MS $3053-$ Problem Set #$6$

Due Date: October $31,2024$

#$1$

The technical support department of a large corporation is developing the shift schedule for the overnight period on weekdays, which lasts from $1$:$00$ a$.$m$.$ to $9$:$00$ a$.$m$.$ The table below indicates the number of technicians who must be on call during each hour, based on the average number of support requests.

$\backslash$table$[[$Time$,$Technicians Needed$],[1$:$00$ a$.$m$.-2$:$00$ a$.$m$.,4],[2$:$00$ a$.$m$.-3$:$00$ a$.$m$.,5],[3$:$00$ a$.$m$.-4$:$00$ a$.$m$.,6],[4$:$00$ a$.$m$.-5$:$00$ a$.$m$.,8],[5$:$00$ a$.$m$.-6$:$00$ a$.$m$.,9],[6$:$00$ a$.$m$.-7$:$00$ a$.$m$.,8],[7$:$00$ a$.$m$.-8$:$00$ a$.$m$.,6],[8$:$00$ a$.$m$.-9$:$00$ a$.$m$.,5]]$

Support personnel are classified into two groups, lead technicians and assistant technicians. During the overnight period, each lead technician starts on the hour and is on call for $3$ hours, takes a mandatory break of $1$ hour, and then is on call another $2$ hours. Each assistant technician starts on the hour and is on call for a continuous $4$ hours. Corporate policy mandates that technicians must work their complete designated shifts, that no technician is allowed to work an extra shift, and that at least one lead technician must be on site at all times.

The number of technicians is measured using a metric known as "labor units," which factors in salaries, benefits, and other costs. Each lead technician counts as $1\mathrm{.}80$ labor units, while each assistant technician counts as $1\mathrm{.}35$ labor units.

Let $L_i$ and $A_i$ be the number of lead technicians and assistant technicians, respectively, who start at hour $i,i=1,2,$dots. Formulate an integer programming model to develop a schedule that will satisfy the corporation's staffing requirements while minimizing the total number of labor units. Include only those decision variables that are necessary, taking into account the latest time at which each type of technician may start. Provide all constraints in standard form. You do not need to solve the $LP.$