“Strategic Planning Models using Mathematical
Programming Techniques”
Presented at
METSOC
Canadian Institute of Mining, Metallurgy and Petroleum (CIM)
COPPER 2003-COBRE 2003
November 30 to December 3, 2003, Santiago, Chile
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Strategic Planning Models using Mathematical Programming
Techniques
R. Jerez
Mineral Industry Consultants
5730 East Princeton Ave.
Englewood, Colorado 80111, USA
rjmic@ecentral.com
R. Featherstone and L. Scheepers
Large Scale Linear Programming Solutions
P.O. Box 145790
Brackenhurst 1452
South Africa
lou@global.co.za
ABSTRACT
Since World War Two, the depletion of the most accessible of the world's high
grade reserves has taken place, forcing the mining industry into working with material of
declining grade. As a result, three aspects of mineral technology have become critical:



Improved mining methods
More efficient metallurgical extraction technologies
Advanced strategic planning
Recent advances in mathematical programming and computer technology are now
providing Top Management with extraordinary strategic planning and decision support
power from the point of view of maximization of NPV over multi-time horizons whilst at
the same time avoiding sub-optimization within the organization. This holistic approach
to strategic planning and decision support, if fully implemented within a mining and
metallurgical complex, is the only effective way to optimally exploit mineral resources
and to remain competitive.
This paper illustrates the holistic and optimization approach to strategic planning
using mathematical programming techniques. The benefits of this approach to the
executive and managerial levels are highlighted.
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INTRODUCTION
During the past six decades, mineral complex business modelers (mathematicians,
geo-statisticians, mining and metallurgical engineers and accountants) have been actively
applying their intellect to finding more effective and comprehensive mathematical
models. These models would not only solve the operational planning problems relating to
mining and metallurgical complexes, but also provide optimal answers in terms of
maximum profitability over the life of the mine, the associated metallurgical
beneficiation and final product manufacturing (where applicable), i.e. from rock face to
metal order book.
As far back as May 1964, Lerchs and Grossmann (10) realized this and reported
that "a mathematical model taking into account all possible alternatives simultaneously
would provide optimal answers in terms of maximum profitability, however, it would be
of formidable size and its formulation and solution would be beyond the means of present
know-how". Although this was stated in the context of an open-pit mining problem, the
generalization of this statement in which the entire integrated mining and metallurgical
complex is included, is equally true.
Since the 1960's, meteoric developments have taken place in the computing
sciences and today, with modern day computers and workstations, mathematical
programming models constituting hundreds of thousands of variables and/or integer
variables in hundreds of thousands of constraints, are successfully solved within a finite
time period.
The consequence of these latest advances within the computing sciences is that
the “present know-how” limitations of the Lerchs and Grossmann “mathematical model”
in terms of “formulation and solution”, are now surmountable.
In contrast to mathematical programming models, input-output modeling, which
are typically spreadsheet based and highly user-friendly, fail to optimize the complexity
of the interactions between the various process units. As a direct consequence, large
integrated mining and metallurgical complexes, which utilize planning by combining the
outputs of the various plants’ spreadsheet model results, are in fact being planned suboptimally, even though each spread-sheet model solution may be optimal for each plant.
The sum-total of the optima of the individual plants can at most be equal to the global
optimum, but it is highly unlikely to be so in modern day large mining and mineral
processing complexes. “The whole is more than the sum of its parts” by Aristotle (11).
Strategic planning requires strong corporate governance that can successfully
bring together and optimize the combined performances of individual plants with varying
objectives in order to accomplish the most effective overall course of action. It requires
thorough analysis and knowledge of each plant’s economics, processes, distribution, and
markets. Mathematical programming techniques that are illustrated in this paper find the
OPTIMAL COMPROMISE plan among the divergent objectives of the different plants.
This level of sophisticated strategic planning and decision support is not attainable with
input-output modeling tools, such as, for example, spreadsheet applications.
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MATHEMATICAL PROGRAMMING - LINEAR (LP), MIXED INTEGER (MIP)
AND RECURSIVE (RP) PROGRAMMING.
LP-models find the optimal solution to problems that are formulated in terms of
an objective function and subject to constraints.
