165
RELIABILITY ANALYSIS OF SUGAR MANUFACTURING PLANT USING BOOLEAN FUNCTION TECHNIQUE
1S.C. Agarwal, 2Deepika & 3 Neha Sharma
1Reader, Dept. of Mathematics, M.M. (P.G.) College, Modinagar 2Lecturer, Dept. of Mathematics, KNMIET, Modinagar
3Lecturer, Dept. of Mathematics, Aryan Inst. Gzb.
ABSTRACT
In this paper, an industrial problem related to sugar manufacturing plant for its reliability assessment by employing Boolean function technique is analyzed.
The sugar plant under consideration is a complex system which consists of various subsystems viz., feeding system, cane cutters, crushing system, fileration system, purifier, boilers and the mills. Reliability of sugar manufacturing plant is obtained in three different cases. Mean time to failure of the system has also been determined. A numerical illustration and its graphical representation have also been appended in the end to highlight important results.
KEY WORDS: B.F. technique, Exponential time distribution, Weibull time distribution, M.T.T.F 1. INTRODUCTION
The sugar manufacturing plant consists of seven subsystems. The first subsystem is Feeding subsystem and it takes input. The second subsystem is sugar cane cutter, where cutters start cutting of cane in fine small pieces and then the pieces go to the third subsystem, namely crushing subsystem. The pieces are crushed and juice is obtained. The juice goes to the fourth Refining subsystem, which has an identical unit in standby redundancy. After filtration, this juice goes to the Purification subsystem which has an identical unit in parallel redundancy. When these impurities are removed, purification is completed by sulphonation or carbonation process. The juice so obtained goes to the subsystem Boiler where juice is heated up to get its concentrated from the boiler has an identical unit in parallel redundancy that follows online through prefect switching device by crystallizing the concentrated juice, a coloured crystalline sugar, called Raw sugar, is obtained. This raw sugar goes to the last subsystem. In this subsystem the raw sugar is directly packed. All the subsystems are connected in series.
2. ASSUMPTIONS
1. Initially, the complete system is in operable state.
2. Failure rates of components are s-independent.
3. There is no repair facility for a failed component.
4. Reliability of each component is known in advance.
5. This system is in 1-out-of-7:F arrangement.
3. LIST OF NOTATIONS
x
1 : State of feeding of sugarcane.x
2 : State of sugarcane cutter.x
3 : State of crushing.5 4
, x
x
: State of filtrations.7 6
, x
x
: State of Purification.10 8
, x
x
: State of Boiler.x
9 : State of perfect switching device.x
11 : State of mills.) 11 2
, 1
( i
x
i : 1 in good state0 in bad state
x
i : Negation ofx
i.
166
R
i : Reliability of component corresponding to system state ofx
i.Q
i :( 1 R
i)
unreliability of ith unit of the system.
/
: Conjunction/disjunction.R
S : Reliability of the system as a whole.) ( / )
( t R t
R
SW SE : Reliability functions, when failures follow Weibull/exponential time distribution.Fig -1 . 4. FORMULATION OF MATHEMATICAL MODEL
By using Boolean Function Technique, the conditions of capability for the successful operation of the system in terms of logical matrix are expressed as:
11 10 9 7 5 3 2 1
11 10 9 6 5 3 2 1
11 8 7 5 3 2 1
11 8 6 5 3 2 1
11 10 9 7 4 3 2 1
11 10 9 6 4 3 2 1
11 8 7 4 3 2 1
11 8 6 4 3 2 1
11 2
1
, ,
x x x x x x x x
