In this video, we'll talk about how to perform the sensitivity analysis and how to explain
the shadow price for LP problems.
In this video, sensitivity analysis is used to answer two types of questions.
The first one is, how do the changes in the available resources affect the optimal solution?
The available resources are represented by the right hand side in the constraints.
The second one is, how do the changes in the unit profit or unit cost affect the optimal
solution?
The unit profit or unit cost are represented by the coefficients in the objective function.
Here is an example.
Two machines are used to produce two types of products.
A unit of Product 1 requires 2 hours of processing time on Machine 1 and 1 hour of processing
time on Machine 2.
A unit of Product 2 requires 1 hour of processing time on Machine 1 and 2 hours of processing
time on Machine 2.
Daily processing time available on each machine is 8 hours.
Selling one unit of Product 1 will generate $300 net profit.
Selling one unit of Product 2 will generate $200 net profit.
Now, we let x1 be the number of Product 1 produced, and x2 be the number of Product
2 produced.
We can formulate a LP problem to maximize the daily profit, which is the total profit
of selling products 1 and 2.
The first constraint means that the total processing time on machine 1 should not exceed
8 hours.
The second constraint means that the total processing time on machine 2 should not exceed
8 hours.
This problem has only two variables.
We can solve it graphically.
This is the constraint of Machine 1, and this is the constraint of Machine 2.
This shaded area is the feasible region.
This point A is the optimal solution.
The objective function value at this point is 4000/3.
Now, let's perform sensitivity analysis on the RHS of the constraints.
For Machine 1, we want to know, if we change the available time by one hour, from 8 to
9, how much will the z value change?
In this graph, the constraint for machine 1 will be changed to M1', and the optimal
solution will be changed to point B.
The objective function value at this point is 4400/3.
The shadow price of machine 1 is defined as follows: The new z value minus the old z value,
divided by Machine 1's capacity change.
So, the shadow price of machine 1 is $133/hour.
It means, each hour increase on Machine 1, will bring in $133 more profit.
For Machine 2, we can do the same.
We want to know, if we change the available time by one hour, from 8 to 9, how much will
the z value change?
In this graph, the constraint for machine 2 will be changed to M2', and the optimal
solution will be changed to point C. The objective function value at this point
is 4100/3.
The shadow price of machine 2 is the new z value minus the old z value, divided by Machine
2's capacity change.
So, the shadow price of machine 2 is $33/hour.
It means, each hour increase on Machine 2, will bring in $33 more profit.
Now we have the shadow prices of these two machines.
Based on these shadow prices, let's answer these questions.
First question, if we can increase the capacity of both machines, which machine should receive
priority?
The answer is Machine 1.
Because if we increase both machines' processing time by one hour, the increased profit from
Machine 1 is much higher than the increased profit from Machine 2.
Second question, should we increase the availability of Machine 1 at an additional cost of $50/hr?
The answer is yes, because $133 is greater than $50.
We will still make more money by increasing the availability of Machine 1.
What about Machine 2?
The answer is No, because $33 is less than $50.
We will lose money and it is not worth it.
Now, let's perform sensitivity analysis on the objective coefficients.
For a general maximization problem z = c1x1+c2x2, the slope of the objective function is -c1/c2.
This slope should be between the slope of these two constraints so that the optimal
solution will be unchanged.
That means the optimality range of c1/c2 is between 1/2 and 2.
With that in mind, let's answer this question.
Suppose that the unit profit for Product 1 is changed to $350, and the unit profit for
Product 2 is changed to $250.
Will the current optimal solution remain the same?
To answer this question, let's check the value of c1/c2, which is 350/250, or 7/5.
It is still between 1/2 and 2.
So the answer to this question is Yes.
Another question: suppose that the unit profit of Product 2 is fixed at its current value
c2 = $200.
What is the range of the unit profit for Product 1, c1, that will keep the optimal solution
unchanged?
Now, c1/c2 is c1/200, which should be between 1/2 and 2.
Finally, we have that c1 is between 100 and 400.
If it's in this range, the optimal solution will remain the same.
Okay, that is how to perform sensitivity analysis for LP problems and how to use the results
to make managerial decisions.
Thank you for watching.