See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/273257999 Pushback Design using Genetic Algorithms Thesis · June 2010 DOI: 10.13140/2.1.4065.6169 CITATIONS READS 0 570 1 author: Mwiya Songolo The Copperbelt University 1 PUBLICATION 0 CITATIONS SEE PROFILE All content following this page was uploaded by Mwiya Songolo on 08 March 2015. The user has requested enhancement of the downloaded file. Western Australian School of Mines Pushback Design using Genetic Algorithms Mwiya W Songolo BEng., Dip. (Mining) A thesis submitted in partial fulfillment for the award of the degree of Master of Engineering Science in Mining of Curtin University. 30th June 2010 Pushback Design using Genetic Algorithms Western Australian School of Mines ii Declaration I declare that this thesis is my own work and has not been previously submitted in whole or in part for any academic credit at Curtin University or elsewhere. To the best of my knowledge and belief, the thesis contains no material previously published by any other person except where due acknowledgement has been made. Name: Mwiya W Songolo Signature: ........................................ Supervisor: Jose Saavedra Rosas (Dr) Signature: ........................................ 30th June 2010 Pushback Design using Genetic Algorithms Western Australian School of Mines iii ABSTRACT In mine optimisation, research work has mainly taken root in open pit optimisation particularly the determination of ultimate pit limit. Once the ultimate pit limit has been determined what follows is production scheduling. Production scheduling is based on the underlying pushback mine sequencing which is widely used for long-term production planning. Pushbacks thus play a very important role in mine design and optimisation. However, it will be appreciated that many pushback design algorithms that have evolved since the 1960s employ parameters like the commodity price (metal price), mining cost and processing cost with the metal price in this case being the major economic parameter. Some of the algorithms developed have endeavored to design pushbacks based on minimum stripping ratio criteria to be used in conjunction with traditionally designed pushbacks so that the resulting production schedules may give a higher net present value possible. On the other hand, grade uncertainty has also been recognized as the major parameter that should be incorporated into the pushback design process. This is all in a quest to generate a series of nested pits that conforms not only to the minimum required pushback width but provide the highest dollar value for a particular optimum pit size. The design of pushbacks for ease sequencing and scheduling should be governed by optimised pushbacks which yield schedules with the highest net present value possible. This requires a development of a tool that can guarantee the achievability of an optimised pushback. The tool is however, farfetched with the current algorithms available. Nevertheless, an endeavour to develop a genetic model for pushback design with capacity constraint has been proposed in this project research thesis as the solution for optimised pits. The solution for the genetic model has the capability of handling multiple scenarios of capacity constrained pushbacks. In view of this, the objective of the research was therefore, intended to expand the current genetic optimiser by proposing a genetic model for pushback design with capacity constraint with a goal to achieve an optimised pit that would yield pushbacks which are ease to sequence and schedule. Pushback Design using Genetic Algorithms Western Australian School of Mines iv The scope of the study was limited to the generation of pushbacks with capacity constraint using Genetic Algorithms (GA) and the genetic code was written using Python programming language. The pushback design with capacity constraint using genetic algorithm in this research was applicable to a hypothetical two dimensional (2D) deposit. From the findings of the study, it has been shown that generation of pits with increasing price was possible using the Genetic Open Pit Optimiser. Of significance is the conclusion that the objective of the research was achieved by generating pushbacks with capacity constraint subject to a penalizing constraint. It is envisaged that future research work for pushback design with capacity constraint should be extended further to include sequencing and scheduling of pushbacks and then the model can be applied to a three dimensional (3D) real life deposit. Keywords: Open Pit Optimisation, Pushback Design with Capacity Constraint Pushback Design using Genetic Algorithms Western Australian School of Mines v Dedications First and foremost, this thesis is dedicated to my Lord and God, Jesus Christ who by His divine power paved the way for me to undertake a master’s degree in mining. Secondly, I dedicate this thesis to my lovely family which I so very much cherish: my dear wife Shilla, my lovely daughter Makazo and my dear son Mwiya. To you my dear and beautiful wife, Shilla, thank you for your unwavering support and endurance to suffer pain of my absence. You are so close to my heart and this thesis is dedicated to you. Makazo, my lovely daughter, I stretched the bond between you and us by sending you away to stay with your granny prior to the commencement of my studies. As if that was not enough, at a tender age, you never denied the great responsibility I placed on you to take care of your little brother Mwiya as my wife and I pursued our studies. This work is for you to cherish. My dear son Mwiya, a little baby as you were, you had to miss the fatherly love. You had no option but to learn to be independent and mature. This was done for you and by the enablement of His Mighty, I am proud that through this volume, you have a sense of belonging. Pushback Design using Genetic Algorithms Western Australian School of Mines The sole responsibility to edit this thesis was also bestowed upon my supervisor. vi Acknowledgements My profound acknowledgement goes to the following: • To the Copperbelt University. Dr Mahinda provided words of counsel throughout the duration of my studies. was always available for consultations and also shared the sole responsibility to edit this thesis with my supervisor. My acknowledgement also extends to Dr Mahinda Kuruppu and Bunda Besa. Without his encouragement. this thesis would not have been possible without the guidance and support I received from him. As a postgraduate coordinator. • I would also like to thank Associate Professor Erkan Topal for the advice he rendered me to undertake a research work under the supervision of Dr Jose Saavedra. he had an exceptional tact that not only helped shape my line of thought for this work but he became my mentor as well. I would not have tapped through an endowed pool of knowledge that rests with my mentor. who. Finally. I am so grateful for providing me with the sponsorship to undertake a Master of Engineering Science degree in Mining. To him I express my gratitude. I am indebted to Bunda. In his constructive criticism. • To my supervisor Dr Jose Saavedra. Jose Saavedra (Dr). Pushback Design using Genetic Algorithms Western Australian School of Mines . The outstanding support was timely for it enabled me bring my dream into fruition. despite being in the middle of his demanding PhD thesis. I would like to express my indebtedness to Professor Roger Thompson whose doors were always open for me even without an appointment whenever I required guidance in my studies. ..............................................................................................................................................1 Selection ................................................................................. 2 1.. 4 1.............. 40 3.......................................................................5........ 40 3....................................................................................................................4.................. 16 2.............................................................................................3 Fundamental Genetic Algorithm Operators ..............................2 Genetic Algorithm Structure .....6 Pushback Design Techniques..... 36 3...................................................... 28 3...........................................................................2 Crossover .........3 Scope of the Study ............................. 25 2................................3.................................................... 17 2....................0 GENETIC ALGORITHM METHODOLOGY................................................................ vi List of Figures ..5 Attributes of Pushbacks ..............................................................5...........4 Production Sequencing and Scheduling ..................................................... 11 2............... 35 3............1 Manual design of pushbacks ............3 Production Planning ......................................................2 Selection of the mining width ...................................................................................... 35 3.................................... 41 Pushback Design using Genetic Algorithms Western Australian School of Mines ...................................................................................................................................................................................................................................3 Pushback design using the Lerchs-Grossmann’s Whittle Optimiser .........................................................................................3 Mutation ...4 Variations of Genetic Algorithm.................................... xi 1...1 Background of Genetic Algorithms ................................................3...........................3....... 20 2................. vii Contents Declaration .........................................................................................................................4 Design Tools . ii ABSTRACT ........................1 Genetic algorithms ................1 Introduction ........................................................................... 5 1......................2 Python programming language ...............1 Factors affecting pushback width.......................................................................................................................................................................................................................... iii Dedications.2 The Lerchs-Grossmann algorithm........2 Problem Statement and Research Objective ............................... 20 2....................................................................... 8 2........6...................................................................... 5 2..................................................................................................................................................6............................ 2 1...............................................................................................................................................................0 LITERATURE REVIEW.............................................................................................................................. 39 3............................. 8 2......................... 4 1...........................................................2 Open Pit Optimisation ....................................... 10 2....0 INTRODUCTION........................................................................................................ 8 2......... 5 1............................ 21 2........................... v Acknowledgements .................... 38 3.......................4..................................1 Research Background................................. ix List of Tables ...............6................................... ............... 55 4.....................1 Capacity penalizing function..2......................5..... 53 4............................................0 CONCLUSION AND RECOMMENDATIONS .........................................4. 54 4............................................................................................................5............................1 Chromosome representation...................................1 Generation of pits using the genetic open pit optimiser .......................................2.. 53 4................. 62 4.............................................................................................2 Steady-state genetic algorithm ..................................... 96 APPENDICES ................................ 67 4...........................................................................................................2 Mine scheduling ............................ 44 3..2 The Genetic Design of an Open Pit........ 81 5....... 94 5.............................................. 67 4....................3 Optimisation of mine ventilation systems ..................... 63 4....5........................................................5.................................................4.....................1 Simple genetic algorithm ...................................................... 46 3............................3 Crossover of pushback chromosomes .......... 94 5..............................................................................................................0 GENETIC MODEL FOR PUSHBACK DESIGN WITH CAPACITY CONSTRAINT ....4..........................................2 Random generation of individuals ............... 57 4.....................................3 Struggle genetic algorithm .............4..........5............................................................. 64 4.........1 Pit optimisation ............................................ 53 4........................... 44 3................5 Analysis of Results and Findings ................................................................................ 56 4..........................2 Seed ... 49 3....................2........................................................................................................................................... 99 Appendix B: Python code for Pushback Design with Capacity Constraint ................... 43 3....................1 Conclusion .............................................................................................2 Recommendations .5 Genetic Algorithms: Research Work in Mining ....................................................... viii 3............................ 46 3.................1 Introduction .............2 Generation of pushbacks with capacity constraint ................................................4................................4 Genetic Pushback Design with Capacity Constraint .............................................................................. 102 Pushback Design using Genetic Algorithms Western Australian School of Mines ............................................ 50 4.............................................................................................. 95 REFERENCES.......................... 99 Appendix A: Data Files used and Testing Results .............................3 Genetic Open Pit Optimiser (GOPO) .......... .... Best value graph for a price of $40........ 33 Figure 14............................................................................................................... 55 Figure 20..................................... Best value graph for a price of $80.................................................................. Initial selection of pushbacks designed using Whittle 4X ...................................................................... ............................................. Differences between classical and proposed open pit planning ................................. 75 Figure 42................................................................................. Optimum pit obtained with a price of $30 .. 30 Figure 11.............................................. Pushback design process .................................................................................................................................................................................. A production plan scheduled by pushbacks ......... 17 Figure 4....................................................................................................... 80 Pushback Design using Genetic Algorithms Western Australian School of Mines ........................................... 59 Figure 24................ 27 Figure 9.................. Simple Genetic Algorithm ................................ Best value graph for a price of $20......... 45 Figure 17.......................................................................... An arc .. Hypothetical orebody model ........................................ 62 Figure 27............................................................................. Chromosome representation ...................................... 58 Figure 23........... 54 Figure 19......................................................... Pushback schematic in plan and section views ............. Optimum pit obtained with a price of $80 ........................................................................................................................................................................................... 62 Figure 26................................................................... 68 Figure 31........................................................... A1 and A2 for the waste required to be stripped...... 16 Figure 3.... Optimum pit obtained with a price of $60 ..................................... A simplified view of pit shells or pushbacks .... Chaining of arcs............................................................................ 57 Figure 22.......................................................................................................................................................... Whittle 4X pit by pit graph with modified revenue factors ....... 69 Figure 32........................................ Optimum pit obtained with a price of $70 ............ 57 Figure 21.......................................................................................................................................................... The Genetic Open Pit Optimiser (GOPO) ............................................................................................................................................................................................... 71 Figure 36......................................................... 13 Figure 2.......................... An illustration of an Ultimate Pit Limit .......... 80 Figure 50...................................................... Initial Pushback i ............................ 26 Figure 8........ 24 Figure 6........................................... Whittle 4X specified case after initial pushback selection ............................................... 75 Figure 43.................. Best value graph for a price of $90............ Best value graph for a price of $30........... The Phase I pit with haulage road ........................................... Calculation of the Areas. An illustration of how the capacity penalizing function works ........................ 74 Figure 40........................................................... 78 Figure 47...... 43 Figure 15............................. Circular reasoning in pit design ...................................................................................................................................... Pushback i + 1 ...... 44 Figure 16...................................... Best value graph for a price of $100............................. 70 Figure 34........................ 23 Figure 5..................................................................... 78 Figure 48................................................. 47 Figure 18...................... Best value graph for a price of $50......... Effect of price on best value .............................................................. 26 Figure 7............ Reparation mechanism for crossover.... ix List of Figures Figure 1........................................ Best value graph for a price of $10.......................... 66 Figure 29............. 77 Figure 46.................................................................. Optimum pit with a price of $20 ................................................................................................................................. Optimum pit obtained with a price of $100 .... Final stage of the search by the 3-D Lerchs-Grossman algorithm ............................. 