Question

Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must determine a schedule for nurses to make sure there are enough of them on duty throughout the day. During the day, the demand for nurses varies. Maureen has broken the day in to twelve 2-hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to 6:00 A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively. The demand for nurses steadily increases during the next four daytime periods. Beginning with the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon and evening hours. For the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70, 70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of one of the 2-hour periods and works 8 consecutive hours (which is required in the nurse's contract). Dr. Becker wants to determine a nursing schedule that will meet the hospital's minimum requirement throughout the day while using the minimum number of nurses.

A. Formulate a linear programming model for this problem

B. Solve this model by using the computer - I need to see the excel spreadsheet, formulas, and solver parameters.

Answer 1

Answer:

Linear Programming is a mathematical modeling technique containing linear relationships that represent an institution’s objectives and resource and /or material constraints.

This technique consists of 3 elements:

1. Decision variables: they are symbols that denote the activities such as x1,x2,x3 and so on.

2. Objective function: they represent the relationships between the variables as per the objectives.

3. Model constraints: they represent the constraints on the variables.

Consider the data given below:

Shift timing	Minimum requirement of nurses
12-2 AM	30
2-4 AM	20
4-6 AM	40
6-8 AM	50
8-10 AM	60
10-12 PM	80
12-2 PM	80
2-4 PM	70
4-6 PM	70
6-8 PM	60
8-10 PM	50
10-12 AM	50

Each nurse works 8-hour shifts.

a)

Step 1: construct the decision variables

The decision variables are as follows:

Xij= number of nurses that begins their shift in period ‘i’

i= 12-2am, 2am-4am, 4am-6am, and so on ( for 12 consecutive periods)

Step 2: Define the Objective function and the constraints

The objective is to determine a nursing schedule that will meet the hospital’s requirements by minimizing the number of nurses.

b)

Enter the linear programming model in Excel as shown below:

Open Excel solver and enter the decision variables, objectives function, and the constraints as shown below:

· The objective function is in cell B15.

· The decision variables are in the cells B2 to B13.

Click on solve. The following solution will appear:

Thus, the hospital should have the above nursing schedule to minimize the number of nurses to 170

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