**3. Problem formulation**

account not only the immediate but also the future consequences of decisions in order not to compromise future generations. In such context, decisions are generally ill-defined, the impacts of decisions are uncertain and often difficult to measure, and the acceptability of decisions is more difficult to attain. And so, the need for using structured methods and novel approaches to support sustainable decisions

A state-of-the-art survey on sustainable decision prioritization [2] shows multicriteria decision aid (MCDA) methods are the most popular approach to support sustainable decisions. These methods enable the simultaneous consideration of conflicting criteria as it occurs in a real-world problem under sustainability imperatives. However, although sustainable development tries to reach a balance between the evaluations of actions in the short and the long term, most articles surveyed in [2] did not investigate the long-term perspective related to sustainable development. Only very recently have some researchers proposed novel temporal MCDA methods for application in SD context. But, the state of the art remains limited, and

This paper presents two novel temporal MCDA methods that were applied in SD

Despite the importance of temporal (multi-periods) evaluation of actions for sustainable decisions, only a few articles have dealt with this aspect. Some authors consider the long-term effects as a criterion [3, 4], while others use scenario planning and predictive techniques or fuzzy modeling to deal with future unknowns [5]. In [6], the long-term effects are discounted, and in [4] they are roughly and qualitatively assessed. Very recently, some temporal extensions of MCDA methods have been developed [3, 6, 7–11]. In a forest management context, the long-term impacts were addressed as a specific criterion [3], and the local community was asked to evaluate it. In [6], the authors proposed a sustainable environmental management system (SEMS) where actions are ranked using ELECTRE III. The authors indicate that special care was taken in the assessment of criteria and that expected short- and long-term consequences were considered but without any explanation on how this was achieved. In [9], a multi-period multi-criteria method based on adapting TOPSIS to temporal context is proposed. But, compensation between the decision criteria on which TOPSIS rely (as scoring methods) is not appropriate for sustainability. In [10], authors generalize PROMETHEE to temporal setting. The weighted mean is applied for aggregation of the net flow scores over the periods, and then the method is compensatory. Another PROMETHEE-based model was published in [11] to assess the long-term impact of energy supply technologies. In this research work, different criteria weights were considered depending on the

life cycle steps (from introduction to saturation of the market).

The literature review presented here shows a limited state of the art and an as yet largely undeveloped research area on multi-period aggregation. As discussed earlier, compensation is the main issue behind the few existing temporal proposals.

context. The first is MUPOM method (MUlti-criteria multi-Period Outranking Method) which demonstrates how outranking methods can be used in processing the temporal impacts of decisions. The second method is named PROMETHEE-MP and consists of a temporal generalization of PROMETHEE in a context of random uncertainty. This paper is organized as follows: Section 2 presents the previous work. Section 3 proposes a formulation for decision-making problem in SD context. Sections 4 and 5 expose the MUPOM and PROMETHEE-MP methods. Section 6 provides an illustration of these two methods on the same case study. Finally,

only a few research studies offer temporal aggregation frameworks.

has emerged.

*Sustainability Concept in Developing Countries*

Section 7 concludes the paper.

**2. Previous work**

**20**

In order to formulate the problem, let us consider a set A of N candidate actions (*a*1, … *a*NÞ, a set *T* of K assessment periods ð*P*1, ..., *PK*), a set *C* of M criteria ð Þ *C*1,*C*<sup>2</sup> … ,*CM* , a set *Π* of M criteria weights ð*π*1, ... *πM*), *a set* ð*α*1, … , *αK*) of the K relative importance of periods (*P*1, ..., *PK*), and *gj* ð Þ *a*<sup>i</sup> the evaluation of an action*a*<sup>i</sup> on criterion *j*.

The following assumptions of the model are made. (i) All evaluations are evaluated in the future with no missing evaluations. (ii) Criteria weights may change over time. (iii) Criteria, preference functions, and thresholds can vary over time. (iv) Criteria ð Þ *C*1,*C*<sup>2</sup> … ,*CM* are assumed to be independent.

**Figure 1** displays the decision matrices for multi-period multi-criteria decision problems.

#### **4. MUPOM: multi-criteria multi-period outranking method**

MUPOM (MUlti-criteria multi-Period Outranking Method) is a three-phase temporal outranking MCDA method. In Phase 1, multi-criteria aggregation is performed in order to obtain outranking and preference relations for each period


**Figure 1.**

*Decision matrices for the considered decision problems.*

and for each pair of actions. Then in Phase 2 and for each pair of actions, a measure of distance between preference relations is used for temporal aggregation of the preference relations obtained in Phase 1. A graph showing relations between all pairs of actions illustrates the results of this aggregation. Next, in Phase 3 an exploitation procedure is used to compute the performance of each action *ai*. The following subsections provide details on the three phases. A full version of the mathematical details of the method is provided in [7].

