Mathematical Models for Reliability and Replacement of Agricultural

Mathematical Models for Reliability and
Replacement of Agricultural Tractors
By
SAMI ATTA ELMOULA BAKHIT MUDAWI
B.Sc. (Agric), Honours (1993), University of Khartoum.
M.Sc. (Agric.), 1998, University of Khartoum.
A thesis submitted to the University of Khartoum in
fulfilment for the degree of Doctor of Philosophy in
Agricultural Science
Supervisor:
Dr. Mohamed Hassan Dahab
Co-supervisor:
Dr. Hassan Ibrahim Mohamed
Department of Agricultural Engineering
Faculty of Agriculture
University of Khartoum
July 2009
DECLARATION
I, Sami Atta Elmoula Bakhit, hereby declare that the work
embodied in this thesis is my own original work. It has not been
submitted and it is not currently being submitted for the award of a
similar degree in any other University.
Candidate:………………….
Date……………………..
S. A. Bakhit
College of Agricultural Studies
Sudan University of Science and Technology
ii
DEDICATION
To my beloved country.
To the Soul of my Mother.
To my Father, Brothers, Sisters and Friends
To My Wife, Son and Daughters with love.
iii
Acknowledgements
I am much indebted to my supervisor Dr. Mohamed Hassan
Dahab for his consultation and valuable advice. I wish to acknowledge
my indebtedness to my co-supervisor Dr. Hassan Ibrahim Mohamed
for his guidance and supervision during all stage of this thesis. The
Sudan University of Science and Technology financial support and
fulltime release are appreciated.
Sincere thanks are also extended to my colleague Abdelgani
Ahmed Hussien, Hitham Elramlawi and Maysara A. Mohamed for
their
help
and
assistance
in
computer
programming.
The
encouragement and assistance received from my oldest best friends
specially Salah E. Dafaalla, Mohamed E. Ahmed, Elsadig A. Elhadi,
Abdelhafiz elaabied, Yousif Abdelgani and Sami Abdelgium has been
help and motivating.
Lastly, might as well be the first, this work would have not
successfully completed without the support and encouragement of my
family and friends.
iv
ABSTRACT
The present study was conducted to build up mathematical
model for reliability and replacement of agricultural tractors. The
objective of reliability module was to formulate an algorithm to
estimate reliability of farm tractors, while the replacement module was
used to identify the ideal length of time that a piece of tractor should
be kept before replacement. The formulation of the strategy adopted to
implement the replacement module was based on economic basis
otherwise to develop repair and maintenance plan to keep up the unreplaced tractors.
The data for the study were collected from El Guneid Sugar
Company (30 tractors) and the Agricultural Bank of Sudan. The
models were developed using MS Excel package and Visual Basic
package (version 2003).
For building reliability module, six forms of probability
distribution forms were examined and Gumbel loglog distribution was
chosen as the best alternative. Gumbel loglog distribution was used
for reliability of farm tractor. The module calculates the Gumbel
loglog parameters using the regression analysis of Log probability
method. The values α and β were 1590.91 and - 3294.72 for Belarus
tractor, 194.99 and -2108.27 for John Deere tractor and 223.21 and
1072.43 for Cameco tractor, respectively.
One difficulty in the replacement module formulation was to
select the proper methods to estimate both depreciation and repair and
maintenance costs. Hence, methods of straight line, initial and
subsequent, double declining balance and sum of year-digit were
examined to predict depreciation, while methods of percentage of
v
original price, power and Exponential were used to predict repair and
maintenance costs. On the basis of slope and correlation coefficient
the optimum estimation methods selected were double declining
balance method for depreciation costs and percentage of original price
method for repair and maintenance costs.
Verification and validation of the reliability module was made
by
using
Kolmogorov-Smirnov
Goodness-of-Fit
Test.
The
implementation of the reliability module was made for Belarus, John
Deere and Cameco tractors using actual field data. The results
included identification of the time to reach a threshold level of failure
of 6399.4, 3002.5 and 1656.7 working hours for Belarus, John Deere
and Cameco tractors, respectively. In addition, reliability module was
used to develop maintenance schedule for these tractors. The Schedule
determines when the tractor is maintained and the spare parts needed,
according to the failure type of sub system identified. Also the results
indicated that tires followed by the cooling system and the
transmission system were the subsystems of frequent and early rate of
failure compared to other systems for all tractors.
Implementation of replacement module was made for high (140
hp), medium (75 hp) and low cost tractors (Chinese tractor)) using
economic data. The results indicated that for both medium and high
cost crawler tractors, the optimum age for replacement in Sugar Cane
Company of Sudan was 9 and 10 years without and with taxation
respectively. The optimum replacement age for Chinese tractor was 14
years without taxation and 13 years with taxation.
The results showed that the adjustment of total holding cost to
taxation will decrease the replacement time for low cost tractors.
vi
‫اﻟﻤﺴﺘﺨﻠﺺ‬
‫أﺟﺮﻳﺖ ه ﺬﻩ اﻟﺪراﺳ ﺔ ﻟﺒﻨ ﺎء ﻧﻤ ﻮذج رﻳﺎﺿ ﻲ ﻟﻼﻋﺘﻤﺎدﻳ ﺔ و ﻻﺳ ﺘﺒﺪال اﻟﺠ ﺮارات‬
‫اﻟﺰراﻋﻴﺔ‪ .‬اﻟﻬﺪف ﻣﻦ ﻧﻤﻮذج اﻻﻋﺘﻤﺎدﻳﺔ هﻮ وﺿ ﻊ ﺧﻮارزﻣﻴ ﺔ ﻟﺘﻘ ﺪﻳﺮ اﻋﺘﻤﺎدﻳ ﺔ‬
‫اﻟﺠ ﺮارات اﻟﺰراﻋﻴ ﺔ ‪ ،‬ﻓ ﻲ ﺣ ﻴﻦ أن ﻧﻤ ﻮذج اﻻﺳ ﺘﺒﺪال ﻳﻬ ﺪف اﻟ ﻲ ﺗﺤﺪﻳ ﺪ ﻃ ﻮل‬
‫اﻟﻔﺘﺮة اﻟﺰﻣﻨﻴﺔ اﻟﻤﺜﻠﻲ اﻟﺘﻲ ﻳﻨﺒﻐﻲ إﺑﻘﺎء اﻟﺠﺮار ﻓﻴﻬﺎ ﻗﺒﻞ اﻻﺳﺘﺒﺪال‪ .‬وإﺳ ﺘﺮاﺗﻴﺠﻴﺔ‬
‫ﺗﻨﻔﻴﺬ اﻟﺒﺮﻧﺎﻣﺞ ﺗﻘﻮم ﻋﻠﻰ أﺳﺎس اﻗﺘﺼﺎدي ﻟﻼﺳﺘﺒﺪال وﻣﻦ ﻧﺎﺣﻴ ﺔ أﺧ ﺮي ﻟﻌﻤ ﻞ‬
‫ﺧﻄﺔ ﻹﺻﻼح وﺻﻴﺎﻧﺔ اﻟﺠﺮارات ﻏﻴﺮ اﻟﻤﺴﺘﺒﺪﻟﺔ‪.‬‬
‫ﺟﻤﻌ ﺖ اﻟﻤﻌﻠﻮﻣ ﺎت ﻣ ﻦ ﺷ ﺮآﺔ ﺳ ﻜﺮ اﻟﺠﻨﻴ ﺪ )‪ 30‬ﺟ ﺮار( و اﻟﺒﻨ ﻚ اﻟﺰراﻋ ﻲ‬
‫اﻟﺴ ﻮداﻧﻰ‪ .‬ﺗ ﻢ ﺑﻨ ﺎء اﻟﻨﻤ ﻮذج اﻟﺮﻳﺎﺿ ﻰ ﺑﺎﺳ ﺘﺨﺪام ﺣﺰﻣ ﺔ ) ‪ (MS-Excel‬و‬
‫ﺣﺰﻣﺔ ) ‪.( Visual Basic, ver. 2003‬‬
‫ﻟﻐﺮض ﺑﻨﺎء ﻧﻤﻮذج اﻻﻋﺘﻤﺎدﻳﺔ ﺗﻢ اﺧﺘﺒ ﺎر ﺳ ﺘﺔ أﺷ ﻜﺎل ﻣ ﻦ ﺗﻮزﻳ ﻊ اﻻﺣﺘﻤ ﺎﻻت‪،‬‬
‫وﻣ ﻦ ﺛ ﻢ ﺗ ﻢ اﺧﺘﻴ ﺎر ﺗﻮزﻳ ﻊ ‪ Gumbel loglog‬آﺄﻓﻀ ﻞ ﺗﻮزﻳ ﻊ ﻳﺴ ﺘﺨﺪم‬
‫ﻻﻋﺘﻤﺎدﻳﺔ اﻟﺠﺮارات‪ .‬ﻳﺤﺴ ﺐ اﻟﻨﻤ ﻮذج ﻋﻮاﻣ ﻞ ‪ Gumbel loglog‬ﺑﺎﺳ ﺘﺨﺪام‬
‫ﺗﺤﻠﻴﻞ اﻻﻧﺤﺪار ﻟﻄﺮﻳﻘﺔ ﻟﻮﻏﺮﻳﺜﻢ اﻻﺣﺘﻤﺎﻻت وآﺎﻧ ﺖ ﻗ ﻴﻢ ‪ α‬و‪1590.91 β‬‬
‫و‬
‫‪ - 3294.72‬ﻟﻠﺠ ﺮار ﺑ ﻴﻼروس‪ 194.99 ،‬و‪ -2108.27‬ﻟﻠﺠ ﺮار‬
‫ﺟﻮﻧﺪﻳﺮ و‪ 223.21‬و ‪ 1072.43‬ﻟﻠﺠﺮار آﻴﻤﻜﻮ‪ ،‬ﻋﻠﻲ اﻟﺘﻮاﻟﻲ‪.‬‬
‫واﺣ ﺪة ﻣ ﻦ اﻟﺼ ﻌﻮﺑﺎت ﻋﻨ ﺪ إﻧﺸ ﺎء ﻧﻤ ﻮذج اﻻﺳ ﺘﺒﺪال‪ ،‬اﺧﺘﻴ ﺎر اﻧﺴ ﺐ اﻟﻄ ﺮق‬
‫ﻟﺘﻘﺪﻳﺮ ﺗﻜﺎﻟﻴﻒ اﻻهﻼك و اﻟﺼﻴﺎﻧﺔ واﻻﺻﻼح‪ ،‬ﻟﺬا ﺗﻢ اﺧﺘﺒ ﺎر ﻋ ﺪة ﻃ ﺮق ﻟﺘﻘ ﺪﻳﺮ‬
‫ﺗﻜﺎﻟﻴﻒ اﻹهﻼك اﻟﺴﻨﻮي )اﻟﺨﻂ اﻟﻤﺴﺘﻘﻴﻢ‪ ،‬اﻟﻨﺴﺒﺔ اﻻوﻟﻴﺔ واﻟﻨﻬﺎﺋﻴﺔ ﻟﻠﺴﻨﺔ‪ ،‬اﻟﻘﻴﻤ ﺔ‬
‫اﻟﻤﺘﻨﺎﻗﺼ ﺔ وﻣﺠﻤ ﻮع ارﻗ ﺎم اﻟﺴ ﻨﻴﻦ(‪ ،‬ﺑﻴﻨﻤ ﺎ ﻃ ﺮق اﻟﻨﺴ ﺒﺔ اﻟﻤﺌﻮﻳ ﺔ ﻣ ﻦ اﻟﺴ ﻌﺮ‬
‫اﻻﺳﺎﺳ ﻲ‪ ،‬ﻣﻌﺎدﻟ ﺔ اﻟﻘ ﻮة واﻟﻤﻌﺎدﻟ ﺔ اﻻﺳ ﻴﺔ اﺳ ﺘﺨﺪﻣﺖ ﻟﺘﻘ ﺪﻳﺮ ﺗﻜ ﺎﻟﻴﻒ اﻟﺼ ﻴﺎﻧﺔ‬
‫واﻹﺻ ﻼح‪ .‬اﻋﺘﻤ ﺎدا ﻋﻠ ﻲ اﻻﻧﺤﻨ ﺎء وﻣﻌﺎﻣ ﻞ اﻻرﺗﺒ ﺎط ﺗﺒ ﻴﻦ أن أﻣﺜ ﻞ اﻟﻄ ﺮق‬
‫‪vii‬‬
‫ﻟﺘﻘ ﺪﻳﺮ اﻹه ﻼك اﻟﺴ ﻨﻮي ه ﻲ اﻟﻘﻴﻤ ﺔ اﻟﻤﺘﻨﺎﻗﺼ ﺔ وﻟﺘﻘ ﺪﻳﺮ ﺗﻜ ﺎﻟﻴﻒ اﻟﺼ ﻴﺎﻧﺔ‬
‫واﻹﺻﻼح هﻲ ﻃﺮﻳﻘﺔ اﻟﻨﺴﺒﺔ اﻟﻤﺌﻮﻳﺔ ﻣﻦ اﻟﺴﻌﺮ اﻻﺳﺎﺳﻲ‪.‬‬
‫ﻟﻠﺘﺄآﺪ ﻣﻦ ﺻﺤﺔ ودﻗﺔ ﻧﻤﻮذج اﻻﻋﺘﻤﺎدﻳﺔ ﺗﻢ اﺳﺘﺨﺪام اﺧﺘﺒ ﺎر ‪Kolmogorov-‬‬
‫‪.Smirnov Goodness-of-Fit‬‬
‫ﺗﻢ ﺗﻄﺒﻴﻖ ﻧﻤﻮذج اﻻﻋﺘﻤﺎدﻳ ﺔ ﻋﻠ ﻲ ﺟ ﺮارات اﻟﺒ ﻴﻼروس ‪ ،‬اﻟﺠﻮﻧ ﺪﻳﺮ واﻟﻜﻴﻤﻜ ﻮ‬
‫ﺑﺎﺳﺘﺨﺪام ﻣﻌﻠﻮﻣﺎت ﺣﻘﻠﻴﺔ ﺣﻘﻴﻘﻴﺔ‪ .‬ﺷﻤﻠﺖ اﻟﻨﺘﺎﺋﺞ ﺗﺤﺪﻳﺪ اﻟﻔﺘ ﺮة اﻟﺰﻣﻨﻴ ﺔ ﻟﻠﻮﺻ ﻮل‬
‫اﻟﻰ ﻣﺴﺘﻮى ﺣﺪوث اﻟﻔﺸ ﻞ‪ 3002.5 ، 6399.4 ،‬و ‪ 1656.7‬ﺳ ﺎﻋﺔ ﺗﺸ ﻐﻴﻞ‬
‫ﻟﺠ ﺮارات ﺑ ﻴﻼروس ﺟﻮﻧ ﺪﻳﺮ و آﻴﻤﻜ ﻮ ﻋﻠ ﻲ اﻟﺘ ﻮاﻟﻲ‪ .‬ﺑﺎﻻﺿ ﺎﻓﺔ ﻟ ﺬﻟﻚ اﺳ ﺘﺨﺪم‬
‫ﻧﻤﻮذج اﻻﻋﺘﻤﺎدﻳﺔ ﻟﺘﻄﻮﻳﺮ ﺟﺪول زﻣﻨﻲ ﻟﺼﻴﺎﻧﺔ اﻟﺠ ﺮارات‪ ،‬ه ﺬا اﻟﺠ ﺪول ﻳﺤ ﺪد‬
‫ﻣﺘﻲ ﺗﺘﻢ ﺻﻴﺎﻧﺔ اﻟﺠﺮار وﻧﻮﻋﻴﺔ ﻗﻄﻊ اﻟﻐﻴﺎر اﻟﻼزﻣ ﺔ وﻓﻘ ﺎ ﻟﻨ ﻮع اﻟﻔﺸ ﻞ اﻟﻤﺤ ﺪد‪.‬‬
‫آﻤﺎ اﺷﺎرت اﻟﻨﺘﺎﺋﺞ إﻟﻲ أن اﻹﻃﺎرات ‪ ،‬ﺟﻬ ﺎز اﻟﺘﺒﺮﻳ ﺪ وﺟﻬ ﺎز ﻧﻘ ﻞ اﻟﺤﺮآ ﺔ ﻣ ﻦ‬
‫اﻷﺟﻬ ﺰة اﻟﺘ ﻲ ﺗﺘﻌ ﺮض ﻟﻸﻋﻄ ﺎل ﻣﺒﻜ ﺮا وﺑﺼ ﻮرة ﻣﺘﻜ ﺮرة ﻣﻘﺎرﻧ ﺔ ﺑ ﺎﻷﺟﻬﺰة‬
‫اﻻﺧﺮي ﻟﻜﻞ اﻟﺠﺮارات‪.‬‬
‫آ ﺬﻟﻚ ﺗ ﻢ ﺗﻄﺒﻴ ﻖ ﻧﻤ ﻮذج اﻹﺣ ﻼل ﻋﻠ ﻲ ﺟ ﺮار ﻋ ﺎﻟﻲ )‪ 140‬ﺣﺼ ﺎن( وﻣﺘﻮﺳ ﻂ‬
‫)‪ 75‬ﺣﺼ ﺎن( وﻣ ﻨﺨﻔﺾ اﻟﺴ ﻌﺮ )ﺟ ﺮار ﺻ ﻴﻨﻲ( وذﻟ ﻚ ﺑﺎﺳ ﺘﺨﺪام ﺑﻴﺎﻧ ﺎت‬
‫اﻗﺘﺼ ﺎدﻳﺔ‪ .‬أوﺿ ﺤﺖ اﻟﻨﺘ ﺎﺋﺞ أن اﻟﻌﻤ ﺮ اﻷﻣﺜ ﻞ ﻻﺳ ﺘﺒﺪال اﻟﺠ ﺮارات اﻟﻤﺘﻮﺳ ﻄﺔ‬
‫واﻟﻜﺒﻴ ﺮة ه ﻮ ﻋﻤ ﺮ ‪ 9‬ﺳ ﻨﻮات و‪ 10‬ﺳ ﻨﻮات ﺑ ﺪون ﺿ ﺮﻳﺒﺔ و ﺑﺘﻄﺒﻴ ﻖ ﻗﻴﻤ ﺔ‬
‫اﻟﻀﺮﻳﺒﺔ ﻋﺘﻰ اﻟﺘﻮاﻟﻰ ﻟﺪي ﺷﺮآﺔ ﺳﻜﺮ اﻟﺠﻨﻴﺪ ﺑﺎﻟﺴﻮدان‪ .‬اﻧﺴﺐ ﻋﻤﺮ ﻹﺳﺘﺒﺪال‬
‫اﻟﺠ ﺮار اﻟﺼ ﻴﻨﻲ ه ﻮ أرﺑﻌ ﺔ ﻋﺸ ﺮ ﺳ ﻨﺔ )ﺑ ﺪون ﺿ ﺮاﺋﺐ( وﺛﻼﺛ ﺔ ﻋﺸ ﺮ ﻋﻨ ﺪ‬
‫اﺳﺘﺨﺪام اﻟﻀﺮاﺋﺐ‪.‬‬
‫وأﻇﻬﺮت اﻟﻨﺘﺎﺋﺞ أن إﺧﻀﺎع ﺗﻜﻠﻔﺔ اﻹﻣﺘﻼك ﻟﻠﻀﺮاﺋﺐ ﻳﻘﻠﻞ ﻣﻦ اﻟﻮﻗﺖ‬
‫اﻟﻼزم ﻻﺳﺘﺒﺪال اﻟﺠﺮارات ﻣﻨﺨﻔﻀﺔ اﻟﺴﻌﺮ‪.‬‬
‫‪viii‬‬
TABLE OF CONTENTS
Acknowledgements............................................................................. iv
ABSTRACT ......................................................................................... v
ABSTRACT (Arabic) ....................................................................... vii
TABLE OF CONTENTS................................................................... ix
LIST OF TABLES ............................................................................ xii
LIST OF FIGURES ......................................................................... xiv
LIST OF ABBREVIATIONS ......................................................... xvi
CHAPTER ONE.................................................................................. 1
INTRODUCTION ............................................................................... 1
1.1 Background and Justification:..................................................... 1
1.2 Problem Definition ...................................................................... 4
1.3 Study Objectives: ........................................................................ 6
CHAPTER TWO ................................................................................ 7
LITERATURE REVIEW................................................................... 7
2.1 Computer Aided Program for Machinery Management: ............ 7
2.2 Machine Failure: ......................................................................... 9
2.3 Failure Analysis: ....................................................................... 15
2.4 Reliability of Machinery, Implement and Tractors: ................. 19
2.5 Reliability Analysis: .................................................................. 22
2.5.1 Definition of a Probability Distribution ............................. 23
2.5.2 Related Distributions.......................................................... 25
2.5.3 Families of Distributions.................................................... 32
2.5.4 Location and scale parameters ........................................... 33
2.5.5 Estimating the parameters of a distribution ....................... 38
2.5.6 Gallery of Distributions: .................................................... 40
2.5.6.1 Continuous Distributions: ........................................... 41
ix
2.5.6.1.1 Exponential Distribution: ..................................... 41
2.5.6.1.2 Weibull Distribution ............................................ 42
2.5.6.1.3 Gamma Distribution............................................. 46
2.5.6.1.4 Gumbel Distribution ............................................ 50
2.5.6.2 Discrete Distributions ................................................. 54
2.5.6.2.1 Poisson Distribution ............................................. 54
2.6 Replacement of machinery, implement and tractors: ............... 55
2.6.1 Depreciation ....................................................................... 57
2.6.2 Repair and Maintenance Costs: ......................................... 62
CHAPTER THREE .......................................................................... 65
MATERIALS AND METHODS ..................................................... 65
3.1 Data Collection: ........................................................................ 65
3.2 Data Analysis ............................................................................ 66
3.3 Tractors breakdowns and failures: ............................................ 66
3.4 Model Development:................................................................. 68
3.4.1 Reliability Module: ............................................................ 68
3.4.1.1 Module Description: ................................................... 68
3.4.1.2 Theoretical Development: ........................................... 68
3.4.2 Replacement Module: ........................................................ 74
3.4.2.1 General: ....................................................................... 74
3.4.2.2 Module Description: ................................................... 74
3.4.2.3 Module Structure......................................................... 76
3.4.2.4 Theoretical Development ............................................ 85
3.4.2.4.1 Basic Consideration: ............................................ 85
3.4.2.4.2 Calculation Procedure: ......................................... 86
CHAPTER FOUR ............................................................................. 92
RESULTS AND DISCUSION.......................................................... 92
4.1 RELIBILITY MODULE: ......................................................... 92
x
4.1.1 Selection of the suitable probability distribution for module
building: ...................................................................................... 92
4.1.2 Verification of Reliability Module: ................................... 99
4.1.3 Validation of Reliability Module: ...................................... 99
4.1.4 Application of Reliability Module: .................................. 106
4.1.4.1 Prediction of failure time for tested tractors: ................ 106
4.1.4.2 Comparison of tractors performance under actual field
conditions: ................................................................................. 118
4.1.4.3 Utilization of reliability module for developing
maintenance scheduling plan: ............................................... 120
