## Why schedule optimization requires variable discount rates

Mine Planning Articles

We asked John Battista, General Manager at Minemax, and Leon Blackwell, Senior Software Developer at Minemax, why variable discount rate functionality was recently implemented in Minemax Scheduler 6.2. Their responses reveal that modelling variable discount rates is a critical component that contributes to finding truly optimal schedule optimization results. Let’s find out why…. ## Why NPV Schedule Optimization in the first place

In strategic mine planning and optimization, the primary objective mining companies use to measure the value of their mining projects is Net Present Value (NPV). In mining, NPV is defined as the sum of present values of profit margins over a period of time. We all know that NPV is by no means a perfect measure, but the concept of measuring the “time value of money” is widely accepted and probably the best measure of project value that we have. As a result, advanced strategic mine planning software like Minemax Scheduler is designed to optimize the NPV of the schedule in order to find the best solution. We call this approach NPV Schedule Optimization.

## The role of discount rates in NPV Schedule Optimization

By definition, the discount rate is the interest rate used to discount a stream of future cash flows to their present value. It represents the “time value of money” idea that a dollar today is worth more than a dollar tomorrow due to its potential earning capacity. The discount rate is an important input parameter used for modelling in mine optimization software. Minemax Scheduler uses the entered value(s) to discount future profits and determine the NPV of the schedule. The discount rate is typically an annual figure provided by the corporate or financial department of mining operations, but it can be easily converted into period values for use in monthy or quarterly time periods. &nbsp; $$NPV = sum_{t=0}^{T-1}frac{(Revenue_{t}-Mining Cost_{t}-ProcessingCost_{t}-Fixed Cost_{t})}{(1+d)^{t}}$$

where,

T is the number of the time period   D is the discount rate

In general, the higher the discount rate, the lower the present value of future profits, and the greater the significance of realizing value NOW rather than later. A high discount rate depreciates future profits more, which also makes accessing value in earlier periods in the schedule more important. This directly impacts optimization in Minemax Scheduler. During optimization, Scheduler determines the NPV of future profits per time period, and decides which schedule out of all possible combinations has the highest NPV.

## Difficulties when expressing discount rates

One of the difficulties when solving the “time value of money“ problem is that the number of periods, the discount rate, and the money components in the NPV calculation must all be expressed using the same time frequency. This means that we can use the annual discount rate for yearly periods and yearly profits, but we need to convert the annual discount rate into a discount rate that is appropriate for monthly or quarterly periods. Let’s find out now why this can cause a problem for modelling.

## Minemax Scheduler with fixed discount rate modelling

Previous versions of Minemax Scheduler (before Minemax Scheduler 6.2) used the basic concept of fixed discount rate modelling. This meant that you had the option to enter a fixed discount rate for each scenario and Minemax Scheduler applied the nominated discount rate across all time periods in the scenario. This modelling approach worked great for time periods of the same length such as years, but introduced a challenge if you wanted to model periods of mixed lengths.

## The challenge of modelling time periods of unequal lengths

In schedule optimization, strategic mine planners like their strategic plans to be more detailed at the beginning of the schedule, and to increase the length of the time periods toward the end of the schedule where there is more uncertainty and more unknowns. Let’s take a typical example for a 10-year life of mine. In this example we will model time periods in quarterly increments for the first two years and in yearly increments for the remaining schedule:

In this example, if we use the same annual discount rate of 10% for all periods, it will lead to an invalid NPV calculation and overdiscounting of the schedule. Due to this error, Minemax Scheduler is more likely to push producing higher profit margins earlier in the schedule. This will make other, potentially more lucrative choices unavailable, and will result in a schedule that is no longer optimal.

## Workaround for Minemax Scheduler with fixed discount rate modelling

The ability to model variable discount rates correctly is critical to finding an optimal solution.

To avoid an invalid NPV computation during the optimization, we used a workaround for the fixed discount rate modelling approach. This method involved breaking the optimization into two parts. For our example this meant optimizing the quarterly periods (1st Qtr 2016 to 4th Qtr 2017) with a periodic discount rate of 2.5% (which had been converted from the annual discount rate of 10%). Once we got the schedule for the first 8 periods, we changed the discount rate to 10% and ran the optimization for the remaining yearly periods (years 2018 – 2025). Unfortunately this workaround was far from perfect. Although we were able to achieve a correct NPV calculation, the optimization became a two-step process and could no longer consider the entire schedule globally. As a result, we were more likely to miss the truly optimal solution.

## Benefits of modelling variable discount rates in Minemax Scheduler 6.2

Adding variable discount rate functionality to Minemax Scheduler greatly enhances the usability of the software and improves optimization results for schedules. Let’s take a look at how it all works for our example of time periods of different lengths. To access the configuration of discount rates in Minemax Scheduler 6.2, we select the “scenario > financials” menu and then the “per period” option to view the screen with variable discount rates set up. For our example, we use the quarterly discount rate of 2.5% for the first 8 quarterly periods and the discount rate of 10% for the remaining periods as shown below:

This enables us to run the global optimization strategy to optimize the entire schedule in one single step without compromising the NPV calculation, and find the truly optimal schedule. &nbsp;

## Improving optimization speed by aggregating periods

Modelling variable discount rates can also improve the optimization speed for scenarios with a large number of time periods. In this case, we have the option to aggregate later periods into longer periods and speed up the optimization solve time. Although we lose a bit of granularity for the end of the schedule, this typically has only a marginal impact on the schedule because the later periods are heavily discounted.

## Still not convinced about the benefits? Let’s compare some results.

We used the first example with unequal time periods (quarterly increments for the first 2 years, and yearly increments for the remaining periods) to compare results between a scenario with fixed discount rates using the workaround, and a scenario with variable discount rates. When we optimized the schedule and compared the financial charts showing cumulative discounted cash flows and the scenario summary screen, there was a significant difference between both scenarios. The scenario with variable discount rates produced a total NPV that was more than 233 million dollars greater. By modelling variable discount rates correctly, we increased the profit by 10%. Financial Chart in Minemax Scheduler – Cumulative Discounted Cash Flow Comparison

## Preview Minemax Scheduler

The latest version of Minemax Scheduler 6.2 with variable discount rates is available for download for all maintained users. Would you like to see if you could achieve a greater NPV for your project by incorporating variable discount rates into your optimization model? Contact us and we will be happy to work with you and demonstrate how you can benefit from using Minemax Scheduler for your projects and operations.