Question

Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows:

Category	Product 1	Product 2	Product 3
Profit/unit	$36	$58	$19
Machine 1 time/unit	0.5	2.5	1.5
Machine 2 time/unit	1	1.5	1.5

Two operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 90 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 25% of the units produced.

1. How many units of each product should be produced to maximize the total profit contribution? If required, round your answers to the nearest integer.
   

   Product	# of units
   1
   2
   3

   
   What is the projected weekly profit associated with your solution?
   
   $  
   
2. How many hours of production time will be scheduled on each machine? If required, round your answers to two decimal places.
   
   Machine Hours Schedule: Machine 1:  hours; Machine 2:  hours
   
3. What is the value of an additional hour of labor? If required, round your answer to two decimal places.
   
   $  
   
4. Assume that labor capacity can be increased to 120 hours. Develop the optimal product mix, assuming that the extra hours are made available. If required, round your answers to the nearest integer.
   

   Product	# of units
   1
   2
   3

   
   Profit = $  
   
   Would you be interested in using the additional 30 hours available for this resource?
   
   

Answer 1

a.

Let,

xi = number of product i to produce where i = {1,2,3}

Objective is to maximize profit = Max 36P1+58P2+19P3

subject to,

2*(0.5x1+2.5x2+1.5x3)+x1+1.5x2+1.5x3 <= 90 (Labor hours)

or, 2x1+6.5x2+4.5x3 <= 90

0.5x1+2.5x2+1.5x3 <= 40 (Machine 1)
x1+1.5x2+1.5x3 <= 40 (Machine 2)

x1 <= 0.5*(x1+x2+x3) (max percentage of product 1)
or,0.5x1-0.5x2-0.5x3 <= 0

x3 >= 0.25*(x1+x2+x3) (min percentage of product 3)
or, -0.25x1-0.25x2+0.75x3 >= 0

xi >= 0 (non-negativity constraint)

Solving in solver we get,

Product	# of units
1	12
2	6
3	6

Projected weekly profit = 894

Solver screenshot

Solver formula

Solver window

Sensitivity report

b) Machine Hours Schedule: Machine 1: 30 hours; Machine 2: 30 hours

c) Value of an additional hour of labour = 9.93

d) New optimal product mix :

Product	# of units
1	16
2	8
3	8

Projected weekly profit = 1192

Yes definitely one would be interested in the additional 30 hours available for this resource as profit has increased by using the additional 30 hours