Question

The workers at Olisur Olive Oil Factory, work four days a week, 15 hours a day. Each day of the week, at least the following numbers of workers are needed: on weekdays 105 workers per day; on weekends 45 workers per day. There are 165 workers in the factory. Formulate a linear programming model that will reveal a schedule for the workers to maximize the sum of their consecutive days off (define decision variables; indicate objective function, constraints, and sign restrictions, if any).

Hint: A worker, who is not scheduled to work on Wednesdays, Thursday and Friday has 3 consecutive days off. If a worker is not scheduled to work on Mondays, Wednesdays, and Sundays, then s/he has 2 consecutive days off (Sunday & Monday); but if s/he is not scheduled to work on Tuesday, Thursday, and Saturday, then that worker has no consecutive days off.

Answer 1

Assuming that Friday is a weekday, we require at least 105 workers per day from Monday to Friday and at least 45 workers per day on Saturday and Sunday.

There will be 3 categories of workers with respect to their holidays:

a) 3 days consecutive- let this be variable a (no. of workers)

b) 2 days consecutive- let this be variable b (no. of workers)

c) no consecutive days off

So, the objective function will be to maximize the sum of a and b

Decision variables will be a matrix of days in the week against the worker taking values 1 or 0 (binary- 1 being a working day and 0 being a day off) for all the 165 workers that will give out their working days in the week

There will be 4 constraints:

1) The decision variables have to take a binary value

2) The number of workers available on the weekdays and weekends should satisfy the minimum criterion of availability of 105 workers and 45 workers per day respectively

3) The numbers of workers available on any given day have to take an integral value

4) The sum of the number of workers categorized into 3 different types of working days configuration has to be equal to the number of workers available at Olisur Olive Oil Factory

Thus, a + b + c =165

This is the linear programming model that we have built up for scheduling the workers at Olisur Olive Oil Factory.