**Step 1.1**: For each period *t* and for each pair of actions *ai* ð Þ , *ak* , compute the

*Temporal MCDA Methods for Decision-Making in Sustainable Development Context*

*<sup>j</sup> ai* ð Þ , *ak* . **Step 1.3**: Construct the relational preference systems *<sup>S</sup><sup>t</sup> ai* ð Þ , *ak* for each pair of actions *ai* ð Þ , *ak* and for each period *t* using concordance and discordance thresholds. We deduce that action *ai* strongly outranks *ak* (*aiSFak*Þ or *ai* weakly outranks

**Step 1.4**: For each period t and for each pair of actions *ai* ð Þ , *ak* , convert the

, *P*�<sup>1</sup> � � where *P*, *Q*,*I*, *R* refers, respectively, to strict preference, weak

This phase consists of aggregating the preference relations obtained for each pair of actions and at each period (results of Phase 1). This aggregation is done using a measure of distance between preorders [19]. Thus, the aggregated preference relation which minimizes the distance with the preorders at each period is obtained.

, *P*�<sup>1</sup> � �. This distance is noted

*<sup>t</sup>*¼<sup>1</sup>*αt<sup>Δ</sup> <sup>H</sup>*, *Rt ai* ð Þ ð Þ , *ak* where *<sup>α</sup><sup>t</sup>* is the relative importance of

,*P*�<sup>1</sup> ð Þ*Φ<sup>H</sup> ai* ð Þ , *ak*

**Step 2.1**: For each pair of actions (*ai*, *ak*) and at each period *t*, compute the distance between the preference relation *Rt ai* ð Þ , *ak* resulting from Step 1.4 and each

**Step 2.2**: Aggregate the distances obtained at each period into a mean distance

**Step 2.3**: Assign to the pair of actions *ai* ð Þ , *ak* the preference relation *<sup>H</sup>*<sup>∗</sup> , such as:

n o

¼ *min <sup>H</sup>* <sup>∈</sup> *<sup>P</sup>*,*Q*,*I*,*R*,*Q*�<sup>1</sup>

A graph representing relations between all pairs of actions displays the results.

This phase consists of computing the performance of each action *ai* . Performance calculation is based on the number of actions that are preferred (strictly or weakly) to *ai* and those that *ai* are preferred to (strictly or weakly). The set of "best compromise" action(s) is then deduced based on the computed performance. This set contains the actions with the highest performance and those which are incom-

MUPOM method has important contributions. First, it proposes a generalization of outranking methods based on ELECTRE principles (concordance, discordance, and credibility indexes) to multi-period and temporal settings. Consequently, the method supports partial preferences and partial rankings and confirms that the outranking methods can be generalized to temporal context. In practical terms, MUPOM provides valuable contributions for researchers and practitioners

concerned with decision-making processes under sustainability. Beyond the financial dimension, it enables integration of social and environmental impacts in the

parable to them. Details on the exploitation phase are provided in [19].

*ak* for *akPai*, and

obtained outranking relations to preference relation *<sup>R</sup><sup>t</sup> ai* ð Þ , *ak* <sup>∈</sup>

preference, indifference, and incomparability. We note *aiP*�<sup>1</sup>

The temporal aggregation phase consists of three steps [7]:

possible preference relation *H* ∈ *P*, *Q*,*I*, *R*, *Q*�<sup>1</sup>

P*<sup>T</sup>*

*<sup>H</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>H</sup>*<sup>∗</sup> *<sup>=</sup>Φ<sup>H</sup>*<sup>∗</sup>

**Step 1.2**: For each period *t*, for each pair of actions *ai* ð Þ , *ak* , and for each criterion

concordance index *<sup>C</sup><sup>t</sup> ai* ð Þ , *ak* .

*ak aiSf ak* � �.

*aiQ*�<sup>1</sup>

*P*, *Q*,*I*, *R*, *Q*�<sup>1</sup>

*<sup>Δ</sup> <sup>H</sup>*, *Rt ai* ð Þ ð Þ , *ak* .

period *t*.

**23**

*<sup>Φ</sup><sup>H</sup> ai* ð Þ , *ak :Φ<sup>H</sup> ai* ð Þ¼ , *ak*

**4.3 Phase 3: exploitation**

*ak* for *akQai*.

**4.2 Phase 2: temporal aggregation**

*j*, compute the discordance index *Dt*

*DOI: http://dx.doi.org/10.5772/intechopen.90698*

**Figure 2** graphs the steps of the MUPOM method.
