Vtyjkyuk

I have 3 tables…

$$beginarrayrrr
textquantity & textexpense & textprofit\
hline
0 & 0 & 0 \
1 & 100 & 200 \
2 & 200 & 450 \
3 & 300 & 700 \
4 & 400 & 1000
endarray
qquad
beginarrayrrr
textquantity & textexpense & textprofit\
hline
0 & 0 & 0 \
1 & 100 & 220 \
2 & 200 & 480 \
3 & 300 & 800 \
endarray
qquad
(cdots)$$

All equal structure but with different values and different number of rows.

I want to write a recursive function describing the optimal choices of quantities in order to maximize my profit.

I have a given amount to spend and the total expenses can't exceed this.

Besides, I use the notation that $p_nj$ should the profit in table $n$ and quantity $j$ and $c_nj$ should express the cost of choosing quantity $j$ in table $n$.

I think it's something like

$$
f_n(A) = textmax left[p_nj + f_n+1left(A-c_njright)right]
$$
where $A$ is the amount I can invest.

My intuition is that I'm adding the profit $p_nj$ for each iteration and subtracting the cost from the total available amount ($A-c_nj$).

I don't think it's correct but I guess it's something like that. How can I iterate through all the three different tables with recursion?

If $f_n(A)$ gives the maximum profit from taking at most $n$ objects and at most $A$ cost, the maximum profit for at most $n+1$ objects costing at most $A$ must be
$$f_n+1(A) = max_j p_j + f_n(A-c_j) mid c_j le A cup 0$$
Note that we

• Maximize over all objects that we could chose. If we can't chose any object within budget, the maximum profit is obtained by chosing nothing ($0$).


• Constrain the objects we try to add to only those objects still within budget.


• $f_n+1(A)$ depends on $f_n(A)$, not the other way around.


• $f_0(A) = 0$ for any $A$, since no objects equals no profit.