EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1731-1743 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global The ranking of districts in Ouagadougou by the risk of flood and runoff using the PROMETHEE Jacob Wendpanga Yougbaré Department of Mathematics, Faculty of Sciences and Technologies, University Norbert Zongo, Koudougou, Burkina Faso Abstract. PROMETHEE methods are used for analyzing the risk of flood runoff in urban areas of Ouagadougou in Burkina Faso. 12 criteria are considered to calculate preference indices of the districts. We generated, on one hand, weights associated to these criteria taking into account hypotheses relatively to the importance or to the impact of each criterion evaluating the risk of flood runoff and on the other hand, we used the PROMETHEE methods to analyze this decision’s problem with its diverse criteria. The PROMETHEE I and PROMETHEE II are used ranking the districts in relation with the degree of risk of flood runoff. This ranking is permitted by the calculated preference indices. Our aim is, on one hand showing to the potential decision-maker the importance of such a multi-criteria analysis and on the other hand, of verifying the coherence of the hypotheses with the obtained weights 2010 Mathematics Subject Classifications: 90B50 Key Words and Phrases: Multi-criteria Decision Aiding, PROMETHEE method, preference indices, preference function, urban management 1. Introduction In the current context of global warming and of issues of sustainable flood risk man- agement in urban areas also occurs as a major concern. It appears therefore appropriate to consider issues relating to flood risk in particular by runoff in cities [18]. So, it is important that the town and country planning takes into account questions of streaming. For example, September 1st, 2009, municipality of Ouagadougou in Burkina Faso knew one of his most devastating floods with various victims. This flood was largely connected by the streaming of rainwater. Some studies have clearly highlighted the relationship between changes in land use and changes in volumes of water run-off. The concept of decision-making is widely used. There are many definitions of decision-making process. In any case, there are different levels in DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3562 Email address: jwyougby@yahoo.com (J. W. Yougbaré) http://www.ejpam.com 1731 c© 2019 EJPAM All rights reserved. J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1732 the process of decision-making: identification and definition of problems, determining col- lection of alternatives, determining collection of criteria for alternative evaluation, alter- native evaluation and, finally, alternative sorting, description, choice or ranking. Over the last few decades, new methods have been found and the methodology of decision-making process has been improving. In order for aiding the decision-making process, many math- ematical methods have been suggested. The Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) represents one of the most frequently used methods of multi-criteria decisions. Besides this method, other ones are also available[11] such as ELECTRE, AHP, ORESTE, etc. Yougbaré and Datoloumbeye (2017) [18] used the ORESTE method analyzing the risk of flood runoff in urban areas in Ouagadougou. The ORESTE method is a method of ordinal ranking (based on ranks) which permits to avoid assigning weighs to criteria. Consequently, it comes from a non-compensatory logic. In this paper, the methods PROMETHEE I and II are used for runoff risk comparison of the districts of Ouagadougou, which represent alternative solutions. The urban space management problem is multicriteria analysis problem with diverse conflicting criteria, protect on one side, the low districts of the cities against floods and on the other side the environment against the discharge amounts of pollutants exceeding the absorptive capacity of the natural environment. One solution to this problem is to make a ranking of districts that have a susceptibility to runoff to determine what are vulnerable will be the priority studies to prevent and manage the risk associated with runoff. Our goal in this paper is using PROMETHEE methods calculating preference indices, analyzing and ranking the districts, by comparing the risk of flooding by runoff from different districts of Ouagadougou which represent alternative solutions. According to 12 criteria their mutual risk is evaluated. This paper is organized as follow: firstly, we present Related Literature, next the mathematical model used, then we present the methodology by highlighting the alternatives, criteria and Data used applying the multicriteria method used here finally before concluding we present and analyze the results obtained. 