79 Figure 49......................................................... 31 Figure 12........................ Optimum pit obtained with a price of $50 ............ 74 Figure 41............ Check-Chromosome: Propagation from random position .............................................................................................. 71 Figure 35... Start of the search by a 3-D Algorithm ....................................... Optimum pit obtained with a price of $40 .. Optimum pit with a price of $10 ................. 27 Figure 10... 61 Figure 25............................ 76 Figure 44.................... The Struggle Genetic Algorithm ............. Best value graph for a price of $70.............................. 64 Figure 28.......................... Ore grade representation for the 2D hypothetical orebody .......................................................................................... Optimum pit obtained with a price of $90 ......................................................... 73 Figure 39...................................... 72 Figure 37................ 77 Figure 45.............................. Best value graph for a price of $60................. Steady-State Genetic Algorithm . 73 Figure 38............................................................................................... 67 Figure 30..................................... 32 Figure 13.... 70 Figure 33............................................................. Pushbacks or nested pits ................ ................................................................................................... Best value graph for Pushback 6 (Final pit) .................................................. Variation of best value with the corresponding number of generations ............... 89 Figure 64............................................. 87 Figure 60................. x Figure 51............................................................. 91 Pushback Design using Genetic Algorithms Western Australian School of Mines .................................................................................................................................................................................................... 88 Figure 62..................... 83 Figure 55.................................................................................................................................................................................................................................... Optimised pit with 6 Pushbacks ........................ Pushback 6 ..... 86 Figure 58........... Pushback 3 .......................................................................................................................................................................................................................... 90 Figure 65......... Best value graph for Pushback 4 .......... Best value graph for Pushback 1 .................... 83 Figure 54...................................... 84 Figure 56........................ Variation of best value with capacity penalizing factor.. 89 Figure 63............................................... Best value graph for Pushback 5 ................... Pushback 5 ..... 86 Figure 59........................................................................................................................................................ Final pit when mined out ....... 82 Figure 53............ Pushback 2 .......... 82 Figure 52.................................................. 85 Figure 57. Pushback 4 ................................ Pushback 1 ..................... Variation of best value with capacity ................................... Best value graph for Pushback 3 ............................................................ Best value graph for Pushback 2 ................ 90 Figure 66.......................................................................................... 91 Figure 67.......................................... 87 Figure 61........................ ................... Best value for a price of $40 ............................................................................................................... 99 Table 19.......................................................... Data file for final testing of the genetic model ................................................... Best value for a price of $50 ................................................... 75 Table 8........................................................................................ Data file for validation purposes ................................................ 74 Table 7....... 78 Table 10.......... xi List of Tables Table 1................................................................. 73 Table 6.................................................................................................................... Results for variation of best value with price............. 83 Table 13................................................. Results for Pushback 5 ..................................... Best value and number of generations .................................. Results for variation of best value with capacity ................................................................................................................................................. 101 Table 22........................... Results for Pushback 6 ........................................................................................................... Best value for a price of $80 ........................................ Best value for a price of $70 ..... Results for Pushback 2 ............................................................................. Best value for a price of $10 ........................................................... 87 Table 16.......... 76 Table 9............... Best value for a price of $60 .......... Best value for a price of $90 ................................................................................................................................... 68 Table 2............................................................ 81 Table 12............. Best value for a price of $100 ...................................... 85 Table 15.............................................................. Results for Pushback 4 ............................ Results for Pushback 1 ...... 99 Table 18................................................................................................................... Best value for a price of $30 ........................ Results for variation of best value with capacity penalising function .. 72 Table 5............. 100 Table 21... 100 Table 20............................................................................... 79 Table 11....................................................... 69 Table 3................... Results for Pushback 3 ........................................................ 71 Table 4......................................................................................................... Best value for a price of $20 ..................................................................... 88 Table 17...... 84 Table 14......................... 101 Pushback Design using Genetic Algorithms Western Australian School of Mines . Chapter One INTRODUCTION Pushback Design using Genetic Algorithm Western Australian School of Mines . the Lerchs- Grossmann stands out as a classic algorithm that the mining industry has reliably used in open pit optimisation and generation of pushbacks. the Lerchs-Grossmann (1965) which is a graph maximisation algorithm has found greater application. pushbacks describe how a pit will expand as the value of the recovered mineral increases. The ability to achieve the highest NPV resulting from production schedules is to a greater degree dependent on the extent to which pushbacks with capacity constraint can be designed. limited by the selection of pushbacks which yields an unconstrained size of pushbacks.1 Research Background In open pit mining. The underlying factor in the design of pushbacks is to generate a series of nested pits that conforms to not only the minimum required pushback width but provide the highest dollar value for a particular pit size. The algorithm is the core of Whittle (1999) optimisation package which is most widely used in the mining industry. Pushbacks thus play a very important role in mine design and optimisation. A great work of research has been undertaken in the area of open pit optimisation particularly the determination of the optimum pit limit (or final pit shell). In recognition of the many techniques that have evolved since the 1960s. The generation of pushbacks that should produce schedules with possible highest net present value (NPV) is however. 2 INTRODUCTION 1. The final pit outline or optimised pit shell must in the final analysis reflect the profitability of the mine investment. Once the ultimate pit limit has been determined what follows is production scheduling. Pushbacks are nothing more than a sequence of pit limits based on alternative economic scenarios. pit optimisation is the heart of the mine investment. Of the many algorithms that have evolved since 1965. As far as this study is concerned. Production scheduling is based on the underling pushback mine sequencing which is widely used for long-term production planning. The progression of pushbacks or nested pit shells roughly corresponds to the optimal evolution of the mine over time. Simply put. Pushback Design using Genetic Algorithms Western Australian School of Mines . Pushback Design using Genetic Algorithms Western Australian School of Mines . It will be appreciated. takes the biological environment as neutral in the sense that the concepts can be taken advantage of in other spheres like mining. These best and fittest design solutions may otherwise be difficult to find using other techniques. production planning. the background of the research is given highlighting the problem statement. Among these issues. There are some other areas in mining where genetic algorithm has found application but have not been cited in this manuscript. The first akin to the traditional biological view. research objective. The chapter then provides a definition or description of pushbacks recognising the attributes of nested pits and goes further to look at some of the pushback design techniques with the classical Lerchs-Grossmann taking the centre stage of discussion. and production sequencing and scheduling have been cited. though. open pit optimisation. In Chapter Three. The second lens focuses on the broader range of applications of genetic algorithm to the mining industry. It is through these aspects that the significant role played by a pushback is brought out. The Genetic Model for Pushback Design with Capacity Constraint is a computerised search and optimization method that work very similar to the principles of natural evolution. the concept of Genetic Algorithm (GA) is dealt with in two lenses. The project research is thus divided into five chapters. It has thus been endeavoured in this study to demonstrate the Whittle optimiser using the genetic model for pushback design with capacity constraint. the scope of the study and introduces the pushback design tools. 1975) has been chosen as the underlying heuristic optimisation concept that may be able to enhance the Lerchs-Grossmann algorithm particularly the design of pushback with capacity constraint. Literature review has been discussed in the second chapter focussing on a range of issues that are vital to mining and are dependent on the design of pushbacks. 3 genetic algorithm (Holland. In this chapter. The intelligent search procedure of genetic algorithm finds the best and fittest design solutions based on Darwin’s survival- of-the fittest principles. that a list of the applications of Genetic Algorithm to mining given in the third chapter may not be exhaustive. It is against this background that the objective of the study has attempted to propose a genetic model for pushback design with capacity constraint with a goal to achieve an optimised pit that yields pushbacks which are ease to sequence and schedule. farfetched with the current algorithms available. Pushback Design using Genetic Algorithms Western Australian School of Mines . The tool is however. The design of pushbacks for ease sequencing and scheduling should be governed by optimised pushbacks which yield schedules with the highest NPV possible.3 Scope of the Study The scope of the study is to gain understanding of the concept of GA and use Python software – the available programming tool to enhance the generation of incremental pits that may result in a robust pushback design. Nevertheless. The objective of the research is the outcome of an intention to expand current genetic optimiser for robust pushback design as suggested by Saavedra (2009). The last chapter provides the conclusion and recommendations of the study as drawn from the findings in Chapter four. 1. This requires a development of a tool that can guarantee the achievability of an optimised pushback. The pushback design using genetic algorithm in this research is applicable to a hypothetical 2D deposit. The solution for the genetic model has the capability of handling multiple scenarios of capacity constrained pushbacks. 1. an endeavour to develop a genetic model for pushback design with capacity constraint has been proposed in this research study as the solution for optimised pushbacks that can be easily sequenced and scheduled. 4 Chapter four is the core of the research project. It is envisaged that further future research of pushback design using genetic algorithm will be applicable to a 3D real life deposit.2 Problem Statement and Research Objective Pushbacks play a very important role in open pit mine design and optimisation. The analysis of results and findings given in this chapter also alludes to how the Whittle Optimiser has been emulated for an enhanced optimum pit. It details the concept of the Genetic Model for Pushback Design with Capacity Constraint. Production scheduling is based on the underling pushback mine sequencing which is widely used for long-term production planning. powerful programming language which has efficient high-level data structures and a simple but effective approach to object-oriented programming. they use the same combination of selection. 2001). mutation. selection. 1. 2009) is an easy to learn.4 Design Tools The design tools in this research evolve within the spheres of Genetic Algorithm and Python. scripting. that a brief discussion of these design tools in this section is vital to this research project. Pushback Design using Genetic Algorithms Western Australian School of Mines . GAs are a particular class of evolutionary algorithms (EA) that use techniques inspired by evolutionary biology such as inheritance. not the intention of this study to go into the in-depth of Genetic Algorithm and the ingenuity of Python.1 Genetic Algorithms A Genetic Algorithm is a search technique used in computing to find exact or approximate solutions to optimisation and search problems. which is superior to any other design or solution. though.2 Python programming language This sub section does not attempt to give a comprehensive or cover single features used in Python. numeric computing and system testing. 5 1. Genetic algorithms are categorized as global search heuristics. recombination and mutation to evolve a solution to a problem. 1. Python (Guido & Fred. These best and fittest design solutions may otherwise be difficult to find using other techniques. It is in the author’s view. Genetic algorithms are attractive in engineering design and applications because they are easy to use and they are likely to find the globally best design or solution. The intelligent search procedure of genetic algorithm finds the best and fittest design solutions based on Darwin’s survival-of-the fittest principles. It merely gives reference to Python as the available programming language that has been used in this project research. As genetic algorithms are a way of solving problems by mimicking the same processes that Mother Nature uses. It is however. Python is an interpreted interactive object-oriented programming language suitable (amongst other uses) for distributed application development. and crossover (Wales.4.4. it is encouraged! Pushback Design using Genetic Algorithms Western Australian School of Mines . 6 Because Python’s elegant syntax and dynamic typing. According to Guido and Fred (2009). makes it an ideal language for scripting and rapid application development in many areas on most platforms. it was felt that it would easy the enormous work required to actualize the programming of a pushback design with capacity constraint using genetic algorithm. Making reference to Monty Python skits in documentation is not only allowed. together with its interpreted nature. the language is named after the British Broadcasting Corporation (BBC) show “Monty Python’s Flying Circus” and has nothing to do with reptiles. 7 Chapter Two LITERATURE REVIEW Pushback Design using Genetic Algorithms Western Australian School of Mines . 2. 2006). and • Ore or waste haulage. pit optimisation is the maximisation of the net present value for a given ore deposit subject to a number of mining and economic constraints (Cardu et al. Understandably. and equipment. The size and shape of the pit depends on economic factors and design or production constraints. an effort has been made to point out the significance of pushbacks in mine design and why a pushback design using genetic algorithm could be the solution to open pit optimisation. the distribution of ore within the orebody. such as the geometric outline of the orebody. depends upon the choice of mining ratio. 1992). production rates. • Ore blending and stockpiles. 2008) of optimisation in open pit mine planning include: • Pit optimisation and pushback generation.2 Open Pit Optimisation Open pit optimisation is the determination of the ultimate pit or optimal pit limit for a given deposit under given set of mining and economic constraints (Schofield & Denby. • Open cut production scheduling. 1993). • Cut-off optimisation. Pushback Design using Genetic Algorithms Western Australian School of Mines . maximum allowable slope angles. however. the question that requires to be answered is “how does a pushback with capacity constraint fit in the process of pit design?” In this regard. Before the aspect of the design of a pushback using genetic algorithm can be dealt with. all of which are determined by the mining engineer (Hartman. 8 LITERATURE REVIEW 2. topography. The economics of the mining program. etc. The principal common applications (MEA.1 Introduction There are several stages in which open pit design can be conducted. The most economic final pit design often depends on factors that are largely outside the mining engineer’s control. The economic factors and design/production constraints determine the shape and size of the pit. There are several methods that are available to optimise the design of a pit. among these applications is pit optimisation and generation of pushbacks. This is assuming that all the other factors are kept constant. Drawing our attention to computer methods. the metal content. Whittle 4X is an implementation of the Lerchs-Grossmann (1965) algorithm. equipment selection. However. however. On the other hand a reduction in price would result in a reduced pit size. In the final analysis the optimal pit outline is defined as the one with the highest dollar value. mine planners who must recommend mine plant size. computer methods and computer assisted handy methods. Whittle 4X (1999) using a graphical method. 1992). we find the industry’s acclaimed optimisation software. Open pit planning itself is dependent upon the interaction of contributing factors that lead to maximising the NPV. Pit optimisation using Whittle 4X is an example of the broad scope of understanding that is required when implementing open pit optimisation. The ultimate pit limit design techniques evolve around the hand/manual methods. and the associated amount of waste to be moved during the life of the operation (Kennedy. 1990). and long-range scheduling are usually faced with the problem of how to optimise an ore deposit not only in terms of mechanical efficiency but also in project life (Hartman. Pushback Design using Genetic Algorithms Western Australian School of Mines . It can increase the value of a pit and can also be used to reduce the corporate risk involved in mining. pit optimisation is a tool that can greatly speed and ease the process of pit design. When used properly. Of interest. 