EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 4195-4210 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Comparative Analysis of Four Group Decision-Making Techniques: KEMIRA G-I, KEMIRA G-II, Lon-Zo, and MACASP Naguiesmongho Christian Nana2, Stéphane Aimé Metchebon Takougang1,2,∗, T. Benoît Joseph Batieno3 1 Faculty of Sciences and Technologies, New Dawn University, 06 B.P. 9283 Ouagadougou 06, Ouagadougou, Burkina Faso 2 Laboratory of Numerical Analysis of Computer Science and Biomathematics, Joseph Ki-Zerbo University, 03 B.P. 7021 Ouagadougou 03, Ouagadougou, Burkina Faso 3 National Institute of Environmental and Agricultural Research (INERA), Kamboinsé 03 BP 7047 Ouagadougou 03, Burkina Faso Abstract. Most selection problems are multi-decision and multi-criteria in nature. The group decision (GD) literature presents several methods for solving them. Most of them belong to utilities functions based class. However, the use of any one group decision method of this class for a specific problem is often not appropriate, given the characteristics of the latter. In this the present work is to compare four GD utility functions based methods, two of which are classical (Lon-Zo and MACASP) and two new (KEMIRA G-I and KEMIRA G-II), by examining their suitability for solving two multi-criteria choice problems, namely the selection of a crop variety adapted to the Centre-Est region of Burkina Faso and the selection of a site for the implementation of a waste incineration plant in the city of Vilnius in Lithuania. The results show that group decision methods based on aggregation utility functions are most suitable when the criteria are homogeneous (i.e. when criteria can compensate naturally). However, when the criteria are heterogeneous (i.e. when there is no natural compensation between criteria), these methods can still be successfully applied when the heterogeneous nature of the criteria is taken into account. This explains the good performance of the KEMIRA G-I and KEMIRA G-II methods, which take into account the heterogeneous nature of the criteria, compared with the Lon-Zo and MACASP methods, which do not. 2020 Mathematics Subject Classifications: 90B50, 90B90, 91B10, 91B14. Key Words and Phrases: Group decision, KEMIRA G-I, KEMIRA G-II, Lon-Zo, MACASP, Borda method ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5488 Email addresses: christ92nana@gmail.com (N. C. Nana), metchebon@gmail.com (S. A. Metchebon T.), batieno52@gmail.com (T. B. J. Batieno) https://www.ejpam.com 4195 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4196 1. Introduction Selection problems are generally multi-criteria and multi-decision-maker in nature. To solve them, the literature on group decision support generally presents three categories of methods: Those based on outranking relations[1, 4], multi-attribute utility theory[11, 12] and interactive methods[2]. In this work, we are interested in the class of methods based on multi-attribute utility theory. More specifically in this paper we focus on the classical Lon- Zo [11] and MACASP [11] methods and two new methods KEMIRA G-I[10] and KEMIRA G-II, both extensions of the KEMIRA method[7]. The classical Lon-Zo and MACASP methods are based on the harmonic and arithmetic mean, respectively. The two new methods are based on the KEmeny Median Indicator Ranks Accordance ( KEMIRA[7] ) method and the Borda [8] method. All these methods have their own advantages and disadvantages. The Lon-Zo and MACASP methods are often used for ranking problems, where al- ternatives are ranked from best to worst[11, 15]. The KEMIRA G-I and KEMIRA G-II methods are suitable for solving, in general, the multi-criteria problem when the set of criteria is divided into a few homogeneous sub-groups of criteria, i.e. criteria between which compensation is naturally possible[5, 7, 10]. For example, with two criteria such as average annual economic profit and average annual wage, we could naturally accept compensation between them when aggregating them using a utility