It is a well-known theory proposed by Tversky and Kahneman (1992) and applied in decision-making with uncertain environment. The crucial part of this theory can be constructed as a prospect function V ( x j ), which can be described as the product of two functions: the value function v ( x j ) and the transformed probability weighting function π ( p j ). (1) V ( x j ) = ∑ j = 1 m v ( x j ) π ( p j ) , where m is the number of attributes for alternatives; j expresses the jth attribute; v ( x j ) reflects the perceived gains or losses, which are defined as follows: (2) v ( x j ) = ( x j − x 0 ) α , if x j − x 0 ⩾ 0 , − λ ( x 0 − x j ) β , if x j − x 0 < 0 , where x j shows the value of the jth attribute, while x 0 expresses the reference point perceived by DMs; thereby, x j − x 0 > 00$]]> represents the gain; on the contrary, x j − x 0 < 0 shows the loss; In addition, x j − x 0 = 0 explains that there is no gain or loss relative to the reference point; α and β are the parameters of DMs’ risk attitudes and they are viewed as preference degrees in the domain of gain and loss, respectively; λ is the parameter of loss aversion that is more sensitive to loss than gain. The value function reflects the different risk attitudes for gains and losses.

When x j − x 0 ⩾ 0, the weighting function is determined by: (3) π + ( p j ) = p j γ / ( p j γ + ( 1 − p j ) γ ) 1 γ . Otherwise, (4) π − ( p j ) = p j δ / ( p j δ + ( 1 − p j ) δ ) 1 δ , where p j is probability of x j ; both γ and δ are the parameters describing the curvature of the weighting function and they express the differences of diminishing sensitivity in the domain of gains and losses.

The value function illustrates that the DMs are risk averse when gains occur; however, they are risk seeking when losses occur. Besides, since the weighting function can represent the extent of risk attitude of DMs, it is obvious that the effect of risk aversion is greater than that of risk seeking in most environments, which is consistent with previous studies.

The classical TODIM is applied to MADM through the measurement of relative dominance degree for each alternative over the others. The ranking result is presented from the comparison of the relative dominance degree of each alternative over the others and based on which the DMs will find the optimal one.

A MADM problem can be abstracted as a decision matrix X obtained from DMs, which includes all the available alternatives A = { A 1 , A 2 , … , A n } and all the attributes C = { C 1 , C 2 , … , C m }. It is described as follows: X = x 11 ⋯ x 1 m ⋮ ⋱ ⋮ x n 1 ⋯ x n m = ( x i j ) n × m , ω = ( ω 1 , ω 2 , … , ω m ) , ∑ j = 1 m ω j = 1 , where x i j is the value of the jth attribute for the alternative i from DMs; ω j is the original weight of the jth attribute.

For convenience, let N = { 1 , 2 , … , n } and M = { 1 , 2 , … , m }. The classical TODIM method involves the following steps:

Step 1. Standardize the decision matrix X = ( x i j ) n × m into G = ( g i j ) n × m , i ∈ N, j ∈ M. (5) g i j = x i j , j is benefit atrribute , − x i j , j is cost atrribute .

Step 2. Calculate the relative weight ω j r from (6): (6) ω j r = ω j / ω r , r , j ∈ M , where ω j and ω r are the original weights of the attributes C j and C r correspondingly and ω r = max ( ω j | j ∈ N ); C r is called a reference attribute.

Step 3. Determine the dominance of the alternative A i over each alternative A k ( i , k ∈ N) depending on (7): (7) ψ ( A i , A k ) = ∑ j = 1 m φ j ( A i , A k ) , ∀ ( i , k ) , where (8) φ j ( A i , A k ) = ω j k ( g i j − g k j ) / ∑ j = 1 m ω j k , if g i j > g k j , 0 , if g i j = g k j , − 1 θ ( ∑ j = 1 m ω j k ) ( g k j − g i j ) / ω j k , if g i j < g k j . {g_{kj}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{g_{ij}}={g_{kj}},\\ {} \frac{-1}{\theta }\sqrt{({\textstyle\textstyle\sum _{j=1}^{m}}{\omega _{jk}})({g_{kj}}-{g_{ij}})/{\omega _{jk}}},\hspace{1em}& \text{if}\hspace{2.5pt}{g_{ij}}<{g_{kj}}.\end{array}\right.\]]]>

