# Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments

^{*}

## Abstract

**:**

## 1. Introduction

_{1}(x), g

_{2}(x)]. After that, Ye et al. [15] introduced the neutrosophic functions in expressions for the joint roughness coefficient and the shear strength in the mechanics of rocks. Further, Ye [16] and Chen et al. [17,18] presented expressions and analyses of the joint roughness coefficient using NNs. Ye [19] proposed the use of neutrosophic linear equations and their solution methods in traffic flow problems with NN information.

## 2. Neutrosophic Numbers and Neutrosophic Number Functions

^{L}, p + qI

^{U}] for z ∈ Z (Z is the set of all NNs) and I ∈ [I

^{L}, I

^{U}] for short. In special cases, z can be expressed as the determinate part z = p if qI = 0 for the best case, and, also, z can be expressed as the indeterminate part z = qI if p = 0 for the worst case.

_{1}= p

_{1}+ q

_{1}I and z

_{2}= p

_{2}+ q

_{2}I for z

_{1}, z

_{2}∈ Z, then their basic operational laws for I ∈ [I

^{L}, I

^{U}] are defined as follows [23,24]:

- (1)
- ${z}_{1}+{z}_{2}={p}_{1}+{p}_{2}+({q}_{1}+{q}_{2})I=[{p}_{1}+{p}_{2}+{q}_{1}{I}^{L}+{q}_{2}{I}^{L},{p}_{1}+{p}_{2}+{q}_{1}{I}^{U}+{q}_{2}{I}^{U}]$;
- (2)
- ${z}_{1}-{z}_{2}={p}_{1}-{p}_{2}+({q}_{1}-{q}_{2})I=[{p}_{1}-{p}_{2}+{q}_{1}{I}^{L}-{q}_{2}{I}^{L},{p}_{1}-{p}_{2}+{q}_{1}{I}^{U}-{q}_{2}{I}^{U}]$;
- (3)
- $\begin{array}{ll}{z}_{1}\times {z}_{2}& ={p}_{1}{p}_{2}+({p}_{1}{q}_{2}+{p}_{2}{q}_{1})I+{q}_{1}{q}_{2}{I}^{2}\\ & =\left[\begin{array}{l}\mathrm{min}\left(\begin{array}{l}({p}_{1}+{q}_{1}{I}^{L})({p}_{2}+{q}_{2}{I}^{L}),({p}_{1}+{q}_{1}{I}^{L})({p}_{2}+{q}_{2}{I}^{U}),\\ ({p}_{1}+{q}_{1}{I}^{U})({p}_{2}+{q}_{2}{I}^{L}),({p}_{1}+{q}_{1}{I}^{U})({p}_{2}+{q}_{2}{I}^{U})\end{array}\right),\\ \mathrm{max}\left(\begin{array}{l}({p}_{1}+{q}_{1}{I}^{L})({p}_{2}+{q}_{2}{I}^{L}),({p}_{1}+{q}_{1}{I}^{L})({p}_{2}+{q}_{2}{I}^{U}),\\ ({p}_{1}+{q}_{1}{I}^{U})({p}_{2}+{q}_{2}{I}^{L}),({p}_{1}+{q}_{1}{I}^{U})({p}_{2}+{q}_{2}{I}^{U})\end{array}\right)\end{array}\right]\end{array}$;
- (4)
- $\begin{array}{ll}\frac{{z}_{1}}{{z}_{2}}& =\frac{{p}_{1}+{q}_{1}I}{{p}_{2}+{q}_{2}I}=\frac{[{p}_{1}+{q}_{1}{I}^{L},{p}_{1}+{q}_{1}{I}^{U}]}{[{p}_{2}+{q}_{2}{I}^{L},{p}_{2}+{q}_{2}{I}^{U}]}\\ & =\left[\begin{array}{l}\mathrm{min}\left(\frac{{p}_{1}+{q}_{1}{I}^{L}}{{p}_{2}+{q}_{2}{I}^{U}},\frac{{p}_{1}+{q}_{1}{I}^{L}}{{p}_{2}+{q}_{2}{I}^{L}},\frac{{p}_{1}+{q}_{1}{I}^{U}}{{p}_{2}+{q}_{2}{I}^{U}},\frac{{p}_{1}+{q}_{1}{I}^{U}}{{p}_{2}+{q}_{2}{I}^{L}}\right),\\ \mathrm{max}\left(\frac{{p}_{1}+{q}_{1}{I}^{L}}{{p}_{2}+{q}_{2}{I}^{U}},\frac{{p}_{1}+{q}_{1}{I}^{L}}{{p}_{2}+{q}_{2}{I}^{L}},\frac{{p}_{1}+{q}_{1}{I}^{U}}{{p}_{2}+{q}_{2}{I}^{U}},\frac{{p}_{1}+{q}_{1}{I}^{U}}{{p}_{2}+{q}_{2}{I}^{L}}\right)\end{array}\right]\end{array}$.

