Thursday, December 1, 2011

8.2 - Optimization and discounting

Change the model to take into account the growing future demand and therefore a higher future price for the resource. Introduce another parameter which will describe the growing demand for the resource. Make the price a function of both demand and supply. How does this change the results of optimization? Does the discounting still play an important role?

I assumed demand growth independent of supply or price according to the equation

demand = .5*exp(0.15*TIME)

I incorporated demand into price by changing the price function to

price = a*demand/(1 + mining) + b

The major effect on optimization was to increase the value of d from 2.36 to 3.01, which represents more rapid changes in rate of mining.

Discounting plays a more important role than without the growth in demand. The following table summarizes the effects of demand growth and discounting.

Total profits
Discount rate: 0% 5%
No demand function $105k $66k
Growing demand $699k $68k

Exhaustion of resource. Time to exhaustion or, if no exhaustion, resource remaining after 50 years
Discount rate: 0% 5%
No demand function after 50y, 2.7 remain 20.7y
Growing demand after 50y, 80 remain 20.2y

Discounting clearly drives much faster extraction of the resource, and, at least as I formulated the demand function, growing demand has little consequence in a 5% discount rate world -- the urgency of getting the resource to market to be able to take advantage of the discount rate seems to dominate the system.

8.1 - Optimization, abstractly

1. Imagine that your system is an airplane and you need to land it. Formulate and optimization task that will allow you to safely touch the ground at the airport nearby. What is the objective function? What are the control parameters, and the restrictions?

The optimization task is to bring the plane to height = 0 at the location of the runway, with the plane's vector parallel with the runway and the change in altitude of the plane as close to zero as possible.

The objective function is the change in altitude of the plane at the time of contact. The control parameters are the speed and course of the plane leading up to landing. The constraints are the location and direction of travel of the plane at the time of landing.

2. Think of an optimization of your own. Define the system, the objective function, the controls and restrictions.

Let's say we want to optimize the hydrology below a surface mine. The system includes precipitation, upland runoff, the topography, geology, and vegetation of the mine, and the topography, geology, and ecology--including, potentially, humans--below the mine.

Let's make the objective function the difference between post-mining flow and pre-mining flow. The controls include placement of overburden, compaction of replaced topsoil, vegetation planted, and potential settling ponds and/or dykes downstream of the mine. The constraints include the surrounding topography, precipitation and other climatic variables, and upstream hydrology.