4.2 Replacement Module: ............................................................. 128
4.2.1 Screening of repair and maintenance estimation method:
................................................................................................... 128
4.2.2 Comparison of method to estimate depreciation: ............ 128
4.2.3 Selection of best combination of R&M with depreciation
Methods: .................................................................................... 132
4.2.4 Module Application: ........................................................ 132
CHAPTER FIVE ............................................................................. 137
CONCLUSIONS AND RECOMMENDATIONS........................ 137
5.1 Conclusion............................................................................... 137
5.2 Recommendations ................................................................... 139
REFERENCES ................................................................................ 140
APPENDICES ................................................................................. 147
xi
LIST OF TABLES
Title
Table 2.1 Initial and subsequent depreciation rate for different
machine.
Page
61
Table 2.2 Repair cost functions.
64
Table 3.1 Technical Specification of the Program.
70
Table 4.1 Comparison of different probability distributions using
data of Belarus tractor.
Table 4.2 Comparison of different probability distributions using
data of John Deere tractor.
Table 4.3 Comparison of different probability distributions using
data of Cameco tractor.
Table 4.4 T-test analysis for mean time between failures for the
tested tractors.
Table 4.5 Sample Kolmogorov-Smirnov test for the tested
tractors.
Table 4.6 T-test analysis for time between failure for Kumar et.
Al. (1977) and predicted reliability module.
93
94
95
98
100
105
Table 4.7 Regression analysis of Bell tractor data.
108
Table 4.8 Time between failures of Belarus tractor.
109
Table 4.9 Regression analysis of John Deere tractor data.
112
Table 4.10 Time between failures of John Deere tractor.
113
Table 4.11 Regression analysis of Cemco tractor data.
116
Table 5.12 Time Between Failures of Cemco.
117
Table 4.13 Slope parameter (β) for the tested tractors
118
Table 4.14 Times of occurrence of failures for sub system of
Belarus tractor.
xii
125
Table 4.15 Times of occurrence of failures for Sub System of
John Deere Tractor.
Table 4.16 Times of occurrence of failures for Sub System of
Cameco Tractor.
Table 4.17 Ranking of depreciation methods according to slope.
Table 4.18 Combination of R&M method with depreciation
methods.
xiii
126
127
129
133
LIST OF FIGURES
Title
Page
Figure 2.1 Normal Probability Density Function
26
Figure 2.2 Cumulative Density Function
26
Figure 2.3 Normal Percent Point Function
27
Figure 2.4 Normal Distribution Hazard Function
28
Figure 2.5 Normal Cumulative Hazard
29
Figure 2.6 Normal Survival
30
Figure 2.7 Normal Inverse Survival
31
Figure 3.1 Elguneid Sugar Factory
67
Figure 3.2 Program Start Menu.
69
Figure 3.3 Reliability Module Flow chart.
71
Figure 3.4 Replacement Module Flow chart.
75
Figure 3.5 Replacement Module Main Menu.
78
Figure 3.6 About MORRAM Model menu.
79
Figure 3.7 Machine Data Entry menu.
80
Figure 3.8 Depreciation Data Entry menu
81
Figure 3.9 Repair and maintenance data entry menu.
82
Figure 3.10 View and print results menu.
83
Figure 3.11 View and print graphs.
84
Figure 4.1 Actual and predicted time of failure for Belarus tractor
using normal Weibull distribution
Figure 4.2 Actual and predicted time of failure for John Deere
tractor using normal Weibull distribution
Figure 4.3 Actual and predicted time failure using Gumbel
distribution for Belarus tractor.
Figure 4.4 Actual and predicted time failure using Gumbel
xiv
96
97
101
102
distribution for John Deere tractor.
Figure 4.5 Actual and predicted time failure using Gumbel
distribution for Cameco tractor.
Figure 4.6 Comparison of Kumar et. al. (1977) model Time of
failure and Predicted Reliability module time of failure.
Figure 4.7 Histogram of number of failure in each cell of Belarus
tractor.
Figure 4.8 Histogram of Number of Failure in each Cell of John
Deere Tractor.
Figure 4.9 Histogram of number of failure in each cell of Cameco
tractor.
Figure 4.10 Comparative performance of reliability module for the
tested tractors.
103
104
107
111
115
119
Figure 4.11 Cumulative time between failure of Belarus tractor.
122
Figure 4.12 Cumulative time between failure of John Deere tractor.
123
Figure 4.13 Cumulative time between failure of Cameco tractor.
124
Figure 4.14 Comparison of R&M cost estimated by different
methods of depreciation.
Figure 4.15 Comparison of depreciation cost estimated by different
methods.
130
131
Figure 5.16 Optimum replacement age of medium cost tractor.
134
Figure 4.17 Optimum replacement age of high cost tractor
135
Figure 4.18 Optimum replacement age of low cost tractor (Chinese
tractor).
xv
136
LIST OF ABBREVIATIONS
Z
Nr
Td
Ta
T
M
N
Γ
fixed length of time (hr)
Number of failures
Dwon-time or dead-time (hr)
Available time (hr)
Total time (hr)
Number of observation
Rank of failure
gamma function
Shape parameter
γ
µ or α Location parameter
β or σ Scale parameter
D
Depreciation ($).
P
Purchase Price ($).
S
Salvage Value or selling price ($).
L
Time between selling and purchasing, years.
n
Age of the tractor in year at beginning of year in question,
year.
x
Ratio of depreciation rate used to that of straight line method.
SFP
Sinking fund annual payment ($)
Di
Initial depreciation rate (%).
Ds
Subsequent depreciation rate.
Y
Accumulated repair and maintenance costs as percent of initial
price ($).
X
Tractors or machine cumulative hours.
R
The reliability at any time t (decimal).
λ
The failure rate (%).
MTTF Mean time between failures (hr).
drr
Real discount rate (%).
Dr
Discount rate (%).
drpt
Post-tax discount rate (%).
MT
Marginal tax rate (%).
Rc
repair cost ($).
I
Inflation rate.
xvi
CHAPTER ONE
INTRODUCTION
1.1 Background and Justification:
Agriculture may be one of the oldest professions, but the development
and use of machinery has made the job title of farmer a rarity.
Agricultural machinery is one of the most revolutionary and impactful
applications of modern technology. The truly elemental human need
for food has often driven the development of technology and
machines. Over the last 250 years, advances in farm equipment have
transformed the way people are employed and produce their food
worldwide. With continuing advances in agricultural machinery, the
role of the farmer will become increasingly specialized and rare
(Culpin, 1975).
The ability to manage machinery is an important skill that must be
mastered by farmers and ranchers who want to compete in our
complex worldwide commodity marketplace. Machinery management
must contribute to total management in a cost effective manner. There
are a number of strategies to follow that will enable the farmer to
achieve maximum life from his machinery. A combination of
practices can have a large impact on costs, improve machine reliability
for many years to come and finally, increase profit margins.
On most farms, the cost of owning and operating its machinery
exceeds all other costs except the cost of land use; in some cases it is
the most expensive part of the business. Efficient selection, operation
and maintenance of this machinery are absolutely critical to the
viability of the farm.
1
In Sudan importance of use of machine is realized since mid of last
century. This is derived by the need to produce food from the vast
rainfed areas with low population to cultivate in short rainy season.
Consequently, mechanized Farming Corporation is erected in rainfed
areas which acted as a model that is followed by private sector. For
irrigated sector machinery is used for land preparation and its use
expands by time to include all cultural practices from canal
construction to stalk removal and field cleaning. Expanding
agricultural productivity in urban areas is sought to be a visible tool to
accelerate development of these areas. Unfortunately, the expansion
was not coupled with a suitable training or suitable program of knowhow for machinery management. Other factors such as under
replacement polices, shortage of spare parts and low repair and
maintenance facilities interfered with the efficient utilization of
agricultural machinery in both rainfed and irrigated farming systems.
The end result is deterioration and final collapse. The solution to
correct, under the newly adopted free market economy, is thought to
be through giving more free hands to private sector to take over all
responsibilities in both rainfed and irrigated sector with complete
withdrawal of the government. However, the private sector capability
now is under question. Do the private sectors have the required high
capital to invest? If so are working plans and proper management
program are prepared. Farmers around the world are currently
operating in an environment that is characterized by high volumes of
grain stocks and fresh produce but low economic activities. This put
pressure on most commodity prices and means less money for
repairing or buying agricultural machinery. By postponing his
replacement policy, the farmer is now faced with the dilemma of
2
overusing his machinery and thereby putting pressure on the cost of
maintenance and repairs.
The ability of the farmer to select the proper machinery is a valuable
function as many farm activities relate to it. In the final analysis, the
selection must increase yields and must add value to the total farm
business. This makes machinery management the most complex
function of farm management as it involves owning and operating the
machine. Owning the machine involves capital and this capital must
return a profit. It will only return a profit when the capital is active,
but operating the machine involves costs. This then, is the fine
balancing act of machinery management: invest the correct amount of
capital to do the farm operations in the most effective way at the
lowest cost. This may sound simple but it means that the farmer must
understand all the different cost components and managerial concepts
of machinery management, he must know the physical side of
matching tractors and implements, he must maintain and repair his
machinery and calculate the cost of doing so, he should replace
obsolete machinery at the appropriate time with the best financial
alternative and must always strive to use his machinery more
effectively. The reason for doing so is to add the most value to his
profit. Adding value to profit means that agricultural machinery
management must have an economical approach. If this is not the
case, the farmer doesn’t have an economical approach to farming as a
whole and will not survive the financial challenges of his farm
business.
Modern agriculture demands efficient, cost-effective management of
all of the resources associated with farming. This is particularly true in
3
the context of agricultural machinery, which can account for a very
high proportion of the fixed costs associated with agricultural
businesses. Conversely, machinery costs can be significantly reduced
by the correct specification, selection and procurement of machinery.
The complexity and size of modern agricultural machinery is such that
there are significant legal issues associated with procuring and
operating the equipment. The final aspect of machinery management
and operation is that of operator training, and the unit considers the
identification of training needs and the importance of statutory and
voluntary codes of practice.
1.2 Problem Definition
In agriculture, timeliness of operations is one of the important factors
for obtaining maximum crop yield. Farm equipment failures,
especially during the busiest part of the season, cause delays which
result in yield reduction and inefficient labor utilization. To make
allowance for these breakdowns in planning, one needs to know the
probability of machine failure. Many times these breakdowns minor
and can be repaired in a very short time if spare parts are available.
Unfortunately more time is often wasted in procuring parts than in
making actual repairs. If an estimate of failure frequency can be
beforehand, then sufficient spare parts can be stocked to minimize
down time (Kumar et. al., 1977).
Unlike building machinery must be constantly monitored, maintained
and eventually replaced. How and when equipment is replaced can
mean a difference of large sum of money in annual production costs.
Tractors need to be replaced due to accidental damage, deterioration
due to age (obsolescence), or damage, in adequate capacity, low
4
reliability or the cost of making an anticipated repair and maintenance
would increases the average unit accumulated cost above expected
minimum. One of the difficulties in analyzing costs is that they change
overtime. This cost of operation is influenced by two main factors the
fall in value of the tractor (depreciation) overtime and annual repair
and maintenance (R&M) costs. In practice both R&M and
depreciation is the major source of costs variability. Typically by
assuming running costs are constant, R&M costs are low during the
first few years of tractor life and then rise as the tractors ages.
Depreciation are assumed to follow the reverse pattern. Hence, the
problem is to balance the high depreciation and low repair costs of
early replacement against the low average depreciation but high
average repair costs of keeping the tractor longer so that we could find
an optimum tractor life. The problem is made more complex by the
fact that interest needs to be charged on the average capital investment
in the tractor.
