Incremental analysis for generalized TODIM

Abstract

This study aims to generalize TODIM to eliminate two types of scaling effects by incremental analysis. One effect is due to an inappropriate presentation of the losses part in S-shaped value function in traditional TODIM, in which the criteria with smaller weights will contribute larger dominance values in its computing process. The other effect is derived from aggregating partial dominance measurements among heterogeneous criteria in traditional TODIM (a Portuguese acronym of Interactive and Multi-criteria Decision Making). The proposed approach firstly divides the criteria into two categories, benefits and costs, each of which is manipulated separately so as to facilitate incremental analysis. For each category, the pairwise dominance of one alternative is accumulated and normalized to represent the global dominance of each alternative. The global dominance measurements of alternatives in both categories are then combined according to benefit-cost ratio in order to rank alternatives. In addition, sensitivity analyses on the parameters of the gains/losses value function and the cutoff benefit-cost ratio demonstrate the robustness of the proposed model. An example of the selection of fuel buses is illustrated, and shows that our model is feasible and effective for MCDM problems with risk preference.

Appendices

Appendix 1: TODIM procedure with prospect theory

The TODIM method was proposed by Gomes and Lima (1992) to construct a preference model for risky decisions based on prospect theory (Kahneman and Tversky 1979). The basic principle of TODIM relies on a global multi-criteria value function. This function is built in parts, with their mathematical descriptions reproducing the gains/losses function of prospect theory. The global multi-criteria value function of TODIM then aggregates all measures of gains and losses over all criteria. The global measurement is employed for rank the alternatives. The larger the global measurement is, the better the alternative is.

Fig. 3

S-shaped value function of prospect theory (Kahneman and Tversky 1979)

In TODIM, the gains/losses function is quite similar to a nonlinear S-shaped value function in prospect theory (Fig. 3), which is also a specific form of the two-part power function (Kahneman and Tversky 1979)

$$\begin{aligned} v\left( x \right) =\left\{ \begin{array}{ll} x^{\alpha },&{}\quad \hbox {if}\,x\ge 0 \\ \left( {-\lambda } \right) \left( {-x} \right) ^{\beta },&{}\quad \hbox {if}\,x<0 \\ \end{array} \right. \end{aligned}$$

(6)

where $\alpha$ and $\beta$ are parameters related to the curvature of the value function for gains and losses, respectively. If $\alpha$, $\beta < 1$, the S-shaped value function is concave in the first quadrant (representing risk-aversion over gain) and convex in the third quadrant (representing risk-seeking over loss). Parameter $\lambda (= 1/\theta$, and $\theta = 1$ in TODIM) is the loss-aversion coefficient; when $\lambda > 1$, the parameter is more sensitive to losses than gains.

TODIM procedure involves the following four steps (Gomes and Lima 1992):

Step 1:

From the $m\times n$ normalized decision matrix and criteria weights, compute values of $\emptyset _c \left( {A_i,A_j } \right)$ according to Eq. (8);

Step 2:

Compute values of $\delta \left( {A_i,A_j } \right)$ by using Eq. (7);

Step 3:

Compute values of $\xi _i$ with Eq. (9);

Step 4:

Rank alternatives according to those values computed by (9).

$$\begin{aligned} \delta \left( {A_i,A_j } \right)= & {} \sum \limits _{c=1}^n \emptyset _c \left( {A_i,A_j } \right) \qquad i,j=1,\ldots ,m \end{aligned}$$

(7)

$$\begin{aligned} \emptyset _c \left( {A_i,A_j } \right)= & {} \left\{ \begin{array}{ll} \sqrt{\frac{w_{fc} \left( {r_{ic} -r_{jc} } \right) }{\sum \nolimits _{c=1}^n w_{fc} }}&{}\quad \textit{if}\left( {r_{ic} -r_{jc} } \right) >0 \\ 0&{}\quad \textit{if}\left( {r_{ic} -r_{jc} } \right) =0 \\ \frac{-1}{\theta }\sqrt{\frac{\left( {\sum \nolimits _{c=1}^n w_{fc} } \right) \left( {r_{jc} -r_{ic} } \right) }{w_{fc} }}&{}\quad \textit{if}\left( {r_{ic} -r_{jc} } \right) <0 \\ \end{array} \right. \end{aligned}$$

(8)

$$\begin{aligned} \xi _i= & {} \frac{\sum \nolimits _{j=1}^m \delta \left( {A_i,A_j } \right) -\min \sum \nolimits _{j=1}^m \delta \left( {A_i,A_j } \right) }{\max \sum \nolimits _{j=1}^m \delta \left( {A_i,A_j } \right) -\min \sum \nolimits _{j=1}^m \delta \left( {A_i,A_j } \right) }\qquad i=1,\ldots ,m\qquad \quad \end{aligned}$$

(9)

where:

• $\emptyset _\mathrm{c} \left( {A_i,A_j } \right) =$ dominance measurement of each alternative $A_{i}$ over each alternative $A_{j}$ with respect to criterion c;

• $\delta \left( {A_i,A_j } \right) =$ dominance measurement of an alternative $A_{i}$ in relation to another alternative $A_{j}$;

• $r_{ic}, r_{jc} =$ valuations of alternatives $A_{i}$ and $A_{j}$ with respect to criterion c;

• $w_{fc} =$ the relative weight of criterion c in relation to criterion f. The subscript f identifies a reference criterion for the decision maker. Usually, the criterion with the highest weight is chosen as the reference criterion;

• $\theta =$ attenuation factor of the losses; different choices of $\theta$ lead to different shapes of the prospect theoretical value function in the negative quadrant;

• $\xi _i =$ normalized global performance of alternative $A_{i}$, when compared against all other alternatives.

Appendix 2: The procedure of incremental analysis

In engineering economy, incremental analysis (IA) or marginal analysis is used to examine differences between alternatives from the perspective of benefits and costs (Blank and Tarquin 1989). Newnan et al. (2002) rearranged alternatives in ascending order according to cost, enabling DMs to determine whether marginal or differential costs are adequately compensated by their marginal benefits. After the ratio of marginal benefits to marginal costs is determined, the ranking of alternatives is used to identify the optimal solution.

The IA procedure consists of the following three steps:

• Step 1: Calculate the differences in benefits, $\Delta B_i$, $i = 1,\ldots ,m$, and differences in costs, $\Delta C_i$, of two alternatives with the smallest cost index and next smallest cost index.

This step compares the differences between alternatives from the viewpoint of benefits and costs. If the ratio of differences in benefit and cost, $\Delta B_i /\Delta C_i$, exceeds 1, then the latter is kept; otherwise, the former is preserved. This process enables the DM to decide whether differential costs are adequately compensated by differential benefits (Newnan et al. 2002).

• Step 2: Calculate the differences in benefits, $\Delta B_i$, and differences in costs, $\Delta C_i$, from Step 1 as well as the alternative with the next smallest cost index.

This step examines the differences between the alternative with the next smallest cost index and the alternative from Step 1. If the ratio of differences, $\Delta B_i /\Delta C_i$, is greater than the cutoff benefit-cost ratio, then the latter is kept; otherwise, the former is reserved. This step is repeated with the remaining alternative and the alternative with the next smallest cost index until all alternatives have been compared.

• Step 3: The remaining alternative in Step 2 is the optimal choice.