Mathematical Models for Reliability and Replacement of Agricultural
Transcription
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 REFERENCES Adam, O.E. 1999. Tractors Repair and Maintenance costs estimation in irrigated areas of the Sudan. M. Sc. Thesis. Khartuom University. Aderoba, A. 1989. Farm power and Machinery Selection model for mechanized farms in developing countries. AMA. 20(3): 69-72. Adigun, Y. J. 1987. Maintainability of agricultural machinery in Kwara State of Nigeria. Unpulished B. Eng.(Agric) Project Report, University of Ilorin, Ilorin, Nigeria. Ahmed, F. M. 1985. Prediction Model for Repair and Maintenance Costs. M. Sc. Thesis. Khartoum University. Alcock, R. 1979. Farm Machinery Reliability. A seminar held at Muresk Agricultural College, Western Australia Agricultural Engineer, 34(1) 28. Amjad, S. I. and Chaudhary, A. P. 1988. Field reliability of farm machinery. Journal of Agricultural Mechanization in Asia, Africa and Latin America. Vol.10 No. 1 pp73 - 78. Amstardter, B. L. 1971. Reliability mathematics, fundamentals, practices, procedures. McGrew-Hill Book. Co. NY. Anon, 1972. Reliability of Electronic Equipment and Components. Part 2. Reliability Concept. Australian Standard 1211, Part1972. Archer, R. C. 1962. Reliability Engineering- its application to farm equipment. ASAE paper no. 62-635, ASAE., St. Joseph, MI 49085. 140 ASAE. 1991. Uniform Terminology for agricultural Machinery Management data. ASAE, St. Joseph, MI 49085. Bakhit, S. E., El Shami, O. M and Mohammed, H.I. 2006. An Empirical Simulation Model of Costs Wheat Combine Harvesting Losses In The Rahad Scheme-Sudan. Sudan University of Science and Technology, Journal of Science and Technology, Vol (7) 84-97. Bedewy, M. K., Radwan, M. R. and Hammam, S. A. 1989. A comparative Study on Reliability and Maintainability of Public Transport Vehicles. Reliability Engineering and Safety, 26: 271-277. Bowers, W. and Hunt, D. R. 1970. Application of Mathematical Formula to Repair Cost Data. Transaction of the ASAE. 14(5): 847-859. Bukhari, Sh. B. 1982. Evaluation of farmer’s competence to maintain farm tractors. AMA. Japan 13(1):45-47. Dumbler T. J., Robert O. B., Jr. and Terry L. K. 2000. Use of Alternative Depreciation Methods to Estimate Farm Tractor Values. The Annual Meeting of the American Agricutural economics Association, Florida. USA. Evans, Hastings, and Peacock, 2000. Statistical Distributions, 3rd. Ed., John Wiley and Sons. Fairbanks, G. E., Larson, G. H. and Chung Do-sup. 1971. Cost of using farm machinery. Transaction of the ASAE. 14(1): 98-101. 141 Grazechowaik, R. 1999. Optimization of parameter estimates for agricultural machinery models. Prace-p. Instytutu- RD.1999, 44:1, 36-38. Hederman, Jr., Rea S. 2004. Tax Cuts Boost Business Investment.” The Heritage Foundation Web Memo. Hunt, D. 1971. Equipment Reliability: Indiana and Illinois data. TRANSACTION of the ASAE 14(5): 742-746. Hunt, D. 1983. Farm Power and Machinery Management.8th edition. Iowa State University Press, Ames, Iowa, U.S.A. Ishola, T. A. and Adeoti J. S. 2004. A Study of Farm Tractors Reliability in Kwara State of Nigeria. Proceeding of the annual conference of the Nigerian institution of agricultural engineers. 5th international conference and 26th annual general meeting, Ilorn, Nigeria. Ismail, W. I. 1994. Expert System for Crop Production Machinery Systems. Agricultural Mechanization in Asia, Africa and Latin America, 25(30): 55-62. Johnson, Kotz, and Balakrishnan, 1994. Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons. Johnson, Kotz, and Kemp, 1992. Univariate Discrete Distributions, 2nd. Ed., John Kaul, R. N. and Egbo, C. O. 1985. Introduction to Agricultural Mechanization, Macmillan Intermediate Agriculture Series. General Editor, Ochapa, C. Onazi. 