A novel method for solving the fully neutrosophic linear programming problems: Suggested modifications

Abstract

Abdel-Basset et al. (Neural Computing and Applications, 2018, https://doi.org/10.1007/s00521-018-3404-6) proposed methods for solving different types of neutrosophic linear programming problems (NLPPs) (NLPPs in which some/all the parameters are represented as trapezoidal neutrosophic numbers (TrNNs)). Abdel-Basset et al. also pointed out that as a trapezoidal fuzzy number is a special case of trapezoidal neutrosophic number. Therefore, the fuzzy linear programming problems which can be solved by the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33) can also be solved by thier proposed method. In addition to that, to show the advantages of their proposed method over the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33), Abdel-Basset et al. solved the same fuzzy linear programming problems by their proposed method as well as the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33) and shown that the results, obtained on applying by their proposed method are better than the results, obtained on applying the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33). After a deep study of Abdel-Basset et al. ’s method, it is observed that Abdel-Basset et al. have considered several mathematical incorrect assumptions in their proposed method and hence, it is scientifically incorrect to use Abdel-Basset et al. ’s method in its present form. The aim of this paper is to make the researchers aware about the mathematical incorrect assumptions, considered by Abdel-Basset et al. in their proposed method, as well as to suggest the required modifications in Abdel-Basset et al. ’s method.

Keywords

Trapezoidal neutrosophic number (TrNNs)linear programming neutrosophic set ranking function

1 Introduction

Abdel-Basset et al. [1] claimed that although several methods have been proposed in the literature to find the solution of different types of fuzzy linear programming problems (LPPs) (LPPs in which some/all the parameters are represented as fuzzy numbers) and different types of intuitionistic fuzzy LPPs (LPPs in which some/all the parameters are represented as intuitionistic fuzzy numbers) [2 –22]. However, there is no method in the literature for solving such NLPPs in which some/all the parameters are represented as TrNNs.

To fill this gap, Abdel-Basset et al. [1] proposed a method for solving same type of NLPPs. In Abdel-Basset et al. ’s method, firstly, a neutrosophic linear programming problem (NLPP) is transformed into a crisp linear programming problem (LPP) by replacing each parameter of the NLPP, represented by a trapezoidal neutrosophic number (TrNN), with its equivalent defuzzified crisp value and then the optimal solution of the transformed crisp LPP is used to find the optimal solution and optimal value of the considered NLPP.

Abdel-Basset et al. [1] also pointed out that as a trapezoidal fuzzy number is a special case of trapezoidal neutrosophic number. Therefore, the fuzzy linear programming problems which can be solved by the existing methods [10 , 21] can also be solved by their proposed method. In addition to that, to show the advantages of their proposed method over the existing methods [10 , 21], Abdel-Basset et al. [1] solved the same fuzzy linear programming problems by their proposed method as well as the existing methods [10 , 21] and shown that the results, obtained on applying by their proposed method are better than the results, obtained on applying the existing methods [10 , 21].

In this paper, it is shown that for the ranking function, used by Abdel-Basset et al. [1], to transform a TrNN into its equivalent crisp value, the linearity property is not satisfying. Whereas, Abdel-Basset et al. [1] have used the linearity property in their proposed method, to transform a NLPP into its equivalent crisp LPP. Therefore, Abdel-Basset et al. ’s method [1] is not valid in its present form. Furthermore, the required modification in Abdel-Basset et al. ’s method [1] are suggested.

2 Method for comparing two TrNNs

Abdel-Basset et al. [1] have used the following method for comparing two TrNNs ${\tilde{A}}_{1} = 〈 a_{1}^{i}, a_{1}^{m_{1}}, a_{1}^{m_{2}}, a_{1}^{u}; T_{{\tilde{a}}_{1}}, I_{{\tilde{a}}_{1}}, F_{{\tilde{a}}_{1}} 〉$ , and ${\tilde{A}}_{2} = 〈 a_{2}^{i}, a_{2}^{m_{1}}, a_{2}^{m_{2}}, a_{2}^{u}; T_{{\tilde{a}}_{2}}, I_{{\tilde{a}}_{2}}, F_{{\tilde{a}}_{2}} 〉$ .

Step 1. Check that the considered NLPP is a maximization problem or a minimization problem.

Case (i) If the considered NLPP is a maximization problem, then

${\tilde{A}}_{1} > {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) > R ({\tilde{A}}_{2})$

${\tilde{A}}_{1} < {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) < R ({\tilde{A}}_{2})$

${\tilde{A}}_{1} = {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) = R ({\tilde{A}}_{2})$

where,

\begin{matrix} R ({\tilde{A}}_{1}) & = (\frac{a_{1}^{l} + a_{1}^{u} + 2 (a_{1}^{m_{1}} + a_{1}^{m_{2}})}{2}) \\ + (T_{{\tilde{a}}_{1}} - I_{{\tilde{a}}_{1}} - F_{{\tilde{a}}_{1}}) \end{matrix}

and

\begin{matrix} R ({\tilde{A}}_{2}) & = (\frac{a_{2}^{l} + a_{2}^{u} + 2 (a_{2}^{m_{1}} + a_{2}^{m_{2}})}{2}) \\ + (T_{{\tilde{a}}_{2}} - I_{{\tilde{a}}_{2}} - F_{{\tilde{a}}_{2}}) . \end{matrix}

Case (ii) If the considered NLPP is a minimization problem then

${\tilde{A}}_{1} > {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) < R ({\tilde{A}}_{2})$

${\tilde{A}}_{1} < {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) > R ({\tilde{A}}_{2})$

${\tilde{A}}_{1} = {\tilde{A}}_{2}$ if $R ({\tilde{A}}_{1}) = R ({\tilde{A}}_{2})$

where,

\begin{matrix} R ({\tilde{A}}_{1}) & = (\frac{a_{1}^{l} + a_{1}^{u} - 3 (a_{1}^{m_{1}} + a_{1}^{m_{2}})}{2}) \\ + (T_{{\tilde{a}}_{1}} - I_{{\tilde{a}}_{1}} - F_{{\tilde{a}}_{1}}) \end{matrix}

and

\begin{matrix} R ({\tilde{A}}_{2}) & = (\frac{a_{2}^{l} + a_{2}^{u} - 3 (a_{2}^{m_{1}} + a_{2}^{m_{2}})}{2}) \\ + (T_{{\tilde{a}}_{2}} - I_{{\tilde{a}}_{2}} - F_{{\tilde{a}}_{2}}) . \end{matrix}

3 Abdel-Basset et al. ’s method

In this section, the Abdel-Basset et al. ’s method [1], for solving different type of NLPPs, is discussed.