function because they are naturally expressed in the same monetary unit. However, if we consider two criteria such as average annual economic profit and average annual level of education, aggregating the latter two using a utility function would naturally not be acceptable. which is why the last two criteria are referred to as heterogeneous. The KEMIRA G-I and KEMIRA G-II methods can be used simultaneously to elicit criteria weights and to select the best alternatives by eliminating certain alternatives ac- cording to predefined performance thresholds. Through two case studies, we propose to compare the two methods of group decision, Lon-Zo and MACASP with the two new methods KEMIRA G-I and KEMIRA G-II The rest of our paper is organised as follows. In Section 2, we present the classic Lon-Zo and MACASP methods. Section 3 describes the new KEMIRA G-I and KEMIRA G-II methods. Section 4 is dedicated to the results of applying the four methods to the two case studies. In section 5 we discuss the different results obtained in section 4. We conclude our work and open the door to future work in section 6. 2. Presentation of Lon-Zo and MACASP method In what follows, we adopt the following notations: • D = {d1, d2, · · · , dL} the set of Decision Makers (DMs), (L ≥ 2) with L the number of DMs; • A = {a1, a2, ..., aK} the set of alternatives, (K ≥ 2) and K is the total number of alternatives; N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4197 • G = {g1, g2, · · · , gm} the set of criteria; • wdl j the weight assigned to criterion j par by Decision Maker dl; • gdlj (ak) the partial evaluation of the alternative ak w.r.t. criterion gj by the DM dl. 2.1. Lon-Zo method The Lon-Zo method uses the weighted sum and harmonic mean as aggregation func- tions. By weighted sum, the overall performance gdl(ak) given to each alternative by the decision-maker dl is determined by equation (1): gdl(ak) = j=m∑ j=1 wdl j .gdlj (ak), k = 1, · · · ,K, j = 1, · · · ,m, (1) where m is the total number of criteria. The harmonic mean is used to determine the overall evaluation (or performance) g(ak) of the action ak. It is defined by equation (2): g(ak) = L L∑ l=1 1 gdl(ak) . (2) 2.2. MACASP method MACASP[11] uses the weighted sum and arithmetic mean as aggregation functions. The evaluation that decision-makers give by consensus on alternative ak with regard to criterion j is gj(a k) and it is defined by equation (3): gj(a k) = L∑ l=1 wdl j .gdlj (ak), k = 1, · · · ,K, j = 1, · · · ,m. (3) The overall performance g(ak) of the alternative ak is obtained according to equation (4) : g(ak) = 1 m m∑ j=1 gj(a k) = 1 m m∑ j=1 ( L∑ l=1 wdl j .gdlj (ak)). (4) 3. Description of KEMIRA G-I and KEMIRA G-II methods In this section we introduce the following notations: N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4198 • G denotes the set of criteria that can be partitioned into S groups Gi such that Gi = {(i, 1), (i, 2), · · · (i, ni)} with i ∈ {1, · · · , S}, where ni is the number of criteria inside the group Gi, and (i, j) denotes the criterion gj inside the group Gi. So G = G1 ∪G2 ∪ · · · ∪GS . • ak,dli,j denotes the performance of the alternative ak with respect to criterion gj inside the group Gi given by the DM dl and wdl i,j the weight of criterion gj inside the group Gi, given by the DM dl. We assume that each decision-maker is able to rank the criteria in each group Gi from most to least important[3, 9, 10], as specified by relation (5). Without loss of generality, we also assume that all criteria are to be maximized. (i, 1)d1 ≿ (i, 2)d1 ≿ · · · ≿ (i, n1) d1 (i, 1)d2 ≿ (i, 2)d2 ≿ · · · ≿ (i, ni) d2 ... ... ... (i, 1)dL ≿ (i, 2)dL ≿ · · · ≿ (i, ni) dL . (5) 3.1. KEMIRA G-I method In what follows, we present the main stages of KEMIRA G-I method[10]. 