# MRP-PCI: A Multiple Reference Point Based Partially Compensatory Composite Indicator for Sustainability Assessment

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## Abstract

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## 1. Introduction

- Normalization. In most of the cases, the individual indicators are measured in different scales and therefore, it is necessary to bring them all down to a common scale before building the composite indicator.
- Weighting. We believe that the construction of composite indicators is essentially subjective. The weights indicate the contribution that, according to the decision center, each single indicator should make to the composite measure. In this sense, it is important to point out that even methods regarded as fully objective (for example, some purely statistical schemes) do imply more or less hidden assumptions about the weights of the indicators, which can be as arguable as other more explicit schemes.
- Aggregation. Finally, once normalized and weighted, the individual indicators are aggregated to get the final composite indicator.

## 2. Literature Review

## 3. Methodology: The MRP-PCI Partially Compensatory Indicator

#### 3.1. Step-Wise Description of the MRP-PCI Method

- Initial information: it will be assumed that we are managing a set of J units, for which I indicators are evaluated, which, without loss of generality, will all be assumed to be of type “the more, the better”. Let us denote by ${x}_{ij}$ the value of indicator i for unit j. It will also be assumed that the decision maker has somehow assigned weights ${\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{I}$ to the indicators, which reflect the contribution of each indicator to the final composite measure.It is also assumed that the decision maker can give, for each indicator i, n reference levels, ${q}_{i}^{1},{q}_{i}^{2},\cdots ,{q}_{i}^{n}$, which somehow define the performance levels of indicator i (e.g., very poor, poor, fair, good, very good,...). Let us denote by ${q}_{i}^{0}$ and ${q}_{i}^{n+1}$, respectively, the minimum and maximum values that indicator i can take. Therefore, we obtain the following $(n+2)$-dimensional reference vector for indicator i:$${\mathbf{q}}_{i}=({q}_{i}^{0},{q}_{i}^{1},\cdots ,{q}_{i}^{n},{q}_{i}^{n+1}).$$We will also assume that a set of $n+2$ real values ${\alpha}^{0},{\alpha}^{1},\cdots ,{\alpha}^{n},{\alpha}^{n+1}$ is available (either provided by the decision maker or set to default values by the analysts), which define the common measurement scale. Note that these values are the same for all the I indicators. Therefore, each ${\alpha}^{t}$ ($t=0,\cdots ,n+1$) is the value in the common scale that a given unit has if it achieves value ${q}_{i}^{t}$ in indicator i. In order to turn each indicator i to the scale defined by the values ${\alpha}^{t}$ ($t=0,\cdots ,n+1$), a so-called achievement function is used, which, apart from allowing the normalisation of the indicators, also informs about the relative position of each unit with respect to the reference levels given in the previous step, for the corresponding indicator. These functions were originally defined in [29] for general reference point procedures (with one reference level), and they were afterwards extended to double reference point methods [30], and adapted to the calculation of composite indicators [28,34,35]. This achievement function is generalized to the case when n reference levels are used in the following way:$${s}_{ij}={s}_{i}({x}_{ij},{\mathbf{q}}_{i})={\alpha}^{t-1}+\frac{{\alpha}^{t}-{\alpha}^{t-1}}{{q}_{i}^{t}-{q}_{i}^{t-1}}({x}_{ij}-{q}_{i}^{t-1}),\phantom{\rule{2.em}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}{x}_{ij}\in [{q}_{i}^{t-1},{q}_{i}^{t}],\phantom{\rule{1.em}{0ex}}(t=1,\cdots ,n+1).