DOCTORAL THESIS
DIV. O F MINING ENGINEERING1994:143 D
ISSN 0 3 4 8 - 8 3 7 3ISRN H L U - T H - T - - 1 4 3 - D - - S E
An Approach to Long Range Production
and Development Planning with
Application to the Kiruna Mine, Sweden
T O R G N Y A L M G R E N
1994:143D
AN APPROACH TO LONG RANGE PRODUCTION
AND DEVELOPMENT PLANNING WITH
APPLICATION TO THE KIRUNA MINE, SWEDEN
av
Torgny Almgren
Avdelningen för Bergteknik
AKADEMISK AVHANDLING
som med vederbörligt tillstånd av Tekniska Fakultetsnämnden
vid Högskolan i Luleå, för avläggande av teknisk
doktors-examen, kommer att offentligen försvaras i Högskolans hörsal
ABSTRACT
The mining industry is increasingly oriented towards large scale mining and the planning concept of "Just in Time", which means activities starting and ending as late as possible without
jeopardising production. The buffers, consisting of production blocks and development ahead of production, are then minimized. This, however, leads to a larger dependency on the quality of the output from the production systems.
The Kiruna mine, Sweden, is a large scale mine and has a number of potential ore blocks with a content that varies in quality (Fe, P, K ) and in quantity (ore tonnage). Ore production is restricted by operative and block sequencing constraints as well as by production requirements. A
computerized planning model, based on operations research (multi period scheduling) has been developed to meet future demands on ore qualities and quantities. The model uses long range objectives, but can satisfy temporary short range demands without jeopardising the long range goals. A sub-optimal plan, made using the model, has been compared with a truly optimal plan. It is shown that, due to the unreliable information concerning the ore, the sub-optimal plan performs as well as the optimal one. The simulation process that was developed for this comparison is also used to estimate the need for production buffer blocks.
Another buffer planning model is developed to determine when development work should take place in order to minimize the risk of additional costs, caused by an inability to start production on time. This algorithm considers the uncertainties in activity durations and is based on a Monte Carlo simulation using project networks and estimates the optimum development buffer.
PREFACE
I would like to express my thanks to my supervisor Professor Gunnnar Almgren for his support during the work with this thesis.
I am also indebted to Dr. Eilif Hensvold (Luleå University of Technology) for critically reviewing my work.
I wish to thank Dr. Mark Kuchta (LKAB/Malmberget) and Gösta Stalnacke (LKAB/Kiruna) for their support and valuable advice on the behalf of LKAB.
Thanks are also due to Dr. Bo Forsman (Luleå University of Technology) and Dr. Uday Kumar (Luleå University of Technology) for commenting on my work, and to Martin Warren (Luleå University of Technology) for revision of the english (at least most of it).
Finally I acknowledge the G2000 programme and LKAB for their financial support of this work.
1. I N T R O D U C T I O N 1 1.1. Background 1 1.2. Goal 2 1.3. Disposition of the thesis 3
1.3.1. Summary
1.3.2. An overview of the thesis (chapter by chapter)
2. T H E K I R U N A M I N E 5 2.1. Introduction 5 2.2. The current block grade assessment 6
2.3. The existing mine planning system 9 2.3.1. Production goal (objective)
2.3.2. Mining and Production Constraints
3. PRODUCTION PLANNING USING OPERATIONS RESEARCH TECHNIQUES 13
3.1. Introduction 13 3.1.1. Operations Research in general
3.1.2. Operations Research applied to mine production planning
3.2. Different OR-techniques for production planning 15 3.2.1. Network Techniques
3.2.2. Dynamic Programming (DP) 3.2.3. Linear Programming (LP) 3.2.4. Simulation
3.2.5. Heuristics
3.3. Geostatistics and Ore Reserve Calculations 30 3.4. Comments on OR-techniques in general 30 3 .5 Available suggestions for mine planning applications of OR planning techniques 30
3.5.1. The use of aggregation and dis-aggregation of ore blocks 3.5.2. Ultimate pit design - no time factor present
3.5.3. Multi period production scheduling
3.6. Choice of OR-technique for the Kiruna case 47 3.6.1. Model requirements
3.6.2. Possible approaches to a long range planning model
3.7. Conclusions from the state of art review 52 4. L O N G RANGE PRODUCTION PLANNING M O D E L FOR T H E K I R U N A M I N E ,
T H E O R E T I C A L APPROACH 53
4.1. The problem 53 4.2. Formulation of a "true" optimum MIP (all periods simultaneously) 54
4.2.1. Objective function 4.2.2. Production constraints 4.2.3. Logical constraints 4.2.4. Shaft group constraints 4.2.5. Capacity (Loading) constraints 4.2.6. Continuity constraints
4.2.7. Sequencing constraints 4.2.8. Additional constraints 4.2.9. Final comments
4.3. Method of "simplification" of the MLP-model 65 4.3.1. Number of integer variables
4.3.2. The application of a lagrangean relaxation technique on the MIP-model 4.3.3. Reformulating the model as a sub-optimal multiple single period model
4.4. Formulation of multiple single period MIP with a long range objective 68 4.4.1. Objective function
4.4.2. Production constraints 4.4.3. Logical constraints 4.4.4. Shaft group constraints 4.4.5. Capacity constraints 4.4.6. Continuity constraints 4.4.7. Sequencing constraints 4.4.8. Additional constraints 4.4.9. Comment
4.4.10. The reduction of the number of active blocks
4.4.11. The additional practicality of the multiple single period model
4.5. The formulation of the problem that was actually used 79 4.5.1. The reduction of the number of active blocks
4.5.2. Definitions used in the formulation 4.5.3. The model
4.5.4. Comments
5. T H E IMPORTANCE OF O P T I M A L I T Y 89
5.1. Geostatistical uncertainty 89 5.1.1. Sampling
5.2. Flow mechanical uncertainty 92
5.3. Test of optimality 93 5.3.1. Test procedure
5.4. Test of the production plans sensitivity to unforeseen changes in the B I ratio 97 5.4.1. Determination of appropriate number of iterations
5.4.2. The results
5.5. Conclusions 102 6. E S T I M A T I O N OF T H E NEED FOR PRODUCTION BUFFER CAPACITY 103
6.1. Determination of buffer volume 103 6.2. The allocation of buffer blocks 105
6.2.1. Approach
6.3. Comments 108 7. L O N G RANGE PRODUCTION PLANNING A T T H E K I R U N A M I N E 109
7.1. The results of the Kiruna case 110 7.1.1. The progress of the mine
7.1.2. The production levels
7.1.3. The variations in Phosphorus level 7.1.4. Production schedules
7.2. Other considerations concerning the model formulation and the results 117 7.2.1. Relaxation of the shaft group constraint
7.2.2. Minimizing the number of simultaneous shaft groups
7.2.3. The number of simultaneously active production blocks 7.2.4. The consequences of a changed B I ratio
7.3. Production buffer need 127 7.3.1. Buffer allocation - A test case
7.4. Comments 131 8. D E T E R M I N A T I O N OF O P T I M U M DEVELOPMENT BUFFER 133
9. S U M M A R Y OF T H E PLANNING PROCEDURE 153 9.1. Development buffer (time) for the production and buffer blocks 154
9.1.1. An example 9.1.2. Comment
9.2. Short-medium range planning 157 9.3. How does the replanning of production affect development planning 157
9.4. Summary 157
10. CONCLUSIONS AND RECOMMENDATIONS 159
10.1. The production planning model 159 10.2. Validation of the multiple single period model 160
10.3. The production buffer 161 10.4. The development buffer 161
10.5. Contributions 161 10.6. Recommendations for further research 162
10.6.1. Evaluation of production plan quality 10.6.2. "Optimal" block size
10.6.3. New modelling techniques 10.6.4. Medium range planning model
11. REFERENCES 165
APPENDICES
1. About the possibility to predict flows of blasted rock (submitted for publication) 2. Lagrangean multiplier technique applied to underground mining
3. The planning model in GAMS code, the true optimum MTP version 4 The planning model in GAMS code, single period MIP version 5. The importance of the weight values in the objective function 6. Test to establish whether the 8 parameter is normally distributed 7. Description of Active ore body
8. The PASCAL program used to carry out the production plan simulations 9. Optimal and sub-optimal production plans for fictive ore body
10. The yearly progress of the mine for both loading cases, vertical sections 11. Production plan for loading case 110 ktonnes/month
1. I N T R O D U C T I O N
1.1. Background
During the last few decades, technical development in underground mining has emphasized the concept of large scale. Large scale mining means larger stopes, larger drill hole dimensions and more heavy equipment in order to increase efficiency, cut costs and achieve better working environment.