Objective: The objective function consists of revenue and cost coefficients and is
used by the optimization algorithm to evaluate the “profitability” of the plan. The search
algorithm is able to detect when optimal profitability has been reached and the search will
cease at that point. The objective function can be used to optimize criteria such as “cash
flow before tax”, “NPV – net present value” or even simply, to minimize costs.
Constraints: Each constraint is defined in terms of a range, which the variables
must adhere to. Thus for example the throughput of a plant cannot exceed “x” tons per
time period, or a shaft has a minimum hoisting capacity of “y” tons per month and a
maximum of “z” tons per month. An example of a constraint is for instance the opening
reserve of an ore body in a given time period, less the tons mined in the same time period,
this equals the closing reserve and this closing reserve becomes the opening reserve of
the subsequent time period. Another example of a chemical constraint in a blast furnace
is the reaction FeO + C  Fe + CO, where iron oxide combines with carbon to form
liquid iron and carbon monoxide gas. The FeO is in its turn constrained by the grade and
availability of the iron ore. In order to ensure a practical LP-model derived plan, which
can be implemented, considerable care must be taken in defining achievable constraints.
Variables: These variables can be either continuous or integer. The tons mined
and the tons of iron ore consumed in the blast furnace in the examples above, are
continuous variables. Integer variables can only take on the values of one or zero and are
used to model non-linear relationships (e.g. mineral recovery curves) or go no go
investment decisions (e.g. in what future year must capital be spent on a shaft expansion,
if at all ?).
In modern day large integrated mining and metallurgical complexes, there are ten
of thousands of variables in as many constraints exist and the latest advances in
optimization technology permit automated data driven matrix generation (4,5) (from user
friendly spreadsheet inputs) and high performance optimization (2,3) and report
generation. The solution of large-scale models (1) takes place in minutes rather than
hours on e.g. 2.0 – 2.5 GHz personal computers.
Two of the most important benefits in obtaining a holistic optimal solution to a
large integrated mining and metallurgical complex problem formulation (5,6,7,8), are : A maximized NPV (net present value) plan of the entire metallurgical
supply chain – any change(s) to the plan will result in a lower NPV.
 The avoidance of sub-optimization – some plants may well operate at
lower than maximum capacity rates, but always with good and logically
motivated reason.
The following case study illustrates the above principles.
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TECHNO-ECONOMIC RATIONALIZATION
OF A NUMBER OF
CU/NI MINING AND METALLURGICAL ENTERPRISES
The diagram in Figure 1, on page 13, summarizes a mining and metallurgical
complex with several business centers defined as the sources of production, processing,
smelting, refining, and marketing. Its primary product is nickel and sub-products include
copper, cobalt and platinum group metals (PGM’s). The company has operations in four
countries, involving multiple currencies and exchange rates.
The flow sheet illustrates the material flow from mines to metal markets. The
solid lines represent base cases of current operation and the dashed lines indicate
investment expansions, new processes in the pipeline, and different production modes
such as startup, standby or steady state production.
The mining business unit is composed of several open pit and underground mines.
Some of them are in production, while others are under construction or at a feasibility
stage. For example, the eastern region in country A, is planning to shut down one of the
mines while the southern district in country B, is planning to add a new underground
operation. The southeast area in country C, defined as the leaching center of the
company, has one project at the feasibility stage and another under construction. Two
new leaching technologies are planned for full production parallel to the current
conventional copper extraction.
The company has interests in various nickel operations. The operations comprise
NCK Limited, NIC mine in the southern district, as well as NCC Corporation and NCL
Limited, which has a base metal refinery
The refining, smelting and concentrating units are facing escalating energy costs.
Pressure is mounting on management to develop new ideas to explore lesser expensive
strategies, which may include price protection programs to ease exposure to price
fluctuations.
Over the last few years, the marketing group has seen declining nickel and copper
prices and is now taking advantage of the best opportunities it may find in the contract
portfolio. The latter is composed of global customers and a range of products is offered
from copper cathode, blister, PGM products, concentrates and scrap copper to byproducts. The PGM products are currently getting high prices, however, the volumes
extracted are small due to low grades. Management is facing what has been stated by
Roling in (9) as the old adage in the commodities markets: “gluts create shortages and
shortages create gluts”, referring to the fact that companies are now paying dearly for
their own over- investments in the early nineties.
