DOI : 10.17577/IJERTV2IS110329
- Open Access
- Total Downloads : 290
- Authors : Payel Ghosh, Tapan Kumar Roy
- Paper ID : IJERTV2IS110329
- Volume & Issue : Volume 02, Issue 11 (November 2013)
- Published (First Online): 09-11-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Intuitionistic Fuzzy Goal Geometric Programming Problem (IF G2 P2) based on Geometric Mean Method
Intuitionistic Fuzzy Goal Geometric Programming Problem (IF) based on Geometric Mean Method
Payel Ghosh 1*, Tapan Kumar Roy 2
1Department of Mathematics, Adamas Institute of Technology, Barasat, P.O.-Jagannathpur, Barbaria, 24 Parganas (N), West Bengal – 700126,
2 Department of Mathematics, Bengal Engineering and Science University, Shibpur, P.O.-Botanic Garden, Howrah, West Bengal-711103,
Abstract
In this paper, a new approach of intuitionistic fuzzy goal programming is proposed. A nonlinear intuitionistic fuzzy goal programming is solved here using geometric programming technique. For this we have used geometric mean method. We have applied the proposed method on industrial waste water treatment design problem. Also there is a comparison of results on industrial waste water treatment design problem with other methods to show the benefit of this method.
Keywords: Goal programming, Geometric programming, Non-linear optimization.
-
Introduction
Intuitionistic fuzzy set (IFS) introduced by Atanassov [1], is a growing field of research in different directions. Now a days goal programming in fuzzy environment is common whereas intuitionistic fuzzy goal programming
is rare. In this paper we have worked on intuitionistic fuzzy goal programming problem where equations are non-linear. Geometric programming gives better result than nonlinear programming (K-K-T conditions), which is already described in Ghosh, Roy [2, 3]. Therefore geometric programming is used here to solve nonlinear goal programming problem. Nan et. al. [4], Ghosh, Roy
[5] discussed arithmetic mean in intuitionistic fuzzy environment. In this paper we have used geometric mean in intuitionistic fuzzy goal programming. A numerical example and an application on industrial waste water treatment design problem is taken here as an illustration. Previously Shih, Krishnan [6], Evenson [7], Ecker, McNamara [8], Beightler, Philips [9] have illustrated this design using dynamic programming, geometric programming. Later Cao [10] has discussed the same using fuzzy geometric programming. In this paper, we have compared the results of industrial wastewater treatment design problem with the results in another method. -
Fuzzy Goal Geometric Programming Problem (F)
Find X = (1 , 2 )T (1.1) so as to
0(X), with target value 0, acceptance tolerance 0 .
subject to (X) , r= 1, 2 m, X=(1 , 2 )T>0. Membership functions can be written as follows
0 (X) =
1, 0 X 0
-
Intuitionistic Fuzzy Goal Geometric Programming Problem (I F)
Find X = (1 , 2 .. )T (2.1) so as to
0(X), with target value 0, acceptance tolerance
0 rejection tolerance 0 .
subject to (X) , r= 1, 2 m X=(1 , 2 )T >0.
Membership and non-membership functions are
0 (X) =
1 0 X 0 ,
X
+
0
0
0
0
0
1, 0 X 0
0, 0 X 0 + 0
1 0 X 0 ,
X
+
0
0
0
0
0
Hence the crisp programming from fuzzy goal programming is
Maximize 0 (), (1.2)
0 (X) =
0, 0 X 0 + 0
0, 0 X 0
subject to 0 0 1
0 X 0 ,
X
+
0
0
0
0
0
(X) , r= 1, 2 m, X=(x1 , x2 xq )T>0.
1, 0
X 0
+ 0
Model (1.2) can be written as
Maximize (1.3) subject to 0 ()
() , = 1, 2
0 1, X=(x1, x2 xq )T>0.
