Applied Mathematics for Business and Economics

Lecture Note Linear Programming (LP)

we can represent the farmer’s cost by the linear function Cx= 20 + 30 y

xy 10

and the

nutritional requirements by the linear inequalities^24 + ≥ xy

unct

and^5. The

resulting linear programming problem is to minimize the objective f
Cx= 20 to

16

ysubject

+≥

ion

+ 30

241

51

,0

xy

xy

xy

0

6

+ ≥

+ ≥

≥

Corner

Point

Value of

F=4x+ 3 y^

(0, 16) 480

(3, 1) 90

(5, 0) 100

Since 90 is the smallest of these values, the corresponding corner point (3, 1) must be
an optimal solution. We conclude that to satify the nutritional requirements at the
smallest possible cost, the farmer should feed each animal 3 units of grain I and 1 unit
of grain II each day. The minimal cost is then 90 cents per animal.

Exercises

1 In the following problems, graph the feasibility region of the given system of

linear inequalities. Find the coordinates of all corner points.cUrKUstMbn;EdlTTYYlyk

)anéncemøIyrbs;RbB½n§vismIkarxageRkam RBmTaMgrkkUG½redaenéncMNuckac;RCug.

1. 2. 21 3. 23

23

70

xy

xy

+<

+≥

0

9

6

45

0

xy

xy

y

+<

−≥

≥

7

1

,0

xy

xy

xy

+ ≤

+ ≤

≥

4.

3

20

3

,0

xy

xy

xy

xy

+≤

−≤

+≤

≥

Formulate the given problem as a linear program

cUrsresrrUbmnþcMeNaTkmμviFIlIenEG‘rxageRkamenaH¬min)ac;edaHRsay¦

2 At a local leather shop, 1 hour of skilled labor and 1 hour of unskilled labor are
required to produce a briefcase, while 1 hour of skilled labor and 2 hours of
unskilled labor are required to produce a suitcase. The owner of the shop can
make a profit of $15on each briefcase and $20 on each suitcase. On a particular
day, only 7 hours of skilled labor and 11 hours of unskilled labor are available,
and the owner wishes to determine how many briefcases and how many suitcases
to make that day to generate the largest profit possible. enAkñúghagplitniglk;plit

• 
  
  
  
  • 
    
    
    
    • (5, 0
    
    
    
    
  
  
  
  

(0,16)

( )

)

3,1

516 xy+=

2410 xy+=