跳到主要內容

簡易檢索 / 詳目顯示

回結果列表
研究生： 李旺蒼
Wang-Tsang Lee
論文名稱： 以粒子群最佳化為基礎之混合式全域搜尋演算法求解含凹形節線成本最小成本轉運問題之研究
Hybrid Global Search Algorithm Based on Particle Swarm Optimization for Concave Cost Minimum Cost Network Flow Problems
指導教授： 顏上堯
Shang-Yao Yan
口試委員:
學位類別： 碩士
Master
系所名稱： 工學院 - 土木工程學系
Department of Civil Engineering
畢業學年度： 94
語文別： 中文
論文頁數： 142
中文關鍵詞： 最小成本轉運問題 、遺傳演算法 、粒子群最佳化演算法 、全域搜尋 、凹形節線成本
外文關鍵詞： global search, concave arc cost, particle swarm optimization, genetic algorithm, minimum cost network flow problem
相關次數： 點閱：17  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 傳統上，最小成本網路流動問題的運送成本常以線性方式來定義，藉以簡化問題的複雜度。在實務上，貨物運送的單位成本常隨數量的增加而遞減，其成本函數曲線為凹形。近期在求解含凹形節線成本之特殊最小成本網路流動問題上，有學者以新近鄰近搜尋法求解問題，達到較大範圍的搜尋方式，期能找到較優於傳統啟發解法的解。然而，此等鄰近搜尋法，容易面臨退化的問題，且是否可快速探尋全域，則不得而知。因此近期有學者發展類遺傳演算法與類螞蟻族群演算法以求解含凹形節線成本最小成本網路流動問題。另外，新近的粒子群最佳化演算法目前在各領域的問題求解上效果頗佳，甚至有發現較遺傳演算法為佳，但尚未發現有應用於含凹形節線成本最小成本網路流動問題。緣此，本研究針對含凹形節線成本之最小成本網路流動問題，以粒子群最佳化演算法之搜尋概念為基礎，並結合遺傳演算法、門檻值接受法與凹形成本網路啟發解法之特點，以節線及路徑為基礎發展兩種不同之混合式全域搜尋法，以有效的求解含凹形節線成本之最小成本網路流動問題。
    本研究演算法的設計如下：在初始解的產生上，以隨機方式及配合凹形成本網路特性發展二初始解法，產生初始的粒子群。在可行解的產生上：以節線為基礎之演算法，以前ㄧ回合伸展樹為基礎，引進遺傳演算法之輪盤法選擇策略，發展數種方式以挑選節線加入伸展樹，並透過流量推擠法產生可行伸展樹；至於以路徑為基礎之演算法，則發展數種供給點指派之順序選擇策略及供需節點對之路徑選取方式，以重新指派供需節點間的運送量，並藉由流量推擠法以形成可行伸展樹。另外再發展路徑集合更新策略，以加入較佳伸展樹中新的供需節點路徑。另外，在粒子的速度更新法則上，參考引入連續型粒子群最佳化之慣性權重值（w）與可行伸展樹之特性，發展與以往不同的速度演算法則。在全體與個體最佳解的更新方式上，則引入門檻值接受法之求解機制。最後在搜尋的深度上，結合區域搜尋能力較強的鄰近搜尋法，以提升演算績效。最後，為測試本研究演算法在不同規模及參數的網路問題之求解績效，本研究設計一隨機網路產生器，產生大量的隨機網路，在個人電腦上以C++語言撰寫所有相關的電腦程式，並測試新近發展之遺傳演算法、門檻值接受法、大洪水法及類螞蟻族群演算法，以評估本研究兩種演算法之求解績效，進而提供實務界求解此類實際的網路運送問題之參考。

    Traditionally, the minimum cost transshipment problems were simplified as linear cost problems in order to reduce problem complexity. In practice, the unit cost for transporting freight usually decreases as the amount of freight increases. Hence, in actual operations the transportation cost function can usually be formulated as a concave cost function. Recently, some advanced neighborhood search algorithms have been used to solve concave cost network problems. However, they were easily encountered degeneracy problems or confined to local regions, resulting in decreased solution efficiency. Recently, there has been research adopting the genetic algorithm (GA) and the ant colony system algorithm (ACS) for solving concave cost network problems, and obtaining better solutions than some neighborhood search algorithms. The particle swarm optimization algorithm (PSO), a global search algorithm, has led to good results in many applications. In some applications, PSO was even more effective than GA. Since there has not yet been any research applying PSO to minimum concave cost network flow problems, we employ PSO, coupled with the techniques of GA　and TA, to develop two global search algorithms for efficiently solving minimum concave cost network flow problems.
