How can I find constraints in a linear programming word problem?

Question

A toy manufacturer makes wooden blocks and horses.  The production process involves twobasic types of labor: carpentry and design. A block requires 2 hours of carpentry and 1 hour of design, and a horse requires 1hour of carpentry and 3 hours of design.  The profit is $10 per block and $14 per horse.  The manufacturer's employeescan supply a maximum of 10 hours of carpentry work and 15 hours of design work per day.  How many blocks and horses should be made each day to maximize profit?

Answer 1

 

Hi Ben,

 

If x and y represent the number of blocks and horses to be made, respectively.

 

The problem is to find out a pair of x and y to miximize

 

     10x + 14y                       (Eq. 1)

 

under the set of conditions

 

     x ≥ 0                              (Eq. 2)

 

     y ≥ 0                              (Eq. 3)

 

     2x + y ≤ 10                    (Eq. 4)

 

     x + 3y ≤ 15                    (Eq. 5)

 

Out of 4 vertices from the region which satisfies Eq. 2 through Eq. 5,

the point (x, y) = (3, 4) maximizes Eq. 1.

 

The maximum profit is

 

     10x + 14y = 10·3 + 14·4

 

                     = 30 + 56

 

                     = 86

 

when x = 3 and y = 4.

 

So, the manufacturer needs to make

 

3 blocks and 4 horses each day.