2. Related Literature Yougbaré and Datoloumbeye (2017) [18] used the ORESTE method analyzing the risk of flood runoff in urban areas in Ouagadougou. The ORESTE method is a method of ordinal ranking (based on ranks) which permits to avoid assigning weighs to crite- ria. Consequently, it comes from a non-compensatory logic. PROMETHEE methods and their applications have attracted much attention from academics and practitioners. Sev- eral authors have been interested in the development and application of the methods of Multicriteria Analysis in general and the PROMETHEE method in particular. As such, we can note certain works such as: • Morkunaite et al. (2019) [10], used PROMETHEE technique for applying the selec- tion of the most efficient cultural heritage contractor’s alternative; J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1733 • Yan et al. (2019)[17], used PROMETHEE for ranking four alternative energy sys- tems for heating; • Dirutigliano et al. (2018) [5] presented an application of the PROMETHEE method for ranking different alternatives of building retrofitting at the building and district level; • In the work of Zindani and Kumar (2018) [19] , material selection for seal strips used in turbines has been done applying PROMETHEE; • He and Xu (2018) [7] applied the PROMETHEE II method to obtain the decision results as the standard of the group preferences; • Coban et al (2018) [3] used PROMETHEE I and II for 8 solid waste disposal scenarios evaluation; • Lopes et al. (2018) [9], applied the PROMETHEE and GAIA methods within a competitiveness study of eight tourist destinations located in the Northern Region of Portugal; • In the paper of Inamdar et al. (2018) [8], PROMETHEE II method is used for evaluating and ranking of framework selects potential stormwater harvesting sites; • In Andreopoulou et al. (2018) [2], PROMETHEE II is used for ranking websites to support market opportunities. They ranked websites and proposed to use the best website to ”form a conceptual content model while designing an enhanced website for a Renewable Energy Sources (RES) enterprise” ; • Vulevic̀ and Dragovic̀ (2017) [16], used the PROMETHEE II method ranking of nine sub-watersheds delineated in the Topciderska river watershed, Serbia; • Silas and Rajsingh (2016) [12] applied different multi-criteria approaches ELECTRE, PROMETHEE and AHP for health care service applications. They concluded that the PROMETHEE method is best suited for solving multi-criteria decision making problem in the selection of desired health care services; • In (2015) [14], PROMETHEE method is used by Veza et al. for ranking industrial enterprises based on enterprise’s competences; • Babaee et al. (2015) study allowed ranking 55 drivers, aged 70 and older by using PROMETHEE II; • Vetschera and Teixeira de Almeida (2012) [13] studied the use of PROMETHEE outranking methods for portfolio selection problems; • Vinodh and Girubha(2012) [15] PROMETHEE is used in the study to select the best sustainable concept considering the criteria from social, economic and natural perspectives; J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1734 • Ghafghazi et al. (2010) [6] used PROMETHEE ranking energy sources available for a case of district heating system in Vancouver, Canada. This list cannot be exhaustive. All these studies clearly show the importance of the PROMETHEE approach in everyday applications. 3. Mathematical model PROMETHEE methods are developed by Brans and Vinck in 1982 [1, 4]. The principle of PROMETHEE method is based on mutual comparison of each alter- native pair with respect to each of the selected criteria. In order to perform alternative ranking by the PROMETHEE method, it is necessary to define preference function ph(x, y) for alternatives x and y after defining the criteria gh. Alternatives x and y are evaluated according to the criteria functions. It is considered that alternative x is better than alter- native y according to criterion gh, if gh(x) > gh(y) for the criterion gh to be maximized. The decision maker has possibility to assign the preference to one of the alternatives on the basis of such comparison. The preference can take values on the scale from 0 to 1, and relation combinations are possible to represent using following relations: • If