Page 1

Deepika Garg, Kuldeep Kumar & Jai Singh

International Journal of Engineering (IJE), Volume (3) : Issue (2) 201

Availability Analysis of A Cattle Feed Plant Using Matrix Method

Deepika Garg

Research scholar,

Dept of Mathematics

N.I.T., Kurukshetra,

India

Kuldeep Kumar

Prof. and Chairman,

Dept of Mathematics,

N.I.T., Kurukshetra,

India

Jai Singh

Principal, M.I.E.T,

Mohri, Kurukshetra,

India

deepikanit@yahoo.in

kuldeepnitk@yahoo.com

jaisinghgurjar@gmail.com

ABSTRACT

A matrix method is used to estimate the probabilities of complex system events by simple

matrix calculation. Unlike existing methods, whose complexity depends highly on the system

events, the matrix method describes the general system event in a simple matrix form.

Therefore, the method provides an easy way to estimate the variation in system performance

in terms of availability with respect to time.

Purpose- The purpose of paper is to compute availability of cattle feed plant .A Cattle feed

plant consists of seven sub-systems working in series. Two subsystems namely mixer and

palletiser are supported by stand-by units having perfect switch over devices and remaining

five subsystems are subjected to major failure.

Methodology/approach- The mathematical model of Cattle feed plant has been developed

using Markov birth – death Process.The differential equations are solved using matrix method

and a C-program is developed to study the variation of availability with respect to time.

Findings- The study of analysis of availability can help in increasing the production and

quality of cattle feed. To ensure the system performance throughout its service life, it is

necessary to set up proper maintenance planning and control which can be done after

studying the variation of availability with respect to time.

Originality/value- Industrial implications of the results have been discussed.

Keywords: Availability, Differential Equations, Markov Process, Matrix Method.

Page 2

Deepika Garg, Kuldeep Kumar & Jai Singh

International Journal of Engineering (IJE), Volume (3) : Issue (2) 202

1 INTRODUCTION

Modern engineering systems like process and energy systems, transport systems, offshore structures, bridges,

pipelines are design to ensure the successful operation throughout the anticipated service life. Unfortunately

there is a threat of deterioration of processes, so it is necessary to study the variation of availability with respect

to time. The objective of the present paper is to analysis the availability of cattle feed plant. Cattle feed plant

mainly consists of seven subsystems namely Elevator, Grinder, Hopper, Mixer, Winch, Palletiser and Screw

conveyor. These units are arranged in series. Failure and repair rates of each machine are assumed to be

constant. The mathematical model of cattle feed plant has been developed using Markov birth – death Process.

The differential equations have been developed on the basis of probabilistic approach using transition diagram.

Matrix method is used to solve these equations and calculations are done with the help of c-program. Won-Hee

Kang, Junho Song and Paolo Gardoni [18] discussed the matrix based system to calculate system reliability. The

findings of the present paper can be considered to be useful for the analysis of availability and for determining

the best possible maintenance strategies for a cattle feed plant concerned.

2. LITERATURE SURVEY

The last decades has witnessed a growing interest in the development and application of reliability methods in

the field of various industrial sectors related with maintenance engineering and management. Recently, many

researchers have discussed reliability of different process industries using different techniques. Kumar and

Singh [2] analyzed the Availability of a washing system of paper industry. Singh, Kumar and Pandey [3, 5]

discussed the reliability and availability of Fertilizer and Sugar industry .Dayal and singh [4] studied reliability

analysis of a system in a fluctuating environment. Zaho [6] developed a generalized availability model for

repairable component and series system including perfect and imperfect repair. Michelson [7] discussed the use

of reliability technology in process industry. Singh and Mahajan [8] examined the reliability and long run

availability of a Utensils Manufacturing Plant using Laplace transforms. Günes and Deveci [9] have studied the

reliability of service systems and its application in student office and Habchi [10] discussed and improved the

method of reliability assessment for suspended test . Jain [11] discussed N-Policy for redundant repairable

system with additional repairman. Gupta, Lal, Sharma and Singh [12] discussed the reliability, long term

availability and MTBF of cement industry with the help of Runga – Kutta method. Kiureghian and Ditlevson [13]

analyzed the availability, reliability & downtime of system with repairable components. Kumar, Singh and

Sharma [15] discussed the availability of an automobile system namely “scooty”. Tewari, Kumar, Kajal and

Khanduja [16] discussed the availability of a Crystallization unit of a sugar plant. In these papers, authors used

either Laplace transforms method or Lagrange’s or runge-kutta method to solve differential associated with

particular problem. Jussi K.Vaurio [17] discussed current research and application related to the modeling,

optimization and application of maintenance procedures for ageing and deteriorating engineering and structural

systems. It has been observed that these methods involve complex computations and it is very difficult to

calculate availability/reliability of the system by these methods. In fact, problem of calculating variation of

availability with time has not satisfactorily been tackled till now. This leads to the development of matrix method

in order to calculate reliability of the system. In the present paper, matrix method is used and then computer

program is developed to calculate the value of availability at various interval of time. The variation in the

availability of cattle feed plant is also shown with the help of graph.

3. THE SYSTEM

The Cattle feed plant mainly consists of seven subsystems namely Elevator, Grinder, Hopper, Mixer, Winch,

Palletiser, Screw conveyor. Initially Elevator lifts the material and put it into the Grinder. Grinder grinds the raw

material and then the material is put into the Hopper. Hopper is used for the storage and cooling of material.

Cooling is done by the fans present in the Hopper. Then the material is put into the Mixer for proper mixing of

certain additives in specified ratio. This mixture is lift by Winch which put this mixture into the Palletiser.

