American Institute of Mathematical Sciences

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January  2022, 18(1): 439-456. doi: 10.3934/jimo.2020162

Solving fuzzy linear fractional set covering problem by a goal programming based solution approach

Ali Mahmoodirad 1, , Harish Garg 2,, and  Sadegh Niroomand 3,

1. 

Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran
2. 

School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala-147004, Punjab, India
3. 

Department of Industrial Engineering, Firouzabad Institute of Higher Education, Firouzabad, Fars, Iran

*Corresponding author: Harish Garg

Received  May 2020 Revised  August 2020 Published  January 2022 Early access  December 2020

In this paper, a fuzzy linear fractional set covering problem is solved. The non-linearity of the objective function of the problem as well as its fuzziness make it difficult and complex to be solved effectively. To overcome these difficulties, using the concepts of fuzzy theory and component-wise optimization, the problem is converted to a crisp multi-objective non-linear problem. In order to tackle the obtained multi-objective non-linear problem, a goal programming based solution approach is proposed for its Pareto-optimal solution. The non-linearity of the problem is linearized by applying some linearization techniques in the procedure of the goal programming approach. The obtained Pareto-optimal solution is also a solution of the initial fuzzy linear fractional set covering problem. As advantage, the proposed approach applies no ranking function of fuzzy numbers and its goal programming stage considers no preferences from decision maker. The computational experiments provided by some examples of the literature show the superiority of the proposed approach over the existing approaches of the literature.

Keywords: Set covering problem, Fractional objective function, Fuzzy theory, Goal programming.
Mathematics Subject Classification: Primary: 68T35; 90B50; 62A86; 03E72.
Citation: Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 439-456. doi: 10.3934/jimo.2020162
References:
[1]

M. Akram, A. Bashir and H. Garg, Decision-making model under complex picture fuzzy Hamacher aggregation operators, Computational and Applied Mathematics, 39 (2020), Paper No. 226, 38 pp. doi: 10.1007/s40314-020-01251-2.

[2]

M. Akram, G. Muhammad and T. Allahviranloo, Bipolar fuzzy linear system of equations,, Computational and Applied Mathematics, 38 (2019a), Paper No. 69, 29 pp. doi: 10.1007/s40314-019-0814-8.

[3]

M. Akram, D. Saleem and T. Allahviranloo, Linear system of equations in m-polar fuzzy environment, Journal of Intelligent & Fuzzy Systems, 37 (2019), 8251-8266.  doi: 10.3233/JIFS-190744.

[4]

M. Arana-Jimz, Nondominated solutions in a fully fuzzy linear programming problem, Mathematical Methods in the Applied Sciences, 41 (2018), 7421-7430.  doi: 10.1002/mma.4882.

[5]

E. Cakita and W. Karwowskib, A fuzzy overlay model for mapping adverse event risk in an active war theatre, Journal of Experimental & Theoretical Artificial Intelligence, 30 (2018), 691-701.  doi: 10.1080/0952813X.2018.1467494.

[6]

A. Charnes and W. W. Cooper, Programming with Linear Fractional Functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.

[7]

T. P. Dao and S. C. Huang, Design and multi-objective optimization for a broad self-amplified 2-DOF monolithic mechanism, Sadhana, 42 (2017), 1527-1542.  doi: 10.1007/s12046-017-0714-9.

[8]

W. A. De Oliveira, M. A. Rojas-Medar, A. Beato-Moreno and M. B. Hernez-Jimz, Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems, Journal of Global Optimization, 74 (2019), 233-253.  doi: 10.1007/s10898-019-00766-1.

[9]

J. Frenk and S. Schaible, Fractional Programming, Handbook of Generalized Convexity and Generalized Monotonicity, In: N. Hadjisavvas, S. Komlosi, S. Schaible, editors, Nonconvex Optimization and its Applications, , Springer-Verlag, Berlin, 2005.

[10]

H. Garg, Exponential operational laws and new aggregation operators for intuitionistic multiplicative set in multiple-attribute group decision making process, Information Sciences, 538 (2020), 245-272.  doi: 10.1016/j.ins.2020.05.095.

