CN103455852A - Power transmission and distribution cost allocation method based on DEA cooperative game - Google Patents

Abstract

The invention discloses a power transmission and distribution cost allocation method based on the DEA cooperative game. The data envelopment analysis (DEA) method and thoughts of the game theory in the economics are combined with the power transmission and distribution cost collation problem, the characteristic that the DEA theory has multiple factors is well reserved, comprehensive consideration to multiple influencing factors of power transmission business and uniform balance of multiple characteristics of users are realized, an allocation result is made to be more stable, the defect that in a traditional method, only one single factor can be considered in an emphasizing mode is overcome, and the problem of cross subsidization among the users in a traditional allocation process is well solved; meanwhile, the idea of the cooperative game is utilized, through game competition among different users, joint interest maximization is realized, and complete recovery of cost is guaranteed.

Description

A kind of power transmission and distribution expense allocation method based on the DEA cooperative game

Technical field

The invention belongs to power transmission and distribution expense allocation technical field, relate to a kind of power transmission and distribution expense allocation method based on the DEA cooperative game.

Background technology

In the last few years, along with constantly carrying out of countries in the world electric Power Reform, and the deepening continuously of electricity market, also more and more and deepen continuously for the research of electricity market related fields.Power transmission and distribution allocation of expenditures problem determines the problem of how to be distributed between the undertaker of power transmission and distribution expense and these expenses undertaker different at each, it is the prerequisite of formulating whole transmission & distribution electricity price, being the foundation that grid company exists and runs, is also the Important Problems that every country mechanism and scholar are concerned about.

The starting of western developed country power market reform early, carry out a series of effective research work on power transmission and distribution expense allocation problem, mainly formed Allocation Theory and the method for two cover comparative maturities: the integrated cost method based on accounting cost and the marginal costing based on economic theory.The integrated cost method mainly comprises postage stamp method, power flow tracing method, contract path method, megawatt kilometer method etc., these class methods are the costs according to enterprise's record, give different types of user by these expense allocation, thereby calculate the Transmission Service expense that the user should bear, have and calculate simply, price steadiness, the advantage that wananty costs reclaims.Marginal costing mainly comprises long-run marginal cost method and short range margin cost method, these class methods are that the concept of marginal cost is applied on power transmission and distribution expense allocation problem, determine according to the change of the caused total product expense of the change of unit product supply the expense that the product of increment should be shared, from economics point, the expansion investments benefit that reflects following electrical network, have certain economic directive significance.

No matter be integrated cost method or marginal costing, it is in sharing the Rulemaking process, often can only stress transmission of electricity business index or user's characteristic in a certain respect in a certain respect, as operational throughput, fed distance etc., make and last share result and only can realize a part of user's justice is shared, alternative subsidy phenomenon between the user is serious, to the most of user in zone all more or less caused infringement to a certain degree.

Summary of the invention

Deficiency for existing methodology, the invention provides a kind of power transmission and distribution expense allocation method based on the DEA cooperative game, be intended to realize the power transmission and distribution expense between all users just, fair, reasonably share.

The present invention is achieved through the following technical solutions:

A kind of power transmission and distribution expense allocation method based on the DEA cooperative game comprises the following steps:

1) determine the transmission of electricity operational indicator that affects power transmission and distribution expense allocation result, and calculate the desired value of corresponding transmission of electricity operational indicator;

2) according to the cooperative game theory, the user is set up to different alliances, obtain the minimum overhead cost of different alliances by the DEA model that solves the transmission of electricity expense;

3) share the expenses that utilizes the shapley value method in the cooperative game theory to calculate each user;

4) solve the optimal weights vector that all users unify approval, obtain the optimum power transmission and distribution expense that the user shares.

In described step 1), definite transmission of electricity operational indicator comprises load index on power consumption, power network line is utilized to share index and marginal cost index;

Wherein, the load index on power consumption be for weigh the electric weight transmitted by electrical network that the user consumes the number;

Power network line utilizes the share index to characterize each user for transmission line of electricity using portion situation in electrical network;

The impact of marginal cost index reflection transmission of electricity business on the electrical network economy aspect.

