Fuzzy unconstrained Parametric Geometric programming problem and its application

In this paper, we have proposed fuzzy unconstrained geometric programming (GP) problem and modified geometric programming (MGP) problem with negative or positive integral degree of difficulty. Geometric programming technique provides a powerful tool for solving optimization problems. Here we use nearest interval approximation method to convert a triangular fuzzy number to an interval number. In this paper, we transform this interval number to a parametric interval-valued functional form and then solve the parametric problem by geometric programming technique. Here some necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications.


Introduction
Since late 1960's, Geometric Programming (GP) used in various field (like OR, Engineering science etc.).Geometric Programming (GP) is one of the effective methods to solve a particular type of Non linear programming problem.The theory of Geometric Programming (GP) first emerged in 1961 by Duffin and Zener.The first publication on GP was published by Duffin and Zener on (1967).There are many references on applications and methods of GP in the survey paper by Ecker.They describe GP with positive or zero degree of difficulty.But there may be some problems on GP with negative degree of difficulty.Sinha, et al., proposed it theoretically.Abot-El-Ata and his group applied modified form of GP in inventory models.S. Islam, T.K. Roy [17] (2006) presented modified geometric programming (MGP) and its applications, they also presented Multi-Objective Geometric-Programming Problem and its Application.http://www.ispacs.com/journals/jfsva/2016/jfsva-00301/International Scientific Publications and Consulting Services In 1990 R.K.Varma has studied fuzzy programming technique to solve geometric programming problem.Biswal (1992) developed fuzzy programming with non linear membership functions approach to multiobjective GP problems.Cao [2] (1992), is the first one to transform Geometric programming problem (GP) to its corresponding fuzzy state and has shown that Fuzzy programming is an useful method to solve multiobjective optimization problem.In this filed a paper named geometric programming problem with fuzzy parameters and its application to crane load sway by S. Yousuf, N. Badra and T.G Abu-El Yazied has been published in world applied science journal in 2009.Bit developed fuzzy programming with hyperbolic member functions to solve GP with several objective functions.A solution method of posynomial geometric programming with interval exponents and coefficients was developed by Liu [12] (2008).Kotba, Halla, Fergancy [8] (2011), presented Multi-item EOQ model with both demand depended unit cost and varying Lead time via Geometric Programming.Samir Dey and Tapan Kumar Roy [14] (2015) presented Optimum shape design of structural model with imprecise coefficient by parametric geometric programming.In this paper we have proposed fuzzy unconstrained GP and MGP problem with negative or positive integral degree of difficulty.Here we transform interval number to parametric interval-valued functional form and then fuzzy geometric programming problem becomes fuzzy parametric geometric programming problem, for that purpose some necessary theorems have been derived.Finally, these are illustrated by numerical examples and applications.
2 Fuzzy number and its nearest interval approximation: 2.1.Fuzzy number A real number  ̃ described as fuzzy subset on the real line ℛ whose membership function   ̃() has the following characteristics with Where   ̃(): [ is the center and   = is the half-width of A.

Nearest interval approximation
Here we want to approximate a fuzzy number by a crisp model.Suppose  ̃ and  ̃ are two fuzzy numbers with α-cuts are [AL(α), AR(α)] and [BL(α), BR(α)], respectively.Then the distance between  ̃ and  ̃ is Given  fuzzy number  ̂, We have to find a closed interval   ( ̃), which is closest to  ̂ with respect to some metric.We can do it since each interval is also a fuzzy number with constant α-cut for all α ∈ [0 (a 1 + 2a 2 + a 3 ), 1 4 (a 3 −a 1 )〉.

