Mehar approach to solve neutrosophic linear programming problems using possibilistic mean

Khatter (Soft Comput 24:6847–16,867, 2020) pointed out that although several approaches are proposed in the literature to solve single-valued neutrosophic linear programming problems (SVNLPPS) (linear programming problems in which all the parameters except decision variables are either represented by single-valued triangular neutrosophic numbers (SVTNNS) or single-valued trapezoidal neutrosophic numbers (SVTrNNS)). However, all the methods for comparing single-valued neutrosophic numbers (SVNNS), used in existing approaches, are independent from the attitude of the decision-maker towards the risk. To fill this gap, Khatter (2020), firstly, proposed a method for comparing two SVNNS by considering the attitude of the decision-maker towards the risk. Then, using the proposed comparing method, Khatter (2020) proposed an approach to solve SVNLPPS. In this paper, it is pointed out that a mathematical incorrect result is considered in Khatter’s approach. Hence, it is inappropriate to use Khatter’s approach. Also, it is pointed out that some mathematical incorrect results are considered in other existing approaches for solving SVNLPPS. Hence, it is inappropriate to use other existing approaches for solving SVNLPPS. Furthermore, to resolve the inappropriateness of Khatter’s approach and other existing approaches, a new approach (named as Mehar approach) is proposed to solve SVNLPPS. Finally, correct optimal solution of some existing SVNLPPS is obtained by the proposed Mehar approach.


