Simplex Method Example

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  • Topic: Harshad number, Semi-detached, Highly composite number
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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 = 120, y = 200, w = 11200, P = 18000 Find the next pivot element. 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

Here’s the next pivot element. What is the next row operation? COPYRIGHT © 2006 by LAVON B. PAGE

x
! 1 # # 2 # # 1 # # 2 # # # 1 # # # -15 # "

y
0 1 0 0

u
1 0 0 0

v

w P
0 0 0 1 $ & 120 & & & 200 & & & & 140 & & & 18000& & %

-1 100 0 1 100 0 -1 1 200 80 9 10 0

The pivot element has been made into 1. Specify the remaining row operations in this step. COPYRIGHT © 2006 by LAVON B. PAGE

x
! # # # # # # # # # # # # # " 0 0 1 0

y
0 1 0 0

u
1 0 0 0

-3 -1 400 160 1 -1 80 160 -1 1 200 80 33 3 40 16

v

w

P
0 0 0 1 $ & 50 & & & 130 & & & & 140 & & & 20100 & & %

After the second step this is what the tableau looks like. Interpret it now. COPYRIGHT © 2006 by LAVON B. PAGE

x
! # # # # # # # # # # # # # " 0 0 1 0

y
0 1 0 0

u
1 0 0 0

-3 -1 400 160 1 -1 80 160 -1 1 200 80 33 3 40 16

v

w

P
0 0 0 1 $ & 50 & & & 130 & & & & 140 & & & 20100 & & %

After the second step: v=0, w=0, u=50, y=130, x=140, P=$20,100. This is the optimal solution since there are no negative numbers in the bottom row.

COPYRIGHT © 2006 by LAVON B. PAGE

Planning Subdivision
A developer is planning to build a new subdivision consisting of townhouses, single-story detached houses and two-story detached houses. On one acre he can put 6 townhouses or 4 single-story houses or 2 two-story houses and he has 60 acres available. It costs him $40,000 to build each townhouse, $50,000 to build each single-story house and $60,000 to build each two-story house. He makes a profit of $15,000 on each townhouse, $18,000 on each single-story house and $20,000 on each two-story house and he has $2,880,000 of capital available. Townhouses require 2500 hours of labor,...
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