# Managing Interacting Criteria: Application to Environmental Evaluation Practices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Evaluating Corporate Environmental Practices

#### 2.2. Evaluation Methodologies

## 3. A Multi-Criteria Decision Integral Model for Evaluating Environmental Practices

#### 3.1. Evaluation Framework

#### 3.1.1. Criteria Selection

- Management performance criteria:$${C}^{M}={C}_{1}^{M}+{C}_{2}^{M},\phantom{\rule{1.em}{0ex}}{C}^{M}=\{{c}_{k}^{M}\mid {c}_{k}^{M}\in {C}_{1}^{M}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{or}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{M}\in {C}_{2}^{M}\},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k=1,\dots ,p.$$
- Operational performance criteria:$${C}^{O}={C}_{1}^{O}+{C}_{2}^{O},\phantom{\rule{1.em}{0ex}}{C}^{O}=\{{c}_{k}^{O}\mid {c}_{k}^{O}\in {C}_{1}^{O}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{or}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{O}\in {C}_{2}^{O}\},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k=1,\dots ,q.$$

#### 3.1.2. Reviewers’ Selection

- A set of internal reviewers:
- -
- A set of company’s internal experts: ${A}^{E}=\{{a}_{1}^{E},\dots ,{a}_{m}^{E}\}.$
- -
- A set of company’s internal non-expert (such as managers, staff, employees, etc.): ${A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}=\{{a}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E},\dots ,{a}_{r}^{N\phantom{\rule{-1.42262pt}{0ex}}E}\}.$

- A set of external reviewers:
- -
- A set of company’s external experts such as auditors: ${B}^{E}=\{{b}_{1}^{E},\dots ,{b}_{s}^{E}\}.$
- -
- A set of company’s external non-experts evaluators, ${B}^{NE}$, which is split in two types depending on their relation to the company:
- A set of general stakeholders (shareholders, suppliers, government regulators, local communities, intermediate customers, large retailers, final consumers): ${B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}=\{{b}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G},\dots ,{b}_{t}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\}.$
- A set of social constituents (community groups, trade associations, labor unions, environmental groups): ${B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}=\{{b}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S},\dots ,{b}_{u}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}\}.$

Therefore, ${B}^{NE}={B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\bigcup {B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}.$

#### 3.2. Gathering Information

- Let ${a}_{i,j,k}^{E}$ and ${a}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E}$ be the internal reviewers’ evaluations, experts and non-experts respectively, on the facility site ${x}_{i}$ by the j-th reviewer regarding the criterion ${c}_{k}^{-}$. Abusing notation, on occasions we refer to criterion k as ${c}_{k}^{-}$, where superscript denotes M or O, the criterion type.$$\begin{array}{ll}{a}_{i,j,k}^{E}\in \left\{\begin{array}{c}{\mathbb{R}}^{+},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{1}^{-},\hfill \\ {S}_{{A}^{E}}^{k},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{2}^{-}.\hfill \end{array}\right.& \phantom{\rule{3cm}{0ex}}{a}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E}\in \left\{\begin{array}{c}{\mathbb{R}}^{+},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{1}^{-},\hfill \\ {S}_{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}^{k},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{2}^{-}.\hfill \end{array}\right.\end{array}$$
- In the same way, let ${b}_{i,j,k}^{E}\phantom{\rule{0.166667em}{0ex}}$, ${b}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\phantom{\rule{0.166667em}{0ex}}$, and ${b}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}$ be the external reviewers’ evaluations, experts and non-experts from stakeholders and social constituencies, respectively, on the facility site ${x}_{i}$ by the j-th reviewer with regard to the criterion ${c}_{k}^{-}$.$$\begin{array}{lll}{b}_{i,j,k}^{E}\in \left\{\begin{array}{c}{\mathbb{R}}^{+}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{1}^{-},\hfill \\ {S}_{{B}^{E}}^{k}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{2}^{-}\hfill \end{array}\right.& {b}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\in \left\{\begin{array}{c}{\mathbb{R}}^{+}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{1}^{-},\hfill \\ {S}_{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}}^{k}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{2}^{-}\hfill \end{array}\right.& {b}_{i,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}\in \left\{\begin{array}{c}{\mathbb{R}}^{+}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{1}^{-},\hfill \\ {S}_{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}}^{k}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{k}^{-}\in {C}_{2}^{-}\hfill \end{array}\right.\end{array}$$

