“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
Strategic Planning Models using Mathematical Programming Techniques
R. Jerez
Mineral Industry Consultants
5730 East Princeton Ave.
Englewood, Colorado 80111, USA
R. Featherstone and L. Scheepers
Large Scale Linear Programming Solutions
P.O. Box 145790
Brackenhurst 1452
South Africa
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.
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 sub-optimally, 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.
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 à 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.
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 by- products. 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.
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.
· 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 LP-model 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