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By K. G. Lomberg

The integration of various highly sophisticated IT applications has been implemented at Bafokeng Rasimone Platinum Mine, a new platinum mine in the Anglo Platinum Group. This system is required to assist with compiling the necessary geological and grade information together with safety, environmental, ventilation, rock engineering and logistical constraints, to allow the design of the appropriate mining layout, to schedule this design, provide the necessary output for technical review and costing as well as budgeting to allow medium- and long-term business planning. In addition the production management and short-term planning needs must be met. This includes the plotting of all underground and surface infrastructure and excavations, giving the necessary mining and best cut instructions, daily production management, ore reserve and grade management as well as assisting with budgetary management. In addition all safety, health, environment and risk assessment requirements and legal requirements have to be met. The system at Bafokeng Rasimone Platinum Mine has met these criteria. Keywords: Platinum, Bushveld Complex, Mine Planning, Production Management

Jan 1, 2003

A knowledge of the steady-state mass and energy requirements of a particular process is essential to a determination of the economic feasibility of that process. A predictive simulation technique can be of great value in this respect. A versatile computer program based on a fundamental thermodynamic approach has been developed for the prediction of process requirements. The equilibrium composition of a multiphase system is determined by the technique of free energy minimization. Energy-balance calculations are performed using the appropriate thermodynamic data for the chemical species present in the composite materials used. All models for the process units under consideration are written in modular form, the output being calculated from the input and the specified operating conditions. The computer program is written in such a way that it can be used easily by anyone familiar with the processes involved, even though he may have no previous experience with a program of this type. The program is structured so as to facilitate the varying of one parameter without any alteration to the others, thus simplifying the investigation of the effects of any parameters considered to be important. The program has the advantage of being able to compare existing and new processes on the same basis, namely from the ores to the molten metal.

Jan 1, 1987

By A. J. Roets

Despite its growth over the years, necessitating ever¬larger office accommodation, the Gold Fields group changed its head office address in Johannesburg only twice in 94 years. The first move was at the turn of the century, from the then select suburb of Doornfontein to the corner of Fox and Simmonds Streets. The second move was in the thirties, from that address-now the Reserve Bank building-to the well-known address of 75 Fox Street. With the advent of the first mines of the West Wits Line by the end of the thirties, 75 Fox Street proved too small, and the eastern section of the Commissioner Street side of the present head office was added on. In the fifties, when space once more became a serious problem, the old Exploration building was purchased and demolished, and the present west wing on Commissioner Street was built on the site. In the early seventies, cramped conditions again made life difficult and, despite a reorganization of the interior layouts, the need for expansion remained critical. Eventually, the adjacent Hollard Place and parking area became available, and plans for an extended new building complex could he considered. The building project was

Jan 1, 1985

By J. H. Selby

The Southern African Institute of Mining and Metallurgy hosted the 6thInternational Heavy Minerals Conference at the Zulu Nyala Game Lodge near Hluhluwe in Zululand, during September 2007. This series of conferences has run for 10 years and has been held on three continents. The interest that has been shown in the conference over the years has demonstrated that the field of heavy minerals is a dynamic and profitable one. It is perhaps appropriate that the tenth conference was held in Zululand, South Africa, where it had its roots. The conference theme was ?back to basics?, and this implied revisiting many of the technologies that made this industry a great one. In doing so, we got a view of how those basic technologies have been put to new applications. There were 39papers accepted for the conference, covering all aspects of the industry. These were presented over a period of four days. This edition of the Conference Proceedings provides the reader with a permanent record of these papers. We hope this volume will join the other five, which this series of conferences has produced, and give a lasting record of the development of our industry.

Jan 1, 2007

By S. Ramatsoma, D. N. C. Phillpotts, B. Whitehead

Changes in flotation performance have been correlated to changes in froth structure, velocity and stability. Literature has shown that when the aeration rate is varied for a specific float cell, the recovery of the valuable mineral reaches a peak which also coincides with a peak in the air recovery – a measure of froth stability. Peak air recovery optimisation methodology is a way to use direct readings of physical phenomena by froth cameras to optimise the metallurgical performance of a flotation plant. The aim of this paper is to show the transient nature of the peak air recovery value and the variety of conditions required to attain the peak air recovery on an industrial flotation plant with tight base-layer control of the feed and internal conditions. Industrial site tests that were conducted under controlled conditions showed different peak values at significantly different corresponding aeration rates during normal operation. The measured peak air recovery points corresponded to maximum PGM recovery during normal flotation, as expected. This dynamic nature of the peak air recovery point conditions has not previously been discussed in literature and indicates the need for online measuring equipment and informs the design of process control algorithms required to exploit the phenomenon for optimisation. Keywords: Froth flotation, flotation control, flotation optimisation, peak air recovery

