**5. PROMETHEE-MP: a generalization of PROMETHEE for multi-period evaluations under uncertainty**

PROMETHEE-MP is a recently developed temporal outranking method that allows aggregation of multi-periods and uncertain evaluations. It consists of three phases. Phase 1 aggregates the criteria, at each period of the horizon, based on PROMETHEE outgoing and incoming flows and Monte Carlo simulations. Binary relations are computed for each pair of actions. Phase 2 consists of aggregating the binary relations obtained over the periods using the measure of distance between preorders [19] as is done with MUPOM. Finally, in Phase 3 the performance of each action *ai* is computed, based on the number of actions that are preferred (strictly or weakly) to *ai* and those that *ai* are preferred to (strictly or weakly). **Figure 2** presents all the steps of PROMETHEE-MP. The following subsections provide details on the three phases. A full version of the mathematical details of the method is found in [8] (**Figure 3**).

**Figure 3.** *Steps of PROMETHEE-MP [8].*

### **5.1 Phase 1: multi-criteria aggregation and Monte Carlo simulations**

In Phase 1, the criteria at each period of the horizon are aggregated. The method looks at a representation of uncertainty with probability distributions for uncertain parameters (evaluations and weights) and uses Monte Carlo simulation to generate numerical values for each uncertainty scenario. In this illustration and without loss of generality, uniform distributions using intervals are simulated for each parameter and for each period *t*. For each scenario of uncertainty *s*, we generate from the interval a specific value for evaluations and weights. Then at each period *t* and for each scenario*s*, we use the PROMETHEE method, and we compute outgoing ∅<sup>þ</sup> *t*,*s* ð Þ *ai* and incoming flows ∅� *t*,*s* ð Þ *ai* for each action *ai*. As part of the model, we propose a generalization of PROMETHEE III that associates an interval to the outgoing and incoming flows for each action and deduces a partial preorder for the actions. The multi-criteria aggregation and Monte Carlo simulation phase consists of these steps [8]:

**Steps 1.1 and 1.2**: For each period t, we conduct a Monte Carlo simulation *s*. Each simulation generates, for each criterion *j*, a specific evaluation of action *ai* noted *g t*,*s <sup>j</sup>* ð Þ *ai* in the interval [*g*� *<sup>j</sup>* ð Þ *ai* ,*g*<sup>þ</sup> *<sup>j</sup>* ð Þ *ai* ]. Also, each simulation considers a different value for criteria weights for each criterion *j*, noted *π<sup>t</sup>*,*<sup>s</sup> <sup>j</sup>* ð Þ *ai* .

**Step 1.3**: For each scenario*s*, action *i* and period *t*, we apply PROMETHEE and compute outgoing and incoming flows ∅<sup>þ</sup> *t*,*s* ð Þ *ai* and ∅� *t*,*s* ð Þ *ai* .

**Step 1.4**: In this step, the outgoing and incoming flow distributions are defined by computing the mean ∅<sup>þ</sup> *<sup>t</sup>* ð Þ *ai* and ∅� *<sup>t</sup>* ð Þ *ai* and the standard deviations *σ* ∅<sup>þ</sup> *<sup>t</sup>* ð Þ *ai* and *σ* ∅� *<sup>t</sup>* ð Þ *ai* .

**Step 1.5**: The resulting interval limits of the outgoing and incoming flows∅<sup>þ</sup> *max* ,*t* ð Þ *ai* , ∅<sup>þ</sup> *min* ,*t* ð Þ *ai* , ∅� *max* ,*t* ð Þ *ai* , ∅� *min* ,*t* ð Þ *ai* are deduced.

**Step 1.6**: Preference relations *St*(*ai*, *ak*Þ*ϵ*f g *I*, *P*, *Q*, *R* are deduced, depending on the values of ∅<sup>þ</sup> *max* ,*t* ð Þ *ai* , ∅<sup>þ</sup> *min* ,*t* ð Þ *ai* , ∅� *max* ,*t* ð Þ *ai* , ∅� *min* ,*t* ð Þ *ai* (see [8]).

#### **5.2 Phase 2: temporal aggregation**

Here the temporal aggregation procedure of MUPOM (Section 4.2) is used to aggregate the preference relations obtained over the periods in Step 1.6. As with the MUPOM method, the measure of distance between preorders developed in [19] is used.

#### **5.3 Phase 3: exploitation**

The temporal exploitation procedure of MUPOM (Section 4.3) is used in this phase. It computes the performance of each action *ai* based on the number of actions that are preferred (strictly or weakly) to *ai* and those that *ai* are preferred to (strictly or weakly).

#### **6. Case study**

In this section, MUPOM and PROMETHEE-MP are applied in the context of sustainable forest management. Sustainable forest management is a well-suited application context since it considers conflicting and heterogeneous criteria that should be assessed on about 150 years ahead. Actually, the selection of sustainable forest management options should arrive at a balance between biodiversity, soil and

short, medium, and long term. By taking into account immediate and future consequences of actions, it guarantees decisions are not made that compromise future

**5. PROMETHEE-MP: a generalization of PROMETHEE for multi-period**

PROMETHEE-MP is a recently developed temporal outranking method that allows aggregation of multi-periods and uncertain evaluations. It consists of three phases. Phase 1 aggregates the criteria, at each period of the horizon, based on PROMETHEE outgoing and incoming flows and Monte Carlo simulations. Binary relations are computed for each pair of actions. Phase 2 consists of aggregating the binary relations obtained over the periods using the measure of distance between preorders [19] as is done with MUPOM. Finally, in Phase 3 the performance of each action *ai* is computed, based on the number of actions that are preferred (strictly or weakly) to *ai* and those that *ai* are preferred to (strictly or weakly). **Figure 2** presents all the steps of PROMETHEE-MP. The following subsections provide details on the three phases. A full version of the mathematical details of the method

generations.

**evaluations under uncertainty**

*Sustainability Concept in Developing Countries*

is found in [8] (**Figure 3**).

**Figure 3.**

**24**

*Steps of PROMETHEE-MP [8].*

water conservation, forest productivity, socioeconomic benefits, and the population's values and needs. Second, the impact of each decision has to be assessed long term over the period of forest regeneration (about 150 years).

Five options are for consideration: (*a*1Þ a reference option corresponding to the terms of the intervention standards regulation; (*a*2) a removal of protected areas for wood production; ð*a*3) a specific plan for protecting the caribou habitat; (*a*4) a reforestation program; and (*a*5) a variable-level harvesting strategy which accelerates the harvest rate for the near periods. For evaluating these forest management options, we consider five criteria assessed every 5 years: (*C*1) the 5-year exploitable volume, ð*C*2) index of caribou habitat, (*C*3) good habitat for moose, (*C*4) old forest areas, and (*C*5) carbon footprint. **Figure 4** provides an example of the evolution over time (30 periods of 5 years) of criteria *C*2.

The AHP method was used to model the preferences in terms of criteria weights. A questionnaire was presented to an expert asking for pairwise comparisons between pairs of criteria and for the indifference, preference, and veto thresholds for each criterion, as well as the most appropriate criteria functions to be used with PROMETHEE. Also requested was the relative importance of periods. **Tables 1**–**3** present the weights and an overview of the data used for option *a*3, respectively. Used weights and data for MUPOM are crisp and for PROMETHEE-MP are intervals.

To start, Phase 1 of MUPOM and PROMETHEE-MP is applied. Results are obtained in terms of binary relations (P, Q, I, R, *Q*�<sup>1</sup> , *P*�<sup>1</sup> ) for each pair of actions

and for each period. **Table 4** shows the results of Phase 1 of MUPOM and

whereas PROMETHEE-MP shows f*a*5} as the only best compromise solution.