Introduction
In the last few years, several approaches are proposed in the literature to solve mathematical programming problems under neutrosophic environment (Smarandache 1998). In this section, some recently proposed approaches are discussed in a brief manner. Hussian et al. (2017) proposed an approach to solve single-valued triangular neutrosophic linear programming problems (SVTNLPPS). In Hussian et al.'s approach (2017), firstly, a single-valued triangular neutrosophic linear programming problem (SVTNLPP) is transformed into its equivalent crisp multi-objective linear programming problem (CrMOLPP). Then, the obtained CrMOLPP is transformed into its equivalent crisp linear programming problem (CrLPP). Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTNLPP. Hussian et al. (2018) proposed an approach to solve single-valued triangular neutrosophic linear fractional programming problems (SVTNLFPPS). In Hussian et al.'s approach (2018), firstly, a single-valued triangular neutrosophic linear fractional programming problem (SVTNLFPP) is transformed into its equivalent crisp multiobjective linear fractional programming problem (CrMOLFPP). Then, the obtained CrMOLFPP is transformed into its equivalent CrMOLPP. After that, the obtained CrMOLPP is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTNLFPP. Abdel-Basset et al. (2019a), firstly, proposed a method for comparing two SVTrNNS. Then, using the proposed comparing method, Abdel-Basset et al. (2019a) proposed an approach to solve single-valued trapezoidal neutrosophic linear programming problems (SVTrNLPPS). In Abdel-Basset et al.'s approach (2019a), firstly, a singlevalued trapezoidal neutrosophic linear programming problem (SVTrNLPP) is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTrNLPP. Singh et al. (2019) pointed out that some mathematical incorrect results are considered in Abdel-Basset et al. 's approach (2019a). Hence, it is inappropriate to use Abdel-Basset et al. 's approach (2019a) in its present form. Singh et al. (2019) also suggested some modifications to resolve the inappropriateness of Abdel-Basset et al. 's approach (2019a). Abdel-Basset et al. (2019b) proposed an approach to solve SVTNLFPPS. In Abdel-Basset et al.'s approach (2019b), firstly, a SVTNLFPP is transformed into its equivalent CrMOLFPP. Then, the obtained CrMOLFPP is transformed into its equivalent CrMOLPP. After that, the obtained CrMOLPP is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTNLFPP. Nafei and Nasseri (2019), firstly, proposed a method for comparing two SVTNNS. Then, using the proposed comparing method, Nafei and Nasseri (2019) proposed an approach to solve single-valued triangular neutrosophic integer programming problems (SVTNIPPS). In Nafei and Nasseri's approach (2019), firstly, a single-valued triangular neutrosophic integer programming problem (SVTNIPP) is transformed into its equivalent crisp integer programming problem (CrIPP). Finally, it is assumed that an optimal solution of the transformed CrIPP also represents an optimal solution of SVTNIPP.  pointed out that it is inappropriate to use Hussian et al.'s approach (2017) for solving SVTNLPPS.  also suggested to use Nafei and Nasseri's approach (2019) for solving SVTNLPPS.  pointed out that a mathematical incorrect result is considered in Nafei and Nasseri's approach (2019). Hence, it is inappropriate to use Nafei and Nasseri's approach (2019).  also proposed an approach to solve SVTNIPPS. In Das and Edalatpanah's approach (2020), firstly, a SVTNIPP is transformed into its equivalent CrIPP. Finally, it is assumed that an optimal solution of the transformed CrIPP also represents an optimal solution of SVTNIPP. Khatter (2020) pointed out that although several approaches are proposed in the literature to solve SVNLPPS. However, all the methods for comparing SVNNS, used in existing approaches, are independent from the attitude of the decision-maker towards the risk. To fill this gap, Khatter (2020), firstly, proposed a method for comparing two SVNNS by considering the attitude of the decision-maker towards the risk. Then, using the proposed comparing method, Khatter (2020) proposed an approach to solve SVNLPPS. In Khatter's approach (2020), a SVNLPP is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVNLPP. Badr et al. (2020), firstly, proposed a method for comparing two SVTrNNS. Then, using the proposed comparing method, Badr et al. (2020) generalized the crisp two-phase simplex algorithm for solving SVTrNLPPS.  proposed an approach to solve SVTNLFPPS. In this approach, firstly, a SVTNLFPP is split into its equivalent two neutrosophic linear programming problems. Then, the obtained neutrosophic linear programming problems are transformed into their equivalent crisp linear programming problems (CrLPPS). Finally, it is assumed that both optimal solutions of the transformed CrLPPS also represents an optimal solution of SVTNLFPP. Abdelfattah (2021) proposed an approach to solve SVTNLPPS. In Abdelfattah's approach (2021), firstly, a SVTNLPP is split into two CrLPPS. Then, the obtained CrLPPS are solved independently. Finally, it is assumed that both optimal solutions of the transformed CrLPPS also represents an optimal solution of SVTNLPP. Kar et al. (2021) proposed a simplex algorithm for solving SVTNLPPS, Badr et al. (2021) proposed a simplex algorithm for solving SVTrNLPPS and Rabie et al. (2021) proposed a two-phase simplex algorithm for solving SVTrNLPPS. Das et al. (2021) proposed an approach to solve SVTrNLPPS. In this approach, firstly, a SVTrNLPP is transformed into its equivalent CrMOLPP. Then, using a lexicographic approach, the transformed CrMOLPP is solved. Finally, it is assumed that an efficient solution of the transformed CrMOLPP also represents an optimal solution of SVTrNLPP. ElHadidi et al. (2021a), firstly, proposed a method for comparing two SVTrNNS. Then, using the proposed comparing method, ElHadidi et al. (2021a) proposed an approach to solve SVTrNLPPS. In ElHadidi et al.'s approach (2021a), firstly, a SVTrNLPP is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTrNLPP. ElHadidi et al. (2021b) proposed an approach to solve single-valued trapezoidal neutrosophic linear fractional programming problems (SVTrNLFPPS). In ElHadidi et al.'s approach (2021b), firstly, a single-valued trapezoidal neutrosophic linear fractional programming problem (SVTrNLFPP) is transformed into its equivalent CrMOLFPP. Then, the obtained CrMOLFPP is transformed into its equivalent CrMOLPP. After that, the obtained CrMOLPP is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTrNLFPP. Das and Edalatpanah (2022) proposed an approach to solve SVTNLFPPS. In Das and Edalatpanah's approach (2022), firstly, a SVTNLFPP is transformed into its equivalent crisp linear fractional programming problem. Then, the obtained crisp linear fractional programming problem is transformed into its equivalent CrLPP. Finally, it is assumed that an optimal solution of the transformed CrLPP also represents an optimal solution of SVTNLFPP.
In this paper, it is shown that some mathematical incorrect results are considered in all existing approaches for solving mathematical programming problems under neutrosophic environment. Hence, it is inappropriate to use existing approaches for solving mathematical programming problems under neutrosophic environment. Also, a new approach (named as Mehar approach) is proposed to solve SVNLPPS.
This paper is organized as follows. In Sect. 2, some basic concepts related to neutrosophic set theory are reviewed. In Sect. 3, it is pointed out that it is inappropriate to use existing approaches for solving mathematical programming problems under neutrosophic environment. In Sect. 4, a new approach (named as Mehar approach) is proposed to solve SVNLPPS. In Sect. 5, correct optimal solution of some existing SVNLPPS are obtained by the proposed Mehar approach. Section 6 concludes the paper.

