In this 21st century, computer has become one of the most essential electronics gadgets in our daily life. It makes our life easier and faster as compared to 15 years ago. In 2018, approximately 259.39 million of PCs were shipped around the world (Holst, 2019). Table 1 shows the global PC units’ shipment from 2006 to 2018 (Statista, 2019).
Moreover, with the technological development lap-tops are replacing the desktop computers and occupying the markets more rapidly because of their portability and ease of use. At present, there are six major laptop manufacturing companies in the market, that is, Lenovo, HP, Dell, Acer, Asus, and Apple. These brands have thousands of models out there in the market with different specifications, so it is very much important to select the appropriate laptop model so that most of our requirements can be fulfilled. Such an initiative is taken and presented in this article, where the best laptop model is proposed among six different models actually available in the market from different brands having different specifications by applying outranking methods the Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE) I and II. The selected six laptop models and their specifications are presented in Table 2.
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 |
(
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 |
(
In this present analysis, six different laptop models having different configurations are selected from the market, which are presented in Table 2, and the aim of this research work is to propose the best model among them based on seven major criteria, that is, processor, hard disk capacity, operating system, RAM, screen size, brand, and color. After doing some research work, it is found that these models are in high demand and are generally preferred by most of the customers. The weightages of the criteria are found by analytic hierarchy process (AHP) and then PROMETHEE is applied by using those weightages to select the best model. Two categories of PROMETHEE are used in this article, where PROMETHEE I depicts the preferences of one model over another and PROMETHEE II helps to propose the complete ranking of the alternatives indicating from best to worst model.
For many years, multiple criteria decision-making (MCDM) proves to be a very useful tools for taking effective decision in the fields of water management (Qin, et al., 2008), waste management (Chandrakar and Limje, 2018), industries (Bentes, et al., 2012; Bulut, et al., 2012; Celik, et al., 2009; Duran and Aguilo, 2008; Roistamzadeh and Sofian, 2011; Sri Krishna, et al., 2014), business and finance (Albadvi, et al., 2007; Korhonen, et al., 2006; Lee, et al., 2008), medical and health sector (Buyukozkan and Cifci, 2012), education (Bhattacharya and Chakraborty, 2014; Melon, et al., 2008), environmental management (Geldermann, et al., 2000; Marzouk and Abdelakder, 2019; Vaillancourt and Waaub, 2004), energy management (Kowalski, et al., 2009; Tsoutsos, et al., 2009), and so on. Many researchers implemented various MCDM techniques in different areas for strategic decision making. Some of the most popular MCDM techniques include AHP (Saaty, 1980; 2001; 2008; 2010), PROMETHEE (Brans, 1982; Brans, et al., 1984; 1986; Brans and Vincke, 1985), ELimination Et Choix Traduisant la REalité (ELECTRE) (Roy, 1968), Multi-Objective Optimization on the basis of Ratio Analysis (MOORA) (Brauers and Zavadskas, 2006; 2009), The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) (Hwang, et al., 1993; Hwang and Yoon, 1981; Yoon, 1987), COmplex PRroportional ASsessment (COPRAS) (Zavadskas, et al., 2008), Additive Ratio ASsessment (ARAS) (Zavadskas, et al., 2010; Zavadskas and Turkis, 2010), and Fuzzy Analytic Hierarchy Process (FAHP) (Ayhan, 2013; Buckley, 1985; Chang, 1996; Zadeh, 1965). This section mainly includes some literatures (Behzadian, et al., 2010; Jayant and Sharma, 2018) of the researchers adopting PROMETHEE and AHP methods for decision-making purposes in various fields.
Geldermann, et al. (2000) considered a case study in iron- and steel-making industry where the out-ranking method PROMETHEE was adopted for environmental assessment. Goumas and Lygerou (2000) applied PROMETHEE method for the evaluation and ranking of alternative energy exploitation schemes of a low-temperature geothermal field under fuzzy environment. Ngai and Chan (2005) applied analytic hierarchy process to select the most appropriate support knowledge management tool. Bertolini, et al. (2006) selected the best discount in defining a proposal for a public work contract by implementing AHP as a tool. Albadvi, et al. (2007) developed a decision-making model in their article for selecting superior stocks in stock exchange by using PROMETHEE method. Wang and Yang (2007) evaluated the information systems (IS) outsourcing problems using hybrid approach of AHP and PROMETHEE by considering six factors, where AHP was used to analyze the structure of the outsourcing problem and determine weights of the criteria and PROMETHEE method was used for final ranking of the alternatives.