**x**, I): Z

^{n}→ Z for

**x**= [x

_{1}, x

_{2}, …, x

_{n}]

^{T}∈ Z

^{n}and I ∈ [I

^{L}, I

^{U}], which is either an NN linear or an NN nonlinear function. For example, ${F}_{1}(\mathit{x},I)={x}_{1}^{}-I{x}_{2}^{}+1+2I$ for

**x**= [x

_{1}, x

_{2}]

^{T}∈ Z

^{2}and I ∈ [I

^{L}, I

^{U}] is an NN linear function, and ${F}_{2}(\mathit{x})={x}_{1}^{2}+{x}_{2}^{2}-2I{x}_{1}^{}-I{x}_{2}^{}+3I$ for

**x**= [x

_{1}, x

_{2}]

^{T}∈ Z

^{2}and I ∈ [I

^{L}, I

^{U}] is an NN nonlinear function.

**x**, I): Z

^{n}→ Z, we can define an NN inequality g(

**x**, I) ≤ (≥) 0 for

**x**= [x

_{1}, x

_{2}, …, x

_{n}]

^{T}∈ Z

^{n}and I ∈ [I

^{L}, I

^{U}], where g(

**x**, I) is either an NN linear function or an NN nonlinear function. For example, ${g}_{1}(\mathit{x},I)=2{x}_{1}^{}-I{x}_{2}^{}+4+3I\le 0$ and ${g}_{2}(\mathit{x},I)=2{x}_{1}^{2}-{x}_{2}^{2}+2+5I\le 0$ for

**x**= [x

_{1}, x

_{2}]

^{T}∈ Z

^{2}and I ∈ [I

^{L}, I

^{U}] are NN linear and NN nonlinear inequalities in two variables, respectively.

**x**, F(

**x**, I), and g(

**x**, I) are NNs (usually but not always). In this study, we mainly research on NN-NP problems and their general solution methods.

## 3. Neutrosophic Number Nonlinear Programming Problems

- (1)
- Unconstrained NN optimization model:min F(
**x**, I),**x**∈ Z^{n},**x**= [x_{1}, x_{2}, …, x_{n}]^{T}∈ Z^{n}, F(**x**, I): Z^{n}→ Z, and I ∈ [I^{L}, I^{U}]. - (2)
- Constrained NN optimization model:min F(
**x**, I)

s.t. g_{i}(**x**, I) ≤ 0, I = 1, 2, …, m

h_{j}(**x**, I) = 0, j = 1, 2, …, l**x**∈ Z^{n},_{1}(**x**, I), g_{2}(**x**, I), …, g_{m}(**x**, I), h_{1}(**x**, I), h_{2}(**x**, I), …, h_{l}(**x**, I): Z^{n}→ Z, and I ∈ [I^{L}, I^{U}].

_{j}(

**x**, I) = 0 without inequality constraints, g

_{i}(

**x**, I) ≤ 0, then the NN-NP problem is called the NN-NP problem with equality constraints. If the NN-NP problem only contains the restrictions g

_{i}(

**x**, I) ≤ 0, without constraints h

_{j}(

**x**, I) = 0, then the NN-NP problem is called the NN-NP problem with inequality constraints. Finally, if the NN-NP problem does not contain either restrictions, h

_{j}(

**x**, I) = 0 or g

_{i}(

**x**, I) ≤ 0, then the constrained NN-NP problem is reduced to the unconstrained NN-NP problem.

**x**and I ∈ [I

^{L}, I

^{U}]. In this case, the value of the NN objective function is an optimal possible interval (NN) for F(

**x**, I).

## 4. General Solution Methods for NN-NP Problems

#### 4.1. One-Dimension Unconstrained NN Nonlinear Optimization

^{L}, I

^{U}]. Then, for a differentiable NN nonlinear objective function F(x, I), a local optimal solution x

^{*}satisfies the following two conditions:

- (1)
- Necessary condition: The derivative is dF(x
^{*}, I)/dx = 0 for I ∈ [I^{L}, I^{U}]; - (2)
- Sufficient condition: If the second derivative is d
^{2}F(x^{*}, I)/dx^{2}< 0 for I ∈ [I^{L}, I^{U}], then x^{*}is an optimal solution for the maximum F(x^{*}, I); if the second derivative is d^{2}F(x^{*}, I)/dx^{2}> 0, then x^{*}is an optimal solution for the minimum F(x^{*}, I).