The large scale approach can, however, result in disadvantages concerning the quality control of rock stability, unit operations and ore products. For instance, a decreased number of loading places means a decrease in flexibility with fluctuations in the quality of the ore product as a result. This can be a serious problem, for example, for the iron ore producers of today, who are subjected to high demands from the market. These demands are, from the mine's point of view, primarily related to ore quality, quantities and delivery times, where ore quality depends on parameters such as the content of phosphorus and alkali-metals.
The Kiruna iron ore mine in northern Sweden has reached the stage in its development where it has been necessary to move towards a larger scale mining system. This development has been followed by difficulties in keeping up with the required production levels of the different ore qualities. A problem that has become even more severe as they are also cutting down on buffers to reduce costs.
What then is the definition of buffers ? The mines usually have to create them to be able to satisfy production requirements, underground as development or production buffers (i.e. blocks of ore that are ready for production but do not produce) and above ground as stock piles. These buffers, in their turn, are considerably costly, and therefore should be kept as low as possible without jeopardising punctual ore deliveries.
There are different ways to approach the problem of buffer minimization, depending on which step in the planning procedure is concerned. First it has to be decided which "package" of ore, or ore blocks, should be delivered at a certain time to meet with the demands at that particular time. Secondly, it has to be decided how this should be done, i.e. primarily how to plan the development work.
I f the first part is considered, it means that a practically feasible plan should be constructed. The plan must indicate which blocks to mine at every point in time within the planning span.
The "sum" of these blocks must equal the demands for that particular time period. The blocks should also be chosen in a way that minimizes the need for development work, but also offers enough flexibility to counter unpredicted disturbances, e.g. rock mechanical instabilities, deviations from the expected ore content, change in demand etc.. The plan should also be as reliable as possible so that a minimum number of "spare" blocks have to be developed.
The second part is primarily concerned with the time planning of the development work. The ideal situation is when development is concluded the same day as production starts, i.e. to maintain the development buffer at zero. However, to use this ideal as a planning goal would be very risky as it is general knowledge that almost nothing is ready on time. It is therefore desirable to have a certain margin which takes into account delays, as failure to deliver on time might be very expensive. However, this margin should not be too large. The issue is therefore to estimate the optimum time margin, i.e. with respect to capital costs versus costs caused by demands that are not fulfilled.
Nemitz (1994) states that 40 % of quality assurance lies within the actual planning, while material and performance represents smaller amounts, namely 15 % and 30 %. So the need for good planning methods and reliable tools is clearly of great importance in efforts to reduce the required buffer need.
1.2. Goal
The work presented in this thesis aims to develop an approach to minimize the size of the buffers needed in the Kiruna mine. This is done in two parts. The first part concerns the actual long range ore production planning and aims to develop a planning tool which will supply reliable plans and in that way also decrease the need for buffers, i.e. to minimize the need for production buffer. The second part aims to optimize the size of the development buffer, i.e. based on the assumption that no spare blocks (production buffer) are available when needed. The planning tool should also be able to estimate the need for production buffer blocks.
1.3. Disposition of the thesis
13.1. Summary
This thesis introduces first the problem in general, and then applies the given "definitions" to the Kiruna mine in northern Sweden. The next step is to look into the production scheduling problem, i.e. in order to achieve part one of the goal. What are the objectives and the constraints for this particular case and what is the current state of the art in this area ? The thesis outlines an approach to solve the problem, and shows how this approach has successfully been used in practise, i.e. a full scale case study.
A short study on the elusive concept of optimality is performed. It is shown that it is not always justified to try to obtain an optimal plan in the case where the information concerning the available ore is not very good. A procedure to estimate the required number of buffer blocks (i.e. the production buffer) is given. This new technique is then applied to the Kiruna mine.
The thesis also presents a new way to determine the optimum development buffer, i.e. part two of the goal. This is primarily a method to plan a time scale of the development work in a probabilistic way that ensures that the extra costs anticipated will be minimised. Thus completing the study on how to decrease buffer levels (i.e. both for production and development) in a mine to a low, but realistic, level.
13.2. A n overview of the thesis (chapter by chapter)
Chapter 1 describes the purpose, and the disposition, of the thesis.
Chapter 2 describes the Kiruna mine and outlines the requirements of the production schedule, and also the constraints that are imposed on the planning model.
Chapter 3 offers an extensive literature review of production planning in mining. Different approaches are discussed, and most of them are rejected for a variety of reasons.
Chapter 4 introduces an optimal, but impractical, way to solve the production planning problem This initial model will then be further developed into a sub-optimal model that works for a large scale problem such as production planning modelling at the Kiruna mine. It is also explained why a sub-optimal model is better suited to be used interactively than an optimal model.
Chapter 5 contains the validation of using the sub-optimal model, instead of the optimal model, presented in chapter 4. This validation is based on the understanding that there are significant discrepancies between the ore database and what is actually are loaded from the loading points. It has been shown in appendix 1 that it is not possible to predict the outcome from the gravity flow with the required precision, which strengthens the impression that the ore database offers only a limited understanding of what will actually be loaded from the loading points. An optimal and a sub-optimal plan, the latter is based on the model developed in this thesis, are compared in a Monte Carlo simulation study, where the different plans' sensitivity to deviations from the expected output from the loading points has been tested. It has been shown that the sub-optimal plan works almost as well as the optimal plan, which is considered adequate as the sub-optimal model is much more practical than the optimal model.