#### 3.3. Rating Process

#### 3.3.1. Normalization Phase

**Definition**

**1.**

#### 3.3.2. Aggregation Phase

- Computing EPIs for each reviewers’ collective and each criterion.Since we assume that the reviewers give their evaluations individually, we propose to use the 2-tuple OWA operator from [46]. We reproduce its definition below.
**Definition****2.**Let $\phantom{\rule{0.166667em}{0ex}}\left(({l}_{1},{\alpha}_{1}),\dots ,({l}_{m},{\alpha}_{m})\right)\in {\langle \overline{S}\rangle}^{m}\phantom{\rule{0.166667em}{0ex}}$ be a vector of linguistic 2-tuples and $\phantom{\rule{0.166667em}{0ex}}W=({w}_{1},\dots ,{w}_{m})\in {[0,1]}^{m}\phantom{\rule{0.166667em}{0ex}}$ be a weighting vector such that ${\sum}_{i=1}^{m}{w}_{i}=1$. The 2-tuple OWA operator associated with $\phantom{\rule{0.166667em}{0ex}}w\phantom{\rule{0.166667em}{0ex}}$ is the function $\phantom{\rule{0.166667em}{0ex}}{G}^{\mathbf{w}}:{\langle \overline{S}\rangle}^{m}\u27f6\langle \overline{S}\rangle \phantom{\rule{0.166667em}{0ex}}$ defined by$${G}^{\mathbf{w}}\left(({l}_{1},{\alpha}_{1}),\dots ,({l}_{m},{\alpha}_{m})\right)={\Delta}_{\overline{S}}\left(\sum _{i=1}^{m}{w}_{i}\phantom{\rule{0.166667em}{0ex}}{\beta}_{i}^{*}\right),$$In order to apply this operator, the weighting vector can be computed using the well-known non-decreasing quantifiers proposed by Yager (see [49]). It is important to note that each concrete aggregation procedure with OWA operators can use a different quantifier, in other words, a different weighting vector. This adds flexibility to the model.The reviewers’ assessments are aggregated for each criterion and each collective (see Figure 3) by means of a 2-tuple OWA operator, $\phantom{\rule{0.166667em}{0ex}}{G}_{-}^{-}$. Then, for each collective and for every criterion ${c}_{k}^{-}$, the process is conducted as follows:- -
- For internal reviewers (experts and non-experts, respectively):$$\begin{array}{ccc}\hfill {I}_{k}^{{A}^{E}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{A}^{E}}}\phantom{\rule{0.166667em}{0ex}}({\tilde{a}}_{1,j,k}^{E}\phantom{\rule{0.166667em}{0ex}},\dots ,{\tilde{a}}_{m,j,k}^{E}\phantom{\rule{0.166667em}{0ex}}),\hfill \\ \hfill {I}_{k}^{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}}\phantom{\rule{0.166667em}{0ex}}({\tilde{a}}_{1,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E}\phantom{\rule{0.166667em}{0ex}},\dots ,{\tilde{a}}_{r,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E}\phantom{\rule{0.166667em}{0ex}}).\hfill \end{array}$$
- -
- For external reviewers (experts and non-experts, respectively):$$\begin{array}{ccc}\hfill {I}_{k}^{{B}^{E}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{B}^{E}}}({\tilde{b}}_{1,j,k}^{E}\phantom{\rule{0.166667em}{0ex}},\dots ,{\tilde{b}}_{s,j,k}^{E}\phantom{\rule{0.166667em}{0ex}}),\hfill \\ \hfill {I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}}\left({I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j}),{I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j})\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}}}({\tilde{b}}_{1,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\phantom{\rule{0.166667em}{0ex}},\dots ,{\tilde{b}}_{t,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j})& =& \phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}}}({\tilde{b}}_{1,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}\phantom{\rule{0.166667em}{0ex}},\dots ,{\tilde{b}}_{u,j,k}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}).\hfill \end{array}$$

- Computing EPIs for experts/non-experts reviewers and each criterion.As in the preceding step, the OWA operator is used. The previous environmental performance indicators for the ${x}_{j}$ facility site: ${I}_{k}^{{A}^{E}}({x}_{j})$, ${I}_{k}^{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j})$, ${I}_{k}^{{B}^{E}}({x}_{j})$ and ${I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j})$ are aggregated for each criterion taking into account if the reviewers are experts or not (see Figure 3). The previous indicators belonging to the experts reviewers are then aggregated by means of an OWA operator for each criterion ${c}_{k}^{-}$.$${I}_{k}^{E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{E}}\left({I}_{k}^{{A}^{E}}({x}_{j}),{I}_{k}^{{B}^{E}}({x}_{j})\right).$$Analogously to the experts reviewers, an environmental performance indicator is computed for each criterion ${c}_{k}^{-}$ by aggregating the opinions of all non-experts reviewers.$${I}_{k}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{G}_{k}^{{W}_{N\phantom{\rule{-1.42262pt}{0ex}}E}}\left({I}_{k}^{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j}),{I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}\phantom{\rule{-2.84526pt}{0ex}}({x}_{j})\right).$$
- Computing global EPIs.The proposed environmental integral evaluation model puts forward the computation of three global environmental performance indicators: an overall that includes all the issues, a management one relative to management issues and an operational one for the operational issues considered. In this way, the major recommendations issued by the ISO 14001 is followed and the model takes the most advantage of the gathered information.From the previous step, there are two values for every single criterion, one from experts reviewers and another from non-experts reviewers for each facility site.Now to aggregate the values corresponding to different criteria, we propose to use the discrete Choquet integral as aggregation operator. It allows for consideration of the interrelations among criteria through the choice of an specific fuzzy measure.
- (a)
- An overall global EPI.In order to cover better the possible interdependences among criteria when we compute an overall global EPI, we do not distinguish between management and operational criteria because we could have interdependences among some management criteria with some operational criteria. Therefore, we do not limit to consider the interdependence among management criteria on the one hand, or only among operational criteria on the another hand.We propose to determine a fuzzy measure, $\mu $, over the set of all criteria,$$C={C}^{M}\bigcup {C}^{O}=\{{c}_{1}^{M},\dots ,{c}_{p}^{M},{c}_{1}^{O},\dots ,{c}_{q}^{O}\}.$$And then, this fuzzy measure is used to calculate the associated Choquet integrals. Theoretical aspects of Choquet Integral are included in Appendix A. We must do twice, one for the values coming from the experts reviewers and another one for the values coming from the non-experts reviewers. Afterwards, a convex linear combination of both is computed which is the overall global EPI. The parameter $\beta \in (0,1)$ is chosen arbitrary taking into account the interest of the company. It stands for the relative importance assigned for the expert reviewers assessments versus the non-expert reviewers assessments.$${I}^{E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC\left({I}_{{1}^{M}}^{E}({x}_{j}),\dots ,{I}_{{p}^{M}}^{E}({x}_{j}),{I}_{{1}^{O}}^{E}({x}_{j}),\dots ,{I}_{{q}^{O}}^{E}({x}_{j})\right).$$$${I}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC\left({I}_{{1}^{M}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j}),\dots ,{I}_{{p}^{M}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j}),{I}_{{1}^{O}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j}),\dots ,{I}_{{q}^{O}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})\right).$$$$I({x}_{j})=\beta {I}^{E}({x}_{j})+(1-\beta ){I}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})).$$
- (b)
- Management and operational global EPIS.At this point we are interested in calculating two more global indicators, a management global EPI and an operational global EPI, with the aim of conforming to the standard ISO 14001. The criteria are divided according to their type, management and operational. Each type of criteria is aggregated separately by means of two discrete Choquet integrals because it can have interrelations (see Figure 3).In this case, we take the fuzzy measure previously built on the set of all criteria, C, and we derive from it two new fuzzy measures, one on the set of management criteria, ${C}^{M}$, and another on the set of operational criteria, ${C}^{O}$, called ${\mu}^{M}$ and ${\mu}^{O}$, respectively. So, they are coherent to the previous one, $\mu $. These are defined as follows$$\begin{array}{cc}{\mu}^{M}:\mathcal{P}({C}^{M})\u27f6[0,1]\hfill & \hfill \phantom{\rule{0.7cm}{0ex}}{\mu}^{M}(T)=\frac{\mu \left(T\bigcap {C}^{M}\right)}{\mu ({C}^{M})}\phantom{\rule{11.38109pt}{0ex}}\mathrm{where}\phantom{\rule{11.38109pt}{0ex}}T\subseteq {C}^{M}\end{array}$$$${\mu}^{O}:\mathcal{P}({C}^{O})\u27f6[0,1]\phantom{\rule{0.7cm}{0ex}}{\mu}^{O}(S)=\frac{\mu \left(S\bigcap {C}^{O}\right)}{\mu ({C}^{O})}\phantom{\rule{11.38109pt}{0ex}}\mathrm{where}\phantom{\rule{11.38109pt}{0ex}}S\subseteq {C}^{O}$$Now, from these fuzzy measures we proceed to calculate the mentioned indicators using a discrete Choquet integral for each one.
- Management global environmental performance indicator (MEPI). In order to calculate this indicator we aggregate the experts and non-experts indicators for management criteria, ${c}^{M}=\{{c}_{1}^{M},\dots ,{c}_{p}^{M}\}$ by means of a Choquet integral based on the fuzzy measure ${\mu}^{M}$ previously defined. We then compute a convex linear combination of both of them. $\gamma \in (0,1)$ is selected depending on the interest of the company (see Figure 3).$${I}_{M}^{E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{{\mu}^{M}}I{C}^{M}\left({I}_{{1}^{M}}^{E}({x}_{j}),\dots ,{I}_{{p}^{M}}^{E}({x}_{j})\right).$$$${I}_{M}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{{\mu}^{M}}I{C}^{M}\left({I}_{{1}^{M}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j}),\dots ,{I}_{{p}^{M}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})\right).$$$${I}^{M}({x}_{j})=\gamma {I}_{M}^{E}({x}_{j})+(1-\gamma ){I}_{M}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})).$$
- Operational global environmental performance indicator (OEPI). Analogously to management global EPI, an operational global environmental performance indicator is computed for operational criteria, ${c}^{O}=\{{c}_{1}^{O},\dots ,{c}_{q}^{O}\}$ adopting the same strategy to aggregate the experts and no-experts indicators for such criteria. We use the Choquet integral based on the fuzzy measure ${\mu}^{O}$ defined above and a convex linear combination of both with a constant $\delta \in (0,1)$ chosen according of the interest of the company (see Figure 3).$${I}_{O}^{E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{{\mu}^{O}}I{C}^{O}\left({I}_{{1}^{O}}^{E}({x}_{j}),\dots ,{I}_{{q}^{O}}^{E}({x}_{j})\right).$$$${I}_{O}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})=\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{{\mu}^{O}}I{C}^{O}\left({I}_{{1}^{O}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j}),\dots ,{I}_{{q}^{O}}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})\right).$$$${I}^{O}({x}_{j})=\delta {I}_{O}^{E}({x}_{j})+(1-\delta ){I}_{O}^{N\phantom{\rule{-1.42262pt}{0ex}}E}({x}_{j})).$$

#### 3.3.3. Rating Phase

## 4. An Illustrative Application

- ${c}_{1}^{M}$: Extension in pollution control initiatives.
- ${c}_{2}^{M}$: Green purchasing. Assessing how far they incorporate environmental considerations in the purchasing process.
- ${c}_{3}^{M}$: Proportion of research and development funds applied to projects with environmental significance.
- ${c}_{1}^{O}$: The amount of CO${}_{2}$ emissions measured in Kg of CO${}_{2}$/m${}^{2}$.
- ${c}_{2}^{O}$: The scope of use of renewable power sources.
- ${c}_{3}^{O}$: The electric power consumption in KWh/m${}^{2}$.

- Internal reviewers. This reviewers’ collective is made up of:
- -
- A set of three company’s internal experts: ${A}^{E}=\{{a}_{1}^{E},{a}_{2}^{E},{a}_{3}^{E}\}.$
- -
- A set of two company’s internal non-experts: ${A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}=\{{a}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E},{a}_{2}^{N\phantom{\rule{-1.42262pt}{0ex}}E}\}.$

- External reviewers. This collective consists of:
- -
- A set of two company’s external experts such as auditors: ${B}^{E}=\{{b}_{1}^{E},{b}_{2}^{E}\}$.
- -
- A set of five company’s external non-experts evaluators which is made up of two different reviewers’ collectives:
- A set of three general stakeholders: ${B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}=\{{b}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G},{b}_{2}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G},{b}_{3}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}\}$.
- A set of two social constituents: ${B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}=\{{b}_{1}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S},{b}_{2}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}\}$.

