University of Antwerp
Operations Research Group
ANT/OR
Analysis of different cost allocation
methods in a collaborative transport
setting
Christine Vanovermeire
University of Antwerp – ANT/OR
February 7, 2013
Dries Vercruysse
Kenneth Sörensen
Collaborative transport
How does it differ from 3PL bundling?
I
Both responsible for creating benefits
I
I
Has to be rewarded
Sharing the risks and the benefits
I
equal share? (small vs. large volume)
Methodology
I
I
Choose set of cost allocation methods
Test them on various scenarios
I
I
I
Different number of partners
Partners differ in size (order size, #orders)
Partners differ in time window sizes
Objective
How do these cost allocation react in when partners are not equal?
Cost allocation methods
Used in practice
I
Stand-aloneP
(SA)
wi = c(i)/( j∈N c(j))
xi = wi c(N)
I
Volume
Cost allocation methods
Used in game-theory
I
Shapley
xi =
X |S|!(|N| − |S| − 1)!
× (c(S ∪ i) − c(S))
|N|!
S⊆N\i
Cost allocation methods
I
Nucleolus
P
Maximize smallest excess e(x; S) = i∈S vi − v (S)
Cost allocation methods
I
Equal Charge Method (ECM)
mi = c(N) − c(N
P \ i)
xi = mi +
c(N)−
|N|
j
mj
I
Alternative Cost Avoided Method (ACAM)
P
i
xi = mi + (c(N) − j mj ) Pc(i)−m
c(j)−mj
I
Cost Gap Method
P
ai = minS|i∈S (c(S) − j∈S mj )
j
Cost allocation methods
I
Equal profit method (EPM)
min f
s.t.
xj
xi
−
f ≥
c(i) c(j)
X
xj ≤ c(S)
∀i, j
∀S ⊆ N
j∈S
X
xj = c(N) −
j∈N
xi ≥ 0
X
c(i)
∀xi ≥ 0,
∀i
i∈N
∀i
Results: Number of partners
I
The smaller the number of partners, the smaller the difference
between cost allocation methods
I
For two partners, none of the allocation methods are adequate
to incentivize flexibility
Results: Total number of pallets
Allocated cost to company B (4073 pallets), collaborating with
partners A (967 pallets) and C (1375 pallets)
·104
Shapley
Nucleolus
Volume
SA
ECM
ACAM
CGM
EPM
ABC
WRSM
4.2
4
3.8
3.6
Stand-alone cost
Allocated collaborative cost
Results: Total number of pallets
I
Do not use the Volume method
I
Dividing cost using stand-alone cost (SA) : least benefit for
large companies
I
Dividing profit using stand-alone cost (EPM): most benefit for
large companies
Shapley, ACAM, Nucleolus and CGM uses stand-alone cost +
costs of subgroups
I
I
I
Shapley, ACAM: slight benefit for small companies
Nucleolus, CGM: slight benefit for large companies
Results: Flexible time windows
Allocated cost to company A (967 pallets), collaborating with partners B (4073 pallets) and C (1375 pallets)
·104
1.4
1.2
1
Shapley
Nucleolus
Volume
SA
ECM
ACAM
CGM
EPM
0.8
Stand-alone cost Alloc. coll. costAlloc. cost(A flexible)
Results: Flexible time windows
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Based on stand-alone costs (EPM, SA): incentives to lower own
cost, not collaborative costs
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Marginal costs (ECM, CGM, ACAM, Shapley) or all the costs
of the subcoalitions (Nucleolus)
Strong subcoalitions
Most of the benefit can already be created with a subset of the
partners
S ∈ N and T ∈ N for which v (S) + v (T ) ≥ v (N)
I
Stability becomes important
I
Nucleolus, EPM
Conclusion
Stand-alone, EPM, WRSM, but none really adequate
Yes
Different flexibility?
2
No
No
Yes1
Different number of
pallets?
Volume, Shapley, Nucleolus, ECM, ACAM, CGM, EPM, WRSM
Stand-alone, EPM
Yes 2
ECM, ACAM, CGM, Shapley, Nucleolus
No
Yes1
Different number of
pallets?
How many partners?
Volume, Shapley, Nucleolus, ECM, ACAM, CGM, EPM, WRSM
ACAM, Shapley
Yes 2
No
EPM, WRSM, Nucleolus
>2
No
Different flexibility?
Yes
No
Yes1
Different number of
pallets?
Strong subcoalitions?
CGM, ACAM, Shapley, Nucleolus, WRSM
ACAM, Shapley
Yes 2
Nucleolus
s
Ye
Different flexibility?
No
EPM (if possible), Nucleolus
Yes
WRSM (if possible), Nucleolus
Conclusion
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There is no best method
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I
Dependent on type of coalition
A difficult method is not always necessary