0:10

In our transportation problem for example, we had considered that we know the demand

with certainty, or the demand is not going to deviate much from the estimated values.

In reality, demand could be deviate from average historical values.

In fact, even constant capacity values may also be uncertain.

If the model does not consider uncertainty, then what-if analysis can be

used to observe the effect of changes in these input values.

One simple way of doing what-if analysis is to change a value, and

then resolve the problem to observe changes in the solution.

For example, let's suppose that it is possible to increase the supply capacity

by ten units in any of the five suppliers in our transportation example.

Where would we put these extra ten units?

Well, it seems reasonable to put them in a place that will result in lowering

the total transportation cost the most.

If we don't have any other information,

we could just add ten units to the capacity of all suppliers.

Resolve the problem, and observe the change in the shipments and

the total cost.

This is fairly simple to do in a small problem like ours, but

there is an easier way that is scalable to large models.

Let's take a look at it.

1:31

Locate and open the Excel file titled Transportation Problem-Optimization.

This file has the optimization model that we built in the last video.

If you click on the ASP tab, the silver options panel is displayed.

We can verify that the model is there.

Click on the play button to rerun the software.

Now click on the reports tool in the analysis group or the ASP tab, and

choose sensitivity within the optimizations report.

A sensitivity report one tab is added to the workbook.

This is the same as the sensitivity report tab

that I already included in the workbook, but

in which I have changed the names of the cells to identify them better.

So let's use a sensitivity report that I created.

The first table includes information about the decision variables.

2:24

The second table includes information about the constraints.

Scroll down to the information about the capacity constraints.

The final value corresponds to the total that is shipped from each supplier.

For example, cell G16 has a value of 60, indicating that in the optimal solution,

60 units are shipped from supplier A to O customers.

The table also shows the right hand side of the constraints.

The right hand sides for

the capacity constraints indicate the supply limits and the right hand sides for

the demand constraints indicate the quantities required by each customer.

The shadow prices are the key information in this table.

A shadow price is the change in the objective function for

a unit of increase in the right hand side of the constraints.

Since we're minimizing costs, a negative shadow price means savings.

Therefore, the shadow price of -4 for supplier A, means that if we increase

the capacity of this supplier by one unit, the total cost will decrease by $4.

We can see that we don't save

any money by increasing the capacity of suppliers B and C.

3:40

This makes sense,

because in the optimal solution, these suppliers have extra capacity.

The savings for increasing the capacity of suppliers D and E are $2.

Now the question is, can we keep saving money if we keep increasing capacity?

In other words, are these saving unlimited?

The answer is no.

The last two columns of the table tells us the operational

range of these shadow prices.

For example, for supplier A,

we can increase capacity by 20 units and save $4 per unit.

After 20 units, we simply don't know what will happen,

the report is not able to give us this information.

If we increase the capacity of supplier A by more than 20 units,

we need to resolve the problem to see what happens.

If we decrease capacity,

the cost will increase at the rate given by the shadow price.

That is, if we reduce capacity by one unit, then the cost will increase by $4.

This is true for reductions of up to 15 units.

So to answer our original question using the sensitivity report,

we examine all the shadow prices for the suppliers, and

we find the one with the most negative value.

This is Supplier A, which has a shadow price of -4.

This is where we should add the ten units of capacity,

and reduce the total cost from 890 to 850.

Let's make the change and resolve the model to verify

that our interpretation of the sensitivity report is correct.

Go to cell G6 in the model, and change the value from 60 to 70.

Click on the play button.

5:27

As predicted by the sensitivity report, the cost went down by $40 to 850.

The one thing that you must keep in mind about the sensitivity report,

is that the information that it gives you is only valid for one change at a time.

That is, the shadow price of -4 for

supplier A is only valid if all the other data in the model remains unchanged.

Let's restore the capacity for Supplier A to 60, and go back to

the sensitivity report to interpret the shadow prices for the main constraints.

Know that all the shadow prices for the demand constraints are positive.

These values are the cost of sending additional units to these customers if all

other data remains unchanged, that is, if the capacities and the cost don't change.

For example, if customer four wants one more unit of product,

the transportation costs for that unit will be $8.

This is very valuable information because without resolving the problem,

we can tell the increase in the costs to the additional units of demand.

Here again, these values are only true within the ranges specified in the table.

For customer four, this means that the shadow price is valued for

an increase of up to 15 more units or for a decrease of 20 units.

Let's change the demand in F11 from 65 to 75, and

verify that the cost will increase by $80, to $907.

As predicted, the cost did change to $970, change the demand back to 65.

Now let's take a look at one more thing in the sensitivity report.

There is a reduced cost for each decision variable.

This value is the amount by which the original cost needs to be reduced in

order for the solution to include a shipment in that particular cell.

For the cells that are already shipping products, the reduced cost is zero.

This makes sense,

the cost doesn't need to be reduced since the cell is already being used.

7:36

Let's take C4 as an example.

The current cost is $9, and the reduced cost is $1.

So, the reduced cost is saying that we need to reduce the cost

of $9 by more than $1 for C4 to be used.

Remember that C4 represents shipments from supplier C to customer four.

Let's try this change.

If we go from nine to eight, nothing is going to change.

The reduction needs to be of more than the reduced cost.

So we can enter 7.9 in cell F8 and resolve.

8:13

The new solution includes a shipment of 20 units from C to 4.

The allowable increase and decrease values for the decision variables

give us an indication of how precise the cost estimates need to be.

If a cost has a very wide range,

then we know that we don't need a very precise estimate.

Because modest changes in this value will not change the ultimate solution.

The sensitivity report provides very useful information for pricing decisions.

The value of resources and the robustness of the optimal solution.

For complex what if questions involving multiple changes in the model,

there might not be any other option than making the changes in the model and

resolving the model.

However, for direct questions about single items,

such as the demand of a customer, or the capacity of a supplier,

the sensitivity report is all you need to answer what if questions.