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Deviation from the Goal

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By doing the graphical method or using Lindo to the problem found in Page 1, you would have found that the solution to the problem is

L = 4 and E = 0. This means 400 gallons of latex paint is produced and 0 gallons of enamel paint

Lets look back at each of our goals.

 

Goal 1 (Overtime Labour)

10 L + 15 E ≤ 40   

10 L + 15 E + Y1-    Y1+ = 40

Thus if we evaluate our goal we have

10(4) + 15 (0) = 40 hours                                   

Therefore, our goal has been met exactly. We have not used any overtime labour. There is no deviation from our goal, i.e. Y1- =  Y1+= 0

Inserting these values we have:

10(4) + 15 (0) + 0 + 0 = 40 hours  

 

Goal 2 (Profit)

100 L + 100 E ≥ 1000

100 L + 100 E + Y2- Y2+ = 1000

Evaluating our goal

100(4) + 100(0) = 400

Our goal is a $1000, but we have only achieved $400. Thus, we have fallen short of our goal, by $600 (i.e. $1000 - $400 = $600). Since, we have not exceeded our goal, then Y2+ = 0. The corresponding variable for  fallen short of our goal is our slack deviation variable, and thus Y2- = $600.

Therefore inserting these values we have:

 100 (4) + 100 (0) + 600 0 = 1000

 

Goal 3 (Enamel Paint Production)

E ≥ 7  

E + Y3-   Y3+  =  7

Thus if we evaluate our goal:

Goal 3 = 0

Therefore our goal is 700 gallons of enamel paint, however we are producing 0 gallons of enamel paint. This means we have fallen short of our goal by 700 (i.e. 700 - 0 = 700 enamel paint). We have not exceeded our goal, therefore, Y3+ = 0. We have however, fallen short of goal, which corresponds to the slack variable, Y3- = 7. Remember that the deviation variables are measured in 100 gallons of paint.

Therefore inserting these values we have

0 + 7 0 = 7

 

Here is some additional information:

Objective function

If we look back at our objective function

Min      P1Y1+ + P2Y2- + P3Y3-

Let us insert the values we found for the deviation variables

  P1(0) + P2(600) + P3(7) = 600 P2 + 7 P3

This equation summarizes the extent to which the goals were met.

 

Constraints

Let us take a look at our constraints

Constraint 1 (Labour)

10 L + 15 E ≤ 70 

Substituting the values

10 (4) + 15 (0) = 40

Therefore the constraint is non-binding and there is a slack of 70 40 = 30 hrs. The 30 hrs corresponds the overtime which we did not use (as found in Goal 1).

 

Constraint 2 (Production Constraint)

L E ≥ 0

Substituting E and L values

4 0 = 4

Therefore this constraint is non-binding and there is an excess of 400 gallons of paint (i.e. 4 0 = 4 100-gallons of paint ).

Note: Remember the minimum required was for E and L to be equal or in other words L E = 0

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