Jan 1, 2020

By C. Mueller

Automation of mining operations is pushed by cost pressure and cyclical product prices. However, until now the different types of equipment were not able to communicate electronically with each other or with central computer systems. Therefore, the International Rock Excavation Data Exchange Standard (IREDES) was launched by major players in the industry. The task of this initiative is to define a ‘common electronic language’ for easy and standardized data exchange. As in other industries, this standard is expected to have significant impact on the use of automation in general and on the markets for third party components and software. The standard’s technical architecture is open and flexible for future applications. It consists of an ‘horizontal’ structure defining ‘Application Profiles’ for different types of data sets. An additional ‘vertical’ structure (‘Equipment Profiles’) implements equipment dependent definitions for each of the applicable Application Profiles. First profiles will be available for Drill Rigs. The standard bases on the XML language thus enabling easy data exchange with commercial databases. IREDES data sets may use storage media as well as networking infrastructures for information exchange. The IREDES work is carried out by the initiative members working jointly together in the IREDES work groups. It is open for participation of all interested organizations. First prototyping applications are expected to start during 2001, the first products may be available in the following year. Key words: Mine Automation; Machine Automation; Data Exchange; Rock Excavation; Standard; Remote Control; Teleoperation; Ethernet; TCP/IP; Internet Technology; Service; Quality Assurance; Control Centre;

Sep 1, 2001

By C. M. Rule

High pressure grinding rolls (HPGR) technology has come a long way in its application for comminution duty in hard rock mineral processing in the last three years. Several major projects in copper, nickel, gold, molybdenum, platinum, and iron ore have opted for this technology. Advances in wear abatement technology have established the HPGR as a reliable machine with the required high utilization and availability. The HPGR captures the imagination as the next step change improvement in ore processing, based on the following benefits: ?Energy efficiency (potentially 10?30% more efficient)?very topical in an increasingly ?green? society, and in times of rapidly increasing energy costs and availability concerns. 2008 has shown this to be especially important in Southern Africa ?High (and stable) throughput with a relatively small footprint ?Lower operation cost ?Liberation benefits ?Creative ?mine-to-metal? system improvements made possible by HPGR. The platinum industry, renowned for its historically conservative approach, has not been left behind with two significant new HPGR installations, one in Platreef application and one in UG2 application, commissioned in 2008. While the decision for the Platreef application was driven primarily by comminution circuit capacity and design imperatives, the UG2 application seeks to develop a flowsheet that will have a significant recovery benefit in addition to capacity increase and reduced operating cost. The Platreef application of HPGR also provides the opportunity to introduce a novel comminution flowsheet and to demonstrate exceptionally low wear rates treating very hard, abrasive and variable ore types. The HPGR lends itself for inclusion in existing SABC circuits to increase the capacity of the plant and mills. With current PGM prices many operations are attracted to the HPGR technology benefits. The HPGR clearly has the potential to transform entrenched opinion and provide decision makers with significant alternative comminution options. This paper aims to provide an overview of the current applications and to speculate on the transforming stimulus that the HPGR exerts on industry.