3.1 NLLP of first type

Abdel-Basset et al. [1], proposed the following method for solving NLPP (P₁)

Maximize/Minimize $[\sum_{j}^{n} = {\tilde{c}}_{j} x_{j}]$

Subject to $\sum_{j = 1}^{n} a_{ij} x_{j} \leq, =, \geq b_{j}, i = 1, 2, . . ., m; (P_{1})$ $x_{j} \geq 0, j = 1, 2, . . ., n .$ where ${\tilde{c}}_{j} = 〈 c_{j}^{l}, c_{j}^{m_{1}}, c_{j}^{m_{2}}, c_{j}^{u}, T_{e_{j}}, I_{e_{j}}, F_{e_{j}} 〉$ is a TrNN.

Step 1. Transform the NLPP (P₁) into its equivalent crisp LPP (P₂).

Maximize/Minimize $[\sum_{j = 1}^{n} R ({\tilde{c}}_{j}) x_{j}]$

Subject to (P₂)

Constraints of the NLPP (P₁).

Step 2. Find the optimal solution {x_j} of the crisp LPP (P₂).

Step 3. Using the optimal solution {x_j}, obtained in Step 2, and using the relation $\begin{matrix} \sum_{j = 1}^{n} 〈 c_{j}^{l}, c_{j}^{m_{1}}, c_{j}^{m_{2}}, c_{j}^{u}, T_{{\tilde{c}}_{j}}, I_{{\tilde{c}}_{j}}, F_{{\tilde{c}}_{j}} 〉 \times x_{j} \\ = 〈 (\sum_{j = 1}^{n} c_{j}^{l}) \times x_{j}, (\sum_{j = 1}^{n} c_{j}^{m_{1}}) \times x_{j} \\ (\sum_{j = 1}^{n} c_{j}^{m_{2}}) \times x_{j} (\sum_{j = 1}^{n} c_{j}^{u}) \times x_{j}; \\ min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}}, max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}}, max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}} 〉, \end{matrix}$ find the optimal value $\sum_{j = 1}^{n} 〈 c_{j}^{l}, c_{j}^{m_{1}}, c_{j}^{m_{2}}, c_{j}^{u}, T_{{\tilde{c}}_{j}}, I_{{\tilde{c}}_{j}}, F_{{\tilde{c}}_{j}} 〉 \times x_{j}$ of the NLPP (P₁).

3.2 NLLP of second type

Abdel-Basset et al. [1] proposed the following method for solving the NLPP (P₃).

Maximize/Minimize $[\sum_{j = 1}^{n} c_{j} x_{j}]$

Subject to $\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j} \leq, =, \geq {\tilde{b}}_{i}, i = 1, 2, . . ., m; (P_{3}) .$ $x_{j} \geq 0, j = 1, 2, . . ., n .$

Step 1. Transform the NLPP (P₃) into its equivalent crisp LPP (P₄).

Maximize/Minimize $[\sum_{j = 1}^{n} c_{j} x_{j}]$

Subject to $\sum_{j = 1}^{n} R ({\tilde{a}}_{ij}) x_{j} \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m; (P_{4})$ $x_{j} \geq 0, j = 1, 2, . . ., n .$

Step 2. Find the optimal solution {x_j} of the crisp LPP (P₄).

Step 3. Using the optimal solution {x_j}, obtained in Step 2, the optimal value of the considered NLPP (P₃) is $\sum_{j = 1}^{n} c_{j} x_{j}$ .

3.3 NLLP of third type

Abdel-Basset et al. [1] proposed the following method for solving NLPP (P₅).

Maximize/Minimize $[\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j}]$

Subject to (P₅) $\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j} \leq, =, \geq {\tilde{b}}_{i}, i = 1, 2, . . ., m .$

Step 1. Transform the NLPP (P₅) into its equivalent crisp LPP (P₆).

Maximize/Minimize $[\sum_{j = 1}^{n} R ({\tilde{c}}_{j}) x_{j}]$

Subject to (P₆) $\sum_{j = 1}^{n} R ({\tilde{a}}_{ij}) x_{j} \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m;$ $x_{j} \geq 0, j = 1, 2, . . ., n .$ .

Step 2. Find the optimal solution {x_j} of the crisp LPP (P₄).

Step 3. Using the optimal solution {x_j}, obtained in Step 2, the optimal value of the considered NLPP (P₃) is $\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j}$ .

4 Origin of Abdel-Basset et al. ’s method

In this section, the origin of Abdel-Basset et al. ’s method [1] is discussed.

4.1 NLPP of first type

Abdel-Basset et al. [1] have used the following methodology to transform the NLPP (P₁) into a crisp linear programming problem (P₂).

Step 1. Using the method for comparing TrNNs, discussed in Section 2, the NLPP (P₁) can be transformed into its equivalent crisp LPP (P₇).

Maximize/Minimize $[R (\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j})]$

Subject to $\begin{matrix} \sum_{j = 1}^{n} a_{ij} x_{j} \leq, =, \geq (b_{i}), i = 1, 2, . . ., m; (P_{7}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

Step 2. Using the property $R (\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j}) = \sum_{j = 1}^{n} [R ({\tilde{c}}_{j})] x_{j}$ , the crisp LPP (P₇) can be transformed into its equivalent crisp LPP (P₂).

4.2 NLPP of second type

Abdel-Basset et al. [1] have used the following methodology to transform the NLPP (P₃) into the crisp linear programming problem (P₄).

Step 1. Using the method for comparing TrNNs, discussed in Section 2, the NLPP (P₃) can be transformed into the crisp LPP (P₈).

Maximize/Minimize $[\sum_{j = 1}^{n} c_{j} x_{j}]$

Subject to $\begin{matrix} R (\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j}) \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m; (P_{8}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

Step 2. Using the linearity property, $R (\sum_{j = 1}^{n}$ ${\tilde{a}}_{j} x_{j}) = \sum_{j = 1}^{n} [R ({\tilde{a}}_{j})] x_{j}$ , the crisp LPP (P₈) can be transformed into the crisp LPP (P₄).

4.3 NLPP of third type

Abdel-Basset et al. [1] have used the following methodology to transform the NLPP (P₅) into the crisp linear programming problem (P₆).

Step 1. Using the method for comparing TrNNs, discussed in Section 2, the NLPP (P₅) can be transformed into the crisp LPP (P₉).

Maximize/Minimize $[R (\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j})]$

Subject to (P₉) $\begin{matrix} R (\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j}) \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m; (P_{9}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

Step 2. Using the linearity property, $R (\sum_{j = 1}^{n}$ ${\tilde{a}}_{ij} x_{j}) = \sum_{j = 1}^{n} [R ({\tilde{a}}_{ij})] x_{j}$ , as well as using the relation R (a) = a, the crisp LPP (P₉) can be transformed into the crisp LPP (P₆).

5 Mathematical incorrect assumptions

The following mathematical incorrect assumptions have been considered by Abdel-Basset et al. [1].