Melon, et al. (2008) adopted AHP to evaluate the proposals for educational innovation projects to help the Institute of Educational Sciences of the Polytechnic University of Valencia to choose the best educational project. Tsoutsos, et al. (2009) determined a set of energy planning alternatives based on economic, technical, social, and environmental criteria factors by implementing PROMETHEE methodology for the sustainable energy planning on the island of Crete in Greece. Alomoush (2010) selected the best optimal location of thyristor-controlled series compensator (TCSC) in a transmission system using PROMETHEE MCDM process. Turcksin, et al. (2011) proposed an integrated approach of AHP and PROMETHEE for selecting the most appropriate policy scenario to stimulate a clean vehicle fleet. Yilmaz and Dagdeviren (2011) used a combined approach of fuzzy-PROMETHEE and zero-one goal programming model for solving welding machine selection problem of a company.
Peng and Xiao (2013) considered a material selection problem for a journal bearing in which PROMETHEE combined with analytic network process (ANP) is presented; ANP was used to identify the weights and PROMETHEE was used to rank alternatives. Zhao, et al. (2013) proposed a modified PROMETHEE method to improve the efficiency and response time in incident management. Kabir and Sumi (2014) considered a case study from Bangladesh where fuzzy-AHP and PROMETHEE were used to select the power substation location. Kucharski (2014) used PROMETHEE method to rank selected types of investment funds offered on the polish market. Maletic, et al. (2014) applied AHP for the selection of maintenance policy. Kilic, et al. (2015) performed an analysis on the enterprise resource planning (ERP) selection problem for the small medium enterprises (SMEs) in Istanbul, Turkey, by combining AHP-PROMETHEE hybrid MCDM methodology.
Polat (2016) used integrated AHP-PROMETHEE methods for selecting the most appropriate subcon-tractor to be worked within an international construction project, where AHP was used to analyze the structure of the subcontractor selection problem and to determine the weights of the criteria and PROMETHEE was used to obtain complete ranking and perform sensitivity analysis by changing the weights of criteria. Polat, et al. (2016) proposed an integrated approach of AHP and PROMETHEE to help a Turkish construction company in selecting the appropriate urban renewal project by finding the weights of the selected criteria and to rank the alternative projects, respectively. Nikouei, et al. (2017) used PROMETHEE based on MCDM approach for selecting the best membrane prepared from sulfonated poly(ether ketone)s and poly(ether sulfone)s, and the final results showed that poly(ether ketone) membranes in selected criteria were better than poly(ether sulfone) membranes. Nourbakhsh and Yousefi (2017) proposed the best indicator water quality parameters using AHP in order to manage the ground water quality. Butowski (2018) developed an integrated AHP and PROMETHEE approach to evaluate the attractiveness of European maritime areas for sailing tourism. Naserizade, et al. (2018) developed a new stochastic model based on conditional value at risk (CVaR) and multi-objective optimization methods for optimal placement of sensors in water distribution system (WDS), and finally, PROMETHEE was used to determine the best solution and to rank the alternatives on the trade-off curve among objective functions.
Apart from the above literatures, there are also few research works in which MCDM techniques are being used for selecting electronics devices, household appliances or other items that are associated with our daily life, for example, selection of desktop (Mitra and Goswami, 2019a; 2019b) and laptop computers (Adali and Isik, 2017), selection of mobiles (Goswami and Mitra, 2020), car selection (Sri Krishna, et al., 2014), refrigerator selection (Mitra and Kundu, 2017; 2018), and air conditioner selection (Adali and Isik, 2016), that have been recorded but there are very few researchers who have adopted and applied PROMETHEE methods for solving decision-making problems associated with our daily life apart from industrial applications. So, such an initiative is taken in this article to apply an integrated AHPPROMETHEE (Butowski, 2018; Kilic, et al., 2015; Polat, et al., 2016; Turcksin, et al., 2011; Wang and Yang, 2007) methodologies to solve the laptop selection problems, where AHP is used for calculating the criteria weightages and PROMETHEE is used for the final ranking of the alternatives.