Jan 1, 2009

By E. Bakhtavar, R. Mikaeil, D. Khademi

"Fuzzy-weighted integer goal programming has been developed and used to select the most suitable site to construct a central processing plant for several adjacent small-scale dimension stone quarries, based on the most effective criteria. In the first step of mathematical modelling, the most important goals and decision variables were defined. Accordingly, an objective function was used in the model for minimizing the sum of undesired fuzzy-weighted deviations. Fuzzy weights were determined by a matrix of pairwise comparisons of various expert judgments in this field and the fuzzy geometric mean. The goal and systemic constraints were also modelled. The most suitable site for the construction of the plant was selected after solving the mathematical model in the ExcelR Solver add-in. The selected site had various advantages over the other sites, such as the shortest average distance to the quarries, the shortest distance to power access and workforce, and the lowest land price. IntroductionSelection of the most suitable site of a plant is one of the most important considerations when constructing an industrial project. A suitable site has a significant effect on a company's competitiveness in the market. Selecting a desirable location for constructing plants such as mineral processing facilities is an essential part of the economic planning that will enable a company to meet the needs of related upstream and downstream operations. These units are considered to be best built near the relevant mining sites because of technical difficulties, the cost of transporting minerals, and the traffic hazard caused by the movement of trucks transporting materials. Two main types of research using multiattribute decision-making models have been reported for selection of a processing plant site (Safari et al. 2010, 2012). In that research, the analytic hierarchy process (AHP) and fuzzy technique for order preference by similarity to ideal solution (TOPSIS) approach were applied to select the suitable locations for the mineral processing plants for phases 1 and 2 of Sangan iron ore mine."

Jan 1, 2017

By J. W. Neale

Mintek was a leading participant in the BioMinE project between 2004 and 2008. This project, which was funded in part by the European Commission, was aimed at the development of biotechnology for the minerals industry in Europe. Mintek?s research programme focused mainly on the development of integrated bioleach-based processes for the recovery of base metals from complex, low-grade sulphide concentrates. Specific European mineral resources were targeted and used in integrated piloting campaigns involving bioleaching, solution purification, and metals recovery. This paper describes the use of thermophilic bioleaching for the recovery of nickel and copper from a low-grade nickel-copper concentrate produced at the Aguablanca Mine in southern Spain. Currently, the Aguablanca Mine produces a bulk nickel-copper concentrate for sale to a smelter, and the proposition is to increase the profitability of the operation by the on-site production of metal or metal intermediate. Initially, bench-scale bioleach tests were conducted to determine the bioleach operating conditions. These tests included an evaluation of mesophilic, moderately thermophilic and thermophilic microorganisms. In order to achieve sufficiently high levels of both copper and nickel extraction, a thermophilic process was selected?this was necessary for leaching of the refractory chalcopyrite that occurs in this concentrate. Additional bench-scale test work was carried out to derive a conceptual process flowsheet for the solution purification and metals recovery circuit. The results of the bench-scale tests were used to design, construct and commission an integrated pilot plant, which was subsequently operated at Mintek for over seven months. During this time, the solution purification and metals recovery processes were optimized, and all recycle loops were closed. The final process flowsheet included the following unit operations: concentrate regrinding, thermophilic bioleaching at 70°C, primary iron removal using limestone, copper solvent extraction and electrowinning, secondary iron removal, nickel hydroxide precipitation using magnesia, and final solution purification using lime. Where applicable, process solutions were recycled to preserve water. The process design data derived from this pilot-plant campaign formed the basis for a conceptual engineering study for the developed process. In the study, mass and energy balances were derived, and a process flowsheet was developed and used as the basis for estimating the capital and operating costs of the process. This enabled a preliminary economic analysis of the process to be undertaken. The findings of this study are discussed.

Jan 1, 2009

By H. Rosenberg

Haldor Topsoe A/S has developed its WSA technology to meet the demand in the market for a simple and reliable process that converts sulphurous emissions from off-gasses into high-grade sulphuric acid. With more than 50 references in 25 countries, the WSA technology has proven its ability to generate acid at 98% concentration, efficiently and economically. The wet gas sulphuric acid (WSA) technology treats the furnace off-gas directly from upstream gas cleaning plants. No further drying is required, since the humidity present in the off-gas is used to hydrate the SO3generated in the converter and produce sulphuric acid. The heat generated during the oxidation, hydration and condensation processes is principally reused in the process to preheat the off-gas to the required temperature level, and also to generate high pressure superheated steam for process use. This advantage of steam production, coupled with lower water and power consumption requirements, is particularly important in the African production environment. Having fewer process components than conventional acid plants, the WSA technology is a simple and compact operation, which makes it easier to retrofit into an existing smelter plant. Further, the WSA technology simultaneously offers the possibility to treat multiple gas streams arising within the production facility due to superior flexibility in terms of SO2concentration and turndown ratio.