Chapter 6 uses the simulation procedure developed in chapter 5 to further investigate the plans' sensitivity to fluctuations from the expected content of high quality ore (BI). The result is an estimate of how large a buffer capacity is needed to ensure constant production of high quality ore. A method for allocating these buffer blocks is also presented.
In chapter 7, the planning procedure developed is applied to the Kiruna mine. Planning is performed for the entire ore body over a time span of 8 years. It is shown that the model successfully fulfils the production requirements over the entire planning range. An
approximation of the buffer need (of production blocks) for the Kiruna mine is made by using the procedure outlined in chapter 6.
Chapter 8 summarizes Almgren's licentiate thesis - Almgren (1989). This work primarily deals with time planning under conditions of uncertainty, especially of the development work, and its effects on production planning are also considered. The concept is then used to calculate the optimum development buffers.
Chapter 9 summarizes the different parts into one concept for long range production and development planning.
2. T H E K I R U N A M I N E
2.1. Introduction
The Kiruna mine is a large underground iron ore mine in northern Sweden. Production in 1992 was 18 Mton, from which 12.4 Mtons of products were produced. The most important products were pellets (56%) and low phosphorus fines (27%). Three main products are produced in the mine; they are B I - which is high quahty ore with low grades in phosphorus and alkali metals; B2 and, finally, D3 which is the low quality ore. The B1 ore is used to produce low phosphorus fines.
The main ore body is disc-shaped and about 80 m wide and about 4 km long. It has two main contaminates, these are phosphorus (P) and potassium (K). The ore type with a higher content of these elements is called D3, and the type with a lower content is called B I .
In the northern part of the ore body, the Bl-ore is mainly situated at the hanging wall contact, and in the southern part at the foot wall. It is therefore rather difficult to extract pure B I in the southern part due to the complexity of the gravity flow. The Bl-ore (in situ) is often rather mixed with D3-ore from the hanging wall, and from the caved areas above. Today, to cope with this problem, mine planning attempts to mine from both the southern and the northern parts simultaneously in order to be able to "mix" the different ore qualities, and in that way obtain the required production levels.
The mine consists of a number of mining blocks, with a width of about 100 m and they reaches from the hanging wall to the foot wall. Each block is divided into three parts for reasons of ore quahty. These parts will be referred to as "ore zones", see figure 2.1.
Figure 2.1. The division of the ore body into blocks and zones. The height of the blocks is determined by the distance between the mining levels.
The ore database follows the convention of describing the ore in blocks and zones. For each particular zone, the amount (tonnage) and grades of the respective ore quality are given, see table 2.1.
Table. 2.1. Content of block Y29 on level 632.
B I B2 D3 Total
Zone 1 0 124 70 194
Zone 2 40 30 0 70
Zone 3 120 66 0 186
2.2. The current block grade assessment
Any production planning system is highly dependent on the accuracy of the block data. The planner, or the planning system, normally has to assume that the given block data are correct. Geostatistics can sometimes be used in order to quantify, or minimize, risk, see Ravenscroft (1992). However, it can be difficult to use geostatistic parameters in decision criteria in OR-models, even though some approaches have been reported, for example Gangwar (1982) and Barnes (1986). This means that OR-models are normally deterministic even though they may be based on grades that have been calculated through the use of kriging or similar techniques. The consequence of this is that the mine that is modelled, and for which the plan is optimal, does not exist. The concept of an optimal production plan can therefore be discussed. The question of unreliable, i.e. not totally reliable or probabilistic, ore block data and the optimality of production plans will be further discussed in this thesis, see chapter 5.
It is a fact that there is a large degree of uncertainty in the grade and tonnage assessments which constitute the ore database of the Kiruna mine. This depends primarily on two things; the geological uncertainty which affects the determination of the in-situ grades; and the uncertainties in the gravity flow of fragmented rock in relation to the loading places.
The in-situ geology is very complex, especially when the distribution of the different ore qualities is considered, an example of a horizontal section is shown in figure 2.2.
Figure 2.2. Horizontal section of ore map, the different ore qualities ( B I etc.) are indicated in the figure.
It is easily understood that the drawing of such maps is very difficult, given that they are normally based on information from previous mining levels, and on diamond drilling in a rather sparse pattern. The distances between the investigation holes are approximately 50 m. Thus, the question is not whether the mine maps are correct or not, but rather "how great are the deviations from the real mine ?".
However, the single most important factor is the influence of the gravity flow on the grades. The mining method is based on gravity flow with the hanging wall caving in on top of the blasted ore. The principle is shown in figure 2.3.
Figure 2.3. Sub-level caving, vertical section, from the side, (LKAB).
It is obvious that there are a number of factors that will influence the mechanics of the gravity flow, the most important are listed below.
• Fragmentation • Blast hole deviation
• Distribution, and choice of, explosives • Rock properties
• Density of the different rock categories
• Surface characteristics, of both blocks (including shape) and adjoining walls • The present situation in adjoining drifts.
It is clearly very hard to predict how the broken rock will behave, and even more so to tell where a certain piece of rock will end up. The unpredictability of the gravity flow has also been demonstrated in Almgren (1994), see also appendix 1. This means that even i f the mine maps were absolutely correct, the actual rock that is loaded will not be what is expected in most cases. This has also been noticed in the mine, where they sometimes succeed in loading 1000% (no misprint) of the ore that was blasted, and sometimes only 20 %. This depends primarily on the gravity flow, and specially on the properties of the broken rock overlying the ore that was last blasted.
What makes this problem even more serious is that even a small dilution (in tons) of high phosphorus ore in the low phosphorus ore is sufficient to contaminate it, and thus render the whole batch as medium- or high phosphorus ore. I f this phenomenon occurs in a couple of blocks simultaneously, it might be impossible for the mine to deliver the amounts required.
2.3. The existing mine planning svstem
Today, long range planning (approximately 5 years) is carried out more or less manually by an experienced mine planner. He only uses the computer as a calculator in which he adds together the tonnages from each block, which are stored in the computer, and checks the sums for each time period. This process is therefore very time consuming. The number of available alternative plans, i f any exist, is therefore very low. Thus, there is a lot of room to improve the long range planning.
2.3.1. Production goal (objective)
The production goal is, simply stated, to produce certain quantities of ore of primarily B l -quality, and a certain total amount is also required. However, the mine has had problems producing the desired amount of Bl-ore due to the strict quality restrictions that are stipulated for the product. It has therefore been considered vital to improve the planning system to avoid this kind of problem in the future. One important part of this is to computerize the actual planning to facilitate the construction of alternative plans, and to simplify replanning as and when more information becomes available to the planner.
