Suppose we want to maximize revenue in the following situation.
We have 3 shipments of products.
Shipment one:
*Has 10 ton of loads available for shipment
*A revenue of 700 euro per ton.
*Occupies 2000 units of Volume per ton
Shipment two:
*Has 12 ton of loads available for shipment
*A revenue of 725 euro per ton.
*Occupies 3500 units of Volume per ton
Shipment three:
*Has 17 ton of loads available for shipment
*A revenue of 685 euro per ton.
*Occupies 3000 units of Volume per ton
The plane which will carry these shipments can hold in total 32 ton
It is also split into 3 sections on the left and right:
Right Side
Hold A: Volume=16,000 units Weight=18% of total ( which I calculated to be 5.76 ton)
Hold B: Volume=20,000 units Weight=25% of total ( which I calculated to be 8 ton)
Hold C: Volume=14,000 units Weight=7% of total ( which I calculated to be 2.24 ton)
Left side
Hold A: Volume=10,000 units Weight=18% of total ( which I calculated to be 5.76 ton)
Hold B: Volume=20,000 units Weight=25% of total ( which I calculated to be 8 ton)
Hold C: Volume=12,000 units Weight=7% of total ( which I calculated to be 2.24 ton).
The Objective funtion
I believe would be
$$700x+725y+685z=R$$
Where $x=$ number of tons of shipment 1, $y=$number of tons of shipment 2, $z=$ number of tons of shipment 3
Constraints
I would have thought the constraints would be
$$0 \leq x \leq 10$$, $$0\leq y\leq 12$$, $$0\leq z\leq17$$ ( as there is a limited amount of each shipment in tons)
$$x+y+z\leq32$$ ( as the total amount of tons that can fit on the plane is 32)
$$2000x+3500y+3000z\leq92,000$$ (As each ton of shipment occupies a specific volume and there is a max Volume that the combination of all the holds on the right and left can hold.
Then I thought separate constraints would be needed for each of the individual containers.
Note: I edited in the subscripts after the advice of a comment. These variables are such that
$x_{1a}+x_{1b}+x_{1c}+x_{2a}+x_{2b}+x_{2c}=x$
$y_{1a}+y_{1b}+y_{1c}+y_{2a}+y_{2b}+y_{2c}=y$
$z_{1a}+z_{1b}+z_{1c}+z_{2a}+z_{2b}+z_{2c}=z$
Right hold A
$$x_{1a}+y_{1a}+z_{1a}\leq 5.76$$
$$2000x_{1a}+3500y_{1a}+3000z{1a}\leq 16,000$$
Right hold B
$$x_{1b}+y_{1b}+z_{1b}\leq 8$$
$$2000x_{1b}+3500y_{1b}+3000z_{1b}\leq 20,000$$
Right hold C
$$x_{1c}+y_{1c}+z_{1c}\leq 2.24$$
$$2000x_{1c}+3500y_{1c}+3000z_{1c}\leq 14,000$$
Left hold A
$$x_{2a}+y_{2a}+z_{2a}\leq 5.76$$
$$2000x_{2a}+3500y_{2a}+3000z_{2a}\leq 10,000$$
Left hold B
$$x_{2b}+y_{2b}+z_{2b}\leq 8$$
$$2000x_{2b}+3500y_{2b}+3000z_{2b}\leq 20,000$$
Left hold C
$$x_{2c}+y_{2c}+z_{2c}\leq 2.24$$
$$2000x_{2c}+3500y_{2c}+3000z_{2c}\leq 12,000$$
After this I thought that there would be a maximum for the revenue for each hold and I could use this to find the overall maximum. However when I put this into Wolfram calculator I found that it says there are no local Maxima. Could someone help me see where I'm going wrong and advise me on what I should do instead ?
Many thanks in advance.