1 Formulate the following problem Clearly identify the decis

1) Formulate the following problem: Clearly identify the decision variables, the objective function and relevant constraints. The police department schedules officers to work 5 out of 7 days a week with 2 successive days off. (ie. An officer works Tuesday-Saturday and gets Sunday and Monday off.) A total of 8 officers must be on duty on Monday, Tuesday, Wednesday and Thursday. 15 officers are required to be on duty on Friday and 18 on Saturday. 10 officers are required to be on duty on Sunday. The department wants to meet the staffing needs with the minimum total number of officers.

Solution

Decision variables

The decision variables usually measure the amounts of resources used or the level of various activities to be performed. Is a factor over which the decision maker has control in this case they have the control to arrange several variables in a defined objective.

These decision variables are the number of officers that have to be on duty around the entire week. (51)

The objective function

Once you\'ve defined the decision variables, the next step is to define the objective, which is normally some function that depends on the decision variables. Is a mathematical function of decision variables use to express the objective of a problem. The goal is to either maximize or minimize the objective function. In this case the decision maker wishes to minimize. They want to know the staffing needs with the minimum total number of officers at week.

So They want to know how many officers have to be on duty for a specific days. (5)

Monday to Thursday = 4 days * 5 days on duty * 8 Officers. You can always make the estimation by separated 1*5*8 and then sum this by 4 times, You\'re going to obtain the same value.

Friday = 1 Day * 5 days on duty * 15 officers

Saturday = 1 Day * 5 days on duty * 18 officers

Sunday = 1 Day * 5 days on duty * 10 officers

The sum of all of these calculations is de number of decision variables of a objective fuction that we can make.

Relevant constraint

Constraints are logical conditions that a solution to an optimization problem must satisfy. They reflect real-world limits, is a restrictive condition that may affect the optimal value for an objective function.

Have you wonder why We just make calculations just for 5 days at week? It\'s because a constraint is presented and are the 2 days off that every officer have to take as a rest of their duties.

I hope, I have cleared your doubts!!