2.3.2. Mining and Production Constraints
There are a number of constraints that have to be satisfied. These can be divided into three main groups, namely;
1. Production requirement constraints 2. Sequencing constraints
3. Practical mining constraints
2.3.2.1. Production requirement constraints
The production goals, for the total production and the production of Bl-ore, should be fulfilled all time periods. This can be described in the manner below.
2 Ton mined (t) = Required total tonnage (t) for all time periods (t)
B1-ton mined (t) = Required amount of Bl-ore (t) for all time periods (t)
2.3.2.2. Sequencing constraints
The ore blocks must be mined in a certain order. The ore blocks are grouped together so that all the blocks in the group use the same shaft group. In all the groups, one of the blocks is considered to be most adjacent to the shafts, this block must be mined last in the group. The other blocks must then be mined with the outer blocks first, and then successively in towards the "last block". The sequencing constraints within a shaft group can thus be described as in figure 2.4.
Shaft group boundary
A Horizontal section of ore body "Last block"
Figure 2.4. Horizontal sequencing constraints within one shaft group.
There are also vertical sequencing constraints, which mean that the over-lying blocks - there are 3 of them as in the "standard" 2D open pit model - must be partially mined before the mining of the underlying block can begin, see figure 2.5.
Level A
Level B
\
7
mined area
Vertical section
Figure 2.5. Vertical sequencing constraints, 30 - 50% of each predecessing block must be mined before the mining of the successor block can start.
2.3.2.3. Mining constraints
There are also some practical mining constraints to consider, these are;
1) The maximum on the number of loaders working in the shaft group simultaneously is set at
two. This is mainly due to the fact that electrical LHDs are used, and there is danger that their cables may run over by other loaders. This constraint will be referred to as the "loaders per shaft group" constraint.
2) A block where mining has begun must be mined out more or less continuously for gravity flow reasons. This means that there should be no time slack between the ore zones, e.g. i f mining of zone 1 is completed at time T, then zone 2 of the same block should start at time T or T+1. This constraint will be referred to as the "continuity constraint".
3) The loading capacity must be within range, this means normal capacity ± 2 0 %. The loading
capacity varies depending mainly on the type of loader and the distance. There are also other important - but unpredictable - parameters which influence the loading capacities, they are primarily related to the gravity flow. The available loaders (mainly TORO) and the mining situation gives an average capacity of about 110 kton/(per month, per loader). This constraint will be referred to as the "capacity constraint".
3. PRODUCTION PLANNING USING OPERATIONS RESEARCH TECHNIQUES
The aim of this chapter is to give a short presentation on operations research (OR), and how it has been applied to mine production planning. This chapter also indicates the need for a new method of approaching the problem of multi-period production scheduling for underground mines. Some scheduling approaches have been considered and discussed. This chapter assumes the reader is not too familiar with OR in mining and serves as an introduction to the issues involved.
Some of the text book material presented in this chapter is taken from Taha (1992) and Hillier and Lieberman (1989), and will not be specifically referred to in the text. A number o f references on production planning in mining are given to show what has been accomplished in this area, but also because it is useful to have them gathered in one place, especially for the reader with limited experience of OR in mining.
3.1. Introduction
The aim of the model is to create a schedule for long range (5-10 years) planning in the Kiruna mine. The schedule is divided into a number of time periods (approximately about 1-3 months) and guarantees that certain production requirements over these periods can be fulfilled.
How could this be achieved? A straightforward, and naive, way to deal with this problem is by exhaustive enumeration, i.e. to compare all the feasible plans and then, by some criterion, choose the best one. It is clear that this task would be an enormous one even for a rather small mine, not to speak of the Kiruna mine.
Consider the following case that concerns how to choose ore zones for just one period. The problem has the same magnitude as the Kiruna case. It is assumed that no practical constraints apply to the problem, i.e. the problem is reduced to choose a number of ore zones to satisfy the production goals. I f 20 ore zones are needed to obtain the required production volume, and a total of 140 ore zones are available, the number of possible combinations is easily calculated, see below.
The number above is large, and remember that only one period is considered. I t seems as it is a good idea to turn to the use of some other approach while waiting for faster computers. The science of operations research offers such approaches to find more effective ways to solve this problem, these techniques will be discussed further below.
3.1.1. Operations Research in general
Operations in the industry of today are frequently very large and highly complex. There is a tendency towards more segmented industrial organisations (companies), which often consist of a number of autonomous parts with their own specific goals. The combinations of these concurrent activities often lead to sub-optimization due to the lack of perspective resulting from the integration to one whole. This somewhat limited vision leads to problems when deciding how to use and allocate resources, which often results in inferior solutions. This is why operations research techniques are needed, to integrate the different parts of the enterprises in an optimum (or close to optimum) way. Thus, operations research is applied to problems concerned with how to conduct and co-ordinate the operations or activities within an organisation.
In principle, an operations research model consists of two parts:
1. The objective of the system
2. The constraints imposed on the system
Both the objective and the constraints must be expressed in terms of decision variables, which represent the courses of action. The analysis of the model should then yield the variable values that give the optimum result in terms of the objective, and satisfy all the system's constraints. The decision variables will thus suggest the best course of action under the given
circumstances.
3.1.2. Operations Research applied to mine production planning
Production planning, in a wide sense, is one of the most important areas in which operations research techniques are applied. Production scheduling in mining, i.e. when and where to do something specific, is one part of that area.
A number of attempts to apply operations research techniques to production planning, using a variety of approaches, have been made. Some have been more successful than others. The major obstacle so far has been the complexity, and the uncertainties, connected with real life problems. Management often expresses a desire to have a model that considers a multitude of restrictions. The enormous complexity of the real world is easier to handle with some types of modelling, which to some extent will be discussed in this section. A model is a simplification of the real world, see figure 3.1, and it is therefore essential that the simplification is made in a way that ensures that the essence of the real world system is preserved.
In the case that is the concern of this thesis, the real life situation causes a combinatorial problem of formidable magnitude. This induces the modeller to make simplifications to make the model tractable, which invariably results in a loss of accuracy. Some modelling techniques will be presented in the following sections.
3.2. Different OR-techniques for production planning
3.2.1. Network Techniques
3.2.1.1. Introduction
It is often very useful to represent an operations research problem as a network, in which the arcs and nodes are assigned certain qualities or meaning. The arcs could for instance represent distances, times or durations, flows, etc., and the nodes could represent similar quantities, or different events. The network then constitutes a framework for the model, and naturally indicates the dependencies between the different arcs and nodes. There exists a number of network applications, some of them are modelled as linear programming problems, others are
modelled in a more straightforward way. Elmaghraby (1977) gives a comprehensive presentation of different network applications with an emphasis on models concerned with time- and resource scheduling.
Network modelling is, as implied above, the most common approach to the time scheduling o f different kinds of projects. The objective is usually just to produce a time plan, with as short a project duration as possible. The techniques most frequently used are CPM and PERT, see also chapter 8, and they are also described in all operations research literature of the type "An introduction to...". The extension of these techniques to incorporate time vs. cost trade off and/or to include resource availability is also rather common.
