We are going to start formulating this staffing decision problem. But before we do that, we need to have a cautionary tale of what can go wrong with such a formulation. One of the common approaches that students take is that they define Xi, where i is index from one to seven signifying the different days of the week. Xi is defined as the number of staff working on day i, so X1 is the number of staff working on day 1, that is Monday. X2 is the number of staff working on day 2, Tuesday and so on. If you define it that way, if the decision variables are defined in that way, then the objective function is expressed as X1 plus X2 plus X3 and so on till X7. Then the constraints are expressed as X1 is greater than equal to 17, which is the Monday's constraint, that is the number of staff who should be working on Monday. X2 is, has to be greater than equal to 13 and so on. This is a formulation that sometimes students would take, but it's a wrong formulation because the objective function that you have is actually wrong. If you have defined your decision variables Xi's as that number of staff working on day i, and then if you add those up across the seven days, then the objective function is not really the number of full-time employees that you need to hire. Why? Because there are two reasons. First of all, when somebody is working on Monday, if they start working on Monday, they have to go work on Monday, Tuesday, Wednesday, Thursday, and Friday. Anybody who starts working on Monday will be working for Monday through Friday. Here if you define a decision variables in this way and set up the problem in this way, you're actually counting that same person five times because that person who is working five days of the week, starting on Monday all the way through Friday, they will be counted in X1, X2, X3, X4, and X5. When you add these items up, you are counting each employee five times. Therefore obviously, this objective function, when you define it as the sum of these decision variables, is not capturing the total number of full-time employees that you need to hire because you're counting them five times each. The second problem is that it doesn't account for the fact that these variables X1, X2, X3, X4 and so on are interrelated because if somebody starts working on Monday, they will be working on Tuesday, Wednesday, and so on. X1 and X2, these two numbers are very much interrelated or correlated if you have defined X1 and X2 as the number of staff working on day 1 and number of staff working on day 2 respectively. You're not capturing those aspects, the fact that a person who starts working on a specific day works for the five consecutive days, including that starting day. That is not being captured in this formulation. Therefore, this is a wrong formulation and you'll get wrong result. What's the solution? What you need to do is to realize that the decision is not how many people should be working on each day, but rather how many people begin working on each day of the week. Because once you have that, if you know, let's say 10 people started working on Monday, you know that that number 10 that you decided is already being added for the next four consecutive days from Monday through Friday. If you have defined X1 as the number of people who begin work on Monday, then that number X1 will already be available in day 2, 3, 4, and 5, and so X2 should exclude those people. X2 should again be the number of people who begin their work on Tuesday and people who are working on Tuesday already includes those that have been working on Monday, and any four days prior to that. You need to think of these decision variables in terms of the number of people who begin to work on each of these days instead of deciding them as the number of people working on each day, if you had gone the other way, as we saw, you would be double-counting or rather counting each person five times and you would be getting a wrong result. The right way to approach this problem is to define the decision variable as Xi being the number of people who begin to work on day i of the week. With that in mind, you may want to approach this problem once again if you hadn't gotten it right the first time. In the next video we're going to see the solution with the right approach.