United Airways is adding more flights to and from its hub airport, so it needs to hire additional customer service agents. However, it is not clear just how many more should be hired. Management recognizes the need for cost control while also consistently providing a satisfactory level of service to customers. Therefore, an OR team is studying how to schedule the agents to provide satisfactory service with the smallest personnel cost.

Based on the new schedule of flights, an analysis has been made of the minimum number of customer service agents that need to be on duty at different times of the day to provide a satisfactory level of service. This has led to compiling the data given in the following table:

The rightmost column of the table shows the number of agents needed for the time periods given in the first column. The other entries in this table reflect one of the provisions in the company’s current contract with the union that represents the customer service agents. The provision is that each agent work an 8-hour shift 5 days per week, and the authorized shifts are

Checkmarks in the main body of the table show the hours covered by the respective shifts. Because some shifts are less desirable than others, the wages specified in the contract differ by shift. For each shift, the daily compensation (including benefits) for each agent is shown in the bottom row. The problem is to determine how many agents should be assigned to the respective shifts each day to minimize the total personnel cost for agents, based on this bottom row, while meeting (or surpassing) the service requirements given in the rightmost column.

(a) Formulate a linear programming model for this problem.

(b) Solve this model by the simplex method.

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Time Period 6:00 A.M. to 8:00 A.M. 8:00 A.M. to 10:00 A.M. 10:00 A.M. to noon Noon to 2:00 P.M. 2:00 P.M. to 4:00 P.M. 4:00 P.M. to 6:00 P.M. 6:00 P.M. to 8:00 P.M. 8:00 P.M. to 10:00 P.M. 10:00 P.M. to midnight Midnight to 6:00 A.M. Daily cost per agent Time Periods Covered Shift 1 2 3 4 5 V V V ✓ ✓ ✓ V V V ✓ V $170 $160 $175 $180 $195 Minimum Number of Agents Needed 48 79 65 87 64 73 82 43 52 15