Transcript Lecture 9 linear programming 3.ppt

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
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
X2
X1 + X2
X1, X2
≥ 30
≥ 20
≥ 60
≥ $0
tons of black-and-white chemical
tons of color chemical
tons total
nonnegativity requirements
Minimization Example
Table B.9
X2
60 –
X1 + X2 = 60
50 –
Feasible
region
40 –
30 –
b
20 –
a
10 –
|–
0
X1 = 30
|
10
|
20
X2 = 20
|
30
|
40
|
50
|
60
X1
Minimization Example
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
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. 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.