The Allocation of Shared Fixed Costs

1
The Allocation of Shared Fixed Costs
Fairness versus Efficiency
H. Paul Williams - London School of
Economics Martin Butler - University College
Dublin
2
The Basic Problem

• Given a Set of Facilities (E.G. Swimming Pools,
  Libraries, Aircraft Runways, Electric Generators,
  Reservoirs Etc.)
• Which Do We Build?
• How Do We Split Their Fixed Costs Between the
  Users Efficiently or Fairly?

3
An Example
Six Potential Facilities 1,2,3,4,5,6
Some of Which are Needed by These Potential
Customers A, B, and C
4
An Example
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
5
A 0-1 Integer Programming Model
Maximise
Subject to
6
Dual of the LP Relaxation
Minimise Subject to
ViX is amount of cost from ith group of
facilities allocated to customer X UX is surplus
benefit accruing to customer X
7

• NB
• 1. Each customer pays within its means
• 2. 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

8
Linear Programming (Fractional) Solution and Cost
Allocation
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
9
Costs Applied to Constraints
Maximise
Subject to Prices - 2
5 6 3 8
6 - 8
10
Linear Programming (Fractional) Solution and Cost
Allocation
Surpluses
Customers
Facilities
8
1
2½
½ x84
A
6
2
1½
0
B
3
½ x84
½ x115 ½
4
4
0
19
1
3
5
C
½ x115½
8
1½
1
4
½ x105½
6
11
Optimal Integer Programming Solution

• Build Facilities 1, 2, 6
• Serve Customers A, B, C
• Revenue - Cost 13
• Is there a cost allocation which will
• Pay for facilities 1, 2, 6
• Leave customers with net revenue of 13
• Make facilities 3, 4, 5 uneconomic?
• NO - Duality Theorem of Linear Programming

12
Dual Values and the Allocation of Costs
If constraint binding in LP satisfied as
equality Hence total cost compensation to
facilities (in equal amounts) equals amount paid
by customer
13
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
14
Possible Methods of Cost Allocation

• (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

15
Possible Methods of Cost Allocation

• Gomory-Baumol Prices obtained by only considering
  multipliers in (i)
• Satisfies only some requirements of cost
  allocation
• e.g. Necessary to subsidise some activities
•  

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Derivation of Price Function

• Append This Cutting Plane To Model
• Resultant Linear programme Yields Integer Solution

17
Corresponding Dual Solution Implies Price
Function on Coefficients

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
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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
19

• Uses for Price Functions
• Charge Customers
• A
  8
• Charge Excess B
  9
• C 19
• Pay for Facilities
• 1 8
• 2
  7 (round up
  necessary)
• 3 7 ½
  (dont build)
• 4 9
  (round up necessary)
• 5 11
• 6 10
• NB Solution is Degenerate. We Build Facilities1,
  2, 6 but Dont 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)

20
Uses for Price Functions 3. Price a New
Facility E.g A New Facility Which Would
Substitute For The 2nd Set of As Needs The
1st Set of Bs Needs The 2nd Set of Cs
Needs Payment Required 16 ½ If Cost
Below This Build If Cost Equal Marginal If
Cost Higher Dont Build
21
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)
22
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
23
Integer Programming Solution and Cost Allocation
Surpluses Customers
Facilities
8
1
8
A
8
2
7
7
11
3
0
4
B
4
0
8
19
5
0
1
C
6
10
10
Facilities 1,2,6 built to serve customers A, B, C
But is this fair?
24
Obtaining a Fair Allocation
The 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
25

A Fair Allocation
Surpluses Customers
Facilities
8
1
8
41/3
A
32/3
7
2
31/3
11
3
0
31/3
41/3
B
4
0
42/3
19
5
0
41/3
C
10
6
10
26
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
27
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

28
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?
29
A Cost of Allocation
Savings 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

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Fair allocation tries to equalise savings over
all possible (including individual) consortia
31
A Fair Cost Allocation
Veterinary 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
32
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.
33
References
M. 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