Lecture 09

Dr. Arshad Zaheer

LINEAR PROGRAMMING (LP)

Minimization

Illustration

Solving Minimization ProblemsSolving Minimization Problems

Formulated and solved in much the same Formulated and solved in much the same way as maximization problemsway as maximization problems

In the graphical approach an iso-cost line In the graphical approach an iso-cost line is usedis used

The objective is to move the iso-cost line The objective is to move the iso-cost line inwards until it reaches the lowest cost inwards until it reaches the lowest cost corner pointcorner point

Minimization ExampleMinimization Example

XX11 = = number of tons of black-and-white picture chemical number of tons of black-and-white picture chemical producedproduced

XX22 = = number of tons of color picture chemical producednumber of tons of color picture chemical produced

Minimize total cost =Minimize total cost = 2,500X2,500X11 ++ 3,000X3,000X22

Subject to:Subject to:XX11 ≥ 30≥ 30 tons of black-and-white chemicaltons of black-and-white chemical

XX22 ≥ 20≥ 20 tons of color chemicaltons of color chemical

XX11 + X + X22 ≥ 60≥ 60 tons totaltons total

XX11, X, X22 ≥ $0≥ $0 nonnegativity requirementsnonnegativity requirements

Minimization ExampleMinimization ExampleTable B.9Table B.9

60 60 –

50 –

40 40 –

30 –

20 20 –

10 –

–| | | | | | |

00 1010 2020 3030 4040 5050 6060XX11

XX22

Feasible region

XX11 = 30 = 30XX22 = 20 = 20

XX11 + X + X22 = 60 = 60

bb

aa

Minimization ExampleMinimization Example

Total cost at aTotal cost at a == 2,500X2,500X11 ++ 3,000X3,000X22

== 2,500 (40)2,500 (40) ++ 3,000(20)3,000(20)== $160,000$160,000

Total cost at bTotal cost at b == 2,500X2,500X11 ++ 3,000X3,000X22

== 2,500 (30)2,500 (30) ++ 3,000(30)3,000(30)== $165,000$165,000

Lowest total cost is at point aLowest 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. 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 constraintsc. Solve graphically, using the corner point method.