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 into 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 at 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 nurses’ contract). Dr. Becker
wants to determine a nursing schedule that will meet the hospital’s minimum requirements
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
12 variables
(one for each time block)
X1 = # of nurses starting at Midnight & working 8 hours
X2 = # of nurses starting at 2am & working 8 hours
X3 = # of nurses starting at 4am & working 8 hours
X4 = # of nurses starting at 6 am & working 8 hours
X5 = # of nurses starting at 8am & working 8 hours
X6 = # of nurses starting at 10 am & working 8 hours
X7 = # of nurses starting at noon & working 8 hours
X8 = # of nurses starting at 2pm & working 8 hours
X9 = # of nurses starting at 4pm & working 8 hours
X10= # of nurses starting at 6pm & working 8 hours
X11 = # of nurses starting at 8pm & working 8 hours
X12= # of nurses starting at 10pm & working 8 hours
Min Z = X1 + X2 + X3 + X4 + X5 + X6 + X7+X8+X9+X10+ X11 + X12
ST
X1+ X10 + X11 + X12 ≥ 30 midn – 2am
X1 + X2+ X11 + X12 ≥ 20 2am – 4am
X1 + X2 + X3 + X12≥ 40 4am – 6am
X1 + X2 + X3 + X4 ≥ 50 6am – 8am
X2 + X3 + X4 + X5 ≥ 60 8am – 10am
X3 + X4 + X5 + X6 ≥ 80 10am–Noon
X4 + X5 + X6 + X7 ≥ 80 Noon – 2pm
X5 + X6 + X7 + X8 ≥ 70 2pm – 4pm
X6 + X7 + X8 + X9≥ 70 4pm – 6pm
X7 + X8 + X9 + X10 ≥ 60 6pm – 8pm
X8 + X9 + X10 + X11≥ 50 8pm – 10pm
X9 + X10 + X11 + X12≥ 50 10pm – midn
1 . (T4. 9) A hospital dietitian prepares breakfast menus every morning for the hospital patients.
Part of the dietitian’s responsibility is to make sure that minimum daily requirements for vitamins
A and Bare met. At the same time, the cost of the menus must be kept as low as possible. The main
breakfast staples providing vitamins A and B are eggs, soy strips, and cereal. The vitamin
requirements and vitamin contributions for each staple follow:
Vitamin Contributions
Vitamin
mg/Egg
mg/Soy Strip mg/Cereal Cup
Minimum Daily Requirements
A
2
4
1
16
B
3
2
1
12
Cost
0.04
0.03
0.02
An egg costs $0.04, a soy strip costs $0.03, and a cup of cereal costs $0.02. The dietitian wants to
know how much of each staple to serve per order to meet the minimum daily vitamin requirements
while minimizing total cost.
a. Formulate a linear programming model for this problem.
b. Solve the model by using Excel.
Let X1= Egg
Let X2= Soy Strip
Let X3= Cereal
Minimize cost in cents= 4X1+3X2+ 2X3
2X1+4X2+1X3≥ 16 (Vitamin A constraint)
3X1+2X2+1X3≥12 (Vitamin B constraint)
X1, X2, X3≥0 (nonnegativity constraint)
Solved: