Approaches to solve linear programming problems with imprecise parameters

Abstract

In this thesis, it is pointed out that (i) The existing results (Marimuthu and Mahapatra, 2021; Jeevaraj, 2022) are not correct. Also, it is pointed out that the existing RM (Marimuthu and Mahapatra, 2021) as well as the existing RM (Jeevaraj, 2022), fails to distinguish two distinct GTrFNs. Hence, the existing RMs (Marimuthu and Mahapatra, 2021; Jeevaraj, 2022) are not appropriate. Furthermore, the correct results, corresponding to the existing results (Marimuthu and Mahapatra, 2021; Jeevaraj, 2022) are stated and proved. Finally, it is shown that the existing RMs (Marimuthu and Mahapatra, 2021; Jeevaraj, 2022) will never fail to distinguish two distinct GTrFNs having same heights. However, both the existing RMs (Marimuthu and Mahapatra, 2021; Jeevaraj, 2022) may fail to distinguish two distinct GTrFNs having different heights. Hence, these existing ranking methods can be used only to solve such LPPs with GTrFCs in which height of all the GTrFNs is same. (ii) The existing methods (Jeevaraj 2021; Bihari et al., 2025) to solve IVFFMCDMPs with known attribute weights and to solve IVFFLPP is not valid. Therefore, to propose an IVFFLPP based method to solve IVFFMCDMPs with unknown attribute weights, firstly, there is a need to resolve the drawbacks of (i) the existing methods to solve IVFFMCDMPs with known attribute weights (ii) the existing method to solve IVFFLPP. Also, modified methods to solve IVFFMCDMPs with known attribute weights are proposed to resolve drawbacks of existing methods to solve IVFFMCDMPs with known attribute weights. Furthermore, it is pointed out that due to some challenges, it is not possible to resolve the drawbacks of existing method to solve IVFFLPP. Hence, it is not possible to propose an IVFFLPP based method to solve IVFFMCDMPs with unknown attribute weights.

(iii) The existing approach (Saghi et al., 2023) fails to find the correct TrHFN (representing the OV of FT HFLPP, LPP in which each element of the ObF is represented by a TrHFN and each of the remaining parameters is represented by a ReN). Also, the reasons for the failure of the existing approach (Saghi et al., 2023) are discussed. Furthermore, to overcome this limitation, a new approach (named as Mehar approach) is proposed to solve FT HFLPP. Finally, the correct TrHFN, representing the OV of the considered FT HFLPP, is obtained by the proposed Mehar approach.

(iv) Much computational efforts are required to solve ST HFLPPs (LPPs in which each decision variable as well as each element of resource vector is represented by a TrHFN and each of the remaining parameters is represented by a ReN) by the existing approach (Saghi et al., 2024). Also, to reduce the computational efforts, an alternative approach is proposed to solve ST HFLPPs. Furthermore, some other advantages of the PrAlApp over Saghi et al.’s approach are discussed. Finally, a ST HFLPP, considered by Saghi et al. to illustrate their proposed approach, is solved by the PrAlApp. (v) The existing approach (Ranjbar et al., 2020) fails to find correct OS of TT HFLPPs (LPPs in which each parameter except decision variable is represented by a TrHFN). Hence, it is inappropriate to use the existing approach (Ranjbar et al., 2020). Also, the reason for this inappropriateness is pointed out. Furthermore, to resolve the inappropriateness of the existing approach (Ranjbar et al., 2020), a modified approach is proposed to solve TT HFLPPs. Finally, the modified approach is illustrated with the help of a numerical example.

(vi) Tamilarasi and Paulraj (2022) have used incorrect definition of a single-valued TNeN to propose their RFn. Therefore, Tamilarasi and Paulraj (2022)’s RFn is not valid and hence, it is inappropriate to use Tamilarasi and Paulraj (2022)’s method for solving LPPs with NCs and CrDVrs. Also, it is shown that if in the method, used by Tamilarasi and Paulraj (2022) to obtain their proposed RFn, the correct definition (Seikh and Dutta, 2022) of a single-valued TNeN is considered then Tamilarasi and Paulraj (2022)’s method fails to find a RFn. Hence, it is not possible to resolve the inappropriateness of Tamilarasi and Paulraj (2022)’s RFn. (vii) Hemalatha and Venkateswarlu (2023)’s ranking approach fails to distinguish two distinct PnFNs. Therefore, it is inappropriate to use Hemalatha and Venkateswarlu (2023)’s ranking approach.