Results show that in a deterministic context and without considering uncertainty, the two options *a*<sup>2</sup> and *a*<sup>5</sup> are both of best compromise and incomparable. However, when considering uncertainty on the evaluations and weights, only *a*<sup>5</sup> is then of best compromise. It should first be noted that by modeling uncertainty on the evaluation and weights, as done with PROMETHEE-MP, the result is more robust because it takes into account the variability of evaluation over the intervals. However, comparison of results given by the two methods needs to take into account that they are not based on the same foundations. MUPOM uses

concordance-discordance principles as ELECTRE methods do, while PROMETHEE-

In future research, it will be important to validate the findings of the two models by comparing the obtained results with those given by a panel of expert in forest management. A Delphi procedure could be applied in order to get the opinion of experts on the results. A level of 70% of agreement between experts will be considered. This validation process will confirm the quality of the results given by the

MP uses outgoing and incoming flows as PROMETHHE methods do.

method.

**27**

**Period C1 (millions of m<sup>3</sup> )**

**Period C1 (millions of m<sup>3</sup> )**

P1 [37.05, 40.95] [0.561,

P2 [34.20, 37.80] [0.558,

P30 [18.05, 19.05] [0.540,

*Decision matrix for option C used with MUPOM.*

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

**Table 2.**

**Table 3.**

**C2 (in [0, 1])**

**C2 (in [0, 1])**

0.620]

0.617]

0.597]

*Decision matrix for option C used with PROMETHEE-MP.*

**C3 (thousands of hectares)**

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

P1 39 0.591 295 361 143,716,919 P2 36 0.588 297 362 145,123,580 ……… … … … ……… … … … P30 19 0.569 453 262 225,395,456

> **C3 (thousands of hectares)**

… …… … … … … …… … … …

**C4 (thousands of hectares)**

**C4 (thousands of hectares)**

[280.25, 309.75] [342.95, 379.05] [136,531,073;

[282.15, 311.85] [343.90, 380.10] [137,867,401;

[430.35, 475.65] [248.9, 275.1] [214,125,683;

**C5 (tons of CO2)**

**C5 (tons of CO2)**

150,902,765]

152,379,759]

236,665,229]

PROMETHEE-MP for the pair (*a*1, *a*2). Then, in Phase 2, the results for each period are aggregated using the temporal aggregation procedure. For each pair of actions, **Table 5** shows the aggregated relation which minimizes the distance between the relations obtained at each period and the set of preference relations (P, Q, I, P-1, Q-1). A graph representing relations between all pairs of actions illustrates the results. Phase 3 consists of exploiting the graph (**Figures 5** and **6**) and determining which action performs better. Results of MUPOM show {*a*2, *a*5} are the best compromise solutions,

#### **Figure 4.**

*Evolution over time of the criteria.*


**Table 1.** *Criteria weight intervals.*


*Temporal MCDA Methods for Decision-Making in Sustainable Development Context DOI: http://dx.doi.org/10.5772/intechopen.90698*

#### **Table 2.**

water conservation, forest productivity, socioeconomic benefits, and the population's values and needs. Second, the impact of each decision has to be assessed long term over the period of forest regeneration (about 150 years).

over time (30 periods of 5 years) of criteria *C*2.

*Sustainability Concept in Developing Countries*

obtained in terms of binary relations (P, Q, I, R, *Q*�<sup>1</sup>

intervals.

**Figure 4.**

**Table 1.**

**26**

*Criteria weight intervals.*

*Evolution over time of the criteria.*

Five options are for consideration: (*a*1Þ a reference option corresponding to the terms of the intervention standards regulation; (*a*2) a removal of protected areas for wood production; ð*a*3) a specific plan for protecting the caribou habitat; (*a*4) a reforestation program; and (*a*5) a variable-level harvesting strategy which accelerates the harvest rate for the near periods. For evaluating these forest management options, we consider five criteria assessed every 5 years: (*C*1) the 5-year exploitable volume, ð*C*2) index of caribou habitat, (*C*3) good habitat for moose, (*C*4) old forest areas, and (*C*5) carbon footprint. **Figure 4** provides an example of the evolution

The AHP method was used to model the preferences in terms of criteria weights.