The φ j ( A i , A k ) explains the contribution of the attribute C j to the function ψ ( A i , A k ) when comparing the dominance of the alternative A i to the alternative A k . The parameter θ shows the attenuation factor of the losses, which can be turned on account of the problem faced with. Three cases will be presented in (8): ① if g i j > g k j {g_{kj}}$]]>, then it states a gain; ② if g i j < g k j , then it describes a loss; ③ if g i j = g k j , then it represents a nil, that is, neither gain nor loss.

Step 4. Obtain the overall value of the alternative A i on the basis of (9): (9) Ψ ( A i ) = ∑ k = 1 n ψ ( A i , A k ) − min i { ∑ k = 1 n ψ ( A i , A k ) } max i { ∑ k = 1 n ψ ( A i , A k ) } − min i { ∑ k = 1 n ψ ( A i , A k ) } , i ∈ N .

Step 5. Rank the overall value Ψ ( A i ), i ∈ N, based on which the promising alternative is then found. The bigger of the overall value Ψ ( A i ) is, the better the alternative A i will be.

Although the classical TODIM considers relative importance of attributes, this method neither provides an appropriate way to determine the weights of attributes nor comprehensively expresses the real perceptions for gains or losses of DMs. Generally, there are two significant hurdles when the classical TODIM is applied in decision-making environment. First, the weight determination of an attribute is presented as an objective probability in the classical TODIM, which is accused of a deviation from decision-making practice by Mattos and Garcia (2011). According to their opinions, the weight of an attribute should be a transformed probability weighting function, driving from CPT to improve the efficiency of decision-making for DMs. Second, although the gains or losses instead of final states of wealth are incorporated in the classical TODIM, they are inconsistent with the perceived gains or losses in value function of the prominent CPT. The real perceptions of gains or losses are well captured by the value function of CPT. Applying the transformed weighting function and value function of the prominent CPT into the extended TODIM can not only make the method more suitable for decision-making environment but also increase the accuracy of decisions for DMs.

In this study, a deep modification of the classical TODIM is proposed, which incorporates prospect function (the product of the transformed weighting function and the value function described in Section 3.1 to respectively identify the weights of attributes and describe the different risk attitudes for gains and losses of DMs) as the relative dominance. Compared with the classical TODIM, this extended TODIM is more appropriate for DMs’ decision-making in both accurate and efficient perspectives. The construction of the extended TODIM is described step by step as follows: We suppose that there are n alternatives i.e. A = { A 1 , A 2 , … , A n }. For each alternative, there are m attributes, i.e. C = { C 1 , C 2 , … , C m }.

Step 1. Identify the decision matrix and attribute values from DMs described as follows: X = x 11 ⋯ x 1 m ⋮ ⋱ ⋮ x n 1 ⋯ x n m = ( x i j ) n × m , w = ( w 1 , w 2 , … , w m ) , ∑ j = 1 m w j = 1 .

Step 2. Work out the transformed probability of the alternative A i to A k , k ∈ M and k ≠ i according to (10) or (11).

When x i j − x k j ⩾ 0, the transformed probability weight is acquired by (10): (10) π i k j + ( w j ) = w j γ / ( w j γ + ( 1 − w j ) γ ) 1 γ .

Otherwise, the transformed probability weight is calculated from (11): (11) π i k j − ( w j ) = w j δ / ( w j δ + ( 1 − w j ) δ ) 1 δ , where γ and δ are the parameters defined in Section 3.1.

