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15-21 

15 

 

 

 

Article 

Sustainable lathe machine selection using 

PROMETHEE 
Ovundah King Wofuru-Nyenke  

Department of Mechanical Engineering, Faculty of Engineering, Rivers State University, Port Harcourt, Rivers State, Nigeria 

               A R T I C L E   I N F O 
 

Article history: 
Received 19 September 2024  
Received in revised form 
23 October 2024 
Accepted 01 November 2024 
 
Keywords:  
Lathe, Machine selection, Multi-criteria decision 
Analysis, PROMETHEE 
 
*Corresponding author 
Email address: 
ovundah.wofuru-nyenke@ust.edu.ng 
 
DOI: 10.55670/fpll.fusus.2.4.3 
 

A B S T R A C T 
 

The manufacturing echelon of supply chains utilizes several machines to 

convert raw materials into finished products. Therefore, during procurement of 

these machines, supply chain managers are usually saddled with the problem 

of obtaining the best machine from a group of similar alternatives, considering 

multiple criteria simultaneously. The main purpose of this study is to utilize the 

Preference Ranking Organization Method for Enrichment Evaluation 

(PROMETHEE) for selecting the best lathe machine from a group of five (5) 

similar alternatives, namely Lathe 1, Lathe 2, Lathe 3, Lathe 4 and Lathe 5. Four 

(4) criteria were used in evaluating the machines, namely power, price, 

complexity, and weight, with preference weights of 0.25, 0.3, 0.25, and 0.2, 

respectively. The results indicated that Lathe 2 is the best alternative because it 

has the highest total net flows of 0.325, followed by Lathe 5, which has total net 

flows of 0.03. Next is Lathe 1, which has a total net flow of -0.0188, followed by 

Lathe 3, having a total net flow of -0.0975, and finally, Lathe 4, which is the worst 

ranking alternative, having a total net flow of -0.2388. Therefore, PROMETHEE 

proved to be a viable multi-criteria decision-making tool for selecting the most 

suitable lathe machine among the group of alternative machines. This study is 

significant because it provides a procedure for aiding supply chain managers in 

selecting the best alternative among a group of similar alternatives using 

PROMETHEE. 

 

1. Introduction 

The Supply chain managers are responsible for 

managing various activities within supply chain networks. 

There are several methods for optimally managing 

production processes [1, 2]. The management process is 

usually tedious when the correct procedure is not 

consistently followed, or the various components of business 

management are not effectively combined. This could lead to 

various forms of waste within the manufacturing supply 

chain, especially when managers cannot predict uncertainty 

within the supply chain [3-8]. The main components of 

business management are money, manpower, materials, 

methods, and machines. Money refers to the capital utilized in 

the production of goods and the offering of services. It is 

important for the acquisition of raw materials, personnel 

hiring, acquisition of machines as well as equipoising costs 

incurred during the operation of the supply chain.  Manpower 

refers to the skilled and unskilled workers involved in 

the production of goods and rendering of services. Materials 

refer to the raw supplies fed into the supply chain and used to 

produce semi-finished or finished goods. Methods refer to the 

usual and recommended procedures for carrying out 

operations within the supply chain by established systems. 

Machines refer to the equipment used in converting raw 

materials into semi-finished or finished products [9, 10]. 

Machines are crucial for the profitability and survival of 

supply chains [11-14]. This is because rapid product output 

from the manufacturing echelon of supply chains is usually a 

result of well-running machinery, which can, in turn, provide 

the entire supply chain with a competitive edge [15, 16]. 