Palletiser allows the mixture to move forward and passes through holes which give them a proper shape. Finally

Screw conveyor carries the final product to the store where it is packed for final delivery

Page 3

Deepika Garg, Kuldeep Kumar & Jai Singh

International Journal of Engineering (IJE), Volume (3) : Issue (2) 203

The Cattle feed plant consists of the following seven main subsystems:

I. Elevator (A) consists of one unit. The system fails when this subsystem fails.

II. Grinder (B) consists of one unit. It is subjected to major failure only.

III. Hopper (C) consists of one unit. It is subjected to major failure only.

IV. Mixer (D) consists of two units, one working and the other is in cold standby. The cold standby unit is of

lower capacity. The system works on standby unit in reduced capacity. Complete failure occurs when both

units fail.

V. Winch (E) consists of one unit. The system fails when this subsystem fails

VI. Palletiser (F) consists of two units, one working and the other is in cold standby. The cold standby unit is of

lower capacity. The system works on standby unit in reduced capacity. Complete failure occurs when both

units fail.

VII. Screw conveyor (G) consists of one unit. The system fails when this subsystem fails

4. ASSUMPTIONS AND NOTATIONS

I.

Repair rates and failure rates are negative exponential and independent of each other.

II. Not more than one failure occurs at a time.

III. A repaired unit is, performance wise, as good as new.

IV. The subsystems D and F fail through reduced states.

V. Switch-over devices are perfect.

A, B, C, D, E, F, G : Capital letters are used for good states.

D , F : Denotes the reduced capacity states.

a, b, c, d, e, f, g : Denotes the respective failed states.

λi : Indicates the respective mean failure rates of Elevator, Grinder, Hopper,

Mixer, Winch, Palletiser, Screw conveyor. i =1,2,3,4,5,6,7,8,9. i = 5 and 8

stands for failure rates of reduced states of D and F respectively.

µi : Indicates the respective repair rates of Elevator, Grinder, Hopper , Mixer,

Winch, Palletiser, Screw conveyor, i =1,2,3,4,5,6,7,8,9. i = 5 and 8 stands

for repair rates of reduced states of D and F respectively.

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

Pi '(t) : Derivative of probability function Pi (t).

5. MATHEMATICAL MODELING

Probabilistic considerations give the following differential equations, associated with the transition diagram as

given by figure 2.

'( )( )( )( )( )( )

p ta p tp tp tp tp t

µµµµ

=++++

111152

µ

6

p

3768994

+

472

µ

( )( )( )

p tp tp t

µµµ

+++

22212122031943618

p

817

p

916

p

7

λ

1

p t

'( )

t

( )( )

t

( )

t

( )

t

( )( )

t

( )

t

( )

t

( )

pa p tppp tpppp t

µµµµµλ

=+++++++

3331282273265256248239 22

( )

4

p t

274

'( )

t

( )( )

t

( )

t

( )

t

( )

t

( )

t

( )

t

( )

t

( )( )

pa p tppppp t

µµµµµµµλ

=+++++++++

4441152143135126119107341

'( )

t

( )( )

t

( )

t

( )

t

( )

t

( )

t

( )

t

( )

pa p tppppppp t

µµµµµµµλ

=++++++++

Where

a

11236947

()

λλλλλλλ

= −++++++

µ

+

2

µ

1

a

23

+

46

+

8

+

9

+

7

+

()

a

λλλλλλλµ

= −+++++++

31235

( )

68947

()

a

λ

µ

λλ

λ

=

λλλλ

= −+++++++

412356794

()

λλλλλλλµ

= −++

1

'( )( )

5,6,7,8,9;1,2,3,6,9;

ijij

p tp tp t

ij

+

==

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Deepika Garg, Kuldeep Kumar & Jai Singh

International Journal of Engineering (IJE), Volume (3) : Issue (2) 204

2

'( )( )( )

16,17,18,19,20,21;

'( )

ji

p t p t

µ

+

9,8,6,3,2,1;

ijij

p tp t p t

ij

µλ

+=

==

3

( )( )

22,23,24,25,26,27,28;

'( )( )

ji

p tp t

µ

+

9,8,6,5,3,2,1;

ij

p t

ij

λ

=

==

4

( )

10,11,12,13,14,15;9,6,5,3,2,1;

ij

p t

i

With initial conditions P1 (0) =1, otherwise zero.

j

λ

=

==

Let p(k, t) denotes the transition probability of the event that the system is in

state k at the time t. Since the number of all the possible transition states of the complex system is‘28’.So the

system of differential difference equations for above equations may be written as;

(θI-A) P (k, t) = 0

Where θ = d/dt, 0 is the null matrix, matrix A is the matrix of coefficients of Pi (t)’s in differential difference

equation

Matrix A=

Page 5

Deepika Garg, Kuldeep Kumar & Jai Singh

International Journal of Engineering (IJE), Volume (3) : Issue (2) 205

17412369

724986321

437

a

9865321

474965321

11

22

00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0

0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

a

a

a

µµ µ µ µ µ µ

λ

µµ µ µ µ µ µ

λλ µ µ µ µ µ µ µ

λµµ µ µ µ µ µ

λµ

−

λµ

−

−

−

−

−

33

66

99

99

66

55

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

λµ

−

λµ

−

λµ

−

λµ

−

λµ

−

λµ

−

33

22

11

99

88

66

3

0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0 0 0

λµ

−

λµ

−

λµ−

λµ

−

λµ

−

λµ

−

λ

3

22

11

99

88

66

55

0 0 0 0 00 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

µ

−

λµ

−

λµ−

λµ

−

λµ

−

λµ

−

λµ

−

33

22

11

0 0 0

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00

0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

λµ−

λµ−

λµ−