Much of the work on estimating and planning for failure, or reliability,
has been done in electronics, missile, aircraft and space research
programs and few in agricultural tractor. However, it is strongly
believed (Archer, 1962) that the application of reliability theory is
both feasible and necessary for good farm equipment planning and
management.
tractor breakdowns are a major source of irritation at any time but may
incur associated losses which are far in excess of the direct repair
costs by delaying critical field operations. Consequently, the quest for
tractor reliability has a substantial influence on tractor replacement
policy. As detailed reliability data on farm equipment is virtually
5
unobtainable, tractor replacement is largely based on economic
pointers to minimize the holding cost of individual tractor and to
eliminate excessive fluctuations in machinery investment from year to
year (Witney, 1988).
In developing countries, a computer models were developed for
reliability and replacement (Mohamed, 2006), which help farmers and
farm manager in decision making. But in Sudan there is a lack of use
of a computer aided program.
1.3 Study Objectives:
The general objective of this study is to improve performance of
enterprise by formulating agricultural machinery management model
to aid workshop machinery manger to develop a policy for deciding
when to replace a piece machinery and how to maintain it if it is not
going to be replaced.
The Specific objective of this study is to formulate, verify and validate
an algorithm to determine when to replace and how to maintain
unreplaced agricultural tractors and to implement the developed model
for the case of tractors in Sudan for purpose of calculations of
replacement year of tractors and time of failure for whole tractors or a
subsystem.
6
CHAPTER TWO
LITERATURE REVIEW
2.1 Computer Aided Program for Machinery Management:
Computer programs for machinery management are most useful when
there is an interaction exchange of information during program
operation between the computer and the program user. They are
becoming increasingly important in making certain type of machinery
management-decisions and employed in some large farming
enterprises. Computer programs are being used to assist farm mangers
and scientist in decision making about how to manage and select their
machine effectively (Oskan and Edward, 1989).
A Computer aided maintenance planning for mechanical equipment
was developed by Sayed et al, (1998). This program was implemented
to optimize extensive maintenance plan at a specific time horizon. A
comparative study on the reliability and maintainability of Public
Transport Vehicles has been developed by Bedeway et al, (1989).
Ntuen, (1990) has presented a simulation study of vehicle
maintenance policies to investigate the most economic age
replacement policies.
Taher (1992) has developed computer aided reliability and
maintainability used for optimum maintenance planning to achieve the
whole maintenance plan for an equipment. Ismail (1994), developed a
crop production machinery system model as a computer interactive
model based on the concept of expert system, which allow the user to
interact with the program. Aderoba (1989) developed a farm selection
model which takes into account value and cost of production, the
7
available machinery mix, timeliness of operation and capital
limitation. In order to plan and design a farm mechanization system,
Konaka (1987) developed a program using a personal computer,
which involved a farm machinery data base, farm operation data base
and farm machinery utilization planning program.
A system of Microsoft Access 97 including Microsoft Visual Basic
was applied. It was concluded that development of this program is
important taking into consideration not only modern techniques and
different forms of farm machinery, but also the graphical presentation
of results. A methodology was presented for determining demand for
agricultural machinery and tractors and calculating the minimum
numbers required to carry out all mechanized work within established
dead lines (Grazechowiak, 1999). ASAE (1991) developed
computer
model
for
agricultural
machinery
a micro
management
(MACHINER). The program consists of three modules: record
keeping, cost estimation and machinery selection. The model was
successfully implemented on a commercial production of agriculture
operations in Honduras and Central America. Major attribute of the
program include a user friendly interface, efficient record keeping and
adaptability to different conditions.
Computer software was developed by Singh, et al. (1992) to optimize
farm machinery systems with the variations in cropping practices,
farm equipment sizes and costs of the users and output. The program
computes the optimum power required for the field, transportation,
irrigation and threshing operation and select optimum power sources.
It computes working hours, required used fixed, operating and
timeliness costs for each of the selected farm operations on an area,
8
crop, seasonal and annual basis. A computer program was designed by
Machackova (1990) as a basis for rational assessment of machinery
requirement in various branches of agriculture. Areas of crops, energy
and transport required to produce these crops and other factors are
listed for the main agricultural regions in Czechoslovakia. These
parameters together with capital costs form the basis for calculations
of machinery requirements. A simulation mathematical model was
developed by Bakhit (2006) for wheat harvesting losses in Rahad
Scheme.
Ishola and Adeoti (2004) studied farm tractor reliability in Kwara
State in Nigeria. He developed a reliability model for field tractors. A
field survey was conducted to assess the repair and maintenance
facilities and reliability functions from the breakdown records of
tractors. They found that the comparison of the reliabilities of the
various tractors revealed that the steering, traction and electrical
systems are more prone to failure than the engine, cooling,
transmission, fuel and hydraulic systems. The Massey Fergusson and
Fiat tractors proved to be more reliable tractors in the state.
2.2 Tractor Failure:
A failure may be referred to as any condition which prevents
operation of a machine or which causes or results in a level of
performance below expectation. There are many causes of tractor
machine failure and their properties are different. Some are depending
on the age of machines and some are purely stochastic as method of
operation condition of work on the field (Witney, 1988).
9
Amjad and Chaudhary (1988) reported that machine failures can be
categorised into: early life failures, random failures and wear-out.
Likewise, Lewis (1987) asserted that reliability considerations appear
throughout the entire life cycle of a system. He claimed that data
collection on field failures are particularly invaluable because they are
likely to provide the only estimate of reliability that incorporates the
loading, environmental effects and imperfect maintenance found in
practice.
According to Hunt (1983) the causes of tractor failures are classified
into direct and indirect causes as follows:
(1) Direct causes of tractor failures:
The direct causes of premature tractor failure, in general, include the
following aspects:
a. Equipment that has been ill-chosen for the tasks required from
it, and for the conditions in which it is to work, will usually fail
prematurely.
b. Linked with the above consideration of faulty specification of
equipment is the mismatching of implements to tractors,
causing failure in one or the other, or even in both.
c. Operators who have not been properly informed and trained, or
who are not machinery oriented, can commit numerous
mistakes that gravely affect the reliability of machinery, and
also its life span. Incorrect handling and setting of machinery
can subject it to stresses for which it was not designed, and
even lead to accidents.
d. Poor and un-cleaning field, that has left rocks and tree stumps
in the ground, can seriously damage equipment. Moreover,
10
working over hard and rough ground increases the fatigue
failure of many components.
e. The use of spurious replacement parts that do not measure up
to manufacturer's original specification. These may fail and, in
doing so, cause damage to other components, thus aggravating
the situation.
(2) Indirect causes for tractor failures:
The indirect causes of premature tractor failure, in general,
include the following aspects:
a. The policies and strategies of governments-or perhaps more
accurately, lack of them in any cases-towards agricultural
mechanization has had a profound effect on the ability of the
agricultural sector to adopt and sustain tractor-based technology.
Tractors and implements may be provided from government and
donor funds, but their operation is open unsustainable because of
poor support services.
b. Technical training has been neglected in many countries,
and at most universally, there have been under-estimations of
the time and resources required to balance the skill imparted by
training institutions with the needs of modern agricultural
equipment.
c. In the majority of developing countries, networks of machinery
distributor and service centers tend to be few in number and
located in urban rather than rural areas. The service center that to
exist often lack capital equipment, skilled and semi-skilled
personnel with diagnostic abilities, service vehicles, and reliable
communication links to the main distributor. It has been argued
11
that these services are under-developed
because there
is
insufficient advanced mechanization in rural areas of low income
countries to warrant, economically speaking, a network of
maintenance workshops.
d. Agricultural machinery mechanics are often poorly trained and
inadequately managed. And, in comparison with other trades,
mechanics are not well treated, the job is considered too very
slowly and it attracts salaries that seldom match those of other,
less skilled trades. It is hardly surprising that the end number of
skilled mechanics in many developing countries often show a
preference for employment in the urban private sector, where
wages, status, and the perception of a better quality of life are
generally higher than in remote e rural areas.
e. Low profit margins in agriculture, often resulting from
government pricing policies for agricultural commodities,
coupled with high fixed costs (caused mainly by low annual
tractor-utilization rates); make it difficult for farmers to pay the
costs of services provided by workshops established in remote
areas.
f. Donors may make their aid conditional to purchases in a
specific country, so called "tied aid", or they may stipulate
procurement through international competitive bidding. In either
case this can lead-and has led in the past-to unsuitable equipment
being purchased and disbursed if the equipment is unsuitable in
the first place, its catty demise is likely to follow. The tying of
aid to products from donor countries has, also, been cited as a
main reason for the multiplicity of makes often found in
12
developing countries, leading to problems of stocking of
replacement parts, specialized training of mechanics, and so on.
g. All farm tractors and implement manufacturers understand the
desirability of the after-sale service and back-up support.
Unfortunately, however, of the dozen or so major manufacturers
selling their products in developing countries, very few show to
all commitment to the idea.
h. From the other-side, government policies do not always
recognize the point of view of the manufacturers that back-up
services must be profitable, or at least break-even.
i. The remote location in which machinery often operates in
developing countries is itself a factor in premature failure. Long
distances to scent nil servicing facilities increase the difficulty in
obtaining replacement parts, and the high costs of transport is
passed on to customer.
Machackova (1990) in a study conducted by the SXVCdlSIi institute
of Agricultural Engineering about the mechanical breakdowns of farm
tractors, indicated flint the two most common causes of breakdowns
were overloading and poor maintenance, particularly in regard to oils
and filters.
Pepi (1994) stated that approximately 25% of tractor breakdowns are
attributed to defects, in the cooling system. He also, described the
principles of water-cooled and air-cooled systems with particular
emphasis on the differences between them and the details of the
maintenance program necessary to avoid problems.
Grisso, (2001) presented some data for two Russian tractors (the TDF55 A and the TB-1) over a 5-year period, which was considered the
13
machine write-off period. The total losses caused by downtime for the
TDT-55A amounted to 46.6% of its whole sale price, and for the TB-I
they were 96.5%.
Dalley (2002) stated that breakdowns were considered to be
unpredictable events which may have arisen from one or more of the
following causes:
(1) Accidents, such as stoking hidden objects, storms, tires, etc.
2) Improper service or maintenance, such as lack of lubrication.
3) Improper machine operation, such as overloading, overturning,
and running too fast.
4) Improper set-up, such as omission of parts, foreign objects,
objects left in the machine, and improper bolt-tightening torques.
5) Inadequate design, such as underestimation of operating
loads and service factors, and the deliberate under design to gain
a price advantage.
Alvarez (2000) stated that a machine wears out with use, but the rate
of wear depends upon the skill of the operator, lubrication and
general maintenance, and design and quality of materials.
According to FAO (1990) the key to reducing premature failures lies
in correcting the causes that related to the frequent deficiencies in
government policies and strategies in respect to mechanization.
Sound policy and well-formulated strategies provide an umbrella
under which almost all of the causes of premature failures can be
eliminated, or at least mitigated. A sound mechanization strategy will
also take the vital foreign exchange issue into account, identifying
not only the foreign exchange requirements for importing new
machinery from abroad, but also allocating foreign exchange to
14
cover the necessary support services for the machinery throughout its
economic life especially with regard to replacement parts. However,
replacement parts require more than foreign exchange allocations,
because their organization and management is a specialized field.
Training of all staff concerned with replacement parts is an essential
element in preventing premature failure. Routine maintenance, if
correctly carried out, will often prevent and sometimes reduces the
effect of catastrophic failures.
2.3 Failure Analysis:
Adigun (1987) claimed that the failure in the farm tractors
components could be classified into the following categories: engine,
cooling, fuel, electrical, transmission, hydraulic, steering and traction.
An important design parameter which deals with minimization of
repair time and which is often affected by the skills of the operator is
machinery maintainability (Oni, 1987). Mishera, (2006) stated that
failure classification may be viewed from different aspects according
to the effect it will have on the overall performance of the
equipment/system. Broadly failures are classified as:
(i) System failure, and
(ii) Component failure.
In some cases failure of component/element may make the
equipment/system completely inoperable and the same cannot be
used without repairing the failed component. This is mainly possible
in case of high risk equipment/system for example airplane
equipment. Even in case of automobiles, the failure of brakes will
make it inoperable and therefore, this will fall in the category of
15
system failure. On the other hand failure of a component/element
may not make it fully inoperable and the equipment/system can be
used with reduced performance. Failure of lights in an automotive
vehicle does not make it fully inoperable; the system can be
effectively used in daylight. When the elements/components are
placed in series, failure of one will make the system completely
inoperable. Whereas, when the same are placed in parallel, failure of
one element/component may not render it completely inoperable.
The system can work but its performance may be reduced. The failure
of one of the cylinders in a multi cylinder engine will fall under this
category. This type of failure can be classified as component failure.
The engineering classification of failures may have:
(i)
Intermittent failure, which may result in lack of some function
of the component only for a very short period of time, and
(ii)
Permanent failure,, where repair/replacement of component
will be required to restore the equipment to operational level.
When considering degree of failures, it can be classified as:
(i)
Complete failure, where equipment/system is inoperative
and cannot be used further, and
(ii)
Partial failure, which leads lack of some functions but the
equipment/system can be used with care, may be with reduced
performance.
Some failures can be sudden and cannot be anticipated in advance,
whereas, the gradual failures can be forecast during inspection/testing,
which follows the part of the condition monitoring. Other classification
of failure can be:
16
(i)
Catastrophic failures, which are both sudden
and complete;
(ii)
Degradation failures, which are both partial
and gradual.
Basically failures are defined as the termination of the ability of a
component/part to perform its required functions. The failure of
component/system can be classified in many ways, which may include
the following:
(i)Catastrophic failures are ones which immediately stop
the working of system/equipment and it cannot be used
without proper repair/maintenance.
(ii) Performance failure: these are related with the
performance of the equipment/system. The system may
remain operative in the failure of some components/parts but
its performance decreases, which is true for the most of
equipment used in engineering application.
(iii) Deliberate failures are: caused either by the neglect of the
operating personnel or by his ill intention to make the
equipment inoperative for sometime/period. In this case
operators make excesses, which are not rational for example
application of brakes in an automobile.
Basically, the failures fall under the following categories:
(i) Infant or early failures. It can be seen from bath-tub curve, that due
to quality of components, some equipment, fail during their initial
life and such failures number can be high. These failures can also
be due to initial turning of the system.
17
(ii) Random failures, which can take place at any time due to unforeseen
reasons and it is difficult to predict them. Their causes could be
extra stress on the component or the quality of material. However,
these failures can be minimized through a proper investigation of load
and quality of material in use.
(iii) Time-dependent failures where mean time to failures can be
predicted
since
the
failures
depend
on
the
usage
of
the
equipment/system. Hence, failure distribution can be plotted to know the
frequency of failures, which can be used to control the rate of system
component failures
All machines will have minor failures from time to time, and troubleshooting charts are provided by manufacturers to help in finding and
correcting such minor problem before they become major ones. Such
charts, as the ones shown in Appendix 1, list the causes of typical failures
of the engine power train, hydraulic system, electrical system, brakes.
Monitoring of machine operation is not necessarily effective. Often bad
maintainability is only improved by early design changes. There are
various types of maintenance, such as time-based maintenance and
condition-based maintenance. It is complicated whether to adopt time
based maintenance or condition-based maintenance. Maintenance
cost very much depends on required maintenance resources and
facilities.
Reliability-Centered Maintenance (RCM) is one of the well
established systematic methods for selecting applicable and
appropriate maintenance operation types. Bukhari (1982) investigated
two types of failure according to RCM; this is failure diagnosis which
can be used to monitoring the fault caused by several levels depend on
18
method of machine or system operating, working hours (age) and type
of failure as partial failure /component or complete failure, other type is
mathematics one named failure analysis which used mathematical
formula to predict sequence of failure behavior during operating time
and find out men time between failure and interval between
maintenance levels.
Mathematically, Alcock (1979) formulated the failures analysis of a
repairable system by using the failure intensity. This is the instantaneous
rate of failure of a system at age t given its previous history of failures
and maintenance interventions. He considered that equipment was not
subjected to preventive maintenance, so that the failure intensity was a
function of equipment age and failure history only. He stated also failure
intensity usually increases with age for mechanical equipment, but may
decrease with age for electronic equipment or software, where defects
are gradually weeded out. Such models of failure intensity as the power
law and log linear Poisson processes are often used.
2.4 Reliability of Machinery, Implement and Tractors:
The reliability of the equipment is defined as the probability that it
will adequately perform its intended function under stated
environmental conditions for a specific time interval (Smith, 1974).
Hence, reliability is a mathematical expression of the likelihood of
satisfactory operation.
Reliability is important to manufacturer and dealer as it is to
purchaser, a product sold by its reputation and in farm machinery,
reputations earned because of reliability (Alcock, 1979). (Amjad and
Chaudhary, 1988) and Anon (1972) defined reliability as “the
probability that the equipment will complete a specific task under
19
specified conditions for a stated period of time”. There is an
increasing interest by all sections of agricultural machinery industry in
U.S.A and Australia on machine reliability. Tullberg et. al. (1984) in
Australia confirmed that reliability is a major problem in some areas,
although they did not choose their tractors randomly, they found
premature failure in more than 30% of the total tractor population
surveyed.
Farm tractors failure especially during the busiest part of the season
cause delays which result in losses and inefficient labour utilization.
As more and more capital in the form of machinery replaces manual
labour on the farm, the reliability of this equipment assumes greater
importance. Indeed, deeper insight into failures and their prevention is
to be gained by comparing and contrasting the reliability
characteristics of systems that make up the tractor (Ishola and Adeoti,
2004).
The failure rate of a population of items for a period of time t1 to t2 is
the number of items which fail per unit time in that period expressed
as a fraction of the number of non-failed items at time t1.Hence, in
reliability, the reciprocal of failure rate is the mean time between
failures [MTBF] (Wingate-Hill, 1981).
