Understand the idea.
More people than jobs
When there are more workers than required jobs, add dummy job columns until the assignment table is square. Assigning a worker to a dummy means that worker is idle. Use zero only if idling is free; otherwise include the stated idle cost.
The four-worker table needs one dummy job. A→1, B→2 and C→3 cost dollars, leaving D idle. The dummy is bookkeeping for unused capacity, so the recommendation must identify the idle worker rather than inventing a fourth real job.
| Worker | Job 1 | Job 2 | Job 3 |
|---|---|---|---|
| A | 4 | 8 | 7 |
| B | 6 | 3 | 8 |
| C | 5 | 7 | 2 |
| D | 9 | 4 | 6 |
More jobs than people
With more required jobs than workers, a dummy worker represents an unfilled job. If every job is compulsory and each worker can do only one, the original model is infeasible. Adding a dummy does not create a person or make the job completed.
When omission is permitted, enter its stated penalty in the dummy row. In the second table, A→1, B→3 and dummy→2 costs dollars. Job 2 remains undone, and its two-dollar penalty belongs in the objective total.
| Resource | Job 1 | Job 2 | Job 3 |
|---|---|---|---|
| A | 6 | 4 | 8 |
| B | 5 | 7 | 3 |
| Unfilled-job penalty | 10 | 2 | 7 |
See the method in action.
One worker stays idle
Solve the four-worker/three-job problem with free idling.
| Worker | Job 1 | Job 2 | Job 3 |
|---|---|---|---|
| A | 4 | 8 | 7 |
| B | 6 | 3 | 8 |
| C | 5 | 7 | 2 |
| D | 9 | 4 | 6 |
Worked solution
- 1
Balance the table with an idle column.
Four workers and three real jobs require one extra column; free idling gives every dummy entry zero.
- 2
Select distinct workers for the real-job minima.
The column minima 4, 3 and 2 occur in different rows, so they can all be attained together.
- 3
Calculate and certify the minimum.
Every allocation must pay at least the sum of the three real-job minima; this feasible one reaches that lower bound.
- 4
Interpret the dummy assignment.
A performs job 1, B job 2 and C job 3, while D is idle at no cost.
Minimum $9; A does job 1, B job 2, C job 3, and D is idle.
Penalties change which job is omitted
Use the two-worker table including unfilled-job penalties. Compare A→2, B→3, omit 1 with A→1, B→3, omit 2.
| Resource | Job 1 | Job 2 | Job 3 |
|---|---|---|---|
| A | 6 | 4 | 8 |
| B | 5 | 7 | 3 |
| Unfilled-job penalty | 10 | 2 | 7 |
Worked solution
- 1
Include the penalty for omitting job 1.
A does job 2 and B does job 3, leaving job 1 unfilled with its ten-dollar penalty.
- 2
Include the penalty for omitting job 2.
A does job 1 and B does job 3, leaving job 2 unfilled with its two-dollar penalty.
- 3
Compare the complete modelled totals.
The second option is six dollars cheaper even though its real-work cost is higher; the omission penalty reverses the comparison.
The second option costs $11 and is preferable to the $17 option.
Try it yourself.
Check your understanding
Five workers must cover three jobs, with each worker doing at most one. How many dummy job columns balance the assignment?
Show answer & reasoning
Two.
Add dummy job columns. The resulting table assigns all three real jobs and identifies the two workers who remain idle.
Keep practising
6 worksheet questions · 15 marks
Read the questions online
- Add the correct type and number of dummy entries to a 4-worker/3-job table with free idling.2 marks
- Solve the displayed four-worker problem and state who is idle.3 marks
Costs ($) for four workers and three required jobs. A worker may remain idle at no cost. Worker Job 1 Job 2 Job 3 A 4 8 7 B 6 3 8 C 5 7 2 D 9 4 6 - Two workers must each do exactly one job, but there are three compulsory jobs. Can a dummy row alone make all three jobs physically complete?2 marks
- Using the penalty table, find the total for A→3, B→1 and job 2 left unfilled.2 marks
Costs ($) for two workers and three jobs, with stated unfilled-job penalties. Resource Job 1 Job 2 Job 3 A 6 4 8 B 5 7 3 Unfilled-job penalty 10 2 7 - Using the penalty table, compare all six assignments and give the minimum.4 marks
Costs ($) for two workers and three jobs, with stated unfilled-job penalties. Resource Job 1 Job 2 Job 3 A 6 4 8 B 5 7 3 Unfilled-job penalty 10 2 7 - Four workers cover three jobs. Idle-worker costs are A:$2, B:$1, C:$4 and D:$3. What should the dummy job column contain?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Add the correct type and number of dummy entries to a 4-worker/3-job table with free idling.2 marks
There is one more worker than required jobs.
The dummy job means one person is idle.
One dummy job column, all zeros. - Solve the displayed four-worker problem and state who is idle.3 marks
Costs ($) for four workers and three required jobs. A worker may remain idle at no cost. Worker Job 1 Job 2 Job 3 A 4 8 7 B 6 3 8 C 5 7 2 D 9 4 6 The real costs are 4,3 and 2.
The dummy assignment costs 0.
Cost $9; A→1, B→2, C→3, D idle. - Two workers must each do exactly one job, but there are three compulsory jobs. Can a dummy row alone make all three jobs physically complete?2 marks
A dummy is a modelling device, not extra labour.
The original capacity restriction makes completion of all jobs infeasible.
No. - Using the penalty table, find the total for A→3, B→1 and job 2 left unfilled.2 marks
Costs ($) for two workers and three jobs, with stated unfilled-job penalties. Resource Job 1 Job 2 Job 3 A 6 4 8 B 5 7 3 Unfilled-job penalty 10 2 7 A/job 3 costs 8; B/job 1 costs 5; job 2 penalty is 2.
.
$15. - Using the penalty table, compare all six assignments and give the minimum.4 marks
Costs ($) for two workers and three jobs, with stated unfilled-job penalties. Resource Job 1 Job 2 Job 3 A 6 4 8 B 5 7 3 Unfilled-job penalty 10 2 7 Totals for A/B/dummy job sequences ,,,,, are 20,11,16,17,15,25.
Choose the smallest feasible total and identify the unfilled job.
Minimum $11: A→1, B→3, leave job 2 unfilled. - Four workers cover three jobs. Idle-worker costs are A:$2, B:$1, C:$4 and D:$3. What should the dummy job column contain?2 marks
The dummy column represents the actual cost of leaving that particular worker idle.
It is not automatically zero.
2,1,4,3 in rows A, B, C, D.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: discrete models, project scheduling and assignment. Activity-on-arrow conventions are used for project networks; technology may verify calculations but does not replace interpreting the model.
Official curriculum source ↗