$$Therefore, the achievement function ${s}_{i}$ of indicator i is a piece-wise linear function that takes values between ${\alpha}^{t-1}$ and ${\alpha}^{t}$ if the unit achieves values between ${q}_{i}^{t-1}$ and ${q}_{i}^{t}$ for indicator i (see Figure 2). It must be noted that the distance-based normalization proposed for the MRP-WSCI methodology is more general than other schemes, meaning that others can be obtained as particular cases of this one. For example the well known range normalization is obtained for $n=0$, ${\alpha}^{0}=0$, ${\alpha}^{1}=1$.All the previously described basic initial elements are shared with the MRP-WSCI methodology, which is described in Appendix A.
- Compensation indices: the main feature of this partially compensatory indicator is allowing the decision maker to provide a compensation index ${\lambda}_{i}$ for each indicator. ${\lambda}_{i}$ is understood as a coefficient, between 0 and 1, which indicates to what extent can a bad value of indicator i be compensated by better values of other indicators.
- Fully compensated values: given all the previous data, let us build now the fully compensated value of each indicator i for each unit j, ${a}_{ij}$, which is the weighted average of ${s}_{ij}$ and the rest of achievement function values that are at least as good as ${s}_{ij}$. As a result, the value of ${s}_{ij}$ is fully compensated by all the better (or equal) values of the rest of indicators:$${a}_{ij}=\frac{{\sum}_{k\in {I}_{ij}}{\mu}_{k}{s}_{kj}}{{\sum}_{k\in {I}_{ij}}{\mu}_{k}},$$$${I}_{ij}=\left\{k\in \{1,\cdots ,I\}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{s}_{kj}\ge {s}_{ij}\right\}.$$
- Partially compensated achievement scalarizing functions: next, we combine the compensation indices given in Step 2 with the fully compensated values of Step 3, to obtain the partially compensated achievement scalarizing functions, ${s}_{ij}^{c}$. These functions measure the partially compensated value of each indicator i for each unit j, taking the compensation indices into account:$${s}_{ij}^{c}={s}_{ij}+({a}_{ij}-{s}_{ij}){\lambda}_{i}.$$${s}_{ij}^{c}$ is a value lying between the original achievement scalarizing function, ${s}_{ij}$, and the fully compensated value, ${a}_{ij}$, of indicator i for unit j, according to the corresponding compensation index of indicator i, ${\lambda}_{i}$. Therefore, if ${\lambda}_{i}=0$, then ${s}_{ij}^{c}$ takes the value of the achievement function ${s}_{ij}$. On the other hand, if ${\lambda}_{i}=1$, then ${s}_{ij}^{c}$ takes the value of the fully compensated value of indicator i, ${a}_{ij}$.
- Partially compensatory composite indicator: finally, the partially compensatory indicator of unit j is defined as the worse (minimum) of the partially compensated achievement scalarizing functions of unit j as follows:$$PC{I}_{j}=\underset{i=1,\cdots ,I}{min}\left\{{s}_{ij}^{c}\right\}.$$
- Successive aggregations: in many real cases, the system of indicators is organized in several levels. For example, in the environmental sustainability assessment reported in Section 4, the indicators are classified in families. Therefore, the process has two stages. First, the indicators of each family are aggregated, and we obtain the partially compensatory composite indicator for each of them. Given the way they have been constructed, $PC{I}_{j}$ take values in the same scale $[{\alpha}^{0},{\alpha}^{n+1}]$ as the original achievement functions and therefore, they can be used as achievement functions in a multi-stage aggregation process. Thus, in a second stage, we use these composite indicators as achievement functions to build the global composite indicators and so on. This is important, because it may make sense to provide different compensation indices depending on the aggregation stage. For example, when assessing sustainability, indicators of the same family may be compensated, but not the families among them.