9 It is not the intention of this study to exhaust the outlined optimisation applications. The limits of the pit must be set as an initial step in long or short-range mine planning so that the ultimate pit limit will represent the maximum boundary of all material within the pit. 1990). Pushbacks are generated within the ultimate pit. These methods try to find a list of blocks which has the maximum total value while still obeying the slope constraints (Kennedy. The mineable material becomes that lying within the pit boundaries. The limits thus define the amount of ore mineable. The available optimisation methods attempt to find the optimal pit outline in terms of a block model. The pit would increase in size if there is an increase in metal price. the mining sequence must be decided upon and then conceptually mine out the pit. it is possible to choose which pit shells are to be used as pushbacks and this is achieved with the aid of a pit by pit graph. In calculating the dollar value. the pit shells or nested pits are defined using revenue factors. • Economics. geomechanics. Once a series of pit shells has been defined. 1990) The impact of the following must however. To this end. interrelated and dependent upon one another. it has been endeavoured in this research to define or design pushbacks with capacity constraint using genetic algorithm in the quest to enhance the optimality of pits. be considered when optimising using Whittle 4X: • The geologic model. equipment selection. progressively accumulating the revenues and costs along the way (Kennedy.3 Production Planning Mine planning is dependent upon the interaction of contributing factors that lead to maximising the NPV. and long-range scheduling is how to optimise a mine property not only in terms of mechanical efficiency but also in project life. it should be realised that the Lerchs-Grossmann (1965) algorithm produces non-optimal pits as the selection process of pit shells to be used as pushbacks does not guarantee the optimality. It entails that the ultimate ore reserves cannot be calculated with the lack of knowledge of the Pushback Design using Genetic Algorithms Western Australian School of Mines . Thus encompassing reserve analysis. economics and mine design. • Cut-off grades selection. The problem facing planners who must recommend mine plant size. processing. however. • Pit slopes and capacities on the mine plan. and • The project’s feasibility. 10 When optimising using Whittle 4X. 2. Although Whittle 4X is the mining industry’s main open pit optimisation software package. Any evaluation of a mine property involves bringing all these parameters together and all the several parameters are. The orebody can also be mined in a way that minimises the cost per unit of metal to be produced. an insured incorporation of sufficient exposed ore to counter this possibility which is particularly true in the early years of a pit cannot be guaranteed. pushback design and sequencing. 11 cutoff grade and without the ore reserves. Unless a robust pushback is designed. So critical to the economic success of a pit is also the need to counter the possibility of miss-estimation of the ore tonnages and grades in the reserve model (Hustrulid & Kuchta. 2006). As cited by Hustrulid and Kuchta (2006). and without the required capacity-generated capital and operating costs. • Long-range production planning: This type of planning supplements pit design and reserve estimation work. Similarly. the cut-off grade and total ore reserves cannot be derived (Hartman. In the long-range production planning for an open pit. Well designed and sequenced pushbacks will ensure deferred waste stripping requirements are met and at the same time provide smooth equipment and manpower build-up.4 Production Sequencing and Scheduling It cannot be over-emphasised that production scheduling is an important facet of mine production planning and can also not go without comment if the essence of a pushback design remains the core of this research project. the final pit limits cannot be established. without any knowledge of the overall tonnages. Once the mill feed grade has been established. the production schedule cannot be selected. It is also an important element in the decision making process as it is done for feasibility or budget studies. maximising the NPV of a property becomes highly dependent on Pushback Design using Genetic Algorithms Western Australian School of Mines . 2. the mine life is dependent upon the determination of production rate. The above two kinds of planning show that planning is an ongoing activity throughout the life of a mine. Couzens (1979) gives two kinds of production planning which correspond to different time spans: • Operational or short range production planning: This type of planning is necessary for the function of an operating mine. 1992). • Man-hour requirements per month. • Location of ore and waste mining areas. a distinction between scheduling and sequencing is vital to this sphere. and sale of some commodity from an ore deposit (Kennedy. Initial production scheduling is based on marginal analysis of ore reserves. operational costs. medium and long term horizons. 2009) refers to the order in which mining blocks are removed from the ore body while scheduling involves specific time when mining activities occur. he further stresses that the procedure used to establish the optimal mining schedule can be divided into three stages. Typical mine schedules may include: • Monthly ore and waste quantities. 1990). • Truck cycle times. the objective of production scheduling is to maximise the net present value and return on investment that can be derived from the extraction. The process of scheduling is complex involving tabulation of quantities. The purpose of scheduling is to determine how the production of the sealable mineral product will be achieved over time. 1992) and the pit design itself has to encompass pushbacks or nested pits. Sequencing (MEA. grades and other values over a time frame. therefore. Mine scheduling is an attempt to predict the future of mining activities in the short. and revenues. concentration. The second defines a cut-off grade strategy that varies through time and will be optimal for Pushback Design using Genetic Algorithms Western Australian School of Mines . the subsequent cash flows including capital requirements. It introduces the time value of money since cash flows will be realised at the time when the product is sold during the life of the mine. including the necessary pre-production development. 12 scheduling. • Machine hour requirements per month. It is scheduling that determines the mine life and. Kennedy doesn’t just end up with an objective of production scheduling. Notably. a haulage study based on a conceptual pit design and overall facility layout (Hartman. The first stage defines the extraction order or mining sequence. To underscore the point. • Average ore grade of the mineral. The first deals with the strip ratio associated with recovering the ore. Direct operating costs can be used to define a breakeven cut-off grade and strip ratio. Figure 1. and refinery will be optimal within the limits placed by logistical. A sequence of nested pits is defined in (a) and (b) shows pushbacks defined by selecting a subset of the nested pits. In order to develop an optimum production schedule. mill. a sequence or extraction order inside of the so-called ultimate pit must first be determined. but the objective of mine planning is to devise a strategy Pushback Design using Genetic Algorithms Western Australian School of Mines . financial. 1990). The bench-phases defined in (c) are then assigned a time period of extraction in (d). the sequencing of pushbacks takes precedence. Figure 1 provides an illustration of a production schedule by pushbacks. marketing. A production plan scheduled by pushbacks (Renaud et al. and other constraints. and the physical location of that ore in respect to availability through time. 2009) The extraction sequence depends on two subsets of parameters. 13 a given set of production parameters while the third defines which combination of production rates of the mine. the grade of that ore. The second subset of parameters consists of costs associated with starting and maintaining the whole operation (Kennedy. While all reasonable steps should be made to follow the optimal pit outline. Production scheduling guidelines provide the basis for alternative comparisons on which the highest NPV is determined. This affects the value of the mine because it determines when various items of revenue and expenditure will occur. The mine sequencing then assures a predictable mill feed (Hartman. • Delayed revenue may not eventuate-one of the risk factors. • Reclamation accountability. • Assuring adequate working room. 14 that will optimise the total investment and the centre stage of this strategy is nothing else but a capacity constrained pushback designed within the optimal pit outline. Proper sequencing is achieved through incremental pit design. the timely exposure of ore grade material hinges on the underlying pushback mine sequencing which is widely used for long-term production planning. When a pit is scheduled. Tentatively. Pushback Design using Genetic Algorithms Western Australian School of Mines . • Timely exposure of ore grade material. Hartman (1992) has categorised the following parameters to provide guidelines for production scheduling of operations: • Minimising preproduction costs. thus reducing the effective revenue. There are various reasons for this: • Delayed revenue may increase the need to borrow funds and pay interest. • Smoothing of the stripping ratios. 1992). mining sequences are established and then analysed to set the most logical development program. Each increment is directly related to mill requirements within specific time constraints. the sequence in which various parts of the pit will be mined and the time interval in which each nested pit is to be mined is planned. • Maximising production. This is important because today’s dollar is more valuable than the dollar that is going to be received or spend in a year's time. Of these outlined parameters. 1990). and equipment requirements are a minimum towards the end of the mine life. It is also unlikely that a balanced production of ore and waste can be achieved given a schedule which is entirely focussed on early ore production and waste deferment. from inside to out implying that the most valuable ore would be recovered as early as possible in the mine life thus maximising cash flow as waste stripping is deferred. The advantages include the operating working space availability. 1992). It greatly reduces the investment risk in waste removal for ore to be mined at a future date. all equipment working on the same level. Each pit shell is mined before starting the next one. however. This sequence relates to the Declining Stripping Ratio Method (Hartman. it involves mining the final pit. is that the overall operating costs are a maximum during the initial years of operation when maximum profits are required to handle interest and repayment of capital (Hartman. There are two limiting schedules between which lies an optimal production schedule. but it indicates the highest possible NPV that can be achieved in an ideal situation. The primary disadvantage of this method. bench by bench from top to bottom. Pushback Design using Genetic Algorithms Western Australian School of Mines . • Something unexpected may go wrong with the operation-another risk factor. This sequence occurs when each of the pit shells is mined one after the other basing on the assumption that the highest valued ore is first accessed in the initial pit. 15 • Delayed expenditure may reduce the need to borrow funds and pay interest. etc. thus reducing the effective expenditure. This schedule is thus in no sense a practical schedule. As for the worst case sequence. no contamination from waste blasting above the ore. the accessibility of the ore on the subsequent bench. The best case scheduling involves mining with many small pushbacks or cutbacks. 1992) where stripping is performed as needed to uncover the ore with the working slopes of the waste faces maintained parallel to the overall pit slope angle. The best case sequence obeys the Increasing Stripping Ratio Method (Hartman. It mines waste earlier than necessary and delays ore production. These schedules are the Best Case and the Worst Case (Kennedy. The drawback is that it is impractical for equipment manoeuvrability as the production benches tend to be narrow with depth thereby resulting in failure to meet production needs. Figure 2 is an illustration of an ultimate pit limit and Figure 3 shows the corresponding pushbacks or nested pits. Therefore. These units are commonly called sequences. the planned pit is large and simple top-down mining would require extensive stripping with associated capital requirements and would probably not provide adequate ore delivery (Crawford. In practical planning process the first step is to break the overall pit reserve into more manageable planning units. been the intention of this study that a pushback design with capacity constraint using genetic algorithm may meet the aspirations of achieving the maximum NPV as well as the ease with which sequencing and scheduling can be done. 16 1992) and the result is lower NPV as compared to the best case scenario. but produces the lowest possible NPV. slices nested pits or pushbacks (Hustrulid & Kuchta. thus. phases. This sequence is usually practical. Thus pushbacks are characterised by a series of intermediate pits. mine planners usually develop the deposit in a series of phases or pushbacks which strip the waste and separate the ore in a more manageable way. In a nutshell. Achievement of this highest NPV can only be realized from well designed pushbacks. an open pit mine is rarely mined to the ultimate pit boundaries. 2. expansions. 2006). 2008) Pushback Design using Genetic Algorithms Western Australian School of Mines . Figure 2. working pits. the aim of production scheduling is to have a balanced production schedule with an NPV as close to the best case as possible.5 Attributes of Pushbacks It will be realised that from the onset of the project. 2001). In most cases. An illustration of an Ultimate Pit Limit (James. It has. factors affecting the design of pushbacks include the following: • The flexibility operation of machinery Flexibility operation of machinery generally favours narrower faces.5. This entails that choosing a wider mining face calls for the consideration of several other factors. the size of the equipment sets the minimum width. One must provide adequate operating room for maneuverability and mining flexibility in ore/waste segregation requirements (Hartman. This calculation is based on the loader or shovel turning radius. haul roads are added. since more working faces are available to provide feed to the mill. According to Crawford (2001). With more faces. Once the nested pits are generated and smoothed. 1992). bank angle. and they are used as pushbacks underlying practical plans from which yearly schedules are generated (Dagdelen. More numerous. truck width. allowing a wider selection of faces with differing mineral properties. Pushbacks or nested pits (James. 2. safety berms and some maneuvering area.1 Factors affecting pushback width As earlier alluded to. a localized problem at one face (such as equipment failure) is unlikely to disrupt production from other faces. Figure 3. narrower faces should allow simplified balancing of waste and ore Pushback Design using Genetic Algorithms Western Australian School of Mines . 17 The starting point for the design of a pushback is the calculation of a minimum mining width. 2008) Usually the incremental mining from the smallest pit to larger pit is referred to as pushback mining and there are cases where production is scheduled from more than one pushback simultaneously. 2001). 18 feeds. Initial mine plans (i. Pushback Design using Genetic Algorithms Western Australian School of Mines . permitting wider pushbacks. The time value of money must be considered in these cases as the mine may incur the cost of stripping sooner than may be needed. and failures may be more likely to occur. • The operating costs of the mine Equipment productivity is often higher with wider faces. Depending on the spatial distribution of the ore types. Wider pushbacks will have a greater vertical extent between pushbacks. narrow working faces may improve opportunities for the blending of ores. for feasibility work) often rely on wider designs for this reason.e. Wider pushbacks are often developed at the ultimate slope design limits. In tabular deposits where the stripping requirements are modest. and can result in lower operating costs. while in tabular deposits the cost of stripping the next phase is similar to the preceding phase. in vertically oriented deposits the cost of stripping the next phase becomes incrementally higher. since narrower pushbacks typically create shallower overall pit slopes with less vertical extent between pushback. • The deposit geometry and stripping ratio There is a close relation between the geometry of the deposit and its stripping ratio. • Slope stability Slope stability is a consideration as well. • The ease with which scheduling can be achieved The ease of scheduling can suffer with narrower working areas because mine planners are required to exercise the necessary caution in order to ensure that the upper levels under development do not overtake the lower levels. Wider pushbacks may be used to test the ultimate slopes early in the project life. It is argued that the high unit-value-per-tonne deposits often have high stripping requirements. and steeper slopes can be excavated. and care must be exercised to place differing sizes of equipment in the most advantageous areas. Equipment movements are more likely in narrow area designs. the time value component is less important. According to Crawford (2001). • The predictability of the ore zones One of the factors that are of considerable importance in pushback design is the continuity or predictability of the ore zones. erratic deposits can pose problems in that there is a possibility that the ore zone may not quite be where it was estimated to occur. little risk would be posed to the production planner. The local predictability of the deposit can be inferred from the variograms used in the resource model development. management will ideally rationalise all of the above attributes of pushbacks into a simplified. while more aggressive stripping can be pursued during periods of better metals prices. For example. Pushback Design using Genetic Algorithms Western Australian School of Mines . during periods of low metal prices. However. outside the scope of this research. narrow pushbacks may mean that there is a risk that the ore is not completely exposed by the stripping. policy. In smaller companies. As a result. The use of scheduling and optimisation programs in mine design will provide due consideration of the discount rate. pushbacks can be tightened to minimise up-front stripping costs. One practice is to consider cash conservation. deposit geometry and stripping ratio. for example. a deposit with predictable characteristics is unlikely to miss the target pay zone after the stripping is completed. 19 • The cost of capital at corporate level Crawford (2001) pointed out that companies that base their economic decisions using a high cost of capital should be particularly concerned about the use of narrow pushbacks in deposits requiring higher stripping ratios. In the larger companies. The detail of how this can be achieved is however. the ‘preferences’ are giving way to more sophisticated simulations of various mining strategies. • The preferences by management In the final analysis. such preferences may go unquestioned. with more extensive engineering (and financial) capabilities. as the staff is occupied with daily operations. but readily defined. Similarly for a well-defined disseminated deposit. All available methods attempt to find the optimal outline in terms of a block model that consists of a regular matrix of blocks in three dimensions. the stripping costs that result from the selection of a mining width can be determined and compared to other options. the more subjective factors must be addressed. 