**Example**

**1.**

^{2}+ 5I for x ∈ Z and I ∈ [I

^{L}, I

^{U}]. Based on the optimal conditions, we can obtain:

^{L}, I

^{U}] according to real situations or actual requirements, then we can discuss its optimal possible value. If I ∈ [1, 2] is considered as a possible interval range, then d

^{2}F(x

^{*}, I)/dx

^{2}> 0, and x

^{*}= 0 is the optimal solution for the minimum F(x

^{*}, I). Thus, the minimum value of the NN objective function is F(x

^{*}, I) = [5, 10], which, in this case, is a possible interval range, but not always. Specifically if I = 1 (crisp value), then F(x

^{*}, I) = 5.

#### 4.2. Multi-Dimension Unconstrained NN Nonlinear Optimization

**x**, I) for

**x**= [x

_{1}, x

_{2}, …, x

^{n}]

^{T}∈ Z

^{n}and I ∈ [I

^{L}, I

^{U}] is considered as an unconstrained differentiable NN nonlinear objective function in n variables. Then, we can obtain the partial derivatives:

**x**, I), are:

**x**, I) is structured as its subsets H

_{i}(

**x**, I) (i = 1, 2, …, n), where H

_{i}(

**x**, I) indicate the subset created by taking the first i rows and columns of H(

**x**, I). You calculate the determinant of each of the n subsets at

**x**

^{*}:

_{i}(

**x**, I) (i = 1, 2, …, n) for I ∈ [I

^{*}^{L}, I

^{U}], as follows:

- (1)
- If H
_{i}(**x**, I) > 0, then H(^{*}**x**, I) is positive definite at^{*}**x**^{*}; - (2)
- If H
_{i}(**x**, I) < 0 and the remaining H^{*}_{i}(**x**, I) alternate in sign, then H(^{*}**x**^{*}, I) is negative definite at**x**^{*}; - (3)
- If some of the values which are supposed to be nonzero turn out to be zero, then H(
**x**^{*}, I) can be positive semi-definite or negative semi-definite.

**x**

^{*}in neutrosophic nonlinear objective function F(

**x**, I) for I ∈ [I

^{*}^{L}, I

^{U}] can be determined by the following categories:

- (1)
**x**^{*}is a local maximum if ∇F(**x**^{*}, I) = 0 and H(**x**^{*}, I) is negative definite;- (2)
**x**^{*}is a local minimum if ∇F(**x**^{*}, I) = 0 and H(**x**^{*}, I) is positive definite;- (3)
**x**^{*}is a saddle point if ∇F(**x**^{*}, I) = 0 and H(**x**^{*}, I) is neither positive semi-definite nor negative semi-definite.

**Example**

**2.**

_{1}and x

_{2}is $F(\mathit{x},I)={x}_{1}^{2}+{x}_{2}^{2}-4I{x}_{1}^{}-2I{x}_{2}^{}+5$ for

**x**∈ Z

^{2}and I ∈ [I

^{L}, I

^{U}]. According to optimal conditions, we first obtain the following derivative and the optimal solution:

**x**

^{*}= [2I, I]

^{T}and the minimum value of the NN objective function is F(

**x**

^{*}, I) = 5(1 − I

^{2}) in this optimization problem.

**x**is x

_{1}

^{*}= [0, 2] and x

_{2}

^{*}= [0, 1] and the minimum value of the NN objective function is F(

**x**

^{*}, I) = [0, 5]. Specifically, when I = 1 is a determinate value, then x

_{1}

^{*}= 2, x

_{2}

^{*}= 1, and F(

**x**

^{*}, I) = 0. In this case, the NN nonlinear optimization is reduced to the traditional nonlinear optimization, which is a special case of the NN nonlinear optimization.

