Problems in Q#1:
In Q#1, you can also define
Xi: the (absolute) amount of Alloy i, i=1,2,3,4,5,
as decision variables. However, if doing so, you must assume a
constant (not a variable) for the
total amount of the new alloy to be produced.
The reason for this is that if the total amount is a variable,
the optimal solution will be Xi = 0 for i = 1, 2, 3, 4, 5,
which is meaningless. The assumed constant can be any positive value,
e.g., 1000kg. Then, the LP formulation will be
Minimize Z = 17X1 + 12X2 + 18.5X3 + 11X4 + 18X5
subject to
X1 + X2 + X3 + X4 + X5 = 1000 (total amount)
0.2X1 + 0.1X2 + 0.5X3 + 0.1X4 + 0.5X5 = 0.3(1000) (lead)
[or 0.2X1 + 0.1X2 + 0.5X3 + 0.1X4 + 0.5X5 - 0.3(X1 + X2 + X3 + X4 +X5) = 0 ]
0.6X1 + 0.2X2 + 0.2X3 + 0.1X4 + 0.1X5 = 0.2(1000) (zinc)
0.2X1 + 0.7X2 + 0.3X3 + 0.8X4 + 0.4X5 = 0.5(1000) (tin)
X1, X2, X3, X4, X5 >= 0
After you have done the formulation, you must state what will be the
proportions (X1/1000, etc.), which are just what the question
requires you to do.
Problems in Q#2:
In Q#2, you can define (relative) fractions of cargoes as decision variables:
Xif : the fraction of cargo i accepted and distributed to
the front compartment;
Xic : the fraction of cargo i accepted and distributed to
the central compartment;
Xir : the fraction of cargo i accepted and distributed to
the rear compartment;
i = 1, 2, 3, 4.
Maximize Z = 100(15)(X1f + X1c + X1r) + 140(9)(X2f + X2c + X2r)
+ 110(18)(X3f + X3c + X3r) + 95(10)(X4f + X4c + X4r)
subject to
15X1f + 9X2f + 18X3f + 10X4f <= 8
15X1c + 9X2c + 18X3c + 10X4c <= 12 (compartment weight capacity)
15X1r + 9X2r + 18X3r + 10X4r <= 7
500(15)X1f + 700(9)X2f + 600(18)X3f + 400(10)X4f <= 5000
500(15)X1f + 700(9)X2f + 600(18)X3f + 400(10)X4f <= 7000 (space)
500(15)X1f + 700(9)X2f + 600(18)X3f + 400(10)X4f <= 3000
X1f + X1c + X1r <= 1
X2f + X2c + X2r <= 1
X3f + X3c + X3r <= 1 (cargo availability)
X4f + X4c + X4r <= 1
Xif >= 0, Xic >= 0, Xir >= 0, i = 1, 2, 3, 4.
Bonus Constraints: Any 2 of the following 3 equations:
12(15X1f + 9X2f + 18X3f + 10X4f) - 8(15X1c + 9X2c + 18X3c + 10X4c) = 0
7(15X1f + 9X2f + 18X3f + 10X4f) - 8(15X1r + 9X2r + 18X3r + 10X4r) = 0
12(15X1r + 9X2r + 18X3r + 10X4r) - 7(15X1c + 9X2c + 18X3c + 10X4c) = 0