During procurement of machines, managers usually 

encounter the problem of deciding which machine is the best 

among alternatives. This is a complex problem because the 

machines must be evaluated simultaneously by considering 

multiple criteria. Multi-criteria decision analysis models have 

proven efficient in solving these decision-making problems 

involving evaluations based on multiple criteria. These 

methods can be employed in supplier selection, materials 

Future Sustainability 

Open Access Journal 

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November 2024| Volume 02 | Issue 04 | Pages 15-21  

Journal homepage: https://fupubco.com/fusus 

 
ISSN 2995-0473 

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OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

16 

 

selection, production scheduling, routing, inventory 

management, pricing strategies, and evaluation of various 

product designs, to name a few. The preference Ranking 

Organization Method for Enrichment Evaluation 

(PROMETHEE) has been utilized in many decision-making 

problems in engineering. The methodology has been applied 

to the complex and strategic problem of selecting a lean 

manufacturing system compared to a computer-integrated 

manufacturing system, considering the benefits to be gained 

and the impact on the organization's stakeholders [17]. It has 

also been applied to rank and select appropriate dispatching 

rules for a Dual-Resource Constrained manufacturing system 

[18]. The methodology has been applied to manufacturing 

scheduling by providing the ranking of alternative schedules 

based on completion times [19, 21]. Furthermore, the 

methodology has been combined with the Bayesian method 

to address equipment failure uncertainty by preventive 

maintenance planning and failure control in the context of 

equipment breakdown [22]. Similarly, the method has been 

used to determine the optimal preventive maintenance 

intervals [23]. The method has been applied to the problem of 

selecting the optimal solution for an inverse electromagnetic 

scattering problem [24]. PROMETHEE has also been used to 

rank alternatives during assembly planning as well as select 

the best equipment combination for individual stations with 

the aid of a multi-objective grouping genetic algorithm [25, 

26]. Moreover, it has been applied to the problem of choosing 

a predictive maintenance program within an automotive 

paint shop [27]. Lathe machines are machine tools that 

operate by rotating a cylindrical workpiece about an axis of 

rotation to perform various operations such as cutting, 

drilling, sanding, knurling, facing, deformation, and turning 

with the aid of a tool applied to the workpiece to create an 

object which is symmetrical about that axis. Lathe machines 

are very important machinery within major metalworking 

plants that produce metal products [28-30]. The main 

purpose of this study is to utilize PROMETHEE to select the 

best lathe machine from a group of similar alternatives. 

PROMETHEE provides the decision maker with a ranking of 

alternatives based on global or total net flows. The following 

sections describe the underlying equations of PROMETHEE, 

and the results of applying the equations to the lathe machine 

selection problem. 

2. Methodology 

In the PROMETHEE method, alternatives are pairwise 

compared in order to find the most appropriate alternative. 

The set of alternatives to be ranked are denoted by A =
{a1, a2, ⋯ , an} and the set of criteria are denoted by F =
{f1, f2, ⋯ , fm}. Also, denoting the evaluation of alternative 

aj on criterion fi by fi(aj) and assuming that fi(aj) is a numeric 

value. A preference matrix is generated from the data, and is 

used in calculating the global flows. The global pairwise 

preference degrees computed between all the ordered pairs 

of alternatives constitute the preference matrix. The global 

preference degrees are obtained from the criterion 

preference degrees by means of the weighted sum and 

provide the basis for deducing the global flows. 

 

 

2.1 Unicriterion preference degrees 

The unicriterion preference degree Pij
k, which can also be 

denoted as Pk(ai, aj), is calculated for each ordered pair of 

alternatives (ai, aj). This unicriterion preference degree, Pij
k, 

depicts how much more preferred alternative ai is to aj based 

solely on criterion fk. Pij
k will be a number between 0 and 1, 

and is a function of fk(ai) −  fk(aj), the more this difference, 

the stronger the unicriterion preference degree. A choice 
between three different types of preference functions has to 
be made by the decision maker, which in turn determines the 
preference degree. Considering the linear preference function 
with q as the indifference threshold and p as the preference 
threshold, the equation for the unicriterion preference degree 
is given by [31, 32]: 

Pij
k = {

0
[fk(ai)−fk(aj)−q]

[p−q]

1

      

if         fk(ai) −  fk(aj) ≤ q 

if  q < fk(ai) − fk(aj) < p

if         fk(ai) −  fk(aj) ≥ p

                   (1) 

However, if a Gaussian preference function is considered the 
equation for the unicriterion preference degree is given by 
[33]. 