[11]

H. Garg, Linguistic interval-valued Pythagorean fuzzy sets and their application to multiple attribute group decision-making process, Cognitive Computation, 12 (2020), 1313-1337.  doi: 10.1007/s12559-020-09750-4.

[12]

H. Garg, Neutrality operations-based Pythagorean fuzzy aggregation operators and its applications to multiple attribute group decision-making process, Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 3021-3041.  doi: 10.1007/s12652-019-01448-2.

[13]

P. Gupta and M. K. Mehlawat, A new possibilistic programming approach for solving fuzzy multiobjective assignment problem, IEEE Transactions on Fuzzy Systems, 22 (2014), 16-34.  doi: 10.1109/TFUZZ.2013.2245134.

[14]

R. Gupta and R. R. Saxena, Fuzzy linear fractional set covering problem with imprecise costs, Rairo Operations Research, 48 (2014), 415-427.  doi: 10.1051/ro/2014015.

[15]

J. Li and R. S. K. Kwan, A meta-heuristic with orthogonal experiment for the set covering problem, Journal of Mathematical Modelling and Algorithms, 3 (2004), 263-283.  doi: 10.1023/B:JMMA.0000038619.69509.bf.

[16]

A. Mahmoodirad, T. Allahviranloo and S. Niroomand, A new effective solution method for fully intuitionistic fuzzy transportation problem, Soft Computing, 23 (2019), 4521-4530.  doi: 10.1007/s00500-018-3115-z.

[17]

A. Mahmoodirad and S. Niroomand, Uncertain location-allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts, Expert Systems, 37 (2020), e12558. doi: 10.1111/exsy.12558.

[18]

A. Mahmoodirad, S. Niroomand, N. Mirzaei and A. Shoja, Fuzzy fractional minimal cost flow problem, International Journal of Fuzzy Systems, 20 (2018), 174-186.  doi: 10.1007/s40815-017-0293-2.

[19]

A. Mahmoodirad, S. Niroomand and M. Shafiee, A closed loop supply chain network design problem with multi-mode demand satisfaction in fuzzy environment, Journal of Intelligent & Fuzzy Systems, 39 (2020), 503-524.  doi: 10.3233/JIFS-191528.

[20]

M. Moula and A. Mekhilef, Quadratic optimization over a discrete pareto set of a multi-objective linear fractional program, Optimization, 2020. doi: 10.1080/02331934.2020.1730834.

[21]

S. Niroomand, A. Bazyar, M. Alborzi and A. Mahmoodirad, A hybrid approach for multi-criteria emergency center location problem considering existing emergency centers with interval type data: A case study, Journal of Ambient Intelligence and Humanized Computing, 9 (2018), 1999-2008. 

[22]

S. Niroomand, H. Garg and A. Mahmoodirad, An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA Transactions, 2020. doi: 10.1016/j.isatra.2020.07.033.

[23]

S. Niroomand, A. Hadi-Vencheh, N. Mirzaei and S. Molla-Alizadeh-Zavardehi, Hybrid greedy algorithms for fuzzy tardiness/earliness minimisation in a special single machine scheduling problem: case study and generalisation, International Journal of Computer Integrated Manufacturing, 29 (2016), 870-888. 

[24]

S. Niroomand, A. Mahmoodirad and S. Mosallaeipour, A hybrid solution approach for fuzzy multiobjective dual supplier and material selection problem of carton box production systems, Expert Systems, 36 (2019), e12341.

[25]

D. Rani, T. R. Gulati and H. Garg, Multi-objective non - linear programming problem in intuitionistic fuzzy environment: Optimistic and pessimistic view point, Expert Systems with Applications, 64 (2016), 228-238.  doi: 10.1016/j.eswa.2016.07.034.

[26]

R. Sahraeian and M. S. Kazemi, A fuzzy set covering-clustering algorithm for facility location problem, IEEE International Conference on Industrial Engineering Management, (2011), 1098–1102.

