Question

The formulation and solution of a certain staff-scheduling problem are shown below: MIN M + T + W + R + F + S + N

T + W + R + F + S >= 14

W + R + F + S + N >= 9 M + R + F + S + N >= 8

M + T + F + S + N >= 6

M + T + W + S + N >= 17

M + T + W + R + N >= 15

M + T + W + R + F >= 18 END

Optimal solution found at step: 4 Objective value: 19.0000000

Variable Value Reduced Cost

   M 5.000000 .0000000

T .0000000 .0000000

W 11.00000 .0000000

R .0000000 .0000000

F 2.000000 .0000000

S 1.000000 .0000000

N .0000000 .3333333

Row Slack or Surplus Dual Price

2 .0000000 -.3333333

3 5.000000 .0000000

4 .0000000 -.3333333

5 2.000000 .0000000

6 .0000000 -.3333333

7 1.000000 .0000000

8 .0000000 -.3333333

where, M, T, W, R, F, S, N is the number of people starting their five-day work week on Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday respectively.

a) How many people are required on duty on Thursday?

b) Suppose that part-time helpers are available who will work the three-day pattern, Thursday, Friday, and Saturday. That is, if you hire one of them, they will work all three days. These people cost 20% more per day than the ordinary folk who work a five-day week. Let P denote the number of part-timers to hire. Show how to modify the formulation to incorporate this option.

c) Using information from the solution report, what can you say about the (economic) attractiveness of the part-time help?

Answer 1

a) Thursday is denoted by R. On the first table, we can see that for the variable R, the value is 0.00. This means the value of variable R is 0 in the optimal solution. This also means that the number of people who will begin their 5 day work shift on Thursday is 0.

b) The part time workers will work only on Thursday, Friday and Saturday. This means in order for them to be optimal, they have to begin their shift on Thursday. The constraint that shows the requirement for Thursday shift beginners is

M + R + F + S + N >= 8

Here we need to include another variable P. Thus this constraint will become

M + R + F + S + N + P >= 8

Similarly the part time workers that begin work on Thursday will also be working on Friday and Saturday. This means their equation will change as well. These will become

M + T + F + S + N + P >= 6

M + T + W + S + N + P >= 17

These three constraints will change and other will remain as it is.

c) Considering that there are no full-time workers beginning their shift on Thursday, we can see that the optimal solution is satisfied. This means there is no need for part time workers to begin working on Thursday with increased cost. Also we can see from the second table that there is a surplus (row 5) on Thursday. This means the attractiveness of the part time worker is very low.