The Allocation of Shared Fixed Costs

Fairness versus Efficiency

H. Paul Williams - London School of Economics

Martin Butler - University College Dublin

The Basic Problem

Given a Set of Facilities (E.G. Swimming Pools, Libraries, Aircraft Runways, Electric Generators, Reservoirs Etc.)

1. Which Do We Build?

2. How Do We Split Their Fixed Costs Between the Users Efficiently or Fairly?

An Example

Six Potential Facilities

{1,2,3,4,5,6}

Some of Which are Needed by These Potential Customers

{A, B, and C}

Customer A requires 1 of {1,2,3} and 1 of {4,5,6}

Customer B requires 1 of {1,4} and 1 of {2,5}

Customer C requires 1 of {1,5} and 1 of {3,6}

Benefits to 3 customers of being catered for 8, 11, 19

Fixed costs of 6 facilities 8, 7, 8, 9, 11, 10

An Example

A 0-1 Integer Programming Model

Maximise 654321cba 1011987819γ11γ8γ

Subject to

aγ

1cγ,bγ,aγ

06δ3δcγ05δ1δcγ05δ2

δbγ04δ1δbγ06δ5δ4δaγ03δ2

δ1δaγ

Dual of the LP RelaxationMinimise

Subject to

0c

6V,

c

5V,

b

4V,

b

3V,

a

2V,

a

1V,

cU,

bU,

aU

10c

6V

a

2V

11c

5V

b

4V

a

2V

9b

3V

a

2V

8c

6V

a

1V

7b

4V

a

1V

8c

5V

b

3V

a

1V

18c

6V

c

5V

cU

11b

4V

b

3V

bU

8a

2V

a

1V

aU

cUbUaU

ViX is amount of cost from ith group of facilities allocated to customer X

UX is surplus benefit accruing to customer X

NB:

1. Each customer pays within its means2. If a facility not totally paid for it is not built (LP dualilty)

This ‘would’ be a satisfactory allocation if a fractional solution were acceptable

(a) No customer pays more than would by alternate provision

(b) Total cost of facilities built met by customers

Build ½ of each of facilities 1,3,5,6 to serve customers A and C and ½ of customer B , Revenue – Cost = 14

Solution is neither integral nor fair

This would be a ‘satisfactory’ cost allocation if associated solution were integral

Linear Programming (Fractional) Solution and Cost Allocation

Costs Applied to Constraints

Maximise 654321cba 1011987819γ11γ8γ

Subject to Prices-25638

6-8

aγ

111

cγbγ

aγ

06δ3δcγ05δ1δcγ05δ2

δbγ04δ1δbγ06δ5δ4δaγ03δ2

δ1δaγ

Surpluses

2½

1½

0

4

13

1½

14

Linear Programming (Fractional) Solution and Cost Allocation

Customers Facilities

2

B 3

4

5

½ x8=4

C

0

½ x8=4

½ x11=5½

½ x10=5½

8

6

½ x11=5 ½

19

8

A

1

6

Optimal Integer Programming Solution

Build Facilities 1, 2, 6Serve Customers A, B, CRevenue - Cost = 13

Is there a cost allocation which will

1. Pay for facilities 1, 2, 62. Leave customers with net revenue of 133. Make facilities 3, 4, 5 uneconomic?

NO - Duality Theorem of Linear Programming

Dual Values and the Allocation of Costs

032 ba

If constraint binding in LP satisfied as equality

Hence total cost compensation to facilities (in equal amounts) equals amount paid by customer

But if constraint binding in IP (non redundant and has positive economic value) will have positive ‘dual value’ but not necessarily satisfied as equality.

Hence Cost allocations may not balance

Possible Methods of Cost Allocation

1. (Sub additive) Price Function instead of Prices

Obtain by appending (Chvátal) Cutting Planes obtained by:

(i) Adding constraints in suitable multiples

(ii) Nested Rounding of resultant right-hand-sides

‘Pays for’ facilities and charges customers appropriately but costs do not ‘balance’

Possible Methods of Cost Allocation

2. Gomory-Baumol Prices obtained by only considering multipliers in (i)

Satisfies only some requirements of cost allocation

e.g. Necessary to subsidise some activities

 

Derivation of Price Function

Append This Cutting Plane To ModelResultant Linear programme Yields Integer Solution

05δ4δ2δ1δcγbγ

15δ

4δ

2δ

1δcγb

γ

04δ0

2δ

1cγ05δ1δcγ05δ2

δbγ04δ1δbγ

1

2

2

32,154321 ,,,,,,

Corresponding Dual Solution Implies Price Function on

Coefficients

35433216542 2

11378111535

2

1

in Each Column of Model.

