Quadratic Fractional Bi-level Fuzzy Probabilistic Programming Problem When 𝒃𝒊 Follows Exponential Distribution Текст научной статьи по специальности «Математика»

Quadratic Fractional Bi-level Fuzzy Probabilistic Programming Problem When bt Follows Exponential

Distribution

1Haneefa Kausar, 2Ahmad Yusuf Adhami, 3*Ahmadur Rahman

Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, UP, India. Email: [email protected], [email protected], "'ahmadur. [email protected] Corresponding Author

Abstract

Some of the actual life decisions are made in decentralized manner under uncertainty. This paper formulates a quadratic fractional bi-level (QFBL) programming problem with probabilistic constraints in both first (leader) and second level (follower) having two parameter exponential random variables with known probability distributions and fuzziness is considered as triangular and trapezoidal fuzzy number. These fuzzy numbers of the membership functions related with the proportional probability density function has been used to introduce a defuzzification approach for finding the crisp values of fuzzy numbers. In the proposed model the problem is first converted into an equivalent deterministic quadratic fractional fuzzy bi level programming model by applying chance constrained programming technique. Secondly, in the suggested model, each objective function of the bi-level quadratic fractional programming problem has its own nonlinear membership function. The fuzzy goal programming (FGP) approach is used to find a compromise solution for the BLQFP problem. Finally, to demonstrate the applicability and performance of the proposed approach an illustrative numerical example is given.

Keywords: Bi-level programming, Quadratic programming, Fractional programming, Two parameter exponential distribution, Fuzzy chance constrained programming, Fuzzy goal programming.

I. Introduction

In actual life decision-making situations, decision-makers are frequently confronted with different kinds of vagueness, the most important of which are randomness and fuzziness. There are two common approaches to dealing with such uncertainties: probability theoretical approach and fuzzy set theoretical approach. Chance constrained programming is a well-defined approach described by Charnes and Cooper [1] for dealing with problems involving probabilistic data (CCP). Gardening, capability planning, banking, forestry, army, manufacture control and arrangement, sports, broadcastings, transport, and eco-friendly management planning are just a few of the fields where it is commonly used.

Quadratic programming (QP) is a technique of solving certain mathematical optimization problems involving quadratic function. Specially a quadratic programming minimized or maximized a multivariate quadratic function subject to linear constraint on the variables.

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62)

PROGRAMMING PROBLEM_Volume 16, March 2021

Quadratic programming is a type of nonlinear programming of the several. A wide range of applications of quadratic programming is portfolio selection, electrical energy growth, agriculture, and harvest selection. Probabilistic quadratic programming is applicable for financial and risk management, and various important related literatures are found in this direction, which are mentioned below.

Mc Carl et al. [2] presented some of the approaches during which QP are often used. Interval parameters are used to represent the cost coefficients, constraint coefficients, and right-hand sides in an interval quadratic programming problem proposed by Liu and Wang [6]. A fuzzy quadratic programming problem was introduced in [7] where fuzzy data represents the cost coefficients, constraint coefficients, and right-hand side values. Kausar and Adhami [18] have given a fuzzy goal programming approach for solving chance constrained bi-Level multi-objective quadratic fractional programming problems. Ammar [9] proposed a multi-objective quadratic programming problem based on the fuzzy random coefficient matrix's goals and constraints where the decision vector interpreted as a fuzzy vector. The problem of fuzzy quadratic programming was introduced by Liu [10] in which convex fuzzy numbers represent the cost coefficients, constraint coefficients, and constraints parameters on the right side. He [11] also specified a stream water quality control solution process. Qin and Huang [12] suggested an inexact chance constrained quadratic programming model. Nasseri [13] outlined a fuzzy quadratic programming problem with trapezoidal and/or triangular fuzzy numbers for the cost coefficients, constraint coefficients, and right-hand parameter values. Guo and Huang [14] developed an incorrect fuzzy-stochastic quadratic programming approach to efficiently distribute waste to a municipal solid waste management scheme while accounting for the nonlinear objective function and various parameter uncertainties in the constraints. Bi-level multi-objective stochastic linear fractional programming with general form of distribution has been developed by Kausar and Adhami [17] In this paper, we are constructing a different approach for solving quadratic fractional bi level programming problems with probabilistic constraints having two parameter exponential distributed fuzzy random variables with known probability distributions. The probabilistic problem is changed into an equivalent deterministic model. Under the quadratic fractional programming outline, both uncertainty and fuzziness are considered. Poularikas [16] proposes a defuzzification technique for determining the crisp values of fuzzy numbers using the Mellin transformation.

