Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 2, 93 - 116 HIKARI Ltd, www.m-hikari.com

Fuzzy Linear Programming Model

for Critical Path Analysis

K. Usha Madhuri1, B. Pardha Saradhi2 and N. Ravi Shankar1

1Dept.of Applied Mathematics, GIS, GITAM University, Visakhapatnam, India

2Dept. of Mathematics, Dr.L.B.College, Visakhapatnam, India

Abstract

In this paper, a new fuzzy linear programming model is proposed to find fuzzy critical path and fuzzy completion time of a fuzzy project. All the activities in the project network are represented by trapezoidal fuzzy numbers. A new representation of trapezoidal fuzzy number is introduced to reduce the constraints in the fuzzy linear programming model. Further, an example is illustrated which shows the advantages of using the proposed representation over the existing representation of trapezoidal fuzzy numbers and will present with great clarity the proposed technique and illustrate its application for fuzzy critical path problems occurring in real life situations. Keywords: Fuzzy Critical Path Problem; Ranking function; Trapezoidal fuzzy number; LR fuzzy number; linear programming

1. Introduction

In today’s highly competitive business environment, project management’s ability to schedule activities and monitor progress strictly in cost, time and performance guidelines is becoming increasingly important to obtain competitive priorities such as on-time delivery and customization. In many situations, projects can be complicated and challenging to manage. Critical Path Method (CPM) is to identify critical activities on the critical path so that resources may be concentrated on these activities in order to reduce the project length time. Further

94 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar implementation of CPM requires the availability of clear determined time duration for each activity. To deal with real life situations, Zadeh[1] introduced the concept of fuzzy set. There is always uncertainty in the network planning of time duration of activities. So, fuzzy critical path method introduced in late 1970s. So many approaches are proposed over the past years to find the fuzzy critical path. The first method called Fuzzy Programming Evaluation and review Technique (FPERT) was proposed by Chanas and Kamburowski [2]. They presented the project completion time in the form of fuzzy set in the time space. In paper [3], Gazdik developed a fuzzy network of unknown project to estimate the activity durations and used fuzzy algebraic operations to calculate the duration of the project and its critical path. A chapter of the book [4] is devoted to the critical path method in which activity times are represented by triangular fuzzy numbers. A new methodology to calculate the fuzzy completion project time is presented in the paper [5]. Nasution [6] proposed a method to compute total floats and the critical paths in a fuzzy project network. Yao and Lin [7] proposed a method for ranking fuzzy numbers without the need for any assumptions and have used both positive and negative fuzzy values to define ordering an it is applied to fuzzy project network. In [8], Dubois et al. extended the fuzzy arithmetic operational model to compute the latest starting time of each activity in a fuzzy project network. Chen[11] proposed an approach based on the extension principle and linear programming formulation to critical path analysis in fuzzy project networks. In [12], Chen and Hsueh presented a simple approach to solve the Fuzzy CPM problems on the basis of linear programming formulation and the fuzzy number ranking method that are more realistic than crisp ones. The problems of determining possible values of earliest and latest starting times of an activity in fuzzy project networks with minimal time intervals and vague durations that are presented by means of fuzzy interval or fuzzy numbers is introduced in the paper[13]. Kumar and Kaur [9] proposed a new fuzzy linear programming model using Jai Mehar Di (JMD) representation of triangular fuzzy numbers. In this paper, we propose a novel fuzzy linear programming model using new representation of trapezoidal fuzzy number. It is observed that the use of proposed representation is better than the existing representations of trapezoidal fuzzy numbers, to find the fuzzy optimal solution of fuzzy critical path problems. To illustrate the fuzzy linear programming model and to show the advantages of the proposed representation of trapezoidal fuzzy numbers, a numerical example is evaluated by representing all the constraints as existing and proposed type of trapezoidal fuzzy numbers. In section 2, the background Information is presented. In that, basic definitions, existing representations of trapezoidal and ranking function are reviewed. In the section 3, linear programming of crisp critical path problems and also the linear programming formulation of fuzzy critical path problems are presented. The section 4 follows the method to find the fuzzy optimal solution of fuzzy critical path problems[9]. In section 5, a new representation of trapezoidal fuzzy number is mentioned. To illustrate the proposed method, a numerical example is taken

Fuzzy linear programming model 95 and solved in section 6. Comparision study of existing model and proposed model is presented in section 7.

2. Background information In this section, basic definitions of fuzzy sets and existing representations of trapezoidal fuzzy numbers, arithmetic operations and ranking functions are reviewed. 2.1. Basic definitions In this section, some basic definition are reviewed [10]. Definition 1: The Characterstic function Aμ of a crisp set XA ⊆ assigns a value either 0 or 1 to each member in X . This function can be generalized to a function A~μ such that the value assigned to the element of the Universal set X fall within a particular range i.e. ]1,0[X:A~ →μ . The allocated value indicates the membership grade of the element in the set A. The function A~μ is called the

membership function and the set }Xx));x(,x{(A~ A~ ∈μ= defined by )x(A~μ for each Xx∈ is called a fuzzy set. Definition 2: A fuzzy set A~ , defined on the Universal set of real numbers R, is said to be a fuzzy number if its membership function has the following characteristics: (i) A~ is convex i.e.,

≥λ−+λμ )x)1(x( 21A~ minimum ]1,0[,Rx,x))x(),x(( 212A~1A~ ∈λ∀∈∀μμ .

(ii) A~ is normal i.e., ,Rx 0 ∈∃ such that .1)x( 0A~ =μ (iii) )x(A~μ is piecewise continuous.

Definition 3: A fuzzy number A~ , is called non-negative fuzzy number if .0x0)x(A~ <∀=μ

2.2. Existing representation of trapezoidal fuzzy numbers Different representations of trapezoidal fuzzy numbers are presented in this section. 2.2.1. General representation of trapezoidal fuzzy numbers with membership function Definition 4: A fuzzy number )d,c,b,a(A~ = where a ≤ b ≤ c ≤ d is said to be a trapezoidal fuzzy number if its membership function is given by

96 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

∞<<

≤<−−

≤<

≤<−−

≤<∞−

=μ

xdif,0

dxcif,cdxd

cxbif,1

bxaif,abax

axif,0

)x(A~

Definition 5 : A trapezoidal fuzzy number )d,c,b,a(A~ = is said to be zero trapezoidal fuzzy number iff .0d,0c,0b,0a ==== Definition 6: A trapezoidal fuzzy number )d,c,b,a(A~ = is said to be non-negative trapezoidal fuzzy number iff .0a ≥ Definition 7: Two trapezoidal fuzzy numbers )d,c,b,a(A~ 1111= and