x x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x
x x x x x x x
x x
x
F
….(1)5. SOLUTION OF THE MODEL
By the application of algebra of logic, equation (1) may be written as
x
1, x
2, x
11 x
1x
2x
3x
11f ( x
1, x
2, x
11)
F
…(2)
167 where
8 7 6 5 4 3 2 1
10 9 7 5
10 9 6 5
8 7 5
8 6 5
10 9 7 4
10 9 6 4
8 7 4
8 6 4
11 2
1
, ,
A A A A A A A A
x x x x
x x x x
x x x
x x x
x x x x
x x x x
x x x
x x x
x x
x
F
…(3)where,
4 6 8
1
x x x
A
…(4)
4 7 8
2
x x x
A
…(5)
4 6 9 10
3
x x x x
A
…(6)
4 7 9 10
4
x x x x
A
…(7)
5 6 8
5
x x x
A
…(8)
5 7 8
6
x x x
A
…(9)
5 6 9 10
7
x x x x
A
…(10)
5 7 9 10
8
x x x x
A
…(11) Using orthogonalization algorithm equation (3) may be written as:
8 7 6 5 4 3 2 1
7 6 5 4 3 2 1
6 5 4 3 2 1
5 4 3 2 1
4 3 2 1
3 2 1
2 1 1
11 2 1
) , , (
A A A A A A A A
A A A A A A A
A A A A A A
A A A A A
A A A A
A A A
A A A
x x x F
…(12)
Now, using algebra of logics
8 6 4
6 4 4
1
x x x
x x x A
8 7 4
7 4 4
2
x x x
x x x A
10 9 6 4
9 6 4
6 4 4
3
x x x x
x x x
x x x A
168
10 9 7 4
9 7 4
7 4 4
4
x x x x
x x x
x x x A
8 6 5
6 5 5
5
x x x
x x x A
8 7 5
7 5 5
6
x x x
x x x A
10 9 6 5
9 6 5
6 5 5
7
x x x x
x x x
x x x A
Now, we have
4 6 8
8 6 4
6 4 4 2
1
x x x
x x x
x x x A
A
x
4x
6 x
7x
8
…(13) …
10 9 8 7 6 4
10 9 8 7 6 4 3 2
1
x x x x x x
x x x x x A x
A
A
…(14)
4 6 7 8 9 10
4 3 2
1
A A A x x x x x x
A
…(15)
4 5 6 8
5 4 3 2
1
A A A A x x x x
A
…(16)
4 5 6 7 8
6 5 4 3 2
1
A A A A A x x x x x
A
…(17)
10 9 8 7 6 5 4
10 9 8 7 6 5 4 7 6 5 4 3 2
1
x x x x x x x
x x x x x x A x
A A A A A
A
…(18)
4 5 6 7 8 9 10
8 7 6 5 4 3 2
1
A A A A A A A x x x x x x x
A
…(19)By making use of equation (13) through (19) equation (3) and then in equation (2)
11 10 9 8 7 6 5 4 3 2 1
11 10 9 8 7 6 5 4 3 2 1
11 10 9 8 7 6 5 4 3 2 1
11 8 7 6 5 4 3 2 1
11 8 6 5 4 3 2 1
11 10 9 8 7 6 4 3 2 1
11 10 9 8 7 6 4 3 2 1
11 10 9 8 7 6 4 3 2 1
11 8 7 6 4 3 2 1
11 8 6 4 3 2 1
11 2
1
, ,
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x
x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x
x x x x x x x
x x
x F
…(20)
169
Finally the probability of successful operation i.e. reliability of the system as a whole is given by
1,
2 ,
11 1
P F x x x
R
s r
10
9 8 7 6 5 4 10 9 8 7 6 5 4
10 9 8 7 6 5 4 8 7 6 5 4 8 6 5 4
10 9 8 7 6 4 10 9 8 7 6 4
10 9 8 7 6 4 8 7 6 4 8 6 4 11 3 2 1
R R Q R Q R Q R R Q R R R Q
R R Q Q R R Q R R Q R Q R R R Q
R R Q R Q R R R Q R R R
R R R Q R R R R Q R R R R R R R R
…(21)
where
R
iis the reliability corresponding to system statex
iand
Q
i 1 R
i 1 1 , 2 11
Thus
10 9 7 5 4 10 9 7 6 5
10 9 8 7 5 10 9 8 7 6 5 4
10 9 6 5 4 10 9 8 6 5 8 7 5 4
8 7 6 5 10 9 8 7 4 8 6 5 4
10 9 8 7 6 4 8 7 6 4 10 9 7 6 5 4
10 9 8 7 5 4 10 9 8 7 6 5 10 9 7 5
10 9 8 6 5 4 10 9 6 5 8 7 6 5 4
8 7 5 10 9 7 4 8 6 5
10 9 8 6 4 8 7 4 8 6 4 11 3 2 1
R R R R R R
R R R R R R R R R R
R R R R R R R R R R R R
R R R R R R R R R R R R R R
R R R R R R R R R R R R R
R R R R R R R R R R R R R R R R
R R R R R R R R R R
R R R R R R R R R R R R R R R
R R R R R R R R R R
R R R R R R R R R R R R R R R Rs
…(22)
6. SOME PARTICUALR CASES
Case I: When reliability of each component is R:
In this case, equation (22) yields
11 10 9 8
7
4 3
4 R R R R R
R
S
…(23) Case II: When all failures follows Weibull time distributionlet
i be the failure rate of component corresponding to system statex
i and it follows Weibull time distribution.Then reliability function of considered system, in this case, at time „t’, is given by:
t t
t
t t
t t
t t
t t
