Welcome!
In this tutorial, I’ll be answering the following questions for this LP model.
I’ll also be interpreting the components of LP sensitivity analysis based on this Sensitivity
Report from Excel.
First note here that these objective coefficients here are displayed here in the output.
We assume these are unit profits for products A, B, and C. And the constraint Right Hand
Side are displayed here.
So let’s begin by examining the top part of the table for optimality ranges.
The optimal solution is represented by the final values here: A = 0, B = 70, and C = 30.
So the optimal objective function value can be found by plugging the optimal solution
into the objective function to obtain 5850.
Now, these Allowable Increase & Decrease values specify how much the objective coefficients
can change before the optimal solution will change.
For example, since the Allowable Increase for A is 7.5, if we increase the objective
coefficient of A, from 50 to any value, up to an upper limit of 57.5, the optimal solution
will not change.
For the Allowable Decrease, Excel usually represents very large values with 1E+30.
So you can think of it as infinity.
Thus the lower limit for the coefficient of A is negative infinity.
For B, the upper limit will be infinity, and the lower limit will be 60 – 5 which gives
55.
For C, we have 60 for the upper limit, and 40 for the lower limit.
So what will happen to the optimal solution if the unit profit on B (that is, the coefficient)
decreases by 20?
We can see here that the Allowable Decrease on B is 5.
Therefore, the optimal solution will change if we decrease it by 20.
That is, these final values will no longer be optimal.
And what will happen if the unit profit on C decreases to 45 from its current value of
55.
You can see here that 45 is between 40 and 60, so the optimal solution will remain optimal,
but the total profit now becomes 5550.
Next, what will happen if unit profits on both A and C are changed to 53?
Note that this sensitivity report only accounts for one change at a time.
So if there are simultaneous changes, as we have here, we have to check if the sum of
the ratio of proposed changes to allowable changes is within 100%.
If the sum of these ratios is over 100%, the optimal solution may no longer be valid.
This is called the 100% Rule.
The proposed increase in A is 53 minus 50 which gives 3 and the allowable is 7.5, giving
a ratio of 0.4.
Whereas for C, the proposed decrease is 55 minus 53 which gives 2, with an allowable
decrease of 15, giving a ratio of 0.133.
The sum of these ratios is 53.3% which is less than 100% so the optimal solution will
remain optimal, and the total profit becomes 5790.
Now let’s discuss the reduced cost for A. The reduced cost of -7.5 here represents the
amount by which profit will be reduced if we include a unit of A in the solution.
In essence, product A is not attracting enough profit to warrant its inclusion in the product
mix.
To include product A (or to make A positive), it’s profit contribution needs to improve
by at least 7.5.
But at its current value of 50, making A positive in the optimal solution will bring a reduction
of 7.5 to Profit, per unit.
Now the bottom part of the table, titled Constraints, addresses the range of feasibility.
That is, the range for the Right Hand Side of a constraint where the shadow price remains
unchanged.
For example, as long as the RHS of constraint 1 is between 93.33 and 110, the shadow price
of 60 will apply.
Shadow Price here refers to the amount of change in the optimal objective function value,
per unit increase in the RHS of a constraint.
So what will happen to optimal profit if the RHS of Constraint 1 increases by 5.
Notice here that the Allowable Increase for constraint 1 is 10.
So an increase of 5 is allowable.
Therefore, the optimal profit will change by 5 (the amount of change) times 60 (the
shadow price) to give 300.
Since this is positive, Profit will increase by 300 from 5850 to become 6150.
And what will happen if the RHS of constraint 2 decreases to 250.
Notice here that the Allowable Decrease for Constraint 2 is 30.
The current value is 300, so decreasing it to 250 is a proposed decrease of 50.
Since the proposed decrease exceeds the allowable decrease, the shadow price is no longer valid.
As a result, we cannot tell what will happen to optimal Profit based on this output.
We need to re-solve the model.
Next, how will the objective function change if the RHS of constraint 4 changes to 44.
The current RHS value for constraint 4 is 60.
Changing it to 44 represents a decrease of 16 units which is less than the Allowable
Decrease if 60.
Therefore, the shadow price applies and the total profit changes by -16*-2.5 to give 40.
Since this value is positive, optimal profit will actually increase by 40 to become 5890.
For Slack and Surplus values, we simply take the differences between the Final Values (that
is, the left side of the constraint) and the RHSides.
Constraints 1 & 2 are less or equal constraints so they will be associated with slack variables
while 3 & 4 are greater or equal constraints and will be associated with surplus variables.
So the slack value for constraint 1 is 0, and for constraint 2, it is 30.
The surplus for constraint 3 is 20, and for constraint 4 it is 0.
Lastly, the binding constraints are the ones that have Final Value equal to the RHSide-
that is, the ones that have 0 slack or surplus values.
So constraints 1 and 4 are binding, while 2 and 3 are non-binding.
Note also that the binding constraints have non-zero shadow prices.
And that’s it for this video.
Thanks for watching.