- Computing environmental performance indicators for each reviewers’ collective and each criterion.In the first step of this process, we apply the 2-tuple OWA operator, which requires a weighting vector. This vector can be chosen in different ways. Particularly, in this example, we use the weighting vector determined by a fuzzy linguistic quantifier (see [50]), the quantifier “most” whose parameters are $(0.3,0.8)$.The aggregation value with these OWA operators is computed for each criterion for the collectives of internal expert and internal non-experts and for each site (${I}_{k}^{{A}^{E}}({x}_{j})$ and ${I}_{k}^{{A}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j})$), see Table 8.The environmental performance indicators for general stakeholder reviewers and for social constituents reviewers are obtained by aggregating the values associated to them for each criterion and for each site ${I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-S}}({x}_{j})$ and ${I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E-G}}({x}_{j})$). They are used to compute the indicators for each criterion for external non-experts and for each site (${I}_{k}^{{B}^{N\phantom{\rule{-1.42262pt}{0ex}}E}}({x}_{j}$). In addition, the aggregation value is obtained for each criterion for the external experts and for each site (${I}_{k}^{{B}^{E}}({x}_{j})$). As noted above, all these calculations have been made using OWA operators with the appropriated weights. The results are showed in Table 9.
- Computing environmental performance indicators for experts/non-experts reviewers and each criterion.In the second step of the process, the 2-tuple OWA operator is also applied using the weighting vector calculated before. It is worth pointing out that there is the possibility to use another one attending to the specific characteristics of a particular case study. After these calculations, an environmental performance indicator is computed for experts and non-experts, for each criterion about each facility site. These results are shown in Table 10. The Figure 5a, displays those values for each criteria.
- Computing global environmental performance indicators.In this step, the values in Table 10 are aggregated for each facility site. At this point, it is needed to aggregate the indicators from different criteria. Therefore, according to the proposed environmental integral evaluation model, the 2-tuple Choquet integral operator is used, which is able to cope with interaction among criteria. Consequently, specific fuzzy measures suited for this example are required and included in Appendix B.
- (a)
- An overall global environmental performance indicator.Now, the fuzzy measures from Table A2 and from Table A4 are employed to calculate the associated 2-tuple Choquet integral for experts and for non experts, respectively, in each site (${I}^{E}$ and ${I}^{NE}$). These computed values are shown in Table 11.Finally, the overall global environmental performance indicator for each facility site is computed from previous indicators as it was established in the Section 3.3.2, using for example $\beta =0.75$. They are shown in Table 12.
- (b)
- Management and operational global environmental performance indicators.In the last step of the process, a management global indicator and an operational global indicator for each facility site are derived. In a similar way to the previous step, the values for several criteria are aggregated using a 2-tuple Choquet integral separately for the management criteria and for operational criteria. In the latter calculation it is used both the fuzzy measure for management ${\mu}^{M}$ and the fuzzy measure for operational criteria ${\mu}^{O}$, respectively. The results are in Table 13. In order to compute the final values by the proposed procedure, both types of criteria are aggregated for experts and non experts using $\gamma =\delta =0.75$. The resultant management and operational global performance indicators are shown in Table 14.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. A Short Survey on Discrete Choquet Integral for Dealing with Dependencies of EPIs

**Definition**

**A1.**

- 1.
- $\mu (\varnothing )=0\phantom{\rule{0.166667em}{0ex}}$ and $\phantom{\rule{0.166667em}{0ex}}\mu (C)=1$,
- 2.
- $A\subseteq B\Rightarrow \mu (A)\le \mu (B)$,

**Definition**

**A2.**

**Definition**

**A3.**

**Remark**

**A1.**

**Definition**

**A4.**

#### Identification of the Fuzzy Measure

## Appendix B. Computing Some Fuzzy Measures for the Illustrative Example

- Firstly, we have a clear orientation to criteria relative to practices in outputs of the company (${c}_{1}^{M}$, ${c}_{1}^{O}$, ${c}_{3}^{O}$). We translate this through some constrains in the optimization problem. Let us consider Table A1 that contains a learning set of possible normalized criteria values. We demand that the aggregated values, with Choquet integral for its rows, satisfy this order$${}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}1)\ge \phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}2)\ge {\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}3)\ge {\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}4)\ge {\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}5)$$Here the aggregation values for rows 1 and 2 are greater than for rows 3, 4, and 5 because the outputs criteria have bigger values whatever are the rest of criteria. Besides, when the output criteria have equal values we prefer to have a greater value in criteria ${c}_{2}^{M}$, ${c}_{2}^{O}$, relative to practices in inputs of the company than in ${c}_{3}^{M}$. This implies that Row 1 has an aggregation value greater than Row 2 and the same happens with Rows 4 and 5.
Output Input ${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ ${\mathit{c}}_{\mathbf{1}}^{\mathit{O}}$ ${\mathit{c}}_{\mathbf{3}}^{\mathit{O}}$ ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ ${\mathit{c}}_{\mathbf{3}}^{\mathit{M}}$ Row 1 G MP MP G MP MP Row 2 G MP MP MP MP G Row 3 MP MP MP G MP G Row 4 MP MP MP G MP MP Row 5 MP MP MP MP MP G This means the following constraints:${}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}1)-{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}2)\ge 0,$ ${}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}2)-{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}3)\ge 0,$${}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}3)-{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}4)\ge 0,$ ${}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}4)-{\phantom{\rule{0.166667em}{0ex}}}_{\mu}IC(Row\phantom{\rule{0.166667em}{0ex}}5)\ge 0$. - Secondly, the decision maker brings some ideas about the interactions of some criteria. The pairs $({c}_{1}^{M},{c}_{1}^{O})$ and $({c}_{1}^{M},{c}_{3}^{O})$ have to interact in a redundancy way. That is, the contribution of the pair should be inferior to the sum of the contribution of the criterion into it. However, the pair $({c}_{1}^{M},{c}_{2}^{O})$ has to interact in a complementary way, i.e., it has a contribution superior to the sum of the contributions of the criteria in it. This means the following constraints on the Shapley indexes:${I}_{\mu}({c}_{1}^{M},{c}_{1}^{O})<0$, ${I}_{\mu}({c}_{1}^{M},{c}_{3}^{O})<0$ and ${I}_{\mu}({c}_{1}^{M},{c}_{2}^{O})>0$
- Thirdly, due to the fact that the ${\mathrm{CO}}_{2}$ emissions are the great importance for the company, it is imposed that the overall importance of this criterion should be at least 0.15 over 1. This adds a new constraint through the Shapley value for this criterion.This implies the constraint on the Shapley value ${I}_{\mu}({c}_{1}^{O})\ge 0.15$.