Preliminaries
In this section, some basic definitions are reviewed.
(a) (b) k reflects the attitude of the decision-maker towards the risk. (c) k 2 0 ½ ; 0:5Þ indicates that the expert is risk taker and gives preference to uncertainty. (d) k ¼ 0:5 indicates that the expert is neutral about deciding the parameters of SVTNLPP problem. (e) k 2 0:5 ð ; 1 indicates that the expert is risk aversive about deciding the parameters of SVTNLPP problem and gives preference to certainty.

Inappropriateness of Singh et al.'s approach
In Singh et al.'s approach (2019), firstly, the SVTrNLPP (P 1 ) is transformed into the CrLPP (P 2 ). Then, the CrLPP (P 2 ) is transformed into the CrLPP (P 3 ). After that, the CrLPP (P 3 ) is transformed into the CrLPP (P 4 ). Finally, it is assumed that an optimal solution of the CrLPP (P 4 ) also represents an optimal solution of the SVTrNLPP (P 1 ).
(ii) n : number of variables.

Inappropriateness of Khatter's approach
In Khatter's approach (2019), firstly, the SVTrNLPP (P 1 ) is transformed into the CrLPP (P 5 ). Then, the CrLPP (P 5 ) is transformed into the CrLPP (P 6 ). Finally, it is assumed that an optimal solution of the CrLPP (P 6 ) also represent an optimal solution of the SVTrNLPP (P 1 ).

Inappropriateness of Abdelfattah's approach
Abdelfattah (2021) claimed that on solving the SVTNLPP (P 7 ), the results presented in Table 1  x 1 ; x 2 ! 0: It is pertinent to mention that as in the problem (P 7 ), x 1 and x 2 are considered as non-negative real numbers. So, the obtained optimal values of x 1 and x 2 should be same for all values of a; b; c. While, it is obvious from Table 1 that the values of x 1 and x 2 are different for different values of a; b;. This clearly indicates that x 1 and x 2 , obtained by Abdelfattah's approach (2021), are not non-negative real numbers. Hence, it is inappropriate to use Abdelfattah's approach (2021).

Inappropriateness of Das et al.'s approach
It is pertinent to mention that in one of the steps of Das et al.'s approach (2021), the scalar multiplication kÃ ¼ ka 1Ã ; ka 2Ã ; ka 3Ã ; ka 4Ã ; kwÃ; kuÃ; kyÃ ; k [ 0, is used to transform the SVTrNLPP (P 1 ) into the SVTrNLPP (P 8 ).

Inappropriateness of Kar et al.'s approach
It pertinent to mention that in one of the steps of Kar  is not satisfying.
Remark 1: It can be easily verified that the shortcoming, pointed out by Singh et al. (2019) in Abdel-Basset et al.'s approach (2019a), also occurs in the existing approaches (Emam et al. 2020;Lachhwani 2021). Hence, it is inappropriate to use the existing approaches (Emam et al. 2020;Lachhwani 2021).

Proposed Mehar approach
In this section, a new approach (named as Mehar approach) is proposed to solve the SVTrNLPP (P 1 ). The proposed Mehar approach can also be used to solve SVTNLPPS.