Parametric Interval-valued function
Let [m, n] be an interval, where m > 0, n > 0. From analytical geometry point of view, any real number can be represented on a line.Similarly, we can express an interval by a function.The parametric interval-valued function for the interval [m, n] can be taken as g(s) = Case-1: For T0 ≥ M+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables.More or unique solution exist for the dual vectors.Case-2: For T0 < M+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables.In this case generally no solution vectors exist for the dual variables.However one can get an approximate solution vector for the system using either the Latest Square (SQ) or Max-Min (MN) method.These are applied to solve such a system of linear equations.Ones optimal dual variable vector  * are known, the corresponding values of the primal variable vector x is found from the following relations: =   *  * ( * , ), (k=1,2, ……..,  0 ).(3.5) Theorem 3.1.If x is a feasible vector for the constraints PGP and δ is a feasible vector for the corresponding DP, then Proof.The expression for g0(x,s) can be written as ).
Case-1: For nT0 ≥n M+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables.More or unique solution exist for the dual vectors.Case-2: For nT0 < nM+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables.In this case generally no solution vector exists for the dual variables.However one can get an approximate solution vector for the system using either the Latest Square (SQ) or Max-Min (MN) method.These are applied to solve such a system of linear equations.Once optimal dual variable vector  * are known, the corresponding values of the primal variable vector x is found from the following relations: Proof.The expression for   (x,s) can be written as Or ∏ ∏ ( i.e.,   (δ,1) ≥   (δ,0).This completes the proof.This problem can be formulated as Sub to  1 ≥ 0,  2 ≥ 0 ,  3 ≥ 0.
Let the input values are
The optimal solution of the model through the parametric approach is given by  * () = ( From primal dual relation we get The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 2.Here the primal problem is subject to , , , .
The optimal solution of the model through the parametric approach is given by From primal dual relation we get The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 3.If we compare crisp solution and fuzzy solution, we see that  0 (x,s) * ≤  0 (x) * for s≤ 4, and  0 (x,s) * ≥  0 (x) * for s≥ 6.

MGP problem (Multi-Grain-box problem)
Suppose that to shift grains from a warehouse to a factory in a finite number (say n) of open rectangular boxes of lengths  1 meters, widths  2 meters and heights  3 meters (i=1,2,….,n).From primal dual relation we get .
The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 5.From primal dual relation we get (75) .The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 6.for all s ∈ [0,1], so the Theorem 3.5 is verified.For s=0, the lower bound of the interval value of the parameter is used to find the optimal solution.For s=1, the upper bound of interval value of the parameter is used for the optimal solution.These results yield the lower and upper bounds of the optimal solution.The main advantage of the proposed technique is that one can get the intermediate optimal result using proper value s.

Conclusion
The advantage of this technique is that we can find directly optimal solution of the objective function without solving two-level mathematical programs.This method is simple and takes minimal time.In this paper, we developed unconstrained fuzzy parametric geometric programming problem.In fuzzy we have considered triangular fuzzy number (T.F.N).In future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number, Trapezoidal Fuzzy Number (TrFN), Parabolic flat Fuzzy Number (PfFN), Parabolic Fuzzy Number (pFN), pentagonal fuzzy number, etc., can be considered to construct the membership function and then model can be easily solved.

Theorem 3 . 5 .
If x is a feasible vector for the constraints PGP and δ is a feasible vector for the corresponding DP, then   (x,s) ≥ n √  (, )  (Primal-Dual Inequality).
optimal solution of the model through the parametric approach is given by  * () = (

3 Unconstrained problem 3.1. Geometric Programming problem with fuzzy coefficient:
This is a parametric geometric programming (PGP) problem.And corresponding dual programming (DP) problem of (3.3) is Max This completes the proof.http://www.ispacs.com/journals/jfsva/2016/jfsva-00301/International Scientific Publications and Consulting Services these maximum and minimum value.i.e., we get an interval of solutions which contained all the values of   * .Here  0 are real numbers and coefficients ̃  are fuzzy triangular numbers, as ̃  =( 1  ,  2  ,  3  ).Using nearest interval approximation method, we transform all triangular fuzzy number into interval number i.e., [   ,    ].The geometric programming problem with imprecise parameters is of the following form http://www.ispacs.com/journals/jfsva/2016/jfsva-00301/Here the weights are  1 ,  2 , … … … ,   0 and positive terms are i.e.,   (x,s) ≥ n √d i (δ, s)  .This completes the proof.

Table 3 :
Optimal Solution of the model (fuzzy method) The bottom, side and http://www.ispacs.com/journals/jfsva/2016/jfsva-00301/InternationalScientificPublications and Consulting Services ends of the box cost $  , $  and $  / 2 respectively.It cost $1 for each round trip of the box.Assuming that the box will have no salvage value, find the minimum cost of transporting    3 of grains.In particular here we assume that the transporting   3of grains by the two different open rectangular boxes whose bottom, sides, and the ends of each box costs are give in table 4.