1 Linear Programming With Sensitivity Analysis Here we look at some graphs to learn some ideas.

2 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 T 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

3 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. T 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

4 T C

5 T C 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.

6 T C 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.

7 T C 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. Note, the total profit of 30 tables and 40 chairs may change, but not the amount of each produced.

8 Also in Ranging section of the output in QM for Windows we get values called dual values. Let’s 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.

9 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.