Step 3. Determine the relative weight π i k j ∗ for the alternative A i to the alternative A k from (12): (12) π i k j ∗ = π i k j ( w j ) / π i k r ( w r ) , r , j ∈ M , ∀ ( i , k ) , where π i k j ( w j ) and π i k r ( w r ) are all acquired from (10) or (11) for the alternative A i to A k depending on the value of x i j − x k j ; while π i k j ( w j ) represents the transformed weight of the jth attribute for the alternative A i ; π i k r ( w r ) refers to the transformed weight of reference attribute for the alternative A i to A k satisfying π i k r ( w r ) = max ( π i k j ( w j ) | j ∈ M ).

Step 4. Calculate the relative prospect dominance of the alternative A i over A k under the attribute j with (13): (13) φ j ∗ ( A i , A k ) = π i k j ∗ ( x i j − x k j ) α / ∑ j ∗ = 1 m π i k j ∗ , if x i j > x k j , 0 , if x i j = x k j , − λ ( ∑ j ∗ = 1 m π i k j ∗ ) ( x k j − x i j ) β / π i k j ∗ , if x i j < x k j , {x_{kj}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ij}}={x_{kj}},\\ {} -\lambda ({\textstyle\textstyle\sum _{{j^{\ast }}=1}^{m}}{\pi _{ik{j^{\ast }}}}){({x_{kj}}-{x_{ij}})^{\beta }}/{\pi _{ik{j^{\ast }}}},\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ij}}<{x_{kj}},\end{array}\right.\]]]> where α, β, and λ are the parameters defined in Section 3.1.

Step 5. Aggregate the relative prospect dominance of the alternative A i over A k under all the attributes depending on (14): (14) ψ ( A i , A k ) = ∑ j ∗ = 1 m φ j ∗ ( A i , A k ) , ∀ ( i , k ) .

Step 6. Obtain the overall prospect dominance of the alternative A i based on (9).

Step 7. Rank the overall prospect dominance Ψ ( A i ), i ∈ N, based on which the optimal alternative is then found. The bigger the overall prospect value Ψ ( A i ) is, the better the project A i will be.

According to the steps above, this extended TODIM includes the transformed probability weighting function and the proper value function in CPT, which is more consistent with reality theoretically. Then, an example is shown in the next section to demonstrate the practical effectiveness of the proposed method.

As the overexploitation of natural resources by humans and the enhanced awareness of sustainable development grows, new energy has attracted a lot of attention of both governments and customers. For instance, the governments subsidize the manufacturers of new automobile energy with reduction of rates and encourage customers to buy them with price support. Thus, the industry of new energy has great prospect and has already attracted many investors, including the Fortune Capital. After preliminary investigation, four VC projects (thermal power A 1 , wind power generation A 2 , hydroelectric power A 3 , solar photovoltaics A 4 ) remain to be further investigated. First, we draw on previous research (Nunes et al., 2014; Dhochak and Sharma, 2015; Widyanto and Dalimunthe, 2015) to find out an appropriate evaluation attributes’ system used by VCs in the selection process as Table 1 shows.

According to Widyanto and Dalimunthe (2015), the weights of attributes are calculated as: ω C = ( ω C 1 , ω C 2 , … , ω C 12 ) = ( 0.098 , 0.098 , 0.092 , 0.087 , 0.085 , 0.0760.080.077 , 0.0740.0750.0740.086 ).2²

This reference introduced investigated data about attributes used by VCs in numerous countries and we comprehensively aggregated those data as the weights of attributes in this paper. We believe that those comprehensive weights are reasonable.

Step 1. We have invited some senior investors to investigate the prospect of the remaining four projects under each attribute. After deliberate thinking and discussion, they have given the consistent evaluating information. It can be seen in Table 2.3³

The VCs give each attribute of each alternative a value. Furthermore, the value ranges from 0 to 100. For the benefit attribute, the higher the value is, the better the alternative will be. It is contrary for the cost attribute.

At this point, Step 1 has already been finished. Next, we take the alternative A 1 for an example to calculate its overall prospect dominance.

Step 2. The transformed probability weight π 1 k j is calculated according to (10) or (11), which depends on the relative value of the alternative A 1 to the others under all attributes and it can be seen in Table 3.