This is equivalent to the following geometric programming problem
,
Minimize 1 (1.4)
0 X
0 X
subject to 1
0 (1 )+ 0
0 0 + 0 0 + 0 X
Fig: 1 Membership and non-membership function
Intuitionistic fuzzy goal programming can be
()
1 , = 1, 2 m,
transformed into crisp programming model using membership and non-membership function as
0 1, X=(x1, x2 xq )T>0. Maximize 0 (), Minimize 0 () (2.2)
subject to 0 0 + 0 () 1
(X) , r= 1, 2 m 0 0 (X), 0 (X) 1, X=(x1, x2 xq )T>0.
Let us take (1 )= v>0, then the above model becomes
Minimize v (2.5)
0 X
0 X
subject to 1
0 × 0 + 0
(X) 1, r= 1, 2 m
This is equivalent to Maximize , Minimize (2.3)
subject to 0 () , 0 ()
() , = 1, 2
0 + 1,0 , 1,
X=(x1, x2 xq )T>0.
It is easily seen that Max is equivalent to Min (1- ) as 0 1. Taking geometric mean, the above model can be written as
Minimize (1 ) (2.4) subject to 0 X 0 × 0 (1 )+ 0
(X) , r= 1, 2 m
0 + 1, 0 , 1,
X=(x1, x2 xq )T>0.
0 + 1, 0 , 1, v (0, 1), X=(x1, x2 xq )T >0.
The above model (2.5) is solved by geometric programming technique with v as parameter.
-
Industrial Wastewater Treatment Design
To optimize the treatment of industrial wastewater, a process flow from a paper and pulp industry has been considered. The treatment units indicate the removal of suspended solid and biological oxygen demand (BOD) from the waste water. In this paper, the design of treatment facilities is based on effluent containing BOD and essentially free from suspended solids. Wastewater treatment is consisted of primary clarification, secondary biological treatment (trickling filter followed by activated sludge or aerated lagoon), sludge disposal and tertiary treatment (coagulation, sedimentation, filtration for effluent of activated sludge and aerated lagoon;
Table-1 Wastewater Treatment Design
Design
Primary
Secondary
Tertiary
1
Primary Clarifier
Trickling Filter & Activated Sludge
Carbon Adsorption
2
Primary Clarifier
Trickling Filter & Aerated Lagoon
Coagulation, Sedimentation, Filtration
3
Primary Clarifier
Activated Sludge
Carbon Adsorption
4
Primary Clarifier
Aerated Lagoon
Coagulation, Sedimentation, Filtration
5
Primary Clarifier
Trickling Filter & Activated Sludge
Coagulation, Sedimentation, Filtration
6
Primary Clarifier
Activated Sludge
Coagulation, Sedimentation, Filtration
7
Primary Clarifier
Activated Sludge
None
8
Primary Clarifier
Trickling Filter & Activated Sludge
None
9
Primary Clarifier
Aerated Lagoon
None
10
Primary Clarifier
Trickling Filter
None
carbon adsorption for effluent of aerated lagoon). There are many combination of wastewater treatment process given in Table-1 (Beightler, Philips (1976)) to remove five day BOD (5 ).
(thousand $) and gives flexibility 200 (thousand $) on this goal.
Then the fuzzy goal programming problem is
1(1 , 2, 3 , 4 ) = 19.41.47 + 16.81.66 +
In our study, we have taken the first design. There are
1
91.5 0.3 + 120 0.33 with targ
300, ac
2
tance
consecutively four processes (Primary Clarifier, Trickling Filter, Activated Sludge, and Carbon
3 4
tolerance 200,
et cep
Adsorption).
Primary Clarifier Trickling Filter Carbon Adsorption Activated Sludge
Let be the percentage of remaining 5 after each step. Then after four processes the remaining percentage of 5 will be 1 2 3 4 . Our aim is to minimize the remaining percentage of 5 with minimum annual
cost as much as possible. The annual cost of 5
2(1, 2 , 3 , 4 ) = 1 2 34 with target 0.015, acceptance tolerance 0.1,
subject to 1, 2 , 3 , 4 > 0.
Membership and non-membership functions are given below
1 (1 , 2 , 3 , 4 )=
1, 1(1 , 2 , 3 , 4 ) 300
1 1 1 , 2 , 3, 4 300 , 300 ( , , , ) 500
removal by various treatments is shown in Table-2.
