Formula for measuring unfairness

Load balancing is a common constraint for many OptaPlanner use cases.
Especially when scheduling employees, the workload needs to be spread out fairly.
Easier said than done. Let’s take a look at this challenging problem.

For example, let’s assign 15 undesirable tasks to 5 employees.
Each task takes a day, but they have different skill requirements.

What is fair?

In a perfectly fair schedule, each employee will get 3 tasks,
because the mean (= average) is 15 / 5 = 3.
Or explained more formally, for the math geeks:

  • Mean (= average): (overline{x} = frac{sum_{i=1}^{n} x_i}{n})

So this schedule is perfectly fair:

Problem solved? Unfortunately not, because there are other constraints.
For example, there are 7 tasks that require a skill which only Ann and Beth posses.
One of them will have to do at least 4 tasks.

Our schedule will not be perfectly fair, but how can we make it as fair as possible?

What is fairer?

Let’s look at two opposing definitions of fairness:

  • A schedule is fair if most users think it is fair to them.

  • A schedule is fair if the employee with most tasks has as little tasks as possible.

Obviously, since we want to treat all employees equally, the second definition is correct.
Besides, if we make almost everyone happy by letting Ann do all the work, she would probably quit on us.
That wouldn’t help.

Number of tasks per employee

Quality

Measurement of fairness

Ann

Beth

Carl

Dan

Ed

Mean

Schedule A

3

3

3

3

3

Fair

3

Sorting solutions by fairness

Let’s look at a few different solutions of the same dataset.
Each one has 15 undesirable tasks:

Number of tasks per employee

Quality

Ann

Beth

Carl

Dan

Ed

Schedule

15

0

0

0

0

Unfair. Ann quits her job.

These are sorted from fair to unfair.
Several of these are probably infeasible, due to hard constraints such as skill requirements.

Ann has most tasks each time. How do we compare 2 schedules in which Ann has the same number of tasks?

Number of tasks per employee

Quality

Ann

Beth

Carl

Dan

Ed

Schedule A

3

3

3

3

3

Fairest

Schedule B

4

4

3

2

2

Schedule C

5

3

3

2

2

Schedule D

5

5

2

2

1

Schedule E

6

3

3

2

1

Schedule F

6

5

2

1

1

Schedule G

11

1

1

1

1

Unfairest

In that case, we look at the second unfairest treated employee: Beth in this case.
And we minimize her tasks.

Now that we’ve defined fairness, how do we implement it?

Measuring unfairness

Ideally, we want to calculate a number for each schedule to measures the fairness of that solution.
The lower, the fairer. How do we calculate that number? Let’s look at some formula’s:

Deviation from the mean

Because a perfectly fair schedule has all employees working the average number of tasks,
what if we simply sum the difference with the mean per employee?

  • Absolute deviation from the mean: (f(n) = sum_{i=1}^{n} |x_i – overline{x}|)

  • Average deviation from the mean: (f(n) = frac{sum_{i=1}^{n} |x_i – overline{x}|}{n})

Number of tasks per employee

Quality

Ann

Beth

Carl

Dan

Ed

Schedule C

5

3

3

2

2

More fair

Schedule D

5

5

2

2

1

Less fair

These measurements are terrible. Both claim that schedule B and C are equally fair. They aren’t.
Both claim that schedule D is worse than E. It’s not: just ask Ann.

Variance and standard deviation

In statistics, variance and standard deviation are used to penalize outliers more.
That sounds like exactly what we need:

  • Variance: (f(n) = frac{sum_{i=1}^{n} (x_i – overline{x})^2}{n})

  • Standard deviation: (f(n) = sqrt{frac{sum_{i=1}^{n} (x_i – overline{x})^2}{n}})

  • Squared deviation from the mean (= variance * n): (f(n) = sum_{i=1}^{n} (x_i – overline{x})^2)

  • Root of squared deviation from the mean: (f(n) = sqrt{sum_{i=1}^{n} (x_i – overline{x})^2})

Number of tasks per employee

Measurement of fairness

Ann

Beth

Carl

Dan

Ed

Absolute deviation

Average deviation

Schedule A

3

3

3

3

3

0

0.00

Schedule B

4

4

3

2

2

4Bad

0.80Bad

Schedule C

5

3

3

2

2

4

0.80

Schedule D

5

5

2

2

1

8Very bad

1.60Very bad

Schedule E

6

3

3

2

1

6

1.20

Schedule F

6

5

2

1

1

10

2.00

Schedule G

11

1

1

1

1

16

3.20

These measurements are good, but still not ideal. They claim that schedule D and E are equally fair.
They aren’t.

Maximum

What if we simply take the maximum of each row?

  • Maximum: (f(n) = underset{0 < i leq n}max x_i)

Number of tasks per employee

Measurement of fairness

Ann

Beth

Carl

Dan

Ed

Variance

Standard deviation

Squared deviation

Root squared deviation

Schedule A

3

3

3

3

3

0.00

0.00

0

0.00

Schedule B

4

4

3

2

2

0.80

0.89

4

2.00

Schedule C

5

3

3

2

2

1.20

1.10

6

2.45

Schedule D

5

5

2

2

1

2.80Bad

1.67Bad

14Bad

3.74Bad

Schedule E

6

3

3

2

1

2.80

1.67

14

3.74

Schedule F

6

5

2

1

1

4.40

2.10

22

4.69

Schedule G

11

1

1

1

1

16.00

4.00

80

8.94

That’s worse than variance: it only looks at one employee.
Furthermore, it completely discards fairness between the remaining employees.
That might be ok if there’s one employee, but not if there are thousands.