Step 3. From the transformed probability weight obtained in Step 2, the relative weight π 1 k j ∗ of the alternative A 1 to the others under each attribute is worked out according to (12). It is shown in Table 4.

Step 4. The relative prospect dominance of the alternative A 1 over the others for each attribute will be determined according to (13) and the result is presented in Table 5.

Step 5. The relative prospect dominance of the alternative A 1 over the others based on (14) is acquired and shown in Table 6.

Step 6. The overall prospect dominance of each alternative will be calculated by repeating Steps 2 to 5 and (9). The results are shown in Table 7.

Step 7. It is known that Ψ ( A 4 ) > Ψ ( A 2 ) > Ψ ( A 3 ) > Ψ ( A 1 )\Psi ({A_{2}})>\Psi ({A_{3}})>\Psi ({A_{1}})$]]>, so A 4 ≻ A 2 ≻ A 3 ≻ A 1 . The alternative A 4 is recognized as the best option among the four alternatives, whereas A 1 is regarded as the worst one.

The results above rely on the degrees of risk attitudes of VCs, that is to say, the results depend on the values of the parameters α, β, λ, γ and δ, but the great difference between the proposed method and the classical TODIM lies in the prospect function which is the product of disparate weighting function and the value function.

In this section, the classical TODIM (Gomes and Lima, 1991) is processed for the sake of comparing it with the extended one. In order to compare those two methods more conveniently, the decision-making information in Table 1 and Table 2 is adopted here as well. Then, the overall dominance of each alternative is calculated depending on the steps in Section 3.2.

Step 1. According to the attribute in Table 1 and the decision-making matrix in Table 2, it is known that there is X = ( x i j ) n × m = G = ( g i j ) n × m due to the fact that all the attributes are the benefit ones. The attribute weights are also calculated via Widyanto and Dalimunthe (2015) as shown in Section 4.1.

Step 2. The relative weight of each attribute is obtained from (6) and it is shown in Table 8.

Step 3. The dominance of each alternative A i over each alternative A k ( i , k ∈ N) shown in Table 9 depends on (7) and (8).

Step 4. The overall value of each alternative is shown in Table 10 based on (9).

Step 5. From Table 10, it is known that Ψ ( A 4 ) > Ψ ( A 2 ) > Ψ ( A 3 ) > Ψ ( A 1 )\Psi ({A_{2}})>\Psi ({A_{3}})>\Psi ({A_{1}})$]]>, that is to say, A 4 ≻ A 2 ≻ A 3 ≻ A 1 .

The ranking result explains that the alternative A 4 is the best choice and A 2 is the suboptimal one, however, A 1 is the worst option.

From the results of Section 4.1 and Section 4.2, it is known that the ranking results from the two methods are the same. The alternatives A 4 and A 1 are recognized as the promising project and the unworthy one respectively in both the extended TODIM and the classical TODIM. The overall dominance derived from the two methods is shown in Table 11, and then, a comparative analysis between the extended TODIM and the classical TODIM is provided in this section.

Although the ranking results of the four projects from the two methods are consistent, the great difference between the proposed method and the classical TODIM lies in the disparate weighting function and value function. From Table 11, it is easy to see that the overall dominance of them is different between the two methods as well. The main reason for such a difference ranking is: the evaluating information of the extended TODIM is presented as prospect values which are the product of value functions (nonlinear gains or losses) and the transformed weights of attributes for VC projects, whereas, the evaluating information of the classical TODIM comes from the product of linear gains or losses and the objective probability that could not reflect the psychological perception of VCs for projects. In theoretical terms, the extended TODIM accompanied with value function and transformed weighting function confirms that the real decision-making situation is more reasonable as the aid for investors. In practical terms, the investors admit such a weighting function in our interview as well. Also, the extended TODIM increases the difference between the alternatives. For instance, the difference of overall dominance between A 2 and A 3 with extended TODIM is larger than with the classical one. This is very useful, especially for the choice among the similar alternatives. To sum up, the extended TODIM is feasible and suitable for investors to make their decisions.