Linear Programming

1. Linear Programming Example 5 Unboundedness Infeasibility

2. 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. Wine Country 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

3. 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 (X3 + X4) should be ≥ twice the number of bottles of imported wine (X1 + X2) can be written as: -2X1 -2X2 + X3 + X4 ≥ 0 • The model is: Max 3.50X1 + 3X2 + 6X3 + 4X4 s.t. X1≥ 1000 X2 ≥ 1000 X3 ≥ 1000 X4 ≥ 1000 -2X1 - 2X2 + X3 + X4 ≥ 0 Note:Since each variable is already restricted to be ≥ 1000, there is no need to add that the variables must also be ≥ 0.

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

5. This means the problem is unbounded!

6. 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: 5X1 + 6X2 + 10X3 + 8X4≤ 40000 • Max # bottles that can be ordered (warehouse space) is 8000. X1 + X2 + X3 + X4 ≤ 8000

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

9. 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) + 1000(6) + 1000(10) + 3000(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.

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

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

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