It is obvious from Section 4 that

Abdel-Basset et al. [1] have used the property $[R (\sum_{j = 1}^{n} {\tilde{c}}_{j} x_{j})] = [(\sum_{j = 1}^{n} R ({\tilde{c}}_{j}) x_{j})]$ to transform the objective function $[R (\sum_{n}^{(j = 1)} {\tilde{c}}_{j} x_{j})]$ of the crisp LPP (P₇) into the objective function $[(\sum_{j = 1}^{n} R ({\tilde{c}}_{j}) x_{j})]$ of the crisp LPP (P₂).

Abdel-Basset et al. [1] have used the property $[R (\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j})] = [(\sum_{j = 1}^{n} R ({\tilde{a}}_{ij}) x_{j})]$ to transform the constraint $R (\sum_{j = 1}^{n} {\tilde{a}}_{ij} x_{j}) \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m;$ of the crisp LPP (P₈) into the constraint $\sum_{j = 1}^{n} R ({\tilde{a}}_{ij}) x_{j} \leq, =, \geq R ({\tilde{b}}_{i}), i = 1, 2, . . ., m;$ of the crisp LPP (P₄).

However, the following clearly indicates that if ${\tilde{A}}_{1} = 〈 a_{2}^{l}, a_{2}^{m_{1}}, a_{2}^{m_{2}}, a_{2}^{u}; T_{{\tilde{a}}_{2}}, I_{{\tilde{a}}_{2}}, F_{{\tilde{a}}_{2}} 〉$ and ${\tilde{A}}_{2} = 〈 a_{2}^{l}, a_{2}^{m_{1}}, a_{2}^{m_{2}}, a_{2}^{u}; T_{{\tilde{a}}_{2}}, I_{{\tilde{a}}_{2}}, F_{{\tilde{a}}_{2}} 〉$ are two TrNNs then $R ({\tilde{A}}_{1} \oplus {\tilde{A}}_{2}) \neq R ({\tilde{A}}_{1}) + R ({\tilde{A}}_{2}) .$ $R ({\tilde{A}}_{1} \oplus {\tilde{A}}_{2}) = R (a_{1}^{l} + a_{2}^{l}), (a_{1}^{m_{1}} + a_{2}^{m_{1}}),$ $(a_{1}^{m_{2}} + a_{2}^{m_{2}}), (a_{1}^{u} + a_{2}^{u}); min {T_{a_{1}}, T_{a_{2}}},$ $max {I_{a_{1}}, I_{a_{2}}}, max {F_{a_{1}}, F_{a_{2}}}$ $= \frac{a_{1}^{l} + a_{2}^{l} + a_{1}^{u} + a_{2}^{u} + 2 (a_{1}^{m_{1}} + a_{2}^{m_{1}} + a_{1}^{m_{2}} + a_{2}^{m_{2}})}{2}$ $+ (min {T_{{\tilde{a}}_{1}}, T_{{\tilde{a}}_{2}}} - max {I_{{\tilde{a}}_{1}}, I_{{\tilde{a}}_{2}}} - max {F_{{\tilde{a}}_{1}}, F_{{\tilde{a}}_{2}}})$ (1)

While, $\begin{matrix} R ({\tilde{A}}_{1}) + R ({\tilde{A}}_{2}) = \frac{a_{1}^{l} + 2 (a_{1}^{m_{1}} + a_{1}^{m_{2}}) + a_{1}^{u}}{2} \end{matrix}$ $+ (T_{{\tilde{a}}_{1}} - I_{{\tilde{a}}_{1}} - F_{{\tilde{a}}_{1}}) + \frac{a_{2}^{l} + 2 (a_{2}^{m_{1}} + a_{2}^{m_{2}}) + a_{2}^{u}}{2}$ $+ (T_{{\tilde{a}}_{2}} - I_{{\tilde{a}}_{2}} - F_{{\tilde{a}}_{2}})$ (2)

It is obvious from (1) and (2) that $R ({\tilde{A}}_{1} \oplus {\tilde{A}}_{2}) \neq R ({\tilde{A}}_{1}) + R ({\tilde{A}}_{2})$ .

Abdel-Basset et al. [1] have assumed that if ‘a’ is a real number then R (a) = a and have used this relation to transform the constraint $R (\sum_{j = 1}^{n} a_{ij} x_{j}) \leq, =, \geq R ({\tilde{b}}_{i})$ of the crisp LPP (P₉) into the constraint $\sum_{j = 1}^{n} a_{ij} x_{j} \leq, =, \geq R ({\tilde{b}}_{i})$ of the crisp LPP (P₆).

However, the following clearly indicates that R (a) ≠ a.

Abdel-Basset et al. [1] have pointed out that if, $T_{\tilde{A}} = 1, I_{\tilde{A}} = 0$ and $F_{\tilde{A}} = 0$ then the TrNN $\tilde{a} = 〈 a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{a}}, I_{\tilde{a}}, F_{\tilde{a}} 〉$ will be transformed into a trapezoidal fuzzy number $\tilde{a} = 〈 a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; 1, 0, 0 〉$ and hence, in this case,

The expression $R (\tilde{a}) = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}})$ is equivalent to the expression

$R (\tilde{a}) = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + 1 .$

The expression $R (\tilde{a}) = (\frac{a^{l} + a^{u} + 3 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}})$ is equivalent to

$R (\tilde{a}) = (\frac{a^{l} + a^{u} + 3 (a^{m_{1}} + a^{m_{2}})}{2}) + 1$ .

Furthermore, it is well known fact that if a^l = a^u = a^{m
₁} = a^{m
₂} then the trapezoidal fuzzy number $\tilde{A} = 〈 a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; 1, 0, 0 〉$ will be transformed into a real number A = (a, a, a, a ; 1, 0, 0) and hence, in this case,

The expression $R (\tilde{A}) = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}})$ is equivalent to the expression R (A) =3a + 1 ≠ a.

The expression $R (\tilde{A}) = (\frac{a^{l} + a^{u} - 3 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}})$ is equivalent to the expression R (A) = -2a + 1 ≠ a.

The TrNN $\tilde{A} = 〈 a^{i}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{A}}$ , $I_{\tilde{A}}, F_{\tilde{A}} 〉$ can also be represented as $\tilde{A} = 〈 a^{m_{1}}, a^{m_{2}}, a, β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ , where, α = a^m
₁ - a¹ and β = a^u - a^m
₂.

It is pertinent to mention that to find $R = 〈 a^{m_{1}}, a^{m_{2}}, a, β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ , firstly, there is need to transform $〈 a^{m_{1}}, a^{m_{2}}, a, β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ into the representation $〈 a^{m_{1}} - α, a^{m_{1}}, a^{m_{2}}, a^{m_{2}}, β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ . However, the following clearly indicates that the value of $R = 〈 a^{m_{1}}, a^{m_{2}}, a, β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ , in the existing NLPP [1, Section 6.1, Example 1], has been obtained by considering a^{m
₁} as a^{l
₁},a^{m
₂} as a^{m
₁}, a as a^{m
₂} and β as a^u, which is mathematically incorrect.

