LINEAR PROGRAMMING

Question

A manufacturer produces two types of grills: Old Smokey and Blaze Away. During production, the grills require the use of two machines, A and B. The number of hours needed on both machines are indicated in the following table:
                           MACHINE A     MACHINE B

OLD SMOKEY       4 HR                6 HR


BLAZE AWAY       5 HR                2 HR


If each machine can be used for 24 hours a day, and the profits of Old Smokey and Blaze Away models are $5 and $7 respectively, how many of each type of grill should be made per day to obtain maximum profit? What is the maximum profit?

Answer 1

A manufacturer produces two types of grills: Old Smokey and Blaze Away. During production, the grills require the use of two machines, A and B. The number of hours needed on both machines are indicated in the following table:
MACHINE A MACHINE B

OLD SMOKEY 4 HR 6 HR


BLAZE AWAY 5 HR 2 HR


If each machine can be used for 24 hours a day, and the profits of Old Smokey and Blaze Away models are $5 and $7 respectively, how many of each type of grill should be made per day to obtain maximum profit? What is the maximum profit?

Solver Engine
Engine: GRG Nonlinear
Solution Time: 0.203 Seconds.
Iterations: 7 Subproblems: 2
Solver Options
Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling
Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds
Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative

Objective Cell (Max)
Cell Name Original Value Final Value
$G$13 5x + 7y       33.6          33


Variable Cells
Cell Name Original Value Final Value Integer
$G$14 x 0                          1 Integer
$G$15 y 4.8                       4 Integer


Constraints
Cell Name Cell Value Formula Status Slack
$G$16 A 24 $G$16<=24 Binding 0
$G$17 B 14 $G$17<=24 Not Binding 10
$G$18 A 0 $G$18>=0 Binding 0
$G$19 B 0 $G$19>=0 Binding 0
$G$14=Integer
$G$15=Integer

There may be two solutions: one making integer numbers of grills an is 1 old smokey and 4 of blaze away with a total of hours worked in machine A of 24 and in machine B of 14 or there may be a second answer but it will have fractions which is not reliable.max profit 33 dollars

 

Answer 2

As it is not evident, I am assuming it does not affect which machine is used first. That is to complete a job, we can either use machine A first or B. 

 

Lets say to get maximum profit company makes x complete units of grill OS and y complete units of grill BA.

 

Now we have

 

 

x,y as non negative integers

4x + 5y  < 24 (Machine A takes 4 hrs for OS grill and 5 hrs for BA grill)

6x + 2y  < 24 (Machine A takes 6 hrs for OS grill and 2 hrs for BA grill)

 

Profit = 5x + 7y

 

Possible solutions for the inequalities based on trial and error

 

1. x=0, y=4

2. x=1, y=4

3. x=2, y=3

4. x=3, y=2

5. x=4, y=0

 

Putting all these values in the profit equation we obtain the maximum value for x=1 and y=4

 

Max profit = 5*1 + 7*4 = $33