List of maximums

What if we don’t use any formula but just store the list of numbers sorted by decreasing size?

Number of tasks per employee

Measurement of fairness

Ann

Beth

Carl

Dan

Ed

Maximum

Schedule A

3

3

3

3

3

3

Schedule B

4

4

3

2

2

4

Schedule C

5

3

3

2

2

5Bad

Schedule D

5

5

2

2

1

5

Schedule E

6

3

3

2

1

6Bad

Schedule F

6

5

2

1

1

6

Schedule G

11

1

1

1

1

11

That will compare perfectly. In OptaPlanner it can be implemented by adding 5 score levels for this dataset.

However, besides obvious memory consumption issues when scaling to thousands of employees,
this isn’t compatible with other soft constraints…​

No constraint is an island

Fairness is typically a soft constraint.
But there are other soft constraints that we’ll need to optimize for too,
so we’ll need to weight them against each other.

An example

For example, presume there’s a soft constraint on priority violations,
that’s 10 times as important as a fairness violation.
Let’s also add a schedule with 1500 tasks, to see how it scales out:

Number of tasks per employee

Measurement of fairness

Ann

Beth

Carl

Dan

Ed

List of maximums

Schedule A

3

3

3

3

3

[3,3,3,3,3]

Schedule B

4

4

3

2

2

[4,4,3,2,2]

Schedule C

5

3

3

2

2

[5,3,3,2,2]

Schedule D

5

5

2

2

1

[5,5,2,2,1]

Schedule E

6

3

3

2

1

[6,3,3,2,1]

Schedule F

6

5

2

1

1

[6,5,2,1,1]

Schedule G

11

1

1

1

1

[11,1,1,1,1]

To calculate the soft score, we sum the fairness violations with 5 times the priority violations and make that negative.

Let’s start the process of elimination…​

Represented by a single number

List of maximums isn’t represented as single number
(because it uses multiple score levels),
so it’s difficult to mix in priority violations:

Number of tasksPriority violationsFairness violationsSoft scoreQuality

Schedule F

15

1

?

?

Best

Schedule C

15

2

?

?

Schedule D

15

2

?

?

Worst

Schedule X

1500

100

?

?

Different dataset

Grow with the number of violations

If we scale out to 1500 employees,
we notice that maximum gets dwarfed by the priority violations:

Number of tasks

Priority violations

Measurement of fairness

Soft score

List of maximums

Schedule F

15

1

[6,5,2,1,1]

ERROR?

Schedule C

15

2

[5,3,3,2,2]

ERROR?

Schedule D

15

2

[5,5,2,2,1]

ERROR?

Schedule X

1500

100

[8,8,7,7,7,7,…​]

ERROR?

Similarly, average deviation from the mean, variance and standard deviation get dwarfed
on bigger datasets too:

Number of tasks

Priority violations

Measurement of fairness

Soft score

Maximum

Schedule F

15

1

6

16

Schedule C

15

2

5

25

Schedule D

15

2

5

25

Schedule X

1500

100

8

1008 – Dwarfed

As the number of fairness violations grow, so should the fairness measurement.

Do not grow exponentially with the number of violations

On the other hand, as the dataset grows, the fairness violations shouldn’t dwarf the other violations either.
Squared deviation does that:

Number of tasks

Priority violations

Measurement of fairness

Average deviation

Variance

Standard deviation

Schedule F

15

1

2.00

4.40

2.10

Schedule C

15

2

0.80

1.20

1.10

Schedule D

15

2

1.60

2.80

1.67

Schedule X

1500

100

1.50 – Dwarfed

2.50 – Dwarfed

1.58 – Dwarfed

Conclusion

That just leaves:

  • absolute deviation from the mean which compares terribly for fairness

  • root squared deviation which isn’t perfect but works well enough

So the recommended approach is:

  • Root of squared deviation from the mean: (f(n) = sqrt{sum_{i=1}^{n} (x_i – overline{x})^2})

Number of tasks

Priority violations

Measurement of fairness

Squared deviation

Schedule F

15

1

80

Schedule C

15

2

6

Schedule D

15

2

14

Schedule X

1500

100 – Dwarfed

10201

Additional notes

Part-time employees

To deal with unequal employees,
for example if some employees work half as many hours as other employees,
multiply their number of tasks by the inverse of their FTE (full time equivalent)
before feeding into this formula.

Other reasons to treat employees unequally (such as disabilities or talent retention)
can be handled in a similar fashion or with separate constraints, depending on the requirement.

Update

There is a formula that seems to compare perfectly,
see this mathexchange answer,
but it might suffer from numerical instability and overflow.

Number of tasks

Priority violations

Measurement of fairness

Soft score

Root squared deviation

Schedule F

15

1

4.69

14.69

Schedule C

15

2

2.45

22.45

Schedule D

15

2

3.74

23.74

Schedule X

1500

100

101.00

1101.00