142 Konaka, T. 1987. Farm machinery utilization planning. Agricultural Mechanization in Asia, Africa and Latin America, 18(3): 75-80. Kumar, R. Goss, J. R. and Studn, H. E. 1977. A Study of Combine Harvester Reliability. Transaction of the ASAE. 20(1): 30-34. Lemense, R. A. 1969. Use of Weibull Distribution in analyzing life test data from vehicle structural components. Proceedings 3rd Aerospace Reliability and Maintainability Conference. Pp. 628638. Lewis, E. E. 1987. Introduction to reliability engineering. John Wiley & Sons Publishers. Liange, T. 1976. A dynamic programming Markov Chain approach to farm machinery preventive maintenance problems. Unpublished Ph. D. Thesis, Dept. of Biological and Agricultural Engineering, Raleigh, NC. Machackova, M. 1990. Standard Methods of determining machinery requirements. Mechanizace-Zemedlstvi (40): 11, 507-509Czechoslovakia. McNeil, Donald, 1977. Interactive Data Analysis, John Wiley and Sons. Mirani, A. N., J. M. Baloc and S. B. Bukhari, 1989. Unit cost of operations of farm tractors. AMA. 20(3): 44-46. Mohamed, M. A. 2003. Expert System for Agricultural Machinery Management. M.Sc. Thesis. Sudan University of Science and Technology. 143 Ntuen, C. A. 1990. Simulation Study of Vehicle Maintenance Policies. Institute of Industrial Engineering, Greensboro, PP. 589-592. Oni, K. C. 1987. Reliability of agricultural machinery in Kwara State. Paper presented at the 11th Annual Conference of Nigerian Society of Agricultural Engineers, University of Nigeria. Oskan, E. and Edward, M. 1989. A farmer- Oriented Machinery Comparison Model. Trans. of the ASAE 29(3): 72-77. Pepi, John W., 1994. Failsafe Design of an All BK-7 Glass Aircraft Window, SPIE Proceedings, Vol. 2286, (Spciety of PhotoOptical Instrumentation Engineers (SPIE), Bellingham, WA). Salih, A. A., 1996. Simulation program to predict farm machinery cost and management parameters under inflated economy. M. Sc. Thesis 1996. U of G. Wad Madani. Sayed M. Metwalli, Mona S. Salama and Rousdy A. Taher. 1998. Computer aided reliability for optimum maintenance planning. Computers Ind. Engng. 35(3-4): 603-606. Schwanhausser, Mark. 2005 . What Can You Depreciate? Here’s a Rarity in Tax Law: Choices. Knight-Ridder Tribune Business News. Available from Dawes Library, Business & Industry database. Singh, G., K. Butani and V. Salokhe. 1992. A decision support system for the selection of agricultural machinery. International Agric. Engineering conference, proceedings of a conference held in Bangkok, Thialand (7-10 DEC. 1992) vol. 1, 167-172. 144 Smith, C. O. 1974. Introduction to Reliability in Design. Mc-Graw Hill, New York. Snedecor, George W. and Cochran, William G. 1989. Statistical Methods, Eighth Edition, Iowa State University Press. Stansberry, Gary. 2004. The Do’s & Don’ts of Depreciation. Rental Equipment Register. Taher, R. A. 1992. Computer Aided Reliability and Maintainability for Optimum Maintenance planning. M. Sc. Thesis, Cairo University. Tullberg, J. N., Richman, J. F. and Doyle, G. J. 1984. Reliability and the Operation of Large Tractors. Agricultural Engineer. 39(1): 10-13. Ullakko, James. 2003. Depreciation Benefits Under the Jobs and Growth Tax Relief Reconciliation Act of: Leasehold Improvements and Office Furnishings. Real Estate News & Articles. Von Bargen, K. 1970. Reliability and the capacity performance of field machines. ASAE paper no. 70-647, ASAE, St. Joseph, MI 49085. William, W. Dorner, 2008. Using Excel for Weibull Analysis. Journal of Quality Technology. Williams, N. 1993. Machinery Replacement timing. SpreadSheets for Agriculture. Longman Scientific and Technical, Longman group UK Limited, Longman House, Burnt Mill, Harlow. 145 Wingate-Hill, R. 1981. The application of reliability to farm machinery. Journal of Agricultural Engineers. Winter edition. pp 109 - 111. Witney, Brian, 1988. Choosing and Using Farm Machines. Longman Scientific and Technical, Longman group UK Limited, Longman House, Burnt Mill, Harlow. 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