The concept of reliability becomes important when failures lead to
some finite length of time associated with repairing, restoring or
replacing the failed item. In the simplest case it may be assumed that,
when an item fails, it is out of action for some fixed length of time (Z)
which represents the repair time or its equivalent (Green and Bourne,
1981). If, using the previous symbolism, it is taken that (N) failures in
20
a total time (T), then the total “down-time” or “dead-time” associated
with the interval (T) is as follows:
Td = NZr
(1)
Similarly, the total available time (Ta) is given by the following
equation:
Ta = T - Nr
(2)
Both equation (1) and (2) represent indices of reliability. However, it
is often more useful to express these times as proportions or fractions
of the total time. A quantity, identified as mean fraction dead time (D)
may therefore be defined, which represents the mean proportion of
dead time over the total time of interest, as follows:
D =
Meantime in failed state
Total time
FAO (1990) summarized the factors, on which reliability indices
depend, as: service, maintenance, operator’s skills, quality of spare
parts and quality of supplies (fuel, oil, etc). For mechanical power,
reliability indices in developing countries will rarely exceed 60% for
engine powered machinery and 80% for implements. On the other
hand, in developed countries, with sophisticated service networks and
easy access to replacement parts and, therefore, reduced downtime,
the indices may be 10-20% higher.
Generally, the reliability of machinery will be increased under the
following conditions (Green and Bourne, 1981):
a- The availability of good maintenance, service and repair failure.
b- The presence of skilled workshop labour.
21
c- Operators are trained and attentive.
d- Services carried out regularly and as recommended.
e- Machinery is protected against damage.
Archer (1962) described problems of reliability prediction in terms of
varying farm conditions under which a particular machine is supposed
to work. Hunt (1971) reported the results of a survey for the incidence
breakdowns, lost time and repair costs experienced by corn and
soybean farmers. His study included the probabilities of breakdowns
for various machines depending on age and use and he concluded that
an average farmer has less than a 50:50 chance of getting through the
season without a breakdown that has timeliness cost associated with it.
Von Bargen (1970) modelled the effect of delays, in terms of
machinery reliability, field environment delays and management
stops, on the capacitative performance of a corn planting system.
Liang (1967) used the Weibull distribution to predict reliability and
formulate preventive maintenance policies for farm machinery.
2.5 Reliability Analysis:
The fundamental definitions of reliability must depend heavily on
concepts from probability theory. In Probability theory and statistics
the probability distributions are a fundamental concept (Johnson and
Kemp, 1992). They are used both on a theoretical level and a practical
level. Some practical uses of probability distributions are:
• To calculate confidence intervals for parameters and to
calculate critical regions for hypothesis tests.
• For univariate data, it is often useful to determine a reasonable
distributional model for the data.
22
• Statistical intervals and hypothesis tests are often based on
specific distributional assumptions. Before computing an
interval or test based on a distributional assumption, we need to
verify that the assumption is justified for the given data set. In
this case, the distribution does not need to be the best-fitting
distribution for the data, but an adequate enough model so that
the statistical technique yields valid conclusions.
• Simulation studies with random numbers generated from using
a specific probability distribution are often needed.
2.5.1 Definition of a Probability Distribution
The mathematical definition of a discrete probability function, p(x), is
a function that satisfies the following properties (McNeil, 1977).
1. The probability that x can take a specific value is p(x). That is
P[X = x] = p(x) = px
2. p(x) is non-negative for all real x.
3. The sum of p(x) over all possible values of x is 1, that is
Σ pj = 1
j
where j represents all possible values that x can have and pj is
the probability at xj.
One consequence of properties 2 and 3 is that 0 <= p(x) <= 1.
What does this actually mean? A discrete probability function is a
function that can take a discrete number of values (not necessarily
finite). This is most often the non-negative integers or some subset of
23
the non-negative integers. There is no mathematical restriction that
discrete probability functions only be defined at integers, but in
practice this is usually what makes sense. For example, if you toss a
coin 6 times, you can get 2 heads or 3 heads but not 2 1/2 heads. Each
of the discrete values has a certain probability of occurrence that is
between zero and one. That is, a discrete function that allows negative
values or values greater than one is not a probability function. The
condition that the probabilities sum to one means that at least one of
the values has to occur.
The mathematical definition of a continuous probability function, f(x),
is a function that satisfies the following properties.
1. The probability that x is between two points a and b is
2. It is non-negative for all real x.
3. The integral of the probability function is one, that is
What does this actually mean? Since continuous probability functions
are defined for an infinite number of points over a continuous interval,
the probability at a single point is always zero. Probabilities are
measured over intervals, not single points. That is, the area under the
curve between two distinct points defines the probability for that
interval. This means that the height of the probability function can in
fact be greater than one. The property that the integral must equal one
24
is equivalent to the property for discrete distributions that the sum of
all the probabilities must equal one.
Discrete probability functions are referred to as probability mass
functions and continuous probability functions are referred to as
probability density functions. The term probability function covers
both discrete and continuous distributions. When we are referring to
probability functions in generic terms, we may use the term
probability density functions to mean both discrete and continuous
probability functions.
2.5.2 Related Distributions
Probability distributions are typically defined in terms of the
probability density function. However, there is a number of
probability functions used in applications (Johnson and Kemp, 1992).
For a continuous function, the probability density function (pdf) is the
probability that the variate has the value x. Since for continuous
distributions the probability at a single point is zero, this is often
expressed in terms of an integral between two points.
For a discrete distribution, the pdf is the probability that the variate
takes the value x.
The following is the plot of the normal probability density function.
25
Probability density
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
x
Figure 2.1 Normal Probability Density Function
The cumulative distribution function (cdf) is the probability that the
variable takes a value less than or equal to x. That is
F(x) = Pr[ X ≤] = α
For a continuous distribution, this can be expressed mathematically as
For a discrete distribution, the cdf can be expressed as
The following is the plot of the normal cumulative distribution
function.
Figure 2.2 Cumulative Density Function
26
The horizontal axis is the allowable domain for the given probability
function. Since the vertical axis is a probability, it must fall between
zero and one. It increases from zero to one as we go from left to right
on the horizontal axis.
The percent point function (ppf) is the inverse of the cumulative
distribution function. For this reason, the percent point function is also
commonly referred to as the inverse distribution function. That is, for
a distribution function we calculate the probability that the variable is
less than or equal to x for a given x. For the percent point function, we
start with the probability and compute the corresponding x for the
cumulative distribution. Mathematically, this can be expressed as
or alternatively
The following is the plot of the normal percent point function.
Figure 2.3 Normal Percent Point Function
27
Since the horizontal axis is a probability, it goes from zero to one. The
vertical axis goes from the smallest to the largest value of the
cumulative distribution function.
The hazard function is the ratio of the probability density function to
the survival function, S(x).
The following is the plot of the normal distribution hazard function.
Figure 2.4 Normal Distribution Hazard Function
Hazard plots are most commonly used in reliability applications. Note
that Johnson and Balakrishnan (1994) refer to this as the conditional
failure density function rather than the hazard function.
28
The cumulative hazard function is the integral of the hazard function.
It can be interpreted as the probability of failure at time x given
survival until time x.
This can alternatively be expressed as
The following is the plot of the normal cumulative hazard function.
Figure 2.5 Normal Cumulative Hazard
Cumulative hazard plots are most commonly used in reliability
applications. Note that Johnson and Balakrishnan (1994) refer to this
as the hazard function rather than the cumulative hazard function.
29
Survival functions are most often used in reliability and related fields.
The survival function is the probability that the variate takes a value
greater than x.
The following is the plot of the normal distribution survival function.
Figure 2.6 Normal Survival
For a survival function, the y value on the graph starts at 1 and
monotonically decreases to zero. The survival function should be
compared to the cumulative distribution function.
30
Just as the percent point function is the inverse of the cumulative
distribution function, the survival function also has an inverse
function. The inverse survival function can be defined in terms of the
percent point function.
The following is the plot of the normal distribution inverse survival
function.
Figure 2.7 Normal Inverse Survival
As with the percent point function, the horizontal axis is a probability.
Therefore the horizontal axis goes from 0 to 1 regardless of the
particular distribution. The appearance is similar to the percent point
function. However, instead of going from the smallest to the largest
value on the vertical axis, it goes from the largest to the smallest
value.
31
2.5.3 Families of Distributions
Many probability distributions are not a single distribution, but are in
fact a family of distributions. This is due to the distribution having one
or more shape parameters. Shape parameters allow a distribution to
take on a variety of shapes, depending on the value of the shape
parameter. These distributions are particularly useful in modeling
applications since they are flexible enough to model a variety of data
sets (Pepi, 1994).
The Weibull distribution is an example of a distribution that has a
shape parameter. The following graph plots the Weibull pdf with the
following values for the shape parameter: 0.5, 1.0, 2.0, and 5.0.
The shapes above include an exponential distribution, a right-skewed
distribution, and a relatively symmetric distribution.
32
Evans and Peacock (2000) stated that the Weibull distribution has a
relatively simple distributional form. However, the shape parameter
allows the Weibull to assume a wide variety of shapes. This
combination of simplicity and flexibility in the shape of the Weibull
distribution has made it an effective distributional model in reliability
applications. This ability to model a wide variety of distributional
shapes using a relatively simple distributional form is possible with
many other distributional families as well.
2.5.4 Location and scale parameters
A probability distribution is characterized by location and scale
parameters. Location and scale parameters are typically used in
modeling applications (Evans and Peacock, 2000).
For example, the following graph is the probability density function
for the standard normal distribution, which has the location parameter
equal to zero and scale parameter equal to one.
Probability density
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
x
33
1
2
3
4
The next plot shows the probability density function for a normal
distribution with a location parameter of 10 and a scale parameter of
1.
Normal PDF (Location =10)
P robability density
0.4
0.3
0.2
0.1
0
6
7
8
9
10
11
12
13
14
x
The effect of the location parameter is to translate the graph, relative
to the standard normal distribution, 10 units to the right on the
horizontal axis. A location parameter of -10 would have shifted the
graph 10 units to the left on the horizontal axis.
That is, a location parameter simply shifts the graph left or right on the
horizontal axis.
The next plot has a scale parameter of 3 (and a location parameter of
zero). The effect of the scale parameter is to stretch out the graph. The
maximum y value is approximately 0.13 as opposed 0.4 in the
previous graphs. The y value, i.e., the vertical axis value, approaches
zero at about (+/-) 9 as opposed to (+/-) 3 with the first graph.
34
Normal PDF (Scale =3)
Probability density
0.2
0.15
0.1
0.05
0
-10
-5
0
5
10
x
In contrast, the next graph has a scale parameter of 1/3 (=0.333). The
effect of this scale parameter is to squeeze the pdf. That is, the
maximum y value is approximately 1.2 as opposed to 0.4 and the y
value is near zero at (+/-) 1 as opposed to (+/-) 3.
Normal PDF (Scale =1/3)
Probability density
1.5
1
0.5
0
-2
-1
0
x
35
1
2
The effect of a scale parameter greater than one is to stretch the pdf.
The greater the magnitude, the greater the stretching. The effect of a
scale parameter less than one is to compress the pdf. The compressing
approaches a spike as the scale parameter goes to zero. A scale
parameter of 1 leaves the pdf unchanged (if the scale parameter is 1 to
begin with) and non-positive scale parameters are not allowed
The following graph shows the effect of both a location and a scale
parameter. The plot has been shifted right 10 units and stretched by a
Probability density
factor of 3.
0.15
0.1
0.05
0
0
5
10
x
15
20
The standard form of any distribution is the form that has location
parameter zero and scale parameter one.
It is common in statistical software packages only to compute the
standard form of the distribution. There are formulas for converting
36
from the standard form to the form with other location and scale
parameters. These formulas are independent of the particular
probability distribution.
The following are the formulas for computing various probability
functions based on the standard form of the distribution. The
parameter (a) refers to the location parameter and the parameter (b)
refers to the scale parameter. Shape parameters are not included.
Cumulative Distribution Function
F(x;a,b) = F((x-a)/b;0,1)
Probability Density Function
f(x;a,b) = (1/b)f((x-a)/b;0,1)
Percent Point Function
G( ;a,b) = a + bG( ;0,1)
Hazard Function
h(x;a,b) = (1/b)h((x-a)/b;0,1)
Cumulative Hazard Function
H(x;a,b) = H((x-a)/b;0,1)
Survival Function
S(x;a,b) = S((x-a)/b;0,1)
Inverse Survival Function
Z( ;a,b) = a + bZ( ;0,1)
Random Numbers
Y(a,b) = a + bY(0,1)
For the normal distribution, the location and scale parameters
correspond to the mean and standard deviation, respectively.
However, this is not necessarily true for other distributions. In fact, it
is not true for most distributions.
37
2.5.5 Estimating the parameters of a distribution
One common application of probability distributions is modelling
univariate data with a specific probability distribution (Snedecor and
Cochran, 1989). This involves the following two steps:
1. Determination of the "best-fitting" distribution.
2. Estimation of the parameters (shape, location, and scale
parameters) for that distribution.
There are various methods, both numerical and graphical, for
estimating the parameters of a probability distribution.
1. Method of moments:
The method of moments equates sample moments to parameter
estimates. When moment methods are available, they have the
advantage of simplicity. The disadvantage is that they are often not
available and they do not have the desirable optimality properties of
maximum likelihood and least squares estimators.
The primary use of moment estimates is as starting values for the
more precise maximum likelihood and least squares estimates.
2. Maximum likelihood:
Maximum likelihood estimation begins with the mathematical
expression known as a likelihood function of the sample data. Loosely
speaking, the likelihood of a set of data is the probability of obtaining
that particular set of data given the chosen probability model. This
expression contains the unknown parameters. Those values of the
parameter that maximize the sample likelihood are known as the
maximum likelihood estimates.
38
3. Least squares:
In least squares (LS) estimation, the unknown values of the
parameters,
β0, β1,…… in the regression function, f (x, β ) are
estimated by finding numerical values for the parameters that
minimize the sum of the squared deviations between the observed
responses and the functional portion of the model. Mathematically, the
least (sum of) squares criterion that is minimized to obtain the
parameter estimates is
As previously noted β0, β1, … are treated as the variables in the
optimization and the predictor variable values, x1, x2, … are treated as
coefficients. To emphasize the fact that the estimates of the parameter
values are not the same as the true values of the parameters, the
estimates are denoted by β0, β1, ….. For linear models, the least
squares minimization is usually done analytically using calculus. For
nonlinear models, on the other hand, the minimization must almost
always be done using iterative numerical algorithms.
To estimate the Gumbel parameters using this method, data on failures
(Time of failure and number of failure) is needed (William, 2008).
These data is accumulated and arranged in descending order and the
rank of each data is obtained. Then from the rank we can find the
probability of failure as:
39
Probability of failure (T) = (Number of Observation + 1)/rank
T = (M+1)/N
Where:
M = Number of observation.
N = the order of failure.
Then calculate the T/T-1
Taking the logarithm of (T/T-1) twice.
Then by plotting the accumulated time of failure with log(log(T/T-1))
we can obtain a regression line from which can find the Gumbel
parameters (α and β). Thus, when we perform the linear regression,
the estimate for the Gumbel α and β parameters comes directly from
the intercept and slope of the regression line.
4. PPCC and probability:
The PPCC plot can be used to estimate the shape parameter of a
distribution with a single shape parameter. After finding the best value
of the shape parameter, the probability plot can be used to estimate the
location and scale parameters of a probability distribution.
2.5.6 Gallery of Distributions:
Detailed information on a few of the most common distributions is
available below. There are a large number of distributions used in
statistical applications (Johnson and Balakrishnan, 1994, and Evans
and Peacock, 2000).
40
2.5.6.1 Continuous Distributions:
2.5.6.1.1 Exponential Distribution:
The exponential distribution is primarily used in reliability
applications (Johnson and Balakrishnan, 1994). The exponential
distribution is used to model data with a constant failure rate
(indicated by the hazard plot which is simply equal to a constant).
The general formula for the probability density function of the
exponential distribution is
where
µ is the location parameter and β is the scale parameter (the
scale parameter is often referred to as which equals 1/β). The case
where
µ = 0 and β = 1 is called the standard exponential
distribution. The equation for the standard exponential distribution is
f (x) = e
–x
for x ≥ 0
The general form of probability functions can be expressed in terms of
the standard distribution. Subsequent formulas in this section are
given for the 1-parameter (i.e., with scale parameter) form of the
function.
The following is the plot of the exponential probability density
function.
The formula for the cumulative distribution function of the
exponential distribution is
41
The formula for the percent point function of the exponential
distribution is
The formula for the hazard function of the exponential distribution is
The formula for the cumulative hazard function of the exponential
distribution is
The formula for the survival function of the exponential distribution is
The formula for the inverse survival function of the exponential
distribution is
2.5.6.1.2 Weibull Distribution
The Weibull distribution is used extensively in reliability applications
to model failure times (Johnson and Balakrishnan, 1994).
The formula for the probability density function of the general
Weibull distribution is
where
is the shape parameter,
is the location parameter and
the scale parameter. The case where
42
= 0 and
is
= 1 is called the
standard Weibull distribution. The case where
= 0 is called the 2-
parameter Weibull distribution. The equation for the standard Weibull
distribution reduces to:
Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this
section are given for the standard form of the function.
The following is the plot of the Weibull probability density function.
The formula for the cumulative distribution function of the Weibull
distribution is
43
The following is the plot of the Weibull cumulative distribution
function with the same values of as the pdf plots above.