Jan 1, 2006

By L. A. Vermeulen, D. D. Howat

New distribution functions are derived to describe the size distribution of grinding elements in ball mils. The formulations are based on the assumption that abrasive as well as impactive interactions occur during bal milling-an assumption that is supported by a large body of experimental evidence. It is shown that the functions can be used in the estimation of the magnitudes of abrasive and impactive components in the total rate of bal wear. The relative magnitudes of the wear components provide a basis for optimization of the chemical and metallurgical properties of the balls in a given milling situation. However, it is contended that these quantities are also useful indicators of the relative intensities of the abrasive and impactive interactions that are operative in the size reduction of mineral particles within ball mils. These quantities can be determined for any industrial ball mil, and their magnitudes provide practical guidelines for mil operation. The theory is used in an analysis of samples of ball charges from two industrial ball mils, and it is also applied to al the data on bal-size distributions found in the literature. The qualitative correlation between the calculated values of the wear components and the reported operating conditions is good for a variety of industrial ball mils, in conformity with the hypothesis that the relative magnitudes of the wear components are related to the milling conditions.

Jan 1, 1986

[The following notations will be used: O an ore body. W a sample of size w located in O; the size of this sample. W a block or a sample, of size W, located in O; the size of this block or sample. z' a point in W. W' a block or a sample, of size W' greater than or equal to W, located in O; the size of this block or sample. z" a point in W'. y (h) point semivariogram in O. s (h) point covariogram in O. In the previous chapter, we saw that the point semivariogram y (h) usually cannot be estimated directly from the data available. Rather, the values of samples of size w are available in O, and from these a regularized semivariogram y w (h) can be estimated. The following relationship has been proved (5.8): Y w(h) = Y (w;w+h) - Y (w;w). y (w;w+h) is the average value of y (z' z") when z' takes all possible positions in a sample w and z" all possible positions in a sample w+h of the same size, h being the distance between w and w+h. y (w;w) is the value of y (w;w+h) for h = 0, i.e. the average value of y (z' z") when z' and z" take all possible positions in w. We also saw that the regularized covariogram aw(h) is related to the point covariogram a (h) as follows (5.9): w(h) Gr In the following chapters, it will be shown that calculation of the average value y of y, or -a of a, in the general case relating to samples or blocks W and W' which need not be of the same size, plays a very important part in all the practical applications of geostatistical theory. The purpose of this chapter is to introduce various methods of calculating y and a-. Furthermore, mathematical equations and graphs are provided which can be used when the point semivariogram can be represented by one of the more common equations, and when the blocks W and W' are of simple shape. By definition (Fig. 6.1): [ ] Y (W;W) = WW f dz' f y (z' z") dz" (6.1) in W z" in W']

Jan 1, 1978

By J. P. Hoffman

In 1988 the Japan Iron and Steel Federation started a research-and-development programme known as DIOS (direct iron-ore smelting). Eight Japanese integrated steel producers and the Coal Mining Research Centre, subsidized by the Ministry of International Trade and Industry, are involved in the project. The aim of the programme is to have a 500 v/d pilot plant for the production of iron via an oxygen-coal in-bath smelting converter in operation by mid 1993. The three main components of the DIOS process are as follows: ?a fluidized bed for the pre-reduction of iron-ore fumes ?in-bath smelting reduction in a converter ?off-gas reforming by the addition of coal fines. The main drive is to provide a smelting process that is more energy-effective, using coal instead of coke and/or electricity. The advantages claimed include inter alia ?direct use of fine iron ore ?direct use of non-coking coals-low coal consumption ?low capital expenditure-replaces costly coke ovens, agglomeration plants, and the blast furnace ?economical production on mini-plant scale ?less of an impact on the environment. This concept of 'direct steel has been carried over to the production of stainless steel direct from the ores, again bypassing such energy-intensive units as the submerged-arc furnace for the production of ferrochromium. A number of Japanese companies have indicated a certain amount of success. Several interacting elements are recognized that could have a productivity-limiting aspect. The possibility of the direct production of stainless steel from chromite is discussed briefly with reference to the Highveld of the eastern Transvaal as a possible site for such a plant

Jan 1, 1992

By Z. Levinson, R. Dimitrakopoulos

A mining complex is an engineering system where raw materials are extracted from several mines, processed, transported, and sold to the market. The uncertainty and local variability of material grades and their locations within the related mineral deposits are critical for determining the optimal life-of-mine extraction sequence, destination policy, operating modes, and capital investment decisions, which define the production schedule of a mining complex. Infill drilling is used to further characterise a mineral deposit and reveals information that changes the representation of the resource model and potentially the resulting optimal long-term production schedule. A self-learning reinforcement learning approach is proposed to determine the optimal drilling locations that maximise the net-present value of the long-term production schedule and the associated financial forecasts by adapting the schedule to additional information. A case study at a copper mining complex demonstrates a $70 M increase in project value by collecting drilling information with a $1 M budget.

