Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow

Affiliation(s)

Department of Mathematics, Ramjas College, University of Delhi, Delhi, India.

Department of Mathematics, Hans Raj College, University of Delhi, Delhi, India.

Department of Mathematics, Ramjas College, University of Delhi, Delhi, India.

Department of Mathematics, Hans Raj College, University of Delhi, Delhi, India.

ABSTRACT

In this paper, a transportation problem with
an objective function as the sum of a linear and fractional function is considered. The linear function
represents the total transportation cost incurred when the goods are shipped
from various sources to the destinations and the fractional function gives the
ratio of sales tax to the total public expenditure. Our objective is to
determine the transportation schedule which minimizes the sum of total
transportation cost and ratio of total sales tax paid to the total public
expenditure. Sometimes, situations
arise where either reserve stocks have to be kept at the supply points, for
emergencies or there may be extra demand in the markets. In such situations,
the total flow needs to be controlled or enhanced. In this paper, a special
class of transportation problems is studied where in the total transportation
flow is restricted to a known specified level. A related transportation
problem is formulated and it is shown that to each basic feasible solution
which is called corner feasible solution to related transportation problem,
there is a corresponding feasible solution to this restricted flow problem. The
optimal solution to restricted flow problem may be obtained from the optimal
solution to related transportation problem. An algorithm is presented to solve
a capacitated linear plus linear fractional transportation problem with
restricted flow. The algorithm is supported by a real life example of a
manufacturing company.

KEYWORDS

Transportation Problem; Linear Plus Linear Fractional; Restricted Flow; Corner Feasible Solution

Transportation Problem; Linear Plus Linear Fractional; Restricted Flow; Corner Feasible Solution

Cite this paper

K. Gupta and S. Arora, "Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow,"*American Journal of Operations Research*, Vol. 3 No. 6, 2013, pp. 581-588. doi: 10.4236/ajor.2013.36055.

K. Gupta and S. Arora, "Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow,"

References

[1] A. Gupta, S. Khanna and M. C. Puri, “A Paradox in Linear Fractional Transportation Problems with Mixed Constraints,” Optimization, Vol. 27, No. 4, 1993, pp. 375387.

http://dx.doi.org/10.1080/02331939308843896

[2] M. Jain and P. K. Saksena, “Time Minimizing Transportation Problem with Fractional Bottleneck Objective Function,” Yugoslav Journal of Operations Research, Vol. 21, No. 2, 2011, pp. 1-16.

[3] F. Xie, Y. Jia and R. Jia, “Duration and Cost Optimization for Transportation Problem,” Advances in Information Sciences and Service Sciences, Vol. 4, No. 6, 2012, pp. 219-233.

http://dx.doi.org/10.4156/aiss.vol4.issue6.26

[4] A. Khurana and S. R. Arora, “The Sum of a Linear and Linear Fractional Transportation Problem with Restricted and Enhanced Flow,” Journal of Interdisciplinary Mathematics, Vol. 9, No. 9, 2006, pp. 373-383.

http://dx.doi.org/10.1080/09720502.2006.10700450

[5] K. Gupta and S. R. Arora, “Paradox in a Fractional Capacitated Transportation Problem,” International Journal of Research in IT, Management and Engineering, Vol. 2, No. 3, 2012, pp. 43-64.

[6] S. Misra and C. Das, “Solid Transportation Problem with Lower and Upper Bounds on Rim Conditions—A Note,” New Zealand Operational Research, Vol. 9, No. 2, 1981, pp. 137-140.

[7] S. Jain and N. Arya, “An Inverse Capacitated Transportation Problem,” IOSR Journal of Mathematics, Vol. 5, No. 4, 2013, pp. 24-27.

http://dx.doi.org/10.9790/5728-0542427

[8] S. R. Arora and K. Gupta, “Restricted Flow in a NonLinear Capacitated Transportation Problem with Bounds on Rim Conditions,” International Journal of Management, IT and Engineering, Vol. 2, No. 5, 2012, pp. 226243.

[9] A. Khurana, D. Thirwani and S. R. Arora, “An Algorithm for Solving Fixed Charge Bi—Criterion Indefinite Quadratic Transportation Problem with Restricted Flow,” International Journal of Optimization: Theory, Methods and Applications, Vol. 1, No. 4, 2009, pp. 367-380.

[1] A. Gupta, S. Khanna and M. C. Puri, “A Paradox in Linear Fractional Transportation Problems with Mixed Constraints,” Optimization, Vol. 27, No. 4, 1993, pp. 375387.

http://dx.doi.org/10.1080/02331939308843896

[2] M. Jain and P. K. Saksena, “Time Minimizing Transportation Problem with Fractional Bottleneck Objective Function,” Yugoslav Journal of Operations Research, Vol. 21, No. 2, 2011, pp. 1-16.

[3] F. Xie, Y. Jia and R. Jia, “Duration and Cost Optimization for Transportation Problem,” Advances in Information Sciences and Service Sciences, Vol. 4, No. 6, 2012, pp. 219-233.

http://dx.doi.org/10.4156/aiss.vol4.issue6.26

[4] A. Khurana and S. R. Arora, “The Sum of a Linear and Linear Fractional Transportation Problem with Restricted and Enhanced Flow,” Journal of Interdisciplinary Mathematics, Vol. 9, No. 9, 2006, pp. 373-383.

http://dx.doi.org/10.1080/09720502.2006.10700450

[5] K. Gupta and S. R. Arora, “Paradox in a Fractional Capacitated Transportation Problem,” International Journal of Research in IT, Management and Engineering, Vol. 2, No. 3, 2012, pp. 43-64.

[6] S. Misra and C. Das, “Solid Transportation Problem with Lower and Upper Bounds on Rim Conditions—A Note,” New Zealand Operational Research, Vol. 9, No. 2, 1981, pp. 137-140.

[7] S. Jain and N. Arya, “An Inverse Capacitated Transportation Problem,” IOSR Journal of Mathematics, Vol. 5, No. 4, 2013, pp. 24-27.

http://dx.doi.org/10.9790/5728-0542427

[8] S. R. Arora and K. Gupta, “Restricted Flow in a NonLinear Capacitated Transportation Problem with Bounds on Rim Conditions,” International Journal of Management, IT and Engineering, Vol. 2, No. 5, 2012, pp. 226243.

[9] A. Khurana, D. Thirwani and S. R. Arora, “An Algorithm for Solving Fixed Charge Bi—Criterion Indefinite Quadratic Transportation Problem with Restricted Flow,” International Journal of Optimization: Theory, Methods and Applications, Vol. 1, No. 4, 2009, pp. 367-380.