A questionnaire was presented to an expert asking for pairwise comparisons between pairs of criteria and for the indifference, preference, and veto thresholds for each criterion, as well as the most appropriate criteria functions to be used with PROMETHEE. Also requested was the relative importance of periods. **Tables 1**–**3** present the weights and an overview of the data used for option *a*3, respectively. Used weights and data for MUPOM are crisp and for PROMETHEE-MP are

To start, Phase 1 of MUPOM and PROMETHEE-MP is applied. Results are

**Criteria Crisp weights for MUPOM Weights intervals for**

C1 5-year exploitable volume 0.1443 [0.137, 0.151] C2 Index of caribou habitat 0.3064 [0.291, 0.322] C3 Good habitat for moose 0.1606 [0.153, 0.169] C4 Old forest areas 0.3063 [0.291, 0.322] C5 Carbon footprint 0.0825 [0.078, 0.087]

, *P*�<sup>1</sup>

) for each pair of actions

**PROMETHEE-MP**

*Decision matrix for option C used with MUPOM.*


#### **Table 3.**

*Decision matrix for option C used with PROMETHEE-MP.*

and for each period. **Table 4** shows the results of Phase 1 of MUPOM and PROMETHEE-MP for the pair (*a*1, *a*2). Then, in Phase 2, the results for each period are aggregated using the temporal aggregation procedure. For each pair of actions, **Table 5** shows the aggregated relation which minimizes the distance between the relations obtained at each period and the set of preference relations (P, Q, I, P-1, Q-1).

A graph representing relations between all pairs of actions illustrates the results. Phase 3 consists of exploiting the graph (**Figures 5** and **6**) and determining which action performs better. Results of MUPOM show {*a*2, *a*5} are the best compromise solutions, whereas PROMETHEE-MP shows f*a*5} as the only best compromise solution.

Results show that in a deterministic context and without considering uncertainty, the two options *a*<sup>2</sup> and *a*<sup>5</sup> are both of best compromise and incomparable. However, when considering uncertainty on the evaluations and weights, only *a*<sup>5</sup> is then of best compromise. It should first be noted that by modeling uncertainty on the evaluation and weights, as done with PROMETHEE-MP, the result is more robust because it takes into account the variability of evaluation over the intervals. However, comparison of results given by the two methods needs to take into account that they are not based on the same foundations. MUPOM uses concordance-discordance principles as ELECTRE methods do, while PROMETHEE-MP uses outgoing and incoming flows as PROMETHHE methods do.

In future research, it will be important to validate the findings of the two models by comparing the obtained results with those given by a panel of expert in forest management. A Delphi procedure could be applied in order to get the opinion of experts on the results. A level of 70% of agreement between experts will be considered. This validation process will confirm the quality of the results given by the method.


#### **Table 4.**

*Preference relations resulting from multi-criteria aggregation for pair (a*1, *a*2*).*


horizon and limit the effect of the aggregation. Results will show if the best compromised option on the whole horizon will differ or not from to the best

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

This paper presents the main results of a recent research program on developing temporal outranking MCDA methods. It presents two generalizations of outranking methods to temporal context to show how outranking methods can be of use in processing the temporal impacts of decisions. The state of the art in this research area still remains limited, and such a proposal is valuable to support sustainable decision-making processes. This paper exposes two recent temporal outranking methods and displays the results of their application in SD context. The MUPOM method demonstrates how outranking methods and, more specifically, the

ELECTRE concordance-discordance principles can be of use in processing temporal impacts of decisions. PROMETHEE-MP consists of a multi-period generalization of PROMETHEE under random uncertainty using Monte Carlo simulations. Their

compromised options in the short, medium, and long term.

application on the same case study shows their applicability.