#### 3.2. Theoretical Properties of the MRP-PCI Indicators

- ${a}_{1j}$ coincides with the weak composite indicator $WC{I}_{j}$, defined in (A2):$${a}_{1j}=\frac{{\sum}_{k=1}^{I}{\mu}_{k}{s}_{kj}}{{\sum}_{k=1}^{I}{\mu}_{k}}=\sum _{k=1}^{I}\left(\frac{{\mu}_{k}}{{\sum}_{v=1}^{I}{\mu}_{v}}\right){s}_{kj}=WC{I}_{j}.$$
- The fully compensated values satisfy:$${a}_{1j}\le {a}_{2j}\le \cdots \le {a}_{Ij}.$$
- For all $i=1,\cdots ,I$, it holds that ${s}_{ij}\le {a}_{ij}$ and, therefore:$${s}_{ij}\le {s}_{ij}^{c}={\lambda}_{i}{a}_{ij}+(1-{\lambda}_{i}){s}_{ij}\le {a}_{ij}.$$

- $PC{I}_{j}\le WC{I}_{j}$ and, if ${\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{I}=1$, then $PC{I}_{j}=WC{I}_{j}$:$$\begin{array}{c}PC{I}_{j}=\underset{i=1,\cdots ,I}{min}\left\{{s}_{ij}^{c}\right\}\le \underset{i=1,\cdots ,I}{min}\left\{{a}_{ij}\right\}={a}_{1j}=WC{I}_{j};\\ \mathrm{if}\phantom{\rule{4.pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{I}=1,\phantom{\rule{1.em}{0ex}}{s}_{ij}^{c}={a}_{ij}\phantom{\rule{1.em}{0ex}}\Rightarrow \phantom{\rule{1.em}{0ex}}PC{I}_{j}=\underset{i=1,\cdots ,I}{min}\left\{{a}_{ij}\right\}={a}_{1j}=WC{I}_{j}.\end{array}$$
- $PC{I}_{j}\ge SC{I}_{j}$ and, if ${\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{I}=0$, then $PC{I}_{j}=SC{I}_{j}$:$$\begin{array}{c}PC{I}_{j}=\underset{i=1,\cdots ,I}{min}\left\{{s}_{ij}^{c}\right\}\ge \underset{i=1,\cdots ,I}{min}\left\{{s}_{ij}\right\}=SC{I}_{j};\\ \mathrm{if}\phantom{\rule{4.pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{I}=0,\phantom{\rule{1.em}{0ex}}{s}_{ij}^{c}={s}_{ij}\phantom{\rule{1.em}{0ex}}\Rightarrow \phantom{\rule{1.em}{0ex}}PC{I}_{j}=\underset{i=1,\cdots ,I}{min}\left\{{s}_{ij}\right\}=SC{I}_{j}.\end{array}$$
- If ${\lambda}_{1}={\lambda}_{2}=\cdots ={\lambda}_{I}=\lambda $, then $PC{I}_{j}=MC{I}_{j}\left(\lambda \right)$:

#### 3.3. A Hypothetical Example

- ${\lambda}_{i}=0$, no compensation: a bad value of indicator i cannot be compensated in any way.
- ${\lambda}_{i}=1$, full compensation: a bad value of indicator i can be completely compensated by better values in other indicators.
- ${\lambda}_{i}=0.5$, middle compensation: a bad value of indicator i can be partially (half) compensated by better values in other indicators.

## 4. Example: Applying the MRP-PCI Method to an Environmental Sustainability Assessment Case

#### 4.1. Aggregation 1

- For three provinces (Córdoba, Granada and Jaén), the partially compensatory indicator coincided with the strong composite indicator. This is because in the three cases, the worst performance of the family took place in the Air quality (I1) indicator, which cannot be compensated at all.
- In four provinces (Cádiz, Huelva, Málaga and Sevilla), the $PCI$ had an intermediate value between $SCI$ and $WCI$, not close to any of them. In this case, the worst performance took place in the Emissions of NO${}_{x}$ (I2) indicator, which was partially compensable.
- Finally, for Almería, the $PCI$ was much closer to the $WCI$, because its worst performance took place in the Emissions of SO${}_{2}$ (I3), which was fully compensable.