    The algorithms are designed as follows: First, two heuristics for generating the initial solutions are proposed: one is the random initial solution algorithm and the other is the concave cost initial solution algorithm. Then, two methods for generating feasible solutions are proposed: In the arc-based method, we employed the roulette wheel method in GA to design several ways for selecting and adding arcs into the spanning tree obtained from the previous iteration. In the path-based method, we developed several methods to select supply/demand node pair paths to form a new feasible solution. The flow augmentation algorithm is then used to adjust the non-spanning trees to become feasible spanning trees for both methods. In addition, for the path-based method we developed a strategy for updating the path set by incorporating the new paths from better feasible spanning trees. Followed by the two steps, we developed a new rule for updating the particle velocities, incorporating the inertia weight method and the spanning tree’s characteristics. We also employed the TA technique for updating the feasible solutions. To enhance the depth search we employed a neighborhood search technique. Finally, to evaluate our algorithms we designed a network generator to generate a sufficient number of problem instances. The C++ computer language was used to code all the necessary programs and the tests was performed on personal computers. To evaluate our algorithms, we also tested the recently designed TA, GDA, GA and ACS that solve minimum concave cost network flow problems.

    摘要..........................................................................I ABSTRACT.....................................................................II 誌謝........................................................................III 目錄..........................................................................V 圖目錄.....................................................................VIII 表目錄.......................................................................IX 第一章 緒論...................................................................1 1.1 研究背景與動機............................................................1 1.2 研究目的與範圍............................................................2 1.3 研究方法與流程............................................................3 第二章 文獻回顧...............................................................5 2.1凹形成本網路流動問題.......................................................5 2.2鄰近搜尋法.................................................................6 2.3全域式演算法...............................................................9 2.4粒子群最佳化(Particle Swarm Optimization, PSO)............................12 2.4.1粒子群最佳化之發展背景..................................................12 2.4.2粒子群最佳化演算法介紹..................................................12 2.4.3粒子群最佳化之相關應用及發展............................................14 2.5 小結 ....................................................................15 第三章 問題描述與求解演算法設計.............................................16 3.1 問題定式及特性...........................................................16 3.2 各求解演算法設計.........................................................19 3.2.1 以節線為基礎之類粒子群最佳化演算法（Arc Base Analogous PSO,AB-APSO)....19 3.2.1.1初始解產生策略........................................................19 3.2.1.2可行解產生策略........................................................25 3.2.1.3機率值更新方式........................................................26 3.2.1.4門檻值個體最佳解與門檻值全體最佳解更新策略............................28 3.2.1.5鄰近搜尋法之改善策略（local search....................................28 3.2.1.6流量推擠法............................................................29 3.2.2以供需節點路徑為基礎之類粒子群最佳化演算法(Path Base Analogous PSO, PB-APSO)..............................................................31 3.2.2.1初始解產生策略........................................................31 3.2.2.2初始供需節點路徑集合及初始機率值之設定................................32 3.2.2.3 可行解產生策略.......................................................32 3.2.2.4 路徑集合更新策略.....................................................34 3.2.2.5 供給點指派順序策略...................................................36 3.2.2.6 機率值更新方式.......................................................36 3.2.2.7 門檻值個體最佳解與門檻值全體最佳解更新策略...........................36 3.2.2.8 鄰近搜尋法之改善策略.................................................36 3.2.2.9 菁英策略(elitist)....................................................37 3.2.3 小結...................................................................37 第四章 