#### 4.3. NN-NP Problem Having Equality Constraints

**x**, I)

s.t. h

_{j}(

**x**, I) = 0, j = 1, 2, …, l

**x**∈ Z

^{n}

_{1}(

**x**, I), h

_{2}(

**x**, I), …, h

_{l}(

**x**, I): Z

^{n}→ Z and I ∈ [I

^{L}, I

^{U}].

_{j}(j = 1, 2, …, l) is a Lagrange multiplier and I ∈ [I

^{L}, I

^{U}]. It is obvious that this method transforms the constrained optimization into unconstrained optimization. Then, the necessary condition for this case to have a minimum is that:

**x**

^{*}= [x

_{1}

^{*}, x

_{2}

^{*}, …, x

_{n}

^{*}]

^{T}and the optimum multiplier values λ

_{j}

^{*}(j = 1, 2, …, l).

**Example**

**3.**

_{1}= −λ/(4I), x

_{2}= −3λ/10, and λ = −12I

^{2}/(1 + 1.8I). Hence, the NN optimal solution is obtained by the results of x

_{1}

^{*}= 3I/(1 + 1.8I) and x

_{2}

^{*}= 18I

^{2}/(5 + 9I). If the indeterminacy I ∈ [1, 2] is considered as a possible interval range, then the optimal solution is x

_{1}

^{*}= [0.6522, 4.2857] and x

_{2}

^{*}= [0.7826, 5.1429]. Specifically, if I = 1 (crisp value), then the optimal solution is x

_{1}

^{*}= 1.0714 and x

_{2}

^{*}= 1.2857, which are reduced to the crisp optimal solution in classical optimization problems.

#### 4.4. General Constrained NN-NP Problems

**x**, I)

s.t. g

_{k}(

**x**, I) ≤ 0, k = 1, 2, …, m

h

_{j}(

**x**, I) = 0, j = 1, 2, …, l

**x**∈ Z

^{n}

_{1}(

**x**, I), g

_{2}(

**x**, I), …, g

_{m}(

**x**, I), h

_{1}(

**x**, I), h

_{2}(

**x**, I), …, h

_{l}(

**x**, I): Z

^{n}→ Z for I ∈ [I

^{L}, I

^{U}]. Then, we can consider the NN Lagrangian function for the NN-NP problem:

_{k}≥ 0 for k = 1, 2, …, m.

**Example**

**4.**

_{1}= μ/(2I), x

_{2}= μ/4, and μ = 4I

^{2}/(2 + I) (μ = 0 yields an infeasible solution for I > 0). Hence, the NN optimal solution is obtained by the results of x

_{1}

^{*}= 2I/(2 + I) and x

_{2}

^{*}= I

^{2}/(2 + I).

_{1}

^{*}= [0.5, 1.3333] and x

_{2}

^{*}= [0.25, 1.3333]. As another case, if the indeterminacy I ∈ [2, 3] is considered as a possible interval range corresponding to some specific actual requirement, then the optimal solution is x

_{1}

^{*}= [0.8, 1.5] and x

_{2}

^{*}= [0.8, 2.25]. Specifically, if I = 2 (a crisp value), then the optimal solution is x

_{1}

^{*}= 1 and x

_{2}

^{*}= 1, which is reduced to the crisp optimal solution of the crisp/classical optimization problem.

^{L}, I

^{U}] and show the flexibility and rationality under indeterminate/NN environments, which is the main advantage of the proposed NN-NP methods.

## 5. Conclusions

^{L}, I

^{U}] is considered as a possible interval range for real situations and actual requirements, and (3) NN-NP is the generalization of traditional nonlinear programming problems and is more flexible and more suitable than the existing unconcerned nonlinear programming methods under indeterminate environments. The proposed NN-NP methods provide a new effective way for avoiding crisp solutions of existing unconcerned programming methods under indeterminate environments.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ye, J.; Cui, W.; Lu, Z.
Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments. *Axioms* **2018**, *7*, 13.
https://doi.org/10.3390/axioms7010013

**AMA Style**

Ye J, Cui W, Lu Z.
Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments. *Axioms*. 2018; 7(1):13.
https://doi.org/10.3390/axioms7010013

**Chicago/Turabian Style**

Ye, Jun, Wenhua Cui, and Zhikang Lu.
2018. "Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments" *Axioms* 7, no. 1: 13.
https://doi.org/10.3390/axioms7010013