For the most part, we had kept uncertainty out of our prescriptive analytic models.

In this last part of our course,

you will learn how to consider uncertainty when developing optimization models.

This video, in particular, deals with the notion of chance constraints and

value at risk.

To illustrate these concepts, let's consider the following situation.

Maintenance at a production facility is an ongoing process that occurs 24

hours a day.

Because it is a long drive from mass residential areas to the facility,

employees do not like to work shifts of fewer than 8 hours.

These 8-hour shifts start every 4 hours throughout the day.

The number of maintenance workers needed at different times

throughout the day varies.

The expected number of workers needed in each time period is given in

the following table.

The facility manager would like to determine the number of employees to

schedule in order to meet the expected staff requirements.

This problem can be solved by setting up an optimization model

using the expected values.

The optimal solution to this problem is a crew of 64 employees that are scheduled,

as shown in this table.

Sees the employees work eight-hour shifts,

the available employees in the table are the ones that start their shift

in the current period plus the ones that start in the previous period.

For example, there are 34 employees available between 4:00 and

8:00 in the morning, 9 that start at midnight and

25 that start at 4:00 in the morning.

The requirements that we have used to come up with these solutions

are expected values.

This means that the actual requirements at a given day are random variables.

Consider the requirement of 30 employees for the period between 4 PM and 8 PM.

Supposed that after some data analysis,

it is determined that there are ten possible requirement scenarios.

The average requirement is 30, but there is some variability.

Because of this variability, the optimal solution found with expected values

results in only 5 times out of 10 where the staff requirements are satisfied.

In other words, there is a 50% chance that the requirement constraint for

the period between 4 and 8 PM is satisfied.

To increase the chance of satisfying the constraint,

we will need to increase the number of employees available in that period.

For instance, if the number of employees that start at 4:00 is changed

from 13 to 15, then the chance that the constraint is satisfied increases to 70%.

This illustrates the notion of a chance constraint.

This special type of constraint is such that it is satisfied

only in a fraction of these scenarios.

This fraction is known as the Value at Risk, or VaR.

Note that Value at Risk doesn't account for the magnitude of the violation.

It just counts the number of times that the constraint was not satisfied.

To account for the magnitude of the validation,

you can use what is called Conditional Value at Risk.

However, here, we're only going to illustrate Value at Risk constraints.

Simulation is the most common mechanism for

developing a model with VaR constraints.

Locate and open the Excel file Maintenance Staff Scheduling.

In this spreadsheet, we use our traditional color scheme where the light

gold indicates decision variables, green indicates uncertainty, and

light orange indicates the objective function.

The difference between this model and the one that uses expected values

is that the staff requirements are modeled with a probability distribution function.

This model uses the Poisson distribution for all the staff requirements.

Click on the ASP tab to access the model panel.

You can see that the objective of the model is to minimize the total staff.

The decision variables are the scheduled employees.

There is a single set of chance constraints.

Double click on the constraints to display the dialogue window.

The form of the constraint is the same as a normal constraint, but

we have chosen VaR as the type of constraint.

We also need to choose a chance value.

In this case, we have chosen 0.6, or 60%.

The optimal solution found with expected values results in probabilities

of satisfying requirement constraints that are less than 60% for

several of the periods during the day.

These values are given in the chance column.

The total number of employees in this solution is 64.