A fuzzy goal programming approach for solving multi-objective minimum cost flow problems with possibilistic coefficients
Keywords
Minimum cost flow, Multi-objective optimization, Possibilistic variables, Fuzzy goal programming approach, α-possibly optimal solution, Goal programming, Compromise solution, Parametric analysis
Abstract
This paper studies a multi-objective Minimum Cost Flow (MCF) with possibilistic objective function coefficients. A necessary and sufficient condition for investigating the α-possibly optimal solution is established. A fuzzy Goal Programming (GP) approach is applied to obtain the α-parametric optimal compromise solution. The parametric study under the concept of α-possibly optimal solution is analyzed without differentiability. Finally, a numerical example is given for the paper to clarify the methodology.
References
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[19] Chalam, G. A. (1994). Fuzzy goal programming (FGP) approach to a stochastic transportation problem under budgetary constraint. <em>Fuzzy sets and systems</em>, <em>66</em>(3), 293–299. https://doi.org/10.1016/0165-0114(94)90096-5
[20] Lai, Y. J., Hwang, C. L., Lai, Y. J., & Hwang, C. L. (1994). <em>Fuzzy multiple objective decision making</em>. Springer. https://doi.org/10.1007/978-3-642-57949-3_3
[21] Cui, J., An, S., & Zhao, M. (2014). A generalized minimum cost flow model for multiple emergency flow routing. <em>Mathematical problems in engineering</em>, <em>2014</em>. https://doi.org/10.1155/2014/832053
[22] Hussein, M. L. (1992). On convex vector optimization problems with possibilistic weights. <em>Fuzzy sets and systems</em>, <em>51</em>(3), 289–294. https://doi.org/10.1016/0165-0114(92)90019-Z
[23] Kassem, M. A. E. H. (1998). Stability of possibilistic multiobjective nonlinear programming problems without differentiability. <em>Fuzzy sets and systems</em>, <em>94</em>(2), 239–246.
[24] Leberling, H. (1981). On finding compromise solutions in multicriteria problems using the fuzzy min-operator. <em>Fuzzy sets and systems</em>, <em>6</em>(2), 105–118. https://doi.org/10.1016/0165-0114(81)90019-1
[25] Abd El-Wahed, W. F., & Lee, S. M. (2006). Interactive fuzzy goal programming for multi-objective transportation problems. <em>Omega</em>, <em>34</em>(2), 158–166. https://ideas.repec.org/a/eee/jomega/v34y2006i2p158-166.html
[26] Bellman, R., & Zadeh, L. (1970). Decision-making in a fuzzy environment. <em>Management science</em>, <em>17</em>(4), B--141. https://doi.org/10.1142/9789812819789_0004
[27] Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. <em>Fuzzy sets and systems</em>, <em>1</em>(1), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3
[28] Sakawa, M. (1993). <em>Fuzzy sets and interactive multiobjective optimization</em>. Springer.
[29] Ghatee, M., & Hashemi, S. M. (2009). Application of fuzzy minimum cost flow problems to network design under uncertainty. <em>Fuzzy sets and systems</em>, <em>160</em>(22), 3263–3289. https://doi.org/10.1016/j.fss.2009.04.004
[30] Bustos, A., Herrera, L., & Jiménez, E. (2014). Efficient frontier for multi-objective stochastic transportation networks in international market of perishable goods. <em>Journal of applied research and technology</em>, <em>12</em>(4), 654–665. https://doi.org/10.1016/S1665-6423(14)70082-3
[31] Alharbi, M. G., Khalifa, H. A. E. W., & Ammar, E. E. (2020). An interactive approach for solving the multiobjective minimum cost flow problem in the fuzzy environment. <em>Journal of mathematics</em>, <em>2020</em>, 1–7. https://doi.org/10.1155/2020/6247423
[2] Jewell, W. S. (1957). Warhousing and distribution of a seasonal product 1. <em>Naval research logistics quarterly</em>, <em>4</em>(1), 29–34. https://doi.org/10.1002/nav.3800040107
[3] Slump, C. H., & Gerbrands, J. J. (1982). A network flow approach to reconstruction of the left ventricle from two projections. <em>Computer graphics and image processing</em>, <em>18</em>(1), 18–36. https://doi.org/10.1016/0146-664X(82)90097-1
[4] Sapountzis, C. (1984). Allocating blood to hospitals from a central blood bank. <em>European journal of operational research</em>, <em>16</em>(2), 157–162. https://doi.org/10.1016/0377-2217(84)90070-5
[5] Hu, Y., Zhao, X., Liu, J., Liang, B., & Ma, C. (2020). An efficient algorithm for solving minimum cost flow problem with complementarity slack conditions. <em>Mathematical problems in engineering</em>, <em>2020</em>, 1–5. https://doi.org/10.1155/2020/2439265
[6] Kumar, A., & Kaur, M. (2014). A new method for solving single and multi-objective fuzzy minimum cost flow problems with different membership functions. <em>Sadhana - academy proceedings in engineering sciences</em>, <em>39</em>(1), 189–206. https://doi.org/10.1007/s12046-014-0228-7
[7] Hamacher, H. W., Pedersen, C. R., & Ruzika, S. (2007). Multiple objective minimum cost flow problems: A review. <em>European journal of operational research</em>, <em>176</em>(3), 1404–1422. https://doi.org/10.1016/j.ejor.2005.09.033
[8] Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (1990). <em>Linear programming and network flows</em>. John Wiley & Sons. https://books.google.com/books/about/Linear_Programming_and_Network_Flows.html?id=AW9v7GyuncsC
[9] Steuer, R. E. (1986). <em>Multiple criteria optimization</em>. Wiley.