I. Probabilistic Fuzzy Quadratic Fractional Bi-Level Programming Problem

In some circumstances, quadratic fractional programming with a quadratic fractional objective function and few linear constraints including fuzziness and randomness is called a probabilistic fuzzy quadratic fractional programming problem. When a number of the input parameters of QFP are described by stochastic and fuzzy parameters, the problem is treated as a probabilistic fuzzy quadratic fractional programming problem. A general probabilistic fuzzy quadratic fractional bilevel programming problem is presented as follows

- 1 T3 ~ f i ci1xj + 9 x "■i1xj + ai1 Max fi (xi, X2) = ff1 =-—2----(1.1)

x

fl2 rv.+l

Ci2Xj + 2 xTdi2Xj + Pi2 where for given X1,X2

f CiXj + iXTdLiiXj + &n

Maxf2(xi,x2)=f=i 2 t_ --(1.2)

2 22 Ci2Xj + 2XT ^i2Xi + Pi2

XEG = [X eRn\P(YIj=1aijXj <bi)>1-Xi, i = 1,2,.....m} (1.3)

Xj>0, j = 1,2,...,n (1.4) where 0<Xt<1 and fy (i=1,2,...,m) represent two parameter exponential distribution fuzzy random variables; ci1,ci2,cti1,cti2 (i=1,2,...,m) and ay are considered as fuzzy number and ^ e[0,1].

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62) PROGRAMMING PROBLEM_Volume 16, March 2021

The decision vector X1 = (x11,x12, ...,x1ni) is controlled by the leader and decision vector X2 =

(x21, x22,..., x2n2) is controlled by the follower; X1U X2 = X = (x-l, x2,..., xn) £ Rn with n1 + n2 = n.

II. Some Preliminaries

In the model formulation the triangular and trapezoidal membership functions are discussed in this section. We can also introduce the Mellin transform to find the expected value of a random variable's function by using proportional probability density function related with membership functions of fuzzy numbers.

Definition 2.1 (triangular fuzzy number): A triangular fuzzy number is one that is represented by the triplet A = (a1, a2,a3) and has a piecewise linear membership function /¿¿(x) is given by

^a(X) = <

x — a-,

a-2 — a1 as—x

ao — a2

, a1<x<a2 , a2 < x < a3

(2.1)

0, otherwise

Definition 2.2 (trapezoidal fuzzy number): A fuzzy number represented by the quadruplet A = (a1, a2,a3, a4) and has a piecewise linear membership function n¿(x) is given by

Wto = <

-, ai <x<a2

a2—a-i

1, a2 < x < a3

0.4-X

a3 < x < a4 otherwise

(2.2)

a4-a3

0,

V

I. Defuzzification with Probability Density Function and Membership Function

Assume that F(R) represent the sum of all fuzzy numbers. In (R) the triangular and trapezoidal fuzzy numbers are (a1,a2,a3) and (a1,a2,a3,a4) respectively. Now the method associated with a probability density function for the membership function of A is defined as follows ([15], [4]).

Proportional probability distribution: describe a probability density function f1 = c^^x) associated with A, where the constant c is obtained by using the property of probability density function, where J<f1(x)dx = 1 and f<c^£(x)dx = 1.

II. Mellin Transform

The Mellin transform ([15], [4]) is used to find this expected value since any probability density function with finite support is associated with expected value.