ph(x, y) = 0 then there is indifference between x and y or no preference of x on y according to the criterion gh; • If ph(x, y) ≈ 0 then there is weak preference of x on y according to the criterion gh; • If ph(x, y) ≈ 1 then there is strong preference of x on y according to the criterion gh; • If ph(x, y) = 1 then there is strict preference of x on y according to the criterion gh. In any case we have 0 ≤ ph(x, y) ≤ 1. For each criterion gh, the decision maker considers certain preference function. The generalized criteria are defined by the preference functions which are in number of six. If we consider that gh is to be maximized, we summarize the six preference functions such as: • Type I: regular criterion: for this preference function, we have ph(x, y) = { 0 if g(x)− g(y) ≤ 0 1 if g(x)− g(y) > 0 (1) • Type II: U-sharpe criterion: for this preference function, we have ph(x, y) = { 0 if g(x)− g(y) ≤ Q 1 if g(x)− g(y) > Q (2) with Q parameter representing the Indifference level; J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1735 • Type III: V-sharpe criterion: for this preference function, we have ph(x, y) =  0 if g(x)− g(y) ≤ 0 g(x)−g(y) P if 0 < g(x)− g(y) ≤ P 1 if g(x)− g(y) > P (3) with P parameter representing the strict Preference level; • Type IV: Level criterion: for this preference function, we have ph(x, y) =  0 if g(x)− g(y) ≤ Q 0.5 if Q < g(x)− g(y) ≤ P 1 if g(x)− g(y) > P (4) with P,Q parameter representing respectively the strict Preference level and Indif- ference level; • Type V: linear criterion: for this preference function, we have ph(x, y) =  0 if g(x)− g(y) ≤ Q g(x)−g(y)−Q P−Q if Q < g(x)− g(y) ≤ P 1 if g(x)− g(y) > P (5) with P,Q parameter representing respectively the strict Preference level and Indif- ference level; • Type VI: Gauss criterion: for this preference function, we have ph(x, y) = 1− e −(g(x)−g(y))2 2σ2 (6) with σ parameter of the function. After defining the type of preference function, it is necessary to determine the value of preference function of alternative x in relation to alternative y for each criterion, and calculate the index of preferences Π(x, y) of alternative x in relation to alternative y. The index preference is calculated in the following way: Π(x, y) = k∑ h=1 whph(x, y) (7) where wh is the weight of criterion gh with 0 < wh < 1 and k∑ h=1 wh = 1. (8) After determining index preference Π(x, y), it is finally possible to calculate alterna- tive flow index, the flow of outclassing which makes it possible to appreciate how each alternative behaves in other faces. If n represents the number of alternatives of A, we calculate: J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1736 • The flow of outgoing outclassing Φ+(x) = 1 n− 1 ∑ y∈A Π(x, y) (9) that provides the index indicating the force of alternative x compared to others alternatives; • The flow of incoming outclassing Φ−(x) = 1 n− 1 ∑ y∈A Π(y, x) (10) that provides the index indicating the weakness of alternative x compared to others alternatives; • The net flow of outclassing Φ(x) = Φ+(x)− Φ−(x). (11) It is the difference between Φ+(x) and Φ−(x). When the alternative x is generally preferred to others it is positive if not it is negative. If the different flow are obtained, it is possible defining the relation of preference according PROMETHEE I or PROMETHEE II. PROMETHEE I The PROMETHEE I method is to classify the actions by decreasing incoming flow and increasing outgoing flow. Consider P I , II and RI respectively the relations of preference, indifference and incomparability. xP Iy ⇐⇒  Φ+(x) > Φ+(y) and Φ−(x) < Φ−(y) or Φ+(x) = Φ+(y) and Φ−(x) < Φ−(y) or Φ+(x) > Φ+(y) and Φ−(x) = Φ−(y) (12) xIIy ⇐⇒ Φ+(x) = Φ+(y) and Φ−(x) = Φ−(y) (13) xRIy othewise. (14) The PROMETHEE I method provides a partial classification of actions. PROMETHEE II The PROMETHEE II method allows us to classify all the actions and leaves no in- comparable action with others. It provides a total pre-order of actions. Consider P II , III and RII respectively the relations of preference, indifference and incomparability. xP IIy ⇐⇒ Φ(x) > Φ(y) (15) xIIIy ⇐⇒ Φ(x) = Φ(y) (16) With PROMETHEE II, there is no incomparability. J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1737 4. Methodology, Alternatives, criteria and Data In order to make a multicriteria analysis of this issue, the PROMETHEE I and II are been chosen. These methods have the ability to take into account the relative importance between different criteria depending on the decision maker. The PROMETHEE I and II methods deal with the