)d,c,b,a(B~ 2222= are said to be equal i.e., =A~ B~ iff .dd,cc,bb,aa 21212121 ====

2.2.2. (m,n,α,β) representation of trapezoidal fuzzy numbers with membership function A trapezoidal fuzzy number )d,c,b,a(A~ = , defined in the section 2.2.1, may also be represented as ),,n,m(A~ βα= , where .cd,ab,cn,bm −=β−=α== Definition 8 : A fuzzy number ),,n,m(A~ βα= is said to be a trapezoidal fuzzy number if its membership function is given by

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

∞<<β+

β+≤<+β−

≤<

≤<α−α−

+

α−≤<∞−

=μ

.xnif,0

nxnif,1xnnxmif,1

mxmif,xm1

mxif,0

)x(A~ where .0, ≥βα

Definition 9: A trapezoidal fuzzy number ),,n,m(A~ βα= is said to be zero trapezoidal fuzzy number iff .0,0,0n,0m =β=α== Definition 10: A trapezoidal fuzzy number ),,n,m(A~ βα= is said to be non-negative trapezoidal fuzzy number iff .0m ≥α−

Fuzzy linear programming model 97 Definition 11: Two trapezoidal fuzzy numbers ),,n,m(A~ 1111 βα= and

),,n,m(B~ 2222 βα= are said to be equal i.e., =A~ B~ iff .,,nn,mm 21212121 β=βα=α==

2.3. Arithmetic fuzzy operations In this section addition and multiplication operations between two trapezoidal fuzzy numbers are presented. 2.3.1. Arithmetic operations between d)c,b,(a, type trapezoidal fuzzy numbers Let )d,c,b,a(A~ 11111 = and )d,c,b,a(A~ 22222 = be two trapezoidal fuzzy numbers, then (i) )dd,cc,bb,aa(A~A~ 2121212121 ++++=⊕ , (ii) ),'d,'c,'b,'a(A~A~ 21 =⊗ where ='a min )dd,cc,bb,aa( 21212121 , 21bb'b = , 21cc'c = ,   

                                                                     ='d max )dd,cc,bb,aa( 21212121 . 2.3.2. Arithmetic Operations between β)α,n,(m, type trapezoidal fuzzy numbers Let ),,n,m(A~ 11111 βα= and ),,n,m(A~ 22222 βα= be two trapezoidal fuzzy numbers, then (i) ),,nn,mm(A~A~ 2121212121 β+βα+α++=⊕ , (ii) ),',','n,'m(A~A~ 21 βα=⊗ where 21mm'm = , 21nn'n = , )'dmin('m' −=α ,                                                              ')'max(' nd −=β  , and

⎟⎟⎠

⎞⎜⎜⎝

⎛ββ+β+β+αβ−β+α−

βα+α−β+αα+α−α−=

2112212121122121

1112212121122121

nnnn,nnnn,mmmm,mmmm

'd . 

2.4. Ranking function of fuzzy number Ranking Function is the most important to compare the fuzzy numbers to our fuzzy approach. ℜ:F(R)→R, where F(R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real space, where a natural order exists.

Let )d,c,b,a( be a trapezoidal fuzzy number. Then ℜ4

dcba)d,c,b,a( +++= .

98 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar Also , Let ),,n,m( βα be a trapezoidal fuzzy number. Then

ℜ ),,n,m( βα =42

nm α−β+

+ .

3. Linear Programming Formulation of Crisp Critical Path and

Fuzzy Critical Path Problems

The fuzzy CPM is a fuzzy project network based method designed to assist in the planning, scheduling and control of the fuzzy project. Its objective is to construct the time scheduling for the fuzzy project. Two basic results provided by fuzzy CPM are the total duration time needed to complete the fuzzy project and the fuzzy critical path. One of the efficient approaches for finding the fuzzy critical paths and total duration time of fuzzy project networks is Linear programming. The fuzzy Linear programming formulation assumes that a unit flow enters the fuzzy project network at the start vertex and leaves at the terminal vertex. In this section, the Linear Programming Formulation of Crisp Critical Path problems is reviewed and also the Linear formulation of Fuzzy Critical Path problems is proposed. 3.1. Linear Programming of Crisp Critical Path Problems The linear programming model discussed in the book written by Taha[14] is reviewed in this section. Consider a project network G=(V,E,T) consisting of a finite set V={1,2,…,n} of n nodes and E is the set of activities (i,j) , T is a function from E to R and tij ∈ R is the time period of activity (i,j). The Linear Programming formulation of Crisp Critical Path Problem is defined as follows: Maximize ∑

∈E)j,i(ijijxt

subject to the constraints 1x

E)j,i(j1 =∑

∈

,

,nk1,i,xxE)k,j(

jkE)j,i(

ij ≠≠= ∑∑∈∈

1xE)n,i(

in =∑∈

, ijx is a non-negative real number E)j,i( ∈∀ .

Fuzzy linear programming model 99 3.2. Linear Programming Formulation of Fuzzy Critical Path Problems Suppose ijt and ijx , E)j,i( ∈∀ are vague and are represented by fuzzy numbers

ijt~ and ijx~ , E)j,i( ∈∀ respectively. Then the Fuzzy Critical Path Problems may be formulated into the following fuzzy linear programming problem: Maximize ∑

∈

⊗E)j,i(

ijij x~t~

subject to 1~x~

E)j,i(j1 =∑

∈

,

,nk,1i,x~x~E)k,j(

jkE)j,i(

ij ≠≠= ∑∑∈∈

1~x~A)n,i(

in =∑∈

, ( 1~ = (1,1,1,1)),

ijx~ is a non-negative real number E)j,i( ∈∀ . 4. Method to find fuzzy critical path and fuzzy completion time of a fuzzy project The steps of the approach are as follows [9]: Step 1: Represent all the parameters of linear programming formulation of Fuzzy Critical Path problems by a particular type of trapezoidal fuzzy number and formulate the given problem, as proposed in the section 3.2. Step 2: Convert the fuzzy objective function and fuzzy constraints into the crisp objective function form by its ranking function. Step 3: Find the critical path and completion fuzzy time for the obtained crisp linear Programming problem by using Tora software. Step 4: Convert crisp solution to fuzzy solution using solution obtained from Step 3. Step 5: Find the critical path and corresponding maximum total completion time in fuzzy sense from Step4.