In the existing NLPP [1, Section 6.1, Example 1], the trapezoidal neutrosophic number (13, 15, 22) has been replaced by the crisp number 19. This crisp number 19 has been obtained by considering, a^l = 13, a^{m
₁} = 15, a^{m
₂} = 2, a^u = 2 in the expression $R 〈 a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u} 〉 = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + 1$ .

While, in actual case, to find a crisp number corresponding to the trapezoidal neutrosophic number (13, 15, 2, 2) is 43, which is obtained as follows:

Since, the trapezoidal neutrosophic number (13, 15, 2, 2) is written in the representation $〈 a^{i}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ . So, firstly, there is need to represent it into the representation $〈 a^{m_{1}} - α, a^{m_{1}}, a^{m_{2}}, a^{m_{2}} + β; T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}} 〉$ . In this representation, the trapezoidal neutrosophic number (13, 15, 2, 2) can be rewritten as (13 - 2, 13, 15, 15 + 2) = (11, 13, 15, 17). Now, as a^l = 11, a^{m
₁} = 13, a^{m
₂} = 15 and a^u = 17, therefore, using the expression $R 〈 a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u} 〉 = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + 1$ , $R (13, 15, 2, 2) = R (11, 13, 15, 17) = \frac{11 + 17 + 2 (13 + 15)}{2} + 1 = 43$ i.e., the actual crisp number corresponding to trapezoidal neutrosophic number (13, 15, 2, 2) is 43 instead of 19.

6 Suggested modification

It is obvious from Section 5 that several mathematical incorrect assumptions have been considered in Abdel-Basset et al. ’s method [1]. Therefore, it is scientifically incorrect to use Abdel-Basset et al. ’s method [1] in its present form.

In this section, the required modifications in Abdel-Basset et al. ’s method [1] are suggested.

Since, $R (\sum_{i = 1}^{m} 〈 a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}, T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}} 〉) = \sum_{i = 1}^{m} R 〈 a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}, T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}} 〉 - \sum_{i = 1}^{m} T_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} I_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} F_{{\tilde{a}}_{i}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}}$ instead of $R (\sum_{i = 1}^{m} 〈 a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}, T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}} 〉) = \sum_{i = 1}^{m} R 〈 a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}, T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}} 〉$

Therefore,

The exact crisp LPP corresponding to the NLPP (P₁) is (P₁₀) instead of the crisp LPP (P₂). Therefore, the optimal solution of the NLPP (P₁) should be obtained by solving the crisp LPP (P₁₀) instead by solving the crisp LPP (P₂).

Maximize/Minimize $[\sum_{j = 1}^{n} R ({\tilde{c}}_{j} x_{j}) - \sum_{j = 1}^{n}$ $T_{{\tilde{c}}_{j}} x_{j} + \sum_{j = 1}^{n} I_{{\tilde{c}}_{j}} x_{j} + \sum_{j = 1}^{n} F_{{\tilde{c}}_{j}} x_{j} + min_{1 \leq j \leq n}$ ${T_{{\tilde{c}}_{j} x_{j}}} - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j} x_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j} x_{j}}}]$

Subject to $\sum_{j = 1}^{n} a_{ij} x_{j} \leq, =, \geq b_{i}, i = 1, 2, . . ., m; (P_{10})$ $x_{j} \geq 0, j = 1, 2, . . ., n .$

The exact crisp LPP corresponding to the NLPP (P₃) is (P₁₁) instead of the crisp LPP (P₄). Therefore, the optimal solution of the NLPP should be obtained by solving the crisp LPP (P₁₁) instead by solving the crisp LPP (P₄).

Maximize/Minimize $[\sum_{j = 1}^{n} c_{j} x_{j}]$

Subject to $\begin{matrix} [\sum_{j = 1}^{n} R ({\tilde{a}}_{ij} x_{j}) - \sum_{j = 1}^{n} T_{{\tilde{a}}_{ij} x_{j}} + \sum_{j = 1}^{n} I_{{\tilde{a}}_{ij} x_{j}} \\ + \sum_{j = 1}^{n} F_{{\tilde{a}}_{ij} x_{j}} + min_{1 \leq j \leq n} {T_{{\tilde{a}}_{ij} x_{j}}} - max_{1 \leq j \leq n} {I_{{\tilde{a}}_{ij} x_{j}}} \\ - max_{1 \leq j \leq n} {F_{{\tilde{a}}_{ij} x_{j}}}] (P_{11}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

Furthermore, it is obvious from Section 5 that R (a) = 3a + 1 (in case of maximization problem) and R (a) = -2a + 1 . (in case of minimization problem).

Therefore, the exact crisp LPP corresponding to the NLPP (P₁₂) (in case of maximization problem) is (P₅) instead of the crisp LPP (P₈). Therefore, the optimal solution of the NLPP(P₅) (in case of maximization problem) should be obtained by solving the crisp LPP (P₁₂) instead by solving the crisp LPP (P₈).

Maximize $[\sum_{j = 1}^{n} R ({\tilde{c}}_{j} x_{j}) - \sum_{j = 1}^{n} T_{{\tilde{c}}_{j} x_{j}} + \sum_{j = 1}^{n} I_{{\tilde{c}}_{j} x_{j}} + \sum_{j = 1}^{n} F_{{\tilde{c}}_{j} x_{j}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j} x_{j}}} - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j} x_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j} x_{j}}}]$

Subject to $\begin{matrix} 3 [\sum_{j = 1}^{n} R ({\tilde{a}}_{ij}) x_{j}] + 1 \leq, =, \geq R ({\tilde{b}}_{i}), \\ i = 1, 2, . . ., m; (P_{12}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

The exact crisp LPP corresponding to the NLPP (P₅) (in case of minimization problem) is (P₁₃) instead of the crisp LPP (P₉). Therefore, the optimal solution of the NLPP (P₅) (in case of minimization problem) should be obtained by solving the crisp LPP (P₁₃) instead by solving the crisp LPP (P₉).

Subject to $\begin{matrix} - 2 [\sum_{j = 1}^{n} R {\tilde{a}}_{ij} x_{j}] + 1 \leq, =, \geq R ({\tilde{b}}_{i}), \\ i = 1, 2, . . ., m; (P_{13}) \\ x_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

7 Correct solution of the existing NLPPs

Abdel-Basset et al. [1] considered some NLPPs as well as a NLPP of a real-life problem to illustrate their proposed method. However, as discussed in earlier sections that several mathematical incorrect assumptions have been considered for the same. Therefore, the solutions, of the NLPPs, obtained by Abdel-Basset et al. [1], are not correct. The correct solutions of the NLLPs, considered by Abdel-Basset et al. [1], are obtained in this section.

7.1 Correct solution of the first NLPP

Abdel-Basset et al. [1], solved the NLPP (P₁₄) to illustrate their proposed method.

