Linear Programming Example 5 Unboundedness Infeasibility

The Problem TwinCon Liquors purchases and distributes wines to retailers. Four wines of particular interest their cost and selling price are given in the table below. During the next purchase cycle TwinCon wants: –To order at least 1000 bottles of each wine –To order at least twice as many bottles of imported wine as domestic wines. WineCountry Cost Selling Price Napa Grove US$ 5.00$ 8.50 Green Valley US$ 6.00$ 9.00 Riviera France$10.00$16.00 Cesarini Italy$ 8.00$12.00

The Model Let the X’s = # bottles of each wine ordered during the next purchase cycle Profit per bottle = (Selling Price) – (Cost) The constraint that # of bottles of imported wine (X 3 + X 4 ) should be ≥ twice the number of bottles of imported wine (X 1 + X 2 ) can be written as: -2X 1 -2X 2 + X 3 + X 4 ≥ 0 The model is: Max 3.50X 1 + 3X 2 + 6X 3 + 4X 4 s.t. X 1 ≥ 1000 X 2 ≥ 1000 X 3 ≥ 1000 X 4 ≥ X 1 - 2X 2 + X 3 + X 4 ≥ 0 Note: Since each variable is already restricted to be ≥ 1000, there is no need to add that the variables must also be ≥ 0.

=SUMPRODUCT($C$3:$F$3,C5:F5) Drag down

This means the problem is unbounded!

Analysis This indicates the problem is unbounded meaning TwinCon will make an infinite profit! –Impossible –Problem formulated correctly to this point –Constraints left out: Budget to purchase wine during next purchase cycle is $40000: 5X 1 + 6X X 3 + 8X 4 ≤ Max # bottles that can be ordered (warehouse space) is X 1 + X 2 + X 3 + X 4 ≤ 8000

=SUMPRODUCT($C$3:$F$3,C5:F5) Drag down

Analysis This time there is no feasible solution. –All constraints were input correctly. –They were just inconsistent. Note that if domestic bottles were at their minimum purchase level– this would amount to 2000 bottles. This would mean that at least 4000 bottles of imported wine must be purchased. If only 1000 on the more expensive (Riviera) were purchased this would mean that 3000 of the less expensive (Cesarini) would need to be purchased. –This solution would cost: 1000(5) (6) (10) (8) = $45,000 This exceeds the budget of $40,000. –Something must be modified. For example, suppose the purchase budget is increased to $50,000.

Change budget from $40,000 to $50,000

Optimal Solution 1040 bottles of Napa Grove 1000 bottles of Green Valley 3080 bottles of Riviera 1000 bottles of Cesarini Profit $29,120

Review Recognizing models that are unbounded Analyzing and modifying unbounded models Recognizing models that are infeasible Analyzing and modifying infeasible models.