2. Ultimately. Due to mining parametric uncertainty. the judgment of the planner. their effects are real and must be considered in selecting the appropriate width for pushbacks. pushbacks are designed so that the deposit is divided into nested pits starting from the smallest pit with highest value per tonne of ore to the largest pit with the lowest value per tonne of ore. This progression roughly corresponds to the optimal evolution of the mine over time. 20 2. The design of the mining phases can be accomplished by rough manual approximation after review of the bench plans and cross sections. Starting from the equipment minimums. While the flexibility and operating cost effects are difficult to enumerate.2 Selection of the mining width While there is no definitive means to calculate the maximum mining width. and the use of the available analytical tools are the best means of reconciling the various trade-offs in the selection of the mining width.5. The pushbacks describe how a pit will expand as the value of the recovered mineral increases. Computer designed phases can be determined by feeding the data developed and stored in a computer block Pushback Design using Genetic Algorithms Western Australian School of Mines . or analytically by computer techniques. Following the analytical work. Within the ultimate pit limits. The different methods try to find the list of blocks which has the maximum possible total value. 2001). pit optimisation and pushback design still poses a challenge to the mining industry.6 Pushback Design Techniques Pushbacks are nothing more than a sequence of pit limits based on alternative economic scenarios. the engineer can develop a number of plans and determine both ore release and comparative NPV for various mine plans. while still obeying the slope constraints. These pushbacks are designed with haul road access and act as a guide during the scheduling of yearly productions from different benches (Dagdelen. the grade-control staff. some manual adjustments will be required. The following manual steps as outlined by Hustrulid and Kuchta (2006) are a series of steps in manual pushback design as provided by Crawford (1989a). 1. This is because the many efficient computerised techniques have been developed from heuristic algorithms that were conceived and birthed as a result of the need to improve on the manual way of design. The manual method is only a first step estimate and. 2. Start with the ultimate pit design: Develop a detailed data of ore grade and stripping distributions for various cutoff grades in zones around the designed pit circumference and in pit shell progression between the beginning surface topography (or pit surface) and the design pit limit. as well as the addition of haul roads out of a phase and if required.6. This is because high stripping reduces the net value of the recovered ore below the net value of medium grade ore in another area with much less stripping. Each surface has to be sufficiently spaced apart to allow adequate room for mining the slices between the surfaces. Pushback Design using Genetic Algorithms Western Australian School of Mines . It would be incorrect for example. therefore. Of particular interest should be locating high ore grade and low stripping zones on level plan maps and cross sections. Although manual planning methods are essentially trial and error approaches. 1990). The objective is to develop three dimensional equal profit potential surfaces throughout the mineral deposit. The data should include locations of ore zones (these vary with cutoff) and the impact of the difference between operating and ultimate pit slopes. 1990).1 Manual design of pushbacks Manual design methods depend on having an experienced engineer review the bench plans and cross sections through the deposit to visually pick out the higher grade targets that have reasonable stripping ratios. to first target high grade areas for mining having very high stripping ratios. 21 model into a set of programs that can be used to calculate an economic phase limit. it will be a great disservice to go un-discussed in this thesis. it will not be as accurate as a computerised technique (Kennedy. access left for the next phase (Kennedy. Since the distance between equal profit potential surfaces will vary. 22 2. Planning goals typically comprise one of the following: (a) maximum NPV economics; (b) provision of stable cash flow patterns; (c) uniform ore grade, grade decline curve, or high grading. Frequently high grading is used during the initial investment payback period. The level(s) of ore grade will be related to the cutoff criteria; (d) uniform stripping ratio or classic stripping ratio curve; (e) uniform total tonnage rate (ore + waste); (e) uniform or variable rate of product output. 3. Operating design criteria for pushback design: These include operating and remnant bench widths, slope between operating benches and roads, overall operating slope, road width and grades, and bench height. It is usually not uncommon to have typical values as follows: remnant bench width equal to bench height; pushback widths normally 60-150 m depending on size of pit and orebody characteristics; minimum pushback widths (single cut phases) about 24 m for small equipment and 40-45 m for 22-27 m (25-30 yd) shovels and 147-190 tonne (150- 200 ton) trucks; road width 15-24 m depending on width of equipment. Maximum road grade 8-12 percent. 4. Laying out of one to several pushbacks: Evaluating whether they satisfy the goals, individually and collectively, is a more or less cut and try process. Normally a pushback is laid out according to the operating criteria in plan and cross-section views (see figure 4). A useful tool is to make a scaled bench crest and toe pattern including operating bench and road widths. 5. The pushbacks are shown on plan view maps as a progression of bench level toes and crests from initial topography to ultimate pit limit (Figure 4). New levels are created as the pushback progresses at the pit rim and at the bottom. New bottom levels are established on the basis of minimum level size and ore grades. Normally, new bottom levels are encouraged by the need to hold stripping at reasonable levels. In addition to pit geometry, ore/waste interface lines for the selected cutoff must be plotted. Pushback Design using Genetic Algorithms Western Australian School of Mines 23 6. Calculations of volumes of ore and waste are done using a planimeter to measure areas, and the average grades within pushbacks are determined. The volumes of material to be removed from each bench are based on the average of the areas encompassed by the movements of the bench crest and toe from initial to new position by pushback, and the average bench height for the zone covered. Ore and waste volumes are calculated separately. The average ore grade is determined by averaging the block values within ore zones. In multiple pushbacks, a pushback serves as the initial location for a subsequent pushback. The calculated values are evaluated against the various goals for acceptability of the individual pushback or series of pushbacks. Figure 4. Pushback schematic in plan and section views (Crawford, 1989a) – an extract from Open Pit Mine Planning and Design (Hustrulid & Kuchta, 2006) 7. For plan view calculations, the planimeter is used to determine the areas of crest and the toe movements; commonly called crest to crest and toe to toe calculations. If the pit geometry is sufficiently regular, only the toe to toe calculations may be necessary. To achieve accurate results, the calculation of volumes for new levels at the pit rim or at the bottom, along irregular pit rim elevations, for roads, at the Pushback Design using Genetic Algorithms Western Australian School of Mines 24 ultimate pit intercept and for irregular bench heights require special planimeter techniques. The key is to divide up the volumes to be calculated into rectangular, parallelogram type solids with flat tops and bottoms and reliable average heights. The areas to be calculated must be closed polygons. The geometric layout prior to calculation is critical for accurate results. All the benches and roads must be described in the form of crests, toe, and average heights. Each bench and its related parts are calculated separately. The manual construction procedure involved in designing a pushback together with the layout of the main haulage road has been described in detail with the aid of an example as highlighted by Hustrulid and Kuchta (2006). Figure 5 is an illustration of the initial stage in pushback design. Figure 5. The Phase I pit with haulage road (Mathieson, 1982) - an extract from Open Pit Mine Planning and Design (Hustrulid & Kuchta, 2006) In order to accelerate the success of pit optimisation and pushback design, the mining industry can no longer afford to be limited to just any feasible solution usually obtained by trial-and error methods. An optimal solution that incorporates all the factors such as the metal price, cost, grade uncertainty, minimum width, safety in terms of slope angle, etc is required. Thus, as far as this study is concerned, genetic algorithm (Holland, 1975) has been chosen as the underlying heuristic optimisation concept. Pushback Design using Genetic Algorithms Western Australian School of Mines an arc from block A to block B indicates that. In their (Lerchs & Grossmann. Pushback Design using Genetic Algorithms Western Australian School of Mines . With this objective. 1965). it has been assumed that the concentration of ores and impurities is known at each point while the problem is to decide what the ultimate contour of the pit will be and in what stages this contour is to be reached. It should be appreciated in this manuscript that a full discussion of the procedure governing the Lerchs-Grossmann algorithm is beyond the scope of the study. the wall slope of the pit must not exceed certain given angles that may vary with depth of the pit or with the material. The objective then is to design the contour of a pit so as to maximise the difference between the total mine value of ore extracted and the total extraction cost of ore and waste. As shown in Figure 6. they recognized the sole restrictions posed by the geometry of the pit.2 The Lerchs-Grossmann Algorithm As published in what has now become a classical paper. The method works with only two types of information i. and 2. A simple dynamic programming algorithm for the two-dimensional pit (or a single vertical section of a mine). if A is to be mined.6. the block values and “arcs” (Lerchs & Grossmann.e. 25 2. block A may or may not be mined. A more elaborate graph algorithm for the general three-dimensional pit. An arc is a relationship between two blocks. then B must be mined to expose A and the reverse is not true. 1965) publication. Lerchs and Grossmann (1965) proposed two numeric methods of pit optimisation: 1. an attempt has been made to highlight the salient points of the algorithm. If block B is to be mined. However. An arc from A to B and that from B to C ensures that C is mined if A is to be mined despite there being no arc from A to C (Figure 7). Chaining of arcs (Lerchs & Grossmann. Figure 7. It uses no other information apart from that given by the arcs and thus it knows nothing about the positions of the blocks or indeed about mining (MEA. 1965) The Lerchs-Grossmann three-dimensional optimisation method achieves its aim by manipulating the block values and the arcs. Because arcs can chain. all the slope requirements are translated into a (large) number of block relationships in the form of arcs. During the optimisation process the algorithm flags each Pushback Design using Genetic Algorithms Western Australian School of Mines . An arc (Lerchs & Grossmann. 1965) In most optimisation packages. 2009). 26 Figure 6. there is no need to have so many arcs. As an illustration of the three-dimensional Lerchs- Grossmann algorithm. Start of the search by a 3-D Algorithm (Whittle. The detailed knowledge of the mathematics that is involved in the algorithm itself has been outlined by Whittle (1999). a two-dimensional block model shown in Figure 8 has been used to show the “start” of the search procedure and the “final” stage is given in Figure 9. A block is flagged to be mined if it currently belongs to a linked group of blocks that have a total value that is positive. 27 block that is to be mined and these flags can be turned on and off many times. Figure 8. 1999) – an MEA (2009) extract Figure 9. 1999) – an MEA (2009) extract Pushback Design using Genetic Algorithms Western Australian School of Mines . Final Stage of the search by the 3-D Lerchs-Grossman algorithm (Whittle. An increased interest in pit limit optimisation has also largely been driven by the success of the graph-maximisation Lerchs-Grossmann algorithm..6. The Lerchs-Grossmann method has been programmed by about four different groups. this issue has attracted the attention of research work because of its increasing relevance. Although the programs produced by these groups differ in their ability to handle slope constraints as well as the machines on which they can run. the underlying principle of the algorithm is the same. 28 2. Considering the block economic value equation. ………. thus.3 Pushback design using the Lerchs-Grossmann’s Whittle Optimiser Notwithstanding the challenge that pit optimisation poses to the mining industry. the issue at hand is discussed with reference to Whittle 4X which provides the fundamental graphical implementation approach that has found appreciable commercial use for pit optimisation in the mining industry. As regards the design of pushbacks using the Lerchs- Grossmann algorithm. It. (1) Where The above variables affect both the value of the blocks and the shape of the pit. sufficed for Whittle to reduce the number of economic variables in the Block Economic Pushback Design using Genetic Algorithms Western Australian School of Mines . Considerable advances in research work have been made since the 1960s leading to the implementation of the Lerchs-Grossmann heuristic approach as used in Whittle 4X. ………. As the revenue factor gets higher. When optimising using Whittle 4X. and . So.. pushbacks are generated by changing λ in each step while implementing the Lerchs-Grossmann algorithm on each converted block model. Whittle 4X works out the cash flow that would result in mining a block whether positive (for ore) or negative (for waste) using a scheduling algorithm. Hence. (2) is the amount of product that should be sold to pay for the mining of a tonne of material.. Equation (2) reduces to ………. the price goes up while the costs stay the same and there will eventually be a point where a block is worth mining because the revenue will be greater than the cost. The aim is to determine the logical progression of economically defined pits between the optimal point of initiating the mine and the ultimate pit. The following is a discussion of generating pushbacks using the Lerchs-Grossmann’s Whittle Optimiser (Whittle. With . one major and the other minor. for smaller revenue factors. the price is low and a block may not be worth mining because of the cost of removing all the waste to get at it. This is the only significant (major) variable. 1999). or a new mining or processing method is introduced (Whittle. The revenue factors are used to calculate different pit shells by varying the prices of the product but keeping the costs the same. Pushback Design using Genetic Algorithms Western Australian School of Mines . 29 Value equation to obtain two factors. 1999). the pit shells or nested pits are defined using revenue factors. (3) V being the dollar value per block. because is not expected to change significantly (thus minor) unless there is a significant change in one of the cost components. This was done by dividing Equation (1) by the cost of mining and removing a tonne of waste (Mc). Whittle 4X simplifies the sequencing and scheduling of pits by grouping all blocks on a bench and within the limits of a pit expansion into a panel. While the basic unit is the individual block. it is possible to choose which pit shells are to be used as pushbacks and this is achieved with the aid of a pit by pit graph (see Figure 11). that can be applied to mining the benches. making it easier to calculate a schedule (Whittle. As shown Figure 10. designated as ‘a’ through ‘t’ as indicated in Figure 10. eight pit shells have been defined. 1999) – an MEA (2009) extract Whittle 4X applies various scheduling heuristics to these panels to give some indication of the relative value of including a pit as a pushback in the scheduling process.b.c. Once a series of pit shells has been defined. A simplified view of pit shells or pushbacks (Whittle. It can be seen that there are many different sequences. Pits that account for a significant increase in volume may not have a significant influence on cash flow. Pushback Design using Genetic Algorithms Western Australian School of Mines .k.d. Figure 10. such as a.e.f.g or a.c. 30 The reason Whittle 4X does this is to create a series of pit shells from the small inner shells to the large outer shells that must be mined in sequence from inside to out and top down.h. 2008. cited in MEA. 2009).g.f. The sequence which is used will determine the cash flow from the project.b. 31 Figure 11. Whittle 4X Pit by pit graph with modified revenue factors (Whittle, 1999) Since each pushback represents a major stage in planning the pit, a relatively small number of pushbacks from among the pits should be selected. The goal is to keep the number of pushbacks as low as possible without losing the resolution necessary to maximise cash flow during the production scheduling process. Whittle 4X includes three different sequencing options that can be used as an aid in selecting pushbacks: the best case, the worst case and a specified case. All of these options use a set of user defined pushbacks. The best case sequence occurs when each of the pit shells is mined one after the other, as in the sequence a-f-b-g-k in Figure 10. Given the assumption that the initial pit accesses the highest valued ore first, this sequence should maximise cash flow by mining ore as early as possible while deferring waste stripping. The worst case sequence of mining occurs when the final pit is mined bench by bench regardless of pit as in the sequence a-b-c-d-e-f. This sequence mines Pushback Design using Genetic Algorithms Western Australian School of Mines 32 waste earlier than necessary and delays ore production. Figure 12 is an illustration of pushbacks selected in the initial stage in Whittle 4X. Figure 12. Initial selection of pushbacks designed using Whittle 4X (Whittle, 1999) Between the two extremes (i.e. the best case and worst case), a specified case (see Figure 13) can be applied which varies the number of panels mined within each pushback before proceeding to mine from the next pushback. The specified case will always have a discounted value somewhere between the best case and the worst case. In the final analysis, the set of pushbacks chosen should result in a discounted value as close to the best case as possible. Pushback Design using Genetic Algorithms Western Australian School of Mines 33 Figure 13. Whittle 4X Specified case after initial pushback selection (Whittle, 1999) Although ultimate pit limit determination can be performed using the many different mining software packages available, there is a similarity in the underlying principle of analysis governed by these packages. The procedure followed to generate nested pits is by changing the commodity price, costs or cut-off grades gradually from a low value to a high value. By changing the commodity price, for example, from a low value to a high value, one can generate a number of pits in increasing size and decreasing average value per tonne of ore contained in the pit. Since the smallest size pit contains the highest valued ore, the production is scheduled by mining the smallest pit first followed by the production in larger pits (Dagdelen, 2001). Pushback Design using Genetic Algorithms Western Australian School of Mines 34 Chapter Three GENETIC ALGORITHMS METHODOLOGY Pushback Design using Genetic Algorithms Western Australian School of Mines . 2001).1 Background of Genetic Algorithms According to Melanie (1998). This view coupled with the capability of the algorithms to evolve populations that are better suited to their environment than the individuals they were created from. These algorithms search or operate on a given population of potential solutions to find those that approach some specification or criteria. To find better and better approximations. mutation. 