Behavior Analysis of a Biscuit Making Plant using Markov Regenerative Modeling

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ISSN : 2319 – 3182, Volume-2, Issue-3, 2013

148

Behavior Analysis of a Biscuit Making Plant using Markov Regenerative Modeling

Parvinder Singh& Atul Goyal

Department of Mechanical Engineering, Lala Lajpat Rai Institute of Engineering & Technology, Moga - 142001, India

Abstract – Modern products ranging from simple components to complex should be designed to be optimal and reliable. The challenge of modern engineering is to ensure that manufacturing costs are reduced and design cycle times are minimized while achieving requirements for performance and reliability. This paper explains a methodology to study the transient behavior of repairable mechanical biscuit shaping System pertaining to a biscuit manufacturing plant. The methodology for determining the availability of the system is based on Markov Modeling. The effect of repair rate on most vulnerable items of the system is examined to realize the highest level of performance. The failure and repair rates of the different subcomponents of the system are taken as constant. Probability considerations at various stages of the system give differential equations, which are solved using Laplace Transform to obtain the state probabilities.

Increase in availability of system confers many benefits such as more profit, improved delivery performance and reduced lead times.

Keywords – Manufacturing system availability, priority performance measures, productivity, reliability.

I. INTRODUCTION

Reliability/availability/maintainability analyses of production systems pertaining to process industries have assumed ever-increasing importance in the recent past. This can benefit the industry in terms of higher productivity and lower maintenance costs. In the recent past quite a good number of studies have been carried out by researchers [1]-[9] in the field of reliability/availability/maintainability, and several methods have been proposed for reliability analysis of industrial systems under preventive/predictive maintenance. A number of performance measures are used in the process industry as indicators to describe the performance of a plant regarding its reliability and maintainability. Commonly used indicators are: on

stream factor, on stream factor with slowdown, availability (inherent, achievable or operational), turnaround rate, annualized turnaround index, routine maintenance cost index etc. Clearly most of these indicators are used mainly in the operational stage while a few of them can be used to evaluate different designs at the early stages of design. Plant availability is commonly considered in the design stage for screening different design alternatives. Availability, in general, is defined as the ability of an item to perform its required function at a stated instant of a time or over a stated period of time.

A biscuit manufacturing plant situated in Ludhiana, India is chosen for study. In this paper a subsystem of the plant, which is a continuous production system, is considered and the availability analysis of the complex mechanical system under preemptive resume priority repair is carried out. Laplace transform is used for solving differential equations to obtain state probabilities. Numerical results based upon the true data collected from industry are presented to illustrate the steady state behavior of the system under different plant conditions.

II. SYSTEM DESCRIPTION

In Biscuit Shaping System, biscuits are shaped from the dough sheet, which is supposed to wound over the dies of a rotary machine. Initially tilter machine serves the purpose of tilting out the dough from the trolleys to the dough hopper. In this hopper, dough is transformed into dough sheets with the help of blades.

The conveyor transfers this dough sheet further to cutter where cutting of dough sheet into predefined size have been done. Further desired size dough sheet is conveyed to rotary machine through metal detector. The metal

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149 detector detects unwanted/foreign particles from the dough sheet before subsequent operation. On rotary machine different sizes and shapes of biscuits can be produced with the help of interchangeable dies. The figure-1 represents the working of this system.

Fig. 1: Process Flow Diagram

III. NOTATIONS AND ASSUMPTIONS NOTATIONS

Subsystem A: Unit namely tilter machine subjected to major failure only.

Subsystem B: Unit namely dough hopper subjected to major failure only.

Subsystem C1&C2: Two units namely conveyor and cutter working parallel in the plant subjected to minor

& major failure.

Subsystem D: Unit namely rotary machine subjected to major failure only.

λ1, λ₂: Failure rates of subsystems C1 & C2resp.

λ3, λ4, λ5 : Failure rates of subsystemsA, B and D resp.

_1, ₂: Repair rates of subsystems C1 & C2resp.

