![]() These values are given in the chance column. 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. In this case, we have chosen 0.6, or 60%. The form of the constraint is the same as a normal constraint, but we have chosen VaR as the type of constraint. Double click on the constraints to display the dialogue window. There is a single set of chance constraints. The decision variables are the scheduled employees. You can see that the objective of the model is to minimize the total staff. Click on the ASP tab to access the model panel. This model uses the Poisson distribution for all the staff requirements. The difference between this model and the one that uses expected values is that the staff requirements are modeled with a probability distribution function. 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. Locate and open the Excel file Maintenance Staff Scheduling. Simulation is the most common mechanism for developing a model with VaR constraints. However, here, we're only going to illustrate Value at Risk constraints. To account for the magnitude of the validation, you can use what is called Conditional Value at Risk. It just counts the number of times that the constraint was not satisfied. Note that Value at Risk doesn't account for the magnitude of the violation. This fraction is known as the Value at Risk, or VaR. This special type of constraint is such that it is satisfied only in a fraction of these scenarios. This illustrates the notion of a chance constraint. 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%. To increase the chance of satisfying the constraint, we will need to increase the number of employees available in that period. In other words, there is a 50% chance that the requirement constraint for the period between 4 and 8 PM is satisfied. 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. The average requirement is 30, but there is some variability. Supposed that after some data analysis, it is determined that there are ten possible requirement scenarios. Consider the requirement of 30 employees for the period between 4 PM and 8 PM. This means that the actual requirements at a given day are random variables. The requirements that we have used to come up with these solutions are expected values. 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. 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. The optimal solution to this problem is a crew of 64 employees that are scheduled, as shown in this table. This problem can be solved by setting up an optimization model using the expected values. The facility manager would like to determine the number of employees to schedule in order to meet the expected staff requirements. The expected number of workers needed in each time period is given in the following table. The number of maintenance workers needed at different times throughout the day varies. These 8-hour shifts start every 4 hours throughout the 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. Maintenance at a production facility is an ongoing process that occurs 24 hours a day. To illustrate these concepts, let's consider the following situation. This video, in particular, deals with the notion of chance constraints and value at risk. In this last part of our course, you will learn how to consider uncertainty when developing optimization models. For the most part, we had kept uncertainty out of our prescriptive analytic models.
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