9.3 a) Optimal solution: (x1, x2) = (2, 3). Profit = 13.
b) The optimal solution to the LP-relaxation is (x1, x2) = (2.6, 1.6). Profit = 14.6.
Rounded to the nearest integer, (x1, x2) = (3, 2). This is not feasible since it violates the third constraint.
Rounded Solution Feasible? (3,2) No (3,1) No (2,2) Yes (2,1) Yes
None of these is optimal for the integer programming model. Two are not feasible and
the other two have lower values of Profit.
Constraint Violated 3rd 2nd & 3rd - - P - - 12 11 9-5
9.4 a) Optimal solution: (x1, x2) = (2, 3). Profit = 680.
b) The optimal solution to the LP-relaxation is (x1, x2) = (2.67, 1.33). Profit = 693.33.
Rounded to the nearest integer, (x1, x2) = (3, 1). This is not feasible since it violates the second and third constraint.
Rounded Solution (3,1) (3,2) (2,2) (2,1)
None of these is optimal for the integer programming model. Two are not feasible and the other two have lower values of Profit.
Feasible? No No Yes Yes Constraint Violated 2nd & 3rd 2nd - - P - - 600 520
9-6
9.5 a)
A123456789101112BCDEFGLong-RangeJetsAnnual Profit ($million)4.2BudgetMaintenance CapacityPilot CrewsMedium-RangeJets3Short-RangeJets2.3ResourceUsed1498<=39.333<=30<=ResourceAvailable15004030Total AnnualProfit ($million)95.6 Resource Used Per Unit Produced6750351.6671.3331111Long-RangeJets14Medium-RangeJets0Short-RangeJets16Purchase
b) Let L = the number of long-range jets to purchase
M = the number of medium-range jets to purchase S = the number of short-range jets to purchase Maximize Annual Profit ($millions) = 4.2L + 3M + 2.3S subject to 67L + 50M + 35S ≤ 1