I have to set up an LP problem based on this situation below:
What I tried:
Let $x_{i,j}$ denote amount of loot i in hold j
for i = 1,2,3 corresponding to materials, gold and spice
for j = 1,2,3 corresponding to forward, center and aft
We want to maximise profit
$$z=16\sum_j x_{1,j} + 36\sum_j x_{2,j} + 3\sum_j x_{3,j}$$
s.t.
Amount:
$$\sum_j x_{1,j} \le 60,000$$
$$\sum_j x_{2,j} \le 30,000$$
$$\sum_j x_{3,j} \le 15,000$$
Weight capacity:
$$\sum_i x_{i,1} \le 19,000$$
$$\sum_i x_{i,2} \le 30,000$$
$$\sum_i x_{i,3} \le 15,000$$
Volume capacity:
$$\sum_i c_i x_{i,1} \le 1,100$$
$$\sum_i c_i x_{i,2} \le 1,340$$
$$\sum_i c_i x_{i,3} \le 450$$
where $c_1 = 2, c_2 = 0.5, c_3 = 0.3$
Ratio for weight (wrong? use volume instead?):
$$\frac{\sum_i x_{i,1}}{\sum_i x_{i,2}} = 19/30$$
$$\frac{\sum_i x_{i,1}}{\sum_i x_{i,3}} = 19/15$$
$$\frac{\sum_i x_{i,2}}{\sum_i x_{i,3}} = 30/15$$
Ratio for volume (wrong? use weight instead?):
$$\frac{\sum_i c_i x_{i,1}}{\sum_i c_i x_{i,2}} = 1,100/1,340$$
$$\frac{\sum_i c_i x_{i,1}}{\sum_i c_i x_{i,3}} = 1,100/450$$
$$\frac{\sum_i c_i x_{i,2}}{\sum_i c_i x_{i,3}} = 1,340/450$$
Nonnegative:
$x_{i,j} \ge 0$
Are those right? To what do capacity ratios refers? Weight? Volume? It can't be both so which is it?
From Chapter 2 here.
I am just posting what I wrote in comments above:
This work looks good as far as I can see. From the problem description, it is not clear if the ratio constraint is with respect to weight or volume. It is reasonable to assume it is for weight.
For general numbers, it cannot be both weight and volume, since: $$ \frac{\mbox{loot $i$ weight}}{\mbox{loot $j$ weight}} = \left(\frac{\mbox{loot $i$ volume}}{\mbox{loot $j$ volume}}\right)\left(\frac{c_j}{c_i}\right)$$ and so, once the weight ratio is fixed for a particular pair, the volume ratio is also fixed.
Also, the weight ratio constraints indeed can be written as linear equality constraints, so it is valid for an LP.