Prescriptive Analytics Integer Programming

At Big Jake$$s Burgers, near downtown Fort Worth, employees hold one of four job categories, cook, order taker, order filler, dish washer$/$janitor$,$ and are scheduled daily. The store is open from $9$ AM to $9$ PM Monday$-$Friday and has customers arriving as outlined in Table $5-7.$ Employees are a mix of part$-$time and full$-$time. Full$-$time employees must start on the hour from $9$ AM to $1$ PM to get in a $40-$hour workweek. Part$-$time employees can start at the top of any hour and are generally hired for a $20-$hour week. On average, full$-$time cooks are paid $$20\mathrm{.}00$ per hour, while part$-$time cooks make $$18\mathrm{.}00.$ The order takes, and order fillers are paid, $$17\mathrm{.}50$ if full$-$time, and $$15\mathrm{.}00$ if part$-$time. Dishwashers$/$janitors are all part$-$time and paid $$10$ per hour.

On average, a single full$-$time cook can handle around $25$ orders per hour, but most part$-$time cooks can only handle about $20$ orders. Order takers can handle around $25$ customer orders per hour. Order fillers typically can support as many as two full$-$time cooks at a time. The store only needs one dishwasher$/$janitor per shift.

In discussions with Big Jake, management does not want to plan at the customer arrival rates listed in Table $5-7.$ Instead, management wants $10\%$ more capacity per hour than customer arrivals suggest. That way, there is some surge capacity built in$,$ and employee stress is contained.

The objective is to help Bis Jake management right size the business. As time passes, management feels like the team composition has become non$-$optimal and might be costing money. Additionally, some jobs appear to be overworked because there are not enough staff members. Develop a model and program to help Big Jake minimize costs while exceeding customer demand. The solution should inform Big Jake management of the number of employees per position, per category $($part$/$full time$),$ and by start hour to obtain an optimal status.