A Word About Simplex Algorithm...

The simplex algorithm (invented by Nelder and Mead) provides an efficient way to find a good approximation of the optimum of a function starting from any place.

A simplex in an n-dimensional space is a figure formed with n + 1 summits. For example, a simplex in a two-dimensional space is a triangle; a simplex in a three-dimensional space is a tetrahedron. The algorithm for finding the optimum of a function with a simplex goes as follows:

1. Pick up n + 1 points in the search space and evaluate the goal function at each of them. Let A be the summit yielding the worst function’s value.

2. If the size of the simplex is smaller than the desired precision, terminate the algorithm.

3. Calculate G, the center of gravity of the n best points – that is, all points except A.

4. Calculate the location of the symmetrical point of A relative to G: A` = 2G - A.

5. If f (A`) is not the best value found so far, go to step 9.

6. Calculate the point A`` = 2A` - G, that is, a point twice as far from G as A`.

7. If f (A``) is a better value than f (A`), build a new simplex with the point A replaced by the point A`` and go to step 2.

8. Otherwise, build a new simplex with the point A replaced by the point A` and go to step 2.

9. Calculate the point B = (G = A ) / 2.

10. If f (B) yields the best value found do far, build a new simplex with the point A replaced by the point A`` and go to step 2.

11. Otherwise, build a new simplex obtained by dividing all edges leading to the point yielding the best value by 2 and go back to step 2.

Implementation goal...

On any mining enterprise, a detailed analysis of individual components of the mineral extraction procedure, provide a worthy groundwork assessment on which to base future management decisions. One key objective in a mining endeavor is to extract mineral at a minimum cost.

The costs incurred in such an operation are naturally directly proportional to the volume of ore extracted. Current market prices have a direct dominance on the volume of mineral to be produced, as well as multiple other factors that vary from geological considerations to labor availability. The grade of the mineral(s) ore and geology vary from place to place within the same mineral formation increasing complexity. Transportation costs , labor, material costs, etc, provide additional constraints.

Our goal is to determine the best possible cost allocations by site as well as optimum grades and volumes of mineral based on parameters analogous to a small mining enterprise. Constraints are explained on the next pages.

Parameters...

A few restrictions to be met by the algorithm are shown on the table below:

How The Matrix Was Built...

Based on known facts, the total cost of mineral production derives from costs associated with the actual mined mineral plus costs of mineral transportation from each work site. With this in mind, we can represent: x1, x2, x3,. . ., xn the volume to be extracted from each site, the purpose being to find the best combination possible to minimize production costs within certain restrictions.

1. The first equation is the goal function. Expressing this situation algebraically:
   f(z) = 3.43x1 + 3.48x2 + 3.57x3 +… + 2.45x13.
   Numeric values represent transportation costs from each work site ( by shaft, rail, etc).

2. The second equation represents the tons of material to be extracted. That is, the sum from all work sites should equal 10,000 tons.
   X1 + x2 + x3 + … +x13 <= 10,000

3. The third and fourth equations refer to shaft (2) capacities.
   X1 + x2 + x3 + … + x13 <= 7,000
   X4 + x5 + x8 + … + x11 <= 2,500

4. From the fifth to the thirteenth are maximum and minimum tonnage that can be extracted from each site.
   X1 <= 1,300
   X1 >= 180
   ……
   x13 <= 1,100
   x13 >= 200

5. Restriction number 14 refers to ore grades in terms of economic indexes. Since this is a complex mineral formation consisting of three different minerals (Sn,Pb,Ag) the concept of economic index identifies simultaneously the grade of all three minerals.
   (For example: Market Price (Sn) = $1.600/lb 100 lbs at (%0.67) = (0.67 x 1.600) = $1.072
   EconomicIndex (Sn) = (2204 lbs x $1.072) = $23.626/ton - similar calculations follow for Ag & Sn)
   
   A total economic index then can be derived:
   54.667x1 + 56.901x2 + 52.008x3 + … + 55.309x13 >= 10,000 x 52.444 (524440)

6. Restrictions 15 to 16 represent transportation costs.
   X1 + x2 + x3 + x5 + x7 + … + <= 7,000
   X6 + x8 + x9 + x10 + x12 + x13 +… + >= 1090

7. Slack variables have been introduced to compensate equalities and inequalities.

Results of the Distribution...

The formulated matrix was input into the program. The table below shows the values obtained based on the constraint parameters introduced.

Calculation of Optimum Values...

Since the optimum mineral distribution is known, it is now possible to calculate the optimum ore grade and associated dollar values:

Calculation of Grade of Ore (Ag)

4599181 gr per 10,000 tons = 459.9181 gr/ton
DM: 459.918 / 100 = 4.599 DM/ton

Calculation of Grade of Ore (Sn)

65.893 / 10,000 = 0.00658 x 100 = %0.658

Calculation of Grade of Ore (Pb)

314.788 / 10,000 = 0.03147 x 100 = %3.147

Calculation of Dollar Value (Ag)

5.079 x 10,000 x 4.599 = $233583.21

Calculation of Dollar Value (Sn)

0.658 x 1.600 = 1.052 x 22.04 = 23.203 x 10,000 = $214680.74

Calculation of Dollar Value(Pb)

3.147 x 0.140 = 0.440 x 22.04 = 9.710 x 10,000 = $97103.83

Total Dollar Value

$545367.78

Next Steps...

Once the optimum distribution volume of material is known, it is also possible to calculate the cost of mineral extraction without associated transportation costs. In this case, it would be possible to introduce other factors into consideration, such as: Labor (salaries, incentives, insurance, etc), Material (explosives, drilling equipment, combustible, wood, etc), Plant Maintenance(electricity, power motors, etc).

In order to implement this second phase, it is necessary to build another matrix taking into account all applicable restrictive parameters and applying the same set of procedures. The new matrix is then input into the Simplex algorithm. This phase has not been implemented.

Other Implementations...