Math 441
Assignment #3
Due Wednesday Oct 7, 2015
1. I’d like you to try the following bus scheduling problem which requires a fair number of iterations
to completion. A bus company requires different numbers of employees on different days of the
week:
Monday: 18, Tuesday: 16, Wednesday: 16, Thursday: 17, Friday: 20, Saturday: 14, Sunday: 8.
Union rules states that each full-time employee must work 5 consecutive days and then receive
two days off. For example, an employee who works Monday to Friday must have Saturday and
Sunday off.
We allow part-time workers and overtime. We allow part-time workers who work half days for
5 consecutive days but at .7 the wage rate of full-time employees because of reduced benefits. Thus
a part-time worker is paid .35 the wage rate of a full-time worker in a week but works 2.5 days.
Also allow the possibility of hiring full-time employees on overtime to work an extra half day on
either or both the days they have off at 1.4 the regular wage of full-time employees. Again this
means that each half day of overtime costs .14 the cost of a fulltime worker.
Union rules specify that at most 1/3 of the work can be done by part-timers Be careful with this
inequality. I mean that paid part-time work only accounts for 1/3 of the total paid work. (Perhaps
compute the total paid work in terms of number of half days).
Seek a schedule (with integer variables) whose weekly labour cost is 20.3 × the cost of a fulltime employee. Such a solution will be obtained relatively quickly but proving optimality by the
software may take a while. If you are using LINDO then under the EDIT tab, select OPTIONS,
and then select Terse Output. When the program has terminated, close the solver window and
return to the input file. Then under REPORTS, click on solution. (My software took just under
200,000 iterations, yours results may differ). Comment on the difficulty of obtaining verification
of optimality using LINDO or other integer linear programming software (some iteration count
should be recorded). Remember to have your input file, output file (no sensitivity analysis for
integer linear programs), and your interpretation of output as an actual schedule.
b) Consider a variation on a) where part-timers get the same pay as full timers. Is there much
payoff to the company from having part-timers in this case?
c) Comment on the 1/3 rule. What would happen if it was removed (from the model in (a))?
Why might it not be removed?
d) Solve a) with the demands for Monday and Wednesday increased by 1 while demands for
Tuesday and Thursday decreased by 1. What do you expect? Comment on what you find. Solve
a) with the demands for Monday and Wednesday increased by 2 while demands for Tuesday and
Thursday decreased by 2. What do you expect? Comment on what you find.
2. Consider adding to an existing LP the following constraint relating x and y where you already
have constrained x to satisfy 0 ≤ x ≤ 1.
y≤
1 2
x + 100x + 20.
10
You can’t add the constraint directly (LP’s don’t allow quadratic constraints) so what do you
suggest?