# Neutrosophic integer programming problem.

1 Introduction

In linear programming models, decision variables are allowed to be fractional. For example, it is reasonable to accept a solution giving an hourly production of automobiles at 64 1/2, if the model were based upon average hourly production. However, fractional solutions are not realistic in many situations and to deal with this matter, integer programming problems are introduced. We can define integer programming problem as a linear programming problem with integer restrictions on decision variables. When some, but not all decision variables are restricted to be integer, this problem called a mixed integer problem and when all decision variables are integers, it's a pure integer program. Integer programming plays an important role in supporting managerial decisions. In integer programming problems the decision maker may not be able to specify the objective function and/or constraints functions precisely. In 1995, Smarandache [1-3] introduce neutrosophy which is the study of neutralities as an extension of dialectics. Neutrosophic is the derivative of neutrosophy and it includes neutrosophic set, neutrosophic probability, neutrosophic statistics and neutrosophic logic. Neutrosophic theory means neutrosophy applied in many fields of sciences, in order to solve problems related to indeterminacy. Although intuitionistic fuzzy sets can only handle incomplete information not indeterminate, the neutrosophic set can handle both incomplete and indeterminate information.  Neutrosophic sets characterized by three independent degrees as in Fig. 1., namely truth-membership degree (T), indeterminacy-membership degree(I), and falsity-membership degree (F), where T,I,F are standard or non-standard subsets of ]-0, I+[. The decision makers in neutrosophic set want to increase the degree of truth-membership and decrease the degree of indeterminacy and falsity membership.

The structure of the paper is as follows: the next section is a preliminary discussion; the third section describes the formulation of integer programing problem using the proposed model; the fourth section presents some illustrative examples to put on view how the approach can be applied; the last section summarizes the conclusions and gives an outlook for future research.

2 Some Preliminaries

2.1 Neutrosophic Set 

Let X be a space of points (objects) and x[member of]X. A neutrosophic set A in X is defined by a truth-membership function (x), an indeterminacy-membership function (x) and a falsity-membership function [F.sub.A](x). (x), I(x) and F(x) are real standard or real nonstandard subsets of ]0-,1+[. That is [T.sub.A](x):X[right arrow]]0-,1+[, [I.sub.A](x):X[right arrow]]0-1+[ and [F.sub.A](x):X[right arrow]]0-,1+[. There is no restriction on the sum of (x), (x) and [F.sub.A](x). so 0-[less than or equal to]sup(x)[less than or equal to]sup[I.sub.A](x)[less than or equal to][F.sub.A](x)[less than or equal to]3+.

2.2 Single Valued Neutrosophic Sets (SVNS) [3-4]

Let X be a universe of discourse. A single valued neutrosophic set A over X is an object having the form A={x, T{x), [I.sub.A](x),[F.sub.A](x)):x[member of]X}.

where [T.sub.A](x):X[right arrow][0,1], [I.sub.A](x)X[right arrow][0,1] and [F.sub.A](x):X[right arrow][0,1] with -[less than or equal to][T.sub.A](x)+ [I.sub.A](x)+[F.sub.A](x)[less than or equal to]3 for all x[member of]X. The intervals T(x), I(x) and [F.sub.A]{x) denote the truth-membership degree, the indeterminacy-membership degree and the falsity membership degree of x to A. respectively.

In the following, we write SVN numbers instead of single valued neutrosophic numbers. For convenience, a SVN number is denoted by A= (a,b,c), where a.b.c[member of][0,1] and a+b+c[less than or equal to]3.

2.3 Complement 

The complement of a single valued neutrosophic set A is denoted by c (A) and is defined by

[T.sub.c](A)(x) = F(A)(x), [I.sub.c](A)(x) = 1 - I (A)(x), [F.sub.c](A)(x) = T(A)(x) for all x in X

2.4 Union 

The union of two single valued neutrosophic sets A and B is a single valued neutrosophic set C, written as C = AUB, whose truth-membership, indeterminacy membership and falsity-membership functions are given by

T(C)(x) = max (T(A)(x), T(B)(x)), I(C)(x) = max (I(A)(x), I(B)(x)), F(C)(x) = min((A)(x),F(B)(x)) for all X in X

2.5 Intersection 

The intersection of two single valued neutrosophic sets A and B is a single valued neutrosophic set C, written as C = A[intersection]B, whose truth-membership, indeterminacy membership and falsity-membership functions are given by

T(C)(x) = min(T(A)(x),T(B)(x)), I(C)(x) = min (I(A)(x),I(B)(x)), F(C)(x) = max((A)(x),F(B)(x)) for all X in X

3 Neutrosophic Integer Programming Problems

Integer programming problem with neutrosophic coefficients (NIPP) is defined as the following:

Maximize Z= [[summation].sup.n.sub.j=1] [??].sub.j][x.sub.j]

Subject to

[[summation].sup.n.sub.j=1] [a.sup.~n.sub.ij, [x.sub.j] [less than or equal to] [b.sub.i] i = 1, ..., m, [x.sub.j] [greater than or equal to] 0, j = 1, ... n, [x.sub.j] integer for j [member of] {0,1, ... n}. (1)

Where [[??].sub.j], [a.sup.~n.sub.ij] are neutrosophic numbres.

