NW Food Ltd needs to store 500 m3 (cubic meters) of wheat in identical rectangular boxes. The top and bottom of each box will be made from a type of material which costs £1.00/m2 (£1.00 per square meter). The four sides of each box will be made from another type of material which costs £1.50/ m2. It costs £0.20 to assemble each box. What should be the dimensions and the total number of boxes to minimise the cost of storing the wheat? You are required to construct a mathematical program to represent the problem. What type of mathematical program is this (linear/non-linear, constrained/non-constrained, single/multi variable, integer, etc.)?
I've attempted to solve it
Assuming, dimensions are: length of base =x, breadth of base =y, height of box =z
Volume of box = xyz
number of boxes required = 500/xyz
Area of Base+Top = 2xy
Area of Sides = 2xz + 2yz
Cost per box = (2xy + 3xz + 3yz +0.2)
Total Cost = (500/xyz)(2xy + 3xz + 3yz +0.2) = (1000/z) + (1500/y) + (1500/x) + (100/xyz)
I've taken the partial derivative of the total with respect to x, y and z and equate them to 0.
However, the results are odd. for the partial derivate with respect to x is (-1500/x^2) - (100/yzx^2) = 0, which can't really be solved.
Where have i gone wrong?