This case focuses on developing a deeper understanding of the personnel scheduling problem.

Big Town Fire Department would like to schedule their human resource requirements in fire and safety operations, aka, firefighters. The town runs two schedules per day, $12$ hours each. Firefighters are compensated hourly at $$30$ per hour plus an additional $33\%$ is added to this amount to pay for the benefits $($health insurance, retirement, etc.$)$ Firefighters who work the night shift received $$5$ more per hour and have a commensurate increase in their benefits cost.

The weekly firefighter personnel requirements are:

Mon

Tues

Weds

Thurs

Fri

Sat

Sun

$5$a$-5$p

$11$

$11$

$12$

$18$

$19$

$19$

$14$

$5$p$-5$a

$16$

$17$

$18$

$20$

$22$

$24$

$20$

The $$normal$$ shift for a firefighter is $4$ days $12$ hours a day followed by $3$ consecutive days off, working on either the day or the night shift $($but not both$).$ In general the managers of Big Town would like to schedule the firefighters in the least cost way, but must make sure the requirements are met?

Questions to Address

What is the fundamental challenge that Big Town faces in this setting? Why is this problem important? What are the implications of failing to address this problem effectively? What assumptions are made when building a model to schedule the fire fighters or in any similar personnel scheduling problem?

Given the above information, and assuming that firefighters stick to a normal schedule, what is the optimal schedule for firefighters that meets $($or exceeds$)$ the personnel needs for Big Town Fire? What is the associated cost of this schedule? Call this solution and the associated total cost the $$benchmark solution$.$

How does the benchmark solution change with increases in the needed number of firefighters on the weekend night shifts $($Friday$,$ Saturday, and Sunday$)$ of $5\%,10\%,$ and $20\%?$

Note: Assume that increases mean that the number of firefighters needed at the specified times go up by $5\%$ or more, $10\%$ or more, etc. Percentage increases lead to fractional numbers, however, it doesn$$t make sense to schedule fractional numbers of firefighters so address these issues in a way that makes the most sense.

How does the benchmark optimal solution change if there are part$-$time employees that are added who work two consecutive days with $5$ days off either day or night but not both day and night? Suppose that these part time employees receive the same hourly pay rates, but that Big Town is not required to pay their associated benefits. How does this solution compare with the original benchmark case? What would you recommend to Big Town managers? Does this make sense in the greater context?

Suppose there is a limit on the total number of hours that part time firefighters can work. $($Why might this be the case?$)$ Suppose these limits are specified as follows: Part timers can work at most X$\%$ of the total hours worked in a week by ALL firefighters. Provide a table with different solutions and solution values for limits of $0\%,10\%,20\%,30\%,40\%,$ of the total hours worked for part time firefighters as defined in part d$.$ For example how does the schedule and associated cost look when the hours worked by part time firefighters is limited to be at most $20\%$ of the total hour firefighter hours worked?

What if the total number of firefighters are limited due to turnover or other challenges in hiring? Starting from the benchmark solution, limit the total number of firefighters by $1,2,3,...,10.$ To enable finding a feasible answer that covers the requirements, allow some firefighters to work $$double$$ shifts $($day and night$).$ Double shift employees would be compensated by $$time and a half$$ for their work on all hours over $48$ hours per week. Report the implications in terms of cost$.$

Note $1$: If the benchmark solution had $22$ firefighters in total $($not the actual answer$),$ then, the goal here would be to find the best schedule with $21$ firefighters, with some overtime involved. Then repeat for $20,19,$ etc.

Note $2$: To include shifts that allow for overtime, you may need to make some assumptions about how much overtime is permitted for a firefighter per week. How safe is it if firefighters are working too much?

Note $3$: Overtime hours include $1\mathrm{.}5$ times the regular pay and benefits.

Summarize and generalize the findings in this case.