Pij
k = {1 − exp (

−(fk(ai)− fk(aj))
2

2s2 )      if fk(ai) − fk(ai)  ≥ 0

0                                                           otherwise

    (2) 

 

where s is the inflexion point. Pij
k and Pji

k are not symmetric 

numbers but respect the condition 0≤ Pij
k + Pji

k ≤ 1. 

2.2 Global preference degree 

After calculating the ordered unicriterion preference 
degrees, the global preference degree, πij, can be computed 

taking the weights of each criterion into account. Denoting wk 
as the weight associated with the criterion fk. If the weight 

respects the condition ∑ wk
q
k=1 =1, then the global preference 

degree of alternative ai on aj is given by [33]: 

π(ai, aj) = πij = ∑ wj
q
k=1 ∙ Pij

k                                                             (3) 

where wj is the weight of a criterion j, and Pij
k is the 

unicriterion preference degree. This global preference degree 
lies between 0 and 1, and respects the constraint 0 ≤  πij +

 πji ≤ 1. Therefore, ∀i ∶  πii = 0. 

2.3 Global flows 

The ordered preference degrees are summarized into a 
unique score for each alternative, using the positive and 
negative flows. Denoting by Φ+(ai) the positive flows of 
alternative ai and Φ−(ai) the negative flows of alternative ai. 
Their values can be computed as follows [33]:  

Φ+(ai)  =
∑ πij

n
j=1

n−1
            (4) 

Φ−(ai)  =
∑ πji

n
j=1

n−1
            (5) 

where πij is the global preference degree of alternative ai on 

aj, and πji is the global preference degree of alternative aj on 

ai.  

2.4 Net flows 

The net flows, Φ(ai), summarizes the positive and 
negative flows with one formula given by [33]:  

 



OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

17 

 

Φ(ai) =  Φ+(ai) − Φ−(ai)                           (6) 

where Φ+(ai) is the positive flow of alternative ai and Φ−(ai) 
is the negative flow of alternative ai. The net flow is a number 
between -1 and 1. The higher this number is, the better the 
alternative will be. 

3. Results and discussion 

This section presents the results of applying the 
PROMETHEE method to the lathe machine selection problem. 
The objective is to rank five (5) different lathe machines 
based on four (4) criteria, namely: power, price, complexity, 
weight. The power criterion is to be maximized, the price 
criterion is to be minimized, the complexity criterion is to be 
minimized, and the weight criterion is to be minimized for 
each of the lathe machines. Table 1 shows the raw 
performance data of the various lathe machines and the 
selection criteria. 

Table 1. The performance of the five (5) lathe machines evaluated on 
four (4) criteria 

 
Power 

(hp) 
Price ($) Complexity 

Weight 
(lb) 

Objective MAX MIN MIN MIN 

Lathe 1 30 1500 Extreme 1000 

Lathe 2 40 1380 Medium 1200 

Lathe 3 50 2500 High 1500 

Lathe 4 60 4750 Medium 2400 

Lathe 5 100 6300 Low 3100 

 

Figure 1 shows the numeric values of the complexity criteria 
on a complexity scale. From Figure 1, low complexity 
corresponds to a numeric value of 2, medium complexity 
corresponds to a numeric value of 4, high complexity 
corresponds to a numeric value of 6, and extreme complexity 
corresponds to a numeric value of 8, on the complexity scale. 

Figure 1. Numeric complexity scale 

Therefore, from the performance data in Table 1 and the 
numerical scale in Figure 1, Lathe 1 has a complexity of 8, 
Lathe 2 has a complexity of 4, Lathe 3 has a complexity of 6, 
Lathe 4 has a complexity of 4, and Lathe 5 has a complexity of 
2. Table 2 shows the preference parameters for all the criteria. 
From Table 2, the power criterion has a linear preference 
function, a weight of 0.25, an indifference threshold of 20, and 
a preference threshold of 40. Also, the price criterion has a 
linear preference function, a weight of 0.3, an indifference 
threshold of 600, and a preference threshold of 1000. 
Furthermore, the complexity criterion has a linear preference 
function, a weight of 0.25, an indifference threshold of 1, and 
a preference threshold of 2. Moreover, the weight criterion 
has a linear preference function, a weight of 0.2, an 

indifference threshold of 500, and a preference threshold of 
1000. Table 3 shows the differences between evaluations of 
the lathes on power criterion. 