The formula for the percent point function of the Weibull distribution is
The formula for the hazard function of the Weibull distribution is
The formula for the cumulative hazard function of the Weibull
distribution is
44
The formula for the survival function of the Weibull distribution is
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.
Mean
where
is the gamma function
Median
Mode
Range
Zero to positive infinity.
Standard Deviation
Coefficient of Variation
Uses of the Weibull Distribution Model
1. Because of its flexible shape and ability to model a wide range
of failure rates, the Weibull has been used successfully in many
applications as a purely empirical model.
2. The Weibull model can be derived theoretically as a form of
extreme value distribution, governing the time to occurrence of
the "weakest link" of many competing failure processes. This
may explain why it has been so successful in applications such
45
as capacitor, ball bearing, and relay and material strength
failures.
3. Another special case of the Weibull occurs when the shape
parameter is 2. The distribution is called the Rayleigh
distribution and it turns out to be the theoretical probability
model for the magnitude of radial error when the x and y
coordinate errors are independent normals with 0 mean and the
same standard deviation
2.5.6.1.3 Gamma Distribution
The general formula for the probability density function of the gamma
distribution is
where
γ is the shape parameter, µ is the location parameter, β is the
scale parameter, and Γ is the gamma function which has the formula
The case where
µ = 0 and β = 1 is called the standard gamma
distribution. The equation for the standard gamma distribution
reduces to
46
Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this
section are given for the standard form of the function.
The following is the plot of the gamma probability density function.
The formula for the cumulative distribution function of the gamma
distribution is
47
Where Γ the gamma function is defined above and
is the
incomplete gamma function. The incomplete gamma function has the
formula
The following is the plot of the gamma cumulative distribution
function with the same values of as the pdf plots above.
The formula for the hazard function of the gamma distribution is
48
The formula for the cumulative hazard function of the gamma
distribution is
where Γ is the gamma function defined above and
is the
incomplete gamma function defined above.
The formula for the survival function of the gamma distribution is
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.
Mean
Mode
Range
Zero to positive infinity.
Standard Deviation
Skewness
Kurtosis
Coefficient of Variation
The method of moments estimators of the gamma distribution are
49
and s are the sample mean and standard deviation,
where
respectively.
2.5.6.1.4 Gumbel Distribution
Gumbel Probability Density Function (pdf) of the Gumbel distribution
is given by:
1
f(t) = σ e
z
z –e
f(T) ≥ 0 , σ > 0
where:
z = T-µ
σ
and:
µ = location parameter.
σ = scale parameter.
The Gumbel mean or MTTF is:
where
(Euler's constant).
The mode of the Gumbel distribution is:
The median of the Gumbel distribution is:
50
The standard deviation for the Gumbel distribution is given by:
The reliability for a mission of time T for the Gumbel distribution is
given by:
R(T) = e
-ez
The unreliability function is given by:
R(T) = 1 - e
-e z
The Gumbel reliable life is given by:
The instantaneous Gumbel failure rate is given by:
λ =
e
z
σ
Some of the specific characteristics of the Gumbel distribution are the
following:
•
The shape of the Gumbel distribution is skewed to the left. The
Gumbel pdf has no shape parameter. This means that the
Gumbel pdf has only one shape, which does not change.
•
The Gumbel pdf has location parameter µ, which is equal to the
mode , but it differs from median and mean. This is because
the Gumbel distribution is not symmetrical about its µ.
•
As µ decreases, the pdf is shifted to the left.
51
•
As µ increases, the pdf is shifted to the right.
•
As σ increases, the pdf spreads out and becomes shallower.
•
As σ decreases, the pdf becomes taller and narrower.
•
For
pdf = 0. For T = µ, the pdf reaches its
maximum point (
•
The points of inflection of the pdf graph are
or
.
•
If times follow the Weibull distribution, then the logarithm of
times follow a Gumbel distribution. If ti follows a Weibull
distribution with β and η, then the Ln(ti) follows a Gumbel
distribution with µ = ln(η) and
.
The form of the Gumbel probability paper is based on a linearization
of the cdf. From Eqn. :
using Eqn.:
Then:
Now let:
52
And:
which results in the linear equation of:
The Gumbel probability paper resulting from this linearized cdf
function is shown next.
For z = 0, T = µ and
(63.21% unreliability).
To read µ from the
For z = 1, σ = T - µ and
plot, find the time value that corresponds to the intersection of the
probability plot with the 63.21% unreliability line. To read σ from the
plot, find the time value that corresponds to the intersection of the
probability plot with the 93.40% unreliability line, then take the
difference between this time value and the µ value.
53
2.5.6.2 Discrete Distributions
2.5.6.2.1 Poisson Distribution
The Poisson distribution is used to model the number of events
occurring within a given time interval.
The formula for the Poisson probability mass function is
is the shape parameter which indicates the average number of
events in the given time interval.
The following is the plot of the Poisson probability density function
for four values of .
The formula for the Poisson cumulative probability function is
54
2.6 Replacement of machinery, implement and tractors:
The replacement policy must be evaluated regularly as it will be
influenced by changing factors such as interest rates, expected repair
or maintenance costs, tax rates, prices and resale value. There are
several strategies that a farmer can follow for replacing machinery
namely replacing it frequently, replace something every year, replace
when cash is available or lastly, keep it forever (Barnard and Nix,
1979).
Machinery replacement decisions have a major influence on the net
profit of a farming enterprise and the cost of owning and operating
farm machinery is in most cases a major cost component. On the other
hand, machinery also generates income and a decision to buy a new
machine influence a farmer’s cash flow over a couple of years. A
wrong decision can have drastic effects on the future of the farm
business (Culpin, 1975).
The replacement cost reflects the present value of a stream of cost and
income over different life spans of a machine. The optimal
replacement age of a machine is therefore that year where the
replacement cost is the lowest (Kay and Edwards, 1994).
Witney (1988) stated that in the absence of detailed component
reliability date, the vehicle replacement policy for fleet operators can
be based on the accumulated repair costs for individual vehicle. On
most farms, however, there are seldom enough tractors or machines of
the same type to provide an accurate comparison of repairs which, by
their very nature, tend to be ‘lumpy’. Some major overhauls and
replacements can be anticipated with a fair degree of accuracy; for
55
example, tire wear is clearly visible and engine life is reasonably
predictable.
The optimum time for replacing a machine is at the point when the
annual ownership cost reaches a minimum (Witney, 1988). The
annual machinery costs from actual cash flow for the full period of
ownership incorporate decremental depreciation, accumulated repairs
as a percentage of the purchase price, interest charges, insurance
premiums, based on the written-down value, and tax allowances. A
separate cash flow analysis is completed for each different period of
ownership. As a machine gets older, the annual depreciation charge
declines but the annual repair cost increases (Witney, 1988).
The economic life of a machine is here defined as the interval of time
during which that machine reaches its minimum average yearly cost.
If a machine is replaced by an exact duplicate with the same annual
costs, replacement occurs when the currently owned machine attains
its economic life. When average cost reaches its minimum, marginal
cost and average cost are equal. This is the same as saying that when
economic life is reached, the actual yearly cost (marginal cost) is
equal to the average yearly cost of the machine. Theoretically,
replacement should occur when marginal cost first crosses average
cost from below (Kletke, 1969). In the real world, yearly machine
costs fluctuate, therefore, some rule of thumb must be used to
implement the theoretical replacement criterion. There are several
alternative rules which may be suggested. First, the machine can be
replaced at the theoretical optimum replacement interval. Second, the
farmer may replace when some average of marginal costs exceeds the
minimum average cost of the proposed replacement. Third,
56
replacement may occur when marginal costs in any year reach a
specified level. The size of repair cost required will be discussed later.
Machines are usually replaced when they reach an age where the costs
of maintaining or operating them have become too high, and it is
economically sound to replace them with newer, more efficient
machines (Chand et. al., 2000).
Williams (1993) developed a model to predict the optimum
replacement time of agricultural machinery. He stated that factors that
will tend to encourage early replacement will include low interest
rates, high second values, high repair costs, rapid technological
improvements in machinery, high level of business, high level of tax
reliefs on machinery investment and high penalties incurred through
machinery breakdowns. Also he stated that the actual decision on
which year is ideal will often depend on other factors that cannot
readily be incorporated in the model because of their qualitative
nature.
2.6.1 Depreciation
Depreciation is the loss in value of a machine due to time and use. It is
usually the largest cost component in machinery operations. Machines
depreciate as result of age, wear and tear and obsolescence (Witney,
1988). Depreciation cost is designed to reflect the reduction in value
over a period of time (Kaul and Egbo, 1985). Hunt (1983) classified
methods of estimating depreciation as:
1- Straight line method: The annual depreciation charge is expressed
by the following equation:
D = (P- S)/L
57
Where:
D = Depreciation ($).
P = Purchase Price ($).
S = Salvage Value or selling price ($).
L = time between selling and purchasing, years.
Hunt (1983) and Mirani, et. al., (1989) described this method as the
simplest method as it charges an easily calculated, constant amount
each year. When using straight line method for depreciation
calculation, Witney (1988) suggested a 10% of the purchase price for
the salvage value.
The straight line depreciation method is not quite accurate for the true
value of a machine at some age short of the end of its assumed life
(Salih, 1996).
2- Estimated value method: may be the most realistic determination of
depreciation. The amount of depreciation is the difference between
the value of the machine at the end of each year and its value at the
start of that year.
3- Declining-balance method: A uniform rate is applied annually to
the remaining value of the tractor or machine at the beginning of the
year, however, the depreciation amount is different for each year of
the tractor’s or machine’s life. Depreciation can be expressed by the
following equation (FMO, 1987):
D
=
Vn -
Vn
=
P[1- /L]n
Vn+1
x
58
=
Vn+1
x
P[1- /L]n+1
Where:
D
= amount of depreciation charged for year n+1 ($).
n
= number presenting age of the tractor in year at beginning
of year in question, year.
V
= remaining value at any time ($).
x
=
ratio of depreciation rate used to that of straight line
method.
4- Sum of the year -digits method: The digits of the estimated number
of years of life are added together. This sum is divided into the num
number of years of life remaining for the tractor or machine
including the year in question, amount of depreciation charged each
years is the fractional part of the difference between purchase price
and the salvage value (Hunt, 1983).
D
=
L – n (P – S)
YD
Where:
YD = sum of the years digits, (1+2+3+…..+L).
n
= age of the tractors or machines in years at the beginning
of the year.
5- The Sinking-fund method: This method considers the problem of
depreciation as one of establishing a fund that will draw compound
interest, uniform annual payments to this fund are of such a size that
by the end of the life of the tractor or machine, the funds and their
interest have accumulated to an amount that will purchase another
equivalent tractor or machine (Hunt, 1983).
SFP =
(P-S)
i….
(1+i)L -1
59
Vn
=
P-S
(1+ i)L – (1+i)n
(1+i)L -1
Where:
SFP = sinking fund annual payment.
Vn
= value at the end of year n.
6- Initial and Subsequent Rate method: Williams (1993) showed that
machines have the average depreciation rates in table (2.1). The initial
depreciation rate refers to the moment of purchase. The subsequent
depreciation rate refers to the following years, including the first.
Depreciation can be expressed by the following equation:
D
=
Vn -
Vn+1
Vn = P* ((1+ i /100) n)*(1-Di/100)*(1-Ds/100) n
Vn+1 =
P* ((1+ i /100) n+1)*(1-Di/100)*(1-Ds/100) n+1
Where:
D
= amount of depreciation charged for year n+1.
n
= number presenting age of the tractor in year at beginning
of year in question.
V
= remaining value at any time.
i
= inflation rate.
Di
= initial depreciation rate.
Ds = subsequent depreciation rate.
60
Table 2.1 Initial and subsequent depreciation rate for different
machines.
Initial
Depreciation
%
Subsequent
Depreciation
%
Baler
Chisel plough
15
40
13
12
Combine harvester-self-propelled
12
12
Combine harvester-trailed
Crop sprayer
Fertilizer distributor
Fertilizer spreader
Field cultivator
Four-wheel drive and crawler
Grain drill
Heavy duty disc
Moldboard plough
20
40
40
40
40
32
40
40
40
13
12
12
12
12
8
12
12
12
Mower
34
11
Mower conditioner
Potato harvester
Roller packer
30
47
40
10
6
12
Rotary cultivator
40
12
Rotary hoe
Row crop cultivator
Spring tooth harrow
Sugar beet harvester
Tandom disc harrow
Two-wheel drive
40
40
40
12
40
15
12
12
12
16
12
10
Trailer
25
14
Machine Type
Source Williams, N. 1993.
61
2.6.2 Repair and Maintenance Costs:
Repair and maintenance are essential in an effort to guarantee a high
standard of machine performance and reliability (Witney, 1988). Hunt
(1983) defined repair and maintenance costs as the expenditure for
spare parts and labour for reinstalling replacement parts after parts
failure and also reconditioning some parts as a result of wear.
Bukhari (1982) concluded that repair and maintenance costs vary with
the variation in tractors and machines make.
Repair and maintenance costs in developing countries accounted for
14-21% of total operating cost and the repair costs may be three to
five times than the corresponding European and North American costs
(Adam, 1999). He established a prediction model for tractor and
machine repair and maintenance costs in the Sudan as follows:
2.4
)10-7
Y = (2.53 X
Where:
Y = Accumulated repair and maintenance costs as percent of initial
price.
X = Tractors or machine cumulative hours.
Ahmed (1985) derived a model to predict the tractor and machine
repair and maintenance costs in the Sudan as follows:
Y = 0.0704
x
740
2.336
Where:
Y = Accumulated repair and maintenance costs as percent of initial
price.
X = Tractors or machine cumulative hours
62
In USA, Bowers and Hunts (1970) derived a function for repair and
maintenance costs for tractors as follows:
Y = 0.076
x
120
1.6
Where:
Y = Total Accumulated repair and maintenance costs as percent of
purchase price ($).
X = accumulated hours of use as percentage of wear-out life (12000 h).
Fairbank et. al., (1971) stated a logarithmic transformation by the least
square method using recent data derived from Kansas survey in USA.
They derived this function:
Y = (1.4x 10-3) x2.19
Where:
Y = Total Accumulated repair and maintenance costs as percent of
initial price ($).
X = accumulated operating hours as percent of total expected operating
hours.
Williams, N. (1993) showed that machinery has the following repair
cost functions as in table (2.2)
63
Table 2.2 Repair cost functions.
Age
Hours
Type 0
Type 1
Type 2
Type 3
Type 4
1
800
0
1
7
2
2
2
1600
0
3
14
5
5
3
2400
0
4
21
9
10
4
3200
0
5
28
15
15
5
4000
0
7
35
21
21
6
4800
0
8
42
28
27
7
5600
0
9
48
36
34
8
6400
0
11
55
44
41
9
7200
0
12
62
53
49
10
8000
0
14
69
63
57
11
8800
0
15
76
73
66
12
9600
0
16
83
84
75
13
10400
0
18
90
96
84
14
11200
0
19
97
108
94
15
12000
0
20
104
120
104
16
12800
0
22
111
133
114
17
13600
0
24
118
145
124
18
14400
0
26
124
156
135
19
15200
0
28
131
168
146
20
16000
0
30
140
179
159
64
CHAPTER THREE
MATERIALS AND METHODS
3.1 Data Collection:
The required input data for reliability model was collected from
relevant records and reports found in the Elguneid Sugar Factory. It
was taken for several types of tractors and machines (John Deere,
Belarus and Cameco). The data being collected included the number
and type of tractor, type of machine, working hours per season,
number of breakdowns during the working season of tractor and
machine and number of breakdowns of some parts of tractor and
machine during the working season. The data was collected for
seasons 2006-2008.
Regarding the input data for replacement model, it was collected from
the Agricultural Bank of Sudan (Appendix 2). This data included
purchase price ($) for tractors and machinery, interest rate of
investment (%), tax rate, inflation rate (%), writing down allowance
(%) and R&M rate %.
Elguneid Sugar Factory: Elguneid lies between latitudes 14O 19-15O
00 north and longitude 33O 19-33O 27 east. Elguneid was established
in 1964 as a governmental Scheme. Two German companies (Buchau
Wolf and BMA) designed and built the factory. The construction
commenced in 1958 and was completed in 1962.
Elguneid Scheme, with its pumps, canalization, land tenancy was a part
of the Gezira Scheme. It lies about 120 km south of Khartoum, on the
East Side of the Blue Nile. The scheme is connected to the main road
of Khartoum Medani by a ferry to cross the Blue Nile. Being unique
65
with the tenancy system, Elguneid administration and the tenants’
production relations are governed by an agreement implemented in
1991. A six-course rotation is adopted namely; (fallow, cash crop, plant
cane, first ratoon, second ratoon and third ratoon).
The scheme is
irrigated by pumps from the Blue Nile. The farm area of the Scheme is
16.8 thousand hectares. The number of tenants is 2507 and the number
of tenancy (hawasha) is 2518. The size of the holding is 6.3 hectares
(15 fedans). The Scheme encompasses 1576 employees, as well as
3000 seasonal and temporary workers.
Elguneid Sugar Factory is
shown in figure (3.1).
Agricultural Bank of Sudan: Agricultural Bank of Sudan is a one of
the largest and most important governmental bank in Sudan that
specialized in agriculture activities. The bank helps small farmers to
buy tractors, machinery and agricultural inputs with loan or fund. On
the other hand the bank buy the farmers product with subsidize price.
3.2 Data Analysis
Simple descriptive tools: regression, t-test and standard deviation and
Kolmogorov-Smirnov goodness-of-Fit Test for homogeneity were
employed using Excel and SPSS software.