Definition 2.3: The Mellin transform Mx(t) of a probability density function f(x), where x denote the positive, is given as

n <

Mx(t) = I xt-1f(x)dx (2.3)

Jo

Now in terms of expected values we find the Mellin transform. Remember that the expected value of any function g(X) of the random variable X whose probability density function is f(x), is given as

Efo(X)] = I g(x)f(x)dx (2.4)

Thus, it follows that Mx(t) = E[Xt-1] = j<xt-1f(x) dx.

Hence, E[Xf] = Mx(t + 1). Thus the expected value of random variable X is E[X] = Mx(2).

For example, if the triangular and trapezoidal fuzzy numbers are Ji1 = (a1,a2,a3) and A2 =

(a1,a2,a3,a4) respectively, then their crisp values are determined by finding expected values

using the probability density function that corresponding to the membership functions of the given fuzzy number.

Thus, the probability density function corresponding to triangular fuzzy number AA1 = (a1,a2,a3) is given as

(2.5)

fAi(x) = Ci^Ai(x) where ¡iAi1(x) is defined as

m, CO = <

x — -a1

a2 — a1

a3 — x

a3 — a2

0,

a1 < x < a2

a2 < x < a3 o therw ise

Now is calculated as

I fA1(x)dx = l

J — on

that is

that is

f

■J — (

c^z (x)dx = l

f

a2 x —

x — a.

-dx + c-

f

°3 a3 — x

dx = l

a2 a3 a2

(2.6)

(2.7)

(2.8) (2.9)

(2.10)

a, a2 a1

On integration, we get 2

C1 = -

1 a3 — a1

The proportional probability function corresponding to triangular fuzzy number A is given by

2(x — a1)

T-77-r, a1 < x <a2

( a2 — a1)( a3 — a1) 1 2

(2.11)

a2 < x < a3

fx,(x) = \

2 — a1)( a3 — a1)

2( a3 — x)

(a3 — a2)(a3 — a1Y

0, o therwise Graphically it is presented in Figure 1

v

0.4

---------------------

_^

Figure 5.1: Proportional probability density function of triangular fuzzy number

By using the Mellin transform, we obtain

Mx(t)=rxt-1fx„ (x) dx = fa2xt-1 2(x-ai) dx + px*-1 2(x-ai)

xw J0 ]xAiy ' Jai (a2-a1)(a3-a1) Ja2 (a2-a1)(a3-

On integration, we obtain

2 \a3(at3—at2) a^—aD

Mx~ (t) = 7--tt

Xai (a3 — a1)t(t + 1)

(a3 — a2) (a2 — aJ

(2.12) (2.13)

1

0

0.3

0

Figure 5.2: Proportional probability density function of trapezoidal fuzzy number

Thus the random variable X^ has mean (p.x_ ) and variance ) can be obtained as

vx^=E[XAi]=MX1i(2) =

a-, + a2+ a3

T

ax, =

^Kd =■

2 a2 + a2 + a2 — a1a2 — a2a3 — a3a1

(2.14)

Further, the probability density function corresponding to trapezoidal fuzzy number A2 = (a1, a2,a3, a4) is given as f^2 (x) = (x), where (x) is defined as

Va2(X) = *

x — a

1

a1<x<a2

a2 —

1, a2 <x <a3

4

aA — x

(2.16)

a4 — a

3

0,

Now c-, is calculated as

a3 < x < a4

otherwise

That is,

I-<< I-<<

fji2(x)dx = 1

C2^a, (x)dx = 1

That is,

L2

J'

a.2 — a.1

■dx + c2

ra3 ra,

I dx + C2 I

Ja2 ^a3

a4 — a3

■dx = 1

(2.17)

(2.18)

(2.19)

(2.20)

On integration, we get

2

c2 =-

a4 + a3-a1-a2

The proportional probability density function corresponding to triangular fuzzy number Ji1 is given by

fx~A*) =

2(x — a1)

(a2 — a1)(a4 + a3 — a1 — a2)' 2

(a4 + 0.3 — 0.1 — 02)' 2(x — a{)

a1<x < a2, a2 < x < a3

(2.21)

(a2 — a1)(a4 + a3 — a1 — a2) ^3<x < a4

0, otherwise Graphically it is shown in figure 5.2 Using the Mellin transform, we get