The computing method parameter value of index that described utilization is given, form decision matrix; The calculating formula of described load index on power consumption value is:

$r_{i1}=\frac{E_i}{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{E}_i}$

In formula, r _i1desired value for the load power consumption of user i; E _ithe power consumption that passes through grid power transmission for user i;

Described power network line utilizes the desired value calculating formula of share to be:

${\mathrm{<figure-callout\; id="2"\; label="r\; i"\; filenames="HDA0000373440240000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_i=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}{\mathrm{cl}}_j$

In formula, r _i2for the power network line of user i utilizes the desired value of share; N is that network system is propped up way; f _ijfor the using portion of user i to branch road j; Cl _jfor the cost of branch road j, this cost comprises conversion annual cost and year operation and maintenance cost of construction cost;

The calculating formula of marginal cost index is:

r _i3=TC′-TC

In formula, r _i3desired value for the marginal cost index of user i; TC is the MW-km transmission cost that is transferred to user i; TC ' is for after user i increases the 1Mw power load, the MW-km transmission cost of the user i that is transferred to.

Described user is when setting up alliance, and all participants wish to add an alliance, sharing while to acquisition, being less than own independent operating, and set the participant and gather N={1,2 ..., any one subset of n} all forms an alliance.

User's number in described alliance does not limit, and is that to take the overhead cost of alliance's minimum be objective function, sets up that linear programming model solves by combined optimization and finally determines.

Step 2) try to achieve the minimum overhead cost of each different alliances in by following DEA model:

$\left\{\begin{array}{c}c\left(S\right)=\min \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_j\left(S\right)\\ s.t.\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}=1\\ \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}\&GreaterEqual;0,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n\\ b_{\mathrm{ij}}\&le;\frac{w_i}{w_j}\&le;a_{\mathrm{ij}}(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,m-1;i<j)\end{array}\right.$

In formula, the minimum overhead cost that c (S) is the S of alliance; M is the total number of transmission of electricity operational indicator; w _jfor weighted value corresponding to index j; r _j(S) be j the index normalized value of the S of alliance; N is total number of users; r _ijj the index normalized value for user i;

B _ij, a _ijfor weight ratio amount, the relative significance level between characteristic index; Wherein, j the index normalized value r of the S of alliance _j(S) calculating formula is:

$r_j\left(S\right)=\underset{i\&Element;S}{\overset{n}{\mathrm{\&Sigma;}}}{r}_{\mathrm{ij}}(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m).$

The share the expenses that utilizes the shapley value method in the cooperative game theory to calculate respectively each user in described step 3), it is calculated as follows:

$c_i=\underset{i\&Element;S}{\mathrm{\&Sigma;}}\frac{(n-|S\left|\right)!\left(\right|S|-1)!}{n!}[c\left(S\right)-c(S-\{i\left\}\right)]$

In formula, c _ishare the expenses for user i; | S| is the participating user quantity in the S of alliance; The minimum overhead cost that c (S) is the S of alliance; S-{i} is illustrated in the S of alliance the new alliance removed after participating user i.

Described step 4), by solving following optimal weights inearized model, is found the weight vectors that all users unify approval:

$\left\{\begin{array}{c}\min p=\mathrm{\&Sigma;}(s_i^{+}+s_i^{-})\\ s.t.w^T{x}_i+s_i^{+}-s_i^{-}=c_i(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ w_1+\&CenterDot;\&CenterDot;\&CenterDot;+w_m=1\\ 0\&le;s_i^{+}\&le;p,0\&le;s_i^{-}\&le;p(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ w_j\&GreaterEqual;0(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}\right.$

In formula: w is the optimal weights vector; w _jit is the optimal weights vector of j index; w ^tfor its transposed matrix, x _iindicator vector for user i;

be respectively the positive and negative side-play amount of user i; c _ioptimal distribution expense for user i.

Compared with prior art, the present invention has following useful technique effect:

Power transmission and distribution expense allocation method based on the DEA cooperative game provided by the invention, game theoretic thought in the method for DEA (DEA) and economics is combined with power transmission and distribution allocation of expenditures problem, effectively inherited the theoretical multifactorial characteristic of DEA, realized to a plurality of influence factors of transmission of electricity business consider and the unified of a plurality of characteristics of user weighed, make the result of sharing more stable, overcome the obstacle that classic method can only stress to consider single factors, the fine tradition that solved is shared the alternative subsidy problem between the user in process; Simultaneously, utilize the viewpoint of cooperative game, by the competition of the game between different user, realize that the associating interests maximize, the recovery fully of wananty costs.

