Outranking Methods: Promethee I and Promethee II

Shankha Shubhra Goswami

Open Access

Outranking Methods: Promethee I and Promethee II

Shankha Shubhra Goswami

| Jul 09, 2020

Foundations of Management

Volume 12 (2020): Issue 1 (January 2020)

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Page range: 93 - 110

DOI: https://doi.org/10.2478/fman-2020-0008

Keywords
PROMETHEE I, PROMETHEE II, AHP, MCDM, laptop models

© 2020 Shankha Shubhra Goswami, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

AHP methodology (Source:Kolios, et al., 2016)

PROMETHEE methodology (Source:Kolios, et al., 2016)

Aggregate preference function matrix

Models	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6
Model 1	-	0.06554	0.02185	0.07647	0	0.04370
Model 2	0.36323	-	0	0.01092	0	0.23526
Model 3	0.55586	0.23632	-	0.05462	0.04697	0.44973
Model 4	0.92353	0.56030	0.36767	-	0.39967	0.79555
Model 5	0.52386	0.22617	0.03683	0.07647	-	0.43958
Model 6	0.12798	0.02185	0	0.03277	0	-

Determination of leaving and entering outranking flow (PROMETHEE I)

Models	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Leaving flow ϕ+
Model 1	-	0.06554	0.02185	0.07647	0	0.04370	0.20756
Model 2	0.36323	-	0	0.01092	0	0.23526	0.60942
Model 3	0.55586	0.23632	-	0.05462	0.04697	0.44973	1.34350
Model 4	0.92353	0.56030	0.36767	-	0.39967	0.79555	3.04672
Model 5	0.52386	0.22617	0.03683	0.07647	-	0.43958	1.30291
Model 6	0.12798	0.02185	0	0.03277	0	-	0.18260
Entering flow ϕ−	2.49446	1.11018	0.42635	0.25125	0.44664	1.96382

Pair-wise comparison matrix

Comparisons	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color
Processor	1	5	7	3	2	7	9
Hard disk capacity	1/5	1	3	1/3	1/3	2	5
Operating system	1/7	1/3	1	1/5	1/3	1/3	3
RAM	1/3	3	5	1	2	3	9
Screen size	1/2	3	3	1/2	1	3	7
Brand	1/7	1/2	3	1/3	1/3	1	5
Color	1/9	1/5	1/3	1/9	1/7	1/5	1
Total	2.43016	13.03333	22.33333	5.47778	6.14286	16.53333	39

Randomly generated consistency index (RI) values

n	1	2	3	4	5	6	7	8	9	10	11	12
RI	0.00	0.00	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49	1.51	1.58

Hwang ang Yoon comparison scale

Qualitative Estimation	Bad	Good	Average	Very Good	Excellent	Types of Criteria
Quantitative Estimation	1	3	5	7	9	Max
Quantitative Estimation	9	7	5	3	1	Min

Evaluation matrix

Models	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color
Model 1	3	5	3	5	3	9	3
Model 2	5	7	5	5	7	3	3
Model 3	5	9	9	7	7	7	5
Model 4	7	9	9	9	9	2	9
Model 5	5	7	9	7	7	9	9
Model 6	3	5	5	5	7	5	3
Max	7	9	9	9	9	9	9
Min	3	5	3	5	3	2	3

Determination of leaving and entering outranking flow (PROMETHEE II)

Models	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Leaving flow ϕ+
Model 1	-	0.06554	0.02185	0.07647	0.00000	0.04370	0.04151
Model 2	0.36323	-	0.00000	0.01092	0.00000	0.23526	0.12188
Model 3	0.55586	0.23632	-	0.05462	0.04697	0.44973	0.26870
Model 4	0.92353	0.56030	0.36767	-	0.39967	0.79555	0.60934
Model 5	0.52386	0.22617	0.03683	0.07647	-	0.43958	0.26058
Model 6	0.12798	0.02185	0.00000	0.03277	0.00000	-	0.03652
Entering flow ϕ−	0.49889	0.22204	0.08527	0.05025	0.08933	0.39276

