Fuzzy Least-Squares Linear Regression Approach to Ascertain Stochastic Demand in the Vehicle Routing Problem

ABSTRACT

Estimation of stochastic demand in physical distribution in general and efficient transport routs management in particular is emerging as a crucial factor in urban planning domain. It is particularly important in some municipalities such as Tehran where a sound demand management calls for a realistic analysis of the routing system. The methodology involved critically investigating a fuzzy least-squares linear regression approach (FLLRs) to estimate the stochastic demands in the vehicle routing problem (VRP) bearing in mind the customer's preferences order. A FLLR method is proposed in solving the VRP with stochastic demands: approximate-distance fuzzy least-squares (ADFL) estimator ADFL estimator is applied to original data taken from a case study. The SSR values of the ADFL estimator and real demand are obtained and then compared to SSR values of the nominal demand and real demand. Empirical results showed that the proposed method can be viable in solving problems under circumstances of having vague and imprecise performance ratings. The results further proved that application of the ADFL was realistic and efficient estimator to face the sto- chastic demand challenges in vehicle routing system management and solve relevant problems.

Estimation of stochastic demand in physical distribution in general and efficient transport routs management in particular is emerging as a crucial factor in urban planning domain. It is particularly important in some municipalities such as Tehran where a sound demand management calls for a realistic analysis of the routing system. The methodology involved critically investigating a fuzzy least-squares linear regression approach (FLLRs) to estimate the stochastic demands in the vehicle routing problem (VRP) bearing in mind the customer's preferences order. A FLLR method is proposed in solving the VRP with stochastic demands: approximate-distance fuzzy least-squares (ADFL) estimator ADFL estimator is applied to original data taken from a case study. The SSR values of the ADFL estimator and real demand are obtained and then compared to SSR values of the nominal demand and real demand. Empirical results showed that the proposed method can be viable in solving problems under circumstances of having vague and imprecise performance ratings. The results further proved that application of the ADFL was realistic and efficient estimator to face the sto- chastic demand challenges in vehicle routing system management and solve relevant problems.

Cite this paper

nullF. Torfi, R. Farahani and I. Mahdavi, "Fuzzy Least-Squares Linear Regression Approach to Ascertain Stochastic Demand in the Vehicle Routing Problem,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 64-73. doi: 10.4236/am.2011.21008.

nullF. Torfi, R. Farahani and I. Mahdavi, "Fuzzy Least-Squares Linear Regression Approach to Ascertain Stochastic Demand in the Vehicle Routing Problem,"

References

[1] R. J. Petch and S. Salhi, “A Multi-Phase Constructive Heuristic for the Vehicle Routing Problem with Multiple Trips,” Discrete Applied Mathematics, Vol. 133, No. 1-3, 2004, pp. 69-92. doi:10.1016/S0166-218X(03)00434-7

[2] D. Mester and O. Braysy, “Active Guided Evolution Strategies for Large-Scale Vehicle Routing Problems with Time Windows,” Computers and Operations Research, Vol. 32, No. 6, 2005, pp. 1593-1614. doi:10.1016/j.cor. 2003.11.017

[3] A. S. Maria, F. Elena and L. Gilbert, “Heuristic and Lower Bound for a Stochastic Location-Routing Problem,” European Journal of Operational Research, Vol. 179, No. 3, 2007, pp. 940-955. doi:10.1016/j.ejor.2005. 04.051

[4] F. Torfi, R. Z. Farahani and S. Rezapour, “Fuzzy AHP to Determine the Relative Weights of Evaluation Criteria and Fuzzy TOPSIS to Rank the Alternatives” Applied Soft Computing, Vol. 10, No. 2, 2010, pp. 520-528. doi: 10.1016/j.asoc.2009.08.021

[5] F. Torfi, R. Z. Farahani and N. Hedayat, “A New Model to Determine the Weights of Multiple Objectives in Combinational Optimization Problems,” Proceedings of ICCESSE International Conference on Computer, Electrical, Systems Science and Engineering, 07, Paris, 2010, pp. 1933-1955.

[6] H. Tanaka, S. Vegima and K. Asai, “Linear Regression Analysis with Fuzzy Model,” IEEE Transactions on Systems, Man, and Cybernet, Vol. 12, No. 6, 1982, pp. 903-907.

[7] H. Tanaka and H. Ishibuchi, “Identification of Positivistic Linear Systems by Quadratic Membership Functions of Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 41, 1991, pp. 145-160. doi:10.1016/0165-0114(91)90218-F

[8] H. Tanaka, H. Ishibuchi and S.Yoshikawa, “Exponential Possibility Regression Analysis,” Fuzzy Sets and Systems, Vol. 69, No. 3, 1995, pp. 305-318. doi:10.1016/0165-0114(94)00179-B

[9] M. Sakawa and H. Yano, “Multiobjective Fuzzy Linear Regression Analysis for Fuzzy Input-Output Data,” Fuzzy Sets and Systems, Vol. 47, 1992, pp. 173-181. doi: 10.1016/0165-0114(92)90175-4

[10] M. Albrecht, “Approximation of Functional Relationships to Fuzzy Observations,” Fuzzy Sets and Systems, Vol. 49, No. 3, 1992, pp. 301-305. doi:10.1016/0165-0114(92) 90281-8

[11] M. S. Yang and C.H. Ko, “On Cluster-Wise Fuzzy Regression Analysis,” IEEE Transactions on Systems, Man, and Cybernet, Vol. 27, No. 1, 1997, pp. 1-13. doi: 10.1109/3477.552181