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Environmental issues are obligations that play a major role in strategic decisions.
They have to be included in the modeling process, in particular when incremental costs
arise as the number of standby or shutdown facilities grow.
In the middle 1990’s a project was undertaken to review long-term strategies.
Apart from the bearish outlook for metal prices, the concerns were the mature stage of
some of the operations (for example, NCK and NIC complexes), declining nickel output,
the quality of the ore reserves, and future utilization of smelter and refinery capacity.
The approach taken by the company was to classify the problem into three
distinctive courses of action:
1.
Analysis and optimization that were to start with the existing business
system as base case, and to determine the flexibility and maneuverability
within it’s confines.
2.
Investigation of the future of the ore resources; what ore quantities from
which sources should be processed in order to extend the reserves for as
long as possible.
3.
Toll treatment possibilities – the opportunity, the price and quantity that
each operation could charge and process. For example, the company was
treating metal through two local refineries, one in-house, and one abroad.
The shipment routes between smelters and refineries were also open to
analysis.
A comprehensive analysis of the above problem showed clearly that input-output
spreadsheet type tools are not sufficiently adequate in function to provide the level of
decision support required by management in order to strategize widely. It was
subsequently decided to follow the holistic and optimization approach to planning using
linear and mixed integer programming techniques.
LP model decision support
The objective of setting up a linear programming model in the above case was to
integrate all the enterprises (each consisting of mines and plants) in which the company
has an interest, into one holistic optimization model as graphically represented in the
diagram of figure 1 and to permit Top Management to strategize as follows:

Suspending all capital expenditure and assuming no changes in the current
metal markets and prices, what should the entire operation look like if it
were to maximize cash flow before tax ?
A run of the LP-model clearly showed which operations were to be operated
at minimum capacities and which were to be operated at maximum capacities
and which were to be shut down.
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
Relaxing the constraints on capital expenditure, and allowing expansions
to take place at all the activities that the LP-model indicated should be run
at upper limits, revealed a totally different picture with an increased cash
flow before tax.
By permitting capital expenditures (expansions) to take place one by one
instead of simultaneously, the effect that each expanded shaft or plant had on
the entire operation was quantified and the implications were studied.
Optimal sizing of expansion projects were analyzed in the manner that each
expansion took place in steps up to that point where no further gains were
registered by the LP-model.

Permitting shut downs to take place at all the activities which the LPmodel indicated to be run at lower limits, revealed once again a totally
different picture and cash flow before tax also increased.
By permitting shut downs to take place one by one instead of simultaneously,
the effect that each shut down had on the entire operation was quantified and
the implications were studied. (The LP-model identified shut downs that
caused minimum harm to cash flow).

Permitting both, capital expenditure (expansions) and /or shut downs
(contractions) to take place simultaneously, based on OPTIMAL
TECHNO-ECONOMICS, registered a dramatic improvement in cash flow
before tax by the LP-model.
These runs produced optimal plans that were communicated to the Managing
Directors of the participating enterprises and the General Managers of the
various mines and plants with a view to obtaining their comments and
enabling them to invest in and participate in this innovative approach.
Valuable feedback was obtained and re-runs of the LP-model took place to
accommodate this feedback. However, as was to be expected, each of the
subsequent LP-model runs registered decreased cash flows, compared to the
OPTIMAL TECHNO-ECONOMIC RUN above. This had great value in itself
in that Top Management was enabled to understand the cost of making the
new strategy practical and acceptable to all the Managements of the
participating mines and plants.

Applying sensitivity analysis to the integrated model in terms of :
o Timing of start-ups
o De-bottle-necking of activities at upper limits
o Determining the techno-economic worths of the ores and
concentrates at various metal prices
o Optimization of raw material intakes; renegotiating contract deals
with suppliers
o Optimization of concentrate mixes
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o Optimization of haulage routes from the most mature mines
o Optimization of intermediate products to the smelters and
refineries (toll smelting; TC/RC and freight rates)
It is stressed that the above sensitivity analyses are done by running the LP-model in its
entirety or holistically, so as to achieve results within the context of the entire operation
and not on a stand-alone and therefore sub-optimized basis.