2003). Simply put. It has been acknowledged (Scott. just as in natural adaptation is the underlying factor for this research to explore the behaviour of capacity constrained pushback using genetic algorithms. selection. the algorithm applies the principle of survival of the fittest using techniques inspired by evolutionary biology such as inheritance. and crossover. genetic algorithm can be defined as a search technique used in computing to find exact or approximate solutions to optimisation and search problems (Wales. Scott (2004) stresses that genetic algorithms offer an interesting alternative to the typical algorithmic solution methods. Further development and formal introduction of genetic algorithms was during the 1970s at the University of Michigan in the United States by Holland and the algorithm was introduced in Germany around the same period by Rechenberg (1973). Genetic Algorithms (GAs) are search methods that employ processes found in biological evolution (Townsend. genetic algorithms operate on more than one solution at once making the algorithm good at both the exploration and exploitation of Pushback Design using Genetic Algorithms Western Australian School of Mines . As stochastic heuristic search method. It is envisaged that the diversity of genetic algorithms would be able to yield results that are well comparable to the Lerchs-Grossmann (1965) algorithm. 35 GENETIC ALGORITHM METHODOLOGY 3. 2004) that genetic algorithms are particularly useful for problems where it is extremely difficult or impossible to get an exact solution or for difficult problems where an exact solution may not be required. the invention of Genetic Algorithms as stochastic search method is accredited to Holland (1975) who together with his students discovered the algorithms during the 1960s at the University of Michigan in the United States. and are thus highly customizable. 36 the search space. The most significant differences between the more traditional search and optimisation methods and those of evolutionary and genetic algorithms are that these algorithms: 1. Use a set or population of points spread over the search space to conduct a search. rather than with the actual parameters themselves. but other encodings are also possible. not deterministic ones. 5. Use probabilistic transition rules. 6. In this case strings representing the binary values would be used. in the belief that it is the global optimum point. This provides genetic algorithms with the power to search spaces that contain many local optimum points without being locked into any one of them. the genetic algorithm would not deal directly with or values. This is one of the most powerful features of genetic algorithms. Are inherently parallel. For example if the minimum of the objective function is to be found. Instead it would work with strings that encode these values. 2. 3. 3. allowing them to deal with a large number of points (strings) simultaneously. many of the implementation details for using genetic algorithms with various data types was described by Michalewicz (1994). Are generally more straightforward to apply. Work with a coded form of the function values (parameter). Can provide a number of potential solutions to a given problem.2 Genetic Algorithm Structure The description of the structure in this section is a simplification of genetic algorithms as given by Townsend (2003). Traditionally. The only information a genetic algorithm requires is the objective function and the corresponding fitness levels to influence the directions of the search. 7. Whereas a comprehensive description of the basic principles at work in genetic algorithms was provided by Goldberg (1989). it can use this to continue searching for the optimum. solutions are represented in binary as strings of 0s and 1s. 4. Once the genetic algorithm knows the current measure of goodness about a point. Pushback Design using Genetic Algorithms Western Australian School of Mines . not just a single point on the space. If the optimisation criteria are not met. are then recombined in a way that guides the search to only the most promising areas of the state or search space. The application of genetic algorithm operators causes information from the previous generation to be carried over to the next.e. the creation of a new generation starts. but to be carried over. In some cases. where the parent already has a high fitness factor. Assuming chromosomes in the initial population. This initial population is then compared against the specifications or criteria (i. Usually at the beginning of a run of genetic algorithm a large population of random chromosomes is created. those with the highest fitness factor. This cycle is performed until the optimisation criteria are reached. Individuals are selected (parents) according to their fitness for the reproduction of the offspring. that is. The fitness of the offspring is then computed. A genetic algorithm has the ability to create an initial population of feasible solutions (or number of individuals). It starts with the generation of a random population of individuals (chromosomes) which are randomly initialised at the beginning of a computation. The fitness of the chromosome determines its ability to survive and produce offspring. each chromosome is tested to see how good it is at solving the problem at hand and a fitness score is assigned accordingly. Each one when decoded will represent a different solution to the problem at hand. Parent chromosomes are combined to produce superior offspring chromosomes (crossover) at some crossover point (locus). All offspring will be mutated with a certain probability. 37 8. The offspring are then inserted into the population replacing the parents. Each feasible solution is encoded as a chromosome (string) also called a genotype and each chromosome is given a measure of fitness (fitness factor) via a fitness (evaluation or objective) function. Thus the first or initial generation is produced. producing a new generation. the fitness) and the individuals that are closest to the criteria. it is better not to allow this parent to be discarded when forming a new generation. Mutation ensures the entire state-space will be searched (given enough time) and is an effective way of leading the population out of a local minima trap. Pushback Design using Genetic Algorithms Western Australian School of Mines . Test: if the end condition is satisfied. At each generation. The chance of being selected is proportional to the chromosome’s fitness (i.e. 3. Selection can be done with replacement meaning that the same chromosome can be selected more than once to become a parent. the best solution is returned. crossover and mutation until the new population is complete as is outlined below. the better the fitness. Replace: before the old generation can be replaced with the new generated population for a further run of the algorithm. • Mutation: with a mutation probability. and in turn new solutions are formed by breeding the best solutions from the population’s members to form a new generation (Scott. • Selection: select two parent chromosomes from the current population according to their fitness. The loop is formed by going back to step 2 (selection) if the end condition is not met. crossover the parents at a randomly chosen point to form two new offspring (children). an offspring is the exact copy of the parent. a new set of approximations is created by the process of selecting Pushback Design using Genetic Algorithms Western Australian School of Mines . mutate two new offspring at each locus (position) in the chromosome. A genetic algorithm is one of a class of algorithms that searches a solution space for the optimal solution to a problem. the first thing in GA is to create a new population by repeating the steps for selection. This search mimics the operation of evolution where a population of possible solutions is formed. 38 In summary. If no crossover is performed. the better chance to be selected). stop and return the best solution in the current population. genetic algorithms are search methods that can be used to solve search and optimisation problems over a period of generations on the basis of the principles of natural selection and survival of the fittest. it is important to accept and place new offspring in the new population. The population evolves for many generations and when the algorithm finishes. 2004). • Crossover: with a crossover probability.3 Fundamental Genetic Algorithm Operators As already been alluded to. 3. The particular genetic encoding of an organism is called its genotype while the resulting collective physical characteristics of an organism are called its phenotype. Reproductive fitness is a measure of how many surviving offspring an organism can produce. only organisms with high fitness survive. Natural selection is based on the fitness of each individual. 39 individual potential solutions (individuals) according to their level of fitness in the problem domain and breeding them together using operators as in natural genetics (Townsend. Pushback Design using Genetic Algorithms Western Australian School of Mines . It determines which individuals are chosen for breeding (recombination) and how many offspring each selected individual produces. 3. 2003). In genetic algorithms. just as in natural adaptation. Those organisms less adapted to their environment than competing organisms will simply die out. often encoded as a bit string. the more successful offspring it will create. The organism’s inherited characteristics are represented into a code.1 Selection Selection (Townsend. The process results in the evolution of populations that are better suited to their environment than the individuals they were created from. The fitness of an organism is typically defined as the probability that an organism will live to reproduce or as a function of the number of offspring the organism has (fertility). 2003) is the determining process by which variants are able to persist and therefore also which parts of the space of possible variations will be explored. The operations of genetic algorithms are used to modify the chosen solutions and select the most appropriate offspring to pass on to the succeeding generations. The term chromosome typically refers to a candidate solution to a problem. These operations include selection. Because of competition for limited resources. The better adapted the organism is to its environment. crossover and mutation. It is worth noting that search happens in the genotype. the main feature in this intelligent exploitation of a random search is called a chromosome. The fitness function drives the population toward better solutions and is the most important part of the algorithm. but selection occurs on the phenotype. given two parent chromosomes (Townsend. the above strings could be crossed over after the ninth locus in each to produce the two offspring: Offspring A: 10001001101000011 Offspring B: 01010001010010010 In the case of non-binary chromosomes. Mutation can occur at each bit position in a string with some probability and this value is usually very small.001 for binary encoded Pushback Design using Genetic Algorithms Western Australian School of Mines . Parent X: X1 X2 X3 X4 X5 X6 Parent Y: Y1 Y2 Y3 Y4 Y5 Y6 The swapping of genetic material between the two parents on either side of the selected crossover point.3. replacing the symbol at a randomly chosen locus with a randomly chosen new symbol). 2003) as: Parent A: 10001001110010010 Parent B: 01010001001000011 The crossover operator randomly chooses a locus (i. 1 becomes 0). say 0. say fourth locus.3. 40 3. It consists of flipping the bit at a randomly chosen locus (or for larger alphabets.e. 2003) is the chance that a bit within the chromosomes will be flipped (0 becomes 1. It is the chance that two chromosomes will swap their bits.2 Crossover Crossover typically consists of exchanging genetic material between two single chromosome parents. produces the following offspring: Offspring X: X1 X2 X3 X4 Y5 Y6 Offspring Y: Y1 Y2 Y3 Y4 X5 X6 3.3 Mutation Mutation (Townsend. The concept roughly imitates biological recombination between two single chromosomes where subparts of two chromosomes are swapped. a bit position) along the length of the chromosomes. If position 9 has randomly been chosen. For example. In addition. These methods operate in the problem space.4 Variations of Genetic Algorithm According to Matthew (1996) genetic algorithms operate independently of the problems to which they are applied. they require information about the optimisation in order to search for better optimisations. genetic operators typically operate in the space defined by the actual representation of a solution (representation-space. If 100% mutation was to take place. Genetic algorithms operate in the representation space. They care only about the structure of a solution. Consider the two offspring (individuals or children) from each crossover operation being subjected to the mutation operator in the final step to forming the new generation. the string 10001001101000011 could be mutated in its third position to yield 10101001101000011. then all of the bits in the chromosome would be inverted. the search algorithm is tightly coupled to the optimisation generator. The mutation operator randomly flips or alters one or more bit values at randomly selected locations in a chromosome. For example. but rather than operating in the space defined by the problem itself (the solution-space. or gene-space). Using Townsend (2003)’s words “selection” is the determining process by which variants are able to persist and therefore also which parts of the space of possible variations will be explored. and how the evolution should progress (the overall algorithm). or phenotype-space). which will survive to the next generation (replacement). not about what that structure Pushback Design using Genetic Algorithms Western Australian School of Mines . It should be noted that a sufficient level of genetic variety in the population is required for the entire solution space to be used in the search for the best solution and without the mutation operator enhancing the ability of the genetic algorithm to find a near optimal solution to a given problem. 3. 41 genes. genetic algorithms include other heuristics for determining which individuals will mate (selection). It has been pointed by Matthew (1996) that in traditional optimisation methods. The genetic operators are heuristics. the maintenance of this sufficient level cannot be guaranteed. If the optimum has not been found. A fast convergence of the population would stagnate the search to local optima whereas slower convergence would require a considerably longer time towards sub-optimal solutions (Fatos et al. and objective function. premature convergence. In order to maintain clusters of similar solutions. It is very unlikely once the population has converged in most cases to achieve further improvement. The research work on genetic algorithms has shown that a key issue in GAs is the convergence of the algorithm. 2006). Critical to the performance of a genetic algorithm is proper choice of representation and tailoring of genetic operators. Thus. The performance of each solution is the only information the genetic algorithm needs to guide its search. Although the genetic algorithm actually controls selection and mating. the genetic operators must effectively balance exploration and exploitation. then the convergence is. Many genetic algorithms appear to be more robust than they actually are only because they are applied to relatively easy problems. the representation and genetic operators determine how these actions will take place. By maintaining diversity in the population. therefore. some genetic algorithms introduce another operator to measure similarity between solutions. operators. otherwise the genetic algorithm will perform no better than a random search. All of the individuals typically end up with the same genetic composition after a population has evolved. The convergence of GAs is achieved by means of selection and replacement strategies and it is. by definition. Pushback Design using Genetic Algorithms Western Australian School of Mines . 42 represents. When applied to problems whose search space is very large and where the ratio of the number of feasible solutions to the number of infeasible solutions is low. It has been shown by Gruninger (1996) that in order for the genetic algorithm to be able to avoid both local minima/maxima in the global search and find small improvements in the local search. care must be taken to properly define the representation. premature. the algorithms have a better chance of exploring the search space and avoid a common problem of genetic algorithms. very important to carefully tune these strategies. the individuals have converged to the same structure. Figure 14. Figure 14 shows individual solutions represented by shaded ovals. These include simple genetic algorithm. the three variations of genetic algorithms discussed in this research are an outline by Matthew (1996). However. This is known to as elitism. Typical of this algorithm is that in order for the algorithm not to inadvertently forget the best that it found. 3. if the crossover operator does not maintain genetic material. the darker shading representing a better solution (Matthew.1 Simple Genetic Algorithm The Simple Genetic Algorithm (SGA) is a classical form of genetic search which uses non-overlapping populations. the best individual is more likely to be selected for mating. stead-state genetic algorithm and struggle genetic algorithm. the population will improve. The only “memory” the algorithm has is from the performance of the crossover operator since the entire population is replaced in each generation. On the other hand. Maintaining the best individual also causes the algorithm to converge more quickly. 1996) Pushback Design using Genetic Algorithms Western Australian School of Mines . such scattering will more often generate infeasible rather than feasible solutions.4. As an illustration of simple genetic algorithm. the population will not improve and the genetic algorithm will perform no better than a random search. If the crossover accurately conveys good genetic material from parents to offspring. A crossover operator that generates children that are more often unlike their parents than like them leads the algorithm to do more exploration than exploitation of the search space. The entire population in each generation is replaced with new individuals. In many selection algorithms. 43 Many hybrid genetic algorithms which combine hill-climbing. In search spaces with many infeasible solutions. repair and other techniques which link the search to a specific problem space exist. the best individual is carried over from one generation to the next. Simple Genetic Algorithm (Matthew. 1996). The distinction factor Pushback Design using Genetic Algorithms Western Australian School of Mines . the steady-state algorithm often converges prematurely to a suboptimal solution.3 Struggle Genetic Algorithm The struggle genetic algorithm was developed by Gruninger (1996) in order to adaptively maintain diversity among solutions. 44 3. At one extreme.2 Steady-State Genetic Algorithm The steady-state genetic algorithm uses overlapping populations. Since the algorithm only replaces a portion of the population of each generation.4. a portion of the population is replaced by the newly generated individuals. The amount of overlap (i. the best individuals are more likely to be selected and the population quickly converges to a single individual. Once again. Steady-State Genetic Algorithm (Matthew. Figure 15. a crossover operator that generates children unlike their parents and/or a high mutation rate can delay the convergence. only one or two individuals may be replaced by each generation (close to 100% overlap). the crossover and mutation operators are key to the algorithm performance. The understanding behind this process is illustrated in Figure 15. 1996) 3. the percentage of population that is replaced) may be specified when tuning the genetic algorithm. the steady-state algorithm becomes a simple genetic algorithm when the entire population is replaced (0% overlap).4. There is a similarity between the struggle genetic algorithm and the steady-state genetic algorithm.e. At the other extreme. In each generation. As a result. The Struggle Genetic Algorithm (Matthew. Figure 16 gives an illustration of the struggle genetic algorithm using a form of replace most similar. this only takes place if the new individual has a score better than that of the one to which it is most similar. Matthew (1996) argues that if allowed to evolve long enough. instead of replacing the worst individual. 