1

1

4

P œ

MÂ2(t) = J xt-i fx^(x)dx

= J '

Jo*

+ J X"

J"2

a3

2

(a4 + a3-a1- a2) On integration, we obtain

M^it) =

x

ai dx +

2(x — ai)

i.a-2 — a1)(a4 + a3 — a1 — a2) a4 2(a4—x)

dx

Ca4

J"3

(a4 — a3)(a4 + a3—ai — a.2)

dx

2 (at4+i — a3+i) at2¥i — ati+r

(a4 + a3 — ai — a2)t(t + 1) (au — a3) (a2 — ai) ,

Thus, the random variable X ¿¡2 has mean ) and variance (ax^ ) can be obtained as

/ux„2 = E[XA2] = MXl2 (2) = 1 [(«i + a2+a3 + a4) +

(aia2 — ^4)

Jx~.„ = >«x~Ai(3) — [Mx~A2(2)\

(a4 + a3 — a2 — %)

oL = My

*A2 1

(al + af+af + af) +

4

(ai + a2)(a3 + ai) — (a.3 + a.4)(al + a2)

(a4 + a3 — a2 — ^i)

+ (Mx~J2

"2

(222)

(2.23)

(2.24)

(2.25)

6

III. Probabilistic Fuzzy Quadratic Programming Problem and Its Crisp Model

Let Cj = (ci,cj, cj),j = 1,2,

.,n,ai3 = (a1ij,a2ij,al),i = 1,2, ...,m, j = 1,2,.

,n and qy = (qlj,qlj,ql),

'J V] '"7 ' J J'J ' ' — ••-•"■ij j —,•<■ ^^ "uj Wij'Vij'Vij

i = 1,2,... ,n,j = 1,2,... ,n denote the triangular fuzzy numbers. By using the method of defuzzification the crisp values of these fuzzy numbers can be obtained with probability density function of given membership function given as follows

c/ + cj + cj

3

j = 1,2,., n,

alj + a2 + a3

3

<hj

lh + tâ + tâ

= 1,2,..., m,j = 1,2,..., n,

i = 1,2, ...,m, j = 1,2,.

(2.26)

.,n,

where the crisp value of the given fuzzy number £j is represented by Cj and so on.

Similarly if all the coefficients have trapezoidal fuzzy numbers such as, £j = cj + cj + cj + cj,j =

1,2, ...,n, ai} = (aij + afj + afj + a4j), i = 1,2, ...,m,j = 1,2, ...,n and qy = (qi, qfj, qfj, qfj), i =

1,2,.

.,n,j = 1,2, 1

., n, then the crisp values are given as

c>=3

au = 3

(cj + cj2 + Cj3 + cj4) +

(cjcj—cj cj)

ai + afj + a3

(cJ+cJ "J

4W

i

+ <•) +

cj)

o-la4)

j = 1,2,..., n

<hj

(tij + tij + tij + qfj) +

(aij + a4i — ah — ^ ^irt — qMj)

+ lu — ih — îoOJ

i = 1,2, ...,m,j = 1,2, ...,n

i = 1,2, ...,n,j = 1,2, ...,n

Thus the probabilistic quadratic fractional bi-level programming can be stated as follows

f11 uHAj + "H

Maxf1(x1,x2) = — =

xi

where for given X1,X2

Cnxj + 2 xTdiixj + an

^1 fi2 ~ I 1 T sJ I fi

Ci2Xj+2X ai2Xj + Pi2

- 1 T3

~ f71 ciixj + 9x diixi + aii Maxf2(xi,x2) =ji = - } 2 }

2 ¡22

1

Ci2Xj + 2 xTdi2Xj + Pi2 XEG = [xe Rn\P(Zrj=iàijXj <bi)>1 — Ai, Xj>0, j = 1,2,...,n where 0 < At < 1,i = 1,2,..., m

i = 1,2, .

