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Expert Answer
Let the number of boxes with small gadgets be “a”, with medium gadgets be “b” and with large gadgets be “c”.
Total revenue (R) = 30a+10b+20c. This is our objective function and has to be maximized.
Constraints:
(i) 2a+b+c<=60 (total packing hours cannot exceed the available 60 hours)
(ii) a+b+c<=45 (storage area cannot hold more than 45 boxes)
(iii) a,b,c>=0 (non negativity)
Solving usinge the ‘solver’ function in excel the following solution is obtained:
| Number | ||||
| Small gadget box | 15 | |||
| Medium gadget box | 0 | |||
| Large gadget box | 30 | |||
| Formula | ||||
| 1,050.00 | Revenue | 30a+10b+20c | ||
| Constraints | Formula | |||
| 60.00 | <= | 60.00 | 2a+b+c<=60 | |
| 45.00 | <= | 45.00 | a+b+c<=45 | |
Thus the company will manufacture 15 boxes of small gadgets, 0 box of medium gadgets and 30 boxes of large gadgets. The maximized revenue is $1,050. All the constraints are satisfied.
As we can see that the above solution uses 60 hours that is availabe entirely. It also uses the entire storage area of 45 boxes. Hence all available storage rooms and hours are being used in the above mentioned solution.
Inclusion of slack variables:
We will use slack variables s1 and s2. The equations using the slack variables will be:
2a+b+c+s1 = 60
a+b+c+s2 = 45
For the objective function R = 30a+10b+20c we will move all the variables to the left side of the equal sign. We will get: -30a-10b-20c+R=0.
The augmented matrix for the above equations will be:
| a | b | c | s1 | s2 | z | |
| 2.00 | 1.00 | 1.00 | 1.00 | 0.00 | 0.00 | 60.00 |
| 1.00 | 1.00 | 1.00 | 0.00 | 1.00 | 0.00 | 45.00 |
| -30.00 | -10.00 | -20.00 | 0.00 | 0.00 | 1.00 | 0.00 |
Image for the solver solution in excel:
