Labor planning problems address staffing needs over a specific time period. They are especially useful when managers have some flexibility in assigning workers to jobs that require overlapping or interchangeable talents. Banks and credit unions frequently use LP to tackle their labor scheduling.

Hong Kong Bank of Commerce and Industry is a busy bank that has requirements for between 10 and 18 tellers, depending on the time of day. The lunch time, from noon to 2 p.m., is usually heaviest. Table 8.4 indicates the workers needed at various hours that the bank is open.

The bank now employs 12 full-time tellers, but many people are on its roster of available part-time employees. A part-time employee must put in exactly 4 hours per day but can start anytime between 9 a.m. and 1 p.m. Part-timers are a fairly inexpensive labor pool, since no retirement or lunch benefits are provided for them. Full-timers, on the other hand, work from 9 a.m. to 5 p.m. but are allowed 1 hour for lunch. (Half of the full-timers eat at 11 a.m., the other half at noon.) Full-timers thus provide 35 hours per week of productive labor time.

By corporate policy, the bank limits part-time hours to a maximum of 50% of the day’s total requirement. Part-timers earn $8 per hour (or $32 per day) on average, and full-timers earn $100 per day in salary and benefits, on average. The bank would like to set a schedule that would minimize its total personnel costs. It will release one or more of its full-time tellers if it is profitable to do so.

In formulating this as an LP, the objective is to minimize cost. There is a constraint for each hour of the day, stating that the number of people working at the bank during that hour should be at least the minimum number shown in Table 8.4, so there are eight of these constraints. Another constraint will limit the total number of full-time workers to no more than 12. The last constraint will specify that the number of part-time hours must not exceed 50% of the total hours.

The bank must decide how many full-time tellers to use, so there will be one decision variable for that. Similarly, the bank must decide about using part-time tellers, but this is more complex, as the part-time workers can start at different times of the day, while all full-time workers start at the beginning of the day. Thus, there must be a variable indicating the number of part-time workers starting at each hour of the day from 9 a.m. until 1 p.m. Any worker who starts at 1 p.m. will work until closing, so there is no need to consider having any part-time workers start after that. Let

Objective function:

Constraints:

For each hour, the available labor hours must be at least equal to the required labor hours.

Only 12 full-time tellers are available, so

Part-time worker hours cannot exceed 50% of total hours required each day, which is the sum of the tellers needed each hour:

or

Program 8.5 gives the solution to this, found using Solver in Excel 2016. There are several alternate optimal schedules that Hong Kong Bank can follow. The first is to employ only 10 full-time tellers $\left(F=10\right)$ and to start 7 part-timers at 10 a.m. $\left(P_2=7\right),$ 2 part-timers at 11 a.m. $\left(P_3=2\right),$ and 5 part-timers at noon $\left(P_4=5\right).$ No part-timers would begin at 9 a.m. or 1 p.m.

A second solution also employs 10 full-time tellers but starts 6 part-timers at 9 a.m. $\left(P_1=6\right),$ 1 part-timer at 10 a.m. $\left(P_2=1\right),$ 2 part-timers at 11 a.m. (P₃ = 2), 5 part-timers at noon (P₄ = 5), and 0 part-timers at 1 p.m. $\left(P_5=0\right).$ The cost of either of these two policies is $1,448 per day.