The single valued neutrosophic number ([a.sup.~n.sub.ij]) is donated by A=(a,b,c) where a,b,c [member of] [0,1] And a,b,c [less than or equal to] 3 The truth- membership function of neutrosophic number [a.sup.~n.sub.ij] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The indeterminacy- membership function of neutrosophic number [a.sup.n.sub.ij] defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

And its falsity- membership function of neutrosophic number [a.sup.~n.sub.ij] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Then we find the maximum and minimum values of the objective function for truth-membership, indeterminacand falsity membership as follows:

[f.sub.max] = max{f([x.sup.*.sub.i])} and [f.sup.min] =min{f([x.sup.*.sub.i])} where 1[less than or equal to] i [less than or equal to] k [f.sup.F.sub.min]=[f.sup.T.sub.min] and [f.sup.F.sub.max]=[f.sup.T.sub.max] - R([f.sup.T.sub.max] - [f.sup.T.sub.min]) [f.sup.I.sub.max]=[f.sup.I.sub.max] and [f.sup.I.sub.min]=[f.sup.I.sub.min] - S([f.sup.T.sub.max] - [f.sup.T.sub.min])

Where R,S are predetermined real number in (0,1) The truth membership, indeterminacy membership, falsity membership of objective function as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The neutrosophic set of the jth decision variable [X.sub.j] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Where [d.sub.j], [b.sub.j] are integer numbers.

4 Neutrosophic Optimization Model of integer programming problem

In our neutrosophic model we want to maximize the degree of acceptance and minimize the degree of rejection and indeterminacy of the neutrosophic objective function and constraints. Neutrosophic optimization model can be defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Where [T.sub.(x)]. [F.sub.(x)], [I.sub.(x)] denotes the degree of acceptance, rejection and indeterminacy of x respectively.

The above problem is equivalent to the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Where [alpha] denotes the minimal acceptable degree, [beta] denote the maximal degree of rejection and [theta] denote maximal degree of indeterminacy.

The neutrosophic optimization model can be changed into the following optimization model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The previous model can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

5 The Algorithms for Solving Neutrosophic integer Programming Problem (NIPP)

5.1 Neutrosophic Cutting Plane Algorithm

Step 1: Convert neutrosophic integer programming problem to its crisp model by using the following method:

By defining a method to compare any two single valued triangular neutrosophic numbers which is based on the score function and the accuracy function. Let [??] = <([a.sub.1],[b.sub.1],[c.sub.1]), [w.sub.[??]], [u.sub.[??]], [y.sub.[??]]} be a single valued triangular neutrosophic number, then

S([??]) = 1/16[a + b + c]x(2 + [[mu].sub.[??]] - [v.sub.[??]] - [[lambda].sub.[??]]) (15)

and

A([??]) = 1/16 [a + b + c]x(2 + [[mu].sub.[??]] - [v.sub.[??]] + [[lambda].sub.[??]]) (16)

is called the score and accuracy degrees of [??], respectively. The neutrosophic integer programming NIP can be represented by crisp programming model using truth membership, indeterminacy membership, and falsity membership functions and the score and accuracy degrees of [??], at equations (15) or (16).

Step 2: Create the decision set which include the highest degree of truth-membership and the least degree of falsity and indeterminacy memberships.

Step 3: Solve the problem as a linear programming problem and ignore integrality.

Step 4: If the optimal solution is integer, then it's right. Otherwise, go to the next step.

Step 5: Generate a constraint which is satisfied by all integer solutions and add this constraint to the problem.

Step 6: Go to step 1.

5.2 Neutrosophic Branch and Bound Algorithm

Step 1: Convert neutrosophic integer programming problem to its crisp model by using the following method:

By defining a method to compare any two single valued triangular neutrosophic numbers which is based on the score function and the accuracy function. Let [??] = <([a.sub.1], [b.sub.1], [c.sub.1]), [w.sub.[??]], [u.sub.[??]], [y.sub.[??]]> be a single valued triangular neutrosophic number, then

S([??]) = 1/16 [a + b + c]x(2 + [[mu].sub.[??]] - [v.sub.[??]] - [[lambda].sup.[??]]) (15)

and

A([??]) = 1/16 [a + b + c]x(2 + [[mu].sub.[??]] - [v.sub.[??]] + [[lambda].sub.[??]]) (16)

is called the score and accuracy degrees of [??], respectively. The neutrosophic integer programming NIP can be represented by crisp programming model using truth membership, indeterminacy membership, and falsity membership functions and the score and accuracy degrees of [??], at equations (15) or (16).

Step 2: Create the decision set which include the highest degree of truth-membership and the least degree of falsity and indeterminacy memberships.