3.1.1. Step 1: median ranking of criteria in descending order of preference Applying Borda’s voting method [8] based on relation (5), we obtain the median ranking of criteria for the set D of decision-makers in each group Gi as specified in equation (6) and of course its corresponding weights ranking in equation (7) such that relation (8) holds. (i, 1) ≿ (i, 2) ≿ · · · ≿ (i, ni), ∀i ∈ {1, 2, . . . , S} (6) wi,1 ≿ wi,2 ≿ · · · ≿ wi,ni , ∀i ∈ {1, 2, . . . , S} (7) ni∑ j=1 wi,j = 1,∀i ∈ {1, . . . , S}. (8) 3.1.2. Step 2: calculating average performance The average performance Wi(a k) of each alternative ak with respect to each group of criteria Gi is then determined according to equation (9). Wi(a k) = ni∑ j=1 a∗ki,j .wi,j , (9) where a∗ki,j is the normalized performance of aki,j obtained following the equation (10) a∗ki,j = aki,j −minj a k i,j maxj aki,j −minj aki,j , ∀i ∈ {1, 2, . . . , S}. (10) N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4199 3.1.3. Step 3: optimization problem This stage consists of formulating an optimization problem that elicits the decision-makers’ preferences, i.e., the weights of the criteria, and also determine the best alternatives for the decision-makers as a whole, by first setting the performance thresholds for each group Gi of criteria. This optimization problem is defined by the equation (11). maxwi,j fopt = |B| s.t.  wi,1 ≥ wi,2 ≥ · · · ≥ wi,ni ,∀i ∈ {1, 2, · · · , S},∑ni j=1wi,j = 1, ∀i ∈ {1, 2, · · · , S}, Wi(a k) > αi, ∀i ∈ {1, 2, · · · , S}, (11) where • fopt is the value of the objective function; • αi is the performance threshold associated to the group Gi, set by the decision-maker; • B = {ak : Wi(a k) > αi, i ∈ {1, 2, · · · , S}} denotes the set of best alternatives; • |B| denotes the number of elements of B. 3.1.4. Step 4: Choosing best alternative(s) The choice of a best alternative(s) is based on the following two conditions: • if for the highest possible threshold we have a single alternative, it will be considered the best alternative; • if for the highest possible threshold, we have at least two alternatives, each decision- maker is asked to rank the alternatives according to his preferences. Then the Borda method is used to obtain a median ranking. This median ranking gives the best alternative(s). 3.2. KEMIRA G-II method In contrast to KEMIRA G-I, in KEMIRA G-II each decision-maker completes the process of choosing the best alternative(s). The intersection and reunion of the sets of best solutions found by each decision-maker is then exploited. Formally, under the hypothesis stipulated by relation (5), the main stages of the KEMIRA G-II method are as follows. 3.2.1. Step 1: Calculate average performance For each decision-maker dl the average performance W dl i (ak) of each alternative ak is determined according to equation (12): N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4200 W dl i (ak) = ni∑ j=1 a∗k,dli,j .wdl i,j (12) where a∗k,dli,j is the normalized performance of ak,dli,j obtained following the equation (13) a∗k,dli,j = ak,dli,j −minj a k,dl i,j maxj a k,dl i,j −minj a k,dl i,j ,∀i ∈ {1, 2, . . . , S}, ∀l ∈ {1, 2, . . . , L}. (13) 3.2.2. Step 2: optimization problem The optimization problem used to elicit the weights of the criteria and to select the best alternatives according to the preferences of each decision-maker is defined by equation (14). The performance thresholds αi for each group of Gi must first be set by each decision-maker dl: maxwi,jfopt = |Bdl | s.t.  wdl i,1 ≥ wdl i,2 ≥ · · · ≥ wdl i,ni ,∀i ∈ {1, 2, · · · , S},∑ni j=1w dl i,j = 1, ∀i ∈ {1, 2, · · · , S}, W dl i (ak) > αi, ∀i ∈ {1, 2, · · · , S}, (14) where • αi denotes the performance threshold of the group Gi; • Bdl = {ak : W dl i (ak) > αi, i ∈ {1, 2, · · · , S}} denotes the set of best alternatives; • |Bdl | denotes the number of elements of the set Bdl . 