100 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar 4.1. Proposed method with )d,c,b,a( representation of trapezoidal fuzzy numbers If all the parameters of linear programming formulation of Fuzzy Critical Path problems are represented by )d,c,b,a( type trapezoidal fuzzy numbers then the steps of the proposed method are as follows: Step1: Suppose all the parameters ijt~ and ijx~ are represented by )d,c,b,a( type trapezoidal fuzzy numbers )'''t,''t,'t,t( ijijijij and ),z,y,x( ijijijij γ respectively then the Linear Programming formulation of Fuzzy Critical Path problems, proposed in the section 3.2., this can written as: Maximize ⊗∑

∈E)j,i(ijijijij )'''t,''t,'t,t( ),z,y,x( ijijijij γ

subject to )1,1,1,1(),z,y,x(

E)j,i(j1j1j1j1 =γ∑

∈

,

,nk,1i,),z,y,x(),z,y,x(E)k,j(

jkjkjkjkE)j,i(

ijijijij ≠≠γ=γ ∑∑∈∈

)1,1,1,1(),z,y,x(E)n,i(

inininin =γ∑∈

,

),z,y,x( ijijijij γ is a non-negative trapezoidal fuzzy number E)j,i( ∈∀ . Step2: The fuzzy linear programming of Fuzzy Critical Path problems may be written as:

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡γ⊗∑

∈

),z,y,x()'''t,''t,'t,t( ijijijijE)j,i(

ijijijij

subject to the constraints )1,1,1,1(),z,y,x(

E)j,i(j1j1j1j1 =γ∑

∈

,

,nk,1i,),z,y,x(),z,y,x(

E)k,j(jkjkjkjk

E)j,i(ijijijij ≠≠γ=γ ∑∑

∈∈

)1,1,1,1(),z,y,x(

E)n,i(inininin =γ∑

∈

,

),z,y,x( ijijijij γ is a non-negative trapezoidal fuzzy number E)j,i( ∈∀ .

Now, the crisp linear programming problem becomes :

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡γ⊗∑

∈

),z,y,x()'''t,''t,'t,t( ijijijijE)j,i(

ijijijij

Fuzzy linear programming model 101 subject to the constraints

1xE)j,i(

j1 =∑∈

, 1yE)j,i(:j

j1 =∑∈

, 1zE)j,i(:j

j1 =∑∈

, 1E)j,i(:j

j1 =γ∑∈

,

,nk,1i,xxE)k,j(:j

jkE)j,i(:i

ij ≠≠= ∑∑∈∈

,nk,1i,yyE)k,j(:j

jkE)j,i(:i

ij ≠≠= ∑∑∈∈

,nk,1i,zzE)k,j(:j

jkE)j,i(:i

ij ≠≠= ∑∑∈∈

,nk,1i,E)k,j(:j

jkE)j,i(:i

ij ≠≠γ=γ ∑∑∈∈

1xE)n,i(:i

in =∑∈

, 1yE)n,i(:i

in =∑∈

, 1zE)n,i(:i

in =∑∈

, 1E)n,i(:i

in =γ∑∈

,

0xy ijij ≥− , 0yz ijij ≥− , 0zijij ≥−γ ,

ijx , ijy , ijz , 0ij ≥γ E)j,i( ∈∀ . Step 3:Find the solution ijx , ijy , ijz , ijγ by solving the Crisp Linear Programming problem, which is obtained in Step2. Step 4: Find the fuzzy solution ijx~ by putting the values of ijx , ijy , ijz and ijγ in

ijx~ = ),z,y,x( ijijijij γ and also calculate the maximum total completion fuzzy time

by putting the values of ijx~ in ∑∈

⊗E)j,i(

ijij x~t~ .

Step 5: Find the fuzzy critical path by combining all the activities )j,i( such that )1,1,1,1(x~ij = .

4.2. Proposed method with ),,n,m( βα representation of trapezoidal fuzzy numbers If all the parameters of Fuzzy Critical Path problems are represented by

),,n,m( βα type trapezoidal fuzzy numbers then the steps of the proposed method are as follows: Step1: Suppose all the parameters ijt~ and ijx~ are represented by ),,n,m( βα type trapezoidal fuzzy numbers ),,''t,'t( ijijijij δλ and ),,z,y( ijijijij βα respectively then the Linear Programming formulation of Fuzzy Critical Path problems, proposed in the section 3.2., this can written as: Maximize ∑

∈

βα⊗δλE)j,i(

ijijijijijijijij ),,z,y(),,''t,'t(

subject to the constraints

102 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

)0,0,1,1(),,z,y(E)j,i(:j

j1j1j1j1 =βα∑∈

,

,nk,1i,),,z,y(),,z,y(E)k,j(:j

jkjkjkjkE)j,i(:i

ijijijij ≠≠βα=βα ∑∑∈∈

)0,0,1,1(),,z,y(E)n,i(:i

inininin =βα∑∈

,

),,z,y( ijijijij βα is a non-negative trapezoidal fuzzy number E)j,i( ∈∀ . Step2: Using the ranking function, presented in section 2.4., the Linear Programming of Fuzzy Critical Path problems may be written as:

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡βα⊗δλ∑

∈E)j,i(ijijijijijijijij ),,z,y(),,''t,'t(  

subject to the constraints

)0,0,1,1(),,z,y(E)j,i(:j

j1j1j1j1 =βα∑∈

,

,nk,1i,),,z,y(),,z,y(E)k,j(:j

jkjkjkjkE)j,i(:i

ijijijij ≠≠βα=βα ∑∑∈∈

)0,0,1,1(),,z,y(E)n,i(:i

inininin =βα∑∈

,

),,z,y( ijijijij βα is a non-negative trapezoidal fuzzy number E)j,i( ∈∀ . Now, the Crisp Linear Programming problem becomes :

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡βα⊗δλ∑

∈E)j,i(ijijijijijijijij ),,z,y(),,''t,'t(  

subject to the constraints

1yE)j,i(:j

j1 =∑∈

, 1zE)j,i(:j

j1 =∑∈

, 0E)j,i(:j

j1 =α∑∈

, 0E)j,i(:j

j1 =β∑∈

,

,nk,1i,yy

E)k,j(:jjk

E)j,i(:iij ≠≠= ∑∑

∈∈

,nk,1i,zzE)k,j(:j

jkE)j,i(:i

ij ≠≠= ∑∑∈∈

,nk,1i,E)k,j(:j

jkE)j,i(:i

ij ≠≠α=α ∑∑∈∈

,nk,1i,E)k,j(:j

jkE)j,i(:i

ij ≠≠β=β ∑∑∈∈

1yE)n,i(:i

in =∑∈

, 1zE)n,i(:i

in =∑∈

, 0E)n,i(:i

in =α∑∈

, 0E)n,i(:i

in =β∑∈

0yz ijij ≥− , 0zijij ≥−α ,

Fuzzy linear programming model 103

ijy , ijz , ijα , 0ij ≥β E)j,i( ∈∀ . Step 3: Find the solution ijy , ijz , ijα , ijβ by solving the Crisp Linear Programming problem, which is obtained in Step2. Step 4: Find the fuzzy solution ijx~ by putting the values of ijy , ijz ijα and ijβ in

ijx~ = ),,z,y( ijijijij βα .and also calculate the maximum total completion fuzzy time

by putting the values of ijx~ in ∑∈

⊗E)j,i(

ijij x~t~ .

