Question

Question 1:

Remember the Post Office example: A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in the following Table. Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday. The post office wants to meet its daily requirements using only full-time employees.

DAY	Number of Full-time Employees Required
1 = Monday	17
2 = Tuesday	13
3 = Wednesday	15
4 = Thursday	19
5 = Friday	14
6 = Saturday	16
7= Sunday	11

Suppose that the post office can force employees to work one day of overtime each week. For example, an employee whose regular shift is Monday to Friday can also be required to work on Saturday. Each employee is paid $50 a day for each of the first five days worked during a week and $62 for the overtime day (if any). Formulate an LP whose solution will enable the post office to minimize the cost of meeting its weekly work requirements.

Answer 1

Answer

Question asked to minimize post office labour cost. So, the objective function should formulate in way that it reduces labour cost.

It is given that normal wage is $50 and over time is $62.

Suppose x_i be the number of employees beginning work on day(i)

That is x₁ means number of employees beginning work on day (1) and so on

And y_i be the number of employees forced to work overtime on day(i)

That is y₁ means the number of employees forced to work overtime on day (1) and so on

Employees start normal work on day (1) will continuously work till day (5) and their overtime will the following day after five normal days and it is their first day of over time.

Number of full- time employees required given as Monday (17), Tuesday (13), Wednesday (15), Thursday (19), Friday (14), Saturday (16), Sunday (11).

So, we can formulate the constraints as follows:

Monday          x₁₊x₄₊x₅₊x₆₊x₇₊y₃ ≥ 17

Tuesday          x₁₊x₂₊x₅₊x₆₊x₇₊y₄ ≥ 13

Wednesday    x₁₊x₂₊x₃₊x₆₊x₇₊y₅ ≥ 15

Thursday        x₁₊x₂₊x₃₊x₄₊x₇₊y₆ ≥ 19

Friday              x₁₊x₂₊x₃₊x₄₊x₅₊y₇ ≥ 14

Saturday         x₂₊x₃₊x₄₊x₅₊x₆₊y₁ ≥ 16

Sunday            x₃₊x₄₊x₅₊x₆₊x₇₊y₂ ≥ 11

                         X_i, Y_i ≥0.

Now we have to formulate objective function to minimize the labour cost

Min Z = 50*(x₁₊x₂₊x₃₊x₄₊x₅₊x₆₊x₇ )₊62_*(y₁₊y₂₊y₃₊y₄₊y₅₊y₆₊y₇ )