3.2.3. Step 3: determining the best compromise alternative(s) To choose the best alternative(s), we use intersection and/or reunion and afterwards Borda’s voting method: • for each decision-maker dl, we determine the set Bdl of the best alternatives according to his preferences; • if Bd1 ∩Bd2 ∩ · · · ∩BdL ̸= ∅: each decision-maker ranks the alternatives obtained in the intersection, and the Borda method is applied to select the best alternative(s); • if Bd1 ∩ Bd2 ∩ · · · ∩ BdL = ∅: we consider the reunion Bd1 ∪ Bd2 ∪ · · · ∪ BdL ; each decision-maker ranks the alternatives inside the reunion Bd1 ∪ Bd2 ∪ · · · ∪ BdL and the Borda method is applied to select the best alternative(s). N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4201 4. Case studies: application to two group decision-making case studies 4.1. First case study The first case study concerns the selection of the best cowpea varieties (a bean species) adapted to the Centre-North region of Burkina Faso. The team of stakeholders involved in this research study includes breeders, producers and processors[9, 10]. The structuring phase enabled fifteen (15) cowpea crop varieties to be identified and twelve (12) evaluation criteria in interaction with the stakeholders. The evaluation criteria were divided into three group (see [10] for more details). 4.1.1. Groups of criteria The twelve criteria were divided into three groups of eight, three and one criteria respec- tively. • Group 1 (Production criteria): Type of plant habit (1, 1), Cycle-semi-maturity (1, 2), Yield potential (1, 3), Disease resistance (1, 4), Striga resistance (1, 5), Drought resistance (1, 6), Insect resistance (1, 7), forage potential (1, 8). • Group 2 (Quality criteria): seed size (2, 1), seed color (2, 2), seed taste (2, 3). Group 3 (Processing criteria): cooking time (3, 1). 4.1.2. Decision matrix The normalized decision matrix or evaluation matrix is unique for all decision-makers and is the one obtained from the evaluations of domain experts and the result synthesised in Table 1. 4.1.3. Using KEMIRA G-I and KEMIRA G-II methods As regards the application of the KEMIRA G-I and KEMIRA G-II methods to this first case study, we highlight the following elements. • Firstly the four decision-makers were able to express their preferences on the criteria by ranking them through the respective groups as showed in Table 2. • Secondly, for the KEMIRA G-I method, the median ranking is obtained by applying the Borda voting method algorithm [8] and the result presented in Table 3. • Thirdly, the KEMIRA[7] algorithm is implemented iteratively with the parameters in- dicated in relations (15),(16),(17). The different results obtained using the KEMIRA G-I and KEMIRA G-II methods, including execution times in second (s), are pre- sented in Table 4 and Table 5 respectively. maxiter = 10000 (15) N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4202 Table 1: Normalized Evaluation Matrix Names of the varieties (1,3) (1,2) (1,5) (1,8) (1,4) (1,7) (1,6) (1,1) (2,2) (2,1) (2,3) (3,1) ak 1,1 ak 1,2 ak 1,3 ak 1,4 ak 1,5 ak 1,6 ak 1,7 ak 1,8 ak 2,1 ak 2,2 ak 2,3 ak 3,1 KVx442-3-25SH(Komcallé) a 1 0.32 0.71 0.11 0.07 0.23 0 1 1 1 1 1 1 KVx61-1(Bengsiido) a 2 0.25 0.42 1 0.34 0.23 0 1 1 1 0 1 0.57 KVx745-11P a 3 0 0.42 1 0.82 0.12 0 1 0 1 1 1 0.07 KVx771-10G(Nafi) a 4 0.25 0.42 1 0.04 0.23 1 1 1 1 1 1 0.78 KVx775-33-2G(Tiligré) a 5 0.36 0.42 1 0.46 1 0 0 1 1 1 1 0.57 Moussa Local a 6 0.25 0 0.11 0 0 1 0 1 1 0.5 1 0.14 Teeksongo a 7 0.25 0.57 1 1 1 1 1 0.5 1 0.5 1 0.5 Yipoussi(KVx780-1) a 8 0.41 1 0.11 0.29 0.23 0 0 0.5 1 0 1 0.07 Niizwe a 9 0.11 0.71 1 0.09 1 0 0 1 1 1 1 0.07 Yiss-Yande a 1 0 0.36 0.71 1 0.14 0.23 0 1 0.5 1 1 1 0.14 Gorom local a 1 1 0.17 0.28 0 0.09 0.12 0 0 0 0 0 0 0.14 Makoyin(KVx780-4) a 1 2 0.7 0.57 1 0.34 0.23 0 1 0.5 1 1 0.9 0.07 Issa-Sosso(KVx780-3) a 1 3 0.7 0.71 1 0.34 0.23 0.16 1 0.5 1 1 0.9 0.14 Neerwaya(KVx780-6) a 1 4 0.85 0.57 1 0.46 0.23 0.16 0 0.5 1 1 0.9 0 Gourgou(TZ1-GOURGOU) a 1 5 1 0.28 1 0.58 0.23 0 0 0.5 1 0.5 0.9 0.14 αi = pi%max15k=1Wi(a k), i ∈ {1, 2, 3}, pi ∈ {10, 20, 30, 40, 50, 60, 70, 75, 80} (16) p1 = p2 = p3 (17) where maxiter is the maximum number of iterations. • Looking KEMIRA G-I result as showed in Table 4, we have a single best variety for the highest threshold: KVx771-10G(Nafi) (a4). So we did not need to ask decision- Table 2: Criteria ranking by Decision-Makers Goup 1 Goup 2 Goup 3 Rank d1 d2 d3 d4 d1 d2 d3 d4 d1 d2 d3 d4 1st (1, 3) (1, 3) (1, 3) (1, 3) (2, 