Case Problem River City Fire Department

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 included 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 firefighters absences due to injury, sick leave, vacation, or personal time. This additional staffing involves finding the best mix of adding full-time firefighter 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. A mean of 20 ans a standard deviation of 5 provide a good approximation of the probability distribution for the number of daily absences.

Managerial Report
Develop a report that will enable Fire Chief O. E. Smith to determine the necessary numbers for the Fire Department. Include, at a minimum, the following items in your report.
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. Using a minimum cost criterion, how many additional firefighters should be added to each unit in order to cover the daily absences? These extra daily needs will be Filled by the additional firefighters and, when necessary, the more expensive use of overtime by off-duty firefighters.
3. On a given day, what is the probability that Kelley-day firefighters will be called in to work overtime?
4. Based on the three-unit organization, how many firefighters should be assigned to each unit? What is the total number of full-time firefighters required for the River City Fire Department?