實證分析..............................................................40 4.1 網路產生器設計...........................................................40 4.1.1 隨機網路產生法.........................................................40 4.1.2 供給(需求)節點與供給(需求)量的隨機產生法...............................41 4.2 AB-APSO全域搜尋演算法....................................................42 4.2.1 粒子群群體數量.........................................................43 4.2.2 隨機初始解與凹形初始解之群體型態比較...................................44 4.2.3 輪盤選擇策略與加入節線數...............................................46 4.2.4 速度更新公式相關參數...................................................47 4.2.5 速度範圍限制策略相關參數...............................................50 4.2.6 門檻值全體最佳解與個體最佳解更新策略相關參數...........................51 4.2.7 鄰近搜尋法相關參數.....................................................53 4.2.8 最佳參數組合測試.......................................................54 4.2.9 初始解與最終解之相關性.................................................55 4.2.10 AB-APSO收斂趨勢.......................................................56 4.2.11 小結..................................................................57 4.3 PB-APSO全域搜尋演算法....................................................58 4.3.1 供給點選擇方式.........................................................60 4.3.2 粒子群群體數量.........................................................61 4.3.3 隨機初始解與凹形初始解之群體型態比較...................................62 4.3.4 速度更新公式相關參數...................................................63 4.3.5 速度範圍限制策略相關參數...............................................66 4.3.6 供給點指派順序策略相關參數.............................................67 4.3.7 門檻值全體最佳解與個體最佳解更新策略相關參數...........................69 4.3.8 菁英策略相關參數.......................................................71 4.3.9 鄰近搜尋法相關參數.....................................................73 4.3.10最佳參數組合測試.......................................................74 4.3.11初始解與最終解之相關性.................................................75 4.3.12 PB-APSO收斂趨勢.......................................................76 4.3.13 小結..................................................................78 4.4 AB-APSO、PB-APSO與GA、AACS、各區域搜尋法之求解績效比較...................80 第五章 結論與建議............................................................83 5.1 結論.....................................................................83 5.2 貢獻.....................................................................84 5.3 建議.....................................................................85 參考文獻.....................................................................86 附錄ㄧ AB-APSO各系統參數測試方案.............................................91 附錄二 AB-APSO各參數測試詳細結果.............................................94 附錄三 初始解與最終解目標值關係.............................................105 附錄四 PB-APSO各系統參數測試方案............................................107 附錄五 PB-APSO各參數測試詳細結果............................................110 附錄六 初始解與最終解目標值關係.............................................125 附錄七 各演算法配合方案求解時間.............................................127 附錄八 GA、AACS與各區域搜尋法輸入參數值.....................................128 附錄九 AB-APSO與PB-APSO之各型網路目前最佳解.................................129 附錄十 各型網路之目前最佳解.................................................130

    1. 王政嵐，「類螞蟻族群演算法於求解含凹形節線成本最小成本轉運問題之研究」，中央大學土木工程研究所碩士論文（2005）。
    2. 朱文正，「考量旅行時間可靠度之車輛途程問題 ─ 螞蟻族群演算法之應用」，交通大學交通運輸研究所碩士論文（2002）。
    3. 林依潔，「整合模糊理論與螞蟻演算法於含時窗限制之車輛途程問題」，台北科技大學生產系統工程與管理研究所碩士論文（2003）。
    4. 胡曉暉，「粒子群優化算法介紹」，http://www.swarmintelligence.org/papers/cPSOTutorial.pdf，2002年4月。
    5. 徐育良，「以粒子群最佳化為基礎之電腦遊戲角色設計之研究」，東海大學資訊工程與科學研究所碩士論文（2002）。
    6. 陳建榮，「含凹形節線成本最小成本網路流動問題之全域搜尋演算法研究」，中央大學土木工程研究所碩士論文（2002）。
    7. 張榮芳，「電力用戶負載歸類及整合」，國立中山大學電機工程研究所博士論文（2001）。
    8. 曾俊傑，「一個智慧型指紋辨識系統的設計方法論」，義守大學電機工程研究所碩士論文（2000）。
    9. 葉麗雯，「供應商產能有限及價格折扣下多產品多供應商最佳化採購決策」，元智大學工業工程與管理研究所碩士論文（2002）。
    10. 葉思緯，「應用粒子群最佳化演算法於多目標存貨分類之研究」，元智大學工業工程與管理研究所碩士論文（2003）。
    11. 韓復華、林修竹，「TA與GDA巨集啟發式法在VRPTW問題上之應用」，中華民國第四屆運輸網路研討會，第83-92頁（1999）。
    12. 韓復華、楊智凱，「門檻接受法在TSP問題上之應用」，運輸計劃季刊，第二十五卷，第二期，第163-188頁（1996）。
    13. 韓復華、陳國清、卓裕仁，「成本擾動法在TSP問題之應用」，中華民國第二屆運輸網路研討會論文集，第283-292頁（1997）。
    14. 韓復華、楊智凱、卓裕仁，「應用門檻接受法求解車輛路線問題之研究」，運輸計畫季刊，第二十六卷，第二期，第253-280頁（1997）。
    15. 顏上堯、周容昌、李其灃，「交通建設計畫評選模式及其解法之研究─以中小型交通建設計畫的評選為例」，運輸計畫季刊，第三十一卷，第一期（2002）。
    16. 顏上堯、陳建榮、湯慶輝，「含凹形節線成本最小成本轉運問題鄰近搜尋法之研究」，運輸計劃季刊，第三十三卷，第二期，第277-306頁（2004）。
    17. Abuali, F. N., Wainwright, R. L., and Schoenefeld, D. A., “Determinant Factorization: ANew Encoding Scheme for Spanning Trees Applied to the Probabilistic Minimum Spanning Tree Problem,” Proceedings of The Sixth International Conference on GeneticAlgorithms, pp. 470-477 (1995).
