The Set Covering problem is one the of most important NP-complete 0-1 integer programming problems because it serves as a model for many real world problems like the crew scheduling problem, facility location problem, vehicle routing etc. In this paper, an algorithm is suggested to solve a multi objective Set Covering problem with fuzzy linear fractional functionals as the objectives. The algorithm obtains the complete set of efficient cover solutions for this problem. It is based on the cutting plane approach, but employs a more generalized and a much deeper form of the Dantzig cut. The fuzziness in the problem lies in the coefficients of the objective functions. In addition, the ordering between two fuzzy numbers is based on the possibility and necessity indices introduced by Dubois and Prade [Ranking fuzzy numbers in the setting of possibility theory. Inf. Sci. 30 (1983) 183–224.]. Our aim is to develop a method which provides the decision maker with a fuzzy solution. An illustrative numerical example is elaborated to clarify the theory and the solution algorithm.

Keywords: Set covering Problem, multi objective linear fractional programming, fuzzy mathematical programming, fuzzy objective function

^{1}; Saxena, Ratnesh R

^{2}

@article{RO_2015__49_3_495_0, author = {Upmanyu, Mudita and Saxena, Ratnesh R}, title = {On solving multi objective {Set} {Covering} {Problem} with imprecise linear fractional objectives}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {495--510}, publisher = {EDP-Sciences}, volume = {49}, number = {3}, year = {2015}, doi = {10.1051/ro/2014045}, mrnumber = {3349131}, zbl = {1326.90052}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro/2014045/} }

TY - JOUR AU - Upmanyu, Mudita AU - Saxena, Ratnesh R TI - On solving multi objective Set Covering Problem with imprecise linear fractional objectives JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2015 SP - 495 EP - 510 VL - 49 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2014045/ DO - 10.1051/ro/2014045 LA - en ID - RO_2015__49_3_495_0 ER -

%0 Journal Article %A Upmanyu, Mudita %A Saxena, Ratnesh R %T On solving multi objective Set Covering Problem with imprecise linear fractional objectives %J RAIRO - Operations Research - Recherche Opérationnelle %D 2015 %P 495-510 %V 49 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2014045/ %R 10.1051/ro/2014045 %G en %F RO_2015__49_3_495_0

Upmanyu, Mudita; Saxena, Ratnesh R. On solving multi objective Set Covering Problem with imprecise linear fractional objectives. RAIRO - Operations Research - Recherche Opérationnelle, Volume 49 (2015) no. 3, pp. 495-510. doi : 10.1051/ro/2014045. http://archive.numdam.org/articles/10.1051/ro/2014045/

The Set Covering Problem with linear fractional functional. Indian J. Pure Appl. Math. 8 (1975) 578–588. | MR | Zbl

, and ,Enumeration technique for the Set Covering Problem with a linear fractional functional as its objective function. ZAMM 57 (1977) 181–186. | DOI | MR | Zbl

and ,Cutting plane technique for the multi-objective Set Covering Problem with linear fractional objective functions. IJOMAS 14 (1998) 111–122.

and ,Cutting plane technique for the Set Covering Problem with linear fractional functional. ZAMM 57 (1977) 597–602. | DOI | MR | Zbl

, and ,Set covering and involutory bases. Manag. Sci. 18 (1971) 194–206. | DOI | MR | Zbl

and ,Linear programming with a fractional objective function. Oper. Res. 21 (1973) 22–29. | DOI | MR | Zbl

and ,Fuzzy mathematical programming for multi objective linear fractional programming problem. Fuzzy Sets Syst. 125 (2002) 335–342. | DOI | MR | Zbl

and ,Programming with linear fractional functionals. Nav. Res. Logist. Q. 9 (1962) 181–186. | DOI | MR | Zbl

and ,An interactive algorithm for multicriteria programming. Comput. Oper. Res. 7. (1980) 81–87. | DOI

and ,D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications. Academic (1980) 393 p. | MR | Zbl

Ranking fuzzy numbers in the setting of possibility theory. Inf. Sci. 30 (1983) 183–224. | DOI | MR | Zbl

and ,R.S. Garfinkel and G.L. Nemhauser, Integer Programming. A Wiley Inter Science Publication, John Wiley and Sons (1973) 448p. | MR | Zbl

Multi-criteria integer linear fractional programming problem. Optimization 35 (1995) 373–389. | DOI | MR | Zbl

and ,Attrition games. Nav. Res. Logist. Q. 3 (1956) 71–93. | DOI | MR | Zbl

and ,Multiple objective linear fractional programming. Manag. Sci. 27 (1981) 1024–1039. | DOI | Zbl

and ,Set Covering by single-branch enumeration with linear-programming subproblems. Oper. Res. 19 (1971) 998–1022. | DOI | MR | Zbl

, and ,A fuzzy programming approach to fuzzy linear fractional programming with fuzzy coefficients. J. Fuzzy Math. 4 (1996) 829–834. | MR | Zbl

and ,M.K. Luhandjula Fuzzy approaches for multiple objective linear fractional optimization. Fuzzy Sets Syst. 13 (1984) 11–23. | MR | Zbl

Acceptable optimality in linear fractional programming with fuzzy coefficients. Fuzzy Optim. Decis. Making 6 (2007) 5–16. | DOI | MR | Zbl

, and ,Pareto optimality for multiobjective linear fractional programming problems with fuzzy parameters. Inf. Sci. 63 (1992) 33–53. | DOI | MR | Zbl

, and ,On a fuzzy set approach to solving multiple objective linear fractional programming problem. Fuzzy Sets Syst. 134 (2003) 397–405. | DOI | MR | Zbl

and ,Linear fractional functionals programming. Oper. Res. 13 (1965) 1029–1036. | DOI | Zbl

,Linearization technique for solving quadratic fractional set covering, partitioning and packing problems. Int. J. Eng. Soc. Sci. 2 (2012) 49–87.

and ,Enumeration technique for solving linear fractional fuzzy set covering problem. Int. J. Pure Appl. Math. 84 (2013) 477–496. | DOI

and ,Enumeration technique for solving linear fuzzy set covering problem. Int. J. Pure Appl. Math. 85 (2013) 635–651. | DOI

and ,Multi objective linear set covering problem with imprecise objective functions. Int. J. Res in IT, Management and Engineering 2 (2012) 28–42.

and ,Ranking in integer linear fractional programming problems. ZOR- Methods and Models of Operations Research. 34 (1990) 325–334. | DOI | MR | Zbl

, and ,*Cited by Sources: *