As LP-models are being built, extensive studies are being carried out by each
operating site and process to better understand the contributions of each of these assets to
the long-term value of the company as a whole. These studies within the organization and
across various business units may include, among others, strategizing and decision
support (see Table I below for an illustration of a typical DECISION TABLE),
optimization of planning at each site, techno-economic analysis (see Table 2 below for an
illustration of a typical “techno-economic ore worth” table) and influence of commodity
market trends, sales forecasts, selling prices and exchange rates on the company’s plans.
The LP-model was used to quantify the techno-economic worth of the various
ores. Various ore resources were classified into their respective relative “worth” at
different metal prices. “Ore worth” is defined as the marginal value per ton of ore. As a
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result the objective function of the base case will be changed by a “delta value” for any
increase or decrease in tonnage of that ore resource by one ton. The amount by which the
objective value varies with a change of one ton, is called marginal profit or “shadow”
price or reduced cost of that particular ore.
Table II illustrates the “worth” of participating ores. From all the ore resources
available to the company in Figure 1, Melhi and Alto from the Eastern Region are the
least valuable. These two uneconomic operations are planned to be replaced by SX/EW
projects from the Southeast Area.
TABLE II - Worth of Ore Resources US ($/ton) *
US$
US$
US$
Ore Source
0.65/Lb
0.75/Lb
0.95/Lb
Copper
Copper
Copper
Eastern Region
MEHI
-9.42
0
5.01
ALTO
-3.1
1.53
4.34
JEDI
28.56
36.87
46.1
Southern District
SAHIR
95.71
123.11
138.2
SELBI
6.26
14.08
24.03
South East Area
STON
1.035
2.35
3.85
STON NW
20.34
32.35
64.69
Feasibility
MERK
15.07
16.01
23.65
MERK #2
20.34
32.35
64.69
North District
ATA
13.11
15.45
18.01
HUNT
88.41
94.12
127.87
* For proprietary reasons all numbers and names
have been changed. This table is only an illustration
Case Study Conclusions
The above generic description of the holistic approach to strategic planning within
a number of linked large integrated mining and metallurgical complexes, is based on an
actual development that was undertaken for Anglo American Corporation of South Africa
in the mid-nineties. Model results indicated clear strategies resulting in the more than
doubling of the group NPV compared to the existing planning. Worth-while Southern
African opportunities for pooling Cu/Ni resources, including the shutting down of
production units and the expansion of others, were identified and after presentations to
the Boards of the companies as well as to committees of some of the regional
Governments, the strategic planning of the Companies were adjusted accordingly.
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LP-MODEL DEVELOPMENT AND IMPLEMENTATION
The successful development and implementation of an LP model requires gaining
consensus on the part of the teams and leaders in each discipline. Without the enthusiastic
cooperation of various levels of staff from the various disciplines, an LP-modeling
approach will fail. The model is the culmination of all the joint efforts across the
enterprise.
During the development stage of a model for a large integrated mining and
metallurgical complex, each ore source is defined by its proven, probable and possible
reserves, metal grades, capacities, and yearly schedules. The concentrators are modeled
using the mass pull ratios and metal recoveries. Their capacities are defined in terms of
grinding energy. The smelters are modeled in such a manner that the constituents of the
concentrate are used as basis to calculate the metal recoveries. The metallurgy and slag
chemistry to stoichiometric levels of detail are included together with the heat balances.
The refineries are described by their metal recoveries, product grades and marketing and
distribution costs. All operating units are allocated fixed and variable costs, capital and
ongoing capital replacement expenditures.
The most common assumptions made are:
 The database consists of one opening year and the applicable mine budget
production schedules, operating costs, capital and ongoing capital
replacement expenditures for that particular year.
 A long term forecast of metal prices, sales volumes and exchange rates;
 No taxation is applied to avoid any bias in the model towards the lowest
tax paying company.
 A hurdle rate (e.g.10% per annum, which may be varied by the user) is
used for discounting purposes
 The linear programming objective function is to maximize the before tax
cash flow NPV @ e.g. 10% (which may also be varied.)
