Linear Programming Problems and Solutions
Problem #B-1: Simultaneous Equations for Optimization
a. Maximize Z = 4×1 + 3×2
Subject to:
Material: 6×1 + 4×2 <= 48 lb (x1 = 2)
Labor: 4×1 + 8×2 <= 80 hr (x2 = 9)
x1, x2 >= 0 (Profit = 35)
b. Maximize Z = 6A + 3B
Subject to:
Material: 20A + 6B <= 600 lb (A = 24)
Machinery: 25A + 20B <= 1000 hr (B = 20)
A, B >= 0 (Profit = 204)
Problem #B-2: Microwave Oven Production
An appliance manufacturer produces two models of microwave ovens: H and W. The goal is to determine the quantities of H and W that will maximize profit.
Maximize z = $40H + $30W
Subject to:
Fabrication: 4H + 2W <= 600 hours
Assembly: 2H + 6W <= 480 hours
H = 132, W = 36, Profit = $6,360
Problem #B-3: Excel Solver for Optimization
a. Maximize Z = 4×1 + 2×2 + 5×3
Subject to:
1×1 + 2×2 + 1×3 <= 25 (x1 = 4)
1×1 + 4×2 + 2×3 <= 40 (x2 = 0)
3×1 + 3×2 + 1×3 <= 30 (x3 = 18)
x1, x2, x3 >= 0 (Profit = 106)
b. Maximize Z = 10×1 + 6×2 + 3×3
Subject to:
1×1 + 1×2 + 2×3 <= 25 (x1 = 15)
2×1 + 1×2 + 4×3 <= 40 (x2 = 10)
1×1 + 2×2 + 3×3 <= 40 (x3 = 0)
x1, x2, x3 >= 0 (Profit = 210)
Problem #B-4: Erlanger Manufacturing Company Production Planning
The Erlanger Manufacturing Company produces two products and wants to maximize profits.
a. Maximize 25×1 + 30×2
Subject to:
1.5×1 + 3×2 <= 450 (Dept A)
2×1 + 1×2 <= 350 (Dept B)
0.25×1 + 0.25×2 <= 50 (Dept C)
x1, x2 >= 0
b. Optimal Solution: x1 = 100, x2 = 100, Profit = $5,500
c. Production Time and Slack Time: All capacity used in departments A and C, 50 hours of slack in department B.
Problem #B-5: M&D Chemicals Production Planning
M&D Chemicals produces two products and wants to satisfy production requirements at minimum cost.
Minimize 2×1 + 3×2
Subject to:
x1 >= 125 (Product 1 demand)
x1 + x2 >= 350 (Total production)
2×1 + x2 <= 600 (Processing time)
x1, x2 >= 0
b. Total Product Cost: x1 = 250, x2 = 100, Total cost = $800
c. Surplus Production: 125 gallons of product 1.
Problem #B-6: Photo Chemicals Production Planning
Photo Chemicals produces two types of photograph-developing fluids and wants to minimize production costs while meeting requirements.
Minimize 1×1 + 1×2
Subject to:
1×1 >= 30 (Product 1 minimum)
1×2 >= 20 (Product 2 minimum)
1×1 + 2×2 >= 80 (Raw material)
x1, x2 >= 0
Minimum-Cost Solution: x1 = 30, x2 = 25, Cost = $55
Problem #B-7: Wood Products Firm Production Planning
A wood products firm produces chopping boards and knife holders and wants to determine optimal production quantities.
a. Optimal Quantities: Board = 0, Holder = 50, Profit = $300
b. Unused Resources: Cutting = 16 minutes, Gluing = 0 minutes, Finishing = 210 minutes
Problem #B-8: Linear Programming Model Analysis
This problem analyzes a linear programming model and answers questions about binding constraints, changes in profit, and resource availability.
A) Binding Constraints: The first constraint (machine) and the third constraint (material) are binding.
B) Profit Change on Product 3: Increasing profit to $22 per unit would not change decision variables but would increase the objective function value to $1,128.
C) Profit Change on Product 1: Increasing profit to $22 per unit would not affect decision variables or the objective function value.
D) Labor Time Reduction: Reducing labor time by 10 hours would not affect decision variables or the objective function value but would reduce slack by 10 hours.
E) Product 2 Production Increase: Increasing production to 20 units would not change profit if no additional resources are obtained.
Problem #B-9: Garden Store Mulch Production
A garden store produces various grades of pine bark mulch and wants to optimize production.
A) Shadow Price of Pine Bark: $1.50 per pound, appropriate for 550 lbs. to 750 lbs. of bark.
B) Maximum Price for Additional Pine Bark: $1.50 per pound.
C) Shadow Price of Labor: Zero, in effect from 375 hours to infinity.
D) Additional Machine Time: No additional machine time can be effectively used due to existing excess.
E) Additional Resources: The manager should choose to add 150 pounds of pine bark for a greater expected increase in profit.
F) Profit Change on Chips: Increasing profit to $7 per bag would not change decision variables but would increase the objective function value to $1,200.