1996) In his analysis of the previously discussed variations. Unlike the steady-state algorithm which uses a replace worst strategy for inserting new individuals into the population. both the simple and the steady-state algorithms converge to a single solution. The similarity measure indicates how different two individuals are. Pushback Design using Genetic Algorithms Western Australian School of Mines . Once the population converges in this manner. Conversely. eventually the population consists of many copies of the same individual. either in terms of their actual structure (the genotype) or of their characteristics in the problem-space (the phenotype). a new individual replaces the individual most similar to it. In other words. The struggle algorithm is similar to deterministic crowding and shares some characteristics of restricted tournament selection. This requires the definition of a measure of similarity (often referred to as a distance function). mutation is the only source of additional change. the struggle algorithm unlike other niching methods such as sharing or crowding (Goldberg & Richardson. Figure 16. a population evolving with a struggle algorithm maintains different solutions (as defined by the similarity measure) long after a simple or steady-state algorithm would have converged. As cited by Matthew (1996). Mahfoud. 45 is that. 1987. requires no niching radius or other parameters to tune the speciation performance. 1995). In recent years. if the crossover operator has a very low probability of generating good individuals when mating between or across the species defined by the similarity measure. Based on their simplicity. 1996).5 Genetic Algorithms: Research Work in Mining Optimisation deals with problems of minimising or maximising a function with several variables usually subject to equality and/or inequality constraints. the algorithm will fail. genetic algorithms have received considerable attention regarding their potential as a novel optimisation tool. 3. minimal requirements and parallel and global perspective. It plays a central role in operations research. the struggle algorithm maintains diversity extremely well. Since pit optimisation entails defining the ultimate pit limit for a particular deposit while taking into consideration the mining and economic constraints. management science and engineering design. Figure 17 Pushback Design using Genetic Algorithms Western Australian School of Mines . all of the constraints involved need to be harnessed into a system that does not detach production scheduling from the ultimate pit determination. like the other genetic algorithms. 1993) that leads to the most efficient optimisers producing sub-optimal pits as this aspect is usually ignored. ease of operation. 1995. Just like in the case of the human physiology where it is difficult to separate between bone and marrow. genetic algorithms are so far being widely researched to near application in a variety of problems in the mining industry. For example.1 Pit Optimisation There are many techniques that have been developed to produce “optimum” pit limits among them the Lerchs-Grossmann (1965) has found wider applications in the mining industry. the algorithm will perform only as well as a random search. 46 If the similarity function is properly defined. Much of the research work of genetic algorithms as related to mining problems has been conducted by Denby and Schofield (1993. 3.5. This is because there is a circularity reasoning (Schofield & Denby. These optimisers rely on the assumption that relates to the timing of extraction of individual blocks as pointed out by Schofield and Denby (1993). Many design problems are very complex and difficult to solve using conventional optimisation techniques. performance is tightly coupled to the genetic operators. However. mining being one of them. If the mutation rate is too high. Pushback Design using Genetic Algorithms Western Australian School of Mines . Schofield and Denby (1993) undertook a research that developed a system with the following objectives: • Produce pit design limits and extraction schedules optimised on NPV. Hence the development of “Generally Optimised Pit (GO-PIT)” by Schofield and Denby (1993). • Be flexible and extensible. It will be realised that practical application of the envisaged system to meet these objectives was achieved through the use of genetic algorithms. • Avoid all assumptions that relate to circular reasoning in design and scheduling. Circular reasoning in Pit Design (Schofield & Denby. • Be efficient in terms of processing-to run on a PC if possible. Figure 17. for example to enable use of underground situations or to optimise on other parameters such as risk. As a way of addressing the issues of circular reasoning. 1993) Pit optimisation is closely linked to scheduling. 47 illustrates the strong relationship between optimum pit limit determination and production scheduling. It has also been found (Schofield & Denby. cut-off grade. mining and processing costs. a set of possible pits that saw the production of future generations (pits) was the initial step of the genetic algorithm. A relatively large population of random schedules which also included a small number of highly constrained schedules was used as a strategy that successfully examined the importance of the initial population. which is one of the key advantages of the genetic algorithm approach. particular attention in the use of the algorithm was given to the pit fitness function which is the measure of the efficiency of a particular schedule. 48 In the research. prospering into future generations has high fitness values. In this case the fitness function carries out a discounted cash flow calculation for the simulated pit on the basis of a set of user definable functions that linked a number of factors such as grade. recovery factors. Also. In terms of reproduction from one generation to the next. discount factors and mineral market values. the fitness function is totally independent of the genetic algorithm operators. This being a major difficulty that arises when pits are produced or modified by one of the genetic operators. it is less likely in this way to have a good schedule that dominates the population quickly and reduce the chance of the system from converging quickly to a false optimum. This ensures that the pits are ranked in order of fitness and their probability of reproduction is mapped onto a linearly decreasing scale from high to low probability. It suffices to consider the size of the population because of its significant effect on the performance of the algorithm. the research saw the GO-PIT system having the potential to easily be customised for different situations. 1993) that the scheduling constraints get violated in the system. for example production tonnages during a scheduling period become greater than feasible or block precedence relationships being broken. The good schedules that usually survive. reproduction in GO-PIT is based on a linear normalization technique. extraction timing. Noticeably. Pushback Design using Genetic Algorithms Western Australian School of Mines . Thus. it is pointed out in GO-PIT that there is an intrinsic link between the chances of reproduction and the fitness value of chromosome. Thus there is a possibility of having either simple cost or revenue functions or highly complex fitness functions without modifying the basic optimising system and on this basis. 2 Mine Scheduling Research work in the use of genetic algorithms in mining scheduling is gaining a lot of ground. investigation of Open-Pit Design and Scheduling by Use of Genetic Algorithms (Denby & Schofield. 49 As regards risk incorporation. Recent work in this area involves “Open-Pit Design and Scheduling by Use of Genetic Algorithms (Denby & Schofield. Unlike a number of techniques and algorithms that separately concentrate on either the ultimate pit limit determination or extraction scheduling. There are several ways of defining the Pushback Design using Genetic Algorithms Western Australian School of Mines . 1993) and The Use of Genetic Algorithms in Underground Mine Scheduling.5. 3. With multiple working areas set-up in advance. operational costs. and revenues. also called GO-SCHED (Denby & Schofield. The individual stopes and panels require delineation before the data is fed into the system. therefore. 1993). It is an important facet of mine production planning and can also not go without comment if the essence of a pushback design remains the core of this research project. 1995)”. All what this entails is that the measure of the fitness function can be modified extensively without the principal parts of the genetic algorithm requiring any alteration. Scheduling determines the mine life and. the investigation of Genetic Algorithms in Underground Mine Scheduling (Denby & Schofield. Schofield and Denby (1993) stresses that the fitness function is decoupled from the optimising mechanism making it one of the highly promising features of a genetically based optimising system for pit design and scheduling. the approach by genetic algorithm is a highly efficient heuristic technique. With the recognition that the two fundamental problems facing the mining engineer are the determination of the pit limit and extraction scheduling. the genetic algorithm system works out the order of extraction that maximises the fitness function for a given production rate. 1993) has demonstrated that it is possible to combine pit limit and extraction scheduling simultaneously in order to maximise NPV. It offers the potential to transcend the circular reasoning difficulties of separate pit limit and extraction scheduling systems (Denby & Schofield. 1995) concentrates on the development of a prototype application of a GA to the problem of scheduling the output from a number of multiple production points. the subsequent cash flows including capital requirements. As regards underground mine operations. research in this area has been an ongoing thing. They underline the point that genetic algorithm method has the ability to identify optimal solutions as they apply the successive operations of reproduction. (1998) in their research on “Application of genetic algorithms to the optimisation of large mine ventilation networks”. (ii) underground booster fan pressure. application of genetic algorithms to ventilation systems is so far proving to meet this objective. diesel fumes and particulates and fumes from blasting. the three researchers point out to the fact that genetic algorithm determines the minimum airpower consumption of a given mine network by the selection of the best: (i) main fan pressure. The contaminants may include dust. and is thus readily reduced to a sub-standard (or even dangerous) condition if contaminants produced in the course of operations are not controlled. and mutation. and (iii) booster fan location in the network. In this piece of work. including sulphide dust ignitions.3 Optimisation of Mine Ventilation Systems Like in open pit optimisation. the versatility of genetic algorithms in mining can be attested to the discussed investigations carried out by a number of researchers. 50 fitness in GO-SCHED (Denby & Schofield. if not the most vital consideration when planning an underground mine. 3. as well as gases released from the rock strata.5. The optimal solutions that are found by GA are better than those obtained by the use of deterministic methods. aerosols. In a nut shell. Thus for each schedule the “fitness” is calculated. In the quest to find optimal solutions for large mines. it goes without saying that ventilation in an underground mine is of critical importance and one of. genetic Pushback Design using Genetic Algorithms Western Australian School of Mines . However. 1995) but it is usually defined as the NPV for long-term scheduling. design and analysis of airflow requirements for individual work places is usually a complex issue which requires an optimal solution. Of equal importance to the maintenance of a healthy working environment underground is the need to protect employees against the risk and the consequences of underground fires and unplanned explosions. or safely extracted or diluted to harmless levels. Correct design. In particular. implementation and maintenance of mine ventilation is therefore of fundamental importance. Because of its complexity. crossover. The atmosphere in an underground mine is limited and confined. This is evidenced by the findings by Yang et al. optimal solutions found will be able to reduce the uncertainty surrounding this industry. It is envisaged that as research work continues to investigate the capability of genetic algorithms in mining. 51 algorithms have proved to be a good alternative for solving a wide variety of hard combinatorial optimisation problems. Pushback Design using Genetic Algorithms Western Australian School of Mines . 52 Chapter Four GENETIC MODEL FOR PUSHBACK DESIGN WITH CAPACITY CONSTRAINT Pushback Design using Genetic Algorithms Western Australian School of Mines . These algorithms are implemented as a computer simulation in which a set of possible solutions randomly selected in the solution space as a population evolves towards better solutions. 4. et al. According to Saavedra (2009).. 2006) that abstracts the block model as a set of columns. the fitness of every individual in the population is evaluated. Pushback Design using Genetic Algorithms Western Australian School of Mines . parallel search algorithms based on the theory of natural selection and process of evolution (Zhang. m) are neighbours and xi. and modified (recombined and possibly mutated) to form a new population. j (t) denoting the length of the column extracted by time t. j) and (l. In each generation. for example: when column (i. the classical slope angle constraints have been specified by taking the differences between two neighbouring columns. 2006). multiple individuals are stochastically selected from the current population (based on their fitness). The new population is then used in the next iteration of the algorithm until an optimal solution is obtained. the use of this kind of an abstraction reduces the size of the problem as it actually converts to a two-dimensional (2D) one instead of a three-dimensional (3D) problem.2 The Genetic Design of an Open Pit Saavedra (2009) cites a presentation of a model for an open pit (Goodwin. Each possible solution is represented as a chromosome and evaluated for its fitness calculated using the objective function. This reduction is substantial and also there is no need to use integer variables in the model. The evolution which usually starts from a population of randomly selected individuals happens in generations. 53 GENETIC MODEL FOR PUSHBACK DESIGN WITH CAPACITY CONSTRAINT 4.1 Introduction Genetic algorithms are stochastic. 7% extraction 32.2. consists of determining the final height of each column. this could be interpreted as: 20% extraction 30% extraction 25% extraction 26.1 Chromosome representation The natural chromosome representation for the model is an array (one-dimensional for 2D deposits and two-dimensional for tri-dimensional deposits) that has on each cell the total percentage of extraction of the corresponding column (which is a number between 0 and 1). 4.5% extraction Pushback Design using Genetic Algorithms Western Australian School of Mines . For example. as the benefit obtained from the column depends on the amount of the ore extracted from the column. Figure 18. It suffices that each column has its own characteristic grade curve. 54 The problem however. Figure 18 shows appreciable differences between the actual models in use and the proposed. It does not depend on a definition of a block size which is the main problem of the classical models (Saavedra. Differences between classical and proposed open pit planning (Saavedra. 2009). 2009) It is noted that the proposed alternative provides more flexibility for the planning process. The only problem with this formation is that the resulting problem is non linear. for a 2D deposit a chromosome could be: And as shown in Figure 19. 2. 55 10% extraction 0% extraction Figure 19. If the vector is used in the initial stage and component i chosen (at random) to assign to it a value. The main idea is to generate a random path to visit the columns and assign a value to the column spreading the change to the rest of the deposit.2 Random generation of individuals As the chromosome structure is heavily constrained. The condition for i. the random creation of feasible individuals must be ensured. then it is natural to choose a random value for those columns uniformly distributed in the interval . For achieving that goal an algorithm similar to that of sequential simulation could be considered. consider the 2D case with one-dimensional array representation. Chromosome representation 4. implies in particular that and . then components and will be affected directly. To simplify the discussion. Pushback Design using Genetic Algorithms Western Australian School of Mines . j neighbours. otherwise the slope constraint is not going to be respected. It will then be required to increase or reduce all the values of the component at the interface and then propagate this change on the remaining part of the chromosome. [0 0. After the changes are propagated. in which case it does not require to be changed.3 Crossover of pushback chromosomes Crossover will take two parents and select a crossover point at random.2 0. For that purpose it is noted that any inconsistencies that exist will happen just at the interface or crossover point. the first child is consistent then this does not require modifications whereas the second is not. Then a repairing mechanism is required to preserve the feasibility of the individuals.1 0. 56 The next step in the generation of values is to adjust the neighbours of and . In this case. For a graphical example see Figure 20. it is required to proceed to visit a different component at random and execute the same process again until all the unassigned components have been visited. etc.2. In the case of a one-dimensional representation this will be a chromosome index. In the case of two- dimensional representation this will be a line (either a column or a row). For example. if the neighbouring component already has a value. then there are two possibilities: • The existent value is valid for the constraints.1] is a one dimensional representation.2 0.3 0.3 0.2 0.3 0. 4. the existing value should be increased or decreased until it lies in the range of validity and propagate the change to the neighbours and subsequently the neighbours of the neighbours. The next step is to exchange genetic material between the parents as usual. The second children will have to be modified only at the interface and it is noted that no further change is required to have consistency. If the neighbouring component does not have a value then the previous procedure is repeated. Pushback Design using Genetic Algorithms Western Australian School of Mines . • The existent value lies outside the constraint range.2 0. This propagation effect will spread through the chromosome. If for this example in Figure 20. 57 Figure 20. Pushback design process – synonymous to Go-Pit flowchart (Denby & Schofield. 1993) Pushback Design using Genetic Algorithms Western Australian School of Mines . 2009) 4. Reparation mechanism for crossover (Saavedra.3 Genetic Open Pit Optimiser (GOPO) The pushback design process has been illustrated in Figure 21. Figure 21. Attainability of the process illustrated in Figure 21 is subject to the fitness function. Figure 22. A different geometrical configuration in Figure 29 (see data files used in Appendix A) was however. A1 and A2 are the waste extensions required to be stripped in order to mine out A3 which is the valuable ore for the optimum pit. Each column has different grades at every depth (also refer to the data files used in appendix A). used later for the final experiment. Pushback Design using Genetic Algorithms Western Australian School of Mines . In Figure 23. The orebody width (W) has been subdivided into 10 equal columns of 10m width (w). this study has been conducted on a two dimensional (2D) orebody which is 100m by 100m in depth and width as shown in Figure 22. Figure 23 is considered in relation to the block economic value equation (see Equation 1). 58 An understanding of how the Genetic Open Pit Optimiser (GOPO) works for the design of pushbacks can be drawn from the determination of the fitness function. Initial testing was conducted in this geometry to validate the algorithm. Ore grade representation for the 2D hypothetical Orebody To establish the fitness function. As earlier mention. 