Minimum number of toll collectors

Oakton River Bridge The Oakton River had long been considered an impediment to the development of a certain medium-sized metropolitan area in the southeast. Lying to the east of the city, the river made it difficult for people living on its eastern bank to commute to jobs in and around the city and to take advantage of the shopping and cultural attractions that the city had to offer. Similarly, the river inhibited those on its western bank from access to the ocean resorts lying one hour to the east. The bridge over the Oakton River had been built prior to World War II and was grossly inadequate to handle the existing traffic, much less the increased traffic that would accompany the forecasted growth in the area. A congressional delegation from the state prevailed upon the federal government to fund a major portion of a new toll bridge over the Oakton River and the state legislature appropriated the rest of the needed monies for the project.
Progress in construction of the bridge has been in accordance with what was anticipated at the start of construction.
The state highway commission, which will have operational jurisdiction over the bridge, has concluded that the opening of the bridge for traffic is likely to take place at the beginning of the next summer, as scheduled. A personnel task force has been established to recruit, train, and schedule the workers needed to operate the toll facility. The personnel task force is well aware of the budgetary problems facing the state. They have taken as part of their mandate the requirement that personnel costs be kept as low as possible. One particular area of concern is the number of toll collectors that will be needed. The bridge is scheduling three shifts of collectors: shift A from midnight to 8 A.M., shift B from 8 A.M. to 4 P.M., and shift C from 4 P.M. to midnight.
Recently, the state employees union negotiated a contract with the state which requires that all toll collectors be permanent, full-time employees. In addition, all collectors must work a five-on, two-off schedule on the same shift. Thus, for example, a worker could be assigned to work Tuesday, Wednesday, Thursday, Friday, and Saturday on shift A, followed by Sunday and Monday off. An employee could not be scheduled to work, say, Tuesday on shift A followed by Wednesday, Thursday, Friday, and Saturday on shift B or on any other mixture of shifts during a 5-day block. The employees would choose their assignments in order of their seniority.
The task force has received projections of traffic flow on the bridge by day and hour. These projections are based on extrapolations of existing traffic patterns—the pattern of commuting, shopping, and beach traffic currently experienced with growth projections factored in. Standards data from other state operated toll facilities have allowed the task force to convert these traffic flows into toll collector requirements, that is, the minimum number of collectors required per shift, per day, to handle the anticipated traffic load. These toll collector requirements are summarized in the following table: Minimum Number of Toll Collectors Required per Shift

SHIFT

SUN.

MON.

TUE.

WED.

THU.

FRI.

SAT.

A

8

13

12

12

13

13

15

B

10

10

10

10

10

13

15

C

15

13

13

12

12

13

8

The numbers in the table include one or two extra collectors per shift to fill in for collectors who call in sick and to provide relief for collectors on their scheduled breaks. Note that each of the eight collectors needed for shift A on Sunday, for example, could have come from any of the A shifts scheduled to begin on Wednesday, Thursday, Friday, Saturday, or Sunday.
Discussion Questions
1. Determine the minimum number of toll collectors that would have to be hired to meet the requirements expressed in the table.
2. The union had indicated that it might lift its opposition to the mixing of shifts in a 5-day block in exchange for additional compensation and benefits. By how much could the number of toll collectors required be reduced if this is done?