Step 5: Find the fuzzy critical path by combining all the activities )j,i( such that )0,0,1,1(x~ij = .

5. A Novel representation of trapezoidal fuzzy numbers In this section, a new representation of trapezoidal fuzzy numbers , Arithmetic operations and new ranking function is defined. Using this new representation, fuzzy linear programming model is generated and presented a method to find fuzzy critical path and maximum total fuzzy completion time. Definition12. Let )d,c,b,a( be a trapezoidal fuzzy number then its New representation is ),,y,x( βα New , where 0ab,by,ax ≥−=α== and

0cd ≥−=β Definition13. Let ),,n,m( βα be a trapezoidal fuzzy number then its New representation is ),,y,x( βα New , where β+=α−= ny,mx . Definition14. A trapezoidal fuzzy number ),,y,x(A~ βα= New is said to be zero trapezoidal fuzzy number iff 0,,0,0y,0x =β=α== . Definition 15. A trapezoidal fuzzy number ),,y,x(A~ βα= New is said to be non-negative trapezoidal fuzzy number iff 0y,0x ≥≥ . Definition 16. Two trapezoidal fuzzy numbers ),,y,x(A~ 1111 βα= New

),,y,x(B~ 2222 βα= New, are said to be equal iff .,,yy,xx 21212121 β=βα=α==

5.1. Arithmetic operation between new trapezoidal fuzzy numbers Let )d,c,b,a(A~ 11111 = and )d,c,b,a(A~ 22222 = be two trapezoidal fuzzy numbers and ),,y,x(A~ 11111 βα= New ),,y,x(A~ 22222 βα= New be their new

104 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar representation. Then the addition and multiplication operations are defined as follows : (i) ),,,yy,xx(A~A~ 2121212121 β+βα+α++=⊕ and (ii) ),,,y,x(A~A~ 333321 βα=⊗ where

=3x minimum )dd,cc,bb,aa( 21212121 , 21bb'b = , 21cc'c = , ='d maximum )dd,cc,bb,aa( 21212121 . 5.2. Ranking function for new trapezoidal fuzzy numbers Let ),,y,x( βα New be a trapezoidal fuzzy number then ranking function

ℜ ),,y,x( βα New=42

yx α−β+

+ .

5.3. Proposed method with β)α,y,(x, New representation of trapezoidal fuzzy numbers In this all the parameters of Fuzzy Critical Path problems are represented by

),,y,x( βα New trapezoidal fuzzy numbers then the steps of the proposed method are as follows: Step 1: Suppose all the parameters ijt~ and ijx~ are represented by ),,y,x( βα New , trapezoidal fuzzy numbers ),,''t,'t( ijijijij δλ and ),,y,x( ijijijij βα New, respectively then the Linear Programming formulation of Fuzzy Critical Path problems, proposed in the section 3.2., this can be written as: Maximize ∑

∈A)j,i(

⊗δλ ),,''t,'t( ijijijij ),,y,x( ijijijij βα

subject to the constraints )0,0,1,1(),,y,x(

A)j,i(:jj1j1j1j1 =βα∑

∈

,

,nk,1i,),,y,x(),,y,x(A)k,j(:j

jkjkjkjkA)j,i(:i

ijijijij ≠≠βα=βα ∑∑∈∈

)0,0,1,1(),,y,x(A)n,i(:i

inininin =βα∑∈

,

),,y,x( ijijijij βα is a non-negative trapezoidal fuzzy number A)j,i( ∈∀ . Step 2: Using the ranking function presented in section 2.4., the Linear Programming of Fuzzy Critical Path problems may be written as:

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡βα⊗δλ∑

∈

),,y,x(),,''t,'t( ijijijijE)j,i(

ijijijij  

Fuzzy linear programming model 105 subject to the constraints

)0,0,1,1(),,y,x(E)j,i(:j

j1j1j1j1 =βα∑∈

,

,nk,1i,),,y,x(),,y,x(E)k,j(:j

jkjkjkjkE)j,i(:i

ijijijij ≠≠βα=βα ∑∑∈∈

)0,0,1,1(),,y,x(E)n,i(:i

inininin =βα∑∈

, ),,y,x( ijijijij βα is a non-negative trapezoidal

fuzzy number E)j,i( ∈∀ . The Crisp Linear Programming problem becomes :

Maximize ℜ ⎥⎦

⎤⎢⎣

⎡βα⊗δλ∑

∈

),,y,x(),,''t,'t( ijijijijE)j,i(

ijijijij  

subject to the constraints

1xE)j,i(:j

j1 =∑∈

, 1yE)j,i(:j

j1 =∑∈

, 0E)j,i(:j

j1 =α∑∈

, 0E)j,i(:j

j1 =β∑∈

,

,nk,1i,xx

E)k,j(:jjk

E)j,i(:iij ≠≠= ∑∑

∈∈

,nk,1i,yyE)k,j(:j

jkE)j,i(:i

ij ≠≠= ∑∑∈∈

,nk,1i,E)k,j(:j

jkE)j,i(:i

ij ≠≠α=α ∑∑∈∈

,nk,1i,E)k,j(:j

jkE)j,i(:i

ij ≠≠β=β ∑∑∈∈

1xE)n,i(:i

in =∑∈

, 1yE)n,i(:i

in =∑∈

, 0E)n,i(:i

in =α∑∈

, 0E)n,i(:i

in =β∑∈

ijx , ijy , ijα , 0ij ≥β E)j,i( ∈∀ . Step 3: Find the optimal solution ijx , ijy , ijα , ijβ by solving the Crisp Linear Programming problem, which is obtained in Step2. Step 4: Find the fuzzy optimal solution ijx~ by putting the values of

ijx , ijy , ijα and ijβ in ijx~ = ),,y,x( ijijijij βα and also find the maximum total

completion fuzzy time by putting the values of ijx~ in ∑∈

⊗E)j,i(

ijij x~t~ .

