Linear Programming With Sensitivity Analysis

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Linear Programming With Sensitivity Analysis

• Here we look at some graphs to learn some ideas.

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Say a company, Flair Furniture can make 7 profit
for every table (T) it makes and 5 for every
chair (C)it makes. Then Profit 7T 5C
C
I have graphed here 3 profit lines, where on each
line the profit is the same, just the number of
tables and chairs changes. Lines farther right
have greater profit. The thing this graph does
not show is the amount of tables and chairs the
firm can make from a production point of view
T
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Say tables and chairs have to pass through both a
carpentry area and a painting area. Say in
carpentry a table needs 4 hours and a chair needs
3 hours and the company only makes 240 hours
available to work in carpentry.
C
Also say in painting that a table needs 2 hours
and a chair needs 1 hour and the total number of
hours is 100. The two constraints are shown, but
on the next slide we see how this shows us as the
feasible region of production
T
4
C
T
5
Now, if the profit line(s) has slope flatter than
the lower right portion, but steeper than the
upper left portion of the constraint, then the
solution will occur with some of both items
produced at the point of intersection of the two
constraints. This is shown at the left.
C
T
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We saw before the profit was 7T 5C, and that
shows up as the solid line in the graph. What
would happen to the shape of the profit line if
the profit on tables became 8? Given a total
profit level ( a numerical value), since we get
more profit now, per table, we would need less
tables made to get a certain total profit. The
profit lines become steeper, like the dashed line.
C
T
7
In QM for Windows we are told, in the Ranging
section of the output of the computer work, how
much the profit amount can change for a table or
chair, BUT the solution of 30 tables and 40
chairs will not change. This amounts to having
the profit line changing slope, but not too much.
The dashed line is an illustration of this.
C
Note, the total profit of 30 tables and 40 chairs
may change, but not the amount of each produced.
T
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Also in Ranging section of the output in QM for
Windows we get values called dual values. Lets
think about what they relate to. Remember in
carpentry for the firm we had 240 hours. If we
add an hour, so that we have 241 hours what will
happen? The dual value is an expression of how
much our profit would rise. Similarly, if we had
only 239 hours the dual value is how much our
profit would fall. More hours in carpentry would
result in the constraint being pushed out,
allowing a larger feasible region, and thus
larger profits.
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QM for Windows Linear Programming When QM for
windows is on go to module on the menu and scroll
to linear programming. Then do file, new. Give
title if you want. Then since we have a painting
constraint and carpentry constraint, say you have
two constraints. Since we have to decide on
tables and chair amounts, put two variables. Hit
the maximize button. Hit OK and you see the input
screen. Input coefficients. You re-label X1 and
X1 as tables and chairs, and similarly with
constraints. Hit solve.