Staffing & Scheduling Problem: Certain types of facilities operate seven days each week and face the problem

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Question:

Staffing & Scheduling Problem: Certain types of facilities operate seven days each week and face the problem of allocating person power during the week as staffing requirements change each day of the week. This kind of problem is commonly encountered in public service and transportation organizations. Perhaps the most fundamental staffing problem involves the assignment of days off to full-time employees. In particular, it is regularly the case that each employee is entitled to two consecutive days off per week. If the number of employees required on each of the seven days of the week is given, then the problem is to find the minimum workforce size which will allow these demands to be met and then to determine the days off for the people in this workforce.

To be specific, let us study the problem faced by the City Transit Bus Company. The number of drivers required for each day of the week are as follows:

Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
18	16	15	16	19	14	12

a) How many drivers should be scheduled to start a five-day stint on each day of the week? Formulate this problem as a linear program. What is the optimal solution to it?

b) Completely unintentionally, several important details were omitted above:

Daily pay is $50 per person on weekdays, $75 on Saturday and $90 on Sunday.
There arc up to three people that can be hired who will work part time, specifically a 3-day week consisting of Friday, Sunday, and Monday. Their pay for this 3-day stint is $200.

Modify the formulation appropriately. is it obvious whether the part time people will be used?

This is a cyclic scheduling problem unlike the other personnel scheduling problems we considered earlier which were acyclic. In this case, for example, a driver who starts his working week on Friday will work for the rest of the week (F, Sa and Su) and the first part of the following week (M and Tu) while is taking off W and Th. In other words, his/her schedule spans not just one week (cycle) but two. The decision variables are the number of drivers working on a particular schedule (pattern). In the example above, F can be the number of drivers starting work on Friday. You have to ensure that each day is covered by enough drivers. Refer to the image below.