#### 4.2. Aggregation 2

#### 4.2.1. The MRP-PCI Composite Indicator

#### 4.2.2. MRP-WSCI Composite Indicators

#### 4.2.3. Comparison of the Results

- Let us take a look at Figure 7, where the values of the $PCI$, $W-W$ and $S-W$ indicators (left) and the rankings provided by them (right) are compared. As can be seen, the $PCI$ got an intermediate position between the fully compensatory and non-compensatory indicators, thus showing a less extreme behavior. In [4], it is argued that the joint use of the weak and strong indicators provides useful additional information. However, if a single composite indicator needs to be used, for example for ranking purposes, then the $PCI$ can be a more balanced option.
- If we look at the variation ranges (between the minimum and the maximum values) of the composite indicators in Figure 5 and Figure 6, we can see that, for this example, the variation of the $PCI$ (1.1) was greater than these of the rest ($W-W$: 0.53; $S-W$: 0.91). Therefore, it seems that the $PCI$ hasd a greater dispersion than the others, which allowed a better distinction among the performances of the different provinces.

## 5. Discussion: Interesting Features of the MRP-PCI Method

## 6. Conclusions

- The method allows the possibility to provide a different compensation index for each indicator, or for each of the families into which the indicators are grouped in successive aggregations. To the best of our knowledge, this is the first composite indicator method that allows this possibility. It is sensible to think that the decision center may regard different indicators as differently compensable, and if there are several aggregation levels, the compensations may not be the same at each level. Therefore, this formulation offers a high modelling flexibility, because the decision maker can decide which indicators can be compensated and to which extent. Given that sustainability indicators are usually classified into different areas (economic, social, environmental), and the compensation among them is a critical issue when assessing the global sustainability of a territory, the MRP-PCI method is specially suitable in this field.
- An aggregation method has been designed that takes all these compensation indices into account, to build a partially compensatory composite indicator. Using an illustrative example about the environmental sustainability assessment, we have shown how this approach works when different compensation indices are established in a problem with two aggregation levels. As seen, the results successfully reflect these compensation indices in an intuitive and easy-to-interpret way.
- The MRP-PCI method is based on the multiple reference point, MRP-WSCI, scheme. Therefore, it is assumed that the decision maker can provide reference levels for each indicator, and an achievement function is used to measure the position of each unit with respect to these levels. Nevertheless, the aggregation process followed to build the partially compensatory indicator is different and more general. In fact, it has been seen that the original weak and strong indicators built using the MRP-WSCI methodology can be obtained as particular cases of the partially compensatory composite indicator.
- A series of computational experiments have been carried out to compare the new MRP-PCI composite indicator with the MRP-WSCI weak and strong indicators. As a result, the two following interesting findings have been derived, that make the partially compensatory indicator suited for ranking purposes, when full compensation is not allowed, which is usually the case in sustainability assessment problems:
- -
- The MRP-PCI method tends to produce results with an intermediate position between the weak and the strong composite indicators, thus showing a less extreme behavior.
- -
- The MRP-PCI method very frequently has a greater dispersion than the others, which allows a better distinction among the performances of the different units.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MCI | Mixed composite indicator |

MRP-PCI | Multiple reference point partially compensatory indicator |

MRP-WSCI | Multiple reference point weak and strong composite indicators |

PCI | Partially compensatory composite indicator |

SCI | Strong composite indicator |

S-S | Strong-strong global composite indicator |

S-W | Strong-weak global composite indicator |

WCI | Weak composite indicator |

W-W | Weak-weak global composite indicator |

${a}_{ij}$ | Fully compensated value of of indicator i for unit j |

${\alpha}^{t}$ | t-th value of the common measurement scale |

I | Number of indicators (i is the corresponding index) |

${I}_{ij}$ | Subset of indicators with a value better or equal to indicator i for unit j |

J | Number of units (j is the corresponding index) |

${\lambda}_{i}$ | Compensation index assigned to indicator i |

${\mu}_{i}$ | Weight assigned to indicator i |

n | Number of reference levels (t is the corresponding index) |

$\overline{{\mu}_{i}}$ | Normalized weight of indicator i |

$PC{I}_{j}$ | Partially compensatory composite indicator of unit j |

${q}_{i}^{t}$ | t-th reference level of indicator i |

${s}_{ij}$ | Value of the scalarizing achievement function of indicator i for unit j |