Step 5: Find the fuzzy critical path by combining all the activities )j,i( such that )0,0,1,1(x~ij = .

106 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

6. Numerical Example To show the advantages of New representation over existing representation of fuzzy numbers, the same numerical example is solved by using all three representation of fuzzy numbers. The problem is to find the fuzzy critical path and maximum total completion fuzzy time of the project network, shown in Fig.1, in which the fuzzy time duration of each activity is represented by the following )d,c,b,a( type trapezoidal fuzzy numbers.

12t~ =(2,2,3,4) , 13t~ =(2,3,3,6), 15t~ =( 2,3,4,5), 24t~ =(2,2,4,5),

25t~ =(2,2,5,8), 34t~ =(1,1,2,2), 36t~ =(7,8,11,15), 45t~ =(2,3,3,5), 46t~ =(3,3,4,6), and

56t~ =(1,1,1,2).

Fig.1: Project Network diagram 6.1. Fuzzy optimal solution using )d,c,b,a( representation of trapezoidal fuzzy numbers Step1:Using the section 4.1, the given problem may be formulated as follows: Maximize((2,2,3,4) ⊕⊗ )γ,z,y,x( 12121212 (2,3,3,6) ⊕⊗ )γ,z,y,x( 13131313 (2,3,4,5)

⊕⊗ )γ,z,y,x( 15151515 (2,2,4,5) ⊕⊗ )γ,z,y,x( 24242424 (2,2,5,8)⊕⊗ )γ,z,y,x( 25252525

(1,1,2,2) ⊕⊗ )γ,z,y,x( 34343434 (7,8,11,15) ⊕⊗ )γ,z,y,x( 36363636 (2,3,3,5) ⊕⊗ )γ,z,y,x( 45454545 (3,3,4,6) ⊕⊗ )γ,z,y,x( 46464646 (1,1,1,2)

)γ,z,y,x( 56565656⊗ ) subject to the constraints

)1,1,1,1(),z,y,x(),z,y,x(),z,y,x( 151515151313131312121212 =γ⊕γ⊕γ , ),z,y,x(),z,y,x(),z,y,x( 121212122525252524242424 γ=γ⊕γ , ),z,y,x(),z,y,x(),z,y,x( 131313133636363634343434 γ=γ⊕γ ,

1

2

3

4

5

6

Fuzzy linear programming model 107

),z,y,x(),z,y,x(),z,y,x(),z,y,x( 34343434242424244646464645454545 γ⊕γ=γ⊕γ , ),z,y,x(),z,y,x(),z,y,x(),z,y,x( 45454545252525251515151556565656 γ⊕γ⊕γ=γ and

)1,1,1,1(),z,y,x(),z,y,x(),z,y,x( 565656564646464636363636 =γ⊕γ⊕γ . where ),z,y,x( 12121212 γ , )γ,z,y,x( 13131313 ,etc. are non negative trapezoidal fuzzy numbers. Step2: Using ranking function to the Fuzzy Linear Programming problem, formulated in Step1, may be written as Maximize ℜ[(2,2,3,4) ⊕⊗ )γ,z,y,x( 12121212 (2,3,3,6) ⊕⊗ )γ,z,y,x( 13131313 (2,3,4,5)

⊕⊗ )γ,z,y,x( 15151515 (2,2,4,5) ⊕⊗ )γ,z,y,x( 24242424 (2,2,5,8)⊕⊗ )γ,z,y,x( 25252525

(1,1,2,2) ⊕⊗ )γ,z,y,x( 34343434 (7,8,11,15) ⊕⊗ )γ,z,y,x( 36363636 (2,3,3,5) ⊕⊗ )γ,z,y,x( 45454545 (3,3,4,6) ⊕⊗ )γ,z,y,x( 46464646 (1,1,1,2)

)γ,z,y,x( 56565656⊗ ] subject to the constraints

)1,1,1,1(),z,y,x(),z,y,x(),z,y,x( 151515151313131312121212 =γ⊕γ⊕γ , ),z,y,x(),z,y,x(),z,y,x( 121212122525252524242424 γ=γ⊕γ , ),z,y,x(),z,y,x(),z,y,x( 131313133636363634343434 γ=γ⊕γ ,

),z,y,x(),z,y,x(),z,y,x(),z,y,x( 34343434242424244646464645454545 γ⊕γ=γ⊕γ , ),z,y,x(),z,y,x(),z,y,x(),z,y,x( 45454545252525251515151556565656 γ⊕γ⊕γ=γ and

)1,1,1,1(),z,y,x(),z,y,x(),z,y,x( 565656564646464636363636 =γ⊕γ⊕γ . ),z,y,x( 12121212 γ , )γ,z,y,x( 13131313  etc. are non-negative trapezoidal fuzzy

numbers. The Crisp Linear Programming problem becomes : Maximize(0.5 12x +0.5 12y +0.75 12z + 12γ +0.5 13x +0.75 13y +0.75 13z +1.5 13γ +0.5

15x +0.75 15y + 15z +1.25 15γ +0.5 24x +0.5 24y + 24z +1.25 24γ +0.5 25x +0.5 25y +1.25

25z +2 25γ +0.25 34x +0.25 34y +0.5 34z +0.5 34γ +1.75 36x +2 36y +2.75 36z +3.75 36γ +0.5 45x +0.75 45y +0.75 45z +1.25 45γ +0.75 46x +0.75 46y + 46z +1.5 46γ +0.25 56x +0.25 56y +0.25 56z +0.5 56γ ) subject to the constraints

1xxx 151312 =++ , 1yyy 151312 =++ , 1zzz 151312 =++ , 1151312 =γ+γ+γ ,

122524 xxx =+ , 122524 yyy =+ , 122524 zzz =+ , 122524 γ=γ+γ ,

133634 xxx =+ , 133634 yyy =+ , 133634 zzz =+ , 133634 γ=γ+γ ,

108 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

34244645 xxxx +=+ , 34244645 yyyy +=+ , 34244645 zzzz +=+ ,

34244645 γ+γ=γ+γ ,

45251556 xxxx ++= , 45251556 yyyy ++= , 45251556 zzzz ++= ,

45251556 γ+γ+γ=γ ,

1xxx 564636 =++ , 1yyy 564636 =++ , 1zzz 564636 =++ , 1564636 =γ+γ+γ ,

0xy 1212 ≥− , 0yz 1212 ≥− , 0z1212 ≥−γ , 0xy 1313 ≥− , 0yz 1313 ≥− , 0z1313 ≥−γ 0xy 1515 ≥− , 0yz 1515 ≥− , 0z1515 ≥−γ , 0xy 2424 ≥− , 0yz 2424 ≥− , 0z2424 ≥−γ , 0xy 2525 ≥− , 0yz 2525 ≥− , 0z2525 ≥−γ , 0xy 3434 ≥− , 0yz 3434 ≥− ,0z3434 ≥−γ 0xy 3636 ≥− , 0yz 3636 ≥− , 0z3636 ≥−γ , 0xy 4545 ≥− , 0yz 4545 ≥− , 0z4545 ≥−γ 0xy 4646 ≥− , 0yz 4646 ≥− , 0z4646 ≥−γ , 0xy 5656 ≥− , 0yz 5656 ≥− ,0z5656 ≥−γ

