1 Introduction
2 Prospect Theory
3 Proposed Method
3.1 Framework Definition
-
\( A = \left\{ {A_{1} ,A_{2} , \ldots ,A_{J} } \right\} \): set of alternatives, where \( A_{j} \) denotes the j-th alternative, \( j = 1,2, \ldots ,J \).
-
\( S = \left\{ {S_{1} ,S_{2} , \ldots ,S_{n} } \right\} \): set of different emergency situations, where \( S_{i} \) denotes the i-th situation, \( i = 1,2, \ldots ,n \).
-
\( X = \left\{ {X_{1} ,X_{2} , \ldots ,X_{M} } \right\} \): set of criteria/attributes, where \( X_{m} \) denotes the m-th criterion, \( m = 1,2, \ldots ,M \).
-
\( C_{j} = \left[ {C_{j}^{L} ,C_{j}^{H} } \right],C_{j}^{L} \le C_{j}^{H} \): an interval value, where \( C_{j} \) denotes the cost of the j-th alternative, \( j = 1,2, \ldots ,J \).
-
\( R_{im} = [R_{im}^{L} ,R_{im}^{H} ],R_{im}^{H} > R_{im}^{L} \): an interval value, where \( R_{im} \) denotes the reference point provided by the decision maker with respect to m-th criterion in the i-th possible situation, \( i = 1,2, \ldots ,n;m = 1,2, \ldots ,M \).
-
\( E_{jm} = [E_{jm}^{L} ,E_{jm}^{H} ],E_{jm}^{H} > E_{jm}^{L} \): an interval value, where \( E_{jm} \) denotes the predefined effective control scope of j-th alternative with respect to m-th attribute (Wang et al. 2015), which means that the alternative can prevent the losses from the emergency event regarding \( X_{m} \), \( j = 1,2, \ldots ,J;m = 1,2, \ldots ,M \).
-
\( W_{{X_{m} }} = \left( {w_{{X_{1} }} ,w_{{X_{2} }} , \ldots ,w_{{X_{M} }} } \right) \): the weighting vector of criteria, where \( w_{{X_{m} }} \) denotes the criterion weight of m-th criterion provided by the decision maker, satisfying \( \sum\limits_{m = 1}^{M} {w_{{X_{m} }} } = 1 \), \( w_{{X_{m} }} \in [0,1] \), \( m = 1,2, \ldots ,M \).
3.2 Information Collection
3.3 Calculation of Gains and Losses
Cases | Positional relationship between \( R_{im} \) and \( E_{jm} \) | |
---|---|---|
Case 1 |
\( E_{jm}^{H} < R_{im}^{L} \)
| |
Case 2 |
\( R_{im}^{H} < E_{jm}^{L} \)
| |
Case 3 |
\( E_{jm}^{L} < R_{im}^{L} \le E_{jm}^{H} < R_{im}^{H} \)
| |
Case 4 |
\( R_{im}^{L} < E_{jm}^{L} \le R_{im}^{H} < E_{jm}^{H} \)
| |
Case 5 |
\( E_{jm}^{L} < R_{im}^{L} < R_{im}^{H} < E_{jm}^{H} \)
| |
Case 6 |
\( R_{im}^{L} \le E_{jm}^{L} < E_{jm}^{H} \le R_{im}^{H} \)
|
Cases | Gain \( G_{jm} \) | Loss \( L_{jm} \) | |
---|---|---|---|
Case 1 |
\( E_{jm}^{H} < R_{im}^{L} \)
|
\( R_{im}^{L} - 0.5(E_{jm}^{L} + E_{jm}^{H} ) \)
| 0 |
Case 2 |
\( R_{im}^{H} < E_{jm}^{L} \)
| 0 |
\( R_{im}^{H} - 0.5(E_{jm}^{L} + E_{jm}^{H} ) \)
|
Case 3 |
\( E_{jm}^{L} < R_{im}^{L} \le E_{jm}^{H} < R_{im}^{H} \)
|
\( 0.5(R_{im}^{L} - E_{jm}^{L} ) \)
| 0 |
Case 4 |
\( R_{\text{im}}^{L} < E_{jm}^{L} \le R_{im}^{H} < E_{jm}^{H} \)
| 0 |
\( 0.5(R_{im}^{H} - E_{jm}^{H} ) \)
|
Case 5 |
\( E_{jm}^{L} < R_{im}^{L} < R_{im}^{H} < E_{jm}^{H} \)
|
\( 0.5(R_{im}^{L} - E_{jm}^{L} ) \)
|
\( 0.5(R_{im}^{H} - E_{jm}^{H} ) \)
|
Case 6 |
\( R_{im}^{L} \le E_{jm}^{L} < E_{jm}^{H} \le R_{im}^{H} \)
| 0 | 0 |
Cases | Gain \( G_{jm} \) | Loss \( L_{jm} \) | |
---|---|---|---|
Case 1 |
\( E_{jm}^{H} < R_{im}^{L} \)
| 0 |
\( 0.5(E_{jm}^{L} + E_{jm}^{H} ) - R_{im}^{L} \)
|
Case 2 |
\( R_{im}^{H} < E_{jm}^{L} \)
|
\( 0.5(E_{jm}^{L} + E_{jm}^{H} ) - R_{im}^{H} \)
| 0 |
Case 3 |
\( E_{jm}^{L} < R_{im}^{L} \le E_{jm}^{H} < R_{im}^{H} \)
| 0 |
\( 0.5(E_{jm}^{L} - R_{im}^{L} ) \)
|
Case 4 |
\( R_{im}^{L} < E_{jm}^{L} \le R_{im}^{H} < E_{jm}^{H} \)
|
\( 0.5(E_{jm}^{H} - R_{im}^{H} ) \)
| 0 |
Case 5 |
\( E_{jm}^{L} < R_{im}^{L} < R_{im}^{H} < E_{jm}^{H} \)
|
\( 0.5(E_{jm}^{H} - R_{im}^{H} ) \)
|
\( 0.5(E_{jm}^{L} - R_{im}^{L} ) \)
|
Case 6 |
\( R_{im}^{L} \le E_{jm}^{L} < E_{jm}^{H} \le R_{im}^{H} \)
| 0 | 0 |
3.4 Calculation of Prospect Values
3.5 Calculation of Overall Prospect Values
3.6 Selection of Optimal Alternatives for Different Emergency Situations
4 Examples of Applying the Proposed Method
4.1 Example 1: Petrochemical Plant Fire Emergency
-
X1: The number of casualties.
