Read the case study below:
The River City Fire Department (RCFD) fights fires and provides a variety of rescue operations in the River City metropolitan area. The RCFD staffs 13 ladder companies, 26 pumper companies, and several rescue units and ambulances. Normal staffing requires 186 firefighters to be on duty every day. RCFD is organized with three firefighting units. Each unit works a full 24-hour day and then has two days (48 hours) off. For example, Unit 1 covers Monday, Unit 2 covers Tuesday, and Unit 3 covers Wednesday. Then Unit 1 returns on Thursday, and so on. Over a three-week (21-day) scheduling period, each unit will be scheduled for seven days. On a rotational basis, firefighters within each unit are given one of the seven regularly scheduled days off. This day off is referred to as a Kelley day. Thus, over a three-week scheduling period, each firefighter in a unit works six of the seven scheduled unit days and gets one Kelley day off.
Determining the number of firefighters to be assigned to each unit includes the 186 firefighters who must be on duty plus the number of firefighters in the unit who are off for a Kelley day. Furthermore, each unit needs additional staffing to cover firefighter absences due to injury, sick leave, vacations, or personal time. This additional staffing involves finding the best mix of adding full-time firefighters to each unit and the selective use of overtime.
If the number of absences on a particular day brings the number of available firefighters below the required 186, firefighters who are currently off (e.g., on a Kelley day) must be scheduled to work overtime. Overtime is compensated at 1.55 times the regular pay rate. Analysis of the records maintained over the last several years concerning the number of daily absences shows a normal probability distribution with a mean of 20 and a standard deviation of 5 providing a good approximation of the probability distribution for the number of daily absences. Human Resources has calculated that on any given day, you’ll need 18.14 firefighters (18, really, since we can’t have 0.14 firefighters) to cover absences.
Your assignment is to calculate the three items below (you’ll need only a standard calculator for this) and develop a report that will enable Fire Chief O. E. Smith to determine the necessary numbers for the Fire Department.
1 Assuming no daily absences and taking into account the need to staff Kelley days, determine the base number of firefighters needed by each unit.
2. Tell us how many total firefighters Chief Smith is really going to need for each unit to make sure daily absences are covered.
FYI, HR used the single-period inventory model to determine the number of additional firefighters (18.14 or “18”( to be added to each unit to cover daily absences.
Let Q* = minimum cost number of additional firefighters
If Demand < Q*, Q* has overestimated demand.
The cost of overestimating demand is the daily wage rate (d) for a firefighter.
co = d
If Demand > Q*, Q* has underestimated demand and overtime is needed.
The cost of underestimating demand is the overtime wage (1.55d) minus the wage rate (d) that would have been required if we had hired enough additional firefighters.
cu = 1.55d ? d = 0.55d
Using the single period inventory model,
Using the normal distribution, area = 0.5000 ? 0.3548 = 0.1452. z = ?0.37
Use 18 additional firefighters.
3. Finally, calculate the number of firefighters Chief Smith will need to cover the entire department.
Please submit this as a memo to Chief Smith, with any appropriate/required commentary/analysis/recommendations as a:
Calculations should be annotated in the text of your memo.