${s}_{ij}^{c}$ | Partially compensated achievement scalarizing function of indicator i for unit j |

${x}_{ij}$ | Value of indicator i for unit j |

## Appendix A. The MRP-WSCI Composite Indicators

- The weights ${\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{I}$ for the indicators.
- The reference vector ${\mathbf{q}}_{i}=({q}_{i}^{0},{q}_{i}^{1},\cdots ,{q}_{i}^{n},{q}_{i}^{n+1})$, expressing performance levels of the indicators.
- The common measurement scale ${\alpha}^{0},{\alpha}^{1},\cdots ,{\alpha}^{n},{\alpha}^{n+1}$.
- The corresponding achievement scalarizing functions, as defined in (2).

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**Figure 1.**Steps of the Multiple Reference Point Partially Compensatory Indicator (MRP-PCI) method. Source: own elaboration.

**Figure 2.**Graphical representation of the achievement function. Source: [4].

**Figure 3.**Data of the illustrative example. Source: [33].

**Figure 9.**Times with the best min–max (

**left**) and interquartile (

**right**) range. Source: own elaboration.

Indicator | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ | ${\mathit{I}}_{3}$ | Composite Ind. | |||
---|---|---|---|---|---|---|---|

Comp. Index | 1 | 0.5 | 0 | $\mathit{P}\mathit{C}\mathit{I}$ | $\mathit{W}\mathit{C}\mathit{I}$ | $\mathit{S}\mathit{C}\mathit{I}$ | $\mathit{M}\mathit{C}\mathit{I}\left(0.5\right)$ |

${U}_{1}$ | 0.45 | 1.24 | 2.89 | 1.53 | 1.53 | 0.45 | 0.99 |

${U}_{2}$ | 2.62 | 0.14 | 2.87 | 1.01 | 1.88 | 0.14 | 1.01 |

${U}_{3}$ | 1.68 | 1.97 | 0.57 | 0.57 | 1.41 | 0.57 | 0.99 |

${U}_{4}$ | 0.45 | 2.96 | 1.23 | 1.23 | 1.34 | 0.45 | 1.90 |

${\mathit{a}}_{\mathit{i}\mathit{j}}$ | ${\mathit{s}}_{\mathit{i}\mathit{j}}^{\mathit{c}}$ | |||||
---|---|---|---|---|---|---|

${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | |

${U}_{1}$ | 1.53 | 2.07 | 2.89 | 1.53 | 1.65 | 2.89 |

${U}_{2}$ | 2.75 | 1.88 | 2.87 | 2.75 | 1.01 | 2.87 |

${U}_{3}$ | 1.83 | 1.97 | 1.41 | 1.83 | 1.97 | 0.57 |

${U}_{4}$ | 1.34 | 2.96 | 2.10 | 1.34 | 2.96 | 1.23 |

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Ruiz, F.; Cabello, J.M.
MRP-PCI: A Multiple Reference Point Based Partially Compensatory Composite Indicator for Sustainability Assessment. *Sustainability* **2021**, *13*, 1261.
https://doi.org/10.3390/su13031261

**AMA Style**

Ruiz F, Cabello JM.
MRP-PCI: A Multiple Reference Point Based Partially Compensatory Composite Indicator for Sustainability Assessment. *Sustainability*. 2021; 13(3):1261.
https://doi.org/10.3390/su13031261

**Chicago/Turabian Style**

Ruiz, Francisco, and José Manuel Cabello.
2021. "MRP-PCI: A Multiple Reference Point Based Partially Compensatory Composite Indicator for Sustainability Assessment" *Sustainability* 13, no. 3: 1261.
https://doi.org/10.3390/su13031261