The approach that is most similar to the production planning problem is the resource scheduling (levelling) model that is described below.
3.2.1.2. Resource scheduling (levelling) approach
Resource scheduling on CPM-networks is closely related to the production scheduling problem. The aim in the resource scheduling case is to produce a time plan where the need for the different resources must not exceed a certain level, i.e. to keep the resource requirements under a certain level and to conclude the project in as short a time as possible. The production scheduling case is just the opposite, i.e. to keep the production level above a certain
production rate. This can be achieved to a certain extent by using a technique analogous to the resource levelling technique, i.e. to try to attain as constant a resource usage as possible. However, in the latter case it is preferable if the production rate is as close as possible to the one required, as low resource utilization is not as fatal to the plan as an excessively high production level.
The most common way to approach the resource scheduling (levelling) problem is to use some kind of heuristics, i.e. to use empirical decision rules, examples applied to mining are presented by Almgren (1989) and Andersson (1992). Heuristic approaches are used because analytical solutions to real life cases are not feasible in practice due to their mathematical complexity. Elmaghraby (1977) provides a good description of resource scheduling approaches.
There have also been attempts to apply expert system techniques to the resource allocation problem, see for example Gudes, Kuflik and Meisels (1990).
3.2.2. Dynamic Programming (DP)
Dynamic programming is a technique where the problem is divided into sub-problems, i.e. a technique to solve the problem in stages, where each stage involves exactly one optimizing variable. The computations are linked in a recursive manner that yields a feasible optimum to the entire problem when the last stage is reached. This principle is stated by Bellman (1957) and is generally referred to as "the principle of optimality". The recursion principle in dynamic prograrnming can be described as:
f0( x o ) = 0
fj(xj) = max (min) [ R j + f j - l ( x j . i ) ] , j=l,2,3 (3.1) feas.alt.
Dynamic programming does not seem to be a suitable technique for the problem at hand. It is much more appropriate for open pit planning, which deals with specific block values (even though they vary over time). However, the complexity of the problem is the major obstacle for both types of mine. Consider, in an underground mine, i f the planning periods were chosen as stages. It can easily be seen that the amount of available sets of production blocks to choose from every time period would result in an unreasonably large state vector, especially as it is also possible to mine parts of blocks. It is also hard to find an optimizing variable that works. One alternative that could be imagined is the total deviation from the desired B1-level. However, there is a number of other parameters that are of importance.
Dynamic programming applied to production planning in mines has been described by Onur and Dowd (1993), Muge et al. (1992), Muge and Santos (1989), Wright (1987,1989), Dowd and Elvan (1987), Ribeiro (1982), Riddle (1976), Dowd (1976) and Noren (1969).
3.2.3. Linear Programming (LP)
3.2.3.1. Introduction
A linear programming model consists of one objective function and a number of constraints. Both these categories have to be linear equations or inequalities, and contain common decision variables. The solution to the problem, if one exists and is unambiguous, satisfies the
constraints and gives the best possible value in the objective function. The objective function can be of the maximize or minimize type.
The formulation of a linear programming model can thus be defined as. Maximize z = CX (3.2) Subject to: A X = B X > 0 or Maximize z = 2~^"=i ° jxj £ " = 1 ai Jxj — bi i=l Å . -.m (3.3) xj > 0
The following definitions are often used:
z Value of objective function, e.g. profit bj Available amount of resource i aj j Need of resource i for activity j cj Profit on activity j
xj Number of times activity j is performed
One interpretation of this formulation might be that it represents one of the most common problems to which linear programming is applied, namely the allocation of limited resources to competing activities in the optimum way. The values of the parameters would then represent a problem concerning n activities using m resources.
The standard linear programming formulation and solution method (the simplex algorithm) also gives the modeller the ability to carry out sensitivity analyses on the results. It is also possible to obtain the limits within which the solution is valid using simple methods.
3.2.3.2. Ordinary Linear Programming
A number of problems can be defined as ordinary linear programming models. There is one difficulty that might occur, the size of the problem, large LP-models can sometimes be very time consuming to solve. There are, however, certain types of large linear programming models where it is possible to take advantage of the structure of the model by using a streamlined solution procedure. Such is the case with the "max flow" model, where the labelling technique was introduced by Ford and Fulkerson (1956); also the transportation model is an adapted form of the simplex algorithm.
Another way to handle large models is by decomposition. This is possible with so called multidivisional problems. These are problems where the divisions operate with considerable autonomy, i.e. the problem is almost decomposable into a number of separate sub-problems. However, some overall co-ordination is required in order to divide certain global resources among the divisions, see figure 3.2.
Common part with resources needed by all sub-problems
Sub-problem 1
Sub-problem 2
Sub-problem 3
Sub-problem 4
Figure 3.2. The principle of decomposition of LP-problems, the figure depicts the A-matrix in equation 3.2.
An example of the decomposition of LP-models is described by Dantzig and Wolfe (1960). Decomposition in mining situations is also used by Gershon (1993), Tachefine, Soumis and Vanderstraten (1993) and Dagdelen (1985).
Gershon (1983) and Barnes and Johnson (1989) describe the use of ordinary LP-programming applied to mine production scheduling.
3.2.3.3. Blending to achieve quality control, mostly found in coal mining
The blending problem is one of the classical LP-problems, which linear programming is very well suited to handle. It answers the question as to how much of each ingredient should be taken to satisfy the content requirements of the blend, and at the lowest cost. Linear
programming is extremely useful when it comes to calculating the optimal proportions of the ingredients. It has often been used in mining, especially in coal mining where parameters such as sulphur, ash and BTU must be balanced against each other, see for instance Long (1982), Van Drew (1985) or Gunn and Rutherford (1990). Blending in mining has also been discussed by Gershon (1986).
The blending technique in mining is most useful for the short range planning, when information concerning the content of the ingredients is hopefully sufficient.
3.2.3.4. Integer Programming fMixed Integer Programming - MIPl
Integer programming is the linear programming technique where discrete (integer) variables can be used, which is not the case for standard LP-models. This is the technique that normally has to be used for scheduling. Multi period scheduling cannot be performed with continuous variables in any straightforward manner, it is then only possible to let the execution of the preceding activity be more advanced than that of the succeeding activity (i.e.
X(predecessor,time) > X(successor,time), where X denotes the amount of the activity that has been concluded). It is not possible to postpone the start of the successor until the predecessor has been fully completed. The principle difference between continuous and discrete variables in sequencing is shown in figure 3.3.
Predecessor activity Duration of activity y Time Successor activity Continuous X
Feasible interval for successor activity
Figure 3.3. Principle of the fundamental difference between continuous and discrete variables in sequencing, where X is the sequencing variable.