This section consists of all the step-wise calculation details of AHP and PROMETHEE. The weightages of the criteria are calculated using AHP, and the final ranking of the alternatives are performed using PROMETHEE, which are discussed in Sections 3.1 and 3.2, respectively.
AHP is an MCDM technique for establishing and analyzing complex decisions based on psychology and mathematics (Saaty, 1980; 2001; 2008; 2010). It was first developed by professor Thomas L. Saaty in 1980 (Saaty, 1980). The AHP methodology is illustrated briefly in Fig. 1.
The steps involved in AHP for calculating the criteria weightages are given as follows:
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 |
(
This pair-wise comparison matrix compares all the criteria among each other, and the relative importance of one criterion over other is given according to the Saaty pair-wise comparison scale.
Table 5 presents the Saaty's 9 pair comparison scale. Here, n is the number of criteria considered for the analysis. In this case, seven criteria are considered; hence, n = 7.
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 |
(
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 |
(
The normalized values are calculated and presented in Table 4. All the row averages (priority vector) are found out for every criterion, which are the criteria weightages. The criteria weightages are found out to be as follows: wprocessor = 0.37657 or 37.657%, whard disk capacity = 0.09395 or 9.395%, woperating system = 0.04529 or 4.529%, wRAM = 0.21594 or 21.594%, wscreen size = 0.16932 or 16.932%, wbrand = 0.07647 or 7.647%, and wcolor = 0.02247 or 2.247%.
Table 4 presents the normalized decision matrix along with the criteria weightages. Now the next step is to check the consistency whether the decision maker's judgment is true and consistent.
The consistencies of all the seven criteria are found out. Now, calculate the averages of these seven values to find out the average consistency (λmax).
CI is calculated using Equation 2, and RI value can be obtained from Table 6 according to the number criteria. In this case, n = 7, so the RI value corresponding to n = 7 is 1.32.
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 |
(
Here, λmax is the average consistency and n is the number of criteria. In this case, n = 7.
Using Equations 1 and 2, CI and CR values are found to be 0.06374 (CI value) and 0.04829 (CR value) or 4.829%.
Here it can be seen that the CR value is less than 0.1 (0.04829 < 0.1), that is, the inconsistency is within 10%; hence, it can be concluded that the decision maker judgment is true and consistent.
In early 1980s, for the first time, Brans presented the PROMETHEE outranking method in 1982 at a conference organized by R. Nadeau and M. Landry at the University Laval, Québec, Canada (Brans, 1982). After that, developments have been carried out for more than a decade to bring out various versions of PROMETHEE by Brans along with other researchers (Brans, 1982; Brans, et al., 1984, 1986; Brans and Mareschal, 1992, 1994, 1995; Brans and Vincke, 1985). The PROMETHEE methodology are explained briefly in Fig. 2.
The steps involved in PROMETHEE are described as follows.
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 |
(
Hwang ang Yoon comparison scale
Qualitative Estimation | Bad | Good | Average | Very Good | Excellent | Types of Criteria |
---|---|---|---|---|---|---|
Quantitative Estimation | 1 | 3 | 5 | 7 | 9 | Max |
9 | 7 | 5 | 3 | 1 | Min |
(
For beneficial criteria
For Non-beneficial criteria or cost criteria
All the criteria considered for this analysis are beneficial in nature, that is, whose higher values are desired. So Equation 4 is used for the normalization of the decision matrix. Table 9 presents the normalized decision matrix.
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 |
(
The evaluative differences (deviations) are calculated and presented in Table 10.
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 |
(
The preference function is calculated using the following two conditions given by Equations 7 and 8.
From Equations 7 and 8, it is clear that if the difference, that is, D (Ma − Mb) in Table 10 is less than or equal to zero, then substitute the preference function value as zero and if D (Ma - Mb) value is greater than zero, then the differences, that is, (R(ij)a − R(ij)b) is used as the preference function value.