200
1 1 2 3 4
Table-2: List of annual costs in different treatments
0, 1(1 , 2, 3 , 4 ) 500
Design
Treatment
Annual Cost
1
Primary Clarifier
19.41.47
1
2
Trickling Filter
16.81.66
2
3
Activated Sludge
91.50.3
3
4
Carbon Adsorption
1200.33
4
Design
Treatment
Annual Cost
1
Primary Clarifier
19.41.47
1
2
Trickling Filter
16.81.66
2
3
Activated Sludge
91.50.3
3
4
Carbon Adsorption
1200.33
4
2 (1 , 2 , 3 , 4 )=
1, 2(1, 2 , 3 , 4 ) 0.015
2 1 , 2, 3 , 4 0.015
1
0.1
, 0.015 2(1 , 2 , 3 , 4) 0.115
0, ( , , , ) 0.115
2 1 2 3 4
-
Fuzzy Goal Geometric Programming Problem (F)
Decision maker wants to remove about 98.5% 5 and gives some relaxation 0.1 on this goal. Also he sets another goal as annual cost should be about 300
Following (1.2), (1.3) and (1.4), above model can be written into crisp programming problem as
Minimize 1 (3)
1
1
2
2
3
3
4
4
subject to 19.41.47 + 16.81.66 + 91.50.3+ 1200.33 200 (1-)+300
1 2 3 4 0.1(1 ) + 0.015,
1 , 2 , 3, 4 > 0, (0, 1).
Table 3: Optimal values of decision variables and objective functions of model (3)
Dual Variables
Primal Variables
Optimal objective functions
Membership and non- membership functions
=1,
01
=0.088967278
11
=0.078784276
12
=0.435939662
13
=0.396308784
14
=0.130781899
21
=0.7059559
1
=0.7248393
2
=0.1598653
3
=0.5733523
4
(1, 2, 3, 4)
1
= 363.8048
( , , , )
2 1 2 3 4
=0.0469024
1 (1, 2, 3, 4) = 0.680976
2 (1 , 2, 3, 4) = 0.680976
Solving it using Caos geometric programming method taking as a parameter, where degree of difficulty is 5- (4+1) = 0, we have the results given in Table-3. The
table shows that, here 100-0.0469024×100= 95.30976%
1 (1 , 2, 3 , 4 )=
0, 1 (1 , 2 , 3, 4 ) 300
1 1 , 2 , 3 , 4 300 , 300 ( , , , ) 600
removes with the cost of 363.8048 (thousand $).
300
1 1 2 3 4
5
-
Intuitionistic Fuzzy Goal Geometric Programming Problem (I F)
Let decision maker wants to remove about 98.5% 5
and the tolerances of acceptance and rejection on this
1, 1(1 , 2 , 3 , 4) 600
2 (1 , 2 , 3 , 4 )=
1, 2(1 , 2, 3 , 4) 0.015
0.1
0.1
1 2 1,2,3,4 0.015 , 0.015 2 (1 , 2 , 3, 4 ) 0.115 0, 2(1 , 2 , 3 , 4 ) 0.115
(1 , 2, 3 , 4 )=
goal are 0.1 and 0.2 respectively. Also he wants to 2
0, ( , , , ) 0.015
remove the said amount of 5 within 300 (thousand)
2
, , , 0.
1 2 3 4
$ tolerances of acceptance and rejection on this goal are
2 1 2 3 4
015 , 0.015 ( , , , ) 0.215
200 and 300 respectively. Hence the intuitionistic fuzzy
0.2
1 1 2 3 4
goal programming problem is
1, 1(1 , 2 , 3, 4 ) 0.215
( , , , ) = 19.41.47 + 16.81.66 +
Following (2.1), (2.2), (2.3) and (2.4), the crisp
1 1
91.5 0.3 + 1
2 3 4 1
0. ith targ
2
300, acce tance
programming problem is
4
4
3 20 33 w et p
tolerance 200 and rejection tolerance 300
2(1 , 2 , 3, 4 ) = 1 23 4 with target 0.015, acceptance tolerance 0.1 and rejection tolerance 0.2
subject to 1, 2 , 3 , 4 > 0.