Mixed integer programming (MLP) - solving techniques an applied to linear programming problems where some of the variables are discrete (integer values) and not continuous, which means that the simplex method cannot be applied directly, as continuity of the variables is one of the basic assumptions that is made in the simplex method. This means that the integer variables must be taken out of the LP-model and substituted by constants. The problem is that the values of these constants (i.e. integer variables) are not known, so every possible
combination of values of the integer variables has to be tried, which gives rise to a
combinatorial problem of great magnitude. The trial of each of these combinations means that one LP-model has to be solved. It is possible, however, to reduce the number of calculations by using some clever scheme which utilizes the fact that almost the whole problem stays constant (i.e. there is little difference between two adjacent sub-problems) and thus simply recompute a small portion of it.
Fortunately, there are techniques that are more intelligent than an exhaustive search. Two techniques are of importance.
Branch and Bound
The first one is the "Branch and Bound" technique. The solution space of an integer
prograrnrning problem is in fact divided into a number of separate parts (solution spaces), and it is not known in which part the optimum solution lies. This means that an LP-model has to be solved for each separate part of the solution space in order to find the solution, and to
guarantee that it is optimal. The branch and bound technique, which is in fact a search technique, is divided into two parts, branching and bounding. The solution procedure starts by solving the relaxed problem, i.e. all integer variables are relaxed and can assume "continuous" values. The branching part then takes care of the combinatorial aspects, i.e. it implicitly generates a number of sub-problems (i.e. new nodes in a search tree) from its starting solution where each sub-problem represents a new, more constrained, solution space, see figure 3.4.
X = 0,i,2,3,..., (i.e. integer variable)
x<=3
r r
Relaxed problem x>=0 .x>=4 Solution space to the relaxed problem3 4
The true (discrete) solution space (in 7 parts)
3 4 x
Figure. 3.4. Branch and Bound Technique applied to an integer programming problem, the solution space of the relaxed problem is divided into two new solution spaces.
Each node on the search tree constitutes a new linear programming problem. The branching part then chooses to examine the most promising sub-problem (e.g. best value on the objective function), which will then form new search trees. The branching part is helped during this process by the bounding part. The objective value of each node (sub-problem) offers the best bound for its particular branch, which points out branches (subordinated search trees) that cannot contain the optimum. These branches, and solution spaces, are then "cut o f f ' from the search tree, and need not be considered in subsequent calculations.
The best bound for a sub-tree is thus very useful when an integer solution is found, because it is then possible to exclude all sub-trees with a best bound that is worse than the best solution found. It may therefore be very useful i f a feasible (integer) solution is known prior to the calculations, as this may significantly reduce the size of the search tree. One way to find such a solution is to use some kind of heuristics.
Heuristics
The second technique is to use some type of heuristics. Heuristic rules, together with the linear algebra of the revised simplex method, may often give a much faster, optimum or close to optimum, answer to an MIP-problem. Even in the case when the answer supplied by the heuristic routine is not in the neighbourhood of the value of the relaxed problem, it is still valuable as a best bound for the branch and bound search. An example of a heuristic algorithm that reports good results is presented by Balas and Martin (1980).
Comment
It is far from certain that the optimum solution is found (i.e. proven) in integer programming. Most often, a close to optimum (or not proven optimum) solution is regarded as acceptable.
Applications
Binary integer variables are used in most applications. They are variables that can only assume two values, 0 and 1. Two important areas where integer programming, and usually binary variables, are used is the sequencing (scheduling) problem; and when to make discrete choices from different alternatives (e.g. in discrete systems with a finite set of possible solutions).
In general, the smallest amount of integer variables that are needed for a scheduling (sequencing) model can be calculated as the number of blocks times the number of time periods.
Integer programming in mining applications can be found in Gershon(1983), Barnes (1986), Barnes and Johnson (1989), Chanda(1990:1), Kim and Cai (1990) etc.
3.2.3.5. The use of Lagrangean Multipliers
This technique has been described from the operations research analyst point of view already by Everett i n (1963), however the technique does not seem to have had any major impact on mine planning yet. However, mine planning applications are described by Francois-Bongarcon and Maréchal(1976), Bongarcon and Guibal (1982), Dagdelen &
Francois-Bongarcon (1982), Dagdelen & Johnson (1986), Barnes and Bertrand (1990) and Wang and Sevim (1992, 1993). Zhao and Kim (1993), however, discuss a serious problem (the gap-problem) that is associated with the technique, this problem will be briefly discussed later in this section.
The Lagrange multipliers is a technique for converting optimization problems with constrained resources into unconstrained maximization (minimization) problems. A complicated mixed integer programming problem may for instance be converted to its lagrangean that automatically offers an integer solution to the relaxed problem (unimodularity).
The following theorem was presented by Everett HI (1963), although that particular formulation was stricter:
Consider the real valued payoff function H(x), and it is interpreted as the payoff which accrues from employing the strategy x G S, where S is the set of possible strategies, k denotes the resource, and n the number of resources.
L ( x ) = H ( x ) - 2 L i Åk*Ck(x) (3.4)
According to Everett i n , i f an arbitrary set of non-negative Åy-'s is chosen, and i f an
unconstrained maximum of the modified function L(x) is found, the result will be a solution to a constrained problem. In general, different choices of the Å^s lead to different resource levels, and it may be necessary to adjust them by iterative search (e.g. trial and error) to achieve any given set of constraints stated in advance.
There are primarily two drawbacks with the technique. The first is to find the multipliers that correspond to the optimum, or close to optimum, solution. The iterative search is ineffective for most real life problems, as they are too complex. The second drawback is the gap phenomenon. The gaps are regions of the solution space that are inaccessible using the lagrangean relaxation technique. This problem is discussed by, for example, Everett i n (1963), Wang and Sevim (1992, 1993) and Zhao and Kim (1993).
An approach to find Everett's lagrangean multipliers is given by Brooks and Geoffrion (1966). The use of lagrangean relaxation in integer programming is described by Geoffrion (1974), Fisher (1981) also uses the lagrangean relaxation technique to solve integer programming problems. Lagrangean relaxation to solve resource constrained network scheduling is shown by Fisher (1973).
Reserve parameterization
Reserve parameterization, which is an open pit planning technique, was suggested by Matheron in 1975. It is a lagrangean multiplier technique and was used by Francis-Bongarcon and Maréchal (1976), Bongarcon and Guibal (1982), Dagdelen &
Francois-Bongarcon (1982) and Wang and Sevim (1992, 1993).
A mining project, i.e. a prospective open pit, is defined by three global parameters; V - total material tonnage; T - tonnage of ore; Q - the metal quantity. I f there is more than one feasible pit of a given size (V,T) only those which maximize the objective have the potential to be optimal. One suitable objective would be to maximize the metal content of the pit.
Thus, the objective of reserve parameterization is to find, among all the possible pits with the same V and T, the one that contains the maximum quantity of metal. The pit is referred to as a "technically feasible optimum pit". In practice, it is not possible to look at every conceivable combination of (V,T).