Initial model implementation may follow these steps:
1.
Each production unit is modeled as a stand-alone operation in the first
time period. This is like a snapshot of the operation, i.e. a single period and in
today’s money.
2.
Accuracy tests are applied to the stand-alone models. This step consists of
improving data sources and modeling up to acceptable levels of accuracy.
Historic data and stringent judgmental and reasonability tests are applied.
3.
The stand-alone models are integrated. All these models are combined into
one model. Model accuracy and behavior against previous production log
sheets are thoroughly checked.
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4.
The production planning and income statement sections are
incorporated. Custom reporting with detailed financial and accounting
information is developed.
5.
NPV and capital are added to the model. This phase incorporates the NPV
objective function instead of “today’s money” together with the capital
expenditure schedules.
6.
Multi-time periods are implemented. The model is extended to multi-time
horizons. Running a model without the time element gives only a single
period snapshot of the distribution of production, shipment and capital
expenditure.
The base case is established. Usually the base case is the in-house long term
(e.g. a ten or fifteen year or life of mine plan) budget, force fed to the LPmodel. All further runs of the model is then compared to the base case.
Normally, the first run after having established the base case, is the “open case
strategy run” in which the base case is improved. The base case is used to do a
final accuracy calibration.
7.
8.
Constraints are relaxed and the “open case strategy” found. The open case
strategy is the one that allows the model to improve on the base case, i.e. the
inter-company and inter-mine-plant “political” constraints are removed.
However, constraints are still to be practical in order to enable plan
implementation. This case is normally an eye opener to Top Management as it
is now realized what the inter-mine-plant and inter-company politics are
costing the company. (Typical example is the age old “miner/geologist –
metallurgist” argument of “you will process what I mine” and “what you
mine cannot be processed”).
9.
The model is now ready for implementation. After an operator training
period, strategic planning and sensitivity studies are conducted.
10.
At this stage the LP-model and its optimal plans are ready for
presentation to Top Management by the development team.
During the implementation phase, different cases are built from the original
base case. Fresh perspectives will emerge for the individual business units
related to quality of ore reserves, ore mineralogy, smelting recoveries,
processing costs, refining/smelting contract terms, etc. This happens in the
context of the optimization of the entire business: what is good for the one is
to the detriment of the other and the LP-model finds the optimal compromise:
the holistic approach avoiding sub-optimization.
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CONCLUSIONS
Strategic decisions are concerned with critical time related issues with which
large mining and metallurgical complexes are confronted. Over multiple time horizons
these may involve alternative start-up situations, capital expansions, downsizing and the
implementation of new processes. These decisions are made at the executive level of an
organization with regard to the direction of the company within a constantly changing
environment, and they can have dramatic effects on the profitability of a company and
consequently on shareholder interests.
It has been stated that strategic planning requires strong governance that can
successfully bring together individual business units with varying objectives in order to
accomplish the overall course of action leading to the ultimate corporate-wide objective.
A compromise must constantly be reached as different units review their plans to achieve
their best long-term objectives.
At this level strategic planning often relies heavily on various input-output type
analyses that can only render a once off snapshot like picture of the scenario under
consideration. It has been pointed out that such decision-making tools are severely
limited in their ability to accommodate the variability of the number of factors and their
impact on any particular business over a period of time. Each must be analyzed as a
separate variable. Consequently, the degree of certainty offered by the analysis must be
questioned as there is no guarantee that the answer provided is anywhere close to the
optimum solution.
On the other hand, as has been stressed in this paper, the approach offered by
Linear, Mixed Integer, and Recursive Programming, results in the optimum solution with
every run of the model, subject to the constraints applied. As such, this approach is
considered to be the only method to exploit optimally all the resources at the disposal of a
company, consequently, to empower that company to remain at the competitive edge of
the industry in which it is involved.
ACKNOWLEDGEMENTS
The authors would like to thank Anglo American Corporation of South Africa for
permission to publish this paper in particular Mr. A. Ramsay, now Managing Director of
Anglo American Research Laboratories for permitting the inclusion of the case study.
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