Table 2. Criteria preference parameters 

 
 

Table 3. Differences between evaluations of the lathes on power 
criterion 

 
From Table 3, considering the power criterion that has to 

be maximized, all lathes compared with themselves result in 
a difference of 0. Lathe 1 compared with Lathe 2 results in a 
difference of 10. Lathe 1 compared with Lathe 3 results in a 
difference of 20. Lathe 1 compared with Lathe 4 results in a 
difference of 30. Finally, Lathe 1 compared with Lathe 5 
results in a difference of 70. Table 4 shows the differences 
between evaluations of the lathes on price criterion. 

Table 4. Differences between evaluations of the lathes on price 
criterion 

 
 

Criterion Function 
Weight, 

wi 

Indifference 

Threshold, qi 

Preference 

Threshold, pi 

Power Linear 0.25 20 40 

Price Linear 0.3 600 1000 

Complexity Linear 0.25 1 2 

Weight Linear 0.2 500 1000 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 -10 -20 -30 -70 

Lathe 2 10 0 -10 -20 -60 

Lathe 3 20 10 0 -10 -50 

Lathe 4 30 20 10 0 -40 

Lathe 5 70 60 50 40 0 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 120 -1000 -3250 -4800 

Lathe 2 -120 0 -1120 -3370 -4920 

Lathe 3 1000 1120 0 -2250 -3800 

Lathe 4 3250 3370 2250 0 -1550 

Lathe 5 4800 4920 3800 1550 0 



OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

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From Table 4, considering the price criterion which has 
to be minimized, all lathes compared with themselves result 
in a difference of 0. Lathe 1 compared with Lathe 2 results in 
a difference of 120. Lathe 1 compared with Lathe 3 results in 
a difference of 1000. Lathe 1 compared with Lathe 4 results in 
a difference of 3250. Finally, Lathe 1 compared with Lathe 5 
results in a difference of 4800. Table 5 shows the differences 
between evaluations of the lathes on complexity criterion. 

Table 5. Differences between evaluations of the lathes on complexity 
criterion 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 4 2 4 6 

Lathe 2 -4 0 -2 0 2 

Lathe 3 -2 2 0 2 4 

Lathe 4 -4 0 -2 0 2 

Lathe 5 -6 -2 -4 -2 0 

 

From Table 5, considering the complexity criterion 
which has to be minimized, all lathes compared with 
themselves result in a difference of 0. Lathe 1 compared with 
Lathe 2 results in a difference of 120. Lathe 1 compared with 
Lathe 3 results in a difference of 1000. Lathe 1 compared with 
Lathe 4 results in a difference of 3250. Finally, Lathe 1 
compared with Lathe 5 results in a difference of 4800. Table 
6 shows the differences between evaluations of the lathes on 
weight criterion. 

Table 6. Differences between evaluations of the lathes on weight 
criterion 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 -200 -500 -1400 -2100 

Lathe 2 200 0 -300 -1200 -1900 

Lathe 3 500 300 0 -900 -1600 

Lathe 4 1400 1200 900 0 -700 

Lathe 5 2100 1900 1600 700 0 

 

From Table 6, considering the weight criterion which has 
to be minimized, all lathes compared with themselves result 
in a difference of 0. Lathe 1 compared with Lathe 2 results in 
a difference of 200. Lathe 1 compared with Lathe 3 results in 
a difference of 500. Lathe 1 compared with Lathe 4 results in 
a difference of 1400. Finally, Lathe 1 compared with Lathe 5 
results in a difference of 2100. Table 7 shows the pairwise 
comparison matrix for the power criterion. From Table 7, 
comparing the differences with the preference and 
indifference thresholds based on the power criterion, Lathe 4 
has a preference degree of 0.5 over Lathe 1. Moreover, the 
preference degree of Lathe 5 over Lathe 1, Lathe 2, Lathe 3, 
and Lathe 4 is 1. This means that based on the power 

criterion, Lathe 5 is preferred. Table 8 shows the pairwise 
comparison matrix for the price criterion. 