The problem can instead be solved by maximizing the quantity A, see below, instead of maximizing Q directly, Francois-Bongarcon and Guibal (1982).
max A = Y \ ( Q j - X*T\ - <9*V;) where i e <f> (3.5)
where
V j = Total tonnage in the it n block
Tj = Ore tonnage in the im block
Qi = Quantity of metal in the it n block
X,8 = Lagrangean multipliers
<f> = The set of blocks that provides the required V and T,
Double parameterization is very complicated, and can be avoided by holding one of the parameters constant and performing single parameterization at multiple values of the fixed parameter. This will, however, result in a large number of problems to solve.
3.2.3.6. Stochastic Programming
Stochastic programming is a branch of linear programming that allows constraints of a more probabilistic nature, i.e. uncertainty can, to some extent, be incorporated in the model. Charnes and Cooper (1963) describes the constraints that could be used as
P(AX<B) > a (3.6)
where
P(S) denotes the probability of S or is a vector of probabilities
The equation can be interpreted as; the probability (risk) of exceeding the available amount o f resource (i) must be lower than ( l - a , ) .
Stochastic programming in mining applications is described by Albach (1967), Gangvar (1982) and Barnes (1986).
3.2.3.7. Goal programming
It is not always possible to formulate one single overriding objective, such as maximizing profit. Goal programming is an LP-technique to try to obtain several goals simultaneously. The idea is to specify a numeric goal for each of the objectives and formulate an objective function that minimizes the weighted sum of the deviations from these objectives, i.e. their respective goals. The weights can of course all be set to one, which means that no weighting is performed, i.e. under the assumption that the goals are normalized in this some way. There are two kinds of goal programming, preemptive and nonpreemptive. The iatter will be further discussed, and referred to as goal programming.
The objective function can thus (somewhat simplified) be formulated as
minimize Z = Y \ Wi* | yj | (3.7a) where
W j is the weight of the 1t h goal Gj
yi is the deviation from the 1t h goal
i.e. yj = x; - Gj
However, auxiliary variables have to be introduced as the functions are made up of two linear parts. This can be solved in the following way.
yi = y+i - y"i
where y+i , y"j > q
This means for example that y+j = yj and y , = 0 i f yj > 0, as one of the variables y+; and y ;
will always be zero in the simplex method.
The absolute value of the deviation (Dj) can easily be calculated as below;
Dj = | y i| = y+i + n
which results in the objective function
minimize Z = J \ W j * (y+j + y"j) (3.7b)
However, the two linear parts of the objective function do not need to have the same slope. For example, consider an inventory example. I f Zj represents costs, and y; the size of the stock, which are desired to be as low as possible. I f there is not a sufficient amount of the product available in stock when needed, this will incur a cost. Too large a stock of the product, on the other hand, will incur capital costs. The optimum is thus to have exactly the required amount at hand. The cost (Zj) is thus a function of the deviation from the desired stock size (yj). This is described in figure 3.5 below.
Figure 3.5. Z j = f t j j ) ; the it h component (i.e. goal i ) in the objective function. It is more
expensive to be unable to produce the product when needed, than to have too much in stock.
Thus, i f w j denotes the cost of insufficient stock, and W2 denotes the cost of excessive stock; w j > W2
Zj is calculated as
Z, = w i * n + w 2 * y+i (3 7 c)
The unlinear function Z j has thus been divided into two linear parts, and can be used in a standard LP-formulation
Goal programming in mining applications is described by Jawed (1993), Huang (1993), Chanda(1990:2) and Jordi and Currin (1979).
3.2.4. Simulation
Pure simulation is not believed to be an efficient approach to the multi period production planning problem. Heuristics is a more efficient method if a "practical" approach is desired. Simulation can, however, be used together with some other approach, see Huang (1993) and Chanda (1990:1).
3.2.5. Heuristics
Heuristics can be defined as a method to use empirical decision rules. This leads to better than average results, but not necessarily the best. The distinctions between analytical and heuristic approaches are thus very clear. The difference between a heuristic and a simulation approach is not very apparent as both techniques involve trial and repeated experimentation. However, in the heuristic case there are internal decision rules that are capable of modifying the answer, these kinds of rules are not present in a simulation model. A flow chart of a plausible heuristic model for production planning is given in figure 3.6.
T=l
Are any blocks started ?
No
Choose best candidate block
No
Are period requirements fuiniled '
Yes
Are all periods planned ? No
£ e s
Stop
T=T+1
Figure 3.6. Flow chart of plausible heuristic model for production planning; T = period.
Heuristic approaches to resource scheduling in general are comprehensively described by Elmaghraby (1977). A heuristic algorithm that, to some extent, considers production rates is described by Almgren (1989) and used in Andersson (1992).
3.3. Geostatistics and Ore Reserve Calculations
Geostatistics is not a central subject in this thesis. However, most mine planners are aware of the importance of good ore block information, or i f it is not good, to know how bad it is, i.e. to evaluate the reliability of the available block data. Geostatistical ore block variances may for instance be a good first step to be able to, in a reliable way, apply stochastic programming to mine production planning. It is also useful to have geostatistical data to be able to evaluate the reliability of existing production plans. Ravenscroft (1992) discusses the use of conditional simulation as a means to perform sensitivity analysis on production plans.
There are numerous papers on different aspects of geostatistics, examples of mine planning references can be found in Barnes (1986), Gangwar (1982), Kuchta (1990), Parker (1979), Journel (1979) and Zuzhao and Xiong (1986)
3.4. Comments on OR-techniques in general
A general remark concerning the effectiveness is that LP-based algorithms have more to gain from the rapid development of computers using parallel processing than algorithms based on dynamic programming and graph theory.
Interesting techniques are developed in the area of neural computation. Applications on optimization problems are given by Hertz, Krogh and Palmer (1991).
3.5. Available suggestions for mine planning applications of OR planning techniques
The major applications so far have mainly been in a few specific areas. Two main types of applications can be distinguished, depending on whether the time factor is present or not. The type when the time factor is present is also referred to as multi period planning. Some of the applications when the time factor is not present are presented briefly in this section to give an overview of the situation, and to make it easier to see where the multi period production planning part fits into the larger whole.
The most common production planning techniques are presented below, but first - a short comment on the use of ore blocks in these approaches.
3.5.1. The use of aggregation and dis-aggregation of ore blocks in production planning
The division of the ore into blocks is normally done in mine planning applications. The number (or size) of these blocks is often of great importance, as few blocks make the model easier to solve. The use of the aggregation of blocks is therefore a procedure that is often suggested, for instance by Gangwar (1982), Gershon (1982) and Barnes and Johnson (1989), to increase the size of the blocks and thus decrease the size of the problem (i.e. fewer blocks). This might be a good approach as long as the blocks that are aggregated have a similar content, i.e. the "new" larger blocks are homogeneous. But the approach might lead to misleading results when the new blocks are heterogeneous. For example, consider the following case which is similar to that of the Kiruna mine, see figure 3.7.
new block
high phosphorus grade
medium phosphorus grad
low phosphorus grade
Figure 3.7. Demonstration of aggregation of three small blocks into a "new" large block, observe that the planning grade for the new block might never be correct.