Maximize [(13, 15, 2, 2) x₁ + (12, 14, 3, 3) x₂ + (15, 17, 2, 2) x₃]

Subject to $12 x_{1} + 13 x_{2} + 12 x_{3} \leq (475, 505, 6, 6),$ $14 x_{1} + 13 x_{3} \leq (460, 480, 8, 8), (P_{14})$ $12 x_{1} + 15 x_{2} \leq (465, 495, 5, 5),$ $x_{1}, x_{2}, x_{3} \geq 0 .$

The correct solution of the NLPP (P₁₄) can be obtained as follows:

Step 1. Since, in the NLPP (P₁₄), the TrNNs have been represented in the form 〈a^{m
₁}, a^{m
₂}, α, β〉, where, α = a^{m
₁} - , a² and β = a^u - a^{m
₂}. Therefore, firstly, there is need to replace each TrNN 〈a^{m
₁}, a^{m
₂}, α, β〉 with its another representation (a^{m
₁} - α, a^{m
₁}, a^{m
₂}, a^{m
₁} + β) i.e., 〈a^l, a^{m
₁}, a^{m
₂}, a^u〉 Following the same the NLPP (P₁₄) can be transformed into NLLP (P₁₅).

Maximize [(11, 13, 15, 17) x₁ + (9, 12, 14, 17) x₂ + (13, 15, 17, 19) x₃]

Subject to $12 x_{1} + 13 x_{2} + 12 x_{3} \leq (469, 475, 505, 511),$ $14 x_{1} + 13 x_{3} \leq (452, 460, 480, 488), (P_{15})$ $12 x_{1} + 15 x_{2} \leq (460, 465, 495, 500),$ $x_{1}, x_{2}, x_{3} \geq 0 .$

Step 2. To find the solution of the NLPP (P₁₅) is equivalent to find the solution of the crisp LPP (P₁₆)

Maximize [R (11, 13, 15, 17) x₁ + (9, 12, 14, 17) x₂ + (13, 15, 17, 19) x₃]

Subject to $R (12 x_{1} + 13 x_{2} + 12 x_{3}) R \leq (469, 475, 505, 511),$ $R (14 x_{1} + 13 x_{3}) \leq R (452, 460, 480, 488), (P_{16})$ $R (12 x_{1} + 15 x_{2}) \leq R (460, 465, 495, 500),$ $x_{1}, x_{2}, x_{3} \geq 0 .$

Step 3. Using the expression

$R (\sum_{i = 1}^{m} 〈 a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}$ , $a_{i}^{u}, T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}$ , $F_{{\tilde{a}}_{i}} 〉) = \sum_{i = 1}^{m} R 〈 a_{i}^{l}, a_{i}^{m_{1}}$ , $a_{i}^{m_{2}}, a_{i}^{u}$ , $T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}$ , $F_{{\tilde{a}}_{i}} 〉 - \sum_{i = 1}^{m}$ $T_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} I_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m}$ $F_{{\tilde{a}}_{i}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}}$ and the expression R (a) =3a + 1, the crisp LPP (P₁₆) can be transformed into its equivalent crisp LPP (P₁₇)

Maximize [R ((11, 13, 15, 17) x₁) + R ((9, 12, 14, 17) x₂) + R ((13, 15, 17, 19) x₃) -3 + 0 +0 + 1 -0 - 0]

Subject to $\begin{matrix} 3 [12 x_{1} + 13 x_{2} + 12 x_{3}] + 1 \leq \\ R (469, 475, 505, 511) \end{matrix}$ $3 [14 x_{1} + 13 x_{3}] + 1 \leq R (452, 460, 480, 488) (P_{17})$ $3 [12 x_{1} + 15 x_{2}] + 1 \leq R (460, 465, 495, 500),$ $x_{1}, x_{2}, x_{3} \geq 0 .$

Step 4. Using the expression, $\begin{matrix} R (a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{a}}, I_{\tilde{a}}, F_{\tilde{a}}) \\ = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}}) \end{matrix}$ with $T_{\tilde{a}} = 1$ , $F_{\tilde{a}} = 0, I_{\tilde{a}} = 0$ the crisp LPP (P₁₇) can be transformed into the crisp LPP(P₁₈).

Maximize $[(\frac{11 + 17 + 2 (13 + 15)}{2} + 1 - 0 - 0)$ $x_{1} + (\frac{9 + 17 + 2 (12 + 14)}{2}) + 1 - 0 - 0)$ x₃ - 3 -0 - 0 +1 - 0 -0]

Subject to $\begin{matrix} 3 [12 x_{1} + 13 x_{2} + 12 x_{3}] + 1 \leq \\ \frac{469 + 511 + 2 (475 + 505)}{2} + 1 - 0 - 0 \end{matrix}$ $\begin{matrix} 3 [14 x_{1} + 13 x_{3}] + 1 \leq \frac{452 + 488 + 2 (460 + 480)}{2} \\ + 1 - - 0 - 0, (P_{18}) \\ 3 [12 x_{1} + 15 x_{2}] + 1 \leq \frac{460 + 500 + 2 (465 + 495)}{2} \\ + 1 - 0 - 0 x_{1}, x_{2} x_{3} \geq 0 \end{matrix}$

Step 5. On solving the crisp LPP (P₁₈), the obtained optimal solution is $x_{1} = 0, x_{2} = 0, x_{3} = \frac{245}{18}$ .

Step 6. Using the optimal solution, obtained in Step 5, the optimal value of the NLPP (P₁₄) is $\begin{matrix} (11, 13, 15, 17) x_{1} + (9, 12, 14, 17) x_{2} + \\ (13, 15, 17, 19) x_{3} = (11, 13, 15, 17) (0) + \\ (9, 12, 14, 17) (0) + (13, 15, 17, 19) (\frac{245}{18}) \\ = 13 (\frac{245}{18}) + 15 (\frac{245}{18}) + 17 (\frac{245}{18}) + 19 (\frac{245}{18}) \\ = \frac{7840}{9} \end{matrix}$

7.2 Correct solution of the second NLPP

Abdel-Basset et al. [1] solved the NLPP (P₁₉) to illustrate their proposed method.

Maximize [25x₁ + 48x₂]

Subject to $\begin{matrix} (14, 15, 17, 18) x_{1} + (25, 30, 34, 38) x_{2} \\ \leq (44, 980, 45, 000, 45, 030, 45, 070) \end{matrix}$ $\begin{matrix} (21, 24, 26, 33) x_{1} + (4, 6, 8, 11) x_{2} \\ \leq (23, 980, 24, 000, 24, 050, 24, 060), (P_{19}) \end{matrix}$ $\begin{matrix} (17, 21, 22, 26) x_{1} + (12, 14, 19, 22) x_{2} \\ \leq (27, 990, 28, 000, 28, 030, 28, 040), \end{matrix}$ $x_{1}, x_{2} \geq 0 .$

The correct solution of the NLPP (P₁₉) can be obtained as follows:

Step 1. Since, in the NLPP the TrNNs have been represented in the form a^l, a^{m
₁}, a^{m
₂}, a^u. Therefore, there is no need to change it.