-
X2: Property loss (in RMB 10,000 Yuan).
-
X3: Negative effects on the environment on a scale of 0–100 (0: no negative effect; 100: serious negative effect).
Alternatives | Criteria (weights) | |||
---|---|---|---|---|
X1 (0.4375) | X2 (0.25) | X3 (0.3125) |
C
j
| |
E
j1
|
E
j2
|
E
j3
|
C
j
| |
A
1
| [3,5] | [50, 100] | [40,50] | [30,50] |
A
2
| [6,13] | [100,200] | [50,60] | [60,80] |
A
3
| [14,20] | [200,400] | [60,70] | [90,120] |
A
4
| [21,30] | [500,1000] | [70,80] | [130,160] |
A
5
| [31,50] | [1000,3000] | [80,90] | [170,200] |
-
\( S_{1} \): The local independent production area catches fire;
-
\( S_{2} \): The storage tanks of different oil products will explode in the local independent production area;
-
\( S_{3} \): The entire independent production area catches fire;
-
\( S_{4} \): The nearby production areas catch fire;
-
\( S_{5} \): The whole petrochemical plant catches fire.
Situations | Criteria | ||
---|---|---|---|
X
1
|
X
2
|
X
3
| |
R
i1
|
R
i2
|
R
i3
| |
S
1
| [3,8] | [100,300] | [20,35] |
S
2
| [10,15] | [300,400] | [65,75] |
S
3
| [18,25] | [600,900] | [50,65] |
S
4
| [30,35] | [1000,1500] | [40,50] |
S
5
| [30,35] | [1200,1600] | [55,60] |
O
ij
| Situations | |||||
---|---|---|---|---|---|---|
S
1
|
S
2
|
S
3
|
S
4
|
S
5
| ||
Alternatives (ranking) |
A
1
| 0.0601(5) | − 0.8655(5) | − 0.9024(5) | − 0.6875(5) | − 1.0000(5) |
A
2
| 0.1829(4) | − 0.4039(4) | − 0.4934(4) | − 0.5276(4) | − 0.6735(4) | |
A
3
| 0.3487(3) | 0.0095(3) | − 0.1733(3) | − 0.2777(3) | − 0.4348(3) | |
A
4
| 0.5850(2) | 0.3878(2) | 0.2188(2) | 0.0695(2) | − 0.0088(2) | |
A
5
| 1.0000(1) | 0.7473(1) | 0.7644(1) | 0.4459(1) | 0.4103(1) |
Situations | |||||
---|---|---|---|---|---|
S
1
|
S
2
|
S
3
|
S
4
|
S
5
| |
Existing EDM methods based on PT |
A
5
|
A
5
|
A
5
|
A
5
|
A
5
|
Our proposed method |
A
1
|
A
3
|
A
4
|
A
4
|
A
5
|
4.2 Example 2: Barrier Lake Emergency
-
\( X_{1} \): The number of people affected.
-
\( X_{2} \): Property loss (in RMB 10,000 Yuan).
Alternatives | Criteria (weights) | ||
---|---|---|---|
X1 (0.5333) | X2 (0.4667) |
C
j
| |
E
j1
|
E
j2
|
C
j
| |
A
1
| [3000,3500] | [2500,3500] | [300,350] |
A
2
| [3500,4000] | [3500,4500] | [350,450] |
A
3
| [4000,4500] | [4500,5500] | [450,550] |
A
4
| [5000,5500] | [5500,6500] | [550,650] |
-
\( S_{1} \): 1/3 dam body of the barrier lake will break;
-
\( S_{2} \): 1/2 dam body of the barrier lake will break;
-
\( S_{3} \): 3/4 dam body of the barrier lake will break;
-
\( S_{4} \): The whole dam body of the barrier lake will break;
Situations | Criteria | |
---|---|---|
X
1
|
X
2
| |
R
i1
|
R
i2
| |
S
1
| [2500,3500] | [3500,4000] |
S
2
| [3000,3500] | [4500,5500] |
S
3
| [3500,4000] | [4000,5500] |
S
4
| [4000,5000] | [5000,5500] |
O
ij
| Situations | ||||
---|---|---|---|---|---|
S
1
|
S
2
|
S
3
|
S
4
| ||
Alternatives (Ranking) |
A
1
| − 0.3684(4) | − 0.4667(4) | − 0.8048(4) | − 1.0000(4) |
A
2
| 0.1677(3) | − 0.0755(3) | − 0.1304(3) | − 0.4408(3) | |
A
3
| 0.5027(2) | 0.2509(2) | 0.1273(2) | − 0.0689(2) | |
A
4
| 1.0000(1) | 0.5960(1) | 0.6243(1) | 0.1212(1) |
Optimal alternative | Situations | |||
---|---|---|---|---|
S
1
|
S
2
|
S
3
|
S
4
| |
Existing EDM methods based on PT |
A
4
|
A
4
|
A
4
|
A
4
|
Our proposed method |
A
2
|
A
3
|
A
3
|
A
4
|