Step 3: At the first node let the solution of linear programming model with integer restriction as an upper bound and the rounded-down integer solution as a lower bound.

Step 4: For branching process, we select the variable with the largest fractional part. Two constrains are obtained after the branching process, one for [less than or equal to] and the other is [greater than or equal to] constraint.

Step 5: Create two nodes for the two new constraints.

Step 6: Solve the model again, after adding new constraints at each node.

Step 7: The optimal integer solution has been reached, if the feasible integer solution has the largest upper bound value of any ending node. Otherwise return to step 4.

The previous algorithm is for a maximization model. For a minimization model, the solution of linear programming problem with integer restrictions are rounded up and upper and lower bounds are reversed.

6 Numerical Examples

To measure the efficiency of our proposed model we solved many numerical examples.

6.1 Illustrative Example #1

max [??][x.sub.1] + [??][x.sub.2]

[??][x.sub.1] + [??][x.sub.2] [less than or equal to] [??]

subject to [??][x.sub.1] + [??][x.sub.2] [less than or equal to] [??]

[x.sub.1], [x.sub.2] [greater than or equal to] 0 and integer

where

[??]= <(4,5,6), 0.8, 0.6,0.4>

[??]= <(2.5,3,3.5), 0.75,0.5,0.3>

[??]= <(3.5,4,4.1), 1,0.5, 0.0>

[??]= <(2.5,3,3.5), 0.75,0.5,0.25>

[??]= <(0,1,2), 1,0.5,0>

[??]= <(2.8,3,3.2), 0.75,0.5,0.25>

[??]= <(11,12,13), 1,0.5,0>

[??]= <(5.5,6,7.5), 0.8,0.6,0.4>

Then the neutrosophic model converted to the crisp model by using Eq. 15, Eq. 16.as follows:

max 5.6875[x.sub.1] + 3.5968[x.sub.2] 4.3125[x.sub.1] + 3.625[x.sub.2] [less than or equal to] 14.375

subject to 0.2815[x.sub.1] + 3.925[x.sub.2] [less than or equal to] 7.6375 [x.sub.1],[x.sub.2] [greater than or equal to] 0 and integer

The optimal solution of the problem is [x.sub.*] = (3,0) with optimal objective value 17.06250.

6.2 Illustrative Example #2

max [??][x.sub.1] + [??][x.sub.2] 15[x.sub.1] + 30[x.sub.2] [less than or equal to] 45000 24[x.sub.1] + 6[X.sub.2] [less than or equal to] 24000

subject to 21[x.sub.1] + 14[x.sub.2] [less than or equal to] 28000 [x.sub.1], [x.sub.2] [greater than or equal to] 0 and integer

where

[??]= <(19,25,33), 0.8,0.5,0 >; [??]= <(44,48,54), 0.9,0.5,0 >

Then the neutrosophic model converted to the crisp model as:

max 27.8875x1 + 55.3[x.sub.2] 15[x.sub.1] + 30[x.sub.2] [less than or equal to] 45000 24[x.sub.1] + 6[X.sub.2] [less than or equal to] 24000

subject to 21[x.sub.1] + 14[x.sub.2<] [less than or equal to] 28000 [x.sub.1], [x.sub.2] [greater than or equal to] 0 and integer

The optimal solution of the problem is [x.sub.*] = (500,1250) with optimal objective value 83068.75.

7 Conclusions and Future Work

In this paper, we proposed an integer programming model based on neutrosophic environment, simultaneously considering the degrees of acceptance, indeterminacy and rejection of objectives, by proposed model for solving neutrosophic integer programming problems (NIPP). In the model, we maximize the degrees of acceptance and minimize indeterminacy and rejection of objectives. NIPP was transformed into a crisp programming model using truth membership, indeterminacy membership, falsity membership and score functions. We also give numerical examples to show the efficiency of the proposed method. Future research directs to studying the duality theory of integer programming problems based on Neutrosophic.

References

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 Abdel-Baset, M., Hezani, I. M., & Smarandache, F. (2016). Neutrosophic Goal Programming. Neutrosophic Sets & Systems, 11.

 Smarandache, F. "A Geometric Interpretation of the Neutrosophic Set-A Generalization of the Intuitionistic Fuzzy Set. " arXiv preprint math/0404520(2004).

 R. Sahin, and Muhammed Y. "A Multi-criteria neutrosophic group decision making metod based TOPSIS for supplier selection." arXiv preprint arXiv: 1412.5077 (2014).

Received: January 6, 2017. Accepted: January 30, 2017.

Mai Mohamed (1), Mohamed Abdel-Basset (1), Abdel Nasser H Zaied (2) and Florentin Smarandache (3)

(1) Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt.

E-mail: analystmohamed@yahoo.com

(2) Department of information system, Faculty of Computers and Informatics, Zagazig University, Egypt. Email:nasserhr@gmail.com

(3) Math & Science Department, University of New Mexico, Gallup, NM 87301, USA. E-mail: smarand@unm.edu

Caption: Figure 1: Neutrosophication process
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