以往文獻以集合涵蓋問題求解車輛路線問題時，多以變數產生法放鬆二元決策 變數限制，再以分支定限法解決放鬆後非整數解的方式來進行求解。雖然，如此可 獲得較佳的求解品質，但複雜的求解過程導致求解時間較長，本研究希望可以加以 調整，改以放鬆涵蓋限制的方式來發展啟發式解法，取得求解效率上的提升。在相 關參數部份，目前參數設定以穩定求解品質，以經驗法則暫訂之，未作相關深入探 討。
有鑒於VRP 問題精確解求解不易，而本研究問題存在自有車隊與委外貨運兩個 求解集合，若要直接以數學規劃模式求取精確解更顯困難，求解時間也更加冗長，
面對直接獲得精確解仍有無法排除的困難情況下，僅以本研究所發展啟發式解法求 解於委外貨運價格高昂時的特例情況，在此設定下，問題形同一般車輛路線問題，
對照題庫中目前最佳解，檢驗本研究所發展演算法是否可取得合適答案。另舉出 n22-k4 例題為例，透過圖形方式呈現委外價格高低對自有車隊路線規劃所產生的合 理演變，並以此間接說明本研究所發展的啟發式解法可獲得合理解答，彌補無法以 精確解直接驗證的缺失。
未來可持續深入探討議題：
1. 在研究過程中，曾嘗試運用 LINGO 套裝軟體，直接求算精確解，然而車輛路 線問題本身複雜度已相當高，13 個節點的例題，使用 LINGO 求解的運算時間 已十分冗長，可考慮運用其他套裝軟體，找出精確解作為確切的比較基準。
2. 在求解集合分割解時，考慮本研究相較一般車輛路線問題，具有委外貨運可供 選擇的優勢，自有車隊無法涵蓋者都可以委外貨運滿足，又為求運算精簡快速，
只選取cr(u)值為負的路線，從中搜尋集合分割解，而不是以整個解題空間進行 找尋，且僅以前次運算遞迴中最後選擇保留的第 m 條路線解的cm(ut-1)值，作 為判斷是否為集合分割解的依據，在解題過程中，某幾次的運算很有可能無法 獲得適當的解，當前解的跳動較為頻繁，相對的也需要較多的運算次數才可找 到合適的解。雖然目前的作法，已可獲得近似解，也能把求解時間控制在相當 程度之內，但若能使當前解的波動程度減少，也可再提升求解的精確度與效率。
3. 以更有效的下限值來取代目前的常數設定，以有效加快求解效率與求解品質，
如以解目標式的線性規劃解為下限值取代目前常數的假設。
參考文獻
Agarwal, Y., Mathur, K., and Salkin H.M. (1989). A Set-Partition-Based Exact Algorithm for the Vehicle Routing Problem. Networks, 19, 731-749.
Altinel, I.K., & Öncan, T. (2005). A new enhancement of the clarke and wright savings heuristic for the capacitated vehicle routing problem. Journal of the Operational Research Society, 56 (8): 954-961.
Balas, E., Carrera, M.C. (1996). A Dynamic Subgradient-based Branch-and-Bound Procedure for Set Covering. Operations Research, 44: 875-890.
Ball, M.O., Golden, B.L., Assad, A.A., & Bodin, L.D. (1983). Planning for truck fleet size in the presence of a common-carrier option. Decision Sciences, 14 (1): 103-120.
Balinski, M., & Quandt, R. (1964). On an integer program for a. delivery problem.
Operations Research, 12: 300–304.
Beasley, J.E. (1990). A Lagrangian Heuristic for Set-Covering Problems. Naval Research Logistics, 37: 151-164.
Berman, P., & Das, S.K. (2005). On the vehicle routing problem. Lecture Notes in Computer Science (3608): 369-371.
Bodin, L. (1990). Twenty years of routing and scheduling. Operations Research, 38 (4):
571-579.
Bodin, L., Golden, B., Assad, A., & Ball, M. (1983). Routing and scheduling of vehicles and crews: The state of the art. Computers & Operations Research, 10 (2): 67-211.
Caprara, A., Fischetti, M. and Toth, P. (1999). A heuristic method for the set covering problems. Operations Research, 47: 730-743.