Evaluative difference of ith alternatives with respect other alternatives

Evaluative difference	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color
Model 1
D (M1-M2)	−0.5	−0.5	−0.33333	0	−0.66667	0.85714	0
D (M1-M3)	−0.5	−1	−1	−0.5	−0.66667	0.28571	−0.33333
D (M1-M4)	−1	−1	−1	−1	−1	1	−1
D (M1-M5)	−0.5	−0.5	−1	−0.5	−0.66667	0	−1
D (M1-M6)	0	0	−0.33333	0	−0.66667	0.57143	0
Model 2
D (M2-M1)	0.5	0.5	0.33333	0	0.66667	−0.85714	0
D (M2-M3)	0	−0.5	−0.66667	−0.5	0	−0.57143	−0.33333
D (M2-M4)	−0.5	−0.5	−0.66667	−1	−0.33333	0.14286	−1
D (M2-M5)	0	0	−0.66667	−0.5	0	−0.85714	−1
D (M2-M6)	0.5	0.5	0	0	0	−0.28571	0
Model 3
D (M3-M1)	0.5	1	1	0.5	0.66667	−0.28571	0.33333
D (M3-M2)	0	0.5	0.66667	0.5	0	0.57143	0.33333
D (M3-M4)	−0.5	0	0	−0.5	−0.33333	0.71429	−0.66667
D (M3-M5)	0	0.5	0	0	0	−0.28571	−0.66667
D (M3-M6)	0.5	1	0.66667	0.5	0	0.28571	0.33333
Model 4
D (M4-M1)	1	1	1	1	1	−1	1
D (M4-M2)	0.5	0.5	0.66667	1	0.33333	−0.14286	1
D (M4-M3)	0.5	0	0	0.5	0.33333	−0.71429	0.66667
D (M4-M5)	0.5	0.5	0	0.5	0.33333	−1	0
D (M4-M6)	1	1	0.66667	1	0.33333	−0.42857	1
Model 5
D (M5-M1)	0.5	0.5	1	0.5	0.66667	0	1
D (M5-M2)	0	0	0.66667	0.5	0	0.85714	1
D (M5-M3)	0	−0.5	0	0	0	0.28571	0.66667
D (M5-M4)	−0.5	−0.5	0	−0.5	−0.33333	1	0
D (M5-M6)	0.5	0.5	0.66667	0.5	0	0.57143	1
Model 6
D (M6-M1)	0	0	0.33333	0	0.66667	−0.57143	0
D (M6-M2)	−0.5	−0.5	0	0	0	0.28571	0
D (M6-M3)	−0.5	−1	−0.66667	−0.5	0	−0.28571	−0.33333
D (M6-M4)	−1	−1	−0.66667	−1	−0.33333	0.42857	−1
D (M6-M5)	−0.5	−0.5	−0.66667	−0.5	0	−0.57143	−1

Calculating the aggregated preference, π (Ma, Mb)

Weights	0.37657	0.09395	0.04529	0.21594	0.16932	0.07647	0.02247	1
								Aggregated preference, π (Ma, Mb)
Model 1
π (M1, M2)	0	0	0	0	0	0.06554	0	0.06554
π (M1, M3)	0	0	0	0	0	0.02185	0	0.02185
π (M1, M4)	0	0	0	0	0	0.07647	0	0.07647
π (M1, M5)	0	0	0	0	0	0	0	0
π (M1, M6)	0	0	0	0	0	0.04370	0	0.04370
Model 2
π (M2, M1)	0.18828	0.04697	0.01510	0	0.11288	0	0	0.36323
π (M2, M3)	0	0	0	0	0	0	0	0
π (M2, M4)	0	0	0	0	0	0.01092	0	0.01092
π (M2, M5)	0	0	0	0	0	0	0	0
π (M2, M6)	0.18828	0.04697	0	0	0	0	0	0.23526
Model 3
π (M3, M1)	0.18828	0.09395	0.04529	0.10797	0.11288	0	0.00749	0.55586
π (M3, M2)	0	0.04697	0.03019	0.10797	0	0.04370	0.00749	0.23632
π (M3, M4)	0	0	0	0	0	0.05462	0	0.05462
π (M3, M5)	0	0.04697	0	0	0	0	0	0.04697
π (M3, M6)	0.18828	0.09395	0.03019	0.10797	0	0.02185	0.00749	0.44973
Model 4
π (M4, M1)	0.37657	0.09395	0.04529	0.21594	0.16932	0	0.02247	0.92353
π (M4, M2)	0.18828	0.04697	0.03019	0.21594	0.05644	0	0.02247	0.56030
π (M4, M3)	0.18828	0	0	0.10797	0.05644	0	0.01498	0.36767
π (M4, M5)	0.18828	0.04697	0	0.10797	0.05644	0	0	0.39967
π (M4, M6)	0.37657	0.09395	0.03019	0.21594	0.05644	0	0.02247	0.79555
Model 5
π (M5, M1)	0.18828	0.04697	0.04529	0.10797	0.11288	0	0.02247	0.52386
π (M5, M2)	0	0	0.03019	0.10797	0.00000	0.06554	0.02247	0.22617
π (M5, M3)	0	0	0	0	0	0.02185	0.01498	0.03683
π (M5, M4)	0	0	0	0	0	0.07647	0	0.07647
π (M5, M6)	0.18828	0.04697	0.03019	0.10797	0	0.04370	0.02247	0.43958
Model 6
π (M6, M1)	0	0	0.01510	0	0.11288	0	0	0.12798
π (M6, M2)	0	0	0	0	0	0.02185	0	0.02185
π (M6, M3)	0	0	0	0	0	0	0	0
π (M6, M4)	0	0	0	0	0	0.03277	0	0.03277
π (M6, M5)	0	0	0	0	0	0	0	0

