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# Optimal Workforce Schedule - Integer Linear Programming

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The number of transaction requests received by a bank peaks around 1 PM. For efficient use of resources, the personnel available should therefore vary correspondingly. A variable capacity could be achieved by employing part-time personnel.

The problem is to find the optimal workforce schedule that would meet the personnel requirements at any given time and also be economical.

By corporate policy part-time personnel hours are limited to a maximum of 40% of the day's total requirement.
Full-time employees work for 8 hours (1 hour for lunch included) per day. Thus their productive time is 35 hours per week.
Part-timers work for at least 4 hours per day but less than 8 hours, and are not allowed lunch break.
50% of full-timers go to lunch between 11 am and noon and the remaining 50% go between noon and 1 PM.
The shift starts at 9 AM and ends at 7 PM (overtime is limited to 2 hours). Any work left over at 7 PM is considered holdover for the next day.
A full time employee is not allowed to work more than 5 hours overtime per week. He is paid at normal rate for over time hours; no lunch hours are applicable during over time hours.

The following are the costs:
The average cost per full-time personnel hour (fringe benefits included) is \$10.11 .
The average cost per overtime personnel hour (for full timers), starting rate excluding fringe benefits is \$8.08 .
The average cost per part-time personnel hour is \$7.82 .

Personnel hours required by hour of day are as given below:
9 AM -10 AM - 14
10 AM -11 AM - 25
11 AM -12 PM - 26
12 PM -1 PM - 38
1 PM -2 PM - 55
2 PM -3 PM - 60
3 PM -4 PM - 51
4 PM -5 PM - 29
5 PM -6 PM - 14
6 PM -7 PM - 9

The bank's goal is to achieve minimum possible personnel cost subject to meeting or exceeding the hourly workforce requirements as well as the constraints on the workers listed above.

https://brainmass.com/math/linear-programming/optimal-workforce-schedule-integer-linear-programming-363677

#### Solution Preview

We define the decision variables as given below.

Full time employees can work overtime (up to two hours). Therefore, they start at 9:00 AM and work up to 5:00 PM, 6:00 PM, or 7:00 PM.

F1 = Full time workers working from 9:00 AM to 5:00 PM
F2 = Full time workers working from 9:00 AM to 6:00 PM (one hour overtime)
F3 = Full time workers working from 9:00 AM to 7:00 PM (two hours overtime)

Part-time workers can start at any time during the day, and should work a minimum of four hours, but less than eight hours. Therefore, if they start at 9:00 AM, they can work up to 1:00 PM, 2:00 PM, 3:00 PM, or 4:00 PM

P1 = Part-time workers working from 9:00 AM to 1:00 PM
P2 = Part-time workers working from 9:00 AM to 2:00 PM
P3 = Part-time workers working from 9:00 AM to 3:00 PM
P4 = Part-time workers working from 9:00 AM to 4:00 PM

Similarly, those starting at 10:00 AM, 11:00 AM, 12:00 PM, 1:00 PM, 2:00 PM, and 3:00 PM.

P5 = Part-time workers working from 10:00 AM to 2:00 PM
P6 = Part-time workers working from 10:00 AM to 3:00 PM
P7 = Part-time workers working from 10:00 AM to 4:00 PM
P8 = Part-time workers working from 10:00 AM to 5:00 PM

P9 = Part-time workers working from 11:00 AM to 3:00 PM
P10 = Part-time workers working from 11:00 AM to 4:00 PM
P11 = Part-time workers working from 11:00 AM to 5:00 PM
P12 = Part-time workers working from 11:00 AM to 6:00 PM

P13 = Part-time workers working from 12:00 PM to 4:00 PM
P14 = Part-time workers working from 12:00 PM to 5:00 PM
P15 = Part-time workers working from 12:00 PM to 6:00 PM
P16 = Part-time workers working from 12:00 PM to 7:00 PM

P17 = Part-time workers working from 1:00 PM to 5:00 PM
P18 = Part-time workers working from 1:00 PM to 6:00 PM
P19 = Part-time workers working from 1:00 PM to 7:00 PM

P20 = Part-time workers working from 2:00 PM to 6:00 PM
P21 = Part-time workers working ...

#### Solution Summary

This problem provides an example of a real life situation that can be modeled as an integer programming problem. The workforce schedule for a bank is to be planned. There are some conditions to be met, including the work requirement and limitations on who can do overtime and how many hours of overtime can be done. Initially the problem seems big and complicated, but by systematically working through it, a simple mathematical formulation can be obtained in the form of a integer linear programming problem. Here the formulation process is shown, with full explanation of each step. The final solution is also included.

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