A $24/7$ customer support center needs to determine the number of customer service

agents required during different time periods to meet customer demand. The following table

shows the minimum number of agents needed during six time periods throughout the day:

PERIOD TIME NUMBER OF AGENTS REQUIRED

$112$ a$.$m$.4$ a$.$m$.5$

$24$ a$.$m$.8$ a$.$m$.10$

$38$ a$.$m$.12$ p$.$m$.20$

$412$ p$.$m$.4$ p$.$m$.15$

$54$ p$.$m$.8$ p$.$m$.18$

$68$ p$.$m$.12$ a$.$m$.8$

Once an agent starts to work, $($s$)$he is available for $2$ periods. For example, if an agent starts at

period $1,($s$)$he will work period $1$ and $2.$ If an agent starts at period $6,($s$)$he will work period $6$ and

$1.$ The customer support center wants to determine how many customer service agents should

be scheduled to minimize the total number of agents required for a day's operation. $($Hint: This is

a scheduling problem. Let Xi be the number of agents beginning work in time period i$,$ where i$=1,$

$2,3,4,5,6.)$

a$.$ Formulate a linear programming model for this scheduling problem, including decision

variables, objective function, and constraints.

b$.$ Solve the linear programming model to find the optimal number of agents to schedule for

each time period.

c$.$ What is the minimum total number of agents required for a day's operation based on your

solution?