Christofides, N., Mingozzi, A., & Toth, P. (1979). The vehicle routing problem. In:
Christofides, N., Mingozzi, A., Toth, P., & Sandi, C. (eds). Combinatorial Optimization.
Wiley, 315-338.
Chu, C. W. (2004). A heuristic algorithm for the truckload and less-than-truckload problem. European Journal of Operational Research, 165 (3): 657-667.
Cordeau, J.F., Gendreau, M., Laporte, G., Potvin, J.Y., & Semet, F. (2002). A guide to vehicle routing heuristics. The Journal of the Operational Research Society, 53 (5):
512-522.
Fisher, M. (1995). Vehicle routing, Chapter 1 in Ball, M.O., Magnanti, T.L., Monma, C.L., & Nemhauser, G.L. Handbooks in operations research and management science, vol. 8: Network routing.
Gaskell, T.J. (1967). Basis for the vehicle fleet scheduling. Operations Research, 18, 281-295.
Gendreau, M., Laporte, G., & Potvin, J. (2001). Metaheuristics for the capacitated VRP.
In: Toth, P., & Vigo, D. (eds). The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications. SIAM Publishing, Philadelphia, PA, 129–154.
Gendreau, M. (2005). Metaheuristics in combinatorial optimization. Annals of Operations Research, 140 (1): 189-213.
Gendreau, M., Hertz, A., & Laporte, G. (1994). A tabu search heuristic for the vehicle routing problem. Management Science, 40 (10): 1276-1290.
Glover, F. (1989). Tabu search-Part . Ⅰ ORSA Journal on Computing, 1 (3): 190-206.
Held, M. , Karp, R.M. (1970). The traveling salesman problem and minimum spanning trees. Operations Research, 18: 1138-1162.
Kelly, J.P., & Xu, J. (1999). A Set-Partitioning-Based Heuristic for the Vehicle Routing Problem. INFORMS Journal on Computing, 11 (2): 161-172.
Klincewicz, J.G., Luss, H., & Pilcher, M.G. (1990). Fleet Size Planning When Outside Carrier Services Are Available. Transportation Science, 24 (3): 169-182.
Laporte, G., Nobert, Y. (1987). Exact algorithms for the vehicle routing problems. Annals of Discrete Mathematics, 31: 147-184.
Laporte, G., Gendreau, M., Potvin, J., & Semet, F. (2000). Classical and modern heuristics for the vehicle routing problem. Int Trans Opns Res, 7: 285–300.
Laporte, G., & Semet, F. (2001). Classical heuristics for the vehicle routing problem. In:
Toth, P., & Vigo, D. (eds). The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications. SIAM Publishing, Philadelphia, PA, 109–128.
Lenstra, J., & Rinnooy, K. A. (1981). Complexity of vehicle routing and scheduling problems. Networks, 11: 221–227.
Osman, I.H. (1991). Metastrategy simulated annealing and tabu search algorithms for combinatorial optimization problem. Ph.D. Dissertation, The Management School, Imperial Collage, London.
Osman, I.H. (1993a). Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Annals of Operational Research, 41: 421-451.
Pureza, V.M., & Franca, P.M. (1991). Vehicle routing problem via tabu search
metaheuristic. Publication CRT-747, Centre de recherche sur les transports, Montreal.
Savelsbergh, M. (1985). Local search in routing problems with time windows.
Operations Research, 4: 285-305.
Taillard, E. (1993). Parallel Iterative Search Methods for Vehicle Routing Problems.
Networks, 23: 661-673
Toth, P., & Vigo, D. (eds). (2001). The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications . SIAM publishing: Philadelphia, PA.
Van Breedam, A. (2001). Comparing descent heuristics and metaheuristics for the vehicle routing problem. Computers & Operations Research, 28: 289-315.
Willard, J.A.G. (1989). Vehicle Routing Using r-Optimal Tabu Search. M.Sc.
Dissertation. The Management School, Imperial College, London.
Yellow, P. (1970). A computation modification to the savings method of vehicle scheduling. Operations Research, 21: 281-283.
吳泰億，民國95 年，以修正之拉氏鬆弛啟發式解法求解車輛路線問題，國立交通大 學，碩士論文。
Solomon website: http://neo.lcc.uma.es/radi-aeb/webvrp/index.html?/resultsSolom.htm