    18. Ahuja, R. K., Maganti, T. L., and Orlin, J. B., Network Flows, Theory, Algorithms, and Applications, Prentice Hall, Englewood Cliffs (1993).
    19. Alfa, A. S., Heragu, S. S., and Chen, M. “A 3-opt Based Simulated Annealing Algorithm for Vehicle Routing Problem,” Computers and Industrial Engineering, Vol. 21, pp. 635-639 (1991).
    20. Amiri, A., and Pirkul, H., “New Formulation and Relaxation to Solve A Concave Cost Network Flow Problem,” Journal of the Operational Research Society, Vol. 48, pp. 278-287 (1997).
    21. Balakrishnan, A., and Graves S. C., “A Composite Algorithm for a Concave-Cost Network Flow Problem,” Networks, Vol. 19, pp. 175-202 (1989).
    22. Blumenfeld, D. E., Burns, L. D., Diltz, J. D., and Daganzo, C. F., “Analyzing Trade-offs Between Transportation, Inventory, and Production Costs on Freight Network,” Transportation Research, Vol. 19B, pp. 361-380 (1985).
    23. Booker, L. B., “Improving Search in Genetic Algorithms,” Genetic Algorithms and Simulated Annealing (L. Davis, editor), Pitman, London, pp. 61-73 (1987).
    24. Charon, I., and Hurdy, O., “The Noising Method: A New Method for Combinatorial Optimization,” Operations Research Letters, Vol. 14, pp. 133-137 (1993).
    25. Costa, D., and Hertz, A., “Ants Can Color Graphs,” Journal of the Computer Science (JORBEL), Vol.34, pp 39-53 (1994).
    26. Davis, L., “Genetic Algorithm and Simulated Annealing,” Morgan Kaufman Publishers, Los Altos, CA (1987).
    27. Davis, L., “Adapting Operator Probabilities in Genetic Algorithms,” Proceedings of the Third International Conference on Genetic Algorithms, pp. 61-69 (1989).
    28. Dorigo, M., and Gambardella, L. M., “A Study of Some Properties of Ant-Q,” Proceedings of PPSN IV-Fourth International Conference on Parallel Problem Solving From Nature, September 22-27, 1996, Berlin, Germany, Berlin: Springer-Verlag, 656-665. (1996).
    29. Dorigo, M., and Gambardella, L. M., “Ant Colonies for the Traveling Salesman Problem,” (1996).
    30. Dorigo, M., and Gambardella, L. M., “Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem,” IEEE Transactions on Evolutionary Computation, 1(1): pp. 53-66. (1997a).
    31. Dorigo, M., and Gambardella, L. M., “Ant Colonies for the Traveling Salesman Problem,” BioSystems, 43: pp. 73-81. (1997b).
    32. Dorigo, M., Maniezzo, V., and Colorni, A., “The Ant System: An Autocatalytic Optimizing Process,” Technical Report No. 91-016 Revised, Politecnico di Milano, Italy (1991).
    33. Dorigo, M., Maniezzo, V., and Colorni, A., “The Ant System: Optimization by a Colony of Cooperating Agents,” IEEE Transactions on Systems, Man, and Cybernetics-Part B, 26(1): pp. 29-41 (1996).
    34. Dueck, G., “New Optimization Heuristics: The Great Deluge Algorithm and the Record-to-Record Travel,” Journal of Computational Physics, Vol. 104, pp. 86-92 (1993).
    35. Dueck, G., and Scheuer, T., “Threshold Accepting: A General Purpose Optimization Algorithm Appearing Superior to Simulated Annealing,” Journal of Computational Physics, Vol. 90, pp.161-175 (1990).