Such a Price Function is Known as a ‘Chvátal Function’

These are the Discrete Analogy of Dual Values (Shadow Prices) for Linear Programmes

A Typical Chvatál Function

b1 b2 b2

Multiply 2 1

Divide & Round Down

Multiply 3 1

Divide & Round Down 2

Multiply 3

Chvátal Function is

Relaxation is

Would be Linear Programming Dual Values

(Shadow Prices)

17

2212

17

13

2

13 bbb

2117

30

17

9bb

17

30,

17

9

Uses for Price Functions

1. Charge Customers

A: =8

Charge & Excess B: =9

C: =19

2. Pay for Facilities

1: =8

2: =7 (round up necessary)

3: =7 ½ (don’t build)

4: =9 (round up necessary)

5: =11

6: =10

NB Solution is Degenerate. We Build Facilities1, 2, 6 but Don’t Build 4, 5 (Although Just Paid For.)

In Order to Recover Full Cost of Facility 4 We Need to Round Up.

Applying Gomory-Baumol Prices (Ignoring Rounding) we Would Need to Subsidise Facility 4: Without Subsidy Charge to Customers (21 ¾ ) Falls Short of Cost of Facilities (25)

1152

1

111371532

12

1

11133

2

12

1

1131

2

12

1

152

1

1135

2

12

1

1113315

2

12

1

1552

1

1113812

12

1

Uses for Price Functions

3. Price a New Facility

E.g A New Facility Which Would Substitute For:

The 2nd Set of A’s Needs

The 1st Set of B’s Needs

The 2nd Set of C’s Needs

Payment Required = 16 ½

If Cost Below This:Build

If Cost Equal: Marginal

If Cost Higher: Don’t Build

113155

2

1

2

1

Optimal Solution is to Build Facilities 1, 2, 6

(Facility 3 ‘Priced Out,’ Facilites 4, 5 ‘Just’ Not

Worth Building (Degenerate Solution) )

Total Cost of Facilities 25

Supply Customers A, B, C

Total Price Paid (Benefits Less Excess)

A More Satisfactory Cost Allocation

Only include facilities to be built (with hindsight) in model i.e. Facilities 1, 2, 6

Solve LP relaxation to give integer solution

Hence dual solution will be ‘sensible’

Integer Programming Solution and Cost Allocation

A

Surpluses Customers Facilities

8

1

B

6

5

4

3

2

8

4

1

11

19

8

7

0

0

0

10

Facilities 1,2,6 built to serve customers A, B, C

But is this fair?

7

8

10

C

Obtaining a Fair AllocationThe allocation given lies in the core of possible allocations

i.e. no customer pays more towards facilities than they would by alternate provision

The dual solution (to restricted LP) will, however, be an extreme solution in core

To be fair we could instead

Minimise maximum surplus

Such a solution should lie at the centre of the core

i.e. in the Nucleolus

A ‘Fair’ Allocation

Surpluses Customers Facilities

A41/3

1

B

6

5

4

3

2

41/3

11

19

8

7

0

0

1010

C

031/3

42/3

8

32/3

31/3

41/3

`

Allocating the Cost of Computing Provision

Faculties Cost of Provision (£100k)

Veterinary Science 6

Medicine 7

Architecture 2

Engineering 10

Arts 18

Commerce 30

Agriculture 11

Science 29

Social Science 7___

120

Allocating the Cost of Computing ProvisionPossible Consortia

Faculties Cost of Provision (£100k)

(Veterinary Science, Medicine) 11

(Architecture, Engineering) 14

(Arts, Social Science) 22

(Agriculture, Science) 37

(Veterinary Science, Medicine, Agriculture, Science) 46

(Arts, Commerce, Social Science) 50

All Faculties (Central Provision) 96

It was decided that all faculties should use central provision.

How do we split the cost of 96 between the faculties?

NB

Savings over sum of individual provision is 24

How do we ‘share the savings’?

A Cost of AllocationSavings

Veterinary Science 6 0

Medicine 3 4

Architecture 2 0

Engineering 0 10

Arts 11 7

Commerce 30 0

Agriculture 8 3

Science 29 0

Social Science 7 0

__ __

96 24

Fair allocation tries to equalise savings over all possible (including individual) consortia

A Fair Cost AllocationVeterinary Science 4 2

Medicine 1 6

Architecture 0 2

Engineering 8 2

Arts 15 3

Commerce 28 2

Agriculture 8 3

Science 27 2

Social Science 5 2

__ __

96 24

Experiments in Social Choice Theory suggest that when allocating limited resources subject to need minimising maximum excess (i.e. trying to equalise benefits) is most acceptable to most people.

ReferencesM. Butler & H.P. Williams, Fairness versus Efficiency in Charging for the Use of Common Facilities, Journal of the Operational Research Society, 53 (2002)

M. Butler & H.P. Williams, The Allocation of Shared Fixed Costs, European Journal of Operational Research, 170 (2006)

J. Broome, Good, Fairness and QALYS, Philosophy and Medical Welfare, 3 (1988)

J. Rawls, A Theory of Justice, Oxford University Press, 1971

J. Rawls & E. Kelly Justice as Fairness: A Restatement Harvard University Press, 2001

M. Yaari & M. Bar-Hillel, On Dividing Justly, Social Choice Welfare 1, 1984