**7. Conclusion**

**29**

*Exploitation graph with PROMETHEE-MP.*

**Figure 6.**

**Figure 5.**

*Exploitation graph with MUPOM.*

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

#### **Table 5.**

*Preference relations resulting from temporal aggregation for each pair of actions.*

Besides, for stronger interpretation of results, future work will focus on applying the proposed methods on different horizons. For instance, in our case study, we can apply MUPOM and PROMETHEE-MP on the short-term horizon (aggregation of evaluations of the first 20 years), the medium term (aggregation of evaluations of year 20 to year 50), and finally the long term (aggregation of evaluations of year 50 to year 150). By doing so, we can compare the different results depending on the

*Temporal MCDA Methods for Decision-Making in Sustainable Development Context DOI: http://dx.doi.org/10.5772/intechopen.90698*

**Figure 5.** *Exploitation graph with MUPOM.*

**Figure 6.** *Exploitation graph with PROMETHEE-MP.*

horizon and limit the effect of the aggregation. Results will show if the best compromised option on the whole horizon will differ or not from to the best compromised options in the short, medium, and long term.

### **7. Conclusion**

This paper presents the main results of a recent research program on developing temporal outranking MCDA methods. It presents two generalizations of outranking methods to temporal context to show how outranking methods can be of use in processing the temporal impacts of decisions. The state of the art in this research area still remains limited, and such a proposal is valuable to support sustainable decision-making processes. This paper exposes two recent temporal outranking methods and displays the results of their application in SD context. The MUPOM method demonstrates how outranking methods and, more specifically, the ELECTRE concordance-discordance principles can be of use in processing temporal impacts of decisions. PROMETHEE-MP consists of a multi-period generalization of PROMETHEE under random uncertainty using Monte Carlo simulations. Their application on the same case study shows their applicability.

Besides, for stronger interpretation of results, future work will focus on applying the proposed methods on different horizons. For instance, in our case study, we can apply MUPOM and PROMETHEE-MP on the short-term horizon (aggregation of evaluations of the first 20 years), the medium term (aggregation of evaluations of year 20 to year 50), and finally the long term (aggregation of evaluations of year 50 to year 150). By doing so, we can compare the different results depending on the

**Period Preference**

**Table 4.**

**Table 5.**

**28**

**Pair Aggregated**

**relation with MUPOM**

**relation resulting from MUPOM**

*Sustainability Concept in Developing Countries*

**Preference relation resulting from PROMETHEE-MP**

*Preference relations resulting from multi-criteria aggregation for pair (a*1, *a*2*).*

**Aggregated relation with PROMETHEE-MP**

*Preference relations resulting from temporal aggregation for each pair of actions.*

(*a*1, *a*2) Q-1 P-1 (*a*2, *a*4)R P (*a*<sup>2</sup> ,*a*1)Q P (*a*4, *a*2) R P-1 (*a*<sup>1</sup> ,*a*3)P P (*a*2, *a*5) R P-1 (*a*<sup>3</sup> ,*a*1) P-1 P-1 (*a*5, *a*2)R P (*a*<sup>1</sup> ,*a*4) R Q-1 (*a*3, *a*4) Q-1 P-1 (*a*<sup>4</sup> ,*a*1)R Q (*a*4, *a*3)Q P (*a*<sup>1</sup> ,*a*5) R P-1 (*a*3, *a*5) R P-1 (*a*<sup>5</sup> ,*a*1)R P (*a*5,*a*3)R P (*a*<sup>2</sup> ,*a*3)Q P (*a*4, *a*5) R P-1 (*a*3, *a*2) Q-1 P-1 (*a*5, *a*4)R P

 P-1 P-1 16 R P-1 P-1 P-1 17 Q-1 P-1 Q P-1 18 Q-1 P-1 Q P-1 19 Q-1 P-1 P Q-1 20 Q-1 Q-1 P I 21 Q-1 Q-1 P I 22 Q-1 Q-1 R I 23 Q-1 I R Q-1 24 Q-1 I Q Q-1 25 Q-1 I R P 26 Q-1 P-1 R R 27 Q-1 P-1 Q Q 28 Q-1 P-1 Q R 29 Q-1 I R Q-1 30 Q-1 P-1

**Period Preference**

**Pair Aggregated**

**relation with MUPOM**

**Aggregated relation with PROMETHEE-MP**

**relation resulting from MUPOM**

**Preference relation resulting from PROMETHEE-MP**

*Sustainability Concept in Developing Countries*