Consider the situation when production is carried out on too many blocks which are in unfavourable positions (e.g. the phosphorus grade is underestimated) simultaneously. This may result in significant difficulties in achieving the required production goals. The situation might get even worse i f there is cyclical behaviour in the plan, so that many actual block grades deviate from the planning grades in the same way simultaneously period after period. This means that the required production grade will never be achieved, as the actual production grade is always too high or too low.
The approach presented by Barnes and Johnson (1989) is an interesting attempt to get around this problem to some extent. This approach will be further discussed in this chapter.
The result of this "homogenization process", caused by the aggregation, is that some kind of slack has to be introduced to secure the production of sufficient amount of the required grade. Such slack will render the plan sub-optimal. It is also difficult to allocate this slack in an efficient way. The conclusion is that the aggregation of blocks should be avoided as far as possible.
Closely connected to the issue about aggregation of blocks is the question concerning what block size to choose. All blocks are, in a sense, the result of an aggregation process, i.e. aggregation is always performed. The smallest realistically conceivable block in the Kiruna case would be the content of one bucket from the loader. It is, however, impossible to predict the characteristics of such a small quantity of ore with any accuracy. Larger blocks are therefore required. I t is likely that there is one block size that is "optimal", which in this case means that the variations in ore qualities are reflected without loosing too much of accuracy in prediction of the block contents. A balance between these two characteristics is desirable. This issue is obviously more complicated than it appear at first sight.
3.5.2. Ultimate pit design - no time factor present
Ultimate pit design deals mainly with the task of designing the ultimate pit limits that maximize the net present value of the pit. The mining blocks are given net values, and are subjected to pit slope constraints. The most well-known techniques are:
3.5.2.1. Floating (moving) cone
Floating cone is a heuristic procedure to estimate the ultimate pit limits. The procedure cannot guarantee that an optimum solution is obtained. The moving cone algorithm is presented in Lemieux (1979).
3.5.2.2. The revised Korobov algorithm
Dowd and Onur (1992, 1993) describe an ultimate pit limit method that was initially devised by David, Dowd and Korobov (1974). The original technique, which is reported to be sub-optimal, uses an attractive scheme (heuristic) that let the positive blocks take out negative ones. The main idea of the algorithm is to allocate a cone to every positive block throughout the pit, and to allocate positive blocks within the cone against negative blocks within the cone, until no negative block remains. A correction to the original method has been made, which is claimed to make the algorithm "optimal". The correction is as follows: " I f two or more cones have blocks in common, the blocks not in common must be paid for first; common blocks are only paid for after all blocks not in common have been paid for". This new algorithm is reported to be faster than the Lerchs-Grossman, a method that is presented below, see figure 3.8. 160 120 100
00
<D 3 _c £ 80 v—' 03i
6 0 p 40•ii
5 10 15 20 25 30 35 40 45Number of blocks x io
3Figure 3.8. Comparison of computing times for the Koborov algorithm and the Lerchs-Grossman algorithm, the coding made by Dowd and Onur (1992).
3.5.2.3. Lerchs and Grossman
The most generally used algorithm was originally described by Lerchs and Grossman (1965), and is based on graph theory. The algorithm offers an optimum solution to the 3D ultimate pit limit problem.
All blocks are represented as nodes, with the block value as node values. They are all
connected to a root. Arcs initially connected to positive nodes are classified as strong and the others as weak. They are overlaid by each other in a way that corresponds to the sequencing constraints, see figure 3.9. By using a scheme that forces nodes on strong arcs to carry overlaying weak nodes, the graph is transformed until all strong arcs carry their weak
predecessors, or until there are no strong arcs left. All strong arcs must emanate from the root, strong arcs emanating from blocks (i.e. not the root) that arise during the process, are moved down to the root by a procedure called normalization. Nodes (blocks) on strong arcs are mined. j = 1 k = 1
Block value
s - strong arc w = weak arcResearch in the area of ultimate pit limits primarily aims to increase the efficiency of the Lerchs-Grossman algorithm. For instance, Zhao and Kim (1992) present a new way to solve the 3D ultimate pit problem more effectively using the Lerchs and Grossman approach.
3.5.2.4. Integer programming
This straightforward ultimate pit limit problem can easily be modelled as a simple IP problem.
1-1, j , k
M,j,k+1
Figure 3.10. The structure of the block model. Maximize £ .J j t V y k* X y> k
Subject to
xr,s,k-l - xi j , k ^ 0
where
X is binary variable and
(3.8)
(sequencing constraints) r = i - l , i , i+1
s = j - l , j > j+1
X= 1 the block is part of the ultimate pit X= 0 the block is not part of the ultimate pit
V the net value of the block
i j the co-ordinates of the block in the plane
The above problem is formulated as an integer programming problem. However, this formulation has the so called unimodularity property, which means that even the relaxed problem (i.e. 0 < X < 1) always results in an integer solution. The problem can thus be solved as an ordinary LP, and will still yield an integer solution. However, it still requires an
enormous amount of calculating power for an ordinary open pit. A number of attempts have therefore been made to find a more efficient technique to calculate the optimum ultimate pit limits. Many of the attempts have divided the pit into a number of two-dimensional sections, which are optimized and then connected in some way.
It is also possible to exclude a number of blocks from the model, (i.e. blocks that are obviously in, or outside, the pit), An example of this is given by Barnes and Johnson (1982). However, there exist other formulations based on LP-technique that calculate the true 3D optimum ultimate pit. Most famous is Johnson's max flow algorithm, another approach which is based on the transportation algorithm is presented by Huttagosol and Cameron (1992).
3.5.2.5. The Max Flow Algorithm
Johnson (1968) recognises the network structure in the ultimate pit problem and formulates it as a max flow problem. Two sets of nodes are used, the first set contains nodes with positive block values, and the second set contains negative blocks. Arcs connect the source with the first set, where the block values constitute arc values (upper boundary of flow), another set of arcs connects the negative blocks with the sink, the absolute value of the blocks indicates the arc values. The two sets of nodes are then connected with arcs corresponding to the sequential constraints in the pit model, these arcs allow infinite flow. The flow problem may then be solved using a labelling technique devised by Ford and Fulkerson (1956), see Johnson and Barnes (1989), where the nodes labelled after the last iteration represent the ultimate pit. An example of the max-flow technique is given in figure 3.11.
k= i
-2 -2 -1
-3-"
4
5
Block value
Dependencies (infinite capacities)
Positive block values
Negative block values
Figure 3.11. The ultimate pit problem formulated as a max flow network.
3.5.2.6. The Transportation Model
Another LP approach based on the transportation algorithm is presented by Huttagosol and Cameron (1992). They use the special case of the simplex method adapted to the
transportation problem. The blocks with positive block values constitute supply nodes and negative blocks constitute demand nodes. Excess supply indicates which blocks should be mined.
3.5.2.7. Other approaches
Dynamic programming approaches are described in a number of papers, e.g. Koenigsberg (1982) and Wright (1987,1989).