The proof analysis result shows, postage stamp method is shared owing to only considering load, causes large load user 1 to share apparently higher than other load users; The power flow tracing method is shared the impact that process is only considered transmission distance, makes load user 2 and load user 4 because of too small near the power supply point amortization ratio; The marginal costing emphasis is considered the power transmission network investment situation in future, loads 4 because near the power supply point power consumption growth can't meet, and causes amortization ratio sharply to raise; And the present invention has considered the each side influence factor in sharing process, gathered the advantage of various classic methods, taken into account the factor of practicality and economy aspect simultaneously, realized just between all users, fair, reasonably share.

The accompanying drawing explanation

The realization flow figure that Fig. 1 is the power transmission and distribution expense allocation method based on the DEA cooperative game that proposes of the present invention;

Fig. 2 is IEEE14 node system structural drawing.

Embodiment

Below in conjunction with specific embodiment, the present invention is described in further detail, and the explanation of the invention is not limited.

Referring to Fig. 1, the power transmission and distribution expense allocation method based on the DEA cooperative game provided by the invention, combine game theoretic thought in the method for DEA (DEA) and economics with power transmission and distribution allocation of expenditures problem, comprise the steps:

1) determine the transmission of electricity operational indicator that affects power transmission and distribution expense allocation result, and calculate the desired value of corresponding transmission of electricity operational indicator;

2) according to the cooperative game theory, the user is set up to different alliances, obtain the minimum overhead cost of different alliances by the DEA model that solves the transmission of electricity expense;

3) share the expenses that utilizes the shapley value method in the cooperative game theory to calculate each user;

4) solve the optimal weights vector that all users unify approval, obtain the optimum power transmission and distribution expense that the user shares.

Concrete, considering and affect the each side of transmission of electricity business factor, choose the most representative three indexs in step 1): load index on power consumption, power network line utilize share index and marginal cost index.

Then utilize the computing method parameter value of the index of giving, form decision matrix;

Wherein, the load index on power consumption be mainly for weigh that the user consumes by electrical network transmit electric weight the number, calculating formula is:

$r_{i1}=\frac{E_i}{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i}$

In formula, r _i1first index (being index on power consumption) value for user i; E _ipower consumption for user i.

Wherein, power network line utilizes each user of share index main for transmission line of electricity using portion situation in electrical network, and calculating formula is:

${\mathrm{<figure-callout\; id="2"\; label="r\; i"\; filenames="HDA0000373440240000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_i=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}{\mathrm{cl}}_j$

In formula, r _i2second index (being that power network line utilizes the share index) value for user i; N is the system branch number; f _ijfor the using portion of user i to branch road j; Cl _jfor the cost of branch road j, this cost comprises conversion annual cost and year operation and maintenance cost of construction cost.

Wherein, the marginal cost index mainly reflects the impact of transmission of electricity business on the electrical network economy aspect, and calculating formula is:

r _i3=TC′-TC

In formula, r _i3the 3rd index (being the marginal cost index) value for user i; TC is the Mw-Km transmission cost that is transferred to user i; TC ' is for after user i increases the 1Mw power load, the new Mw-Km transmission cost that is transferred to user i.The electric power that the Mw-Km transmission cost is every megawatt is carried the expense of every kilometer.

The user is when setting up alliance, and all participants wish to add an alliance, sharing while to acquisition, being less than own independent operating, and set the participant and gather N={1,2 ..., any one subset of n} all forms an alliance.

User's number in alliance does not limit, and is that to take the overhead cost of alliance's minimum be objective function, sets up that linear programming model solves by combined optimization and finally determines.

Concrete, step 2) try to achieve the minimum overhead cost of each different alliances in by following DEA model.

$\left\{\begin{array}{c}c\left(S\right)=\min \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_j\left(S\right)\\ s.t.\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}=1\\ \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}\&GreaterEqual;0,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n\\ b_{\mathrm{ij}}\&le;\frac{w_i}{w_j}\&le;a_{\mathrm{ij}}(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,m-1;i<j)\end{array}\right.$

In formula, the minimum overhead cost that c (S) is the S of alliance; M is the total number of index; w _jfor weighted value corresponding to index j; r _j(S) be j the index normalized value of the S of alliance; N is total number of users; r _ijj the index normalized value for user i; b _ij, a _ijfor weight ratio amount, the relative significance level between characteristic index.

Wherein, j the index normalized value (r of the S of alliance _j(S) calculating formula) is:

$r_j\left(S\right)=\underset{i\&Element;S}{\overset{n}{\mathrm{\&Sigma;}}}{r}_{\mathrm{ij}}(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m)$

In formula, r _ijj the index normalized value for the participating user i of the S of alliance.