[27]

M. Sanei, A. Mahmoodirad and S. Niroomand, Two-stage supply chain network design problem with interval data, International Journal of e-Navigation and Maritime Economy, 5 (2016), 74-84.  doi: 10.1016/j.enavi.2016.12.006.

[28]

R. R. Saxena and S. R. Arora, A linearization technique for solving the quadratic set covering problem, Optimization, 39 (1997), 35-42.  doi: 10.1080/02331939708844269.

[29]

R. R. Saxena and R. Gupta, Enumeration technique for solving linear fractional fuzzy set covering problem, International Journal of Pure and Applied Mathematics, 84 (2013), 477-496.  doi: 10.12732/ijpam.v84i5.3.

[30]

S. Schaible and J. Shi, Fractional programming: The sum-of-ratios case, Optimization Methods and Software, 18 (2003), 219-229.  doi: 10.1080/1055678031000105242.

[31]

S. Schaible, A note on the sum of a linear and linear-fractional function, Naval Research Logistics Quarterly, 24 (1977), 691-693. 

[32]

H. Shavandi and H. Mahlooji, Fuzzy hierarchical queueing models for the location set covering problem in congested systems, Scientia Iranica, 15 (2008), 378-388. 

[33]

R. H. Walia, U. Mishra, H. Garg and H. P. Umap, A nonlinear programming approach to solve the stochastic multi-objective inventory model using the uncertain information, Arabian Journal for Science and Engineering, 45 (2020), 6963-6973. 

[34]

Z. Yang, H. Garg, J. Li, G. Srivastava and Z. Cao, Investigation of multiple heterogeneous relationships using a Q-order neighbor pair fuzzy multi-criteria decision algorithm, Neural Computing and Applications, (2020). doi: 10.1007/s00521-020-05003-5.

[35]

D. Yousri M. Abd Elaziz S. Mirjalili, Fractional-order calculus-based flower pollination algorithm with local search for global optimization and image segmentation, Knowledge-Based Systems, 1975 (2020), 105889. doi: 10.1016/j.knosys.2020.105889.

[36]

K. Zimmermann, Fuzzy set covering problem, International Journal of General Systems, 20 (1991), 127-131.  doi: 10.1080/03081079108945020.

show all references

References:
[1]

M. Akram, A. Bashir and H. Garg, Decision-making model under complex picture fuzzy Hamacher aggregation operators, Computational and Applied Mathematics, 39 (2020), Paper No. 226, 38 pp. doi: 10.1007/s40314-020-01251-2.

[2]

M. Akram, G. Muhammad and T. Allahviranloo, Bipolar fuzzy linear system of equations,, Computational and Applied Mathematics, 38 (2019a), Paper No. 69, 29 pp. doi: 10.1007/s40314-019-0814-8.

[3]

M. Akram, D. Saleem and T. Allahviranloo, Linear system of equations in m-polar fuzzy environment, Journal of Intelligent & Fuzzy Systems, 37 (2019), 8251-8266.  doi: 10.3233/JIFS-190744.

[4]

M. Arana-Jimz, Nondominated solutions in a fully fuzzy linear programming problem, Mathematical Methods in the Applied Sciences, 41 (2018), 7421-7430.  doi: 10.1002/mma.4882.

[5]

E. Cakita and W. Karwowskib, A fuzzy overlay model for mapping adverse event risk in an active war theatre, Journal of Experimental & Theoretical Artificial Intelligence, 30 (2018), 691-701.  doi: 10.1080/0952813X.2018.1467494.

[6]

A. Charnes and W. W. Cooper, Programming with Linear Fractional Functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.

[7]

T. P. Dao and S. C. Huang, Design and multi-objective optimization for a broad self-amplified 2-DOF monolithic mechanism, Sadhana, 42 (2017), 1527-1542.  doi: 10.1007/s12046-017-0714-9.

[8]

W. A. De Oliveira, M. A. Rojas-Medar, A. Beato-Moreno and M. B. Hernez-Jimz, Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems, Journal of Global Optimization, 74 (2019), 233-253.  doi: 10.1007/s10898-019-00766-1.