Membership and non-membership functions are given below
1 (1 , 2 , 3 , 4 )=
1, 1(1 , 2 , 3 , 4 ) 300
1 1 1 , 2, 3 , 4 300 , 300 ( , , , ) 500
Minimize v ` (4)
subject to
1 2 3 4
1 2 3 4
19.4 1.47 + 16.8 1.66 + 91.5 0.3 + 120 0.33 1
200 ×300 +300
1 23 4 1,
0.1×0.2+0.015
1 , 2, 3 , 4 > 0, v (0, 1)
Solving it using Caos geometric programming method having degree of difficulty 5-(4+1) = 0, we have the results given in table 4.
200 1
1 2 3 4
0, 1(1 , 2 , 3, 4 ) 500
Table 4: Optimal values of decision variables and objective functions of model (4)
Dual Variables
Primal Variables
Optimal objective functions
Membership and non-membership functions
=1,
01
=0.088967278
11
=0.078784276
12
=0.435939662
13
=0.396308784
14
=0.130781899
21
=0.6380199
1
=0.6627170
2
=0.09737155
3
=0.3653206
4
(1 , 2 , 3 , 4)
1
=422.1483
( , , , )
2 1 2 3 4
=0.01504072
1 (1 , 2 , 3 , 4 ) =0.3892583
1 (1 , 2 , 3 , 4) =0.4071612
2 (1 , 2 , 3 , 4 ) =0.9995928
2 (1 , 2, 3 , 4 ) =0.00020358
The table shows that membership and non-membership functions satisfy all the restrictions as in model (2.2). The percentage of 5 removed from the wastewater is (100 0.01504072× 100) = 98.495928% which attains
the set quota by the national standard and the annual total cost is 422.1483 (thousands $).
-
Comparison
Here is a comparison of results between other method and our proposed method.
Method
Total Annual Cost (Thousnd
$)
Remaining
in waste water
Removed BOD5
F2 2
363.8048
0.0469024
95.30976%
IF2 2
Arithmetic mean (Ghosh, Roy (2013b))
359.2533
0.04919147
95.080853%
IF2 2
Geometric mean
422.1483
0.01504072
98.495928%
Method
Total Annual Cost (Thousand
$)
Remaining
in waste water
Removed BOD5
F2 2
363.8048
0.0469024
95.30976%
IF2 2
Arithmetic mean (Ghosh, Roy (2013b))
359.2533
0.04919147
95.080853%
IF2 2
Geometric mean
422.1483
0.01504072
98.495928%
Table-5: Comparison of results using different methods
-
Conclusion
-
We have applied fuzzy and intuitionistic fuzzy goal geometric programming on industrial waste water treatment design. We have compared the results of various methods on industrial waste water treatment design. We have seen that in fuzzy goal geometric programming and intuitionistic fuzzy goal geometric programming with arithmetic mean, percentage of 5
removal is almost same. But 4551.5 $ is saved in
intuitionistic fuzzy goal geometric programming with arithmetic mean. In the proposed method, intuitionistic fuzzy goal geometric programming with geometric mean, 98.495928% 5 is removed. Hence for better purification intuitionistic fuzzy goal geometric
programming with geometric mean is more appropriate.
Acknowledgement
Its my privilege to thank my respected guide for his enormous support and encourage for the preparation of
this research paper. The great support of my family members makes me possible to do it. Last but not the least I am thankful to Prof. (Dr.) Chanchal Majumder, Department of Civil Engineering, Bengal Engineering and Science University, Shibpur.
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P. Ghosh and T. K. Roy, Intuitionistic Fuzzy Goal Geometric Programming Problem and its application on Industrial waste treatment plant, Accepted in International Conference on Facets of Uncertainties and Applications (ICFUA 2013), Operational Research Society of India, Kolkata Chapter and Department of Applied Mathematics Calcutta University, Kolkata, 2013.