模糊軸式三維指派問題是軸式三維指派問題的延伸。在指派成本模糊化的情況下，我們無法直接使用針對軸式三維指派問題的方法進行求解。在回顧過去的文獻過程中，我們發現分支界限法是很適合求解此問題的精確解演算法。然而分支界限法的正確率雖然很高，但卻有著如窮舉法一般的低效率。因此，我們針對模糊軸式三維指派問題建構了一個改良式的分支界限法，稱之為投影分支界限法。本論文提出投影分支界限法將三維空間投影至二維的平面，再利用求解二維指派問題的匈牙利法得到分支的界限，最後再將所得到的界限轉換回三維空間做運算。如此一來維持了正確率高的特性，同時也提高了演算的效率。藉由測試題組的結果顯示，本研究所提出的投影分支界限法優於先前文獻中所提出的演算法。

Fuzzy axial three-dimensional assignment problem is an extension of axial three-dimensional assignment problems. In this situation, when the assignment cost is fuzzied, we cannot use the traditional algorithms for solving axial three-dimensional assignment problems to solve the variant assignment problem. In reviewing the literatures, we find out that the B&B algorithm is a suitable way to find the exact solution of this kind of problem. However, the existing B&B algorithm has low efficiency as the complete enumerational procedure. For this reason, we propose an improved B&B algorithm for solving fuzzy axial three-dimensional assignment problem called the projected B&B algorithm. We first transfer the fuzzy axial three-dimensional assignment into two dimensions, then use the projection method followed by the Hungarian algorithm to obtain the upper bound of the resulting problem. In this way, we maintain the high accuracy and improve the efficiency. In order to demonstrate the procedure of the proposed algorithm, we present a numerical example in this thesis. Besides, computational results show that the performance of the proposed algorithm is superior to the existing algorithm for solving the fuzzy axial three-dimensional assignment problem.

Alan, F., Gregory, B. S. (2013). Efficient algorithms for three-dimensional axial and planar random assignment problems, Data Structures and Algorithms.
Arrow, K.J., Hurwicz, L. (1956). Reduction of constrained maxima to saddle-point problem. In: Neyman, J. (Ed.), Proceedings of the Third Berkeley Symposium on Probability, University of California Press, Berkeley, CA, vol. 5, pp. 1–20.
Balas, E., Saltzman, M. J. (1991). An Algorithm for the Three-Index Assignment Problem, Operations Research, vol. 39, pp. 150-161.
Bellman, R. E., Zadeh L.A. (1970). Decision-Making in a Fuzzy Environment, Management Science, vol. 17, pp. 141-164.
Chanas, S., Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems, Vol. 82, pp. 299–305.
Chanas, S., Kuchta, D. (1998). Fuzzy integer transportation problem, Fuzzy Sets and Systems, Vol. 98, pp. 291–298.
Chen, M. S. (1985). On a Fuzzy Assignment Problem, Tamkang Journal, vol. 22, pp. 407-411.
Feng, Y., Yang, L. (2006). A Two-Objective Fuzzy K-Cardinality Assignment Problem, Journal of Computational and Applied Mathematics, Vol. 197, pp.233-244.
Gilbert, K. C., Hofstra, R. B. (1987). An Algorithm for a Class of Three-Dimensional Assignment Problems Arising in Scheduling Applications, IIE Transactions, pp. 29-33.
Gilbert, K. C., Hofstra, R. B. (1988). Multidimensional Assignment Problems, Decision Sciences, Vol. 19 , pp. 306-321.
Geetha, S., Vartak, M. N. (1989). Time-cost Tradeoff in a Three Dimensional Assignment Problem, European Journal of Operational Research, Vol. 38, pp. 255-258.
Goossens, D., Polyakovskiy, S., Spieksma, F. C. R., Woeginger, G. J. (2009). The approximability of three-dimensional assignment problems with bottleneck objective, Optimization Letters, Vol. 4, pp. 7-16.
Hahn, P. M., Kim, B. J., Stutzle, T., Kanthak, S., Hightower, W.L., Samra, H., Ding, Z., Guignard, M. (2008). The quadratic three-dimensional assignment problem: Exact and approximate solution methods, European Journal of Operational Research, Vol. 184, pp. 416-428.
Hansen, P., Kaufman L. (1973). A Primal-dual Algorithm for the Three-dimensional Assignment Problem, Cahiers du CERO, Vol. 15, pp. 327-336.
Huang, L. S., Xu, G. H. (2005). Solution of assignment problem of restriction of qualification, Operations Research and Management Science, Vol. 14, pp. 28-31.
Kim, B. J., Hightower, W. L., Hahn, P. M., Zhu, Y. R., Sun, L. (2009). Lower bounds for the axial three-index assignment problem, European Journal of Operational Research, Vol. 202, pp. 654-668.
Kuhn, H. W. (1955). The Hungarian Method for the Assignment Problem, Naval
Research Logistics Quarterly, Vol. 2, pp. 83-97.
Lin, C. J., Wen, U. P. (2004). A Labeling Algorithm for the Fuzzy Assignment Problem, Fuzzy Sets and Systems, Vol. 142, pp. 373-391.
Lin, H. (2009). Model Formulation and Algorithms for a Fuzzy Axial Three-Dimensional Assignment Problem. Master Thesis, National Tsing Hua University, Hsinchu, Taiwan, ROC.
Pandian, P., Natarajan, G. (2010). A New Algorithm for Finding a Fuzzy Optimal
Solution for Fuzzy Transportation Problems, Applied Mathematical Sciences, Vol. 4, pp. 79-90.
Pandian, P., kavitha, k. (2012). Support Set Invariant Cost Sensitivity Analysis in Degenerate Transportation Problem – A Parametric Approach, International Journal of Math. Sci. & Engg. Appls., Vol. 6, pp. 301-309.
Pierskalla, W. P. (1967). The Tri-Substitution Method for the Three-Dimensional Assignment Problem, Journal of the Canadian Operational Research Society, Vol. 5, pp. 71-81.
Pierskalla, W. P. (1968). The Multidimensional Assignment Problem, Journal of the
Canadian Operations Research, Vol. 16, pp. 422-431.
Sakawa, M., Yauch, K. (2001). An Interactive Fuzzy Satisficing Method for Multiobjective Nonconvex Programming Problems with Fuzzy Numbers through Coevolutionary Genetic Algorithms. IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, Vol. 31, pp. 59-67.
Tada, M., Ishii, H. (1996). An integer fuzzy transportation problem, Comput. Math. Appl., Vol. 31, pp. 71-87.