    36. Eberhart, R. C. and Kennedy, J. A new optimizer using particle swarm theory. Proceedings of the sixth international symposium on micro machine and human science pp. 39-43. IEEE service center, Piscataway, NJ, Nagoya, Japan (1995).
    37. Eberhart, R. C. and Kennedy, J. , “Particle Swarm Optimization,” In proceedings of IEEE International Conference on Neural Networks, Vol. IV, pp.1942-1948 (1995).
    38. Eberhart, R. C. and Shi, Y. Comparison between genetic algorithms and particle swarm optimization. Evolutionary programming vii: proc. 7th ann. conf. on evolutionary conf., Springer-Verlag, Berlin, San Diego, CA(1998).
    39. Eberhart, R. C. and Shi, Y. Particle swarm optimization: developments, applications and resources. Proc. congress on evolutionary computation 2001 IEEE service center, Piscataway, NJ., Seoul, Korea. (2001).
    40. Gallo, G., Sandi C., and Sodini, C., “An Algorithm for the Min Concave Cost Flow Problem,” European Journal of Operation Research, Vol. 4, pp. 248-255 (1980).
    41. Gallo, G., and Sandi, C., “Adjacent Extreme Flows and Application to Min Concave Cost Flow Problems,” Networks, Vol. 9, pp. 95-121 (1979).
    42. Gambardella, L.M., and Dorigo, M., “Solving Symmetric and Asymmetric TSPs by Ant Colonies,” Proceedings of IEEE International Conference on Evolutionary Computation, IEEE-EC 96, May 20-22, 1996, Nagoya, Japan, IEEE Press, pp. 622-627 (1996).
    43. Glover, F., and Laguna, M., “Tabu search, Kluwer Academic Publishers,” Massachusetts (1997).
    44. Glover, F., “Tabu Search, Part I,” ORSA Journal on Computing Vol. 1, No. 3, pp.190-206 (1989).
    45. Glover, F., “Tabu Search- Part II,” ORSA Journal on Computing, Vol. 2, No. 1, pp. 4-32 (1990).
    46. Gen, M., and Cheng, R., “Genetic Algorithms and Engineering Design,” Wiley Interscience Publication, MA (1997).
    47. Guisewite, G. M., and Pardalos, P. M., “A Polynomial Time Solvable Concave Network Flow Problems,” Networks, Vol. 23, pp. 143-147 (1993).
    48. Gu, J., and Huang, X., “Efficient Local Search with Search Space Smoothing: A Case Study of the Traveling Salesman Problem (TSP),” IEEE Transaction on Systems, Man and Cybernetics, Vol. 24, pp. 728-739 (1994).
    49. Goldberg, D. E., “Genetic Algorithms in Search, Optimization, and Machine Learning,” Addison-Wesley, Reading MA (1989).
    50. Golden, B. L., and Skiscim, C. C., “Using Stimulated Annealing to Solve Routing and Location Problems,” Naval Research Logistic Quarterly, Vol. 33, pp. 261-279 (1986).
    51. Hall, R. W., “Direct Versus Terminal Freight Routing on Network with Concave Costs,” GMR-4517, Transportation Research Dept., GM Research Laboratories (1983).
    52. Jordan, W. C., “Scale Economies on Multi-Commodity Networks,” GMR-5579, Operating Systems Research Dept., GM Research Laboratories (1986).
    53. Kennedy, J. and Eberhart, R. C. , “A Discrete Binary Version of the Particle Swarm Optimization,” In proceedings of IEEE International Conference on Neural Networks, Vol. V, pp.4104-4108 (1997).
    54. Kennedy, J., Eberhart, R.C. and Shi, Y., “Swarm Intelligence,” Morgan Kaufmann division of Academin Press (2001).
    55. Kennedy, J. and Spears, W., “Matching algorithms to problems: an experimental test of the particle swarm and some genetic algorithms on the multimodal problem generator,” In IEEE World Congress on Computational Intelligence, pp. 74–77 (1998).
    56. Kershenbaum, A., “When Genetic Algorithms Work Best,” INFORMS Journal of Computing, Vol. 9, No. 3, pp.253-254 (1997).
    57. Kirkpatrick, S., Gelatt, C. D., and Vecchi, M.P., “Optimization by Simulated Annealing,” Science, Vol. 220, pp. 671-680 (1983).
    58. Kuhn, H. W., and Baumol, W. J., “An Approximate Algorithm for the Fixed-Charge Transportation Problem,” Naval Res. Logistics Quarterly, Vol. 9, pp. 1-16 (1962).
