A company manufactures gadgets that come in small, medium, and large. The storage area can hold a

A company manufactures gadgets that come in small, medium, and large. The storage area can hold a total of 45 boxes. Each box can hold 200 small. 100 medium, or 50 large gadgets. They have have enough hours to make 5000 gadgets. A box of small gadgets takes 2 hours to pack, white the boxes of medium or large require only 1 hour. They have 60 hours available for packing. Small gadgets sell for $30 a box, medium for $10. and large for $20. How many boxes of each should be made to maximize their revenue? Have they used all available storage room and hours? Tell the story behind each basic feasible solution sure to include the slack variables and specify if this is a maximum.

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.

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(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:

Small gadget box 15
Medium gadget box 0
Large gadget box 30
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:

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