Step 2. To find the solution of the NLPP (P₁₉) is equivalent to find the solution of the crisp LPP (P₂₀).

Maximize [25x₁ + 48x₂]

Subject to $\begin{matrix} R [(14, 15, 17, 18) x_{1} + (25, 30, 34, 38) x_{2}] \\ \leq R (44980, 45000, 45030, 45070) \end{matrix}$ $\begin{matrix} R [(21, 24, 26, 33) x_{1} + (4, 6, 8, 11) x_{2}] \\ \leq R (23980, 24000, 24050, 24060), (P_{20}) \end{matrix}$ $\begin{matrix} R [(17, 21, 22, 26) x_{1} + (12, 14, 19, 22) x_{2}] \\ \leq R (27990, 28000, 28030, 28040), \end{matrix}$ $x_{1}, x_{2} \geq 0 .$

Step 3. Using the expression $R (\sum_{i = 1}^{m} (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}}))$ $\begin{matrix} = \sum_{i = 1}^{m} R (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{l}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}}) \end{matrix}$ $\begin{matrix} - \sum_{i = 1}^{m} T_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} I_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} F_{{\tilde{a}}_{i}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}} \\ - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}} \end{matrix}$ and the expression R (a) = 3a + 1, the crisp LPP (P₂₀) can be transformed into its equivalent crisp LPP (P₂₁).

Maximize [25x₁ + 48x₂]

Subject to $\begin{matrix} R [(14, 15, 17, 18) x_{1}] + R [(25, 30, 34, 38) x_{2}] \\ - 2 - 0 - 0 + 1 - 0 - 0 \\ \leq R (44980, 45000, 45030, 45070), (P_{21}) \end{matrix}$ $\begin{matrix} R [(21, 24, 26, 33) x_{1}] + R [(4, 6, 8, 11) x_{2}] \\ - 2 - 0 - 0 + 1 - 0 - 0 \\ \leq R (23980, 24000, 24050, 24060) \end{matrix}$ $\begin{matrix} R [(17, 21, 22, 26) x_{1}] + R [(12, 14, 19, 22) x_{2}] \\ - 2 - 0 - 0 + 1 - 0 - 0 \\ \leq R (27990, 28000, 28030, 28040) \end{matrix}$ $x_{1}, x_{2} \geq 0 .$

Step 4. Using the expression, $\begin{matrix} R (a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{a}}, I_{\tilde{a}}, F_{\tilde{a}}) \\ = (\frac{a^{l} + a^{u} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}}) \end{matrix}$ with $T_{\tilde{a}} = 1, F_{\tilde{a}} = 0, I_{\tilde{a}} = 0$ the crisp LPP (P₂₁) can be transformed into the crisp LPP (P₂₂).

Maximize [25x₁ + 48x₂]

Subject to $\begin{matrix} (\frac{14 + 18 + 2 (15 + 17)}{2} + 1 - 0 - 0) x_{1} \\ + (\frac{25 + 38 + 2 (30 + 34)}{2} + 1 - 0 - 0) x_{2} \\ - 2 - 0 - 0 + 1 - 0 - 0 \end{matrix}$ $\begin{matrix} \leq (\frac{44980 + 45070 + 2 (45000 + 45030)}{2} \\ + 1 - 0 - 0), \end{matrix}$ $\begin{matrix} (\frac{21 + 33 + 2 (24 + 26)}{2} + 1 - 0 - 0) x_{1} \\ + (\frac{4 + 11 + 2 (6 + 8)}{2} + 1 - 0 - 0) x_{2} \\ - 2 - 0 - 0 + 1 - 0 - 0 \\ \leq (\frac{23980 + 24060 + 2 (24000 + 24050)}{2} \\ + 1 - 0 - 0) (P_{22}) \\ (\frac{17 + 26 + 2 (21 + 22)}{2} \\ + 1 - 0 - 0) x_{1} + (\frac{12 + 22 + 2 (14 + 19)}{2} \\ + 1 - 0 - 0) x_{2} - 2 - 0 - 0 + 1 - 0 - 0 \\ \leq (\frac{27990 + 28040 + 2 (28000 + 28030)}{2} \\ + 1 - 0 - 0) x_{1}, x_{2} \geq 0 . \end{matrix}$

Step 5. On solving the crisp LPP (P₂₂), the obtained optimal solution is $x_{1} = \frac{5790432}{1810}, x_{2} \frac{943916}{763}$ .

Step 6. Using the optimal solution, obtained in Step 5, the optimal value of the NLPP (P₁₉) is

$25 x_{1} + 48 \frac{943916}{763} x_{2} = 25 (\frac{579043}{1810}) + 48 = \frac{10174256}{151}$ .

7.3 Correct solution of the third NLPP

Abdel-Basset et al. [1] solved the NLPP (P₂₃) to illustrate their proposed method.

Minimize (6x₁ + 10x₂)

Subject to $2 x_{1} + 5 x_{2} \geq (5, 8, 3, 13) (P_{23})$ $3 x_{1} + 4 x_{2} \geq (6, 0, 4, 16)$ $x_{1}, x_{2} \geq 0 .$

The correct solution of the NLPP (P₂₃) can be obtained as follows:

Step 1. Since, in the NLPP the TrNNs have been represented in the form a^L, a^{m
₁}, a^{m
₂}, a^U. Therefore, there is no need to change it.

Step 2. To find the solution of the NLPP (P₂₃) is equivalent to find the solution of the crisp LPP (P₂₄)

Minimize [6x₁ + 10x₂]

Subject to $R [2 x_{1} + 5 x_{2}] \geq R (5, 8, 3, 13)$ $R [3 x_{1} + 4 x_{2}] \geq R (6, 0, 4, 16) (P_{24})$ $x_{1}, x_{2} \geq 0 .$

Step 3. Using the expression $\begin{matrix} R (\sum_{i = 1}^{m} (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}})) \\ = \sum_{i = 1}^{m} R (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}}) \\ - \sum_{i = 1}^{m} T_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} I_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} F_{{\tilde{a}}_{i}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}} \\ - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}} \end{matrix}$ and the expression R (a) = -2a + 1, the crisp LPP (P₂₄) can be transformed into its equivalent crisp LPP (P₂₅).

Minimize [6x₁ + 10x₂]

Subject to $\begin{matrix} - 2 [2 x_{1} + 5 x_{2}] + 1 \geq \frac{5 + 13 - 3 (8 + 3)}{2} \\ + 1 (P_{25}) \end{matrix}$ $\begin{matrix} - 2 [3 x_{1} + 4 x_{2}] + 1 \geq \frac{6 + 16 - 3 (0 + 4)}{2} \\ + 1 x_{1}, x_{2} \geq 0 . \end{matrix}$

Step 4. On solving the crisp LPP(P₂₅), the obtained optimal solution is $x_{1} = 0, x_{2} = \frac{3}{4}$ .