$Subset$ | {${c}_{1}^{M}$} | {${c}_{2}^{M}$} | {${c}_{3}^{M}$} | {${c}_{1}^{O}$} | {${c}_{2}^{O}$} | {${c}_{3}^{O}$} |

${m}_{\mu}(subset)$ | 0.503 | 0.203 | 0.036 | 0.266 | 0.109 | 0.222 |

$Subset$ | {${c}_{1}^{M}$, ${c}_{2}^{M}$} | {${c}_{1}^{M}$, ${c}_{3}^{M}$} | {${c}_{1}^{M}$, ${c}_{1}^{O}$} | {${c}_{1}^{M}$, ${c}_{2}^{O}$} | {${c}_{1}^{M}$, ${c}_{3}^{O}$} | {${c}_{2}^{M}$, ${c}_{3}^{M}$} |

${m}_{\mu}(subset)$ | −0.003 | −0.003 | −0.219 | 0.100 | −0.197 | 0.131 |

$Subset$ | {${c}_{2}^{M}$, ${c}_{1}^{O}$} | {${c}_{2}^{M}$, ${c}_{2}^{O}$} | {${c}_{2}^{M}$, ${c}_{3}^{O}$} | {${c}_{3}^{M}$, ${c}_{1}^{O}$} | {${c}_{3}^{M}$, ${c}_{2}^{O}$} | {${c}_{3}^{M}$, ${c}_{3}^{O}$} |

${m}_{\mu}(subset)$ | −0.047 | −0.099 | −0.025 | 0.000 | −0.010 | 0.000 |

$Subset$ | {${c}_{1}^{O}$, ${c}_{2}^{O}$} | {${c}_{1}^{O}$, ${c}_{3}^{O}$} | {${c}_{2}^{O}$, ${c}_{3}^{O}$} | |||

${m}_{\mu}(subset)$ | 0.006 | 0.028 | 0.000 |

**Table A3.**Mobius for ${\mu}^{M}$ and ${\mu}^{O}$ for management and for operational criteria, respectively (for experts).

$Subset$ | {${c}_{1}^{M}$} | {${c}_{2}^{M}$} | {${c}_{3}^{M}$} | {${c}_{1}^{M}$, ${c}_{2}^{M}$} | {${c}_{1}^{M}$, ${c}_{3}^{M}$} | {${c}_{2}^{M}$, ${c}_{3}^{M}$} |

${m}_{{\mu}^{M}}(subset)$ | 0.58 | 0.23 | 0.04 | −0.00 | −0.00 | 0.15 |

$Subset$ | {${c}_{1}^{O}$} | {${c}_{2}^{O}$} | {${c}_{3}^{O}$} | {${c}_{1}^{O}$, ${c}_{2}^{O}$} | {${c}_{1}^{O}$, ${c}_{3}^{O}$} | {${c}_{2}^{O}$, ${c}_{3}^{O}$} |

${m}_{{\mu}^{O}}(subset)$ | 0.42 | 0.17 | 0.35 | 0.01 | 0.04 | 0.00 |

$\mathit{Subset}$ | {${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$} | {${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$} | {${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$} | {${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$, ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$} | {${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$, ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$} | {${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$, ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$} |
---|---|---|---|---|---|---|

${m}_{\mu}(subset)$ | 0.29 | 0.35 | 0.29 | −0.01 | 0.10 | −0.01 |

${m}_{{\mu}^{M}}(subset)$ | 0.47 | 0.56 | - | −0.02 | - | - |

${m}_{{\mu}^{O}}(subset)$ | - | - | 1 | - | - | - |

**Table A5.**Shapley values for fuzzy measure $\mu $, ${\mu}^{M}$ and ${\mu}^{M}$ for experts (right) and non-experts (left).

Non-Experts | Experts | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{1}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ | ||

