LINEAR PROGRAMMING (LP) Lecture 09 Dr. Arshad Zaheer

Minimization Illustration

Solving Minimization Problems Formulated and solved in much the same way as maximization problems In the graphical approach an iso-cost line is used The objective is to move the iso-cost line inwards until it reaches the lowest cost corner point It might be useful at this point to discuss typical equipment utilization rates for different process strategies if you have not done so before.

Minimization Example X1 = number of tons of black-and-white picture chemical produced X2 = number of tons of color picture chemical produced Minimize total cost = 2,500X1 + 3,000X2 Subject to: X1 ≥ 30 tons of black-and-white chemical X2 ≥ 20 tons of color chemical X1 + X2 ≥ 60 tons total X1, X2 ≥ $0 nonnegativity requirements It might be useful at this point to discuss typical equipment utilization rates for different process strategies if you have not done so before.

Minimization Example X2 X1 + X2 = 60 Feasible region b a X1 = 30 60 – 50 – 40 – 30 – 20 – 10 – – | | | | | | | 0 10 20 30 40 50 60 X1 X2 Table B.9 X1 + X2 = 60 Feasible region b It might be useful at this point to discuss typical equipment utilization rates for different process strategies if you have not done so before. a X2 = 20 X1 = 30

Minimization Example Lowest total cost is at point a Total cost at a = 2,500X1 + 3,000X2 = 2,500 (40) + 3,000(20) = $160,000 Total cost at b = 2,500X1 + 3,000X2 = 2,500 (30) + 3,000(30) = $165,000 It might be useful at this point to discuss typical equipment utilization rates for different process strategies if you have not done so before. Lowest total cost is at point a

The Queen City Nursery manufactures bags of potting soil from compost and topsoil. Each cubic foot of compost costs 12 cents and contains 4 pounds of sand, 3 pounds of clay, and 5 pounds of humus. Each cubic foot of topsoil costs 20 cents and contains 3 pounds of sand, 6 pounds of clay, and 12 pounds of humus. Each bag of potting soil must contain at least 12 pounds of sand, at least 12 pounds of clay, and at least 10 pounds of humus. Plot the constraints and identify the feasible region. Graphically or with corner points find the best combination of compost and topsoil that meets the stated conditions at the lowest cost per bag. Identify the lowest cost possible.

Rienzi Farms grows sugar cane and soybeans on its 500 acres of land Rienzi Farms grows sugar cane and soybeans on its 500 acres of land. An acre of soybeans brings a $1000 contribution to overhead and profit; an acre of sugar cane has a contribution of $2000. Because of a government program no more than 200 acres may be planted in soybeans. During the planting season 1200 hours of planting time will be available. Each acre of soybeans requires 2 hours, while each acre of sugar cane requires 5 hours. The company seeks maximum contribution (profit) from its planting decision. a. Algebraically state the decision variables, objective and constraints. b. Plot the constraints c. Solve graphically, using the corner point method.