Solve Linear Programming: Find Max Value & Calculate Bounds

[Yankel]

Hello all,

I hope I am in the right forum, I have a question regarding linear programming (optimization).

I have this target function:

\[Z=0.045X+0.06Y-2\]

subject to the following constraints:
1.
\[X+Y\leqslant 500\]
2.
\[X\geqslant 0.2(X+Y)\]
3.
\[Y\geqslant 0.5(X+Y)\]
4.
\[X\leqslant 300\]
5.
\[Y\leqslant 300\]

\[X,Y\geq 0\]

And I have some questions related to this problem. The first one was to find the maximum value. I did this fairly easy graphically, by finding the region, looking at the "edges" and calculating values of 4 points.According to the graph the max value is at X=200 and Y=300.

Now for my question. In the solution attached to this problem, there were two tables, one for the variables X and Y and one for the constraints. In the table, there were values of LOWER BOUND and UPPER BOUND for each variable and each constraint. I do not understand where these values came from and how they were calculated.

For variable X and lower bound is 0 and the upper is 0.06. For variable Y the lower bound is 0.045 and the upper is infinity.

For constraint 1 the lower bound is 375 and the upper is 600. For constraint 2 the lower is minus infinity while the upper is 100. For constraint 3 the lower is -50 and the upper is infinity. For constraint 4 the lower is 200 and upper infinity. For constraint 5 the lower is 250 and the upper is 400.

For constraints 1 and 5 there was also something called "dual value", for con.1 it was 0.045 and for cons5. it was 0.015.

Can you please help me figure out how there values were calculated ?

thank you !

 

[I like Serena]

Hi Yankel,

Those numbers are the results of a so called Sensitivity Analysis or a What-if-analysis.
There is an option in Excel that will generate such a report.

The boundaries of the constraints indicate how far the constraint can be changed before the point of the optimal solution shifts to another intersection.

Similarly the boundaries of X and Y indicate how far the objective coefficients (0.045 resp. 0.06) can be changed before the point of the optimal solution shifts to another intersection.

Part of the analysis is also how much the target function increases when a constraint value is changed. This is your "dual value", which is also known as the Shadow Price.