Simplex Method Example
Simplex Method — more examples with standard problems
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem (once again)
A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy?
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem x y u v w P
0 0 1 0 0 0 0 1 20000 & & 19200 & 0 % & 320 $ &
! 1 1 1 0 # # 50 100 0 1 # #100 40 0 0 # # -60 -90 0 0 # "
& &
Pick the first pivot element.
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem x y u v w P
0 0 1 0 0 0 0 1 20000 & & 19200 & 0 % & 320 $ &
! 1 1 1 0 # # 50 100 0 1 # #100 40 0 0 # # -60 -90 0 0 # "
& &
Here’s the first pivot element. Now determine the first row operation.
COPYRIGHT © 2006 by LAVON B. PAGE
x y u v w P ! 1 1 1 0 0 0 320 $ # & # & 1 1 # & 1 0 100 0 0 200 & # 2 # & #100 40 0 0 1 0 19200 & # & # -60 -90 0 0 0 1 " % 0 & The pivot element has been made into 1. What are the next row operations?
COPYRIGHT © 2006 by LAVON B. PAGE
x
! 1 # # 2 # # 1 # # 2 # # # 80 # # # -15 # "
y
0 1 0 0
u
v
w P
0 0 1 0 0 0 0 1 $ & 120 & & & 200 & & & & 11200 & & & 18000 & & %
-1 1 100 1 0 100 -2 0 5 9 0 10
This completes the first step. Now give an interpretation of this tableau.
COPYRIGHT © 2006 by LAVON B. PAGE
x
! 1 # # 2 # # 1 # # 2 # # # 80 # # # -15 # "
y
0 1 0 0
u
1 0 0 0
v
-1 100 1 100 -2 5 9 10
w P
0 0 1 0 0 0 0 1 $ & 120 & & & 200 & & & & 11200 & & & 18000 & & %
This completes the first step: x = 0, v = 0, u
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem (once again)
A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy?
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem x y u v w P
0 0 1 0 0 0 0 1 20000 & & 19200 & 0 % & 320 $ &
! 1 1 1 0 # # 50 100 0 1 # #100 40 0 0 # # -60 -90 0 0 # "
& &
Pick the first pivot element.
COPYRIGHT © 2006 by LAVON B. PAGE
Farmer problem x y u v w P
0 0 1 0 0 0 0 1 20000 & & 19200 & 0 % & 320 $ &
! 1 1 1 0 # # 50 100 0 1 # #100 40 0 0 # # -60 -90 0 0 # "
& &
Here’s the first pivot element. Now determine the first row operation.
COPYRIGHT © 2006 by LAVON B. PAGE
x y u v w P ! 1 1 1 0 0 0 320 $ # & # & 1 1 # & 1 0 100 0 0 200 & # 2 # & #100 40 0 0 1 0 19200 & # & # -60 -90 0 0 0 1 " % 0 & The pivot element has been made into 1. What are the next row operations?
COPYRIGHT © 2006 by LAVON B. PAGE
x
! 1 # # 2 # # 1 # # 2 # # # 80 # # # -15 # "
y
0 1 0 0
u
v
w P
0 0 1 0 0 0 0 1 $ & 120 & & & 200 & & & & 11200 & & & 18000 & & %
-1 1 100 1 0 100 -2 0 5 9 0 10
This completes the first step. Now give an interpretation of this tableau.
COPYRIGHT © 2006 by LAVON B. PAGE
x
! 1 # # 2 # # 1 # # 2 # # # 80 # # # -15 # "
y
0 1 0 0
u
1 0 0 0
v
-1 100 1 100 -2 5 9 10
w P
0 0 1 0 0 0 0 1 $ & 120 & & & 200 & & & & 11200 & & & 18000 & & %
This completes the first step: x = 0, v = 0, u