$\mu $ | 0.34 | 0.18 | 0.10 | 0.15 | 0.11 | 0.12 | $\mu $ | 0.33 | 0.33 | 0.33 |

${\mu}^{M}$ | 0.58 | 0.31 | 0.12 | ${\mu}^{M}$ | 0.45 | 0.55 | ||||

${\mu}^{O}$ | 0.45 | 0.18 | 0.37 | ${\mu}^{O}$ | 1 |

## References

- Sarkis, J. Evaluating environmentally conscious business practices. Eur. J. Oper. Res.
**1998**, 107, 159–174. [Google Scholar] [CrossRef] - Darnall, N.; Henriques, I.; Sadorsky, P. Do environmental management systems improve business performance in an international setting? J. Int. Manag.
**2008**, 14, 364–376. [Google Scholar] [CrossRef] - Stanwick, P.; Stanwick, S. CEO compensation: Does it pay to be green? Bus. Strategy Environ.
**2001**, 10, 176–182. [Google Scholar] [CrossRef] - Rivera, J. Assessing a voluntary environmental initiative in the developing world: The Costa Rican Certification for Sustainable Tourism. Policy Sci.
**2002**, 35, 333–360. [Google Scholar] [CrossRef] - Iraldo, F.; Testa, F.; Frey, M. Is an environmental management system able to influence environmental and competitive performance? The case of the eco-management and audit scheme (EMAS) in the European union. J. Clean. Prod.
**2009**, 17, 1444–1452. [Google Scholar] [CrossRef] - Jørgensen, S.; Burkhard, B.; Müller, F. Twenty volumes of ecological indicators: An accounting short review. Ecol. Indic.
**2013**, 28, 4–9. [Google Scholar] [CrossRef] - Rahdari, A.; Rostamy, A. Designing a general set of sustainability indicators at the corporate level. J. Clean. Prod.
**2015**, 108, 757–771. [Google Scholar] [CrossRef] - Jasch, C. Environmental performance evaluation and indicators. J. Clean. Prod.
**2000**, 8, 79–88. [Google Scholar] [CrossRef] - Lin, B.; Jones, C.; Hsieh, C. Environmental practices and assessment: A process perspective. Ind. Manag. Data Syst.
**2001**, 101, 71–79. [Google Scholar] [CrossRef] - Handfield, R.; Walton, S.; Sroufe, R.; Melnyk, S. Applying environmental criteria to supplier assessment: A study in the application of the Analytical Hierarchy Process. Eur. J. Oper. Res.
**2002**, 141, 70–87. [Google Scholar] [CrossRef] - Sarkis, J. A strategic decision framework for green supply chain management. J. Clean. Prod.
**2003**, 11, 397–409. [Google Scholar] [CrossRef] - Humphreys, P.; Wong, Y.; Chan, F. Integrating environmental criteria into the supplier selection process. J. Mater. Process. Technol.
**2003**, 138, 349–356. [Google Scholar] [CrossRef] - Zhu, Q.; Sarkis, J.; Lai, K.H. Confirmation of a measurement model for green supply chain management practices implementation. Int. J. Prod. Econ.
**2008**, 111, 261–273. [Google Scholar] [CrossRef] - Jabbour, A.B.L.S.; Jabbour, C.J.C. Are supplier selection criteria going green? Case studies of companies in Brazil. Ind. Manag. Data Syst.
**2009**, 109, 477–495. [Google Scholar] [CrossRef] - Herva, M.; Roca, E. Review of combined approaches and multi-criteria analysis for corporate environmental evaluation. J. Clean. Prod.
**2013**, 39, 355–371. [Google Scholar] [CrossRef] - Olsthoorn, X.; Tyteca, D.; Wehrmeyer, W.; Wagner, M. Environmental indicators for business: A review of the literature and standardisation methods. J. Clean. Prod.
**2001**, 9, 453–463. [Google Scholar] [CrossRef] - Niemeijer, D.; de Groot, R.S. A conceptual framework for selecting environmental indicator sets. Ecol. Indic.
**2008**, 8, 14–25. [Google Scholar] [CrossRef] - Brunklaus, B.; Malmqvist, T.; Baumann, H. Managing Stakeholders or the Environment? The Challenge of Relating Indicators in Practice. Corp. Soc. Responsib. Environ. Manag.
**2009**, 16, 27–37. [Google Scholar] [CrossRef] - Gandhi, S.; Mangla, S.; Kumar, P.; Kumar, D. Evaluating factors in implementation of successful green supply chain management using DEMATEL: A case study. Int. Strateg. Manag. Rev.
**2015**, 3, 96–109. [Google Scholar] [CrossRef] - Tseng, M.L. Using a hybrid MCDM model to evaluate firm environmental knowledge management in uncertainty. Appl. Soft Comput.
**2011**, 11, 1340–1352. [Google Scholar] [CrossRef] - Xie, S.; Hayase, K. Corporate Environmental Performance Evaluation: A Measurement Model and a New Concept. Bus. Strategy Environ.
**2007**, 16, 148–168. [Google Scholar] [CrossRef] - Tsoulfas, G.T.; Pappis, C.P. A model for supply chains environmental performance analysis and decision making. J. Clean. Prod.
**2008**, 16, 1647–1657. [Google Scholar] [CrossRef] - Ho, W.; Xu, X.; Dey, P.K. Multi-criteria decision making approaches for supplier evaluation and selection: A literature review. Eur. J. Oper. Res.
**2010**, 202, 16–24. [Google Scholar] [CrossRef] - Arfi, B. Fuzzy decision making in politics: A linguistic fuzzy-set approach. Political Anal.
**2005**, 13, 23–56. [Google Scholar] [CrossRef] - Martínez, L. Sensory evaluation based on linguistic decision analysis. Int. J. Approx. Reason.
**2007**, 44, 148–164. [Google Scholar] [CrossRef] - Grabisch, M. The application of fuzzy integrals in multicriteria decision making. Eur. J. Oper. Res.
**1996**, 89, 445–456. [Google Scholar] [CrossRef] - Darnall, N.; Seol, I.; Sarkis, J. Perceived stakeholder influences and organizations’ use of environmental audits. Acc. Organ. Soc.
**2009**, 34, 170–187. [Google Scholar] [CrossRef] - King, A.; Lenox, M.; Terlaak, A. The strategic use of decentralized institutions: Exploring certification with the ISO 14001 management standard. Acad. Manag. J.
**2005**, 48, 1091–1106. [Google Scholar] [CrossRef] - Kain, J.H.; Saderberg, H. Management of complex knowledge in planning for sustainable development: The use of multi-criteria decision aids. Environ. Impact Assess. Rev.
**2008**, 28, 7–21. [Google Scholar] [CrossRef] - Nawrocka, D.; Parker, T. Finding the connection: Environmental management systems and environmental performance. J. Clean. Prod.
**2009**, 17, 601–607. [Google Scholar] [CrossRef] - International Organization for Standardization (ISO). Environmental Management Systems–Requirements with Guidance for Use (ISO 14001:2004); ISO: Geneva, Switzerland, 2004. [Google Scholar]
- To, W.; Lee, P. Diffusion of ISO 14001 environmental management system: Global, regional and country-level analyses. J. Clean. Prod.