In more easy words, if the value in any of the cell in Table 10 is less than or equal to zero (i.e., negative values), then substitute the value as zero and if the value in any of the cell in Table 10 is greater than zero, then leave it untouched. So, by substituting all the negative values of Table 10 by zeroes and keeping all the positive values as it is, thus obtaining the Table 11 as shown above.
Preference function, Pj (Ma, Mb)
Weights (wj) | 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 |
(
The summation of the criteria weightages is always equal to 1; hence, the denominator of Equation 9 becomes one. Now, multiplying all the criteria weightages with their respective column elements of Table 11 and using Equation 9, find all aggregated preference values π (Ma, Mb).
Table 12 presents the calculated aggregated preference values.
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 |
(
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 | - |
(
For π (M1, M2), the aggregated preference value is 0.06554; this means the aggregated preference value of Model 1 with respect to Model 2 is 0.06554 (from Table 12). So this value (i.e., 0.06554) is allotted in the cell 1, 2 as shown in Table 13.
Similarly, for π (M1, M3), the aggregated preference value of Model 1 with respect to Model 3 is 0.02185 (from Table 12), so this value (i.e., 0.02185) is allotted in the cell 1, 3 as shown in Table 13.
In this way, the aggregated preference function value from Table 12 is allotted in all the cells of the above matrix, thus forming Table 13. When an alternative is compared to itself, no values should be assigned.
The leaving and entering outranking flow in case of PROMETHEE I is calculated and presented in Table 14 using Equations 10 and 11 given below.
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 |
(
The leaving and entering outranking flow in case of PROMETHEE II is calculated and presented in Table 15 using the Equations 12 and 13 given below.
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 |
(
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 |
(
The best alternatives chosen by PROMETHEE method depends on the leaving and entering outranking flows of each alternatives, which are calculated in Section 3 and presented in Table 14 for PROMETHEE I and in Table 15 for PROMETHEE II.
In case of PROMETHEE II, the net outranking flow is also calculated as shown in Table 16.
As mentioned earlier, PROMETHEE 1 provides the partial ranking and PROMETHEE II provides the complete ranking of the alternatives. The outcome results from both the methods are explained in Sections 4.1 and 4.2, respectively.
In case of PROMETHEE I, partial ranking is performed and the decision made by the decision maker is based on the three conditions given below.
In PROMETHEE I, instead of defining the rank, preferences are calculated.
aPb if:
ϕ+ (a) > ϕ+ (b) and ϕ− (a) < ϕ− (b); or ϕ+ (a) > ϕ+ (b) and ϕ− (a) = ϕ− (b); or ϕ+ (a) = ϕ+ (b) and ϕ− (a) < ϕ− (b)
aIb if:
ϕ+ (a) = ϕ+ (b) and ϕ− (a) = ϕ− (b)
aRb if:
ϕ+ (a) > ϕ+ (b) and ϕ− (a) > ϕ− (b); or ϕ+ (a) < ϕ+ (b) and ϕ− (a) < ϕ− (b)
On the basis of the above three conditions, all the alternatives are compared among each other and the decisions are made comparing one laptop model over another, which is presented in Table 17.
As there are 6 alternatives selected for this research purposes, there will be 15 comparisons in total. Table 17 presents the choices and preferences of one model over another.
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 |
(
Ranking is performed according to the decreasing order of net flow values. Table 18 shows the complete ranking of the laptop models. The preference ranking order of the models are given as follows:
Model 4 > Model 3 > Model 5 > Model 2 > Model 6 > Model 1
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 |
(
At the end of this analysis, it can be concluded that Model 4 is the best laptop model among these six available models followed by Model 3 and Model 5. Model 4 is coming out with the highest net flow value, that is, 0.55909, which enables it to occupy the rank 1 position, and similarly, Model 1 is coming out with the lowest net flow value, that is, −0.45738, so it occupies the last position, which denotes that it is the worst model among these group.
Although seven main important criteria are considered for this analysis, there are other specifications, both technical and nontechnical, that can also be included along with these, for example, display resolution, service center, weight, customer support, and graphics card for making these decision-making problem more precise and accurate. However, the same problem can also be carried out by applying other MCDM methods and the results can be compared with these results.