Sep 1, 2021

By W. C. Joughin

A risk evaluation model has been developed, that enables the quantification of expected injuries and economic losses resulting from rockfalls associated with a support system in a given geotechnical environment. This enables the comparison of different support systems based on the expected frequency of injuries and the total cost of the support system, including the expected economic losses. The model takes into consideration the variability in joint orientations and joint strength within the geotechnical environment and the variability and quality of support systems. The quantification of the benefits of other rockfall risk-mitigating strategies, such as the effectiveness of barring or monitoring, are also incorporated in the model. The risk model is described in this paper and the validation of the model and case studies are described in Part II.

Jan 1, 2012

By K. Böde

Hochtief AG Germany has developed a comprehensive management and information system for the construction industry named Aristoteles. This system is an efficient supporting tool to manage a variety of construction sites as well as operating business units with required economic targets. As such, Aristoteles offers wide-ranging control of complex economic processes in construction in a consistent manner. In this paper the process oriented functions of the system will be described with special emphasis on finance accounting and cost control. The basic elements in this respect are tender estimation, creation of initial project budgets and running budgets, regular determination of work done and result forecast to the end of the project. Further, cost accounting, monthly reporting, and all balance-related postings are included. On the basis of the extensive reporting (targeted/actual cost comparison, cost manager, project report, actual/targeted hours worked comparison, overhead cost report, amendment report or project (result) lists of all projects realized by one business unit), the responsible managers can make justified project evaluations, project controls and variance analyses during the project period and derive appropriate management measures. Thus, Aristoteles includes all substantial elements for the risk management of the operational business. In big construction joint-stock companies it is of crucial importance to satisfy the information requirements of the investors. In this respect the system provides a transparent presentation of the financial, investment and profitability situation of the company. In the international construction industry the balancing rules according to IAS (International Accounting Standards) have already been used to a larger extent. In accordance with these rules the partial profits of the work during execution can be shown as realized even before completion of a long-term contract. The validity of Aristoteles will be outlined briefly. Major advantages using the system are highly enhanced accuracy of project controlling, systematic overhead allocation to projects, change of attitude of the project personnel due to increased cost consciousness and improved flow of information within the company.

Jan 1, 2003

Notices COMPETITION FOR STUDENT MEMBERS OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY Each year the Institute offers a prize (or prizes should the entries warrant it) of up to RlOO for the best paper or dissertation on a topic appropriate to the interests of the Institute. The competition is open to all Student Members of the Institute. APPLICATION AND PRACTICAL OPERATION OF HYDROCYCLONES Dr H. F. Trawinski of Amberger Kaolinwerke, Hirschau, West Germany, gave a very interesting lecture on this subject to an audience of over one hundred, composed largely of members of the Institute, on 15th August, 1973, at the University of the Witwatersrand. ROTARY INTERNATIONAL Applications are invited before 15th March, 1974, for the following awards of The Rotary Foundation: Post-graduate Fellowships Under-graduate Scholarships Awards for Technical Training Awards for Teachers of the Handicapped. NIM REPORTS Report No. 1509 The instrumentation developed for electrochemical studies of the le(Whing of uranium dioxide. Report No. 1550 A fundamental study of the dissolution in (Wid solutions of uranium minerals from South African ores. Report No. 1552 The preparation and analysis of minerals for use as reference material.Progress report No. 2. Report No. 1556 The spectral analysis of solutions.2. The analysis of leach liquors, concentrates, slags, and residues. Report No. 1559 A cost assessment of the production of elemental sulphur by chlorination roasting. Report No. 1562 Application of a flotation model to an industrial plant

