The Allocation of Shared Fixed Costs Fairness versus Efficiency H. Paul Williams - London School of Economics Martin Butler - University College Dublin
Dec 18, 2015
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
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