In this video, we're going to look at the right way or the right approach to formulating this problem. We're going to define the decision variables, constraints, and the objective function. In terms of the decision variables, as we discussed in the last video, the decision variable Xi should be the number of people or employers beginning their work on day i. Where i is indexed by one through seven, for the seven days of the week. X1 will be the number of people beginning work on Monday, X2 will be the number of people beginning their work on Tuesday and so on for the seven days of the week. Once you have defined the decision variables this way, then we can specify that the objective function, which is the number of full-time employees that you need to hire. That will be equal to the number of employees who start their work on Monday plus the number of employees who start their work on Tuesday, plus the number of employees who start their work on Wednesday and so on, to the number of employees who start their work on Sunday. Because each employee begins their work on exactly one day of the week, they have to start either on Monday or Tuesday or Wednesday and so on. Since it's unique for each employee, there's no double counting of the term when you add up X1 to X7. Z which is equal to X1 plus X2 plus X3, all the way to X7, which is the number of employees who start working on Monday plus number of employees who start working on Tuesday, plus number of employees who start working on Wednesday, plus number of employees you start working on Thursday, and so on. That number is going to be your total number of employees that you need, and you need to minimize that. Minimize Z, which is the sum of the number of people starting work on each day of the week for the seven days. That's your objective function. Now that we have defined the objective function, let's turn our focus to the constraints. The constraints are that there is a certain number of staff or employees that are needed on each of the days of the week as specified in the table. Let's begin by trying to figure out who are the people that are working on Monday based on the decision variables here. Who is working on Monday? Well, those who began their work on either Thursday, Friday, Saturday, Sunday, or Monday, all these people will still be working on Monday. Folks that had started their work on Tuesday or Wednesday, they would not be active on Monday because they would have worked for the five consecutive days starting from the day that they started, and they would end either on Saturday or Sunday. They won't be active on Monday. The folks that are working on Monday are those that began their work schedule on Thursday, Friday, Saturday, Sunday, and Monday. If we now express this in terms of the variables that we defined earlier, the number of people is going to be X1 for the number of people who started their work on Monday, plus X4 which is the number of people that started their work on Thursday, plus X5 which is the number of people who started their work on Friday, plus X6, plus X7. That's going to be the total number of people that are working on Monday. Similarly, you can try to figure out who are working on Tuesday. Folks that are working on Tuesday should have began their work either on Monday. If they had started work on Monday, they would be active on Tuesday since they have to work for five consecutive days. Similarly, folks that started their work on Tuesday would be active on Tuesday. Folks who had started their work on Friday, so Friday, Saturday, Sunday, Monday, Tuesday. Those guys are also in the five consecutive days starting on Friday, so those will be active. Similarly, folks who had their work on Saturday and Sunday, they would be also active on Tuesday. These are the people who would be working on Tuesday. As you can see, people that had started their work on Wednesday, they would not be working on Tuesday because those who start working on Wednesday, will be working on Wednesday, Thursday, Friday, Saturday, Sunday. Then they will not be working on Monday and Tuesday. Those folks won't be active on Monday, they're on Tuesday. Similarly, folks who had started their work on Thursday, they will be working on Thursday, Friday, Saturday, Sunday, and Monday. They would be taking the day off on Tuesday and Wednesday. Those folks also won't be active on Tuesday. X4 and X3 will be absent. As you can see here, if you express this in terms of the variables, who are working on Tuesday? These are the number of people that had started their work on Monday, or Tuesday, or Friday, Saturday, Sunday, which is going to be X1 plus X2 plus X5 plus X6 plus X7. That's the total number of people that are working on Tuesday. These are the left-hand side of the constraints for Monday and Tuesday. We have the minimums that are needed for these days. You can apply this same logic to write down the expression of who are working on each day of the week all the way from Monday through Sunday. That is going to look something like this. For Monday, the folks that are working are X1 plus X4 plus X5 plus X6 plus X7. That is people who had started their work on Monday, or Thursday, Friday, Saturday, Sunday. This total number has to be greater than equal to 70. Similarly, folks who will be working on Tuesday are those who have started their work on Monday or Tuesday, and on Friday, Saturday, Sunday. Those who had been working on Wednesday and Thursday, they won't be active on Tuesday. You apply that same logic to write down the left-hand side of the expression for all the seven days of the week. The right-hand side are the values that are in the table for the number of staff required for each of the days of the week. These values, X1 through X7 have to be non-negative. Since number of people has to be integer, these values have to be integer variables. So far the problems that we had been working on are those where the values of the decision variables are real numbers. They could be decimal value. But here, since it's number of people we're going to require them to be integer value. You can first solve the problem in Excel without putting an integer constraints. Let it be solved in the normal way. The numbers for the decision variables will likely be real in that case. But then we're going to see how to implement the integer constraints so that these numbers for the variables X1 through X7 are forced to be integer value so that it reflects the number of people. Let's get started with the Excel formulation. Once you have this formulation here, let's set it up in Excel. To summarize, this is going to be the full formulation for this problem with the objective function defined as we have earlier, and the constraints for the seven days. That in addition, we have non-negativity and integer constraints on the values of the decision variables.