.m}

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

au =

3

1

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62) PROGRAMMING PROBLEM_Volume 16, March 2021

IV. Deterministic Model of the Probabilistic Quadratic Fractional Programming

Problem

A technique for converting the quadratic fractional bi-level fuzzy probabilistic programming into its equivalent quadratic fractional bi-level fuzzy programming is discussed. We assume that bi ( i = 1,2, ...,m) in the model (2.28)-(2.31) are independent random variables following two-parameter exponential distribution [3] with parameters 0i, oi where mean and variance of random variable bi are given by:

E(bd = 0i + ffi i = 1,2,..., m (232)

V(b-) = a2 i = 1,2,...,m (2.33)

The probability density function of the ith two-parameter exponential variable bi is given by

, 1 (-(hi - di)\

f(bi) = —exp (——--), i = 1,2,...,m (2.34)

Oi \ Oi J

where bi > 8i,oi > 0

To solve the problem (2.28)-(2.31), the deterministic form of the problem is established. Then from the chance-constraint (2.30), we have

n

Pr\

^ a^Xj < bij >(1- Yi)

n \

Pr( bi>YdaijXjj > (1-Yi)

f oo

I f(bi)dbi >(1- Yi)

h1}

U J = i

r 1 (-(bi - B:)\

I — exp (——-ll)dbi>(1-Yi) (2.35)

After integrating the above equation, following result is obtained

n

^ aijxj <6i-oi ln(1 - Yi) (2.36)

i=i

Hence the quadratic fractional bi-level fuzzy probabilistic programming into its equivalent deterministic quadratic fractional bi-level fuzzy programming by using the derived methodology is given as follows

~ 1 T3

~ cnxj + 2 x <ilXi + Max f (x^ x2) =-2---

1 Ci2xj + 2xT<^i2Xj + Pi2

where for given Xr, X2

~ Cnxj + 2 Xdnxj + an Max f2 (Xi, x2) =-2-:-

2 Ci2Xj + 2XT(di2Xj + ßi2

Subject to

^ äijXj < &i — a^n (1 - Yi) (2.37)

j=i

0 < Yi < 1, i = 1,2,....,m

Xj > 0, j = 1,2, ....,m

n

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62) PROGRAMMING PROBLEM_Volume 16, March 2021

III. Transformation of bi-level quadratic fractional programming problem to

nonlinear programming problem

The objective function of each decision maker is transformed from fractional form to nonlinear form with the procedure described below in order to find the best solution for each level without considering other levels. Consider the most basic model of the quadratic fractional programming problem, which is defined as follows.

~ fi1 + 2xT^iixj + an

Mf^ =f\2=~~-^-

1 12 Ci2Xj + 2XT(^i2Xj + Pi2

Subject to

^ auxj <6i- ai\n (1 - yd

j=1

0 <yi<1, i = 1,2, ....,m Xj > 0, j = 1,2, ....,n

To convert the above type of quadratic fractional programming problem into the nonlinear programming problem, take -1 = ---— = y^ f^yi = 1

fl2 Ci2Xj+-X< dfrXj +Pi2

Thus, problem becomes as the following:

Maxfi = fiiyi = (cnxj +1xTdnxj + a^yi o

( Ci2Xj +1XTdi2Xj +pa)yi = 1 n

^ aijXj <di- ffiln (1 - yi)

atjXi < d; — aj\n \

=1

0 <yi<1, i = 1,2, ....,m Xj > 0, j = 1,2, ....,n

IV. Formulation of fuzzy goal programming approach for bi-level quadratic

fractional programming problem

The decision makers of a bi-level quadratic fractional programming problem (BLQFPP) are fundamentally cooperative and make sequential decisions. By optimizing each decision maker individually for the given set of constraints for each objective functions fi, i = 1,2,.m of the problem the fuzzy goal are taken as the maximum and minimum value of each decision maker. . Let f"l,fL is the maximum and minimum value of each objective function which are obtained by optimizing them as individually i.e. f" = maxfi with the solution set of decision variables as

/ Vl sy^l sy^l

(X1 , X2 , .■■ , Xn )

f l = min fi with the solution set of decision variables as Xl,X2l.....X„l)

