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246 CHAPTER 5 LINEAR PROGRAMMING: THE SIMPLEX METHOD

We cannot use the graphical method here as we have three decision variables so we shall use the Simplex.
Following the steps set out earlier, using x 1 , x 2 and x 3 as the decision variables for the quantities of the three
products to be produced we have a formulation:

Max 1x 1 þ 1:2x 2 þ 2x 3
s:t:

1x 1 þ 2x 2 150
1x 1 þ 2x 3 150
2x 1 þ 1x 2 80
2x 1 þ 3x 2 þ 1x 3 225
x 1 25

x 1 ; x 2 ; x 3 0
Putting the problem into standard form we then have:

Max 1x 1 þ 1:2x 2 þ 2x 3 þ 0s 1 þ 0s 2 þ 0s 3 þ 0s 4

s:t:

1x 1 þ 2x 2 þ 1s 1 150
1x 1 þ 2x 3 þ 1s 2 150
2x 1 þ 1x 2 þ 1s 3 80
2x 1 þ 3x 2 þ 1x 3 þ 1s 4 225
x 1 1s 5 25

x 1 ; x 2 ; x 3 ; s 1 ; s 2 ; s 3 ; s 4 0

We note that we have one constraint which takes the form so we will require an artificial variable for this
constraint and using the M method we then have:

Max 1x 1 þ 1:2x 2 þ 2x 3 þ 0s 1 þ 0s 2 þ 0s 3 þ 0s 4 Ma 5

s:t:

1x 1 þ 2x 2 þ 1s 1 150
1x 1 þ 2x 3 þ 1s 2 150
2x 1 þ 1x 2 þ 1s 3 80
2x 1 þ 3x 2 þ 1x 3 þ 1s 4 225
x 1 1s 5 þ Ma 5 25

x 1 ; x 2 ; x 3 ; s 1 ; s 2 ; s 3 ; s 4 ; s 5 ; a 5 0

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