & Pang, Z. (2017). Dynamic booking control for car rental revenue management: A decomposition approach. European Journal of Operational Research, 256(3), 850-867. doi:10.1016/j.ejor.2016.06.044
[12] Li, Z. L., & Tao, F. (2010). On determining optimal fleet size and vehicle transfer policy for a car rental company. Computers & Operations Research, 37(2), 341-350. doi:10.1016/j.cor.2009.05.010
[13] Liu, R. M., Li, S. K., & Yang, L. X. (2020). Collaborative optimization for metro train scheduling and train connections combined with passenger flow control strategy. Omega-International Journal of Management Science, 90. doi:ARTN 10199010.1016/j.omega.2018.10.020
[14] Oliveira, B. B., Carravilla, M. A., & Oliveira, J. F. (2017). Fleet and revenue management in car rental companies: A literature review and an integrated conceptual framework. Omega-International Journal of Management Science, 71, 11-26. doi:10.1016/j.omega.2016.08.011
[15] Oliveira, B. B., Carravilla, M. A., & Oliveira, J. F. (2018). A Dynamic Programming Approach for Integrating Dynamic Pricing and Capacity Decisions in a Rental Context. Operational Research, 223, 297-311. doi:10.1007/978-3-319-71583-4_20
[16] Oliveira, B. B., Carravilla, M. A., Oliveira, J. F., & Costa, A. M. (2019). A co-evolutionary matheuristic for the car stochastic problem. European Journal of Operational Research, 276(2), 637-655. doi:10.1016/j.ejor.2019.01.015
[17] Oliveira, B. B., Carravilla, M. A., Oliveira, J. F., & Toledo, F. M. B. (2014). A relax-and-fix-based algorithm for the vehicle-reservation assignment problem in a car rental company. European Journal of Operational Research, 237(2), 729-737. doi:10.1016/j.ejor.2014.02.018
[18] Pachon, J. E., Iakovou, E., Ip, C., & Aboudi, R. (2003). A synthesis of tactical fleet planning models for the car rental industry. IIE Transactions, 35(9), 907-916. doi:Doi 10.1080/07408170304409
[19] Pazour, J. A., & Roy, D. (2015). Analyzing rental vehicle threshold policies that consider expected waiting times for two customer classes. Computers & Industrial Engineering, 80, 80-96. doi:10.1016/j.cie.2014.10.030
[20] Roy, D., Pazour, J. A., & De Koster, R. (2014). A novel approach for designing rental vehicle repositioning strategies. IIE Transactions, 46(9), 948-967. doi:10.1080/0740817x.2013.876129
[21] You, P. S., & Hsieh, Y. C. (2014). A study on the vehicle size and transfer policy for car rental problems. Transportation Research Part E-Logistics and Transportation Review, 64, 110-121. doi:10.1016/j.tre.2014.01.007
[22] Yu, Y. G., Dong, Y. X., & Guo, X. L. (2018). Pricing for sales and per-use rental services with vertical differentiation. European Journal of Operational Research, 270(2), 586-598. doi:10.1016/j.ejor.2018.03.035