Concrete, the share the expenses that utilizes the shapley value method in the cooperative game theory to calculate respectively each user in step 3), calculating formula is as follows:

$c_i=\underset{i\&Element;S}{\mathrm{\&Sigma;}}\frac{(n-|S\left|\right)!\left(\right|S|-1)!}{n!}[c\left(S\right)-c(S-\{i\left\}\right)]$

In formula, c _ishare the expenses for user i; | S| is the participating user quantity in the S of alliance; The minimum overhead cost that c (S) is the S of alliance; S-{i} is illustrated in the S of alliance the new alliance removed after participating user i.

The preferred a kind of embodiment of the present invention is: step 4) solves following optimal weights inearized model, finds the weight vectors that all users unify approval.

$\left\{\begin{array}{c}\min p=\mathrm{\&Sigma;}(s_i^{+}+s_i^{-})\\ s.t.w^T{x}_i+s_i^{+}-s_i^{-}=c_i(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ {\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA0000373440240000012.png"\; state="\{\{state\}\}">w</figure-callout>}}_1+\&CenterDot;\&CenterDot;\&CenterDot;+w_m=1\\ 0\&le;s_i^{+}\&le;p,0\&le;s_i^{-}\&le;p(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ w_j\&GreaterEqual;0(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}\right.$

In formula: w is the optimal weights vector; x _iindicator vector for user i;

be respectively the positive and negative side-play amount of user i; c _ioptimal distribution expense for user i.

The weight vectors of described unified approval, refer to that the user, when selecting the index of alliance separately, although be the consideration for private interests, has still obtained more consistent result; Illustrating that established index system is reasonable for the result fairness of expense allocation, is the recognition and acceptance of user institute.

Below in conjunction with drawings and Examples, the present invention is further described.

Adopt the IEEE14 node system to carry out the analysis of power transmission and distribution expense allocation, as shown in Figure 2, in figure, 1～14 is different nodes to system architecture.System line parameter, load bus parameter and generator node parameter are respectively as table 1, shown in 2,3.

Table 1IEEE14 node system line parameter circuit value

Table 2 load bus parameter

Table 3 generator node parameter

Try to achieve by step 1) user's normalization desired value of respectively loading as shown in table 4.

Table 4 normalization desired value

According to step 2), step 3) tries to achieve the share situation of power transmission and distribution expense between each customer charge, solve the optimization weight model according to step 4), can try to achieve the optimization weight vectors of sharing that native system is corresponding is: w=(0.6576,0.2261,0.1163), thereby the power transmission and distribution expense of acquisition optimal distribution.

And contrasted with the result of sharing of traditional postage stamp method, power flow tracing method and marginal costing, as shown in table 5.

Table 5 Computation for apportionment result

Postage stamp method is shared owing to only considering load as can be seen from Table 5, causes large load user 1 to share apparently higher than other load users; The power flow tracing method is shared the impact that process is only considered transmission distance, makes load user 2 and load user 4 because of too small near the power supply point amortization ratio; The marginal costing emphasis is considered the power transmission network investment situation in future, loads 4 because near the power supply point power consumption growth can't meet, and causes amortization ratio sharply to raise; And the present invention has considered the each side influence factor in sharing process, gathered the advantage of various classic methods, share result more stable, fair, rationally.

Finally should be noted that: above embodiment is only in order to illustrate technical scheme of the present invention; protection scope of the present invention is not limited to this; anyly be familiar with those skilled in the art in the scope that the present invention discloses; be equal to replacement or changed according to technical scheme of the present invention and inventive concept thereof, all being belonged to protection scope of the present invention.

Claims (8)

1. the power transmission and distribution expense allocation method based on the DEA cooperative game, is characterized in that, comprises the following steps:

1) determine the transmission of electricity operational indicator that affects power transmission and distribution expense allocation result, and calculate the desired value of corresponding transmission of electricity operational indicator;

2) according to the cooperative game theory, the user is set up to different alliances, obtain the minimum overhead cost of different alliances by the DEA model that solves the transmission of electricity expense;

3) share the expenses that utilizes the shapley value method in the cooperative game theory to calculate each user;

4) solve the optimal weights vector that all users unify approval, obtain the optimum power transmission and distribution expense that the user shares.

2. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 1, is characterized in that, in step 1), definite transmission of electricity operational indicator comprises load index on power consumption, power network line are utilized to share index and marginal cost index;

Wherein, the load index on power consumption be for weigh the electric weight transmitted by electrical network that the user consumes the number;

Power network line utilizes the share index to characterize each user for transmission line of electricity using portion situation in electrical network;

The impact of marginal cost index reflection transmission of electricity business on the electrical network economy aspect.

3. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 2, is characterized in that, utilizes the computing method parameter value of the index of giving, and forms decision matrix; The calculating formula of described load index on power consumption value is:

$r_{i1}=\frac{E_i}{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i}$

In formula, r _i1desired value for the load power consumption of user i; E _ithe power consumption that passes through grid power transmission for user i;

Described power network line utilizes the desired value calculating formula of share to be:

$r_{i2}=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}{\mathrm{cl}}_j$

In formula, r _i2for the power network line of user i utilizes the desired value of share; N is that network system is propped up way; f _ijfor the using portion of user i to branch road j; Cl _jfor the cost of branch road j, this cost comprises conversion annual cost and year operation and maintenance cost of construction cost;

The calculating formula of marginal cost index is:

r _i3=TC′-TC

In formula, r _i3desired value for the marginal cost index of user i; TC is the MW-km transmission cost that is transferred to user i; TC ' is for after user i increases the 1Mw power load, the MW-km transmission cost of the user i that is transferred to.

4. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 1, it is characterized in that, the user is when setting up alliance, all participants wish to add an alliance, sharing while to acquisition, being less than own independent operating, and set the participant and gather N={1,2,, any one subset of n} all forms an alliance.

5. the power transmission and distribution expense allocation method based on the DEA cooperative game as described as claim 1 or 4, it is characterized in that, user's number in alliance does not limit, and is that to take the overhead cost of alliance's minimum be objective function, sets up that linear programming model solves by combined optimization and finally determines.

6. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 1, is characterized in that step 2) in try to achieve the minimum overhead cost of each different alliances by following DEA model:

$\left\{\begin{array}{c}c\left(S\right)=\min \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_j\left(S\right)\\ s.t.\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}=1\\ \underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{w}_j{r}_{\mathrm{ij}}\&GreaterEqual;0,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n\\ b_{\mathrm{ij}}\&le;\frac{w_i}{w_j}\&le;a_{\mathrm{ij}}(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,m-1;i<j)\end{array}\right.$

In formula, the minimum overhead cost that c (S) is the S of alliance; M is the total number of transmission of electricity operational indicator; w _jfor weighted value corresponding to index j; r _j(S) be j the index normalized value of the S of alliance; N is total number of users; r _ijj the index normalized value for user i;

B _ij, a _ijfor weight ratio amount, the relative significance level between characteristic index; Wherein, j the index normalized value r of the S of alliance _j(S) calculating formula is:

$r_j\left(S\right)=\underset{i\&Element;S}{\overset{n}{\mathrm{\&Sigma;}}}{r}_{\mathrm{ij}}(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m).$

7. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 1, is characterized in that, the share the expenses that utilizes the shapley value method in the cooperative game theory to calculate respectively each user in step 3), and it is calculated as follows:

$c_i=\underset{i\&Element;S}{\mathrm{\&Sigma;}}\frac{(n-|S\left|\right)!\left(\right|S|-1)!}{n!}[c\left(S\right)-c(S-\{i\left\}\right)]$

In formula, c _ishare the expenses for user i; | S| is the participating user quantity in the S of alliance; The minimum overhead cost that c (S) is the S of alliance; S-{i} is illustrated in the S of alliance the new alliance removed after participating user i.

8. the power transmission and distribution expense allocation method based on the DEA cooperative game as claimed in claim 1, is characterized in that, step 4), by solving following optimal weights inearized model, is found the weight vectors that all users unify approval:

$\left\{\begin{array}{c}\min p=\mathrm{\&Sigma;}(s_i^{+}+s_i^{-})\\ s.t.w^T{x}_i+s_i^{+}-s_i^{-}=c_i(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ w_1+\&CenterDot;\&CenterDot;\&CenterDot;+w_m=1\\ 0\&le;s_i^{+}\&le;p,0\&le;s_i^{-}\&le;p(i=1,\&CenterDot;\&CenterDot;\&CenterDot;,n)\\ w_j\&GreaterEqual;0(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}\right.$

In formula: w is the optimal weights vector; w _jit is the optimal weights vector of j index; w ^tfor its transposed matrix, x _iindicator vector for user i;

be respectively the positive and negative side-play amount of user i; c _ioptimal distribution expense for user i.

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