[9]

J. Frenk and S. Schaible, Fractional Programming, Handbook of Generalized Convexity and Generalized Monotonicity, In: N. Hadjisavvas, S. Komlosi, S. Schaible, editors, Nonconvex Optimization and its Applications, , Springer-Verlag, Berlin, 2005.

[10]

H. Garg, Exponential operational laws and new aggregation operators for intuitionistic multiplicative set in multiple-attribute group decision making process, Information Sciences, 538 (2020), 245-272.  doi: 10.1016/j.ins.2020.05.095.

[11]

H. Garg, Linguistic interval-valued Pythagorean fuzzy sets and their application to multiple attribute group decision-making process, Cognitive Computation, 12 (2020), 1313-1337.  doi: 10.1007/s12559-020-09750-4.

[12]

H. Garg, Neutrality operations-based Pythagorean fuzzy aggregation operators and its applications to multiple attribute group decision-making process, Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 3021-3041.  doi: 10.1007/s12652-019-01448-2.

[13]

P. Gupta and M. K. Mehlawat, A new possibilistic programming approach for solving fuzzy multiobjective assignment problem, IEEE Transactions on Fuzzy Systems, 22 (2014), 16-34.  doi: 10.1109/TFUZZ.2013.2245134.

[14]

R. Gupta and R. R. Saxena, Fuzzy linear fractional set covering problem with imprecise costs, Rairo Operations Research, 48 (2014), 415-427.  doi: 10.1051/ro/2014015.

[15]

J. Li and R. S. K. Kwan, A meta-heuristic with orthogonal experiment for the set covering problem, Journal of Mathematical Modelling and Algorithms, 3 (2004), 263-283.  doi: 10.1023/B:JMMA.0000038619.69509.bf.

[16]

A. Mahmoodirad, T. Allahviranloo and S. Niroomand, A new effective solution method for fully intuitionistic fuzzy transportation problem, Soft Computing, 23 (2019), 4521-4530.  doi: 10.1007/s00500-018-3115-z.

[17]

A. Mahmoodirad and S. Niroomand, Uncertain location-allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts, Expert Systems, 37 (2020), e12558. doi: 10.1111/exsy.12558.

[18]

A. Mahmoodirad, S. Niroomand, N. Mirzaei and A. Shoja, Fuzzy fractional minimal cost flow problem, International Journal of Fuzzy Systems, 20 (2018), 174-186.  doi: 10.1007/s40815-017-0293-2.

[19]

A. Mahmoodirad, S. Niroomand and M. Shafiee, A closed loop supply chain network design problem with multi-mode demand satisfaction in fuzzy environment, Journal of Intelligent & Fuzzy Systems, 39 (2020), 503-524.  doi: 10.3233/JIFS-191528.

[20]

M. Moula and A. Mekhilef, Quadratic optimization over a discrete pareto set of a multi-objective linear fractional program, Optimization, 2020. doi: 10.1080/02331934.2020.1730834.

[21]

S. Niroomand, A. Bazyar, M. Alborzi and A. Mahmoodirad, A hybrid approach for multi-criteria emergency center location problem considering existing emergency centers with interval type data: A case study, Journal of Ambient Intelligence and Humanized Computing, 9 (2018), 1999-2008. 

[22]

S. Niroomand, H. Garg and A. Mahmoodirad, An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA Transactions, 2020. doi: 10.1016/j.isatra.2020.07.033.

[23]

S. Niroomand, A. Hadi-Vencheh, N. Mirzaei and S. Molla-Alizadeh-Zavardehi, Hybrid greedy algorithms for fuzzy tardiness/earliness minimisation in a special single machine scheduling problem: case study and generalisation, International Journal of Computer Integrated Manufacturing, 29 (2016), 870-888. 

[24]

S. Niroomand, A. Mahmoodirad and S. Mosallaeipour, A hybrid solution approach for fuzzy multiobjective dual supplier and material selection problem of carton box production systems, Expert Systems, 36 (2019), e12341.