[12] P. Diamond, “Fuzzy Least Squares,” Information Science, Vol. 46, 1988, pp. 141-157. doi:10.1016/0020-0255(88) 90047-3

[13] L. A. Zadeh, “Fuzzy Sets,” Information Control, Vol. 8, 1965, pp. 338-53. doi:10.1016/S0019-9958(65)90241-X

[14] T. Yang and H. Chih-Ching, “Multiple-Attribute Decision Making Methods for Plant Layout Design Problem,” Robotics and computer-integrated manufacturing, Vol. 23, No. 1, 2007, pp. 126-137. doi:10.1016/j.rcim.2005. 12.002

[15] C. T. Chen, “Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment,” Fuzzy Sets and Systems, Vol. 114, No. 1, 2000, pp. 1-9. doi:10.1016/ S0165-0114(97)00377-1

[16] M. S. Yang and T. S. Lin, “Fuzzy Least-Squares Linear Regression Analysis for Fuzzy Input-Output Data,” Fuzzy Sets and Systems, Vol. 126, No. 3, 2002, pp. 389-399. doi:10.1016/S0165-0114(01)00066-5

[17] D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, New York, 1980.

[1] R. J. Petch and S. Salhi, “A Multi-Phase Constructive Heuristic for the Vehicle Routing Problem with Multiple Trips,” Discrete Applied Mathematics, Vol. 133, No. 1-3, 2004, pp. 69-92. doi:10.1016/S0166-218X(03)00434-7

[2] D. Mester and O. Braysy, “Active Guided Evolution Strategies for Large-Scale Vehicle Routing Problems with Time Windows,” Computers and Operations Research, Vol. 32, No. 6, 2005, pp. 1593-1614. doi:10.1016/j.cor. 2003.11.017

[3] A. S. Maria, F. Elena and L. Gilbert, “Heuristic and Lower Bound for a Stochastic Location-Routing Problem,” European Journal of Operational Research, Vol. 179, No. 3, 2007, pp. 940-955. doi:10.1016/j.ejor.2005. 04.051

[4] F. Torfi, R. Z. Farahani and S. Rezapour, “Fuzzy AHP to Determine the Relative Weights of Evaluation Criteria and Fuzzy TOPSIS to Rank the Alternatives” Applied Soft Computing, Vol. 10, No. 2, 2010, pp. 520-528. doi: 10.1016/j.asoc.2009.08.021

[5] F. Torfi, R. Z. Farahani and N. Hedayat, “A New Model to Determine the Weights of Multiple Objectives in Combinational Optimization Problems,” Proceedings of ICCESSE International Conference on Computer, Electrical, Systems Science and Engineering, 07, Paris, 2010, pp. 1933-1955.

[6] H. Tanaka, S. Vegima and K. Asai, “Linear Regression Analysis with Fuzzy Model,” IEEE Transactions on Systems, Man, and Cybernet, Vol. 12, No. 6, 1982, pp. 903-907.

[7] H. Tanaka and H. Ishibuchi, “Identification of Positivistic Linear Systems by Quadratic Membership Functions of Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 41, 1991, pp. 145-160. doi:10.1016/0165-0114(91)90218-F

[8] H. Tanaka, H. Ishibuchi and S.Yoshikawa, “Exponential Possibility Regression Analysis,” Fuzzy Sets and Systems, Vol. 69, No. 3, 1995, pp. 305-318. doi:10.1016/0165-0114(94)00179-B

[9] M. Sakawa and H. Yano, “Multiobjective Fuzzy Linear Regression Analysis for Fuzzy Input-Output Data,” Fuzzy Sets and Systems, Vol. 47, 1992, pp. 173-181. doi: 10.1016/0165-0114(92)90175-4

[10] M. Albrecht, “Approximation of Functional Relationships to Fuzzy Observations,” Fuzzy Sets and Systems, Vol. 49, No. 3, 1992, pp. 301-305. doi:10.1016/0165-0114(92) 90281-8

[11] M. S. Yang and C.H. Ko, “On Cluster-Wise Fuzzy Regression Analysis,” IEEE Transactions on Systems, Man, and Cybernet, Vol. 27, No. 1, 1997, pp. 1-13. doi: 10.1109/3477.552181

[12] P. Diamond, “Fuzzy Least Squares,” Information Science, Vol. 46, 1988, pp. 141-157. doi:10.1016/0020-0255(88) 90047-3

[13] L. A. Zadeh, “Fuzzy Sets,” Information Control, Vol. 8, 1965, pp. 338-53. doi:10.1016/S0019-9958(65)90241-X

[14] T. Yang and H. Chih-Ching, “Multiple-Attribute Decision Making Methods for Plant Layout Design Problem,” Robotics and computer-integrated manufacturing, Vol. 23, No. 1, 2007, pp. 126-137. doi:10.1016/j.rcim.2005. 12.002

[15] C. T. Chen, “Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment,” Fuzzy Sets and Systems, Vol. 114, No. 1, 2000, pp. 1-9. doi:10.1016/ S0165-0114(97)00377-1

[16] M. S. Yang and T. S. Lin, “Fuzzy Least-Squares Linear Regression Analysis for Fuzzy Input-Output Data,” Fuzzy Sets and Systems, Vol. 126, No. 3, 2002, pp. 389-399. doi:10.1016/S0165-0114(01)00066-5

[17] D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, New York, 1980.