Correct optimal solution of some existing SVNLPPS
In this section, the correct optimal solution of some existing SVNLPPS is obtained by the proposed Mehar approach.
A ; k 2 0; 1 ½ :  5.1 Correct optimal solution of some existing SVTNLPPS Hussian et al. (2017) as well as Khatter (2020) have considered the following real-life problem to illustrate their proposed approach. A Pottery Company, run by a Native American tribal council, desires to find the number of bowls and mugs to be produced each day in order to maximize the profit by considering.
(i) The data presented in Table 2.
(ii) The data presented in Table 3. (iii) The data presented in Table 4.
However, as some mathematical incorrect results are considered in Hussian et al.'s approach (2017) as well as in Khatter's approach (2020), the existing optimal solution (Hussian et al. 2017;Khatter 2020) is not correct. In this section, a correct optimal solution of this real-life problem is obtained by the proposed Mehar approach.

Second illustrative example
If the data, presented in Table 3, are considered, then to find an optimal solution of the real-life problem is equivalent to find an optimal solution of the SVTNLPP (P 16 ). x 1 ; x 2 ! 0: Using the proposed Mehar approach, an optimal solution of the SVTNLPP (P 16 ) can be obtained as follows: Step 1: Using Step 1 of the proposed Mehar approach, the SVTNLPP (P 16 ) can be transformed into its equivalent SVTNLPP (P 17  x 1 ; x 2 ! 0: Step 2: Using Step 2 of the proposed Mehar approach, the SVTNLPP (P 17 ) can be transformed into its equivalent SVTNLPP (P 18 ).

Third illustrative example
If the data, presented in Table 4, are considered, then to find an optimal solution of the real-life problem is equivalent to find an optimal solution of the SVTNLPP (P 20  21x 1 þ 14x 2 28000; x 1 ; x 2 ! 0: Using the proposed Mehar approach, an optimal solution of the SVTNLPP (P 20 ) can be obtained as follows: Step 1: Using Step 1 of the proposed Mehar approach, the SVTNLPP (P 20 ) can be transformed into its equivalent SVTNLPP (P 21 ).
Step 2: Using Step 2 of the proposed Mehar approach, the SVTNLPP (P 21 ) can be transformed into its equivalent SVTNLPP (P 22 ).
Step 3: Using Step 3 of the proposed Mehar approach, the SVTNLPP (P 22 ) can be transformed into its equivalent CrLPP (P 23 ).

Subject to
Constraints of the problem (P 20 ) where Step 4: The obtained optimal solution of the CrLPP (P 23 ) for some values of k 2 ½0; 1 are shown in Table 7. It is pertinent to mention that according to Step 4 of the proposed Mehar approach, the obtained optimal solution also represents an optimal solution of the SVTNLPP (P 20 ). Das et al. (2021) have considered the following real-life problem to illustrate their proposed approach. An electric cable maker desires to find the number of cable 1 and cable 2 to be produced each day in order to maximize the profit by considering the data presented in Table 8.

Correct optimal solution of an existing SVTrNLPP
However, as some mathematical incorrect results are considered in Das et al.'s approach (2021). So, the existing optimal solution  is not correct. In this section, a correct optimal solution of this real-life problem is obtained by the proposed Mehar approach.
If the data, presented in Table 8, are considered, then to find an optimal solution of the real-life problem is equivalent to find an optimal solution of the SVTrNLPP (P 24 ).

Conclusions and future work
It is shown that some mathematical incorrect results are considered in all existing approaches for solving mathematical programming problems under neutrosophic environment. Hence, it is inappropriate to use any existing approach to solve mathematical programming problems under neutrosophic environment. Also, a new approach (named as Mehar approach) is proposed to solve SVNLPPS. Furthermore, correct optimal solutions of some existing real-life problems under neutrosophic environment (Hussian et al. 2017;Khatter 2020;Das et al. 2021) are obtained by the proposed Mehar approach.
The following work may be considered as a future work.