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.
