A Hospital Needs to Determine How Many Nurses to Hire

question 22

Essay

A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
$\begin{array}{lcc}\text { Time period } & \text { Required \# of Nurses } & \text { Wage }(\$ / \mathrm{hr}) \\\hline 12 \mathrm{am}-4 \mathrm{am} & 20 & 15 \\4 \mathrm{am}-8 & 30 & 16 \\\text { am } & & \\8 \mathrm{am}-12 \mathrm{pm} & 40 & 13 \\12 \mathrm{pm}-4 \mathrm{pm} & 50 & 13 \\4 \mathrm{pm}-4 & 40 & 14 \\\mathrm{pm}\\8 \mathrm{pm}-12 \mathrm{am}&30 & 15 \end{array}$
What are the key formulas for this Excel spreadsheet implementation of the following formulation?
Let $\quad X _ { i } =$ mumber of nurses warking in thme period $i ; i = 1,6$
$$\begin{array} { l l } \text { MiN: } & 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { 2 } } + 1 \mathbf { X } _ { \mathbf { 3 } } + 1 \mathbf { X } _ { 4 } + 1 \mathbf { X } _ { \mathbf { 5 } } + 1 \mathbf { X } _ { \mathbf { 6 } } \\\text { Subject ta: } & 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { z } } \geq 30 \\& 1 \mathbf { X } _ { \mathbf { 2 } } + 1 \mathbf { X } _ { \mathbf { 3 } } \geq 40 \\& 1 \mathbf { X } _ { \mathbf { 3 } } + 1 \mathbf { X } _ { 4 } \geq 50 \\& 1 \mathbf { X } _ { 4 } + 1 \mathbf { X } _ { \mathbf { 5 } } \geq 40 \\& 1 \mathbf { X } _ { 5 } + 1 \mathbf { X } _ { \mathbf { 1 } } \geq 30 \\& 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { 1 } } \geq 20 \\& \mathbf { X } _ { \mathrm { i } } \geq 0\end{array}$$

---

Definitions:

Goodwill Messages

Communications intended to foster positive relationships and express appreciation or encouragement.

Personal Tone

A communication style that uses a more individualized and informal approach, often to create a sense of connection or empathy.

Handwritten Note

A personal message or document written by hand, often perceived as more thoughtful or genuine than typed communication.

Routine Requests

Simple, standard inquiries or appeals for information, services, or assistance that follow established protocols.