issue of ranking or classification. These methods are advantageous because they can compare complex situations, naturally incomparable situations in includ- ing criteria of different natures (objective and subjective). We considered as alternatives the districts: BAS: Baskuy, BOG: Bogodogo, BOU: Boulmiougou, NON: Nongre-maasom and SIG: Sig-Nonghin. The districts will constitute our set of alternatives. The criteria make it possible to assess the alternatives with regard to the respect for the principles of the town and country planning, and the principles of ecology. The combination of these factors (sub-criteria) was achieved by taking into account the planning principles and the principles of ecology [18]. The criteria built to meet, these planning principles and ecology, are: • z1, whose ecological objective is to limit soil sealing. The sub-criteria considered are: g1, modern proportion of habitat (high buildings, medium, low or mixed); g2, areas of economic activities (industrial and commercial areas, the formal market, stations, airport, landfill and mining materials); g3, proportion of traditional housing (rural housing and low buildings); g4, agricultural areas (rainfed crops, horticulture and agriculture); • z2, whose objective is to use management techniques with less impact (negative). The sub-criteria considered are: g5, area green space and planting (plantations and green spaces); g6, recreational areas (sports and recreation grounds); • z3, whose objective is to minimize the development (urbanization) in sensitive areas. The sub-criteria considered are: g7, the natural vegetation surfaces (gallery forest, shrubland, grassland and orchards); g8, surfaces of marshy meadows; • z4, with the goal of sustainability is to preserve as much as possible the natural areas in an undisturbed state. The sub-criteria considered are: g9, urbanized area in flood areas (temporary water bodies and wetlands); g10, free water surface (courses and waterways, natural and artificial lake); g11, soil type (hydromorphic soils); • z5, which aims to improve prevention. The sub-criterion considered is: g12, capacity of hospitals. By doing operations in order to make a ranking of alternatives by the PROMETHEE method we used, the data in the table 2 and the sub-criteria (gh, h = 1, . . . , 12). Using the criteria ranking provided by the expert [18], we have the results with are summarized in the table 1 J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1738 Table 1: Criteria average rank. Min/max z1 z2 z3 z4 z5 Criterion min max min max max rank 1 3.5 2 5 3.5 This ranking gives a relationship linking the weights (Λj , j = 1, . . . , 5) of these different criteria (zj , j = 1, . . . , 5), 5∑ j=1 Λj = 1, Λ1 > Λ3 > Λ2 = Λ5 > Λ4. (17) Using the sub-criteria (of the five criteria) ranking (provided by the expert [18]) we have the results with are summarized as follows table 2. This ranking gives a relationship linking the weights (λh, h = 1, . . . , 12) of these differ- ent sub-criteria (gh, h = 1, . . . , 12), Λ1 = 4∑ h=1 λh, Λ2 = 6∑ h=5 λh, Λ3 = 9∑ h=7 λh, Λ4 = 11∑ h=10 λh, Λ5 = λ12. (18) λ2 > λ1 > λ3 > λ4. (19) λ6 > λ5. (20) λ7 > λ8 > λ9. (21) λ11 > λ10. (22) The relations (17), (18), (19), (20), (21)and (22) linking the weights between them allowed us to generate weights associated with the different criteria. It must be empha- sized that in our case, the determination of weights could not be explicit. We obtained the weights of criteria summarized in the table 3. J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1739 Table 2: Performance matrix. Criteria Sub-criteria Rang Min/max BAS BOG BOU NON SIG z1 g1 2 min 52.15 72.72 42.089 30.22 53.98 g2 1 min 29.77 3.2 2.383 3.58 4.53 g3 3 min 0 0.12 14.953 6.23 2.83 g4 4 min 1.59 15.07 25.721 23.88 23.75 z2 g5 2 max 0.86 0.47 1.168 9.03 1.81 g6 1 max 3.78 0.93 0.546 0.24 1.63 z3 g7 1 min 2.75 0.55 0.93 0.40 0.31 g8 2 min 4.6 0.74 4.706 5.17 3.56 g9 3 min 98.76 87.47 97.29 38.60 77.97 z4 g10 2 max 0 4.57 6.488 18.22 5.74 g11 1 max 1.04 0.07 0.247 1.19 0.78 z5 g12 1 max 985.98 1771.54 801.51 1795.14 53.98 Table 3: Weights of criteria. λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12 0.0029 0.0040 0.0022 0.0019 0.0345 0.1542 0.3203 0.0788 0.0596 0.0850 0.0679 0.1887 5. Results The essence of the problem is, using mathematical support, to find indices Φ+(x),Φ−(x),Φ(x) of the five districts of Ouagadougou. These indices evaluate performances, strengths and weaknesses in terms of risk of flooding by runoff. According to these indices, it is possible