For i = 1,2,., m now, for the given model, our objective was to maximize the objective function of each decision maker such that maximum value of objective function for first decision maker is f at the point (x"1,x"1, ...,xVi ), similarly maximum value of objective function for second decision maker is f"2 at the point (x"2,x"2, ...,x"2 ). It can be assumed reasonably that the value of fi > f" are acceptable and all values less than fLl = min fi are absolutely unacceptable. Then the membership function ^ifi for the ith fuzzy goal can be formulated as

n

1, if fi > f"

Pifi =

f — fL

-f-^ iffiLi<fi<flUii = 1,2,...m fi - fi

0 if fi< fL

Each decision maker seeks to maximize his or her own objective function when making a decision. When each decision maker's optimum solution is computed separately, it is considered as the best solution, and the associated value of the objective function is regarded as the aspiration level of the corresponding fuzzy goal. In fuzzy programming approach, the highest degree of membership is one Mohamed [5]. The flexible membership goal having the aspired level unity can be represented as follows:

^fi(fd + d- - d+ =1, i = 1,2,....,m Or equivalently as f—fLi

¡i. i,. + d- — d+ = 1, i = 1,2, ....,m,

fui fLi i i ' h ~H

where dF, d+ > 0, with dF x d+ = 0, represent the under and over deviations, respectively, from the aspired levels Pramanik and Roy [8]

In the formulation of fuzzy goal programming, the under and over deviational variables in the achievement function for minimizing them depends up on the type of objective functions to be optimized. To reach the aspiration level in the proposed fuzzy goal programming approach, the sum of under deviational variables must be minimized. The proposed fuzzy goal programming model for BLQFP problem follows as:

= ^Widt

MinZ = y w, i=1 f — fLi

7U—bn + dF — d+ = 1

fi"i — fiL

( Ci2Xj +1xTdi2Xj + Pi2)yi =

n

y a^Xj <Bi — (ln (1 — Yi)

=1

0 <Yi<1, i = 1,2, ....,m Xj = x* j = 1,2,...,n

and dF, d+ = 0,dF,d+ > 0,Vi = 1,2, ...,k

where Z is the fuzzy goal's achievement function which is made up of the weighted under -deviational variables. The numerical weights wi represent the relative importance of achieving the aspired levels of the respective fuzzy goals. The values of wi are determined as Mohamed [5]:

1

w;=^-r i = 1,2,...,m

i fui fLi fi —fi

V. Numerical Examples

To demonstrate the proposed FGP approach, consider the following BLQFP problem with

probabilistic nature in the constraints.

[FLDM]

6X + 3X? +—1X2 +—1X2 + 6 MaxF1 =—-^—^-2-

1 " i av2

1X2 + 1X2 + 4

where x2 solves [SLDM]

v.

1

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62) PROGRAMMING PROBLEM_Volume 16, March 2021

1x1 + 5x2 + —1x2 + 8

MaxF2 = , , ,-2-

2 1x2 + 1x2 + 6

Subject to

P(1x1 + 1x2 < b1) > 0.99 P(3x1 + 2x2 < b2) > 0.95 P(2x1 + 1x2 > b3) > 0.90

x1,x2 > 0 (5.1)

Here, we assume that bt (i = 1,2,3) are random variables following two parameter exponential distributions with following parameters:

E(b1) = 161, E(b2) = 144, E(b3) = 106 Var(b1) = 25, Var(b2) = 36, Var(b3) = 64 Using (2.32) and (2.33), the parameters are calculated as follows:

d1 = 156, a1 = 5, d2 = 138, a2 = 6, d3 = 98 ,a3=8 The coefficients of the objectives are taken as triangular fuzzy number with the values -1 = (-1.5,-1,-0.5), 6 = (4,6,9), —1 = (-1.1,-1, -0.15), 5 = (4.5,5,5.8), 1 = (0.95,1,1.05),3 = (2,3,5), -11 = (-1.04, -1,0.03), -2(-1.5, -2, -2.5),-2 = (-1.8,-2, -2.2), 3 = (4.2,5,5.8), 3 = (0.8,1,1.2), 3 = (2,3,4)

The coefficients of the probabilistic constraints are taken as trapezoidal fuzzy number with the values

1 = (0.2,0.8,1,1.2),1 = (0.4,1,1.6,2), 3 = (1,2,3,4),2 = (0.5,1.5,2.5,3),2 = (1,1.8,2.2,3),1 = (0.5,1,1.5,2).