2) (2, 1) (2, 2) (2, 2) (3, 1) (3, 1) (3, 1) (3, 1) 2nd (1,2) (1,2) (1,2) (1,2) (2,3) (2,3) (2,1) (2,1) 3th (1,1) (1,5) (1,4) (1,8) (2,1) (2,2) (2,3) (2,3) 4th (1,8) (1,6) (1,5) (1,7) 5th (1,5) (1,8) (1,7) (1,5) 6th (1,6) (1,7) (1,8) (1,4) 7th (1,4) (1,4) (1,6) (1,6) 8th (1,7) (1,1) (1,1) (1,1) Table 3: Median criteria ranking Group 1 median ranking Group 2 median ranking Group 3 median ranking (1, 3) ≿ (1, 2) ≿ (1, 5) ≿ (1, 8) ≿ (1, 4) ≿ (1, 7) ≿ (1, 6) ≿ (1, 1) (2, 2) ≿ (2, 1) ≿ (2, 3) (3, 1) N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4203 Table 4: KEMIRA G-I results pi Criteria weights best varieties Time (s) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (2,1) (2,2) (2,3) (3,1) w1,1 w1,2 w1,3 w1,4 w1,5 w1,6 w1,7 w1,8 w2,1 w2,2 w2,3 w3,1 10 0.19 0.17 0.15 0.15 0.13 0.13 0.05 0.02 0.66 0.25 0.09 1.0 {a1, a2, a4, a5, a7, a6, a10, a13, a15} 19.9298 20 0.2 0.16 0.15 0.13 0.11 0.11 0.1 0.04 0.5 0.36 0.14 1.0 {a1, a2, a4, a5, a7, a6, a10, a13} 23.5859 30; 40;50;60 0.26 0.22 0.20 0.10 0.09 0.02 0.02 0.02 0.42 0.32 0.25 1.0 {a1, a2, a4, a5, a7} 20.5216 70 0.31 0.29 0.06 0.06 0.05 0.04 0.04 0.04 0.37 0.37 0.25 1.0 {a1, a4} 21.2741 75;80 0.18 0.18 0.12 0.08 0.08 0.06 0.04 0.04 0.56 0.23 0.20 1.0 {a4} 20.8315 makers to rank the best varieties. Table 5: KEMIRA G-II results DMs pi Criteria weights Best varieties Time (s) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (2,1) (2,2) (2,3) (3,1) w1,1 w1,2 w1,3 w1,4 w1,5 w1,6 w1,7 w1,8 w2,1 w2,2 w2,3 w3,1 10 0.18 0.18 0.17 0.13 0.1 0.09 0.09 0.06 0.39 0.35 0.26 1.0 {a1, a2, a4, a5, a7, a6, a10, a13, a15} 21.30 20 0.22 0.22 0.19 0.13 0.1 0.06 0.05 0.05 0.38 0.35 0.27 1.0 {a1, a2, a4, a5, a7, a6, a10, a13} 19.8321 d1 30; 40;50;60 0.23 0.22 0.17 0.15 0.12 0.05 0.00 0.00 0.37 0.33 0.28 1.01 {a1, a2, a4, a5, a7} 20.0806 70;75;80 0.21 0.2 0.17 0.14 0.1 0.09 0.05 0.04 0.55 0.31 0.13 1.0 {a1, a4} 20.3378 90 0.24 0.23 0.21 0.14 0.09 0.05 0.05 0.0 0.64 0.32 0.03 1.0 {a4} 10 0.19 0.18 0.16 0.12 0.12 0.09 0.07 0.06 0.52 0.31 0.17 1.0 {a1, a2, a4, a5, a7, a6, a10, a13, a15} 20.0149 20 0.21 0.21 0.13 0.13 0.1 0.08 0.07 0.06 0.58 0.38 0.04 1.0 {a1, a2, a4, a5, a7, a6, a10, a13} 20.0825 d2 30; 40;50;60 0.24 0.20 0.17 0.08 0.08 0.04 0.04 0.02 0.23 0.19 0.19 1. {a1, a2, a4, a5, a7} 20.2741 70 0.24 0.19 0.13 0.13 0.06 0.05 0.05 0.05 0.59 0.39 0.02 1.01 {a1, a4} 19.9214 75;80 0.17 0.17 0.16 0.13 0.12 0.11 0.09 0.06 0.5 0.31 0.19 1.0 {a4} 19.8384 90 ∅ 10 0.18 0.18 0.17 0.16 0.16 0.06 0.05 0.03 0.99 0.0 0.0 1.0 {a1, a2, a4, a5, a7, a6, a10, a13, a15} 20.4354 20 0.25 0.16 0.15 0.12 0.08 0.05 0.05 0.04 0.43 0.38 0.17 1. {a1, a2, a4, a5, a7, a6, a10, a13} 20.2265 30; 40;50;60 0.27 0.22 0.15 0.13 0.11 0.05 0.05 0.01 0.54 0.37 0.07 1 {a1, a2, a4, a5, a7} 20.0531 d3 70 0.25 0.24 0.19 0.06 0.06 0.04 0.03 0.02 0.52 0.44 0.04 1.01 {a1, a4} 20.0882 75;80 0.22 0.15 0.14 0.12 0.12 0.09 0.08 0.06 0.38 0.33 0.27 1 {a4} 20.0554 90 ∅ 10 0.23 0.18 0.16 0.13 0.1 0.08 0.07 0.06 0.44 0.32 0.25 1.0 {a1, a2, a4, a5, a7, a6, a10, a13, a15} 20.0593 20 0.18 0.17 0.17 0.16 0.14 0.11 0.04 0.03 0.58 0.39 0.03 1.0 {a1, a2, a4, a5, a7, a6, a10, a13} 20.1220 d4 30; 40;50;60 0.27 0.15 0.11 0.09 0.05 0.05 0.05 0.05 0.68 0.18 0.14 0.99 {a1, a2, a4, a5, a7} 20.0279 70 0.24 0.23 0.09 0.07 0.07 0.07 0.07 0.04 0.72 0.16 0.11 0.99 {a1, a4} 20.2027 75;80 0.15 0.14 0.14 0.14 0.10 0.07 0.07 0.03 0.54 0.24 0.20 1. {a4} 20.0774 90 ∅ • Looking KEMIRA G-II result as showed in Table 5 we note that: – for pi = 80 we have Bd1 ∩Bd2 ∩Bd3 ∩Bd4 = {a4}; – for pi = 70 we have Bd1 ∩Bd2 ∩Bd3 ∩Bd4 = {a1, a4}; – for pi = 90 we have Bd1 = {a1} and Bd2 = Bd3 = Bd4 = ∅. Considering particularly the case where pi = 70, the set of the best varieties is B = {a1, a4}. To choose the best alternative among the element of B, we also use the Borda’s rule as illustrated in Table 6. These results show that Vx771-10G(Nafi) a4 is the best variety for KEMIRA G-II method, as for the results obtained with pi = 80. 