1 in 1 slope angle or 45o). A3 is the area defining the total amount of ore for the optimum pit while on the cost side. (4) Where G is the optimum pit’s total average grade.1). A3 is the area associated with the cost of mining the ore. ………. When the random chromosome is initialised. θ (for example. The Genetic Open Pit Optimiser (GOPO) In terms of the Fitness function. On the revenue side. The first thing in GA is the creation of random individuals. it transforms into an individual chromosome of 10 columns: Pushback Design using Genetic Algorithms Western Australian School of Mines . A1 and A2 are the areas representing the cost of stripping the corresponding waste to uncover the ore in A3 subject to the pit slope angle. Once this is done chromosomes are assigned. The size is the number of columns.. R is the recovery and P the commodity price. It suffices to mention that the depth of the orebody is divided into step size of a tenth (0. 59 Figure 23. the equation for the block economic value equates to. if the initialised chromosome is the percentage depth of extraction for Ccolumn1 is extracted which is of the total depth extracted (i. Column 2 would be extracted or . 20% of 100m or 20m) or 20m of ore extraction. In the case of A1 and A2. For Column 2. Pushback Design using Genetic Algorithms Western Australian School of Mines . Similarly for: Column : 25% extracted or 25m of ore extraction Column : 25% extracted or 25m of ore extraction Column : 30% extracted or 30m of ore extraction Column : 40% extracted or 40m of ore extraction Column : 50% extracted or 50m of ore extraction Column : 40% extracted or 40m of ore extraction Column : 40% extracted or 40m of ore extraction Column : 35% extracted or 35m of ore extraction Column : 30% extracted or 30m of ore extraction The area for Column 1 is . Thus for the chromosome Column1 area equals 25m x w which is 25m x 10m.. 60 ………. The sum total area of the 10 columns must equate to A3. the area would be . (5) This means that the percentage depth of ore extraction for Column 1 is or of the total orebody depth . the area of each triangle is calculated with respect to the 45o slope angle. For example. The rest of the columns will follow a similar pattern.e. xi. 61 Figure 24. Calculation of the Areas.. (7) By increasing the commodity price in the fitness function. Thus ……….. (6) ………. A1 and A2 for the waste required to be stripped. Pushback Design using Genetic Algorithms Western Australian School of Mines . the optimum pit will become bigger and bigger replicating or emulating the Whittle Optimiser as developed by Lerchs-Grossmann (1965).D is the percentage depth extracted depending on the particular triangle with D being the 100m orebody depth. The progression of pushbacks or nested pit shells roughly corresponds to the optimal evolution of the open pit mine over time. Taking the Genetic Open Pit Optimiser and modifying the fitness function. 62 4. Pushbacks describe how a pit will expand as the value of the recovered mineral increases. Figure 25. the pit shell transforms from the shape shown in Figure 25 to the shape in Figure 26. Initial Pushback i Figure 26.4 Genetic Pushback Design with Capacity Constraint By definition. Pushback i + 1 Pushback Design using Genetic Algorithms Western Australian School of Mines . pushbacks are nothing more than a sequence of pit limits based on alternative economic scenarios. 4. the random creation of individuals is also modified since the chromosome has to take a starting seed (refer to Figure 27 and section 4.. The initial “pushback i” is more or less like the pit for the Genetic Open Pit Optimiser and the fitness function is equal to ………. ……….4. How Pushback Design using Genetic Algorithms Western Australian School of Mines . Pc For the final algorithm. 4. (9) Where is A3 extra or which is what is added to the previous topography Pi. 63 From Figure 25. for we create such that each one of these pushbacks is constrained by capacity (measured in tonnes) subject to capacity penalising function .1 Capacity penalizing function. (8) As illustrated in Figure 26. the new fitness function for the pushback with capacity constraint is: …. Thus.2). S0 is the starting point or the surface profile. For the implementation of the new fitness function. As illustrated in Figure 27 the capacity penalising function.. Pushback i (Pi) is the initial profile or topography for the first pushback. (10) where is a factor. is a conditionality that restricts the capacity of the pushback required from going beyond the target size.. Capacity Penalising function. The seed (Figure 27) in this case is the preceding profile or topography of the previous pushback. crossover and mutation are modified. Figure 27. 64 this works is that the new profile of Pi+2 can wander about in either direction (Figure 27) as much as it wants up to a certain extent but must respect the original pushback profile. Take for instance the starting profile is Where Si is the percentage of the column and when initialised the random chromosome yields Pushback Design using Genetic Algorithms Western Australian School of Mines . the chromosome has to take a starting seed as a parameter. An illustration of how the Capacity Penalizing Function (Pc) works 4.4. Bearing in mind that we cannot go back in time for we are constrained by pushback i if we fix a deficient chromosome.2 Seed As earlier alluded to. . the crossover is performed by randomly choosing a position at random and then propagating to both the left and right exchanging the genes of two parents at selected position (Figure 28). Bearing in mind that we cannot go back in time for we are constrained by pushback i if we fix a deficient chromosome.. 65 5 Columns Col1: 20% extracted Col2: 45% extracted Col3: 50% extracted Col4: 35% extracted Col5: 30% extracted Based on . a new pushback (P1. Pn) has to satisfy the condition: It is hence imperative to also modify crossover and mutation... Pushback Design using Genetic Algorithms Western Australian School of Mines . The crossover operation in this case will help to reproduce good offspring from two fit parents as a better offspring shall have more opportunities to reproduce even better offspring. P2. It adds diversity to the genetic characteristics of the population. This operation is aimed to introduce new material into the existing individual. Pushback Design using Genetic Algorithms Western Australian School of Mines . CHECK-CHROMOSOME: Propagation from random position As shown in Figure 28. 66 Figure 28. the condition has to be satisfied such that where the range of values is: for and Mutation is performed by randomly choosing a gene and re-encoding the same gene. The corresponding maximum number of generations was 100 for a size individual of 10. The variable input parameters included the commodity price. Pits of different sizes were generated with an increasing price.1. 10 runs (iterations) were performed to obtain the results and findings in sections 4. the code was written in Python (a script programming language) for the running of the program. Hypothetical Orebody model Pushback Design using Genetic Algorithms Western Australian School of Mines . the geometry for the final experiment was changed to show that the algorithm is capable of dealing with any depth or shape and not only with square deposits Figure 29 is a rectangular hypothetical orebody (see Appendix A for the data files used) of 100m by 120m depth and width used for the final testing of the genetic model. crossover rate of 0. the capacity and capacity penalizing factor. 67 4. 4. a size population of 200 was found to be appropriate to work with.8 and mutation rate of 0.5 Analysis of Results and Findings Following the genetic design of pushbacks as has been explained in the previous section. The best value together with the associated best fitted chromosome and the minimum number of generations were recorded.5.2 respectively.5. As the 100m by 100m depth and width hypothetical orebody was used for validation purposes. On testing the program.5. Figure 29. For this 2D hypothetical orebody under-study.1 Generation of Pits using the Genetic Open Pit Optimiser A demonstration of the Genetic Open Pit Optimiser in this section is an emulation of the Whittle Optimiser which works on the Lerchs-Grossmann algorithm.1 and 4. 837 10 5 5 22 100 -17 -17 Pit profile 0. A negative best value is an indication of a loss attributed to large costs outweighing the revenue as there is more waste to strip than is the ore to mine.0 0.0 2x10 0.0 0. 68 The tables and figures that follow show the testing results of increasing price from $10 to $100.837 (Table 1) and the graph in Figure 30 was obtained. Best value for a price of $10 BEST PRICE CAPACITY Min Max VALUE ($) ($) (t) Pc Generations Generations -14. Table 1.0 0.0 % extraction 0 0 0 2x10-15 0 2x10-15 0 0 0 0 Figure 30.0 0.0 0.0 2x10 0. Testing results with a price of $10 An input price of $10 yielded the best value of minus $14. Best value graph for a price of $10 Pushback Design using Genetic Algorithms Western Australian School of Mines . there is hardly any pit obtained with a price of $10.043 at a price of $20).0 0.1 0.043 (see Table 2 and Figure 32) was realised and a very small pit obtained as shown in Figure 33.0 0. Refer to Figure 50 for an observation of this trend.0 0.837 at a price of $10 to –$30.1 0.0 0. Table 2. 69 As can be seen from Figure 31.043 20 5 5 19 100 Pit profile 0. Although the pit is increasing with an increase in price. the best value is decreasing (moving from –$14. Figure 31.0 % extraction 0 10 0 10 0 0 0 0 0 0 Pushback Design using Genetic Algorithms Western Australian School of Mines . Best value for a price of $20 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) -30.0 0. the best value of minus $30.0 0 0. Optimum pit with a price of $10 Testing results with a price of $20 On running the program with an input price of $20. it is worth noting from Figure 35 that the pit is getting bigger and bigger as the price increases. Optimum pit with a price of $20 Testing results with a price of $30 The testing results with a price of $30 produced a graph shown in Figure 34. 70 Figure 32. Best value graph for a price of $20 Figure 33. However. The best value as read from the graph is getting smaller and smaller (Table 3). Pushback Design using Genetic Algorithms Western Australian School of Mines . 3 0.0 0.1 % extraction 0 10 20 30 20 30 20 30 20 10 Figure 34.442 30 5 5 25 100 Pit profile 0. Best value for a price of $30 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) -40. Best value graph for a price of $30 Figure 35. Optimum pit obtained with a price of $30 Pushback Design using Genetic Algorithms Western Australian School of Mines . 71 Table 3.3 0.3 0.2 0.2 0.2 0.1 0.2 0. When compared with the previous value for a price of $30.4 0.1 0. There is also a corresponding increase in waste to be stripped.3 0.2 0. we see that the best value has started to improve.777 40 5 5 9 100 Pit profile 0. The best value as read from the graph in Figure 36 and recorded in Table 4 is –$36.3 0.4 0. Best value graph for a price of $40 Pushback Design using Genetic Algorithms Western Australian School of Mines . we see the pit is getting deeper and deeper (Figure 37). Best value for a price of $40 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) -36.3 0. Table 4.4 0.777. 72 Testing results with a price of $40 When testing the program with a price of $40.2 % extraction 10 20 30 40 30 40 40 40 30 20 Figure 36.4 0. Table 5. Best value graph for a price of $50 Pushback Design using Genetic Algorithms Western Australian School of Mines .4 0.4 0. Best value for a price of $50 BEST PRICE CAPACITY Min Max VALUE ($) ($) (t) Pc Generations Generations -28.2 0.3 0.2 % extraction 20 30 40 40 40 40 40 40 30 20 Figure 38. 73 Figure 37.4 0.545 50 5 5 9 100 Pit profile 0.4 0.4 0.3 0. the pit (in Figure 39) is expanding gradually as the best value (Table 5 and Figure 38) is tending to increase. Optimum pit obtained with a price of $40 Testing results with a price of $50 With a price of $50.4 0. 4 0.853 60 5 5 11 100 Pit profile 0.4 0.4 0. Best value graph for a price of $60 Pushback Design using Genetic Algorithms Western Australian School of Mines .3 0. Table 6. Best value for a price of $60 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) -17.853 (Table 6).4 0.4 0.3 % extraction 20 30 40 40 40 40 40 40 40 30 Figure 40.2 0. Optimum pit obtained with a price of $50 Testing results with a price of $60 A further pit expansion (Figure 41) is seen with a price of $60 and the best value as depicted in Figure 40 has continued to increase to –$17. 74 Figure 39.4 0.4 0. 6 0. Best value graph for a price of $70 Pushback Design using Genetic Algorithms Western Australian School of Mines . Best value for a price of $70 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) -668 70 5 5 14 100 Pit profile 0. there is a sudden pit expansion as well as a tremendous increase in depth (Figure 43).5 0. A corresponding sudden increase of the best value depicted in Figure 42 is seen.6 0.4 0.6 0.3 0.6 0.4 0.5 0. Optimum pit obtained with a price of $60 Testing results with a price of $70 When an input price of $70 was used. 75 Figure 41.3 % extraction 30 40 50 60 60 60 60 50 40 30 Figure 42. Table 7. the best value has greatly increased from minus $668 (see Table 7) to positive $16.843 (see Table 8). However.4 0. Best value for a price of $80 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) 16. The positive best value as depicted by the graph in Figure 44 is a sign that at a price of $80.6 0. The strange shift from a loss making pit to a profit making pit without any increase in pit size can be well understood from Figure 50.3 % extraction 30 40 50 60 60 60 60 50 40 30 Pushback Design using Genetic Algorithms Western Australian School of Mines .6 0. the pit is now profitable.4 0.5 0. 76 Figure 43. Optimum pit obtained with a price of $70 Testing results with a price of $80 A close observation on the results obtained with a price of $80 indicates that there is neither any pit expansion nor an increase in depth (compare Figure 43 and Figure 45).6 0.843 80 5 5 14 100 Pit profile 0. Table 8.5 0.3 0.6 0. It is observed also that the waste to be stripped has also increased in conformity to the slope angle. 77 Figure 44. The corresponding pit is shown in Figure 47. Pushback Design using Genetic Algorithms Western Australian School of Mines . Best value graph for a price of $80 Figure 45. θ.274. Optimum pit obtained with a price of $80 Testing results with a price of $90 The test results with a price of $90 shows that the pit has begun to expand as the best value (Table 9) continues to increase. The graph of the best value can be seen in Figure 46. The best value associated with this pit is $35. 4 0.6 0. Best value graph for a price of $90 Figure 47.274 90 5 5 17 100 Pit profile 0.5 0.6 0.4 % extraction 30 40 50 60 60 60 60 60 50 40 Figure 46.6 0. 78 Table 9.6 0.6 0. Optimum pit obtained with a price of $90 Pushback Design using Genetic Algorithms Western Australian School of Mines .5 0. Best value for a price of $90 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) 35.3 0. 6 0.5 0. The best value of $54. Figure 49 shows a pit size bigger than that shown in Figure 47. Best value graph for a price of $100 Pushback Design using Genetic Algorithms Western Australian School of Mines .6 0.6 0.4 % extraction 30 40 50 60 60 60 60 60 50 40 Figure 48.480 (Table 10) as depicted by the graph in Figure 48 also shows a further increase of the gains. the program was further tested with a price of $100 and there is an indication that the pit continues to increase in size as the price increases.6 0.6 0.480 100 5 5 13 100 Pit profile 0. 79 Testing results with a price of $100 As it was necessary to further observe the trend of pit size with an increasing price.4 0.5 0. Best value for a price of $100 BEST PRICE CAPACITY Min Max VALUE ($) (t) Pc Generations Generations ($) 54.6 0. Table 10. In the case of the Genetic Open Pit Optimiser however. Optimum pit obtained with a price of $100 Variation of best value with price It has been observed from the generation of pits with an increasing price that the Genetic Open Pit Optimiser compares well with the Lerchs-Grossmann’s Whittle Optimiser. This can be done by observing the variation of the best value with the corresponding price increase as shown in Figure 50 (refer to Table 19 in Appendix A for the results used for this plot). Effect of Price on Best value Pushback Design using Genetic Algorithms Western Australian School of Mines . Figure 50. 80 Figure 49. the profitable price to work with can be decide upon. 2 0.1 0. however. The seed is the best fit chromosome. we can determine the breakeven price as approximately $72 and then decide on the price for the ore that can yield profit when the pit is mined.2 Generation of pushbacks with capacity constraint To generate pushbacks all the parameters (i. Pushback 1 For pushback 1. capacity and capacity penalizing function) were kept constant.1 SEED 1 – SEED 0 0 0.2 0.e. the initial seed (SEED 0) was the surface topography which is the starting point. in the figure shown is the shape of the graph below the horizontal axis.1 0 0. Strangely.1 0.042 $90 5 500 21 100 SEED 0 (%) 0 0 0 0 0 0 0 0 0 0 SEED 1 (%) 0 0. It is noticed in Figure 51 that the initial pit identifies the optimal point at which to open up the deposit. 81 From the graph in Figure 50.1 0.e.1 0.1 0. Suffice to mention that the portion below the breakeven point indicates that the cost of mining is more than the revenue from the investment. The behaviour of the genetic model as portrayed by the graph requires further investigation. Pushback 1) the percentage pushback depth extracted is the same as the profile of the first pushback. Table 11.1 (%) Pushback Design using Genetic Algorithms Western Australian School of Mines .1 0. Results for Pushback 1 BEST VALUE PRICE CAPACITY Pc Min Max ($) (t) Generations Generations -93. In this case (i. The difference between seed 0 and seed 1 gives the percentage pushback depth extracted (approximate pushback target size).5. When the test was done the best fit chromosome obtained was SEED 1 (Table 11).1 0. The only variable in this case was the seed (profile). 4.1 0.1 0.1 0 0.1 0. price. Pushback 1 Figure 52. Pushback Design using Genetic Algorithms Western Australian School of Mines . the difference between seed 2 and seed 1 gave the percentage pushback depth extracted as in Table 12. seed 2 is now the profile for the new pushback (Pushback 2). In this case. Figure 53 shows the generated pushback 2 and the profile for mined out pushback 1. 82 Figure 51. Best value graph for Pushback 1 Pushback 2 For the second pushback. 1 0.186 $90 5m 500 57 100 SEED 1 (%) 0 0. Best value graph for Pushback 2 Pushback Design using Genetic Algorithms Western Australian School of Mines .1 0. Results for Pushback 2 BEST VALUE ($) PRICE CAPACITY Pc Min Max (t) Generations Generations -53.1 0.1 0.1 0.2 0.3 0.1 0.2 0. 83 Table 12.2 0.2 0 0 (%) Figure 53.1 0.1 0 0. Pushback 2 Figure 54.1 SEED 2 (%) 0.1 0.1 0.1 0.1 0.1 0.2 0.1 SEED 2 – SEED 1 0.1 0 0 0 0.1 0. 2 0. Table 13. 2 and 3).1 0.1 0.3 0. seed 2 which is the profile for pushback 2 was subtracted from seed 3 (profile for pushback 3) and the results have been shown in Table 13.2 0.2 0.1 0.1 0.2 0.1 0. It has been observed that the best value for pushback 3 indicates no payback yet just as is the case for pushback 1 and pushback 2 (see best value graphs for pushbacks 1.2 0.1 0. the profile is that for pushback 2 when it has been mined out.1 0.2 0.4 0.2 SEED 3 – SEED 2 0 0 0. These successive pushbacks identify the optimal directions in which to expand the pit. 84 Pushback 3 To generate pushback 3.859 $90 5m 500 36 100 SEED 2 (%) 0.2 0. As can be seen from Figure 55.3 0.3 0.1 0.1 0. Results for Pushback 3 BEST VALUE ($) PRICE CAPACITY Pc Min Max (t) Generations Generations -11.1 (%) Figure 55.1 0. This kind of scenario signifies that in the early stages of mining there is large capital of investment required before any profit can be realised.2 0.1 SEED 3 (%) 0.1 0. Pushback 3 Pushback Design using Genetic Algorithms Western Australian School of Mines .1 0.1 0. 1 0 0.1 0.1 0.2 SEED 4 (%) 0.428 (Figure 58).3 SEED 4 – SEED 3 0 0 0 0.2 0.428 $90 5m 500 29 100 SEED 3 (%) 0.2 0.2 0.1 (%) Pushback Design using Genetic Algorithms Western Australian School of Mines .3 0. Worth noting for this pushback is that the profit is now being realised as shown by the best value graph for pushback 4.2 0.1 0.1 0.4 0. Pushback 4 has been shown in Figure 57.4 0.3 0. The best value for pushback 4 is $18.3 0.4 0.2 0.4 0.4 0. Best value graph for Pushback 3 Pushback 4 Like for the first three pushbacks. 85 Figure 56. pushback 4 was generated in the similar manner and the results have been shown in Table 14.2 0. Results for Pushback 4 BEST VALUE ($) PRICE CAPACITY Pc Min Max (t) Generations Generations 18. Table 14.3 0.2 0.1 0.2 0. Best value graph for Pushback 4 Pushback 5 Table 15 shows the results for pushback 5 which has been generated in a similar manner as the preceding pushbacks. Pushback 4 Figure 58. Interesting is that the best value is steadily increasing. 