Global PC units’ shipment from 2006–2018

Year	Shipments in millions
2018	259.39
2017	262.68
2016	269.72
2015	287.68
2014	313.68
2013	316.46
2012	351.06
2011	365.36
2010	350.90
2009	308.34
2008	290.80
2007	272.45
2006	239.21

Normalized pair-wise comparison matrix

Comparisons	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color	Priority Vector	Weight %
Processor	0.41150	0.38363	0.31343	0.54767	0.32558	0.42339	0.23077	0.37657	37.657
Hard disk capacity	0.08230	0.07673	0.13433	0.06085	0.05426	0.12097	0.12821	0.09395	9.395
Operating system	0.05879	0.02558	0.04478	0.03651	0.05426	0.02016	0.07692	0.04529	4.529
RAM	0.13717	0.23018	0.22388	0.18256	0.32558	0.18145	0.23077	0.21594	21.594
Screen size	0.20575	0.23018	0.13433	0.09128	0.16279	0.18145	0.17949	0.16932	16.932
Brand	0.05879	0.03836	0.13433	0.06085	0.05426	0.06048	0.12821	0.07647	7.647
Color	0.04572	0.01535	0.01493	0.02028	0.02326	0.01210	0.02564	0.02247	2.247
Total	1	1	1	1	1	1	1	1	100

Comparisons of the laptop models among each other

Sl. No.	Comparisons	Sl. No.	Comparisons
1	Model 2 P Model 1	9	Model 2 P Model 6
2	Model 3 P Model 1	10	Model 4 P Model 3
3	Model 4 P Model 1	11	Model 3 P Model 5
4	Model 5 P Model 1	12	Model 3 P Model 6
5	Model 1 R Model 6	13	Model 4 P Model 5
6	Model 3 P Model 2	14	Model 4 P Model 6
7	Model 4 P Model 2	15	Model 5 P Model 6
8	Model 5 P Model 2

Saaty's pair-wise comparison scale

Saaty's pair wise comparison scale	Compare factor of i & j
1	Equal Importance
3	Moderate Importance
5	Strong Importance
7	Very Strong or Demonstrated Importance
9	Extreme Importance
2, 4, 6, 8	Intermediate values when compromise is needed

Selected laptop models and their configurations

Models	Processor	Hard Disk Capacity	Operating System	RAM	Screen Size	Brand	Color
Model 1	I3	512GB	DOS	4GB	14 Inch	HP	Black
Model 2	I5	1TB	Linux	4GB	15.6 Inch	Acer	Black
Model 3	I5	2TB	Windows	8GB	15.6 Inch	Lenovo	Gold
Model 4	I7	2TB	Windows	16GB	17.3 Inch	Asus	Silver
Model 5	I5	1TB	Windows	8GB	15.6 Inch	HP	Silver
Model 6	I3	512GB	Linux	4GB	15.6 Inch	Dell	Black

Preference function, Pj (Ma, Mb)

Weights (w_j)	0.37657	0.09395	0.04529	0.21594	0.16932	0.07647	0.02247
Evaluative difference	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color
Model 1
P (M1, M2)	0	0	0	0	0	0.85714	0
P (M1, M3)	0	0	0	0	0	0.28571	0
P (M1, M4)	0	0	0	0	0	1	0
P (M1, M5)	0	0	0	0	0	0	0
P (M1, M6)	0	0	0	0	0	0.57143	0
Model 2
P (M2, M1)	0.5	0.5	0.33333	0	0.66667	0	0
P (M2, M3)	0	0	0	0	0	0	0
P (M2, M4)	0	0	0	0	0	0.14286	0
P (M2, M5)	0	0	0	0	0	0	0
P (M2, M6)	0.5	0.5	0	0	0	0	0
Model 3
P (M3, M1)	0.5	1	1	0.5	0.66667	0	0.33333
P (M3, M2)	0	0.5	0.66667	0.5	0	0.57143	0.33333
P (M3, M4)	0	0	0	0	0	0.71429	0
P (M3, M5)	0	0.5	0	0	0	0	0
P (M3, M6)	0.5	1	0.66667	0.5	0	0.28571	0.33333
Model 4
P (M4, M1)	1	1	1	1	1	0	1
P (M4, M2)	0.5	0.5	0.66667	1	0.33333	0	1
P (M4, M3)	0.5	0	0	0.5	0.33333	0	0.66667
P (M4, M5)	0.5	0.5	0	0.5	0.33333	0	0
P (M4, M6)	1	1	0.66667	1	0.33333	0	1
Model 5
P (M5, M1)	0.5	0.5	1	0.5	0.66667	0	1
P (M5, M2)	0	0	0.66667	0.5	0	0.85714	1
P (M5, M3)	0	0	0	0	0	0.28571	0.66667
P (M5, M4)	0	0	0	0	0	1	0
P (M5, M6)	0.5	0.5	0.66667	0.5	0	0.57143	1
Model 6
P (M6, M1)	0	0	0.33333	0	0.66667	0	0
P (M6, M2)	0	0	0	0	0	0.28571	0
P (M6, M3)	0	0	0	0	0	0	0
P (M6, M4)	0	0	0	0	0	0.42857	0
P (M6, M5)	0	0	0	0	0	0	0

