Subsidize-free cost allocation method for infrastructure ... - Ozyrys

subject to
∑
i∈S
x i ≥ C(I) − C(I \ S) ∀S ⊆ I (15)
x i ≥ 0 ∀i ∈ I (16)
For formulas simplication, further we will use
transformed objective max(−x 1 , −x 2 , . . . , −x n ) =
min(x 1 , x 2 , . . . , x n ) and variables ¯x = −x =
(−x 1 , −x 2 , . . . , −x n ) = (¯x 1 , ¯x 2 , . . . , ¯x n ). Solution of
minimization function (14) over simplex formed by set
of incremental cost test inequalities usually is not unique.
Simple scalarizing function, e.g. minimization of mean or
maximum allocation, can result with an allocation easy to
question from the point of view of general acknowledged
allocation fairness. Particulary, two players with the same
impact on the global costs may be charged different
values which minimize scalarizing function. If optimal
solution equally treating both of the players exists, it
should be strictly preferred. This lead to equitable rational
preference relation concept based on equitable Pigou-Dalton
shifts axiom. Equitable shift consists in worsening better
(lower) allocation ¯x i and simultaneously decreasing higher
allocation ¯x j by relatively small value ε > 0. Allocation
vector ¯x − εe i + εe j arising from equitable shift is strictly
preferred then original vector ¯x. Allocation vector ¯x ′
equitable dominates vector ¯x ′′ if it is strictly preferred
according to the rational equitable preference relation
¯x ′ ≻ w ¯x ′′ .
Let Θ : R n → R n denotes an operator of non-decreasing
ordering the coordinates of vector ¯x, it means that Θ(¯x) =
(Θ 1 (¯x), Θ 1 (¯x), . . . , Θ n (¯x)), where Θ 1 (¯x) ≤ Θ 2 (¯x) ≤ · · · ≤
Θ n (¯x). Let ¯Θ = ( ¯Θ 1 , ¯Θ 2 , . . . , ¯Θ n ) be an operator of cumulative
ordering, where ¯Θ i = ∑ i
l=1 Θ l(¯x) for i = 1, 2, . . . , n.
Successive coordinates of vector ¯Θ(¯x) signify the highest
allocated value, sum of two highest allocated values, sum of
three highest allocated values and so on. Then admissible
solution x of task (14)-(16) is equitable effective if and only
if it is solution of the following multi-criteria problem [7]:
max{ ¯Θ(¯x)} (17)
subject to constraints (15), (16) and ¯x = −x. This problem
can be solved by transforming it into single criteria problem
by weighing the criteria. This approach is equivalent to OWA
aggregation (Ordered Weighted Average) applied to task of
Θ(x) maximization subject to (15) and (16) [15]. OWA aggregation
can be depicted in computational convenient form
of maximization linear combination of cumulative ordered
criteria which can be expressed by linear formulas. Finally,
the following linear programme can be formulated:
MASIT OWA problem:
max
n∑
w k (kv k −
k=1
n∑
i=1
d ki ) (18)
∑
i∈S
x i ≥ C(I) − C(I \ S) ∀S ⊆ I (19)
v k + x i ≤ d ki ∀i, k ∈ I (20)
d ki , x i ≥ 0 ∀i, k ∈ I (21)
where w k are nonnegative coefcients, v k are unlimited
variables and d ki are nonnegative variables which represent
bottom deviation form v k .
Notice, that in the consequence of equitable rational preference
relation properties any solution of (18)-(21) satises
allocation symmetry (anonymous) condition. Thus, the allocation
is not sensitive for players (constraints) renumbering.
Constraints (19) result in incremental cost test satisfaction.
Also property of positive cost allocation on inuential player
and no cost allocation on insignicant player (dummy player)
are satised. Therefore, according to the condition (7), a
solution of problem (18)-(21) is an allocation free from
subsidizing.
IV. ILLUSTRATIVE EXAMPLE
We consider simple example of single commodity turnover
among four sellers and one buyer. Offer data are presented
in table I. Decisions related to the market game result
from solving model OPT (1)-(5), where constraints (3) are
following:
−p 1 ≥ −50 (22)
p 3 ≥ 100 (23)
p 4 ≥ 50 (24)
p 1 + p 3 ≥ 140 (25)
If constraints (22)-(25) are relaxed, the economical benets
are equal 13 thousands. When resource constraints are
considered, benets fall to 9 thousands. Thus, the joint
infrastructure cost of resource constraints is 4 thousands.
An allocation according to the Shapley value is (1; 1.066;
1.8; 0.133)∗10 3 respectively to the successive constraints
(22)-(25). Let us notice, that relaxing third constraint causes
economical benets increase by 2 thousands, while charge
for this constraint is only 1.8 thousand. Incentives for this
constraint intensication appear, if positive part of difference
0.2 thousand can be directly or indirectly (as result of collusion)
intercepted by player responsible for this constraint.
Notice also, that there are many allocation vectors which
are equitable effective solutions of multi-criteria task MASIT,
e.g. allocations (1; 1.2; 2; 0.8), (1.2; 1; 2; 0.8), (1.1; 1.1; 2;
0.8). Because rst and second constraints of (22)-(25) are
redundant, ona may expect that their costs will not differ
signicantly between them. Therefore, an allocation (1.1;
1.1; 2; 0.8) is preferred to the rest two allocations. This
allocation is also a solution of problem MASIT OWA (18)-
(21) and is symmetric equitable solution for any values of
w k .
V. CONCLUSIONS
In various practical problems, including infrastructure cost
allocation in the competitive market conditions, there is