$5/5$: 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 works an $8-$

hour shift $5$ days per week, and the authorized shifts

are:

Shift $1$: $6$:$00$ a$.$m$.$ to $2$:$00$ p$.$m$.$

Shift $2$: $8$:$00$ a$.$m$.$ to $4$:$00$ p$.$m$.$

Shift $3$: Noon to $8$:$00$ p$.$m$.$

Shift $4$: $4$:$00$ p$.$m$.$ to midnight

Shift $5$: $10$:$00$ p$.$m$.$ to $6$:$00$ a$.$m$.$

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

OPERATIONS RESEARCH TECHNIQUES

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. Formulate a linear programming model for this problem.