MS $3053$ Problem Set #$6$

Due Date: March $25,2024$

#$1$

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.

Time Technicians Needed

$1$:$00$ a$.$m$.2$:$00$ a$.$m$.3$

$2$:$00$ a$.$m$.3$:$00$ a$.$m$.4$

$3$:$00$ a$.$m$.4$:$00$ a$.$m$.6$

$4$:$00$ a$.$m$.5$:$00$ a$.$m$.8$

$5$:$00$ a$.$m$.6$:$00$ a$.$m$.7$

$6$:$00$ a$.$m$.7$:$00$ a$.$m$.9$

$7$:$00$ 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 $1\mathrm{.}6$ labor units, while each

assistant technician counts as $1\mathrm{.}1$ labor units.

Let Li and Ai be the number of lead technicians and assistant technicians, respectively, who start

at hour i$,$ i $=1,2,....$ 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. You do not need to solve the LP$.$

MS $3053($Panda$)$ Problem Set #$6($Spring $2024)1$

#$2$

A construction company must transport materials from its storage facility to a job site. The table

below provides information on the travel times, in minutes, between locations. The storage facility

is indicated as node $1$ and the job site is indicated as node $7$; the transshipment nodes represent

road intersections. Note that some connections are unidirectional and some are bidirectional; in

the table, the row number represents the $$from$$ node and the column number represents the $$to$$

node. Connections that do not exist are indicated by a dashed line. For example, the travel time

from node $1$ to node $2$ is $13$ minutes, but travel from node $2$ to node $1$ is infeasible.

$1234567$

$1131716$

$281026$

$3787$

$48119$

$5811414$

$67416$

$7$

Let

Xij $=$

$($

$1,$ the route from i to j is used

$0,$ otherwise $,$

i$,$ j $=1,2,...,7.$ We wish to develop a linear programming model to determine the route from the

storage facility to the job site that minimizes the total travel time.

$($a$)$ For the LP for the shortest route problem, write the supply $($node $1)$ constraint.

$($b$)$ For the LP for the shortest route problem, write the balance constraint for node $4.$

$($c$)$ For the LP for the shortest route problem, write the demand $($node $7)$ constraint.

You do not need to draw the network diagram, provide the complete LP formulation, or solve the

LP$.$