Question

1. The world renown Chef Beaujolais Restaurant in New Orleans is open 24 hours a day. Each waitperson works an 8-hour shift and can report for duty at midnight, 4 am, 8am, noon, 4pm or 8 pm. The table below shows the minimum number of waitpersons needed during each 4 hour period into which the day is divided.

Time period	Waitstaff needed.
Midnight to 4am	4
4am to 8am	3
8am to noon	11
Noon to 4pm	10
4pm to 8pm	15
8pm to midnight	12

Write the LP formulation to determine the minimum total number of operators the company needs to fulfill the schedule requirements.

Answer 1

Let us first divide the time period into the 24hr clock starting from midnight at 0 hrs. This will make the calculations easier to perform.

Time Period	Waitstaff
0-4	4
4-8	3
8-12	11
12-16	10
16-20	15
20-24	12

Since each person works for an 8-hour shift. We have to make sure only the required number of the waitstaff are maintained all the time.

The following illustration describes the situation.

We observe that Shift 1 starts at t = 0hrs and t = 24hrs. For simplification, we'll take the time of start for Shift 1 to be 0hrs.

Now, we can start to define an LP model for the given problem. For that, we need three things.

1) Variables

2) Objective function

3) Constraints

Let's start defining each one of these starting with variables. The only thing that we'll be changing in this situation is the number of the waitstaff. And they are all non-negative, since, no. of people can't be in negative.

Secondly, our objective is to minimize the total number of waitstaff. So, we are looking at minimizing the sum of all of our variables in the objective function.

Thirdly, from the table, we observe that we require the minimum number of the waitstaff at different time periods. So that forms the constraints of our LP problem.

Algebraically we can write,

Similarly, the objective function will be :

Since the question is regarding the formulation of the LP problem, therefore we'll not solve for the variables.

In case you have any doubts regarding the procedure followed, then do mention it in the comments.

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