1 Introduction
1.1 Raise Alternatives
1.2 Choose Alternatives
1.3 Innovation and Structure
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Propose a model that can solve the above problems, which mainly includes three steps: raising, filtering, and choosing alternatives (see Fig. 1 for details).
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Based on the Bankruptcy theory, we propose a novel distribution rule (the AMO rule) that takes into account the different characteristics of participants while ensuring fairness.
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Propose a new constrain measure, the Security Restriction, to find the feasible solutions, together with the “Core” Solution in the Cooperative Game Theory (CGT).
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Five voting methods, base on SCT, are launched to aggregate preferences and to obtain a “win” alternative in different situations
2 Methodology
2.1 Bankruptcy Allocation Rules
2.1.1 Basic Scene
2.1.2 Classic Rules
2.1.3 The Proposed Rule
2.2 “Core” Solution and Security Restrictions
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\(x_i\) represent the benefit when agent i cooperate with others;
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\(x_i^*\) represent the benefit when agent i act alone;
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\(S \in N\) is a “coalition” , when S=N is the “grand coalition” ;
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v(S) is the benefit linked to the coalition S;
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v(N) is the benefit linked to the grand coalition.
2.3 Aggregate Preference under Social Choice Theory
2.3.1 Basic Scene
2.3.2 Voting Methods
Voting Methods
|
Advantages
|
Disadvantages
|
---|---|---|
PV | • Clear and straightforward, widely used voting method. | • The “win” alternative is not guaranteed to be the “best” (optional) solution. |
• The “win” alternative is the first choice for most voters | • Ranking results do not reflect actual preference gaps. | |
HS | • The voting process is also a process of reaching consensus. | • Multiple voting rounds are required, which becomes further complicated as alternatives increase. |
• The “Win” alternative tends to be widely accepted. | • Inefficient voting process, resulting in relatively high management costs. | |
BC | • Relatively clear and straightforward, which can be seen as an improvement to PV. | • The “win” alternative may not be the first choice for most voters. |
• Scores reflect actual preference between alternatives. | • There may be two (or more) “win” alternatives. | |
PC | • Differences in preferences are fully considered | • The “win” alternative may not be the first choice for most voters. |
• Head-to-head comparisons from multiple perspectives make it easier to reach consensus. | • The process is relatively complex and difficult to understand, making it difficult to promote. | |
AV | • Easy to reach consensus and get implementable solutions. | • Determining the right number of approvals is a challenge for managers. |
• The number of approvals can be adjusted based on the problem, providing better flexibility. | • Ranking results do not reflect actual preference gaps. |
3 Case Study
3.1 Study Area
3.2 Problem Description
Regions | Agriculture | Industry | Domestic | Claims |
---|---|---|---|---|
Urban Area | 7.66 | 105.97 | 100.58 | 214.21 |
Gedian DZ | 45.79 | 412.34 | 35.82 | 493.94 |
Huarong | 55.06 | 52.47 | 18.61 | 126.14 |
Echeng | 92.48 | 62.11 | 66.45 | 221.04 |
Liangzihu | 103.75 | 16.90 | 20.87 | 141.52 |
Total Claims | 304.74 | 649.79 | 242.32 | 1,196.85 |
3.3 Water Allocation and Simulation
4 Results and Discussion
4.1 Bankruptcy Alternatives
Regions | PRO | AP | CEA | CEL | TAL | AMO ( The proposed ) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(x_i\) | deficit | \(x_i\) | deficit | \(x_i\) | deficit | \(x_i\) | deficit | \(x_i\) | deficit | \(x_i\) | deficit | |
Urban Area | 193.30 | 9.8% | 190.85 | 10.9% | 214.21 | 0.0% | 190.85 | 10.9% | 190.85 | 10.9% | 201.60 | 5.9% |
Gedian DZ | 445.73 | 9.8% | 470.58 | 4.7% | 377.12 | 23.7% | 470.58 | 4.7% | 470.58 | 4.7% | 415.98 | 15.8% |
Huarong | 113.83 | 9.8% | 102.77 | 18.5% | 126.14 | 0.0% | 102.77 | 18.5% | 102.77 | 18.5% | 120.17 | 4.7% |
Echeng | 199.46 | 9.8% | 197.68 | 10.6% | 221.04 | 0.0% | 197.68 | 10.6% | 197.68 | 10.6% | 207.63 | 6.1% |
Liangzihu | 127.71 | 9.8% | 118.15 | 16.5% | 141.52 | 0.0% | 118.15 | 16.5% | 118.15 | 16.5% | 134.64 | 4.9% |
4.2 Feasible Test
Regions | Core Solutions | Security Restriction | ||
---|---|---|---|---|
Lower | Upper | Lower | Upper | |
Urban Area | 97.39 | 214.21 | 189.46 | 214.21 |
Gedian DZ | 377.12 | 493.94 | 407.41 | 493.94 |
Huarong | 9.31 | 126.14 | 89.81 | 126.14 |
Echeng | 104.21 | 221.04 | 162.16 | 221.04 |
Liangzihu | 24.69 | 141.52 | 86.07 | 141.52 |
4.3 Social Choice Selection
Voting methods | Preference order | ||
---|---|---|---|
1st | 2nd | 3rd | |
PV | AMO | AP, CEL and TAL | PRO |
HS | AMO | AP, CEL and TAL | PRO |
BC | AMO | PRO | AP, CEL and TAL |
PC | AMO | PRO | AP, CEL and TAL |
AV | PRO | AMO | AP, CEL and TAL |