**2014**, 66, 489–498. [Google Scholar] [CrossRef] - International Organization for Standardization (ISO). Standards 14031. Environmental Management Environmental Performance Evaluation: Guidelines; ISO: Geneva, Switzerland, 1999. [Google Scholar]
- Henri, J.F.; Journeault, M. Environmental performance indicators: An empirical study of Canadian manufacturing firms. J. Environ. Manag.
**2008**, 87, 165–176. [Google Scholar] [CrossRef] [PubMed] - Thoresen, J. Environmental performance evaluation—A tool for industrial improvement. J. Clean. Prod.
**1999**, 7, 365–370. [Google Scholar] [CrossRef] - Lee, A.H.; Kang, H.Y.; Hsu, C.F.; Hung, H.C. A green supplier selection model for high-tech industry. Expert Syst. Appl.
**2009**, 36, 7917–7927. [Google Scholar] [CrossRef] - Awasthi, A.; Chauhan, S.S.; Goyal, S. A fuzzy multicriteria approach for evaluating environmental performance of suppliers. Int. J. Prod. Econ.
**2010**, 126, 370–378. [Google Scholar] [CrossRef] - Tseng, M.L.; Chiu, A.S. Evaluating firm’s green supply chain management in linguistic preferences. J. Clean. Prod.
**2013**, 40, 22–31. [Google Scholar] [CrossRef] - Espinilla, M.; de Andres, R.; Martinez, F.J.; Martinez, L. A Heterogeneous Evaluation Model for the Assessment of Sustainable Energy Policies; MeloPinto, P., Couto, P., Serodio, C., Fodor, J., DeBaets, B., Eds.; Advances in Intelligent and Soft Computing; Springer: Berlin/Heidelberg, Germany, 2011; Volume 107, pp. 209–220. [Google Scholar]
- Tseng, M.L.; Chiang, J.H.; Lan, L.W. Selection of optimal supplier in supply chain management strategy with analytic network process and choquet integral. Comput. Ind. Eng.
**2009**, 57, 330–340. [Google Scholar] [CrossRef] - Buyukozkan, G.; Cifci, G. A novel fuzzy multi-criteria decision framework for sustainable supplier selection with incomplete information. Comput. Ind.
**2011**, 62, 164–174. [Google Scholar] [CrossRef] - Herrera, F.; Martínez, L.; Sánchez, P. Managing non-homogeneous information in group decision making. Eur. J. Oper. Res.
**2005**, 166, 115–132. [Google Scholar] [CrossRef] - Likert, R. A Technique for the measumerement of attitudes. Arch. Psychol.
**1932**, 140, 1–55. [Google Scholar] - Herrera, F.; Herrera-Viedma, E.; Martínez, L. A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst.
**2000**, 114, 43–58. [Google Scholar] [CrossRef] - Martínez, L.; Herrera, F. An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges. Inf. Sci.
**2012**, 207, 1–18. [Google Scholar] [CrossRef] - Herrera, F.; Martínez, L. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 8, 746–752. [Google Scholar] - Grabisch, M.; Kojadinovic, I.; Meyer, P. A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package. Eur. J. Oper. Res.
**2008**, 186, 766–785. [Google Scholar] [CrossRef] [Green Version] - Klement, E.; Mesiar, R. Discrete Integrals and Axiomatically Defined Functionals. Axioms
**2012**, 1, 9–20. [Google Scholar] [CrossRef] - Yager, R. On Ordered Weighted Averaging Operators in Multicriteria Decision Making. IEEE Trans. Syst. Man Cybern.
**1988**, 18, 183–190. [Google Scholar] [CrossRef] - Zadeh, L. A Computational Approach to Fuzzy Quantifiers in Natural Languages. Comput. Math. Appl.
**1983**, 9, 149–184. [Google Scholar] [CrossRef] - Marichal, J.L. Behavioral analysis of aggregation in multicriteria decision aid. In Preferences and Decisions under Incomplete Knowledge; Fodor, J., De Baets, B., Perny, P., Eds.; Studies in Fuzziness and Soft Computing; Physica Verlag: Heidelberg, Germany, 2000; Volume 51, pp. 153–178. [Google Scholar]
- Beliakov, G.; Pradera, A.; Calvo, T. Aggregation Functions: A Guide for Practitioners; Springer: Berlin, Germany, 2007. [Google Scholar]
- Yang, W.; Chen, Z. New aggregation operators based on the Choquet integral and 2-tuple linguistic information. Expert Syst. Appl.
**2012**, 39, 2662–2668. [Google Scholar] [CrossRef] - Meyer, P.; Roubens, M. Choice, Ranking and Sorting in Fuzzy Multiple Criteria Decision Aid. In Multiple Criteria Decision Analysis: State of the Art Surveys; Figueira, J., Greco, S., Ehrgott, M., Eds.; International Series in Operations Research and Management Science; Springer: New York, NY, USA, 2005; Volume 78, pp. 471–503. [Google Scholar]
- Wu, J.Z.; Zhang, Q. 2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle. Fuzzy Optim. Decis. Mak.
**2010**, 9, 435–453. [Google Scholar] [CrossRef] - Grabisch, M.; Kojadinovic, I.; Meyer, P. Kappalab: Non-Additive Measure and Integral Manipulation Functions, R package version 0.4-7; Repository CRAN; 2015. Available online: https://cran.r-project.org/web/packages/kappalab/index.html (accessed on 20 September 2017).
- R Development Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2017; ISBN 3-900051-07-0. [Google Scholar]