4.1.4. Resolution using Lon-Zo and MACASP methods The application of these two methods requires the weights of the criteria to be determined. To do this, we used the revised Simos[13, 14] card method, called SFR[3]. The different N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4204 Table 6: KEMIRA G-II results on the set of best alternatives Rank d1 d2 d3 d4 1st a1 a4 a4 a4 2nd a4 a1 a1 a1 weight values found are shown in Table 7. Table 7: Criteria weights for Lon-Zo and MACASP methods in study case 1 Criteria g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 w di 1 w di 2 w di 3 w di 4 w di 5 w di 6 w di 7 w di 8 w di 9 w di 10 w di 11 w di 12 d1 5.25 6.13 7.30 2.04 3.79 2.92 1.45 4.08 10.99 10.99 10.99 33 d2 1.03 7.21 6.89 2.97 3.30 2.65 3.63 5.27 10.99 10.99 10.99 33 d3 0.83 6.54 7.53 4.80 4.55 1.58 4.31 2.82 13.34 16.84 2.80 33 d4 2.00 5.47 6.01 2.90 5.29 4.36 3.10 3.82 14.13 7.06 11.80 33 • Resolution using Lon-Zo method: Applying the Lon-Zo method to the first case study gives the results shown in Table 8. Looking the overall performance, the Table 8: Lon-Zo results g d1 (a k ) = g d2 (a k ) = g d3 (a k ) = g d4 (a k ) = Time (s) j=12∑ j=1 w d1 j .g d1 j (a k ) j=12∑ j=1 w d2 j .g d2 j (a k ) j=12∑ j=1 w d3 j .g d3 j (a k ) j=12∑ j=1 w d4 j .g d4 j (a k ) g(ak) = 4 4∑ l=1 1 gdl (ak) a1 82.0401 78.4368 77.2826 79.7055 79.3273 a2 59.0424 55.0413 52.1441 55.1176 55.2289 a3 29.9938 29.0018 34.5060 25.5707 29.4315 a4 77.2033 75.0291 75.8826 78.1417 76.5455 a5 69.9877 67.0653 68.7719 68.2416 68.5005 a6 41.0759 38.8767 38.4859 37.7534 39.0096 a7 66.2537 68.2096 66.9445 67.5662 67.2357 a8 38.1381 37.4442 34.4434 32.4802 35.4773 0.004998 a9 51.9282 48.9857 51.2404 50.4098 50.6171 a10 58.9875 53.1307 53.0384 55.5417 54.1520 a11 8.1923 8.6440 8.5635 7.8684 8.3053 a12 54.0195 52.1071 52.6589 54.0951 53.2062 a13 57.4217 56.0099 56.5747 57.6689 56.9111 a14 50.6085 49.3857 50.9241 49.2745 50.0376 a15 49.3026 47.4969 47.7555 46.1038 47.6377 KVx442-3-25SH(Komcallé (a1) variety is considered the best. • Resolution using the MACASP method: Applying the MACASP method to the first case study gives the results shown in Table 9. Like the Lon-Zo method, the MACASP method proposes KVx442-3-25SH(Komcallé (a1) as the best variety. 4.2. Second case study The second case study concerns the choice of a site for a non-hazardous waste incinera- tion plant in the Lithuanian capital Vilnius[6, 7]. In this study, five experts were involved in N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4205 Table 9: MACASP results Varieties g(ak) = j=12∑ j=1 j=4∑ l=1 w dl j j.g dl j (a k ) 4 Time(s) a1 79.366 a2 55.3364 a3 29.7681 a4 76.5642 a5 68.5166 a6 39.0480 a7 67.2435 a8 35.6265 a9 50.6410 0.004994 a10 54.1746 a11 8.3171 a12 53.2201 a13 56.9188 a14 50.0482 a15 47.6647 the decision-making process. They also played the role of decision-makers. Seven potential sites were selected on the basis of seven criteria, divided into two groups of criteria. • Group 1 representing criteria related to different engineering infrastructures: (1,1): Distance en km au réseaux de chauffage centralisé; (1,2):Distance in km to power supply networks of 110 kW; (1,3): Distance in km to high-pressure gas pipeline (12 bar); (1,4): Distance in km to water supply networks • Group 2 representing urban planning and social criteria: (2,1):Distance in km to Vilnius city center; (2,2):Average number of people living in the territory within a radius of 1 km2; (2,3):Usable surface owned by people living in the project area in m2. The normalized evaluation matrix is given in Table 10. Table 10: Normalized Evaluation matrix Sites (1, 1) (1, 2) (1, 3) (1, 4) (2, 1) (2, 2) (2, 3) ak 1,1 ak 1,2 ak 1,3 ak 1,4 ak 2,1 ak 2,2 ak 2,3 a1 1.5 0.6 2.5 1.37 9.26 3188.6 55,269 a2 3.5 1.2 4.5 0.5 8.64 497.5 9,327 a3 0.8 0.5 3 0.1 6.44 2.484 50.798 a4 4.8 1.2 1.6 2 11.19 2.676 56.206 a5 5.5 1 1.6 0.3 5.9 3.291 66.807 a6 0.6 0.7 2 0.6 6.09 6.490 132.136 a7 0.3 0.4 2 0.6 5.72 5946.7 123.314 4.2.1. Resolution using the KEMIRA G-I and KEMIRA G-II methods • First, the five decision-makers (Experts) express their preferences by ranking the criteria as showed in the