    59. Kuntz, P. and Snyers, D., “Emergant Colonization and Graph Partitioning,” Proceedings of the 3th International Conference on Simulation of Adaptive Behavior: From Animals to Animate, 3, The MIT Press, Cambridge, MA (1997).
    60. Larsson, T., Migdalas, A., and Ronnqvist, M., “A Lagrangian Heuristic for the Capacitated Concave Minimum Cost Network Flow Problem,” European Journal of Operational Research, Vol. 78, pp. 116-129 (1994).
    61. Nourie, F. J., and Guder, F., “A Restricted-Entry Method for a Transportation Problem with Piecewise-Linear Concave Cost,” Computer & Operations Research, Vol. 21, pp. 723-733 (1994).
    62. Osman, I. H., and Kelly, J. P., “Meta-Heuristics: An overview,” Meta-Heuristics: Theory & Applications, Kluwer Academic Publishers, Boston, London, Dordrecht, pp. 1-21 (1996).
    63. Palmer, C. C., and Kershenbaum, A., “Representing Trees in Genetic Algorithms,” Proceedings of the First IEEE Conference on Evolutionary Computation, Piscataway, NJ: IEEE Service Center, Vol. 1, pp. 379-384 (1994).
    64. Rech, P., and Barton, L. G., “A Non-Convex Transportation Algorithm,” Applications of Mathematical Programming Techniques, E. M. Beale, ed. (1970).
    65. Reeves, C. R., “Improving the Efficiency of Tabu Search for Machine Sequencing Problems,” Journal of the Operation Research Society, Vol. 44, No. 4, pp. 375-382 (1993).
    66. Shyu, S. J., Yin, P. Y.and Lin, B. M. T., “An Ant Colony Optimization Algorithm for the Minimum Weight Vertex Cover Problem,” Manuscript Submitted for Publication. (NSC-90-2213-E-130-001) (2002).
    67. Shi, Y. and Eberhart, R. C. A modified particle swarm optimizer. Proceedings of the IEEE International Conference on Evolutionary Computation pp. 69-73. IEEE Press, Piscataway, NJ (1998a).
    68. Shi, Y. and Eberhart, R. C. Parameter selection in particle swarm optimization. Evolutionary Programming VII: Proc. EP 98 pp. 591-600. Springer-Verlag, New York (1998b).
    69. Shi, Y., “Particle Swarm Optimization,” IEEE Connections , Vol 2, pp.8-13 (2004).
    70. Suwan, R., and Sawased, T., “Link Capacity Assignment in Packet- Switched Networks: The Case of Piecewise Linear Concave Cost Function,” IEICE Trans. Commun., Vol. E82-B, No. 10 (1999).
    71. Taguhi, T., Ida. K., and Gen, M., “A Genetic Algorithm for Optimal Flow Assignment in Computer Network,” Computers ind. Engng, Vol. 35, No3-4, pp. 535-538 (1998).
    72. Thach, P. T., “A Decomposition Method Using A Pricing Mechanism for Min Concave Cost Flow Problems With a Hierarchical Structure,” Mathematical Programming, Vol. 53, pp. 339-359 (1992).
    73. Yaged, B., “Minimum Cost Routing for Static Network Models,” Networks, Vol. 1, pp 139-172 (1971).
    74. Yan, S., Juang, D. H., Chen, C. R., and Lai, W. S., “Global and Local Search Algorithms for Concave Cost Transshipment Problems,” Journal of Global Optimization, Vol. 33, No. 1, pp. 123 – 156 (2005).
    75. Yan, S., and Luo, S. C., “A Tabu Search-Based Algorithm for Concave Cost Transportation Network Problems,” Journal of the Chinese Institute of Engineers, Vol. 21, pp. 327-335 (1998).
    76. Yan, S., and Luo, S. C., “Probabilistic Local Search Algorithms for Concave Cost Transportation Network Problems,” European Journal of Operational Research, Vol.117, pp. 511-521 (1999).
    77. Yan, S., and Young, H. F., “A Decision Support Framework for Multi-Fleet Routing and Multi-Stop Flight Scheduling,” Transportation Research, Vol. 30A, pp. 379-398 (1996).
    78. Zangwill, W. I., “Minimum Concave Cost Flows in Certain Networks,” Management Science, Vol. 14, pp. 429-450 (1968)

    QR CODE
    :::