Step 5. Using the optimal solution, obtained in Step 4, the optimal value of the NLPP (P25) is $6 x_{1} + 10 x_{2} = 6 (0) + 10 (\frac{3}{4}) = \frac{3}{4} .$ .

7.4 Correct solution of the real-life NLPP

Abdel-Basset et al. [1] solved the NLPP (P₂₆) to obtain the solution of a real life problem.

Maximize [(6, 8, 9, 12) x₁ + (9, 10, 12, 14) x₂ + (12, 13, 15, 17) + x₃ (8, 9, 11, 13) x₄]

Subject to $0.5 x_{1} + 1.5 x_{2} + 1.5 x_{3} + x_{4} \leq (1200, 1300, 1500, 1700)$ $3 x_{1} + x_{2} + 2 x_{3} + 3 x_{4} \leq (2200, 2250, 2350, 2400),$ $2 x_{1} + 4 x_{2} + x_{3} + 2 x_{4} \leq (2200, 2400, 2600, 2800)$ $\begin{matrix} 0.5 x_{1} + x_{2} + 0.5 x_{3} + 0.5 x_{4} \leq (1000, 1100, 1200, \\ 1300) \\ x_{1} \geq R(120, 130, 150, 170), \\ x_{2} \geq R(70, 80, 100, 120), \\ x_{3} \geq R(270, 280, 300, 320), \\ x_{4} \geq R(370, 380, 400, 420), \\ x_{1} {, x}_{2} {, x}_{3} {, x}_{4} \geq 0 . (P_{26}) \end{matrix}$

The correct solution of the NLPP (P₂₆) can be obtained as follows:

Step 1. Since, in the NLPP (P₂₆), the TrNNs have been represented in the form <a^l, a^{m
₁}, a^{m
₂}, a^u>. Therefore, there is no need to change it.

Step 2. To find the solution of the NLPP (P₂₆) is equivalent to find the solution of the crisp LPP (P₂₇)

Maximize [R ((6, 8, 9, 12) x₁ + (9, 10, 12, 14) x₂ + (12, 13, 15, 17) x₃ + (8, 9, 11, 13) x₄)]

Subject to $\begin{matrix} R (0.5 x_{1} + 1.5 x_{2} + 1.5 x_{3} + x_{4}) \leq R (1200, 1300, \\ 1500, 1700), \end{matrix}$ $\begin{matrix} R (3 x_{1} + x_{2} + 2 x_{3} + 3 x_{4}) \leq R (2200, 2250, \\ 2350, 2400), (P_{27}) \end{matrix}$ $\begin{matrix} R (2 x_{1} + 4 x_{2} + x_{3} + 2 x_{4}) \leq R (2200, 2400, \\ 2600, 2800), \end{matrix}$ $\begin{matrix} R (0.5 x_{1} + x_{2} + 0.5 x_{3} + 0.5 x_{4}) \leq R (1000, 1100, \\ 1200, 1300), \end{matrix}$ $\begin{matrix} x_{1} \geq R(120, 130, 150, 170), \\ x_{2} \geq R(70, 80, 100, 120), \\ x_{3} \geq R(270, 280, 300, 320), \\ x_{4} \geq R(370, 380, 400, 420), \\ x_{1} {, x}_{2} {, x}_{3} {, x}_{4} \geq 0 . \end{matrix}$

Step 3. Using the expression $\begin{matrix} R (\sum_{i = 1}^{m} (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}})) \\ = \sum_{i = 1}^{m} R (a_{i}^{l}, a_{i}^{m_{1}}, a_{i}^{m_{2}}, a_{i}^{u}; T_{{\tilde{a}}_{i}}, I_{{\tilde{a}}_{i}}, F_{{\tilde{a}}_{i}}) \\ - \sum_{i = 1}^{m} T_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} I_{{\tilde{a}}_{i}} + \sum_{i = 1}^{m} F_{{\tilde{a}}_{i}} + min_{1 \leq j \leq n} {T_{{\tilde{c}}_{j}}} \\ - max_{1 \leq j \leq n} {I_{{\tilde{c}}_{j}}} - max_{1 \leq j \leq n} {F_{{\tilde{c}}_{j}}} \end{matrix}$ and the expression R (a) = 3a + 1, the crisp LPP (P₂₇) can be transformed into its equivalent crisp LPP (P₂₈).

Maximize [R (6, 8, 9, 12) x₁ + R (9, 10, 12, 14) x₂ + R (12, 13, 15, 17) x₃ + R (8, 9, 11, 13) x₄] - 4 + 0 + 0 + 1 - 0 - 0

Subject to $\begin{matrix} 3 [0.5 x_{1} + 1.5 x_{2} + 1.5 x_{3} + x_{4}] + 1 \leq R (1200, \\ 1300, 1500, 1700) \end{matrix}$ $\begin{matrix} 3 [3 x_{1} + x_{2} + 2 x_{3} + 3 x_{4}] + 1 \leq R (2200, 2250, \\ 2350, 2400) (P_{28}) . \end{matrix}$ $\begin{matrix} 3 [2 x_{1} + 4 x_{2} + x_{3} + 2 x_{4}] + 1 \leq R (2200, 2400, \\ 2600, 2800) \end{matrix}$ $\begin{matrix} 3 [0.5 x_{1} + x_{2} + 0.5 x_{3} + 0.5 x_{4}] + 1 \leq R (1000, \\ 1100, 1200, 1300) \end{matrix}$ $\begin{matrix} x_{1} \geq R(120, 130, 150, 170), \\ x_{2} \geq R(70, 80, 100, 120), \\ x_{3} \geq R(270, 280, 300, 320), \\ x_{4} \geq R(370, 380, 400, 420), \\ x_{1} {, x}_{2} {, x}_{3} {, x}_{4} \geq 0 . \end{matrix}$

Step 4. Using the expression, $\begin{matrix} R (a^{l}, a^{m_{1}}, a^{m_{2}}, a^{u}; T_{\tilde{a}}, I_{\tilde{a}}, F_{\tilde{a}}) \\ = (\frac{a^{l} + a^{l} + 2 (a^{m_{1}} + a^{m_{2}})}{2}) \\ + (T_{\tilde{a}} - I_{\tilde{a}} - F_{\tilde{a}}) \end{matrix}$ with $T_{\tilde{a}} = 1, F_{\tilde{a}} = 0, I_{\tilde{a}} = 0$ , the crisp LPP (P₂₈) can be transformed into the crisp LPP (P₂₉).

