Union Airways is adding more flights to andfrom its hub airport and so needs to hire additional customerservice agents. These numbers are shown in the last column of tablefor the time periods given in the first column. The other entriesin this table reflect one of the provisions in the company’scurrent contract with the union that represents the customerservice agents. The provision is that each agent works aneight-hour shift. The authorized shifts are

• Shift 1: 6:00 AM to 2:00 PM.

• Shift 2: 8:00 AM to 4:00 PM.

• Shift 3: Noon to 8:00 PM.

• Shift 4: 4:00 PM to midnight.

• Shift 5: 10:00 PM to 6:00 AM.

The table shows the time periods covered by the respectiveshifts. Because some shifts are less desirable than others, thewages specified in the contract differ by shift. For each shift,the daily compensation (including benefits) for each agent is shownin the bottom row. The problem is to determine how many agentsshould be assigned to the respective shifts each day to minimizethe total personnel cost for agents, based on this bottomrow, while meeting (or surpassing) the service requirements givenin the last column.

a. Formulate algebraic form for this problem. (Please clearlydefine all the decision variables, clearly write down the objectivefunction and each constraints)

b. Formulate and solve this model on a spreadsheet.

TABLE Data for the Union Airways PersonnelScheduling Problem

Time Period	1	2	3	4	5	Minimum Number of Agents Needed
6:00am to 8:00am	X					45
8:00am to 10:00am	X	X				75
10:00am to 12:00pm	X	X				60
12:00pm to 2:00pm	X	X	X			81
2:00pm to 4:00pm		X	X			64
4:00pm to 6:00pm			X	X		73
6:00pm to 8:00pm			X	X		82
8:00pm to 10:00pm				X		42
10:00pm to 12:00am				X	X	52
12:00am to 6:00am					X	16
Daily Cost per agent	$172	$150	$165	$181	$190