Nurses in the Pediatrics Unit at Mott Hospital work shifts in which nurses work

Question

Nurses in the Pediatrics Unit at Mott Hospital work shifts in which nurses work 5 of the 7 days of the week with 2 successive days off. For example, a shift might work Sunday through Thursday and then have Friday and Saturday off. Based on historical data, it is known that a at least 5 nurses are needed on duty Monday, Tuesday, Wednesday, and Thursday; 7 are required for Friday and Saturday; and 6 are needed on Sunday. There are a total of 9 nurses that can be scheduled.

a) Formulate a linear program that maximizes the number of weekend days off received by the nurses (weekend is Saturday and Sunday). Clearly define your decision variables, explain your formulation of the objective function, and describe what each of your constraints means in words.

Explanation / Answer

Solution:-
Now, the objective is to maxmize the number of weekend days off received by the nurses.Since the weekend is a staurday and sunday and the nurses work 5 of the 7 days, The Nurses who start the shift on Monday would work till Friday.All the other Nurses who join their Shift from Tuesday , Wednesday etc would have to work on Sunday /Saturday .

Hence we would need to maximize the number of Nurses starting their shifts on Monday so that they can get their weekends off and still satisfy the constraints.

Now the objective function is to maxmize number of Nurses joining shift on Monday.

Constraints are the Minimum number of Nurses required for each day :

Monday ,Tuesday , Wednesday , Thuersday >= 5

Friday,Saturday >=7

Sunday >= 6

Also the total number of Nurses present in eachc day at the hospital cannot exceed 9 as we have only 9 Nurse in total.

Now formulating the Nurses schedule as shown below:

Let us start by taking random number of Nurses in each of the days of the week at the start:

Now let us create the Nurse Shift Schedule as follows:Nurses joining on Monday will work till Friday.

Hence to create the Nurse Schedule , we build the table as follows:

Now the Minimum Number of Nurses required on each day is given by :

Now we create the Nurse schedule as follows :

Nurses joinin on Monday will work till Friday. The Nurses joining on each day is highlighted in the same colur of the day in the table below:

The Formula used in this table is given by :

B19 = C19 = D19 =E19 =F19 = J7 (Number of Nurse joining on Monday)

Similarly

C20 = D20 = E20 = F20 = G20 = J8 (Number of Nurses joining on Tuesday)

Now the objective function is nothing but to maximize the number of Nurses coming on monday so that they can earn maximum weekend off.

Hence objective function = Maximize (J7)

The Solver Constraints are added as shown below:

On Solving , we get the values and the Nurses schedule as below

Thus to maxmize the weekend offs to the Nurses:

Number of Nurses to join on each day is given below:

Monday : 4

Tuesday : 1

Wednesday : 1

Thursday : 1

Friday : 1

Saturday : 3

Sunday : 0

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