For what ranges of values, holding all else constant, could each of the objectiv

Question

For what ranges of values, holding all else constant, could each of the objective function coefficients be changed without changing the optimal solution? If the profit for Caribbean units was decreased to $75 while at the same time the profit for the Italian product was increased to $80, what would be the new solution values and the new profit? You give up 10 hours of carpentry and get 12 hours of finishing time. (Calculate new profit, show work). Suppose that two additional units of French type are delivered to the distributor, how would the solution change?

Explanation / Answer

1) Current solution is,
Number of units Italian = 72
Number of units French = 65
Number of Caribbean = 78

Range of values for which the optimal solution will not change is (Coeff - Allowable Decrease, Coeff + Allowable Increase)
Range of Number of units Italian = (72 - 3.17, 72 + 21.6) = (68.83, 93.6)
Number of units French = (65 - inf, 65 + 4) = (-inf, 69)
Number of Caribbean = (78 - 4.8, 78 + 3.6) = (73.2, 81.6)

2) Since the profit of Caribbean and Italian units lie in the range (calculated in earlier part), the solution will not change.
The new solution values will be,
Number of units Italian = 116.67
Number of units French = 60
Number of Caribbean = 350

Profit = 116.67* 72 + 60* 65 + 350* 78 = 39600.24

3) Allowable decrease of Carpentry constraint is 45
Allowable increase of Finishing is 15.3
If the constraint R.H. side for Carpentry is decreased by 10 (< 45) and constraint R.H. side for Finishing is increased by 12 (< 15.3)
the new profit will decreased by 10*4 (Decreased value of Carpentry* Shadow Price) and increased by 12*80 (Increased value of Finishing * Shadow Price) )
New profit = 39600.24 - 10*4 + 12*80 = 40520.24

4) Allowable increase of French unit is 348 which is greater than 2.
So, if the number of units of French increased by 2, the profits will changed by -4*2 = -8
New profit = 39600.24 - 8 = 39592.24

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