Normalized decision matrix

Models	Processor	Hard disk capacity	Operating system	RAM	Screen size	Brand	Color
Model 1	0	0	0	0	0	1	0
Model 2	0.5	0.5	0.33333	0	0.66667	0.14286	0
Model 3	0.5	1	1	0.5	0.66667	0.71429	0.33333
Model 4	1	1	1	1	1	0	1
Model 5	0.5	0.5	1	0.5	0.66667	1	1
Model 6	0	0	0.33333	0	0.66667	0.42857	0

Ranking of the laptop models

Models	Net flow ϕ	Ranking
Model 1	−0.45738	Rank 6
Model 2	−0.10015	Rank 4
Model 3	0.18343	Rank 2
Model 4	0.55909	Rank 1
Model 5	0.17125	Rank 3
Model 6	−0.35624	Rank 5

Net outranking flow of the alternatives

Models	Leaving flow ϕ+	Entering flow ϕ−	Net flow ϕ
Model 1	0.04151	0.49889	−0.45738
Model 2	0.12188	0.22204	−0.10015
Model 3	0.26870	0.08527	0.18343
Model 4	0.60934	0.05025	0.55909
Model 5	0.26058	0.08933	0.17125
Model 6	0.03652	0.39276	−0.35624

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Publication timeframe:: Volume Open
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Outranking Methods: Promethee I and Promethee II

Published Online: Jul 09, 2020

Page range: 93 - 110

DOI: https://doi.org/10.2478/fman-2020-0008

Keywords
PROMETHEE I, PROMETHEE II, AHP, MCDM, laptop models

© 2020 Shankha Shubhra Goswami, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Figure 1

Figure 2

Aggregate preference function matrix

Determination of leaving and entering outranking flow (PROMETHEE I)

Pair-wise comparison matrix

Randomly generated consistency index (RI) values

Hwang ang Yoon comparison scale

Evaluation matrix

Determination of leaving and entering outranking flow (PROMETHEE II)

Evaluative difference of ith alternatives with respect other alternatives

Calculating the aggregated preference, π (Ma, Mb)

Global PC units’ shipment from 2006–2018

Normalized pair-wise comparison matrix

Comparisons of the laptop models among each other

Saaty's pair-wise comparison scale

Selected laptop models and their configurations

Preference function, Pj (Ma, Mb)

Normalized decision matrix

Ranking of the laptop models

Net outranking flow of the alternatives

Outranking Methods: Promethee I and Promethee II

Published Online: Jul 09, 2020

Page range: 93 - 110

DOI: https://doi.org/10.2478/fman-2020-0008

KeywordsPROMETHEE I, PROMETHEE II, AHP, MCDM, laptop models

© 2020 Shankha Shubhra Goswami, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Figure 1

Figure 2

Aggregate preference function matrix

Determination of leaving and entering outranking flow (PROMETHEE I)

Pair-wise comparison matrix

Randomly generated consistency index (RI) values

Hwang ang Yoon comparison scale

Evaluation matrix

Determination of leaving and entering outranking flow (PROMETHEE II)

Evaluative difference of ith alternatives with respect other alternatives

Calculating the aggregated preference, π (Ma, Mb)

Global PC units’ shipment from 2006–2018

Normalized pair-wise comparison matrix

Comparisons of the laptop models among each other

Saaty's pair-wise comparison scale

Selected laptop models and their configurations

Preference function, Pj (Ma, Mb)

Normalized decision matrix

Ranking of the laptop models

Net outranking flow of the alternatives

Keywords
PROMETHEE I, PROMETHEE II, AHP, MCDM, laptop models