12x , 12y , 12z , 12γ , 13x , 13y , 13z , 13γ , 14x , 14y , 14z , 14γ , 23x , 23y , 23z , 23γ , 25x , 25y , 25z ,

25γ , 35x , 35y , 35z 35γ , 45x , 45y , 45z , 45γ ≥ 0. Step 3: On solving Crisp Linear Programming Using TORA System, obtained in Step 2, solution is 13x = 13y = 13z = 13γ = 36x = 36y = 36z = 36γ =1 and the remaining all values

12x , 12y , 12z , 12γ , 24x , 24y , 24z , 24γ 25x , 25y , 25z , 25γ , 34x , 34y , 34z , 34γ , 45x , 45y , 45z ,

45γ , 46x , 46y , 46z 46γ , 56x , 56y , 56z , 56γ are zero. Step 4: Putting the values of ijx , ijy , ijz and ijγ in ijx~ = ),z,y,x( ijijijij γ . The solution is

12x~ =(0,0,0,0), 13x~ =(1,1,1,1) , 15x~ =(0,0,0,0), 24x~ =(0,0,0,0), 25x~ =(0,0,0,0), 34x~ =(0,0,0,0),

36x~ =(1,1,1,1), 45x~ =(0,0,0,0), 46x~ =(0,0,0,0), 56x~ =(0,0,0,0). Step 5: Using the fuzzy solution, the fuzzy critical path is 1-3-6. Replacing the values of ijx , ijy , ijz and ijγ in Step1, the maximum total completion fuzzy time is (9,11,14,21). Hence, in this problem, the fuzzy critical path is 1-3-6 and the corresponding maximum total completion fuzzy time is (9,11,14,21) respectively.

Fuzzy linear programming model 109 6.2. Fuzzy optimal solution using ),,n,m( βα representation of trapezoidal fuzzy numbers Using the ),,n,m( βα representation of 12

~t =(2,2,3,4) , 13~t =(2,3,3,6), 15t~ =(2,3,4,5),

24t~ =(2,2,4,5),

25~t =(2,2,5,8), 34t~ =(1,1,2,2), 36t~ =(7,8,11,15), 45t~ =(2,3,3,5), 46t~ =(3,3,4,6),

56t~ =(1,1,1,2). are 12~t =(2,3,0.1) , 13

~t =(3,3,1,3), 15t~ =( 3,4,1,1), 24t~ =(2,4,0,1), 25

~t =(2,5,0,3), 34t~ =(1,2,0,0),

36t~ =(8,11,1,4), 45t~ =(3,3,1,2), 46t~ =(3,4,0,2), and 56t~ =(1,1,0,1). Step 1: Using the section 4.1, the given problem may be formulated as follows: Maximize [(2,3,0,1) ⊕βα⊗ ),,n,m( 12121212 ( 3,3,1,3) ⊕βα⊗ ),,n,m( 13131313 ( 3,4,1,1)

⊕βα⊗ ),,n,m( 15151515 (2,4,0,1) ⊕βα⊗ ),,n,m( 24242424 (2,5,0,3)⊕βα⊗ ),,n,m( 25252525 (1,2,0,0) ⊕βα⊗ ),,n,m( 34343434 (8,11,1,4)⊕βα⊗ ),,n,m( 36363636 (3,3,1,2) ⊕βα⊗ ),,n,m( 45454545 (3,4,0,2)⊕βα⊗ ),,n,m( 46464646 (1,1,0,1) ),,n,m( 56565656 βα⊗ ]

subject to the constraints )0,0,1,1(),,n,m(),,n,m(),,n,m( 151515151313131312121212 =βα⊕βα⊕βα ,

),,n,m(),,n,m(),,n,m( 121212122525252524242424 βα=βα⊕βα , ),,n,m(),,n,m(),,n,m( 131313133636363634343434 βα=βα⊕βα ,

),,n,m(),,n,m(),,n,m(),,n,m( 34343434242424244646464645454545 βα+βα=βα⊕βα),,n,m(),,n,m(),,n,m(),,n,m( 45454545252525251515151556565656 βα⊕βα⊕βα=βα

and )0,0,1,1(),,n,m(),,n,m(),,n,m( 565656564646464636363636 =βα⊕βα⊕βα .

),,n,m( 12121212 βα , ),,n,m( 13131313 βα  etc. are non-negative trapezoidal fuzzy numbers. Step 2:Using ranking formula to the Fuzzy Linear Programming problem, formulated in Step1, may be written as Maximize ℜ((2,3,0,1) ⊕βα⊗ ),,n,m( 12121212 ( 3,3,1,3) ⊕βα⊗ ),,n,m( 13131313 ( 3,4,1,1)

⊕βα⊗ ),,n,m( 15151515 (2,4,0,1) ⊕βα⊗ ),,n,m( 24242424 (2,5,0,3)⊕βα⊗ ),,n,m( 25252525 (1,2,0,0) ⊕βα⊗ ),,n,m( 34343434 (8,11,1,4)

110 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

⊕βα⊗ ),,n,m( 36363636 (3,3,1,2) ⊕βα⊗ ),,n,m( 45454545 (3,4,0,2)⊕βα⊗ ),,n,m( 46464646 (1,1,0,1) ),,n,m( 56565656 βα⊗ )

subject to the constraints )0,0,1,1(),,n,m(),,n,m(),,n,m( 151515151313131312121212 =βα⊕βα⊕βα ,

),,n,m(),,n,m(),,n,m( 121212122525252524242424 βα=βα⊕βα , ),,n,m(),,n,m(),,n,m( 131313133636363634343434 βα=βα⊕βα ,

),,n,m(),,n,m(),,n,m(),,n,m( 34343434242424244646464645454545 βα+βα=βα⊕βα),,n,m(),,n,m(),,n,m(),,n,m( 45454545252525251515151556565656 βα⊕βα⊕βα=βα

and )0,0,1,1(),,n,m(),,n,m(),,n,m( 565656564646464636363636 =βα⊕βα⊕βα .