3.3 Tractors breakdowns and failures:
Tractors breakdowns or failures in this study were considered to be
unpredictable events for whole tractors or sub systems which may
have arisen from one or more of the: Accidents, improper service or
maintenance, improper machine operation, improper set-up and
inadequate design.
66
Figure (3.1): Elguneid Sugar Factory
67
3.4 Model Development:
The program is composed of a start interface (Fig. 3.2) to choose
either the Reliability module or replacement module.
3.4.1 Reliability Module:
3.4.1.1 Module Description:
The Reliability Design module is a simple module that uses Excel and
Visual Basic software. The module presents a spreadsheet-style input
screen that allows the users to inter the data of failure and sort them.
After entering data, the module analyzes the data by performing a
simple linear regression to estimate distribution parameters.
Spreadsheets are either visible or hidden processing sheets. The
software does an excellent job of presenting results in a logical
fashion with very detailed graphics. The program technical
specification is shown in table (3.1) while Figure (3.3) shows the
flowchart of the module.
3.4.1.2 Theoretical Development:
Reliability is usually defined as the probability that a device will
perform as intended without failure for a specified period of time
under given operating conditions. Thus, the definition of failure, the
time period and the operating conditions must be clearly stated. A
failure may be defined as any condition which prevents operation of a
machine or which causes or results in a level of performance below a
specified limit (Kumur et. al., 1977). For example, in a combine,
failure could be a broken elevator chain link (a down-time failure) or
it could be a broken screen in the shoe resulting in a degradation of
performance. The time period can vary with the machine type.
68
Figure 3.2 Program start menu.
69
Table 3.1 Technical specification of the program.
Description
Item
Reliability module
Replacement module
Program language
Excel and Visual basic
Excel and Visual basic
Program type
Button menu driven and Button menu driven
Spreadsheet user
Program flexibility
Program
Inherited from Excel XP Inherited from Excel XP
2003 and visual basic
2003 and visual basic
Work under Window
Work under Window
adaptability
Program interface
One
main
Spreadsheet
macros
menu
and Multi menu with control
contain tools including one main
menu and multi submenu
Units used
SI- system
SI- system
424 kb
Space required on
hard disc
70
Start
Input Data
SubSystem Name, Time between failures, Number
of Failure.
Accumulate and Arrange Time and
Number of failure in Descending
order, find the rank of each failure.
Calculate the probability of Failure from
the Rank and find the loglog of
Probability of failure
Display Regression to Calculate
Gumbel Parameters by linear
regression
Calculate Mean Time between failures and
for Whole system and Subsystem.
End
Figure 3.3 Reliability module flow chart.
71
Environmental and operating conditions and the field conditions under
which a particular machine is designed to operate.
Machine failure can be categorized into three basic types (Amstardter,
1971): early life failures, random failures and wear-out.
The general expression for reliability is given as
t
R
=e
∫ λ dt
(4.1)
0
Where:
R = the reliability at any time t (decimal).
λ = the failure rate (%).
There are many kinds of probability (or failure) distributions in use.
The choice of distribution to use depends on the characteristic failure
rate. In this study, the Gumbel distribution was used. It has been used
in many applications such as failure in vehicle structural components
(Lemense, 1969). An attempt has also been made to use Weibul
distribution for farm equipment (Archer, 1962; Von Bargen, 1970;
Kumar et. al., 1977).
The Gumbel failure probability density function (PDF) is given as:
(4.2)
where:
and:
72
µ = location parameters.
σ = scale parameters.
Integrating the pdf gives the Gumbel cumulative density function
(CDF):
F(t) = α + β (log(log(T+1/T)))
(4.3)
To estimate the Gumbel parameters by Least square method (William,
2008), data on failures is needed. With some effort, the Gumbel
cumulative distribution function can be transformed so that it appears
in the familiar form of a straight line:Y=mX+b: Here's how :
The first step is to bring our function into a linear form. Gumbel
distribution, the cdf (cumulative density function) is:
F(t) = α + β (log(log(T+1/T)))
(4.4)
Comparing this equation with the simple equation for a line, we see
that the left side of the equation corresponds to Y, loglog(T+1/T)
corresponds to X, β corresponds to m, and
corresponds to b. Thus,
when we perform the linear regression, the estimate for the Gumbel α
and β parameter comes directly from the intercept and slope of the
line.
After obtaining estimates for the Gumbel parameters, then we can
predict time between failure for future machine use by the following
equation:
The mean,
, (also called MTTF or MTBF by some authors) of the
Gumbel pdf is given by:
MTTF = α + β (log(log(T+1/T)))
73
(4.7)
3.4.2 Replacement Module:
3.4.2.1 General:
The module works by calculating the total cost of owning a machine
for any number of years. The costs and returns generated in future
years are converted back to their present day values using discount
factors. This automatically allow for the interest charges. The
discounted cost of owning a machine is the sum of the purchase price
and the present value of the stream of running and repair costs less
present value of the machine's resale value. This calculation is
repeated for each year of the machine's life. The total cost of owning
the machine must then be converted to an equivalent annual cost as
this will vary with length of time the machine is owned. The annual
equivalent cost for each of these total costs is calculated using
amortization factors. The year which has the lowest annual equivalent
cost is the point where replacement should take place, other things
being equal, as this gives the lowest average annual cost of owning the
machine.
3.4.2.2 Module Description:
The replacement module is a computer interactive module which
allows the user to interact directly with program (Fig 3.2). The
computer module consist of build in data that outlined by Williams, N.
(1993) and Witney (1988). Figure 3.4 shows the flow chart of the
module.
The user enters the required input data for machinery replacement, the
module compute the depreciation, repair and maintenance costs,
annual holding cost and determine the year of replacement.
74
Start
Input Data
Machine Data, Depreciation Data and
Repair and Maintenance Data
Intermediate Calculation
(real discount rate, post tax discount rate,
depreciation, R & M)
Calculation of total cost of owning a
machine and converted it to an
equivalent annual cost.
Highlight the year in which the
equivalent annuity is at a minimum
End
Figure 3.4 Replacement module flow chart.
75
3.4.2.3 Module Structure
The program is an interactive program where the user is prompt to
enter the required data. The built in data were made to help the user of
lacking of own data. The data was entered in Visual basic forms and
linked to cells in Excel spreadsheets to process intermediate and final
calculation for annual holding costs of tractor or machine. The
program technical specification is shown in Table (3.1). The program
is composed of a main menu and sub-menu. The main menu (Fig. 3.5)
controls the sequence of all program operations. The sub-menus are:
About replacement module menu to give the introductory information
about the program (Fig. 3.6), machine data menu (Fig. 3.7), data entry
for depreciation menu (Fig. 3.8) and Repair and maintenance menu
(Fig. 3.9) and view and print results and graphs (Fig. 3.10 and Fig.
3.11), respectively.
Data required for replacement module:
a) Machine Data:
Machine Type.
Purchase price.
Discount rate.
Inflation rate.
Writing down allowance.
Marginal tax rate.
b) Depreciation Data:
Life span.
Salvage value (%).
76
Salvage value ($).
Depreciation method.
Depreciation rate (%).
Initial depreciation rate (%).
Subsequent depreciation rate. (%).
c) Repair and Maintenance Data:
Repair and Maintenance Method.
Williams (1993) R & M Functions.
Hourly use/season.
77
Figure 3.5 Replacement module main menu.
78
Figure 3.6 About MORRAM model menu.
79
Figure 3.7 Machine data entry menu.
80
Figure 3.8 Depreciation data entry menu
81
Figure 3.9 Repair and maintenance data entry menu.
82
Figure 3.10 View and print results menu.
83
Figure 3.11 View and print graphs.
84
3.4.2.4 Theoretical Development
3.4.2.4.1 Basic Consideration:
The module works by calculating the total cost of owning a machine
for any number of years (Williams, 1993). The costs and returns
generated in future years are converted back to their present day
values using discount factors. This automatically allows for the
interest charges. The discounted cost of owning a machine is the sum
of the purchase price and the present value of the stream of running
repair costs less the present value of the machine’s resale value. This
computation provides the total cost of owning the machine for any
given length of time. This calculation is repeated for each year of
machine’s life.
The total cost of owning the machine must then be converted to an
equivalent annual cost as this will vary with the length of time the
machine is owned (Williams, 1993). The annual equivalent costs for
each of these total costs are calculated using amortization factors. The
year which has the lowest annual equivalent cost is the point where
replacement should take place, other things being equal, as this gives
the lowest average annual cost of owning the machine.
The spreadsheet contains depreciation methods and formula for a
range of machine type. Most of these can be based on (Witney, 1988),
Hunt (1983) and Williams (1993). The module enables the users to
select a depreciation schedule. Also there a choice of repair methods
and function derived from UK survey data. Most of these can be based
on Williams (1993), Ahmed (1985) and Adam (1999).
The module can also be adjusted to allow for taxation and inflation.
85
3.4.2.4.2 Calculation Procedure:
a – Intermediate calculation:
= ((1+dr/100)/(1+i/100)-1)*100
drr
drpt
= dr * (100- MT)/100.
Where:
drr
dr
drpt
MT
= real discount rate (%).
= discount rate (%).
= post-tax discount rate (%).
= marginal tax rate (%).
b- Calculation before Taxation:
Machine value (Vm) : showing the estimated value of the machine
for years of age. It was estimated according to different methods of
depreciation.
- Straight line method:
Machine value
D =
= value at the beginning of year - D
(P-S)/L
Where:
D
P
S
L
=
=
=
=
depreciation ($/year).
purchase price ($).
salvage value or selling price ($).
time between and purchasing, year.
- Declining-Balance method:
D = Vn Vn = P [1Vn+1 = P [1-
Vn+1
x
/L]n
x
/L]n+1
Where:
86
D = amount of depreciation charged for year n+1 ($).
n = number presenting age of machine in year at beginning of
year in question (year).
V = remaining value at any time ($).
x
= ratio of depreciation rate used to that of straight line
method.
- Sum of year-digits method:
D
=
L – n (P – S)
YD
Where:
YD = sum of the years digits, (1+2+3+…..+L).
n = age of the tractors or machines in years at the beginning
of the year.
- The Sinking-fund method:
SFP =
(P-S)
Vn
P-S
=
i….
(1+i)L -1
(1+ i)L – (1+i)n
(1+i)L -1
Where:
SFP = sinking fund annual payment ($).
Vn = value at the end of year n ($).
-Initial and Subsequent Rate method:
D
=
Vn -
Vn+1
Vn = P* ((1+ i /100) n)*(1-Di/100)*(1-Ds/100) n
Vn+1 =
P* ((1+ i /100) n+1)*(1-Di/100)*(1-Ds/100) n+1
Where:
87
D
n
= amount of depreciation charged for year n+1 ($).
= number presenting age of the tractor in year at beginning
of year in question (year).
= remaining value at any time ($).
= inflation rate (%).
V
i
Di
= initial depreciation rate (%).
Ds = subsequent depreciation rate (%).
Discounted secondhand value: showing the present value of the
machine at the chosen discount rate for a given age ($).
Discounted second value
=
secondhand value
inverse discount factor
Vm/ (1+(dr/100))n
=
Repair costs: showing the estimated cost of repairs for each year of
the machine’s life based on the chosen repair cost method and
function adjusted for inflation rate and age.
-Ahmed (1985) method:
Rc
= (0.0704(x/740)2.336)*(1+i/100)n
Where:
R c = repair cost.
i
= inflation rate.
-William (1993)) method: This method has a five function to estimate
repair cost.
R c = (repair function * P/100)*(1+i/100)n
-Adam (1999) method:
Rc
= ((2.53 x
2.4
-7
)*10 )*(1+i/100)
88
n
Discounted repair cost: showing the present value of the repair cost
based on chosen discount rate and the machine age.
Discounted second value
=
repair cost
inverse discount factor
R c / (1+(dr/100))n
=
Discount holding cost: showing the present value of the total cost
of owning the machine for each year of age.
Discount holding cost = discount holding cost of previous year + discount second
hand value in previous year - discount second hand value in this year discount repair cost in this year
Equivalent annuity: showing the average cost in present value
terms of keeping the machine to any particular age.
Equivalent annuity =
Discount holding cost
Annuity factor
n
= Discount holding cost * (dr/100)/ (1-(1+dr/100) )
Then the program module highlights the year in which the equivalent
annuity is at a minimum for year of replacement.
c- Calculation after Taxation:
Machine value (Vm) : As same as value of machine before
taxation.
Written down value: showing the written
down value of the
machine calculated using the writing down allowance.
Written down value
= original cost * writing down factor
= P ((100-twd)/100)
89
n
Where:
twd
= tax writing down allowance as a percentage.
Tax on sale: showing the tax payable (or refundable) on the
difference between the economic value of the machine and the
written down value for each year of age at the marginal tax rate.
Tax on sale = (Vm - written down value)* tax rate
= (Vm - written down value)* (MT/100)
Post-tax writing down allowance: showing the tax relief on the
writing down allowance at the marginal tax rate for that year.
Post-tax writing down allowance = (written down value in the previous
year - written down value in the current year)* tax rate.
Post-tax repair cost: showing the cost of repairs in each year of age
after allowing for tax relief on the cost.
Post-tax repair cost
= repair cost * tax adjustment factor
= R c * (100-MT)/100
Cumulative repairs and writing down allowances: showing present
values of the accumulated post-tax writing down allowances and
post-tax repair cost for each year of age.
90
Cumulative repairs and writing down allowances = value for previous year +
(-post-tax writing down allowances in current year + post-tax repair
costs in the present year)* post-tax discount factor.
= value for previous year + (-post-tax writing down allowances in current
n
year + post-tax repair costs in the present year)*(1+ drpt/100)
Discount holding cost: showing the present value of the total posttax cost of owning the machine for each year of age.
Discount holding cost = original capital cost+ Cumulative repairs and writing
down allowances + ( - resale value + tax on sale) * post-tax discount factor.
Post-tax equivalent annuity: showing the average post-tax cost in
present value terms of keeping the machine to any particular age.
Post-tax Equivalent annuity
=
Discount holding cost
Post-tax annuity factor
n
= Discount holding cost * (drpt /100)/ (1- (1+ drpt /100) )
Then the program module highlight the year in which the equivalent
annuity is at a minimum for year of replacement.
91
CHAPTER FOUR
RESULTS AND DISCUSION
4.1 RELIBILITY MODULE:
4.1.1 Selection of the suitable probability distribution for
module building:
As given by Kumar et. al (1977) regression coefficient is used to
select the suitable type of probability distribution and its parameters
under least square method (Tables 4.1, 4.2 and 4.3). Six probability
distributions namely: Person, Gumbel normal, Gumbel log, Gumbel
loglog, Weibull normal and Weibull logXlogY, were tested for John
Deere, Belarus and Cameco Tractors.
Table 4.1, 4.2 and 4.3 shows that the most accepted distribution is the
Gumbel loglog (R2 = 0.96, 0.89 and 0.99 for John Deere, Belarus and
Cameco tractors respectively). Although, Coefficient of regression
(R2) for Weibull normal distribution for John Deere and Belarus
tractors are high but they results in negative time distribution which is
not logically acceptable (Figure 4.1 and 4.2).
The set of graphs given in Appendix 3 (Six figures per each tractor)
gives more visualization of the actual distribution compared to
predicted ones. Using 2- tailed T test it evident from Table 4.4 that
there is no significant difference at 5% confidence level between the
predicted and actual time of failure.
92
Table 4.1 Comparison of different probability distributions using data of Belarus tractor.
Distribution
Person
Gumbell
Gumbell
Gumell
Weibell
Weibell
Normal
logX
LoglogX
Normal
LogX, logY
(N+1)/M
Plotting Position
N/M
(N+1)/M
M/(N+1)
y = 5247.6x –
y = 226.13x +
y = 4137.6x +
14698
2725.9
2000.7
1632.4
6479.5
3.236
R2
0.8846
0.4125
0.7811
0.8964
0.9774
0.5341
Comment
Rejected
Rejected
Rejected
Accepted
Rejected
Rejected
Gives
Lower
Lower
High
Gives negative
Lower
negative
R2
R2
R2
Number
R2
Linear Regression
93
Equation
Reason
Number
y = -3303.6x + y = -5798.7x +
M/(N+1)
y = 0.6133x +
Table 4.2 Comparison of Different Probability Distributions Using Data of John Deere Tractor
Distribution
Person
Gumbell
Gumbell
Gumbell
Weibell
Weibell
Normal
logX
LoglogX
Normal
LogX, logY
(N+1)/M
Plotting Position
N/M
(N+1)/M
M/(N+1)
M/(N+1)
Linear Regression y = 1746.1x - y = 154.43x + y = 2564x + y = -1986.8x + y = -3372.6x + y = 1.0292x +
Equation
94
R2
Comment
Reason
3751.3
914.21
518.85
326.22
3183.9
2.5914
0.81
0.57
0.89
0.96
0.98
0.54
Rejected
Rejected
Rejected
Accepted
Rejected
Rejected
Gives
Lower
Lower
High
Gives negative
Lower
negative
R2
R2
R2
Number
R2
Number
Table 4.3 Comparison of Different Probability Distributions Using Data of Cameco Tractor
Distribution
Person
Gumbell
Gumbell
Gumell
Weibell
Weibell
Normal
logX
LoglogX
Normal
LogX, logY
(N+1)/M
Plotting Position
N/M
(N+1)/M
M/(N+1)
M/(N+1)
Linear Regression y = 1303.3x - y = 85.389x y = 1386.6x + y = -1055.6x + y = -1761x + y = 0.833x +
2839.1
+ 488.43
281.69
188.62
1691.5
2.465
R2
0.85
0.64
0.95
0.99
0.98
0.69
Comment
Rejected
Rejected
Rejected
Accepted
Rejected
Rejected
Lower
Lower
Lower
High
Lower
Lower
R2
R2
R2
R2
R2
R2
95
Equation
Reason
Time of failure (hr)
3600.00
3100.00
2600.00
2100.00
1600.00
1100.00
600.00
100.00
-400.00
4.000
3.500
3.000
2.500
2.000
1.500
-900.00
Probability of failure
Predicted Time
of Failure
Actual Time
of Failure
Figure 4.1 Actual and predicted time of failure for Belarus tractor
using normal Weibull distribution.