t t
t
t t
t t
t
t t
t t
t t
t t
SW
e e
e
e e
e e
e e
e e
e e
e
e e
e e
e
e e
e e
e e
e e
t R
8 4
1 0 9 7 6 5
8 4 6
1 0 9 7 5
8 7
4 1 0
9 6 5
6 4
8 7 5
4 8
6 5
8 6
1 0 9 7 4
8 1 0
9 7 6 4
7 1 0
9 8 6 4
6 8
7 4 8
6 4 1 1
3 2 1
1 1
1 1
1
1 1
1
1 1
1
1 1
1
1
1 )
(
…(24)
When
1
2
11
then from equation (24)
t t t t t t
SW
t e e e e e e
R ( )
44
3
4 4
5
7 3
6 …(25) Case III: When all failures follows Exponential time distributionExponential time distribution is a particular case of Weibull distribution for
1
.Hence the reliability of whole system at an instant„t‟ in this case, is given by:
170
t t t t t
t
SE
t e e e e e e
R ( )
44
3
4 4
5 3
6
7 …(26) and the expression for M.T.T.F. in this case isM.T.T.F. =
0
) ( dt t R
SE=
11
1 10
3 9
4 8
1 7
4
…(27)7. NUMERICAL COMPUTATION
For a numerical computation, let us consider the values:
(i) Setting
i( i 1 , 2 , 11 ) . 1
and 2
in equation (25) (ii) Setting
i( i 1 , 2 , 11 ) . 1
in equation (26)(iii) Setting
i( i 1 , 2 , 11 ) 0 , 0 . 1 0 . 8
in equation (27),one may compute the table-1 and 2.Corresponding graphs have shown through fig-2 and 3, respectively.
S.No. t
R
SE(t ) R
SW(t )
1 0 1.000 1
2 1 .68150085 .681500858
3 2 .418498476 .135852735
4 3 .242195164 .00570295933
5 4 .135852735 .00005002133
6 5 .074164978 9.7742577X10-8
7 6 .053311867 .45137996X10-11 8 7 .021026067 5.0690434X10-15
Table-1
Reliability Vs Time
0 0 0 1 1 1 1
1 2 3 4 5 6 7 8
Time -->
Reliability -->
RSW(t) RSE(t)
Fig- 2
171
S.No.
M.T.T.F1 0 ∞
2 0.1 2.110794805
3 0.2 1.05537518
4 0.3 .703583453
5 0.4 .52768759
6 0.5 .403968254
7 0.6 .351791726
8 0.7 .3015535765
9 0.8 .263843795
Table-2
MTTF vs Failure rate
0 0.5 1 1.5 2 2.5
1 2 3 4 5 6 7 8
Failure rate Lamda -->
MTTF -->
Fig- 3 8. INTERPRETATION OF THE RESULTS:
1. A critical examination of Table-1 and Graph “Reliability V/s Time” reveals that reliability of the system decreases rapidly when failures follows Weibull distribution but it decreases smoothly when failures follow Exponential time distribution.
2. An inspection of Table-2 and Graph “M.T.T.F. V/s Failure rate” discloses the fact that M.T.T.F. decreases catastrophically in the beginning but later on it decreases approximately at the uniform rate.
9. REFERENCES:
[1]. Yong, C.; Jianghua, J., “Quality reliability chain modeling for system reliability analysis of complex manufacturing process”, IEEE TR on Reliability, Vol. 54, issue-3 pp 425-478, (2005).
[2]. P Gupta, A.K. Lal, R.K. Sharma and J.Singh “ Behavior at study of cement manufacturing plant – A numerical approach”, Journal of mathematics and system science, Vol. 1, No.1, pp. 50-69,(2005).
[3]. Grodin – Perez, B., Benne, M., Bonnecaze, C., Chabriat, J.P., “Industrial multi-step forward predictor of mother liquor purity of the final stage of a cane sugar crystallization plant”, Journal of Food Engineering, 66, 361-367 (2005).
[4]. Goza, O., Kazemian, H., Santana, R. “The affect of impurities on target purity of molasses in cane sugar industry”, Monografia Topicos de ingenieria Qumica. Ciudao Habana, (1) Jae (2008).
[5]. P.C. Tewari, D.Kumar, S.Kajal, R. Khanduja, “Decision support system for the crystallization unit of a sugar plant”, Icfai J. of Science & Technology. Vol. – 4, No.3, pp. 7-16, (2008).
[6]. Panagiotis, H. Tsarauhas, “ Classification and calculation of primary failure modes in bread production line”, Reliability Engineering and system safety, Vol. 94, Issue 2, Pages 551-557 (2009).