On the basis of the method of defuzzification with probability density function and CCP technique the above model (5.1) can be expressed as

[FLDM]

1 + 3.33 x2 - x1

6.33 x + 3.33 x - x2

MaxF1 = ■ ., .,

1 0.8x2 + 0.93x2 + 4

where x2 solves [SLDM]

M a x F2 =

x2 + 5.1x2 - 0.61xx + 10 1.06x2 + 1.2x2 + 6

Subject to

0.78x1 + 1.24x2 < 156.05 2.67x1 + 1.86x2 < 138.308 2x1 + 1.25x2 < 98.41

x1,x2 > 0

The bi-level multi-objective quadratic fractional programming problem is transformed into the bilevel quadratic programming model as follows.

[FLDM]

MaxF1 = (6.33x1 + 3.33x2 - - 0.75x2 + 8)y1 where x2 solves

[SLDM]

MaxF2 = (x2 + 5.1x2 - 0.61x1 + 10)y2

Subject to

(0.8x2 + 0.93x2 + 4)y1 = 1 (1.06x2 + 1.2x2 + 6)y2 = 1 0.78x1 + 1.24x2 < 156.05

0.75 x22 + 8

Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62)

PROGRAMMING PROBLEM_Volume 16, March 2021

2.67x1 + 1.86x2 < 138.3082x1 + 1.25x2 < 98.41

x1 , x2 > 0

We first obtain the value of F1nax = 2.945, F2?ax = 4.088, F1nin = 2,F2?in = 0.924 and ft = 2.945, f2u = 4.088, ft- = 2,f2L = 0.924

To solve FGP models to get x1 = x£. Thus the first level FGP model follows as:

MinZ = 1.058d-Subject to

(6.33x1 + 3.33x2 - x2 - 0.75x2 + 8)y1 + 0.9448d- - 0.9448d+ = 2.945 (0.8 x12 + 0.93 x22 + 4) y1 = 1 (1.06 x12 + 1.2 x2 + 6) y2 = 1 0.78x1 + 1.24x2 < 156.05 2.67x1 + 1.86x2 < 138.3082x1 + 1.25x2 < 98.41 x1 , x2 > 0 d+,d- > 0

Using Lingo software, the compromise solution of first level decision maker problem is obtained as; (x1, x2) = (0.954, 0.478). Then assuming that the FLDM set x^ = 0.954 MinZ = 1.058d- + 0.316d-

(633x1 + 3.33x2 - x2 - 0.75x2 + 8)y1 + 0.945d- - 0.945d+ = 2.945 (x2 + 5.1x2 - 0.61x1 + 10)y2 + 3.164d- - 3.164d+ = 4.0877 (0.8 x12 + 0.93 x22 + 4) y1 = 1

(1.06 x12 + 1.2 x2 + 6) y2 = 1

0.78 x1 + 1.24 x2 < 156.05 2.67 x1 + 1.86 x2 < 138.3082 x1 + 1.25 x2 < 98.41 xl = 0.943 d+,d-,d+,d- > 0

Using Lingo software, the compromise solution of the BLQFP problem is obtained as (x1,x2) = (0.954, 0.798) with the corresponding objective function.

VI. Discussion

In this paper, the technique for solving a Quadratic Fractional Bi-level Fuzzy Probabilistic Programming (QFBLFP) programming is outlined by considering the fact that the random variable follows two parameter exponential distributions and using a combination of probabilistic and fuzzy concepts. Firstly the probabilistic nature of the problem is transformed into an equivalent deterministic problem and then a fuzzy goal programming technique is used to solve the bi-level deterministic model.

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Haneefa Kausar, Ahmad Yusuf Adhami, Ahmadur Rahman

QUADRATIC FRACTIONAL BI-LEVEL FUZZY PROBABILISTIC RT&A, No 2 (62) PROGRAMMING PROBLEM_Volume 16, March 2021

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