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Keywords:
transportation problem; capacitated transportation problem; multi-index problem; linear programming
transportation problem; capacitated transportation problem; multi-index problem; linear programming
Summary:
This study focuses on a particular instance of the capacitated multi-index transportation problem, specifically the four-index variant. It involves transporting various products from multiple sources to multiple destinations using different means of transport chosen depending on reserved charges, while satisfying the constraints associated with each dimension of the model. Although this problem was previously studied by Zitouni and Keraghel (2003), the available methods still suffer from slow convergence and difficulty in handling degeneracy. This work proposes a new procedure that ensures a stable and feasible starting solution. The approach involves introducing dummy allocations with very small values to some null allocations, in order to handle cases of degeneracy. It is based on an extended version of the Vogel's approximation method, where the traditional selection criterion is modified to account for the four dimensions of the problem. This enhancement improves the initial distribution of quantities, reduces ineffective allocations and accelerates convergence. The proposed method maintains full compliance with all problem constraints including supply, demand, capacity and the structural logic of the four-index model. The methodology was evaluated through a comparative numerical study, applying the proposed method to a set of problem instances of varying sizes, and comparing it with Zitouni's approach in terms of convergence speed and number of iterations. The results show that the proposed method achieves a clear advantage in terms of faster convergence required to reach the solution. This highlights the relevance of the approach for complex logistical models and supply chain planning, where efficient solutions must be obtained within limited time. Moreover, the proposed approach lays the foundation for the development of more advanced multi-index optimization algorithms.
References:
[1] Buvaneshwari, T. K., Anuradha, D.: Solving stochastic fuzzy transportation problem with mixed constraints using the weibull distribution. J. Math. 1) (2022), 1-11. DOI
[2] Bisht, M., Ebrahimnejad, A.: Four-dimensional green transportation problem considering multiple objectives and product blending in fermatean fuzzy environment. Complex Intell. Systems 1 (2025), 6, 1-29. DOI
[3] Dahiya, K., Verma, V.: Capacitated transportation problem with bounds on RIM conditions. Europ. J. Oper. Res. 178 (2007), 3, 718-737. DOI
[4] Hakim, M., Zitouni, R.: An approach to solve a fuzzy bi-objective multi-index fixed charge transportation problem. Kybernetika 60 (2024), 3, 271-292. DOI
[5] Hedid, M., Zitouni, R., Labdai, S.: Solving the four index fully fuzzy transportation problem. Croatian Oper. Res. Rev. 11 (2020), 2, 199-215. DOI
[6] Jiménez, F., Verdegay, J. L.: Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. Europ. J. Oper. Res. 117 (1999), 3, 485-510. DOI
[7] Kartli, N.: Hybrid algorithms for fixed charge transportation problem. Kybernetika 61 (2025), 2, 141-167. DOI
[8] Kartli, N., Bostanci, E., Guzel, M. S.: A new algorithm for optimal solution of fixed charge transportation problem. Kybernetika 59 (2023), 1, 45-63. DOI
[9] Kartli, N., Bostanci, E., Guzel, M. S.: Heuristic algorithm for an optimal solution of fully fuzzy transportation problem. Computing 106 (2024), 10, 3195-3227. DOI
[10] Kumar, E., Dhanapal, A.: Solving multi-objective bi-item capacitated transportation problem with fermatean fuzzy multi-choice stochastic mixed constraints involving normal distribution. Contemporary Math. 5 (2024), 4, 4776-4804. DOI
[11] Liu, S.-T.: Fuzzy total transportation cost measures for fuzzy solid transportation problem. Appl. Math. Comput. 174 (2006), 2, 927-941. DOI
[12] Ninh, P.: $n$-index transportation problem. Math. J. 7 (1979), 1, 18-25.
[13] Queyranne, M., Spieksma, F. C. R.: Approximation algorithms for multi-index transportation problem with decomposable costs. Discrete Appl. Math. 76 (2007), 1-3, 239-253. DOI
[14] Pandian, P., Dhanapal, A.: A new approach for solving solid transportation problems. Appl. Math. Sci. 4 (2010), 72, 3603-3610. DOI
[15] Rani, D.: Solving non-linear fixed-charge transportation problems using nature inspired non-linear particle swarm optimization algorithm. Appl. Soft Comput. 146 (2023), 110699. DOI
[16] Reinfeld, N. V., Vogel, W. R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs, N.J. 1958.
[17] Ritzinger, U., Brandstätter, G., Krasel, L. F., Prandtstetter, M.: Collaborative logistics: optimizing fixed-scheduled container trains. Europ. Transport Res. Rev. 16 (2024),1, 71. DOI
[18] Russell, E. J.: Extension of Dantzig's algorithm to finding an initial nearoptimal basis for the transportation problem. Oper. Res. 17 (1969), 1, 187-191. DOI
[19] Sandhiya, S., Dhanapal, A.: Solving neutrosophic multi-dimensional fixed charge transportation problem. Contemporary Math. 5 (2024), 3, 3601-3627. DOI
[20] Shivani, Chauhan, D., Rani, D.: A feasibility restoration particle swarm optimizer with chaotic maps for two-stage fixed-charge transportation problems. Swarm Evolutionary Comput. 91 (2024), 101776. DOI
[21] Singh, B., Singh, A.: Multi-index transportation problem: An overview of its variants, solution techniques and applications. J. Punjab Academy Sci. 23 (2023), 164-174. DOI
[22] Uddin, M., Huynh, N.: Reliable routing of road-rail intermodal freight under uncertainty. Europ. Transport Res. Rev. 16 (2024), 71, 929-952. DOI
[23] Zitouni, R., Keraghel, A.: Resolution of the capacitated transportation problem with four subscripts. Kybernetes 32 (2003), 9/10, 1450-1463. DOI
[24] Zitouni, R., Keraghel, A.: A note on the algorithm of resolution of a capacitated transportation problem with four subscripts. East J. Math. Sci. (FJMS) 26 (2007), 3, 769-778.
[25] Zitouni, R., Achache, M.: A numerical comparison between two exact simplicial methods for solving a capacitated 4-index transportation problem. J. Numer. Anal. Approx. Theory 46 (2017), 2, 181-192. DOI
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