Maximize $[(\frac{6 + 12 + 2 (8 + 9)}{2} + 1 - 0 - 0)] x_{1} +$ $[(\frac{9 + 14 + 2 (10 + 12)}{2} + 1 - 0 - 0)] x_{2} + [(\frac{12 + 17 + 2 (13 + 15)}{2}$ $+ 1 - 0 - 0) x_{3} + (\frac{8 + 13 + 2 (9 + 11)}{2} + 1 - 0 - 0) x_{2}$ $- 4 - 0 - 0 + 1 - 0 - 0]$

Subject to $\begin{matrix} 3 [0.5 x_{1} + 1.5 x_{2} + 1.5 x_{3} + x_{4}] + 1 \leq \\ (\frac{1200 + 1700 + 2 (1300 + 1500)}{2} + 1 - 0 - 0), \end{matrix}$ $\begin{matrix} 3 [3 x_{1} + x_{2} + 2 x_{3} + 3 x_{4}] + 1 \leq \\ (\frac{2200 + 2400 + 2 (2250 + 2350)}{2} + 1 - 0 - 0), \\ (P_{29}) \end{matrix}$ $3 [2 x_{1} + 4 x_{2} + x_{3} + 2 x_{4}] + 1 \leq (\frac{2200 + 2800 + 2 (2400 + 2600)}{2} + 1 - 0 - 0)$ , $\begin{matrix} 3 [0.5 x_{1} + x_{2} + 0.5 x_{3} + 0.5 x_{4}] + 1 \leq \\ (\frac{1000 + 1300 + 2 (1100 + 1200)}{2} + 1 - 0 - 0), \\ x_{1} \geq 426, \\ x_{2} \geq 343, \\ x_{3} \geq 876, \\ x_{4} \geq 1176, \\ x_{1}, x_{2}, x_{3}, x_{4} \geq 0. \end{matrix}$

Step 5. On solving the crisp LPP (P₁₈), the obtained optimal solution is $x_{1} = \frac{3700}{21}, x_{2} = 0, x_{3} = \frac{6200}{7}$ , $x_{4} = 0$ .

Step 6. Using the optimal solution, obtained in Step 5, the optimal value of the NLPP (P₂₉) is $\begin{array}{l} (6, 8, 9, 12) x_{1} + (9, 10, 12, 14) \\ x_{2} + (12, 13, 15, 17) x_{3} + (8, 9, 11, 13) \\ x_{4} = (6, 8, 9, 12) (\frac{3700}{21}) + (9, 10, 12, 14) (0) \\ + (12, 13, 15, 17) (\frac{6200}{7}) + (8, 9, 11, 13) (0) \\ = [6 (\frac{3700}{21}) + 8 (\frac{3700}{21}) + 9 (\frac{3700}{21}) \\ + 12 \frac{3700}{21}] + [12 (\frac{6200}{7}) + 13 (\frac{6200}{7}) \\ + 15 (\frac{6200}{7}) + 17 (\frac{6200}{7})] = \frac{1189700}{21} . \\ = \frac{1343000}{23} . \end{array}$

8 Conclusions

The mathematical incorrect assumptions, used in Abdel-Basset et al. ’s method [1], are pointed out. Also, the required modification in Abdel-Basset et al. ’s method [1] are suggested. Furthermore, the correct results of the NLPPs, solved by Abdel-Basset et al. [1] to illustrate their proposed method, are obtained.

References

Abdel-Basset

, Gunasekaran

, Mohamed

and Smarandache

, A novel method for solving the fully neutrosophic linear programming problems, Neural Comput & Applic (2018). https://doi.org/10.1007/s00521-018-3404-

Zimmermann

H.-J.

, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Syst 1 (1978), 45–55.

Tanaka

and Asai

, Fuzzy solution in fuzzy linear programming problems, IEEE Trans Syst Man Cybern 14(2) (1984)325–328.

Verdegay

J.L.

, A dual approach to solve the fuzzy linear programming problem, Fuzzy Sets Syst 14 (1984)131–141.

Herrera

, Kovacs

and Verdegay

, Optimality for fuzzified mathematical programming problems: A parametric approach, Fuzzy Sets Syst 54 (1993)279–285.

Buckley

J.J.

and Feuring

, Evolutionary algorithm solution to fuzzy problems: Fuzzy linear programming, Fuzzy Sets Syst 109 (2000)35–53.

Maleki

H.R.

, Tata

and Mashinchi

, Linear programming with fuzzy variables, Fuzzy Sets Syst 109 (2000)21–33.

Stanciulescu

C.V.

, Fortemps

, Installé

and Wertz

, Multiobjective fuzzy linear programming problems with fuzzy decision variables, Eur J Oper Res 149 (2003)654–675.

Zhang

, Wu

Y.-H.

, Remias

and Lu

, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Appl Math Comput 139 (2003)383–399.

10.

Ganesan

and Veeramani

, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann Oper Res 143 (2006)305–315.

11.

Hashemi

S.M.

, Modarres

, Nasrabadi

and Nasrabadi

M.M.

, Fully fuzzified linear programming, solution and duality, J Intell Fuzzy Syst 17 (2006)253–261.

12.

Mahdavi-Amiri

and Nasseri

, Duality in fuzzy number linear programming by use of a certain linear ranking function, Appl Math Comput 180 (2006)206–216.

13.

Allahviranloo

, Lotfi

F.H.

, Kiasary

M.K.

, Kiani

and Alizadeh

, Solving fully fuzzy linear programming problem by the ranking function, Appl Math Sci 2 (2008)19–32.

14.

H.-C.

, Using the technique of scalarization to solve the multiobjective programming problems with fuzzy coefficients, Math Comput Model 48 (2008)232–248.

15.

Lotfi

F.H.

, Allahviranloo

, Jondabeh

M.A.

and Alizadeh

, Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl Math Model 33 (2009)3151–3156.

16.

Ebrahimnejad

, Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method, Appl Math Model 35 (2011)4526–4540.

17.

Kumar

, Kaur

and Singh

, A new method for solving fully fuzzy linear programming problems, Appl Math Model 35 (2011)817–823.

18.

Nagoorgani

and Ponnalagu

, A new approach on solving intuitionistic fuzzy linear programming problem, Appl Math Sci 6 (2012)3467–3474.

19.

Ebrahimnejad

and Tavana

, A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers, Appl Math Model 38 (2014)4388–4395.

20.

Bharati

and Singh

, A note on solving a fully intuitionistic fuzzy linear programming problem based on sign distance, Int J Comput Appl 119 (2015)30–35.

21.

Saati

, Tavana

, Hatami-Marbini

and Hajiakhondi

, A fuzzy linear programming model with fuzzy parameters and decision variables, Int J Inf Decis Sci 7 (2015)312–333.

22.

Sidhu

S.K.

and Kumar

, A note on “Solving intuitionistic fuzzy linear programming problems by ranking function”, J Intell Fuzzy Syst 30 (2016)2787–2790.