to determine the level of risk of flooding by runoff of the districts. To calculate these indices, we considered the preference function on the table 4 . Table 4: Preference functions of criteria. Sub-criteria Type Parameter g1 Type III P= 14.07 g2 Type III P= 10.56 g3 Type III P= 5.55 g4 Type III P= 9 g5 Type III P= 3.21 g6 Type III P= 1.27 g7 Type III P= 0.91 g8 Type III P= 1.6 g9 Type III P= 22.02 g10 Type III P= 6.04 g11 Type III P= 0.44 g12 Type V Q= 108.16; P= 652.5 J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1740 The different preference functions considered applying are summarized by the table 4 and indices Φ+(x),Φ−(x),Φ(x) calculated are summarized in the table 5 . Table 5: Flow of districts. BAS BOG BOU NON SIG Φ+(x) 0.2722 0.4007 0.1867 0.5223 0.4054 Φ−(x) 0.5866 0.2256 0.4816 0.1699 0.3235 Φ(x) -0.3144 0.1751 -0.2950 0.3524 0.0819 With PROMETHEE I method the indices obtained provide the following relations (P= Preference, R= Incomparability, I= Indifference) summarized by the table 6. Table 6: Relation with PROMETHEE I. BAS BOG BOU NON SIG BAS R BOG P P R BOU R NON P P P P SIG P R P This table provides the graph of outranking given by the following figure 1. Figure 1: Graph of outranking of the districts by PROMETHEE I. With PROMETHEE I we obtained that the district of Nongr-maasom is best ranked as the others and there are two incomparability relations: Baskuy and Boulmiougou are considered incomparable. The first has a better force measured by the flow Φ+(x) than the second and the latter has a low weakness so desirable than the first. We cannot base J. W. Yougbaré / Eur. J. Pure Appl. Math, 12 (4) (2019), 1731-1743 1741 on these two indices to distinguish them; Sig-Nonghin and Bogodogo are also considered incomparable. The first has a better force measured by the flow Φ+(x) than the second and the latter has a low weakness so desirable than the first. We cannot base on these two indices to distinguish them; As for the district of Nongr-maasom, the application of PROMETHEE I showed that it is better ranked than all others. It appears from this study that Sig- Nonghin is the district less vulnerable in terms of flood runoff and the two incomparable districts, Baskuy and Boulmiougou, the most vulnerable districts. With PROMETHEE II we obtained the following relations summarized by the figure 2. Figure 2: Graph of outranking of the districts by PROMETHEE II. It shows the following result NON � BOG � SIG � BOU � BAS It appears from this study that Nongre-maasom is the district less vulnerable in terms of flood runoff and Baskuy the most vulnerable district. Comments In the paper of Yougbaré and Datoloumbeye (2017) [18], with the same data the results show that Boulmiougou is more vulnerable than the others districts. Here, with PROMETHEE I the results show that Boulmiougou and Baskuy are incomparable and they are the most vulnerable districts than the others. With PROMETHEE II the results show that Baskuy is the most vulnerable district. In the paper of Yougbaré and Datoloumbeye (2017) [18], results show that Bogodogo, Nongre-Massom and Sig-Nonghin are incomparable and by compensatory Sig- Nonghin is the district which is less vulnerable in terms of flood runoff. Here, with PROMETHEE I and PROMETHEE II, the results show that Nongre-Massom is the district which is less vulnerable in terms of flood runoff. These two studies show that the Boulmiougou and Baskuy areas need to be monitored more because they have a higher risk of runoff flooding than the others. 6. Conclusion The purpose of this study is to compare and classify the districts by risk of flooding and to guide the decision-maker by proposing values representing solutions verifying the crite- REFERENCES 1742 ria and constraints set by the experts. The aim of the research in this paper is to obtain preference indices of risk of runoff of the districts of Ouagadougou by using multi-criteria analysis. In this paper, the PROMETHEE method is used, as well as a mathematical tool in order to obtain preference indices. The PROMETHEE method is ranked as one of the most famous and most frequently used methods of multi-criteria decisions. We generated the weights of criteria according the hierarchic relation between these criteria. By using the PROMETHEE method, preference indices are obtained for the districts of Ouagadougou on the basis of twelve criteria. According to these results, Nongre-Massom emerged as the district less vulnerable in terms of flood runoff. 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