Problem 1) Graphically solve the following LPs and identify which of Cases 1-4 (Unique, Many, Infeasible, Unbounded) apply.

Answer:1(i) – Unbounded – since there is no single poiny in feasible region that can maximize the function.

1(ii)-Unique Solution with Z=12(0,6)

1(iii) Many Solutions- since function Z lies on the minimum Z line

1(iv) Unbounded- Since Z has many minimum arbitrary values in the feasible region

Problem 2) Leary Chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B and 1 of C. Running process 2 for an hour costs $1 and produces 1 unit of A and 1 of B. To meet customer demands, at least 10 units of A, 5 of B, and 3 of C must be produced daily. Formulate an LP that determines a daily production plan that minimizes the cost of meeting Leary Chemical’s daily demands.

Answer:

Given:

Product A B CRunning Process 1/hour yields (units) 3 1 1Running Process 2/hour yields (units) 1 1 0

Running Process 1 per hour cost = $4Running Process 2 per hour cost = $1

Also, at least 10 units of A, 5 of B, and 3 of C must be produced daily.

Let X1 be number of hours of running process 1X2 be number of hours of running process 2

Formulation:To minimize cost: Min= 4X1+X2

Constraints:

3X1+X2≥10 (Constraint on chemical A)X1+X2≥5 (Constraint on chemical B)X1≥3 (Constraint on chemical C)X1, X2 ≥ 0 (Sign Restriction)

Problem 3) Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3. A desk contributes $40 to profit, and a chair contributes $25. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. If 20 units of wood are available, formulate an LP to maximize Furnco’s profit.

Answer:

Given:

Product Desk ChairProfit per each unit 40 25Wood Consumption per unit 4 3

Available quantity of wood = 20 units Number of chairs should be at least twice the number of desks produced.

Let X1 be number of units of DeskX2 be number of units of chair

Formulation:To maximize revenue: Max= 40X1+25X2

Constraints:4X1+3X2≤ 20 (Constraint on Wood)2X1≤ X2 => 2X1- X2≤0 (Marketing Restriction)X1, X2 ≥ 0 (Sign Restriction)

Problem 4) Bloomington Breweries produces beer and ale. Beer sells for $5 per barrel, and ale sells for $2 per barrel. Producing a barrel of beer requires 5 lbs of corn and 2 lbs of hops. Producing a barrel of ales requires 2 lbs of corn and 1 lb of hops. Sixty lbs of corn and 25 lbs of hops are available. Formulate an LP that can be used to maximize revenue.

Answer:

Given:

Product Beer AleProfit per barrel 5 2Corn Consumption per barrel (lbs) 5 2Hops consumption per barrel (lbs) 2 1

Available quantity of corn (in lbs) = 60 Available quantity of hops (in lbs) = 25

Let X1 be number of barrels of beerX2 be number of barrels of ale

Formulation:To maximize revenue: Max= 5X1+2X2

Constraints:5X1+2X2≤ 60 (Constraint on Corn)2X1+X2≤25 (Constraint on Hops)X1, X2 ≥ 0 (Sign Restriction)