),,n,m( 13131313 βα ,  ),,n,m( 12121212 βα etc. are non-negative trapezoidal fuzzy numbers. The Crisp Linear Programming problem becomes : Maximize( 12m + 12n -0 12α +0.25 12β +1.5 13m +1.5 13n -0.25 13α +0.75 13β +1.5 15m + 15n -0.25 15α +0.25 15β + 24m +2 24n -0 24α +0.25 24β + 25m +2.5 25n -0 25α +0.75 25β +0.5 34m + 34n -0 34α +0 34β +4 36m +5.5 36n -0.25 36α + 36β +1.5 45m +1.5 45n -0.25 45α +0.5 45β +1.5 46m +2 46n -0 46α +0.5 46β +0.5 56m +0.5 56n -0 56α +0.25 56β ). subject to the constraints

1mmm 151312 =++ , 1nnn 151312 =++ , 0151312 =α+α+α , 0151312 =β+β+β ,

122524 mmm =+ , 122524 nnn =+ , 122524 α=α+α , 122524 β=β+β ,

133634 mmm =+ , 133634 nnn =+ , 133634 α=α+α , 143634 β=β+β ,

34244645 mmmm +=+ , 34244645 nnnn +=+ , 34244645 α+α=α+α ,

34244645 β+β=β+β

45251556 mmmm ++= , 45251556 nnnn ++= , 45251556 α+α+α=α ,

45251556 β+β+β=β ,

1mmm 564636 =++ , 1nnn 564636 =++ , 0564636 =α+α+α , 0564636 =β+β+β ,

0mn 1212 ≥− , 0n1212 ≥−α , 0mn 1313 ≥− , 0n1313 ≥−α , 0mn 1515 ≥− , 0n1515 ≥−α , 0mn 2424 ≥− , 0n 2424 ≥−α , 0mn 2525 ≥− , 0n 2525 ≥−α , 0mn 3434 ≥− , 0n3434 ≥−α , 0mn 3636 ≥− , 0n3636 ≥−α , 0mn 4545 ≥− , 0n 4545 ≥−α , 0mn 4646 ≥− ,0n 4646 ≥−α

Fuzzy linear programming model 111

0mn 5656 ≥− , 0n5656 ≥−α .

12m , 12n , 12α , 12β , 13m , 13n , 13α , 13β , 15m , 15n , 15α , 15β , 24m , 24n , 24α , 24β , 25m , 25n ,

25α , 25β , 34m , 34n , 34α , 34β , 36m , 36n , 36α , 36β , 45m , 45n , 45α , 45β , 46m , 46n , 46α , 46β, 56m , 56n , 56α , 56β ≥0. Step 3: On solving Crisp Linear Programming Using TORA System, obtained in Step2, an optimal solution is 13m = 13n = 13α = 13β = 36m = 36n = 36α = 36β =1 and the remaining all values

12m , 12n , 12α , 12β , 15m , 15n , 15α , 15β , 24m , 24n , 24α , 24β , 25m , 25n , 25α , 25β , 34m , 34n ,

34α , 34β 45m , 45n , 45α , 45β 46m , 46n , 46α , 46β , 56m , 56n , 56α , 56β are zero. Step 4: Putting the values of ijx , ijy , ijα and ijβ in ijx~ = ),,n,m( ijij ijij βα . The solution is

12x~ =(0,0,0,0), 13x~ =(1,1,1,1),

15x~ =(0,0,0,0), 24x~ =(0,0,0,0), 25x~ =(0,0,0,0), 34x~ =(0,0,0,0),

36x~ =(1,1,1,1), 45x~ =(0,0,0,0), 46x~ =(0,0,0,0), 56x~ =(0,0,0,0). Step 5: Using the fuzzy solution, the fuzzy critical path is 1-3-6. Replacing the values of ijx , ijy , ijα and ijβ in Step1, the maximum total completion fuzzy time is (11,14,2,7). Hence, in this problem, the fuzzy critical path is 1-3-6 and the corresponding maximum total completion fuzzy time is (11,14,2,7) respectively.

6.3. Fuzzy optimal solution using β)α,y,(x, representation of trapezoidal fuzzy numbers Using the ),,yx,( βα representation of 12

~t =(2,2,3,4) , 13~t =(2,3,3,6), 15t~ =(

2,3,4,5), 24t~ =(2,2,4,5),

25~t =(2,2,5,8), 34t~ =(1,1,2,2), 36t~ =(7,8,11,15), 45t~ =(2,3,3,5), 46t~ =(3,3,4,6),

56t~ =(1,1,1,2) are

12~t =(2,4,0.1) , 13

~t =(2,6,1,3), 15t~ =( 2,5,1,1), 24t~ =(2,5,0,1), 25

~t =(2,8,0,3), 34t~ =(1,2,0,0),

36t~ =(7,15,1,4), 45t~ =(2,5,1,2), 46t~ =(3,6,0,2), and 56t~ =(1,2,0,1). Step1: Using the section 4.1, the given problem may be formulated as follows: Maximize((2,4,0,1) ⊕βα⊗ ),,y,x( 12121212 ( 2,6,1,3) ⊕βα⊗ ),,y,x( 13131313 ( 2,5,1,1)

112 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar

⊕βα⊗ ),,y,x( 15151515 (2,5,0,1) ⊕βα⊗ ),,y,x( 24242424 (2,8,0,3)⊕βα⊗ ),,y,x( 25252525 (1,2,0,0) ⊕βα⊗ ),,y,x( 34343434 (7,15,1,4)⊕βα⊗ ),,y,x( 36363636 (2,5,1,2) ⊕βα⊗ ),,y,x( 45454545 (3,6,0,2) ⊕βα⊗ ),,y,x( 46464646 (1,2,0,1) ),,y,x( 56565656 βα⊗ )

subject to the constraints )0,0,1,1(),,y,x(),,y,x(),,y,x( 151515151313131312121212 =βα⊕βα⊕βα ,

),,y,x(),,y,x(),,y,x( 121212122525252524242424 βα=βα⊕βα , ),,y,x(),,y,x(),,y,x( 131313133636363634343434 βα=βα⊕βα ,

),,y,x(),,y,x(),,y,x(),,y,x( 34343434242424244646464645454545 βα⊕βα=βα+βα , ),,y,x(),,y,x(),,y,x(),,y,x( 45454545252525251515151556565656 βα⊕βα+βα=βα

and )0,0,1,1(),,y,x(),,y,x(),,y,x( 454545453535353525252525 =βα⊕βα⊕βα .

