Linear programming: simplex question (might belong in precal)

[saching]

Homework Statement

Original question: The Cut-Right Company sells sets of kitchen knives. The Basic Set consists of 2 utility knives and 1 chef’s knife. The Regular Set consists of 2 utility knives, 1 chef’s knife, and 1 slicer. The Deluxe Set consists of 3 utility knives, 1 chef’s knife, and 1 slicer. Their profit is $30 on a Basic Set, $40 on a Regular Set, and $60 on a Deluxe Set. The factory has on hand 800 utility knives, 400 chef’s knives, and 200 slicers. Assuming that all sets will be sold, how may of each type should be produced in order to maximize profit? What is the maximum profit?

Homework Equations

s₁, s₂, s₃ are slack variables
x₁, x₂, x₃ are all non-negative

x₁ = # of basic sets produced
x₂ = # of regular sets produced
x₃ = # of deluxe sets produced

2*x₁ + 2*x₂ + 3*x₃ + s₁ = 800
x₁ + x₂ + x₃ + s₂ = 400
x₂ + x₃ + s₃ = 200

a is the profit function given by:
a = 30*x₁ + 40*x₂ + 60*x₃

The Attempt at a Solution

I circled the pivot element in row3,col3. It's a "1".

I know I chose the pivot correctly(chose the most negative number in the bottom row, then found the column entry that is least when divided into the right-most column). Then I perform row operations to get all the rest of the numbers in the pivot column to be 0. Then I perform a row operation to turn the entry in the last row of the pivot column to 0. This makes the entire last row positive so the simplex algorithm is supposedly done--but it's not. The correct answer requires 200 deluxe sets and 100 basic sets be produced.

I notice that I still have slack variables as basic variables. My guess is that I either performed the row operations incorrectly or there is some other step that I am overlooking. Ideas?

 

[cyeokpeng]

Hi,

Hopefully, I am not late to help.

After performing the row (pivot) operations, remember to update the "a", your cost coefficient of the entering variable x3 into the 3rd row, i.e. now "a" for the 3rd row is updated from 0 to 60!
I think this is what causes the error in your Simplex calculations.
Redo your workings again, and I am sure you can get the correct optimal solution!

If you still can't find it, you can pm me here.