96
4000.00
Time of failure (hr)
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
0.00
1.00000
0.90000
0.80000
0.70000
0.60000
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
-500.00
Probability of failure
Predicted Time
of Failure
Actual Time
of Failure
Figure 4.2Actual and predicted time of failure for John Deere
tractor using normal Weibull distribution.
97
Table 4.4 T-test analysis for mean time between failures for the
tested tractors.
Parameter
Belarus
*
John Deere
Cameco
355.34
201.72
100.96
294.25
169.80
91.35
t value
1.14
0.99
0.60
Significance
0.26
0.33
0.55
Predicted time
*
Actual time
98
4.1.2 Verification of Reliability Module:
To verify statistically the developed Reliability Module KolmogorovSmirnov Goodness-of-Fit Test is used (Kumar et. Al. ,1977). Table 4.5
indicated that by using 2-tailed test of significance and based on
Kolmogorov-Smirnov there is insufficient evidence at 5% confidence
level to reject the assumed distribution as representative of the true
distribution.
Figure 4.3, 4.4 and 4.5 show the comparison between the actual data
points used to estimate the Gumbel loglog parameters and the
theoretical curve using the above distribution for the three studied
tractors.
4.1.3 Validation of Reliability Module:
Data reported by Kumar et. Al., (1997) for combine harvester is used
as input in the Gumbel distribution module to predict the time
distribution. Figure 4.6 shows that there is no difference between
Kumar et. Al., (1997) Model and the developed Gumbel Model.
The T-test gives in Table 4.6 shows that there is no significant
difference between the developed module predicted time distribution
of failure and that of Kumar et. al. (1997).
This can be taken as evidence of the capability of the developed
module to be used as a tool to predict failure time of occurrence in
future.
99
Table 4.5 Sample Kolmogorov-Smirnov test for the tested
tractors.
JohnDeere
Bellarus
20
20
20
Mean
201.72
355.34
100.95
Std. Deviation
108.21
192.32
61.95
Absolute
.23
.28
.26
Positive
.23
.28
.21
Negative
-.21
-.19
-.257
Kolmogorov-Smirnov Z
1.02
1.23
1.15
Asymp. Sig. (2-tailed)
.25
.098
.14
N
Normal
Cameco
Parameters(a,b)
Most Extreme
Differences
a Test distribution is Normal.
b Calculated from field data.
100
7000
Time of Failure
6000
5000
4000
3000
2000
1000
1.00
0.50
0.00
-0.50
-1.00
-1.50
-2.00
0
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure 4.3 Actual and predicted time failure using Gumbel
distribution for Belarus tractor.
101
4000
3500
Time of Failure (hrs)
3000
2500
2000
1500
1000
500
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure 4.4 Actual and predicted time failure using Gumbel
distribution for John Deere tractor.
102
Time of Failure (hrs)
2000
1500
1000
500
1.00
0.50
0.00
-0.50
-1.00
-1.50
-2.00
0
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure 4.5 Actual and predicted time failure using Gumbel
distribution for Cameco tractor.
103
150
100
50
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0
Reliability Model
200
Kumar et al (1977) Model
Figure 4.6 Comparison of Kumar et. al. (1977) model Time of
failure and Predicted Reliability module time of failure.
104
Table 4.6 T-test analysis for time between failure for Kumar et. al.
(1977) and predicted reliability module.
Parameter
Mean Time
of Failure
Kumar et. al. (1977) time
*
Predicted Module time
*
106.34
86.73
T value
0.98
Significance
0.34
105
4.1.4 Application of Reliability Module:
4.1.4.1 Prediction of failure time for tested tractors:
Belarus Tractor: The developed module was used to estimate the
reliability of a Belarus tractor as a whole unit and for each one of its
sub system, using data collected from the repair records of the tractor
for five years.
Considering the whole tractor as a unit, the time between failures was
the time between any two failures, irrespective of component type.
Figure 4.7 shows the histogram of input data as a function of time
between failures. The program calculates the Gumbel parameters
using the regression analysis of Log probability method. The values α
and β were 1590.91 and - 3294.72, respectively (Table 4.7).
Thus, the Gumbel distribution which predicts the time between
failures (MTBF) for the Bell tractors studied is:
MTBF
= 1590.91 - 3294.72 * (Log(Log(T/T-1)))
Table 4.8 shows the output of the module to predict mean time
between failure (MTBF). First column displays failure ranks, while
the second one displays mean time between failures. The value 266.9
means that the first failure should be expected after about 266 hours.
Next, 187.9 means that the second failure should be expected about
188 hrs after the first failure has occurred. These values are of tractor
having above Gumbel distribution.
106
8
Number of Observation
7
6
5
4
3
2
0
403
801
1390
1737
2212
2579
2793
2900
3270
3609
3750
4111
4488
4665
4757
5032
5354
5425
5508
5885
1
Cumulative time between Failures (hr)
Figure 4.7 Histogram of number of failure in each cell of
Belarus tractor.
107
Table 4.7 Regression analysis of Belarus tractor data.
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.95
0.90
0.89
534.91
20
ANOVA
108
Regression
Residual
Total
Intercept (α)
Beta (β)
Df
1
18
19
SS
46254809.9
5150398.96
51405208.9
Coefficients
1590.91
-3294.72
Standard
Error
194.03
259.13
MS
46254809.98
286133.27
F
161.65
Significance F
1.97841E-10
t Stat
8.20
-12.71
P-value
1.7237E-07
1.97841E-10
Lower
95%
1183.26106
-3839.14
Upper
95%
1998.56
-2750.29
Lower
95.0%
1183.26
-3839.14
Upper
95.0%
1998.56
-2750.29
Table 4.8 Time between failures of Belarus tractor.
Failure Rank
1
Mean Time Between Failure
266.9
2
187.9
3
451.3
4
185.4
5
196.7
6
240.2
7
304.7
8
618.0
9
186.2
10
190.0
11
194.4
12
206.8
13
220.9
14
360.9
15
369.7
16
1028.1
17
1191.3
18
206.8
19
228.9
20
270.8
109
John Deere Tractor: The developed module was used to estimate the
reliability of a John Deere tractor as a whole unit and for each one of
its sub system, using data collected from the repair records of the
tractor for five years.
Considering the whole tractor as a unit, the time between failures was
the time between any two failures, irrespective of component type.
Figure 4.8 shows the histogram of input data as a function of time
between failures. The program calculates the Gumbel parameters
using the regression analysis of Log probability method. The values α
and β were 194.99 and -2108.27, respectively (Table 4.9).
Thus, the Gumbel distribution which predicts the time between
failures (MTBF) for the Bell tractors studied is:
MTBF
= 194.99 - 2108.27 * (Log(Log(T/T-1)))
Table 4.10 shows the output of the module to predict mean time
between failures (MTBF). First column depicts failure ranks, while
the second one depicts mean time between failures. The value 119.2
means that the first failure should be expected after about 119 hours.
Next, 120.2 means that the second failure should be expected about
120 hrs after the first failure has occurred. These values are of tractor
having above Gumbel distribution.
Cameco Tractor: The developed module was used to estimate the
reliability of a Cameco tractor as a whole unit and for each one of its
sub system, using data collected from the repair records of the tractor
for five years.
110
7
Number of Observation
6
5
4
3
2
0
58
129
219
320
526
688
758
1022
1160
1256
1353
1533
1700
1850
2091
2390
2607
2755
2948
3397
1
Cumulative time between Failures (hr)
Figure 4.8 Histogram of number of failure in each cell of John
Deere tractor.
111
Table 4.9 Regression analysis of John Deere tractor data.
Regression Statistics
Multiple R
0.99
R Square
0.98
Adjusted R Square
0.98
Standard Error
141.51
Observations
20
ANOVA
112
Regression
Residual
Total
Intercept (α)
Beta (β)
Df
1
18
19
Coefficients
194.99
-2108.27
SS
MS
18939584.66 18939584.66
360471.340
20026.19
19300056
Standard
Error
51.33
68.56
T Stat
3.80
-30.75
F
945.74
Significance F
5.17328E-17
P-value
0.00131
5.17328E-17
Lower
95%
87.14
-2252.29
Upper
95%
302.83
-1964.24
Lower
95.0%
87.14
-2252.29
Upper
95.0%
302.8311
-1964.2365
Table 4.10 Time between failures of John Deere tractor.
Failure Rank
Mean Time Between Failure
1
119.2
2
120.2
3
121.6
4
288.8
5
125.9
6
132.3
7
230.9
8
395.5
9
132.3
10
146.5
11
153.7
12
170.8
13
173.3
14
195.0
15
236.5
16
118.7
17
141.3
18
124.4
19
657.8
20
426.6
113
Considering the whole tractor as a unit, the time between failures was
the time between any two failures, irrespective of component type.
Figure 4.9 shows the histogram of input data as a function of time
between failures. The program calculates the Gumbel parameters
using the regression analysis of Log probability method. The values α
and β were 223.21 and 1072.43 respectively (Table 4.11).
Thus, the Gumbel distribution which predicts the time between
failures (MTBF) for the Bell tractors studied is:
MTBF
= 223.21 + 1072.43 * (Log(Log(T/T-1)))
Table 4.12 shows the output of the module to predict mean time
between failures (MTBF). First column is the failure ranks, where as
the second one is the mean time between failures. The value 61.8
means that the first failure should be expected after about 61 hours.
Next, 78.2 means that the second failure should be expected about 78
hrs after
the first failure has occurred. These values are of tractor
having above Gumbel distribution.
114
5
4.5
Number of Observation
4
3.5
3
2.5
2
1.5
1
0
99
151
268
308
394
476
526
639
706
783
882
924
1011
1086
1155
1263
1401
1515
1696
1827
0.5
Cumulative time between Failures (hr)
Figure 4.9 Histogram of number of failure in each cell of
Cameco tractor
115
Table 4.11 Regression analysis of Cameco tractor data.
Regression Statistics
Multiple R
0.99
R Square
0.99
Adjusted R
Square
0.98
Standard Error
64.18
Observations
20
116
ANOVA
Regression
Residual
Total
Intercept (α)
Beta (β)
Df
1
18
19
SS
4900662.56
74138.43
4974801
Coefficients
223.21
1072.43
Standard
Error
23.28
31.09
MS
4900662.57
4118.80
F
1189.83
Significance F
6.77044E-18
t Stat
9.59
-34.49
P-value
1.69966E-08
6.77044E-18
Lower
95%
174.30
-1137.74
Upper
95%
272.12
-1007.11
Lower
95.0%
174.30
-1137.74
Upper
95.0%
272.12
-1007.11
Table 4.12 Time between failures of Cameco.
1
Mean Time
Between Failure
61.8
2
78.2
3
60.6
4
61.2
5
67.3
6
71.9
7
86.9
8
60.4
9
63.3
10
64.0
11
74.5
12
99.2
13
117.5
14
120.3
15
146.9
16
334.6
17
88.2
18
93.1
19
201.2
20
67.3
Failure Rank
117
4.1.4.2 Comparison of tractors performance under actual field
conditions:
Figures 4.10 show the relation between cumulative time between
failure and probability of failure for Belarus, John Deere and Cameco
tractors. For all tractors probability of failure is linear up to 5.25, this
level is taken as a threshold level and after which there is evident
turning point where the probability start to rise sharply. This indicate
that after a certain operating time values of 6399.4, 3002.5 and 1656.7
hours for Belarus, John Deere and Cameco tractors, the workshop
manger need to inspect these tractors before a serious failure occurs
(Vide: figure 4.10). Also the figure indicates that the tractors may rank
in the order of Cameco, John Deere and Belarus with respect to time
of occurrence of the threshold level of failure.
The slope parameters of the Gumbel distribution (β) can be taken as
indicator of tractor reliability. As value of (β) decrease the probability
level of failure decreases (Kumar et. Al. 1977).
Table 4.13 indicates that the reliability of the three tested tractors
decreases with the order of Belarus, John Deere and Cameco tractor.
Ishola, T, A. and J. S. Adeoti (2004) found that Massey Fergusson and
Fiat tractors proved to be more reliable tractors in Kwara State.
Table 4.13 Slope parameter (β) for the tested tractors
Tractor Make
Gumbel Slope Parameter (β)
Belarus
0.002
John Deere
0.003
Cameco
0.006
118
Cameco: y = 0.0059x - 1.3555
R2 = 0.6085
John Deere: y = 0.0031x - 2.0015
R2 = 0.5954
Bellarus: y = 0.0016x - 1.5757
R2 = 0.5869
25.00
Probability of Failure
20.00
15.00
10.00
(1)
(3)
(2)
5.00
0.00
0.0
1000.0
2000.0
3000.0
4000.0
5000.0
6000.0
7000.0
Cumulative Time Between Failure
John Deere
Bellarus
(1) 1656,7 hr
(2) 3002,5 hr
Cameco
(3) 6399.4 hr
Figure 4.10 Comparative performance of reliability module for
the tested tractors.
119
4.1.4.3
Utilization
of
reliability
module
for
developing
maintenance scheduling plan:
Recall that the replacement module function is to help decision maker
either to keep the machine or to replace it. If the workshop manager
found on economic basis that he should keep the tractor or machine
then how can he maintain its performance at good standards over its
working life. To achieve this objective he needs to develop
maintenance scheduling plan for each tractor. One avenue to reach this
objective is to follow the tractor maintenance manual. Tractor
maintenance chart (Appendix 1) given in the manual is based on
theoretical design ideal conditions. In reality tractors operate in
changing dynamic environment that differ from place to place (Lewis,
1987).
To practically fulfil the said target figures 4.11, 4.12 and 4.13 were
made. They show the time occurrence in working hours of failure
ranked from starting date for each one of the three tested tractors. By
using these graphs it is possible to predict when failure is expected
and consequently plan the human, money and material resources
needed to conduct maintenance.
To have a proper maintenance scheduling plan it is not enough only
just to detect the time of failure but it is essential to help workshop
manager by determining what types of failure is expected and what
subsystem of the tractor is to fail (Ishola, T, A. and J. S. Adeoti 2004).
Consequently, the prediction module was employed to predict time of
occurrence of failure of each subsystem for each tractor. Table 4.14,
4.15 and 4.16 show the time of failure of each subsystem for the three
tested tractors.
120
For developing the maintenance plan it is necessary to read these
tables in conjunction with Figures 4.11, 4.12 and 4.13 for each tractor.
Analysis of failure of subsystem of the tested tractors (Tables 4.14,
4.15 and 4.16) indicate that tires followed by cooling system and the
transmission system are the subsystems of frequent and early rate of
failure compared to other systems for all tractors. However,
occurrence of such problems under conditions of Sugar company
workshops needs to be analyzed. Ishola, and Adeoti (2004) found that
the steering, traction and electrical systems are more prone to failure
than the cooling, transmission, engine fuel and Hydraulic systems.
121
122
Failure Rank
0
5
10
15
20
25
0
2000
4000
5000
Cumulative Time Between Failure
3000
6000
7000
Figure 4.11 Cumulative time between failure of Belarus tractor.
1000
123
Failure Rank
0.00
5.00
10.00
15.00
20.00
25.00
500
1000
2000
2500
3000
Cumulative Time Between Failure
1500
3500
4000
4500
Figure 4.12 Cumulative time between failure of John Deere tractor.
0
124
Failure Rank
0
500
1500
Cumulative Time Between Failure
1000
2000
Figure 4.13 Cumulative time between failure of Cameco tractor.
0.00
5.00
10.00
15.00
20.00
25.00
Table 4.14 Times of occurrence of failures for sub system of
Belarus tractor.
Subsystem
Time of
Failure
Tires
266.9
Tires
454.8
Hitching/Steering/Engine/Exhaust/Transmission/Electrical/Ti
res
906.1
Brake/Fuel/Tires
1091.5
Implement
1288.2
Fuel/Transmission
1528.4
Fuel/Cooling/Transmission/Brake
1833.1
Hitching/Steering/Cooling/Exhaust
2451.2
Cooling/Fuel/Exhaust
2637.4
Implement/Tires
2827.4
Tires
3021.8
Exhaust/Transmission/Seals
3228.6
Hitching/Transmission
3449.5
Brake/Steering/Cooling/Exhaust//Electrical
3810.4
Brake/Fuel/Transmission
4180.1
Fuel/Steering/Tires/Electrical
5208.1
Tires/Lubrication/Engine
6399.4
Hitching/Steering/Tires
6606.2
Engine/Cooling/Transmission/Exhaust
6835.1
Cooling/Exhaust
7105.9
125
Table 4.15 Times of occurrence of failures for Sub System of John
Deere Tractor.