**Figure 4.**

**Left**: label set for criteria ${c}_{2}^{O}$ and its associated fuzzy numbers;

**Right**: label set for criteria ${c}_{1}^{M}$ and ${c}_{2}^{M}$ and its associated fuzzy numbers.

**Figure 5.**EPIs for site ${x}_{1}$ and ${x}_{2}$. (

**a**) EPIs for each criteria for experts and non-experts; (

**b**) Partial and global EPIs.

Criteria | Classification | |||
---|---|---|---|---|

${c}_{1}^{M}$ | Pollution control initiatives | M | Qualitative | Granularity 7 |

${c}_{2}^{M}$ | Green purchasing | M | Qualitative | Granularity 7 |

${c}_{3}^{M}$ | Funds in research projects with environmental significance | M | Quantitative | Benefit |

${c}_{1}^{O}$ | $C{O}_{2}$ emissions | O | Quantitative | Cost |

${c}_{2}^{O}$ | The scope of renewable power source | O | Qualitative | Granularity 5 |

${c}_{3}^{O}$ | The electric power consumption | O | Quantitative | Cost |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{3}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Site ${x}_{1}$ | P | VP | P | VP | P | MP | VP | VP | P | P | P | P |

Site ${x}_{2}$ | P | P | VP | VP | P | P | VP | P | P | VP | MP | P |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{3}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Site ${x}_{1}$ | MG | G | MG | MG | MG | F | MG | G | F | MG | G | G |

Site ${x}_{2}$ | MG | G | G | G | MG | MG | F | MG | MG | F | MG | G |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | |
---|---|---|---|---|---|

Site ${x}_{1}$ | 0.150 | 0.135 | 0.156 | 0.105 | 0.087 |

Site ${x}_{2}$ | 0.291 | 0.300 | 0.300 | 0.243 | 0.219 |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | |
---|---|---|---|---|---|

Site ${x}_{1}$ | 134 | 120 | 134 | 112 | 88 |

Site ${x}_{2}$ | 88 | 112 | 156 | 146 | 200 |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{3}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{G}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{N}\phantom{\rule{-1.42262pt}{0ex}}\mathit{E}-\mathit{S}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Site ${x}_{1}$ | MP | MP | F | MP | F | MG | F | MP | F | F | F | F |

Site ${x}_{2}$ | VP | MP | VP | VP | MP | MP | VP | MP | MP | VP | MP | MP |

${\mathit{a}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{2}}^{\mathit{E}}$ | ${\mathit{a}}_{\mathbf{3}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{1}}^{\mathit{E}}$ | ${\mathit{b}}_{\mathbf{2}}^{\mathit{E}}$ | |
---|---|---|---|---|---|

Site ${x}_{1}$ | 182.4 | 163.4 | 186.2 | 167.2 | 133.0 |

Site ${x}_{2}$ | 357.2 | 372.4 | 380.0 | 326.8 | 330.6 |

${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{1}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{O}}$ | |
---|---|---|---|---|---|---|

${I}^{{A}^{E}}({x}_{1})$ | (P, 0) | (MG, 0.27) | (F, 0.03) | (MP, 0.06) | (MP, −0.09) | (F, 0.21) |

${I}^{{A}^{NE}}({x}_{1})$ | (P, −0.4) | (MG, 0) | (MP, 0.4) | |||

${I}^{{A}^{E}}({x}_{2})$ | (P, 0) | (G, 0) | (VG, 0) | (F, −0.17) | (P, −0.18) | (VP, 0.19) |

${I}^{{A}^{NE}}({x}_{2})$ | (P, −0.4) | (G, −0.4) | (P, 0.13) |

${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{1}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{O}}$ | |
---|---|---|---|---|---|---|

${I}^{{B}^{E}}({x}_{1})$ | (P, 0.2) | (MG, −0.4) | (MP, −0.07) | (F, 0.08) | (MG, −0.1) | (MG, −0.29) |

${I}^{{B}^{NE}}({x}_{1})$ | (P, 0) | (G, −0.29) | (F, 0) | |||

${I}^{{B}^{E}}({x}_{2})$ | (P, −0.4) | (MG, −0.4) | (G, −0.34) | (P, −0.03) | (P, 0.13) | (P, −0.15) |

${I}^{{B}^{NE}}({x}_{2})$ | (P, 0.36) | (MG, 0.36) | (MP, −0.5) |

${\mathit{c}}_{\mathbf{1}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{M}}$ | ${\mathit{c}}_{\mathbf{1}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{2}}^{\mathit{O}}$ | ${\mathit{c}}_{\mathbf{3}}^{\mathit{O}}$ | |
---|---|---|---|---|---|---|

${I}^{E}({x}_{1})$ | (P, 0.12) | (MG, 0) | (F, −0.41) | (F, −0.33) | (F, 0.1) | (MG, −0.49) |

${I}^{NE}({x}_{1})$ | (P, −0.16) | (MG, 0.42) | (F, −0.24) | |||

${I}^{E}({x}_{2})$ | (P, −0.16) | (MG, 0.44) | (G, 0.46) | (MP, 0.08) | (P, 0) | (P, −0.41) |

${I}^{NE}({x}_{2})$ | (P, 0.06) | (G, −0.5) | (P, 0.35) |

Experts | Non Experts | |
---|---|---|

Site ${x}_{1}$ | (F, −0.18) | (F, −0.38) |

Site ${x}_{2}$ | (MP, 0.47) | (MP, 0.34) |

Overall EPI | |
---|---|

Site ${x}_{1}$ | (F, −0.23) |

Site ${x}_{2}$ | (MP, 0.44) |

Site ${\mathit{x}}_{1}$ | Site ${\mathit{x}}_{2}$ | |||
---|---|---|---|---|

Experts | Non Experts | Experts | Non Experts | |

Manag. | (MP, 0.08) | (F, −0.16) | (MP, 0.42) | (F, −0.02) |

Opera. | (F, 0.04) | (F, −0.24) | (P, 0.29) | (P, 0.35) |

MEPI | OEPI | |
---|---|---|

Site ${x}_{1}$ | (MP, 0.27) | (F, −0.03) |

Site ${x}_{2}$ | (F, −0.44) | (P, 0.31) |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

González-Arteaga, T.; De Andrés Calle, R.; Martínez, L.
Managing Interacting Criteria: Application to Environmental Evaluation Practices. *Axioms* **2018**, *7*, 4.
https://doi.org/10.3390/axioms7010004

**AMA Style**

González-Arteaga T, De Andrés Calle R, Martínez L.
Managing Interacting Criteria: Application to Environmental Evaluation Practices. *Axioms*. 2018; 7(1):4.
https://doi.org/10.3390/axioms7010004

**Chicago/Turabian Style**

González-Arteaga, Teresa, Rocio De Andrés Calle, and Luis Martínez.
2018. "Managing Interacting Criteria: Application to Environmental Evaluation Practices" *Axioms* 7, no. 1: 4.
https://doi.org/10.3390/axioms7010004