Table 11. • In a second stage, the median ranking is determined using the Borda voting method and the result presented in Table 12. N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4206 Table 11: Ranking of criteria by the five DMs Group 1 Group 2 Rank d1 d2 d3 d4 d5 d1 d2 d3 d4 d5 1st (1,1) (1,1) (1,1) (1,1) (1,1) (2,2) (2,1) (2,3) (2,3) (2,1) 2nd (1,4) (1,4) (1,2) (1,2) (1,2) (2,3) (2,2) (2,1) (2,1) (2,3) 3rd (1,2) (1,2) (1,3) (1,4) (1,4) (2,1) (2,3) (2,2) (2,2) (2,2) 4th (1,3) (1,3) (1,4) (1,3) (1,3) Table 12: Median ranking of criteria group 1 median ranking group 2 median ranking (1, 1) ≿ (1, 2) ≿ (1, 4) ≿ (1, 3) (2, 3) ≿ (2, 1) ≿ (2, 2) • In a third stage, the KEMIRA algorithm is implemented with the parameters indi- cated in relations (18), (19), (20). The results with different execution times associ- ated are shown in Table 13 and Table 14. maxiter = 10000 (18) αi = pi%max15k=1Wi(a k); i ∈ {1, 2, 3}, pi ∈ {10, 20, 30, 40, 50, 60, 70, 75, 80} (19) p1 = p2 = p3 (20) Table 13: KEMIRA G-I results pi Criteria weights Best varieties Time (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) w1,1 w1,2 w1,3 w1,4 w2,1 w2,2 w2,3 10;20;30;40;50 0.5 0.25 0.23 0.03 0.5 0.35 0.15 {a1, a6, a7} 14.001 60;70;80 0.33 0.3 0.21 0.16 0.39 0.39 0.21 {a7} 14.29 • Looking Table 13, KEMIRA G-I method select a7 as the best location site. • Looking Table 14, for pi = 80: Bd1 = Bd2 = Bd3 = Bd4 = Bd5 = {a7}. So the alternative a7 is selected as the best location site by KEMIRA G-II method. 4.2.2. Resolution by Lon-Zo et MACASP methods Since we have taken this case study from the literature[6], we have considered the weights obtained with the KEMIRA G-II method and assumed that the two groups of criteria are of equal importance. The weights thus determined are presented in table 15. Lon-Zo and MACASP methods also select a7 as the best location site as showed in Table 16 and Table 17 respectively. 5. Discussion In the first case study, we saw that the new methods and the classic methods proposed different results. The Nafi (a4) is the best variety for the highest possible threshold ac- N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4207 Table 14: KEMIRA G-II results DMs pi Criteria weights Best varieties Time (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) w1,1 w1,2 w1,3 w1,4 w2,1 w2,2 w2,3 10;20 0.34 0.34 0.26 0.06 0.44 0.28 0.28 {a1, a2, a3, a4, a6, a7} 14.54 30 0.33 0.30 0.24 0.12 0.35 0.33 0.32 {a1, a2, a4, a6, a7} 14.26 40 0.27 0.26 0.25 0.22 0.42 0.39 0.17 {a1, a4, a6, a7} 14.04 d1 50 0.33 0.32 0.18 0.17 0.46 0.43 0.11 {a1, a6, a7} 14.10 60;70 0.41 0.37 0.13 0.09 0.55 0.29 0.17 {a1, a7} 14.45 80 0.39 0.38 0.12 0.11 0.69 0.18 0.13 {a7} 13.92 10;20 0.36 0.3 0.19 0.15 0.64 0.22 0.13 {a1, a2, a3, a4, a6, a7} 13.64 30;40 0.38 0.27 0.18 0.15 0.37 0.34 0.27 {a1, a2, a4, a6, a7} 13.71 d2 50 0.37 0.33 0.25 0.05 0.54 0.34 0.11 {a1, a6, a7} 14.22 60;70 0.38 0.3 0.22 0.11 0.46 0.42 0.12 {a1, a7} 14.03 80 0.46 0.26 0.17 0.11 0.47 0.39 0.14 {a7} 14.20 d3 10;20;30;40;50 0.56 0.27 0.11 0.06 0.43 0.33 0.24 {a1, a6, a7} 14.69 60;70;80 0.45 0.35 0.16 0.04 0.5 0.36 0.14 {a7} 14.07 d4 10;20;30;40;50 0.5 0.25 0.23 0.03 0.5 0.35 0.15 {a1, a6, a7} 14.001 60;70;80 0.33 0.3 0.21 0.16 0.39 0.39 0.21 {a7} 14.29 10;20;30;40;50 0.56 0.27 0.11 0.06 0.43 0.33 0.24 {a1, a6, a7} 14.69 d5 60;70;80 0.45 0.35 0.16 0.04 0.5 0.36 0.14 {a7} 14.07 Table 15: Criteria weights Criteria g1 g2 g3 g4 g5 g6 g7 w di 1 w di 2 w di 3 w di 4 w di 5 w di 6 w di 7 d1 0.195 0.06 0.055 0.19 0.065 0.345 0.09 d2 0.23 0.085 0.055 0.13 0.235 0.195 0.07 d3 0.225 0.175 0.08 0.02 0.18 0.07 0.25 d4 0.165 0.15 0.08 0.105 0.195 0.11 0.195 d5 0.225 0.175 0.02 0.08 0.25 0.07 0.18 cording to the KEMIRA G-I and KEMIRA G-II methods, i.e., this variety outperforms the highest possible threshold on all the criteria. Unlike the Lon-Zo and MACASP methods, the a1 variety is considered the best. This can be explained by the fact that this variety performs very well on some criteria and very poorly on others without said criteria being homogeneous. However, such