),,y,x( 12121212 βα , ),,y,x( 13131313 βα etc. are non-negative trapezoidal fuzzy numbers. Step2: Using ranking function to the Fuzzy Linear Programming problem, formulated in Step1, may be written as Maximize ℜ [(2,4,0,1) ⊕βα⊗ ),,y,x( 12121212 ( 2,6,1,3) ⊕βα⊗ ),,y,x( 13131313 ( 2,5,1,1)

⊕βα⊗ ),,y,x( 15151515 (2,5,0,1) ⊕βα⊗ ),,y,x( 24242424 (2,8,0,3)⊕βα⊗ ),,y,x( 25252525 (1,2,0,0) ⊕βα⊗ ),,y,x( 34343434 (7,15,1,4)⊕βα⊗ ),,y,x( 36363636 (2,5,1,2) ⊕βα⊗ ),,y,x( 45454545 (3,6,0,2) ⊕βα⊗ ),,y,x( 46464646 (1,2,0,1) ),,y,x( 56565656 βα⊗ ]

subject to the constraints )0,0,1,1(),,y,x(),,y,x(),,y,x( 151515151313131312121212 =βα⊕βα⊕βα ,

),,y,x(),,y,x(),,y,x( 121212122525252524242424 βα=βα⊕βα , ),,y,x(),,y,x(),,y,x( 131313133636363634343434 βα=βα⊕βα ,

),,y,x(),,y,x(),,y,x(),,y,x( 34343434242424244646464645454545 βα⊕βα=βα+βα , ),,y,x(),,y,x(),,y,x(),,y,x( 45454545252525251515151556565656 βα⊕βα+βα=βα

and )0,0,1,1(),,y,x(),,y,x(),,y,x( 454545453535353525252525 =βα⊕βα⊕βα .

),,y,x( 12121212 βα , ),,y,x( 13131313 βα etc. are non-negative trapezoidal fuzzy numbers. The Crisp Linear Programming problem becomes :

Fuzzy linear programming model 113 Maximize ( 12x +2 12y - 0 12α +0.25 12β + 13x +3 13y -0.25 13α +0.75 13β + 15x +2.5 15y -0.25 15α +0.25 15β + 24x +2.5 24y -0 24α +0.25 24β + 25x +4 25y -0 25α +0.75 25β +0.5 34x + 34y -0 34α +0 34β +3.5 36x +7.5 36y -0.25 36α + 36β + 45x +2.5 45y -0.25 45α +0.5 45β +1.5 46x +3 46y -0 46α +0.5 46β +0.5 56x + 56y -0 56α +0.25 56β ). subject to the constraints

1xxx 151312 =++ , 1yyy 151312 =++ , 0151312 =β+α+α , 0151312 =β+β+β

122524 xxx =+ , 122524 yyy =+ , 122524 α=α+α , 122524 β=β+β

133634 xxx =+ , 133634 yyy =+ , 133634 α=α+α , 133634 β=β+β

34244645 xxxx +=+ , 34244645 yyyy +=+ , 34244645 α+α=α+α ,

34244645 β+β=β+β

45251556 xxxx ++= , 56251556 yyyy ++= , 45251556 α+α+α=α ,

45251556 β+β+β=β

1xxx 564636 =++ , 1yyy 564636 =++ , 0564636 =α+α+α , 0564636 =β+β+β

12x , 12y , 12α , 12β , 13x , 13y , 13α , 13β , 15x , 15y , 15α , 15β , 24x , 24y , 24α , 24β , 25x , 25y ,

25α , 25β , 34x , 34y , 34α , 34β , 36x , 36y , 36α , 36β , 45x , 45y , 45α , 45β , 46x , 46y , 46α , 46β ,

56x , 56y , 56α , 56β ≥0. Step 3: On solving Crisp Linear Programming Using TORA System, obtained in Step2, an optimal solution is

13x = 13y = 13α = 13β = 36x = 36y = 36α = 36β =1 and the remaining all values

12x , 12y , 12α , 12β , 15x , 15y , 15α , 15β , 24x , 24y , 24α , 24β , 25x , 25y , 25α , 25β , 34x , 34y ,

34α , 34β , 45x , 45y , 45α , 45β , 46x , 46y , 46α , 46β , 56x , 56y , 56α , 56β are zero. Step 4: Putting the values of ijx , ijy , ijα and ijβ in ijx~ = )β,α,y,x( ijijijij . The solution is

12x~ =(0,0,0,0), 13x~ =(1,1,1,1),

15x~ =(0,0,0,0), 24x~ =(0,0,0,0), 25x~ =(0,0,0,0), 34x~ =(0,0,0,0),

36x~ =(1,1,1,1), 45x~ =(0,0,0,0), 46x~ =(0,0,0,0), 56x~ =(0,0,0,0). Step 5: Using the fuzzy optimal solution, the fuzzy critical path is 1-3-6. Replacing the values of ijx , ijy , ijα and ijβ in Step1, the maximum total completion fuzzy time is (9,21,2,7). Hence, the fuzzy critical path is 1-3-6 and the corresponding maximum total completion fuzzy time is (9,21,2,7) respectively.

114 K. Usha Madhuri, B. Pardha Saradhi and N. Ravi Shankar 7. Comparision of proposed method with existing method The results of the numerical example obtained from the sections 6.1 , 6.2,and 6.3 are presented in table I. From table I, we can easily seen that number of constraints in crisp linear programming problem represented by new representation of trapezoidal fuzzy numbers are less compare to crisp linear programming problem represented by existing representations of trapezoidal fuzzy numbers. Hence, it is better to use crisp linear programming problem represented by new representation of trapezoidal fuzzy numbers.

Table I: Results for existing method and proposed representation of trapezoidal fuzzy numbers

Representation of

Trapezoidal fuzzy number

Number of constraints

in Fuzzy Linear

programming problem

Number of constraints in Crisp Linear

Programming problem

Fuzzy critical

path

Maximum total

completion fuzzy

time

(a,b,c,d) 16 (4×16)+(3×10)=48+30=78

1→3→6

(9,11,14,21)

(m,n,α,β) 16 (4×16)+(2×10)=48+20=68

1→3→6

(11,14,2,7)

(x,y,α,β)New 16 (4×16)=48 1→3→6

(9,21,2,7)

8. Conclusion

A new method has been proposed to find the fuzzy critical path and fuzzy completion time of a fuzzy project. Also a new representation of trapezoidal fuzzy numbers is proposed and it is shown that it is better to use the proposed representation of trapezoidal fuzzy numbers instead of existing representations to find the fuzzy critical path and fuzzy completion time of fuzzy critical path problems.

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