86 Figure 57. Figure 59 shows the generated pushback 5 and the corresponding best value graph in Figure 60. Pushback Design using Genetic Algorithms Western Australian School of Mines . Pushback 5 Figure 60.1 0.3 0.2 0.4 0.2 0.4 0.1 0 0 0 0 0 (%) Figure 59.4 0.3 SEED 5 (%) 0.4 0. Results for Pushback 5 BEST VALUE ($) PRICE CAPACITY Pc Min Max (t) Generations Generations 29. Best value graph for Pushback 5 Pushback Design using Genetic Algorithms Western Australian School of Mines . 87 Table 15.1 0.4 0.4 0.3 SEED 5 – SEED 4 0.4 0.4 0.3 0.4 0.2 0.1 0.1 0.3 0.2 0.4 0.4 0.263 $90 5m 500 20 100 SEED 4 (%) 0. Results for Pushback 6 BEST VALUE ($) PRICE CAPACITY Pc Min Max (t) Generations Generations 41.599 $90 5m 500 26 100 SEED 5 (%) 0.1 0 0 0 (%) Figure 61.1 0.4 0.4 0.2 0.5 0. Pushback 6 Pushback Design using Genetic Algorithms Western Australian School of Mines . the last pushback is generated when there is convergence of the profile (i. best chromosome) towards the optimal solution. Table 16.6 0.4 0.e.4 0. 88 Pushback 6 The generation of pushbacks is a repetitive procedure regardless of the number of pushbacks.4 0.3 0.1 0 0 0. During testing of the genetic model.3 SEED 6 – SEED 5 0.4 0.4 0.4 0.3 0.5 0.3 SEED 6 (%) 0.2 0. Figure 61 shows the generated pushback 6 as well as the profile for pushback 5 once mined out. In this case the optimal solution was found to be $41.4 0.599 for pushback 6.4 0.4 0.4 0.1 0. Best value graph for Pushback 6 (Final Pit) In Figure 63 is the optimised pit with 6 (six) generated pushbacks. the final profile would look like that shown in Figure 64. On mining this optimum pit. 89 Figure 62. Figure 63. Optimised Pit with 6 pushbacks Pushback Design using Genetic Algorithms Western Australian School of Mines . With the capacity as an input variable. Final Pit when mined out Variation of Best value with Capacity To assess the trend of the best value with capacity. All the other variables were kept constant except the capacity. 90 Figure 64. Figure 65. the seed (pit profile) was maintained and 10 runs were done. the corresponding best value was noted (refer to appendix A for the data used to plot the graph in Figure 65). thus an optimum capacity of 40 tonnes can be determined from the same graph. Variation of Best value with Capacity Pushback Design using Genetic Algorithms Western Australian School of Mines . the graph shows that the best value is increasing with an increased capacity and then begins to decrease as the capacity is further increased. As can be seen in Figure 65. It is a variation of the best value with the capacity penalizing factor. 91 Variation of Best value with Capacity Penalizing Factor Figure 66 was plotted from the results in Appendix A. This indicates that when the maximum capacity penalizing factor is reached the best value is breakeven. Variation of Best value with the corresponding number of Generations Pushback Design using Genetic Algorithms Western Australian School of Mines . Figure 67. From the graph it can be seen that the best value decreases with increasing capacity penalizing factor. Figure 66. Variation of best value with capacity penalizing factor (Pc) Variation of Best value with the number of generations (iterations) The variation of the best value with the associated number of generations (iterations) was also done. Figure 67 shows the results obtained. the minimum number of generations ranges between 10 and 30. there are some outliers. Pushback Design using Genetic Algorithms Western Australian School of Mines . it is observed that although the maximum number of generations was set at 100 when running the program. 92 In the scatter graph shown. It can be observed though that in some cases. 93 Chapter Five CONCLUSION AND RECOMMENDATIONS Pushback Design using Genetic Algorithms Western Australian School of Mines . 1 Conclusion Pushbacks play a very important role in open pit mine design and optimisation. With the incorporation of the Genetic Whittle Optimiser in the model. From the findings of the study. dependent on the commodity price and the capacity penalizing factor at play. The study further showed that an optimum capacity constraint can also be determined. the genetic model as a solution tool was capable of emulating the Whittle Optimiser by generating pits with increasing prices. Of significance was that the objective of the research was achieved by generating pushbacks with capacity constraint subject to a penalizing constraint. it has been concluded that the genetic model has the capability to generate pushbacks with capacity constraint. Pushback Design using Genetic Algorithms Western Australian School of Mines . 94 CONCLUSION AND RECOMMENDATIONS 5. This may be an ideal solution tool to enhance the envisaged ease sequencing and scheduling of pushbacks without the subsequent compromise on the highest NPV possible. This is however. An analysis of the findings also indicated that the best value for an optimum pit decreases as the capacity penalizing factor increases. It has been endeavoured in this study to develop a genetic model that provides pushbacks with capacity constraint which results in an optimum pit. the breakeven price can be determined to help guide on the decision for the price to work with during the generation of pushbacks with capacity constraint. In view of the findings of this research. As a solution tool the genetic model has the capability of handling multiple scenarios of capacity constrained pushbacks. the research was however. This would be able to yield pushbacks which may be ease to sequence and schedule without any compromise on the overall highest NPV for the optimum pit. It is therefore recommended that: 1. 2. 95 5. 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Pushback Design using Genetic Algorithms Western Australian School of Mines ... 15-23). 89 70 0 0 0 0 0 0 0 0 0 0 80 0.88 0.634 0.076 0.812 0.311 0.472 2.316 0.8 0.742 0.566 0.86 0.823 1.835 0.8 0.128 1.869 0.449 1.252 1.86 90 0 0 0 0 0 0 0 0 0 0 100 0.869 0.8 0.6 0.835 0.864 0.884 0.455 1.694 0.379 0.8 0.472 0.693 0.853 0.89 0.8 0.833 0.465 0.012 1.835 0.884 0.51 1.187 0.34 1.224 1.495 1.13 0.851 0.612 0.6 0.835 0.159 0.639 0.61 0.421 20 0.884 Pushback Design using Genetic Algorithms Western Australian School of Mines .881 30 0.356 2.828 80 0.502 0.5 0.042 1.853 0.8 0. 99 APPENDICES Appendix A: Data files used and testing results Table 17.951 0.537 1.8 0.522 0.056 1.993 0.951 0.73 1.499 1.5 0.294 2.797 0.102 0.86 0.8 0.644 0.414 90 0.56 0.527 60 0.211 1.019 0.815 0.87 0.82 110 0 0 0 0 0 0 0 0 0 0 120 0.609 2.794 0.531 0.82 0.447 0.86 0.8 0.635 0.635 0.447 50 0.963 1.45 0.62 0.244 1.057 0.909 0.384 0.713 1.86 50 0 0 0 0 0 0 0 0 0 0 60 0.635 0.21 0.754 0.841 70 1.665 0.88 0.8 0.83 0.429 1.713 40 1.69 0.83 0.546 2.884 0.287 1.89 0.86 0.8 0.184 100 0.8 0.8 0.336 0.423 2.09 Table 18.708 0. Data file for validation purposes Depth Column (m) 1 2 3 4 5 6 7 8 9 10 10 0.442 0.239 0.88 0.597 1.83 30 0 0 0 0 0 0 0 0 0 0 40 0.267 0.132 0.199 0. Data file for final testing of the genetic model Depth Column (m) 1 2 3 4 5 6 7 8 9 10 10 0 0 0 0 0 0 0 0 0 0 20 0.121 0.69 0.87 0.008 0.751 0.09 1.167 0.534 0.56 0.8 0.323 0.879 0.271 0.82 0.815 0.15 1.608 0.82 0.437 0.978 0.183 1. Results for variation of best value with price PRICE BEST VALUE 10 -14837 20 -30043 30 -40442 40 -36777 50 -28545 60 -17853 70 -668 80 16843 90 36209 100 54480 110 74661 120 95547 130 118888 140 147499 150 170460 Table 20. Results for variation of best value with capacity CAPACITY BEST VALUE 0 37854 5 39729 10 41354 15 42729 20 43854 25 44729 30 45354 35 45729 40 45854 45 45729 50 45354 55 44729 60 43854 65 43034 70 41354 75 40334 80 33889 85 32464 90 30789 95 27474 100 27119 Pushback Design using Genetic Algorithms Western Australian School of Mines . 100 Table 19. Best value and number of generations MIN. Results for variation of best value with capacity penalising function Pc BEST VALUE 0 41654 5 39729 10 33604 15 28239 20 23119 25 17999 30 11648 35 9228 40 6808 45 4928 50 2928 55 928 60 -1072 65 -3037 70 -4842 75 -6647 80 -8452 85 -10257 90 -11567 95 -12847 100 -14127 Table 22. GENERATIONS BEST VALUE 17 37854 12 39729 14 41354 11 42729 17 43854 23 44729 14 45354 11 45729 13 45854 16 45729 19 45354 26 44729 24 43854 20 43034 23 41354 21 40334 4 33889 11 32464 13 30789 11 27474 58 27119 Pushback Design using Genetic Algorithms Western Australian School of Mines . 101 Table 21. 1): self.eps = eps self.check_chromosome(temp.chromosome) random_position = random.0) b = min(temp_chrom[random_position+1]+self.seed) #seed argument needs to be compatible with the STEP number.append(0) for i in range(1.size+1)) if random_position == 1 : a = max(temp_chrom[random_position+1]-self.1 STEPS = list(np.eps=0.STEPS)) Pushback Design using Genetic Algorithms Western Australian School of Mines .deepcopy(self.arange(0.random_chromosome(self.seed): temp_chrom = copy.lenth_steps-1) temp[i]=STEPS[rand_steps_index] temp = self.size. # -*.0 self.1) diff = random.size_individual. seed is a list of STEPS def random_chromosome(self.fitness_value = 0.STEP)) #capacity measured in steps CAPACITY = 5 class Individual(object): def __init__(self.1+1e-6.choice(filter(lambda x:a<=x<=b.columns_list = xx2.self.coding: utf-8 -*- """ Created on Thu Jun 03 23:20:24 2010 @author: Mwiya Songolo """ import random import copy import math import numpy as np import xx2 STEP = 0.seed): temp = [] lenth_steps = len(STEPS) #This code add up to size columns.size-1): rand_steps_index = random.data_sort() #columns_list self.size = size_individual self.seed.choice(range(1.randint(int(seed[i]/STEP). initial value will be zero for i in range(size): temp.deepcopy(temp) def mutate(self. 102 Appendix B: Python code for pushback design with capacity constraint The Python code for Pushback design with capacity constraint using GA was programmed with the assistance of Dr Jose Saavedra Rosas.seed) return copy.size+2.eps.eps.chromosome =self. seed) self.random_position.seed) temp_chrom = copy.randint(max(index_steps .len(STEPS)-1)) # temp_list = np. 1.1) temp_list = filter(lambda x: a<=x<=b.choice(temp_list) if temp_arr[index+direction]-temp_arr[index]>self.eps.eps.size : if temp_arr[index+direction].temp_chrom[random_position+1]) b = max(temp_chrom[random_position-1].temp_chrom[random_position+1]) if b-a>2*self.choice(temp_list) index = index + direction return temp_arr def propagate2(self.eps / STEP ).seed) temp = self.seed): # direction = -1 for left and +1 for right temp_arr = copy.array.STEP) # diff = random.eps.1).-1.eps.STEPS) diff = random.arange(max(temp_arr[index]-self.self. seed): # direction = -1 for left and +1 for right temp_arr = [] # the first and the last elements of arr must be equal to 0 temp_arr = copy.0) else : a = max(temp_arr[index]-self. 103 temp_chrom[random_position] = diff elif random_position == self.chromosome = copy.random_position.eps.seed) temp_chrom = self.0) .int(self.seed[random_position]) up = a+self.eps : temp_arr[index+direction] = min(temp_arr[index] + self.seed[index]) b = min(temp_arr[index]+self.deepcopy(temp) else : down = max(b-self.eps.direction.start.eps : temp_arr[index+direction] = max(temp_arr[index] .index(temp_arr[index]) # random_index = random.eps.direction. random_position.temp_arr[index]>self.eps.eps.start.min(temp_arr[index]+self.choice(filter(lambda x:a<=x<=b.eps if down == up : temp_chrom[random_position] = up elif up-down <= STEP : temp_chrom[random_position] = up else : temp_chrom[random_position] = random.1) elif temp_arr[index+direction] .choice(filter(lambda x: down <= x <= up.propagate(temp_chrom. random_position.eps / STEP).0).propagate2(temp_chrom.eps : temp_arr[index+direction] = min(temp_arr[index] + self.propagate(temp_chrom.deepcopy(temp_chrom) def propagate(self.deepcopy(array) index = start while temp_arr[index+direction] != 0 : # index_steps = STEPS.propagate2(temp_chrom.size : a = max(temp_chrom[random_position-1]-self.0) b = min(temp_chrom[random_position-1]+self.temp_arr[index] < -self.1) Pushback Design using Genetic Algorithms Western Australian School of Mines .STEPS)) temp_chrom[random_position] = diff else : a = min(temp_chrom[random_position-1]. -1.array.deepcopy(array) index = start while 1 < index < self.eps : temp = self.1.STEPS)) temp_chrom = self.eps. \ # min(index_steps + int(self.1) diff = random. choice(temp_list) index = index + direction return temp_arr def check_chromosome(self.1) else : a = max(temp_arr[index]-self.chromosome)) cave_detph = [] for c in self.eps.size]**2) / (math.1) index = index + 1 chromosome = copy.1) index = index .STEPS) diff = random.0 : temp_chrom[index+1] = max(temp_chrom[index]-self.seed.cos(theta)* (DEPTH**2) * (self.eps.seed): temp_chrom = copy.0 : temp_chrom[index-1] = max(temp_chrom[index]-self.eps : temp_arr[index+direction] = max(temp_arr[index] .size+1)) # to left index = random_index while index-1>0 : if temp_chrom[index-1].eps.chromosome.seed[index+1]) elif temp_chrom[index+1] .eps.eps < 0.cos(theta)* (DEPTH**2) * (self.chromosome[1]**2) / (math.chromosome[self.deepcopy(temp_chrom) return chromosome def fitness(self.self.sin(theta)*2)) area3 = float(WIDTH * DEPTH * sum(self.temp_chrom[index] + self.eps : temp_chrom[index+1] = min(temp_chrom[index]+self.eps : temp_chrom[index-1] = min(temp_chrom[index]+self.1) temp_list = filter(lambda x:a<=x<=b.0 theta = math.deepcopy(chromosome) #we choose a position at random to start propagation random_index = random.0 recovery = 1.temp_chrom[index] > self.temp_chrom[index] + self.eps < 0.seed[index]) b = min(temp_arr[index]+self.sum_up_caved_column_grade(key.eps.seed[index-1]) elif temp_chrom[index-1] .1 # to right index = random_index while index+1 < self.append(column_sum) Pushback Design using Genetic Algorithms Western Australian School of Mines .size+1: if temp_chrom[index+1].chromosome : cave_detph.temp_chrom[index] > self.columns_list.depth) value.temp_arr[index] < -self.append(c*DEPTH) for key in self.choice(range(1. 104 elif temp_arr[index+direction] .eps.self. area2 and area3 are used to compute the cost area1 = float(math.pi/4 # area1.pressure): value = [] DEPTH=100 WIDTH=10 price = 90 cost = 20.sin(theta)*2)) area2 = float(math.keys(): depth = cave_detph[key] column_sum = self.eps. size_population.mutation_rate=0.columns_list[key][1] sum_value = 0.seed.len(depths)): if depth > depths_sorted[i] : sum_value += grades[depths.seed.eps = eps self.index(depths_sorted[i])]*(10) else : sum_value += grades[depths.chromosome)-1): #chromosome is always bigger than profile by definition difference += self.difference)) def sum_up_caved_column_grade(self.sort() depths_sorted = depths_raw depths = self.columns_list[key][0] grades = self. pressure) #penalisation for capacity constraint violation def penalty(self.crossover_rate=0.c class Population(object): def __init__(self.pressure = 0.size_individual.penalty(seed.total_score = None self.index(depths_sorted[i])]*(depth-depths_sorted[i-1]) break return sum_value # def check(self): # for chrom in self.self.difference)*(CAPACITY .eps=0.crossover_rate = crossover_rate self.seed = seed self.size_individual = size_individual self.len(self.key.selected_individuals = None self.chromosome[i]-seed[i] #Total number of steps difference = difference /STEP return pressure*((CAPACITY .8.0 Pushback Design using Genetic Algorithms Western Australian School of Mines .best_score = [] self.fitness_value = price * sum(value)*WIDTH * recovery .best_individual = [] self.2): self.mutation_rate = mutation_rate self.0 for i in range(1.size = size_population self. 105 self.child_individuals = None self.01.individuals = [] self.columns_list[key][0] depths_raw.pressure): difference = 0 for i in range(0.cost * (area1 + area2 + area3).depth): depths_raw = self.select_times = None self. chromosome[i] chromosome1.size): if (self.size : rand = random.chromosome.append(int(s/sum_scores*self.individual2.select_times=0 total_select_times = sum(self.size_individual): chromosome1.chromosome2 def select(self): smallest_score = 0.check_chromosome(chromosome1.individual1. 106 for i in range(self.append(generation_best_score) self.chromosome[i] = individual1.size): self.pressure) def two_individuals_crossover(self.individuals.self.select_times[i] < 0) : self.fitness < 0.individuals: if individual.check_chromosome(chromosome2.01) # if individual.0 temp = None for individual in self.deepcopy(individual1) chromosome2 = copy.select_times) while total_select_times < self.size)) for i in range(self.randint(1.size_individual.eps)) self.select_times.size_individual-1) for i in range(random_position+1.append(individual.chromosome = individual2.fitness_value < smallest_score : smallest_score = individual.seed) return chromosome1.fitness_value temp = individual individual_scores=[] for individual in self.0 : # continue # self.select_times = [] sum_scores = sum(individual_scores) for s in individual_scores : self.fitness_value-smallest_score+0.seed) chromosome2.chromosome[i] chromosome2.seed.self.individuals : individual_scores.append(temp) self.total_score = 0.0 generation_best_score = -1000000000000000.best_score.fitness_value if generation_best_score < individual.chromosome[i] = individual2.fitness(seed.fitness_value self.seed): chromosome1 = copy.append(Individual(self.chromosome = individual1.self.self.randint(0.deepcopy(individual2) random_position = random.fitness_value : generation_best_score = individual.total_score += individual.size-1) Pushback Design using Genetic Algorithms Western Australian School of Mines .best_individual.self.chromosome.0 self.self. individuals: individual. self.individuals[i].individuals: Pushback Design using Genetic Algorithms Western Australian School of Mines .selected_individuals)-1) temp.individuals[i].seed) def crossover(self.selected_individuals. pressure): for individual in self.individuals[i] = copy.size.individuals[i+1]) value1 = temp1.deepcopy(self.size : self.mutate(seed) value3 = self.individuals[i+1] = copy.len(self.selected_individuals[random_index] if len(temp)==self.deepcopy(temp) else : raise RuntimeError('desc') def mutate(self.individuals[i+1].individuals: individual.deepcopy(self.individuals[i+1] = copy.individuals[i+1].seed): rand = random.individuals[i+1].individuals[index])) temp = [] for i in range(len(self.size%2==0 : for i in range(0.size) : for j in range(self.selected_individuals[random_index]) del self.append(self.deepcopy(self.chromosome.seed) self.deepcopy(temp1) if (value3 < value2): self.deepcopy(temp2) def check(self): for individual in self. 107 self.fitness_value value4 = self.random() if rand <= self.individuals[i].seed): rand = random.check() def fitness(self.mutate(seed) individual.select_times[rand] += 1 total_select_times = sum(self.select_times[index]): self.individuals[i] = copy.mutate(seed) self.seed.self.fitness_value if ((value3 < value1) or (value3 < value2)): if (value3 < value1): self.individuals[i]) temp2 = copy.2): temp1 = copy.mutation_rate : for individual in self.selected_individuals)): random_index = random. self.individuals = copy.selected_individuals = [] for index in range(self.deepcopy(temp1) if (value4 < value2): self.crossover_rate and self.chromosome = individual.deepcopy(temp2) if ((value4 < value1) or (value4 < value2)): if (value4 < value1): self.individuals[i+1] = self.fitness_value self.select_times) self.random() if rand <= self.fitness_value value2 = temp2.check_chromosome(individual.randint(0.two_individuals_crossover(self.append(copy.individuals[i]. select() my_population.seed.5.0.5.0.pressure = generations best_score = my_population.best_score.1 crossover_rate = 0.0.pyplot as plt plt.0.plot(x.5.chromosome: print caved_percentage x = np.5.fitness(seed.0.show() Pushback Design using Genetic Algorithms Western Australian School of Mines View publication stats .ylabel("best_value") plt.pressure = 500 while generations < MAXGENERATIONS : # print generations my_population.8 #seed has to be specified as percentages compatible with the step size seed = [0.pressure) def get_best_individual(self): return self.len(best_score)) import matplotlib.3.0.5.mutate(seed) generations = generations + 1 #my_population.arange(0.crossover(seed) #my_population.4.mutation_rate = mutation_rate my_population.0.pressure) my_population.0.best_individual if __name__ == "__main__" : size_population = 200 size_individual = 10 mutation_rate = 0.0.best_individual[-1] #if best_score[-1]>0: for caved_percentage in best_individual. 108 individual.3] my_population = Population(size_population.0.4.size_individual.fitness(seed.0.5.crossover_rate = crossover_rate MAXGENERATIONS = 100 generations = 0 my_population.'r') plt.my_population.eps=0.get_best() best_individual = my_population.xlabel("generations") plt.1) my_population.best_score print best_score print max(best_score) #print my_population.