compensation are generally allowed only when the criteria are all homogeneous, i.e. when natural com- pensation are allowed between criteria. This is not the case, for example, when considering in group 1, the criterion (1,4): disease resistance and in group 2, the criterion (2,2): seed color. These two criteria are said to be heterogeneous. The KEMIRA G-I and KEMIRA G-II methods are designed to avoid such compensation between heterogeneous criteria, unlike the Lon-Zo and MACASP methods where this is not the case Table 16: Lon-Zo results g d1 (a k ) = g d2 (a k ) = g d3 (a k ) = g d4 (a k ) = g d5 (a k ) = Time(s) j=7∑ j=1 w d1 j .g d1 j (a k ) j=7∑ j=1 w d2 j .g d2 j (a k ) j=7∑ j=1 w d3 j .g d3 j (a k ) j=7∑ j=1 w d4 j .g d4 j (a k ) j=7∑ j=1 w d5 j .g d5 j (a k ) g(ak) = 5 5∑ l=1 1 gdl (ak) a1 0.5562 0.6055 0.6202 0.5935 0.6343 0.6007 a2 0.3447 0.4006 0.3248 0.3707 0.4147 0.3680 a3 0.4358 0.4300 0.406 0.4106 0.3848 0.4126 a4 0.2017 0.3859 0.4770 0.4460 0.4814 0.3580 0.0039987 a5 0.3148 0.2700 0.3257 0.3669 0.3168 0.3158 a6 0.5168 0.5126 0.6875 0.6247 0.6373 0.5875 a7 0.8994 0.7118 0.7612 0.7353 0.7124 0.7538 N. C., Nana , S. A. M. Takougang , T. B. J. Batieno / Eur. J. Pure Appl. Math, 17 (4) (2024), 4195-4210 4208 Table 17: MACASP results Variétés g(ak) = j=12∑ j=1 l=5∑ l=1 w dl j .g dl j (a k ) 5 Time(s) a1 0.6019 a2 0.3711 a3 0.4134 a4 0.3984 0.0030000 a5 0.3188 a6 0.5958 a7 0.7580 In the second case study, all four methods selected a7 as the best site. All criteria can be considered in a single group. In fact, the criteria in both groups are all distance-related, except for one criterion which refers to the number of people (and therefore has no unit). Consequently, all the criteria of the two groups can naturally compensate for each other, i.e., the criteria can all be considered as homogeneous. Based on the partial or incomplete information given by the decision makers (we have only asked to the DMs to rank the alternatives from best to the worst in each group as showed in Table 3 and Table 11), KEMIRA G-I and KEMIRA G-II methods have proposed an elicitation of criteria weights as showed in Table 13 and Table 14. This is an advantage in a decision-making process, as the process of weighting criteria is generally tedious for decision-makers. Of course, applying the Lon-Zo and MACASP methods assumes that you have previously determined the weights of all the criteria. With regard to the execution times of the KEMIRA G-I and KEMIRA G-II methods on the one hand, and Lon-Zo and MACASP on the other, we note that the latter two are faster. This is also an advantage to use Lon-Zo or MACAP methods in case where all the criteria can be considered as homogenous. The relatively long runtimes of the KEMIRA G-I and KEMIRA G-II methods can be explained by the fact that, due to incomplete information on criteria weights, more iterations are needed to stabilize the corresponding algorithms, as the search space for suitable weights that allow an alternative to be selected as the best is too large. 6. Conclusion By solving two multi-criteria choice problems, we were able to demonstrate the effec- tiveness of the two new methods, KEMIRA G-I and KEMIRA G-II, when the criteria are heterogeneous. However, when the criteria can be considered as homogeneous, the Lon-Zo and MACASP methods seem to be better suited, given their speed. Another advantage of the two new methods, KEMIRA G-I and KEMIRA G-II, is the elicitation of criteria weights based on incomplete information, which is not the case with Lon-Zo and MACASP methods. In our future work, we intend to implement a user friendly computer system that integrates these four decision group methods to better handle multi-criteria group prob- lems. On the other, we want to see how we can reinforce the process of eliciting the weights of the criteria proposed by the new KEMIRA G-I and KEMIRA G-II methods by adding information on the intensity between criteria in their respective algorithms. 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