1) The problem is to excavate within a permissible area in order to obtain valua

Question

1) The problem is to excavate within a permissible area in order to obtain valuable ore. The volume to be excavated is divided into blocks (50 foot cubes). Each block has a certain net income obtained by subtracting the cost of extracting it from the revenue obtained after refining and selling the ore. For blocks near the surface (overburden) the net income is small or even negative, but becomes larger (and positive) for richer ore deeper down. There is a slip angle restriction on excavation. A block can only be excavated if the upturned cone of blocks above it have already been removed. Referring to the figure below, block 1 cannot be excavated unless blocks 2 and 3 have already been removed 10 In this particular example, there are ten blocks arranged as shown, each with an estimated net income of R,i-1,2,...,10. Formulate a problem whose objective is to maximize net income subject to the logical conditions on when a block can be mined.

Explanation / Answer

In the arrangement shown above, we have to consider two things:

1. the time factor, we have to reach the deeper and more rich ore as early as possible.

2. the sequence of mining considering the slip angle so that the above blocks are removed 1st.

We have to start with blocks no. 7 and 8 which would provide us R7 and R8 net income.

Then we mine block 4 giving R4 net income.

Then 9 the top block giving R9 net income.

Then 5 giving R5 net income.

This opens up block no. 2 which would give high net income of R

Now the sequence will be block no. 10, 6, 3 and finally 1.

This sequencing will help us maintain slip angle constraint and also maximize net income by mining the high net income blocks as soon they are open to mine.

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