Subsystem
Time of Failure
Tires
119.2
Cooling/Transmission
239.4
Cooling/Engine
361.0
Steering/draw/Tires
649.8
Electrical/Exhaust/Cooling
775.6
Engine/Fuel
908.0
Engine/Electrical/Exhaust
1138.9
Cooling/Tires
1534.4
Transmission/Tires/draw
1666.7
Engine/Tires/Seals
1813.2
Engine/Tires
1966.9
rical/Tires
2137.7
Electrical/Exhaust
2311.0
Cooling/Steering/Electrical/Transmission/Fuel
2506.0
Cooling
2742.5
Engine/Lubrication/Cooling
2861.2
Cooling/Lubricating
3002.5
Engine/Fuel/draw
3126.9
Transmission/Tires/Cooling/draw
3784.7
Engine/Steering/Brake/cooling/Electrical
4211.3
126
Table 4.16 Times of occurrence of failures for Sub System of
Cameco Tractor.
Subsystem
Time of Failure
Tires
61.8
Drawbar
140.0
Engine
200.6
Engine/Drawbar
261.8
Draw/Tires
329.1
Brake/Tires/Fuel
401.0
Body/Tires/
487.9
Hitching/Engine
548.2
Electrical/Body
611.5
Tires
675.6
Hitching/Transmission/Exhaust/Engine
750.1
Tires/Cooling/Electrical
849.2
Hitching/Cooling
966.7
Tires/Body/Electrical
1087.0
Draw/Body/Tires/Electrical
1233.9
Lubrication/fuel/Tires
1568.6
Tires/Body/Cooling
1656.7
Brake/Transmission
1749.8
Implement/Tires
1951.0
Electrical/Tires/Body
2018.3
127
4.2 Replacement Module:
4.2.1 Screening of repair and maintenance estimation
method:
In the module building the user is given three options to choose the
method to estimate the repair and maintenance costs. Economic data
collected from the Agricultural Sudanese Bank for medium tractor
was taken as module input to calculate R&M costs by the module
(Vide: Chapter three material and methods).
Figure 4.14 shows the change of R&M costs with machine age
generated by Ahmed (1985), Williams (1993) and Adam (1999).
Taking ordinary least square method (linear model) these estimation
techniques can be ranked using slope values (β) (349.4, 159.4 and
4.61) in descending order as Ahmed (1985), Williams (1993) and
Adam (1999), respectively.
4.2.2 Comparison of method to estimate depreciation:
In the module the user has the option to select his preferred method to
estimate depreciation from an option of four methods: Straight line
method (Mirani et. al., 1989), double declining balance method
(FMO, 1987), sum of year digit method (Hunt, 1983) and initial and
subsequent method (Williams, 1993).
Table 4.17 shows the ranking of depreciation methods according to
slope (β). The general rule is that the method with the largest slope is
most preferred. As shown in figure 4.15 double declining balance
(Accelerated method) is most preferred. This result is in agreement
with Dumbler et. al., (2000).
128
Table 4.17 Ranking of depreciation methods according to slope.
Rank
slope values (β)
Depreciation Methods
1
204.5
Double Declining Double
2
139.5
Initial and Subsequent
3
102.86
Sum of year-digit
4
0
Straight line
From Figure 4.15 it is clear that several gains may be obtained by
using accelerated depreciation methods (Double declining balance and
Initial and Subsequent methods) as opposed to the straight line
method. One benefit is related to changing future expenses in up peak
of the tractor. This approach is logical since the annual benefit from
the tractor's use decreases with age and tractor repair and maintenance
cost increases (as will be shown later in Figure 4.15). By offsetting the
increased repair and maintenance costs, the accelerated method
equalizes the combined charge of both repairs and depreciation.
Another benefit would be for selecting accelerated depreciation will
be how to make allowance in case of inflation. In such a case by
expensing more of the cost of tractor at the beginning, then in the
future when inflation causes expenses to be higher, the amount of
expense will be lower. This because accumulated expenses increases
with time with increase of inflation hence, it is better to deduct
depreciation earlier and this is in agreement with Stansberry (2004).
Finally, there also tax benefit from using accelerated depreciation
method because when deductions are accelerated the business holder
129
can be save the additional money and apply it twoards future growth
(Schwanhausser, 2005, Ullakko, 2003 and Hederman and Rea, 2005).
8000.0
Y= 349.36x – 164.19
R2 = 0.9946
7000.0
6000.0
R&M cost ($)
5000.0
4000.0
Y= 159.44x – 679.52
R2 = 0.91
3000.0
2000.0
1000.0
Y=4.6111 – 19.278
R2 = 0.91
0.0
-1000.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
Ahmed
Adam
William
Figure 4.14 Comparison of R&M cost estimated by different
methods of depreciation.
130
Double Declining: y = -204.53x + 3333.7
Initial and Subsequent: y = -139.49x + 2540.7
6000
Sum of year digit: y = -102.86x + 2160
Straight line:y = 1080
Depreciation Cost ($)
5000
4000
3000
2000
1000
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
Straight line
Declining balance
Sum of year-digit
Initial and Subseqent
Figure 4.15 Comparison of depreciation cost estimated by
different methods.
131
4.2.3 Selection of best combination of R&M with depreciation
Methods:
To check the applicability of the different R&M methods with their
respective depreciation methods, data for high (240 hp) tractor,
medium (72 hp) tractor cost and a low tractor cost (Chinese tractor) is
used as input to the module for purpose of determining the optimum
replacement age.
Table 4.18 shows that Williams (1993) method for estimating repair
and maintenance and double declining balance and initial and
subsequent methods for estimating depreciation can be predict
replacement age for the entire machine used. Adam (1999) method is
suitable only at low machine price. In contrast Ahmed (1985) method
failed to predict replacement age. This could be due to the low value
of estimated R&M cost by this method (vide: Figure 4.14).
4.2.4 Module Application:
Taking medium tractor data the module was used to calculate the total
holding cost (THC) using double declining balance method and
Williams (1993) R&M estimating method which is considered as
preferred methods (Section 4.2.3).
From Figure 4.16 the optimum age to replace without taxation is 9
year and 10 year for the case with taxation for medium tractors
working under sugar cane company- Sudan. For crawler tractor the
optimum replacement age is 9 and 10 year for case of without and
with taxation respectively (Figure 4.17). Figure 4.18 shows that the
132
optimum replacement age for Chinese tractor is 14 and 13 year
without taxation and with taxation respectively.
133
Table 4.18 Combination of R&M method with depreciation methods.
Machine
Straight Line Method
Crawler
Tractor
Chinese
Williams
Declining Balance
Method
No Tax
with tax
No Tax
with tax
-
4
9
-
3
-
Sum of year digit
Init. And Sub
No Tax
with tax
No Tax
with tax
10
-
-
12
12
9
10
-
-
12
11
3
14
14
15
14
14
13
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
5
9
9
7
7
9
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Tractor
Crawler
133
Tractor
Chinese
Adam
Tractor
Crawler
Tractor
Chinese
Tractor
Ahmed
-
-
-
-
-
-
-
-
8000
Total Holding Cost($)
7000
6000
5000
4000
3000
2000
1000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
a- Without taxation.
8000
Total Holding Cost($)
7000
6000
5000
4000
3000
2000
1000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
Total Holding Cost
Depreciation Declining Balance
Williams (1993) R&M Cost
b- With taxation
Figure 5.16 Optimum replacement age of medium cost tractor.
134
40000
Total Holding Cost($)
35000
30000
25000
20000
15000
10000
5000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
a- Without taxation.
40000
Total Holding Cost($)
35000
30000
25000
20000
15000
10000
5000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
Total Holding Cost
Depreciation Declining Balance
Williams (1993) R&M Cost
b- With taxation
Figure 4.17 Optimum replacement age of high cost tractor
135
1000
Total Holding Cost($)
900
800
700
600
500
400
300
200
100
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
a- Without taxation.
1000
Total Holding Cost($)
900
800
700
600
500
400
300
200
100
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Machine Age (year)
Total Holding Cost
Depreciation Declining Balance
Williams (1993) R&M Cost
b- With taxation
Figure 4.18 Optimum replacement age of low cost tractor
(Chinese tractor).
136
CHAPTER FIVE
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusion
Most suitable probability distribution is the Gumbel loglog (R2 =
0.96, 0.89 and 0.99 for John Deere, Belarus and Cameco tractors,
respectively).
Using 2- tailed T test showed that there is no significant difference
at 5% confidence level between the predicted and actual time of
failure for Gumbel Distribution.
The results obtained by the reliability module are found to be valid
when tested with Kolmogorov-Smirnov Goodness-of-Fit Test. This
can be taken as evidence of the capability of the developed module
to be used as a tool to predict failure time of occurrence in future.
Application of the reliability module in case of Belarus tractor
revealed that the first failure should be expected after about 266
hours. Next, the second failure should be expected about 188 hrs
after the first failure has occurred. For John Deere the first failure
should be expected after about 119 hours, and the second failure
should be expected about 120 hrs after
the first failure has
occurred. For Cameco tractor the first failure should be expected
after about 61 hours, and the second failure should be expected
about 78 hrs after the first failure has occurred.
Inspection to be carried out after operating times of 6399.4, 3002.5
and 1656.7 hours for Belarus, John Deere and Cameco tractors,
respectively.
137
Analysis of failure of subsystem of the three tested tractors
indicates that tires followed by cooling system and the
transmission system are the subsystems of frequent and early rate
of failure compared to other systems for all tractors. This calls for
developing special on job training programs for operators.
Taking ordinary least square method (Linear model) the repair and
maintenance estimation techniques can be ranked using slope
values (β) (349.4, 159.4 and 4.61) in descending order as Williams
(1993) methods, Adam (1999) methods and Ahmed (1985)
methods, respectively which help the user to select his preferred
methods.
The optimum age to replace without taxation is 9 year and 10 year
with taxation for crawler and medium tractors working under
Sugar Cane company- Sudan. The optimum replacement age for
Chinese small tractors is 14 year without taxation and 13 year with
taxation.
5.2 Recommendations
Record for individual subsystems failure could be made separately
in order to quantify the distribution parameters for reliability
modelling.
Analysis of failure of subsystem of the tested tractors indicates that
some subsystem of frequent and early rate of failure compared to
other systems for all tractors. Occurrence of such problems under
conditions of Sugar company workshops needs to be more
investigated.
138
Other functional forms to predict the expected distribution of
machine failure and allows determination of machine availability
such as markov chain need to be examined and compared to
Gumbel method
Other repair and maintenance cost prediction methods as
percentage of purchase price need to be examined and used in
replacement module.
139
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146
APPENDICES
Appendix 1
Maintenance chart from manual of tractor.
147
Appendix 2
a- Machine list price and economic information
Equipment
No.
Unit price
Total cost
(SDG)
($)
Loader
250,000
100,000
Dozer
310,000
124,000
Tractor 120 Hp
160,000
64,000
Tractor 75 Hp
60,000
24,000
Heavy discing
9,000
3,600
Light disc
7,500
3,000
Land plane
4,500
1,800
Ridger
6,000
2,400
Ferti+ridger
5,700
2,280
12,000
4,800
Row planter
8,000
3,200
Broadcaster
7,500
3,000
Combine
15,000
6,000
Harvesting (Thresher)
17,000
6,800
Harvesting (stationary)
15,000
6,000
Harvesting (mower)
14,000
5,600
Digger
14,000
5,600
Baller
15,000
6,000
Trailer
6,000
2,400
Trailed tanker
8,000
3,200
14,250
5,700
Seed drill
Chinese Tractor
Economic information
Marginal Tax
10% -30%
inflation rate
15% -35%
Writing down allowance
25% - 45%
Source: Agricultural Sudanese Bank.
148
Appendix 2
b- Machine list price
Implement
Tractor Striger 234 Hp
Tractor Maxim 170Hp
Tractor Fiat 80 Hp
Heavy Disc plough28''
Light Disc plough 24''
AbuXX Ditcher
AbuVI Ditcher
Ridger
Pneumatic Planter
Fertilizer distribution
Planter
Trailer 2 wheel
Trailer 4 wheel
Thresher
Groundnut Harvester
Disc plough
Baler
149
Price ($)
117,647
98,039
21,176
17,647
13,725
7,843
1,961
1,961
12,941
4,706
2,745
5,882
7,843
3,922
9,804
1,961
9,804
Appendix 3
Sets of graph for different forms of distributions.
3600.00
3100.00
2600.00
2100.00
1600.00
1100.00
600.00
100.00
-400.00
Predicted Time
of Failure
4.000
3.500
3.000
2.500
2.000
1.500
-900.00
Actual Time
of Failure
Figure a. Actual and Predicted Time of Failure of
Belarus Tractor using Person Distribution
150
7000.00
6000.00
5000.00
4000.00
3000.00
2000.00
1000.00
Predicted Time
of Failure
20.00
15.00
10.00
5.00
0.00
0.00
Actual Time
of Failure
Figure b, Actual and Predicted Time of Failure of
Belarus Tractor using Normal Gumbel Distribution
151
4000.00
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
Predicted Time
of Failure
1.00000
0.90000
0.80000
0.70000
0.60000
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
0.00
Actual Time
of Failure
Figure c Actual and Predicted Time of Failure of
Belarus Tractor using Log Gumbel Distribution
152
7000
5000
4000
3000
2000
1000
1.00
0.50
0.00
-0.50
-1.00
-1.50
0
-2.00
Time of Failure
6000
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure d. Actual and Predicted Time of Failure of Belarus
Tractor using LogLog Gumbel Distribution
153
4000.00
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
Predicted Time
of Failure
1.00000
0.90000
0.80000
0.70000
0.60000
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
0.00
Actual Time
of Failure
Figure e Actual and Predicted Time of Failure of
Belarus Tractor using Normal Weibull Distribution
154
4000.0
3500.0
3000.0
2500.0
2000.0
1500.0
1000.0
500.0
Predicted Time
of Failure
1.0000
0.9000
0.8000
0.7000
0.6000
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
0.0
Actual Time
of Failure
Figure f Actual and Predicted Time of Failure of
Belarus Tractor using Log two axes Weibul
distribution
155
3900.00
3400.00
2900.00
2400.00
1900.00
1400.00
900.00
400.00
-100.00
Predicted Time
of Failure
4.000
3.500
3.000
2.500
2.000
1.500
-600.00
Actual Time
of Failure
Figure a. Actual and Predicted Time of Failure of
John Deere Tractor using Person Distribution
156
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
Predicted Time
of Failure
10.00
5.00
0.00
0.00
Actual Time
of Failure
Figure b. Actual and Predicted Time of Failure of
John Deere Tractor using Normal Gumbel
Distribution
157
4000.00
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
Predicted Time
of Failure
1.00000
0.90000
0.80000
0.70000
0.60000
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
0.00
Actual Time
of Failure
Figure c. Actual and Predicted Time of Failure of John
Deere Tractor using Log Gumbel Distribution
158
4000
3500
2500
2000
1500
1000
500
1.0
0.5
0.0
-0.5
-1.0
-1.5
0
-2.0
Time of Failure (hrs)
3000
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure d. Actual and Predicted Time of Failure of John Deere Tractor
using loglog Gumbel Distribution
159
4000.00
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
0.00
Predicted Time
of Failure
1.00000
0.90000
0.80000
0.70000
0.60000
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
-500.00
Actual Time
of Failure
Figure e. Actual and Predicted Time of Failure of John
Deere Tractor using Normal Weibull Distribution
160
4000.0
3500.0
3000.0
2500.0
2000.0
1500.0
1000.0
500.0
Predicted Time
of Failure
1.0000
0.9000
0.8000
0.7000
0.6000
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
0.0
Actual Time
of Failure
Figure f. Actual and Predicted Time of Failure of John
Deere Tractor using log two axes Weibull Distribution
161
45000.00
40000.00
35000.00
30000.00
25000.00
20000.00
15000.00
10000.00
5000.00
Predicted Time
of Failure
3.500
3.300
3.100
2.900
2.700
2.500
2.300
2.100
1.900
1.700
1.500
0.00
Actual Time
of Failure
Figure a. Actual and Predicted Time of Failure of
Cameco Tractor using Person Distribution
162
2500.00
2000.00
1500.00
1000.00
500.00
Predicted Time
of Failure
20.00
15.00
10.00
5.00
0.00
0.00
Actual Time
of Failure
Figure b. Actual and Predicted Time of Failure of
Cameco Tractor using Normal Gumbel Distribution
163
2200.00
1700.00
1200.00
700.00
Predicted Time
of Failure
1.40000
1.20000
1.00000
0.80000
0.60000
0.40000
0.20000
-300.00
0.00000
200.00
Actual Time
of Failure
Figure c. Actual and Predicted Time of Failure of
Cameco Tractor using Log Gumbel Distribution
164
1500
1000
500
1.00
0.50
0.00
-0.50
-1.00
-1.50
0
-2.00
Time of Failure (hrs)
2000
Gumbel Probability
Predicted Time
of Failure
Actual Time
of Failure
Figure d. Actual and Predicted Time of Failure of Cameco Tractor using
Loglog Gumbel Distribution
165
4000.00
3500.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
1.00000
Actual Time
of Failure
0.90000
0.80000
0.70000
0.60000
Predicted Time
of Failure
0.50000
0.40000
0.30000
0.20000
0.10000
0.00000
0.00
Figure e. Actual and Predicted Time of Failure of
Cameco Tractor using Normal Weibull Distribution
166
4000.0
3500.0
3000.0
2500.0
2000.0
1500.0
1000.0
500.0
Predicted Time
of Failure
1.0000
0.9000
0.8000
0.7000
0.6000
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
0.0
Actual Time
of Failure
Figure f. Actual and Predicted Time of Failure of
Cameco Tractor using Log tow axes Gumbel
Distribution
167