_3, _4, ₅: Repair rates of subsystems A, B and Dresp.

g: The subsystem /unit is good but not operative.

r: The subsystem /unit is under repair.

qr: The subsystem/unit is queuing for repair.

Pi (t): State probability that the system is in i^th state at time t.

S: Laplace transform variable

Dash (`) : Represent derivatives w.r.t.„t‟.

ASSUMPTIONS

1. All the units are initially operating and are in good state.

2. Each unit is as good as new after repair.

3. The failure and repair events are statistically independent.

4. In case of the failure of tilter machine, the dough will be fed manually to the dough hopper. So tilter machine is the least priority item for repair action.

The system can work at full capacity even when this subsystem is under repair action.

5. There is single repair facility and thus repair is performed on priority basis if the failed items are more.

6. The subsystem namely Metal Detector (M) never fails.

7. Whenever a unit fails, its repair begins immediately.

8. Maximum two subsystems will come to failed state at the same time because repair of failed unit begins immediately and is carried quite quickly.

IV. AVAILABILITY ANALYSIS OF THE SYSTEM Probability consideration gives the following differential equations associated with the state transition diagram as shown in figure-2.

) ( ) ( ) ( ) ( ) ( ) ( )

( 0 0 1 2 2 3 3 1 4 4 5 5

'

0t

P

t

P

t

P

t

P

t

P

t

P

t

P

^



^



^



^



^



^



Where



0



1



2



3



4



5 ) ( ) ( ) ( ) ( ) ( ) ( )

( ₉

8 5 7 4

6 2 3 1

1 1 '

1t

P

t

P

t

P

t

P

t

P

t

P

t

P

^



^



^o ^



^



^



^



Where



1



3



1



2



4



5

Fig. 2: State Transition Diagram

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150

) ( )

( )

( ₂ ₁ ₀

1 '

2 t

P

t

P

t

P

^



^



) ( )

( )

( ₄ ₄ ₀

4 '

4t

P

t

P

t

P

^



^



) ( )

( )

( ₅ ₅ ₀

5 '

5t

P

t

P

t

P

^



^



) ( )

( )

( ₆ ₁ ₁

1 '

6 t

P

t

P

t

P

^



^



) ( )

( )

( ₁

7 2 2 '

7 t

P

t

P

t

P

^



^



) ( )

( )

( ₈ ₄ ₁

4 '

8 t

P

t

P

t

P

^



^



) ( )

( )

( ₁

9 5 5 '

9 t

P

t

P

t

P

^



^



With initial conditions at time t =0 )

P

i(t = 1 for i=0 )

P

i(t = 0 for i 0

Taking Laplace transform of above equations, the probability transform are:



₀ ₁ ₂



¹

0( )

A



G

^

P

^s



Where

A

⁰

s

^



0^



2

G

3^



3

G

1^



4

G

4^



5

G

5

 )

P

ⁱ(s

G

ⁱ

P

⁰⁽^s⁾

For i = 1 to 9 Where G A

1 3 1





 

1 1 2 

G s

 

2 2

3 

G

s

 

4 4

4 

G

s

 

5 5

5 

G

s

G

⁶^

G

¹

G

²

G G

G

⁷^ ¹ ³

G

⁸^

G

¹

G

⁴

G

G G

⁹^ ¹ ⁵

A

¹

s

^



1^



1

G

2^



2

G

3^



4

G

4^



5

G

⁵

Availability function A (t) for the system is given as )

( ) ( )

(s

P

₀ s

P

₁ s

A

^ ^

^

P

₀(s)^

G

₁

P

₀(s)

 )

A

(s



¹^

G

₁

 P

₀⁽^s⁾

where

P

₀⁽^s⁾^

 A01G21.

