**Deviation from the Goal**

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By doing the graphical method 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 + Y _{1}^{-}_{
}– Y_{1}^{+}* = 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. *Y _{1}^{-}_{ }=_{
} Y_{1}^{+}= 0*

Inserting these values we have*:*

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

Therefore, there is no penalty associated with this goal

**Goal 2 (Profit)**

100 *L* + 100 *E*
≥ 1000

100* L + *100 *E + Y _{2}^{-}
– Y_{2}^{+}* = 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 *Y _{2}^{+}*
= 0. The corresponding variable for fallen short of our goal is our
slack deviation variable, and thus Y

Therefore inserting these values we have:

100* (4) + *100 *(0)
+ 600 – 0* = 1000

Thus, since Y_{2}^{-}
is $600, our penalty associated with this goal is thus

($600) × ($1) = $600

**Goal 3 (Enamel Paint Production)**

*E ≥ *7* *

*E + Y _{3}^{-}
– Y_{3}^{+}* = 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, *Y _{3}*

Therefore inserting these values we have

*0* + *7 –* *0* =
7

Since Y_{3}^{-}_{
}is 7, then we have fallen short of our goal, and our penalty associated
with this goal is

(700) × ($0.50) = $350

**Objective function**

If we look back at our objective function

Min *3Y _{1}^{+}+
100 Y_{2}^{-} + 50 Y_{3}^{-}*

Let us insert the values we found for the deviation variables

*3(0)*
+ 100 *(600)* + *50 (7) = $950*

Thus the total penalties we have suffered for not achieving our goals 2 and 3 is $950.

**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