[10] Luhandjula, M. K. (1987). Multiple objective programming problems with possibilistic coefficients. <em>Fuzzy sets and systems</em>, <em>21</em>(2), 135–145. https://doi.org/10.1016/0165-0114(87)90159-X
[11] Lee, S. M., & Moore, L. J. (1973). Optimizing transportation problems with multiple objectives. <em>AIIE transactions</em>, <em>5</em>(4), 333–338. https://doi.org/10.1080/05695557308974920
[12] Hemaida, R. S., & Kwak, N. K. (1994). A linear goal programming model for trans-shipment problems with flexible supply and demand constraints. <em>Journal of the operational research society</em>, <em>45</em>(2), 215–224. https://doi.org/10.2307/2584127
[13] Tamiz, M., Jones, D., & Romero, C. (1998). Goal programming for decision making: An overview of the current state-of-the-art. <em>European journal of operational research</em>, <em>111</em>(3), 569–581. https://doi.org/10.1016/S0377-2217(97)00317-2
[14] Romero, C. (2014). <em>Handbook of critical issues in goal programming</em>. Elsevier.
[15] Bit, A. K., Biswal, M. P., & Alam, S. S. (1993). An additive fuzzy programming model for multiobjective transportation problem. <em>Fuzzy sets and systems</em>, <em>57</em>(3), 313–319. https://doi.org/10.1016/0165-0114(93)90026-E
[16] Chanas, S., & Kuchta, D. (1998). Fuzzy integer transportation problem. <em>Fuzzy sets and systems</em>, <em>98</em>(3), 291–298. https://doi.org/10.1016/S0165-0114(96)00380-6
[17] Chanas, S., Kołodziejczyk, W., & Machaj, A. (1984). A fuzzy approach to the transportation problem. <em>Fuzzy sets and systems</em>, <em>13</em>(3), 211–221. https://doi.org/10.1016/0165-0114(84)90057-5
[18] Ehrgott, M., & Verma, R. (2001). A note on solving multicriteria transportation-location problems by fuzzy programming. <em>Asia-pacific journal of operational research</em>, <em>18</em>(2), 149–164.
[19] Chalam, G. A. (1994). Fuzzy goal programming (FGP) approach to a stochastic transportation problem under budgetary constraint. <em>Fuzzy sets and systems</em>, <em>66</em>(3), 293–299. https://doi.org/10.1016/0165-0114(94)90096-5
[20] Lai, Y. J., Hwang, C. L., Lai, Y. J., & Hwang, C. L. (1994). <em>Fuzzy multiple objective decision making</em>. Springer. https://doi.org/10.1007/978-3-642-57949-3_3
[21] Cui, J., An, S., & Zhao, M. (2014). A generalized minimum cost flow model for multiple emergency flow routing. <em>Mathematical problems in engineering</em>, <em>2014</em>. https://doi.org/10.1155/2014/832053
[22] Hussein, M. L. (1992). On convex vector optimization problems with possibilistic weights. <em>Fuzzy sets and systems</em>, <em>51</em>(3), 289–294. https://doi.org/10.1016/0165-0114(92)90019-Z
[23] Kassem, M. A. E. H. (1998). Stability of possibilistic multiobjective nonlinear programming problems without differentiability. <em>Fuzzy sets and systems</em>, <em>94</em>(2), 239–246.
[24] Leberling, H. (1981). On finding compromise solutions in multicriteria problems using the fuzzy min-operator. <em>Fuzzy sets and systems</em>, <em>6</em>(2), 105–118. https://doi.org/10.1016/0165-0114(81)90019-1
[25] Abd El-Wahed, W. F., & Lee, S. M. (2006). Interactive fuzzy goal programming for multi-objective transportation problems. <em>Omega</em>, <em>34</em>(2), 158–166. https://ideas.repec.org/a/eee/jomega/v34y2006i2p158-166.html
[26] Bellman, R., & Zadeh, L. (1970). Decision-making in a fuzzy environment. <em>Management science</em>, <em>17</em>(4), B--141. https://doi.org/10.1142/9789812819789_0004
[27] Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. <em>Fuzzy sets and systems</em>, <em>1</em>(1), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3
[28] Sakawa, M. (1993). <em>Fuzzy sets and interactive multiobjective optimization</em>. Springer.
[29] Ghatee, M., & Hashemi, S. M. (2009). Application of fuzzy minimum cost flow problems to network design under uncertainty. <em>Fuzzy sets and systems</em>, <em>160</em>(22), 3263–3289. https://doi.org/10.1016/j.fss.2009.04.004
[30] Bustos, A., Herrera, L., & Jiménez, E. (2014). Efficient frontier for multi-objective stochastic transportation networks in international market of perishable goods. <em>Journal of applied research and technology</em>, <em>12</em>(4), 654–665. https://doi.org/10.1016/S1665-6423(14)70082-3
[31] Alharbi, M. G., Khalifa, H. A. E. W., & Ammar, E. E. (2020). An interactive approach for solving the multiobjective minimum cost flow problem in the fuzzy environment. <em>Journal of mathematics</em>, <em>2020</em>, 1–7. https://doi.org/10.1155/2020/6247423
Publication status
Accepted — production in progress