Inversion of A_(s₎ gives the availability function A_(t₎

STEADY STATE BEHAVIOR OF THE SYSTEM The Steady state behavior of the system can be analyzed by setting ___, _₀_,

ddt

t the state

probabilities are:

P P P P P

P

0 1 2 2 3 3 1 4 4 5 ⁵

0 1

    



^ ^ ^ ^ ^ ^

P P P P P

P

1 3 ^o 1 6 2 7 4 8 5 ⁹

1

    



^ ^ ^ ^ ^

P

² ¹ ⁰

1





^

P P

³ ² ⁰

2





^

P P

⁴ ⁴ ⁰

4





^

P P

⁵ ⁵ ⁰

5





^

P P

⁶ ¹ ¹

1





^

P P

⁷ ² ¹

2





^

P P

⁸ ⁴ ¹

4





^

P P

⁹ ⁵ ¹

5





^

On solving equations recursively, we get

P

ⁱ

H

ⁱ

P

^o For i = 1 to 9 Where

H B

1 3 1





 

1 1 2

H

 

2 2 3

H

 

4 4 4

H

 

5 5 5

H

⁶^

H

¹

H

²

H

⁷^ ¹ ³

H

⁸^

H

¹

H

⁴

H

⁹^ ¹ ⁵

B

¹



1^



1

H

2^



2

H

3^



4

H

4^



5

H

⁵

Using normalizing condition

1

9

0

 

 i

P

i ^{, we get}

P

⁰

9 1

1







 



 



i

H

i

The Overall steady state availabilities of the system running with either full capacity is given by:

P P A

^O^ ^o^ ¹

 P

⁰



1

H

1



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151









9

1 1

i

H

i

H

NUMERICAL RESULTS

The Overall steady state availability of the system by taking ₁=0.01, ₂= 0.01, ₃=0.02, ₄=0.005, ₅= 0.01, ₁= 0.33, ₂ =0.5, ₃= 0.2, ₄ = 0.33, ₅=0.4 is evaluated as

P P A

^O^ ^o^ ¹

^

P

⁰



¹^

H

₁



= 0.9170

V. AVAILABILITY ANALYSIS

The effect of various parameters on availability is studied. If the failure rate and repair rates are altered, the availability is affected. This effect is shown in following tables obtained for availability of biscuit making system.

1. Effect of failure rate of Conveyor-I on availabilityAo: Taking 2 = 0.03, 3 = 0.01, 4

=0.005, ₅=0.04, ₁= 0.33, ₂= 0.5, ₃=0.5, ₄= 0.33, ₅= 0.25.

Table1. Steady state availability versus failure rate of Conveyor-I

1 0.01 0.02 0.04 0.08

A₀ 0.9170 0.8923 0.8465 0.7677 2. Effect of failure rate of Cutter on availability Ao:

Taking ₁= 0.02, ₃= 0.01, ₄=0.005, ₅=0.04, ₁

= 0.33, ₂= 0.5, ₃=0.5, ₄= 0.33, ₅= 0.25.

Table 2. Steady state availability versus failure rate of Cutter.

2 0.01 0.02 0.04 0.08

A₀ 0.9170 0.9005 0.8692 0.8127 3. Effect of failure rate of Dough hopper on

availability Ao: Taking ₁ = 0.02, ₂= 0.03,

₃=0.01, ₅=0.04, ₁= 0.33, ₂= 0.5, ₃=0.5, ₄= 0.33, ₅= 0.25.

Table 3.Steady state availability versus failure rate of dough hopper

₄ 0.005 0.010 0.020 0.040

A₀ 0.9170 0.9045 0.8804 0.8358

4. Effect of failure rate of Rotary machine on availability Ao: Taking ₁ = 0.02, ₂= 0.03,

₃=0.01, ₄=0.005, ₁= 0.33 , ₂= 0.5, ₃=0.5, ₄

= 0.33, 5 = 0.25

Table 4 Steady state availability versus failure rate of rotary machine

₅ 0.01 0.02 0.04 0.08

A₀ 0.9170 0.8965 0.8580 0.7902 5. Effect of repair rate of Conveyor-I on availability

Ao: Taking ₁= 0.02, ₂= 0.03, ₃=0.01, ₄=0.005,

5 =0.04, 2= 0.5, 3 =0.5, 4 = 0.33, 5 = 0.25 Table5. Steady state availability versus repair rate of

Conveyor-I

₁ 0.33 0.66 1.32 2.64

A₀ 0.9170 0.9300 0.9366 0.9399 6 Effect of repair rate of Cutter on availability Ao:

Taking ₁= 0.02, ₂= 0.03, ₃=0.01, ₄=0.005 , ₅

=0.04, 1= 0.33, 3 =0.5, 4 = 0.33, 5 = 0.25.