Jan 11, 1973

Metso Minerals Process Technology Asia-Pacific (MMPT-AP) has been working with many mining companies worldwide performing `Mine to Mill' or Process Integration and Optimisation (PIO) studies. MMPT-AP has developed a proven methodology to improve the efficiency of the mine-mill interface and gain maximum benefit. The PIO methodology involves rock characterisation, mathematical modelling and simulation to generate a list of operational and control changes for both the mine and concentrator. Following the methodology, the list of recommendations is developed relatively easily; the more challenging part depends on the company. Success of a PIO project relies on the ability of an organisation to implement change - both operationally and culturally: the human element. When a PIO project is started, there is initial interest and commitment from the company. But how can this be maintained? How does an operation know if the benefit continues? MMPT AP is now developing long-term relationships with clients to ensure that the increased production and improved efficiency continues. Direct and indirect measurements are used to determine the key factors involved and to understand the cause/effect relationships. Simulation tools also can be used to compare current with recommended practices - so when material properties vary over time, the benefits can still be quantified. PIO is about empowerment: by measuring and understanding what parameters affect concentrator performance, the site personnel now have the knowledge to make a difference. Rather than simply reporting variable or poor concentrator performance, operations can predict when production will be affected and put controls in place to prevent it from happening. This requires support and commitment from the company or `buy in'. Ensuring this buy in continues is the challenge facing every company involved in PIO work - but the benefits are well worth it. This paper will review MMPT-AP's experiences in implementing PIO projects in the long term: from both the consulting and operational viewpoints. MMPT-AP's involvement with a number of companies will be discussed as examples of what works and what does not.

Jan 1, 2007

It is suggested that an increase in the production from thin coal seams in South Africa is necessary, and that a substantial amount of any increase will be mined by the bord-and-pillar method. For the purpose of this study, a thin coal seam was regarded as any seam 1,5 m thick or less to the minimum thickness at which mechanized equipment can operate in bord-and-pillar workings. The equipment being utilized in thin-seam bord-and-pillar layouts includes conventional equipment, load-hauldump cars (scoops), and drum-type and Fairchild-Wilcox continuous miners. Comparisons are made of these types of equipment for bord-and-pillar mining at various seam thicknesses within the thin-seam range. The evaluations involved production rates and the cost of extraction for South African conditions, use being made of an existing computer modelling technique in the generation of production rates. The indications are that, for seam thicknesses greater than 0,9 m, drum-type continuous miners utilizing continuous haulage are the most productive and cost-effective equipment at present operating in thin-seam bord-and- pillar workings. Where seams thinner than 0,9 m are to be exploited, Fairchild-Wilcox continuous miners, particularly the Mark 22, produce the most coal at the lowest cost.

Jan 1, 1984

By M. Berjak, D. M. Hawlins, R. B. MacGillivray

SYNOPSIS The paper describes a method of producing iso line plans using a mathematical technique for interpolating between boreholes, and its computer application. The iso line plans are used for assessing the value parameters of an ore body. The computer programme also produces means and standard deviations to supplement these iso line plans. The mathematical techniques tested for the production of a satisfactory fit of the data were the polynomial, Fourier and moving average technique. The iso line plans are used for assessing the quality of the ore and other salient features such as depth to the floor of the ore body at different points, with a view to deciding how best the ore may be mined. The plans are plotted by the computer to any scale required. INTRODUCTION To assess the economic exploitability of an ore body many tedious and repetitive calculations must be carried out. Observations of borehole or development samples have to be processed yielding measures of ore reserve and quality in each sampled area, and also measures of the worth of the ore body as a whole and of variation in quality from section to section of the ore body. The observations made on the ore (which we shall term 'value parameters') may embrace a variety of characteristics of the ore. Obvious value parameters are those directly related to the economics of mining, viz. assay values, ore body thickness and depth below surface. Other value parameters however may be almost as important. In the case of coal for example a purchasing contract may specify limits of acceptability on other value parameters such as calorific value, ash content or the percentage of volatile matter. In this case, a detailed study of these value parameters will also be crucial. This detailed study must include an assessment, not only of the body as a whole, but also the regional variation in value of the parameter. An obvious and satisfactory way of displaying this variation is by drawing iso lines of constant value parameter on a map of the ore body and supplementing this with the means and standard deviations of the value parameters taken over all boreholes. Such plans, considered together, show up the zones of significance. The relatively stereotyped nature of the operations involved in this process makes it amenable to computer solution. In order to produce iso line plans on the computer we require some mathematical formula for producing a 'valuation surface'-a mathematical