[25]

D. Rani, T. R. Gulati and H. Garg, Multi-objective non - linear programming problem in intuitionistic fuzzy environment: Optimistic and pessimistic view point, Expert Systems with Applications, 64 (2016), 228-238.  doi: 10.1016/j.eswa.2016.07.034.

[26]

R. Sahraeian and M. S. Kazemi, A fuzzy set covering-clustering algorithm for facility location problem, IEEE International Conference on Industrial Engineering Management, (2011), 1098–1102.

[27]

M. Sanei, A. Mahmoodirad and S. Niroomand, Two-stage supply chain network design problem with interval data, International Journal of e-Navigation and Maritime Economy, 5 (2016), 74-84.  doi: 10.1016/j.enavi.2016.12.006.

[28]

R. R. Saxena and S. R. Arora, A linearization technique for solving the quadratic set covering problem, Optimization, 39 (1997), 35-42.  doi: 10.1080/02331939708844269.

[29]

R. R. Saxena and R. Gupta, Enumeration technique for solving linear fractional fuzzy set covering problem, International Journal of Pure and Applied Mathematics, 84 (2013), 477-496.  doi: 10.12732/ijpam.v84i5.3.

[30]

S. Schaible and J. Shi, Fractional programming: The sum-of-ratios case, Optimization Methods and Software, 18 (2003), 219-229.  doi: 10.1080/1055678031000105242.

[31]

S. Schaible, A note on the sum of a linear and linear-fractional function, Naval Research Logistics Quarterly, 24 (1977), 691-693. 

[32]

H. Shavandi and H. Mahlooji, Fuzzy hierarchical queueing models for the location set covering problem in congested systems, Scientia Iranica, 15 (2008), 378-388. 

[33]

R. H. Walia, U. Mishra, H. Garg and H. P. Umap, A nonlinear programming approach to solve the stochastic multi-objective inventory model using the uncertain information, Arabian Journal for Science and Engineering, 45 (2020), 6963-6973. 

[34]

Z. Yang, H. Garg, J. Li, G. Srivastava and Z. Cao, Investigation of multiple heterogeneous relationships using a Q-order neighbor pair fuzzy multi-criteria decision algorithm, Neural Computing and Applications, (2020). doi: 10.1007/s00521-020-05003-5.

[35]

D. Yousri M. Abd Elaziz S. Mirjalili, Fractional-order calculus-based flower pollination algorithm with local search for global optimization and image segmentation, Knowledge-Based Systems, 1975 (2020), 105889. doi: 10.1016/j.knosys.2020.105889.

[36]

K. Zimmermann, Fuzzy set covering problem, International Journal of General Systems, 20 (1991), 127-131.  doi: 10.1080/03081079108945020.

Figure 1.  The fuzzy objective function obtained by the proposed approach and the approach of Gupta and Saxena [14] for Example 1
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Figure 2.  The fuzzy objective function obtained by the proposed approach and the approach of Gupta and Saxena [14] for Example 2
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Table 1.  The goals obtained for Example 1 from Step 3 of the proposed approach
Objective
$ X_1 $ $ X_2 $ $ X_3 $ function value
Model (30) $ 1 $ $ 1 $ $ 0 $ $ Z^{1*}=0.622 $
Model (31) $ 1 $ $ 1 $ $ 0 $ $ Z^{2*}=0.966 $
Model (32) $ 1 $ $ 1 $ $ 0 $ $ Z^{3*}=1.034 $
Model (33) $ 1 $ $ 1 $ $ 0 $ $ Z^{4*}=2.875 $
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Objective
$ X_1 $ $ X_2 $ $ X_3 $ function value
Model (30) $ 1 $ $ 1 $ $ 0 $ $ Z^{1*}=0.622 $
Model (31) $ 1 $ $ 1 $ $ 0 $ $ Z^{2*}=0.966 $
Model (32) $ 1 $ $ 1 $ $ 0 $ $ Z^{3*}=1.034 $
Model (33) $ 1 $ $ 1 $ $ 0 $ $ Z^{4*}=2.875 $
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