Table 6. Steady state availability versus repair rate of Cutter

₂ 0.5 1.0 1.5 2.0 A₀ 0.9170 0.9255 0.9284 0.9298 7 Effect of repair rate of Dough hopper on

availability Ao: Taking ₁= 0.02, ₂= 0.03 ,

3=0.01, 4 =0.005 , 5 =0.04, 1= 0.33, 2 =0.5,

₃=0.5, ₅= 0.25.

Table 7 Steady state availability versus repair rate of Dough hopper

₄ 0.33 0.66 1.32 2.64

A₀ 0.9170 0.9235 0.9267 0.9283 8. Effect of repair rate of Rotary machine on

availability Ao: Taking 1 = 0.02, 2= 0.03,

₃=0.01, ₄=0.005, ₅=0.04, ₁= 0.33, ₂=0.5, ₃

=0.5, ₄= 0.33.

Table 8 Steady state availability versus repair rate of rotary machine

5 0.4 0.8 1.6 2.0 A₀ 0.9170 0.9277 0.9331 0.9342

VI. RESULTS AND DISCUSSION

From analysis part it is being found that increase in the failure rate of conveyor-I, cutter, dough hopper and rotary machine reduces the availability of the system.

Table 1 to 4 highlights the effect of failure rate of

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152 conveyor-I, cutter, dough hopper and rotary machine on the long run availability of the biscuit making system.

On the other hand the repair rates of the constituent component increase the availability of the system. This effect is shown in tables 5 to 8.The respective improvements in the availability of the system are 2.29%, 1.28%, 1.13% and 1.72% on increasing the repair rate of conveyor-I, cutter, dough hopper and rotary machine from 0.33 to 2.64, 0.5 to 2.0, 0.33 to 2.64 and 0.4 to 2.0 repairs per hour respectively.

VII. REFERENCES

[1] Singh I.P., „„A Complex System having Four Types of Components with Preventive Repeat Priority Repairs‟‟, Micron reliability, 1989, Vol.

29, No. 6, pp. 959-962.

[2] I. P. Singh, „„Preemptive repeat priority repairs and failures of non-failed components during system failure of a complex system‟‟, Micron reliability, 1991, pp. 216-221.

[3] R.T. Islamov, „„Using Markov Reliability Modelling for Multiple Repairable Systems‟‟, Reliability engineering and system safety, 1994, pp. 113 – 118

[4] Q. Microelsen, „„Use of Reliability technology in the process industry‟‟, Reliability engineering and system safety, 1998, pp. 179 - 181

[5] Gupta P., Singh J., and Singh I.P., „„Behavioral Analysis of Duplex Casting System under Preventive Resume Priority Repair Discipline a Case Study‟‟, Industrial Engineering Journal Vol. XXXIII No.12, December 2004.

[6] P. Gupta, J. Singh, and I. P. Singh,

„„Maintenance Planning based on Performance Analysis of 7-out-of-14: G Chemical-system: a Case Study,‟‟, International Journal of Industrial Engineering, 2005, pp. 264-274.

[7] P. Gupta, A. Gupta, „„Availability improvement of a production system under preventive maintenance‟‟, Industrial Engineering Journal, India, 2006. Vol. XXXV No.8, pp. 21-26.

[8] D. Dyer, “Unification of Reliability / Availability / Repairability Model for Markov Systems” , IEEE Transactions on Reliability, 1985, Vol. 38.

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