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

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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
l TABLE I OFFER DATA p max l s l 1 100 MWh 80 $/MWh 2 50 MWh 100 $/MWh 3 100 MWh 120 $/MWh 4 100 MWh 140 $/MWh m d max m e m 1 200 MWh 160 $/MWh a need to allocate joint costs onto many agents, where characteristic function C(S) is not subadditive. Then known allocation methods, which assume distributing value C(I) accurately (break-even property), do not prevent the subsidize phenomena. We relax condition for break-even allocation property of total cost C(I), but prevent to break the incremental cost tests which are acknowledged as a subsidize-free allocation conditions. Subsidize-free allocation can be obtained by multi-criteria model MASIT. From the set of effective solutions of the problem MASIT the solutions which equally treat players of the similar impact on infrastructure costs are preferred. These solutions can be obtained due to equitable rational preference relation. In computational practice the problem can be formulated as a multi-criteria linear programme MASIT OWA, for which effective solutions are also equitable solutions of problem MASIT. [15] R.R. Yager, ”On ordered weighted averaging aggregation operators in multicriteria decision making”, IEEE Tr. Sys. Man Cyber., vol. 18, pp. 183–190. [16] H. P. Young, Cost Allocation: Methods, Principles, Applications, Elseviers Science Publishers B.V., 1985. REFERENCES [1] J. Bialek, ”Tracing the ow of electricity”, in IEE Proc.-Gen. Transm. Distrib., vol. 143, 1996, pp. 313–320. [2] J. Bialek, ”Electricity tracing and co-operative game theory”, in Proc. 13th Power System Computation Conference, Trondheim, 1999, pp. 238–243. [3] P. Dubey and A. Neyman and R.J. Weber, Value theory without efciency, Mathematics of Operations Research, vol. 6, no. 1, 1981. [4] E. Faria et al., ”Allocation of rm-energy rights among hydro agents using cooperative game theory: an Aumann-Shapley approach”, in 19th Mini-EURO Conference Operation Research Models and Methods in the Energy Sector (ORMMES 2006), Coimbra, Portugal, 2006, (electronic publication). [5] W. W. Hogan, Contract networks for electricity power transmission, Journal of Regulatory Economics, 4(3), 1992, pp. 211–242. [6] M. Kaleta, Toczyowski E., ”Resource cost allocation in infrastructure market balancing models”, in Proceedings of Krajowa Konferencja Automatyki, Warsaw, vol. 3, 2005, pp. 341–346, (in Polish). [7] M.M. Kostreva and W. Ogryczak, ”Linear Optimization with Multiple Equitable Criteria”, RAIRO Rech. Opér., 33, 1999, pp. 275–297. [8] J. Kuipers, Bin packing games, Mathematical Methods of Operations Research, vol. 47, 1998, pp. 499-510. [9] H. Luss, On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach, Operations Research, 47, 1999, pp. 361– 378. [10] J.-F. Mertens and A. Neyman, A value of 'AN, International Journal of Game Theory, vol. 32, 2003, pp. 109-120. [11] F. C. Schwepe, Spot pricing of electricity, New Kluwer Academic Publishers, New York, 1988. [12] K. Smolira and M. Kaleta and E. Toczyowski, ”Pricing methods for infrastructure utilization on the intra-day electric power markets”, in XIII International Science Conference on Present-day Problems of power engineering, Jurata, Poland, vol. III, 2007, pp. 111-118. [13] X.H. Tan and T.T. Lie, ”Allocation of transmission loss cost using cooperative game theory in the context of open transmission access”, in IEEE Power Engineering Society Winter Meeting, vol.3, 2001, pp. 1215–1219. [14] E. Toczyowski, Optimization of Market Processes under Constraints, EXIT Publisher Company, Warsaw, 2002 (in Polish).
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