Wichita$$s famous Sethi Restaurant is open $24$ hours

a day. Servers report for duty at $3$ a$.$m$.,7$ a$.$m$.,11$ a$.$m$.,3$ p$.$m$.,$

$7$ p$.$m$.,$ or $11$ p$.$m$.,$ and each works an $8-$hour shift. The following

table shows the minimum number of workers needed during the

$6$ periods into which the day is divided:

PERIOD TIME NUMBER OF SERVERS REQUIRED

$13$ A$.$M$.7$ A$.$M$.3$

$27$ A$.$M$.11$ A$.$M$.12$

$311$ A$.$M$.3$ P$.$M$.16$

$43$ P$.$M$.7$ P$.$M$.9$

$57$ P$.$M$.11$ P$.$M$.11$

$611$ P$.$M$.3$ A$.$M$.4$

Owner Avanti Sethi$$s scheduling problem is to determine how

many servers should report for work at the start of each time

period in order to minimize the total staff required for one day$$s

operation. $($ Hint: Let X i

equal the number of servers beginning

work in time period i $,$ where i $51,2,3,4,5,6.)$ P