Table 7. Pairwise comparison matrix for the power criterion 

 

Table 8. Pairwise comparison matrix for the price criterion 

 

From Table 8, comparing the differences with the 
preference and indifference thresholds based on the price 
criterion, Lathe 1 has a preference degree of 1 over Lathe 3, 
Lathe 4 and Lathe 5. Moreover, the preference degree of Lathe 
2 over Lathe 3, Lathe 4, Lathe 5 is 1. Again, the preference 
degree of Lathe 3 over Lathe 4 and Lathe 5 is 1. Furthermore, 
the preference degree of Lathe 4 over Lathe 5 is 1. This means 
that based on the price criterion Lathe 1 and Lathe 2 are 
preferred over Lathe 3, Lathe 4 and Lathe 5. While Lathe 3 is 
preferred over Lathe 4 and Lathe 5, and Lathe 4 is preferred 
over Lathe 5. Table 9 shows the pairwise comparison matrix 
for the complexity criterion. 

Table 9. Pairwise comparison matrix for the complexity criterion 

 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 
1 

0 0 0 0 0 

Lathe 
2 

0 0 0 0 0 

Lathe 
3 

0 0 0 0 0 

Lathe 
4 

0.5 0 0 0 0 

Lathe 
5 

1 1 1 1 0 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 
1 

0 0 1 1 1 

Lathe 
2 

0 0 1 1 1 

Lathe 
3 

0 0 0 1 1 

Lathe 
4 

0 0 0 0 1 

Lathe 
5 

0 0 0 0 0 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 0 0 0 0 

Lathe 2 1 0 1 0 0 

Lathe 3 1 0 0 0 0 

Lathe 4 1 0 1 0 0 

Lathe 5 1 1 1 1 0 



OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

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From Table 9, comparing the differences with the 
preference and indifference thresholds based on the 
complexity criterion, when the preference degree is 0, it 
indicates that the difference in price is lower than the 
indifference threshold, and there is no difference between the 
two lathe machines being compared. On the other hand, when 
the preference degree is 1, it indicates that the difference 
between the two lathe machines being compared is greater 
than the preference threshold; therefore, there is a difference 
between the two lathe machines being compared. Table 10 
shows the pairwise comparison matrix for the weight 
criterion. 

Table 10. Pairwise comparison matrix for the weight criterion 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 0 0 1 1 

Lathe 2 0 0 0 1 1 

Lathe 3 0 0 0 0.8 1 

Lathe 4 0 0 0 0 0.4 

Lathe 5 0 0 0 0 0 

 

From Table 10, comparing the differences with the 
preference and indifference thresholds based on the weight 
criterion, when the preference degree is 0, it indicates that the 
difference in price is lower than the indifference threshold, 
and there is no difference between the two lathe machines 
being compared. When the preference degree is between 0 
and 1, it implies that the difference between the lathe 
machines being compared is between the indifference and 
preference thresholds. On the other hand, when the 
preference degree is 1, it indicates that the difference 
between the two lathe machines being compared is greater 
than the preference threshold. Therefore, there is a difference 
between the two lathe machines being compared. Table 11 
shows the pairwise preference matrix considering all the 
criteria and their weights. 

Table 11. Pairwise preference matrix 

 

 

From Table 11, the total positive flows, total negative 
flows, and total net flows for each lathe machine were 
calculated, and these data are shown in Table 12. 

Table 12. Total positive flows, total negative flows, and total net 
flows 

 

From Table 12, the total positive flows were calculated 
by averaging all the row preference degrees of a lathe 
compared to other lathes, excluding the preference degree of 
the lathe compared with itself. The total negative flows were 
calculated by averaging all the column preference degrees of 
a lathe, excluding the preference degree on the diagonal. The 
total net flows were obtained by subtracting the negative 
flows from the positive flows. Figure 2 is a plot of total net 
flows versus lathe machine type, and it shows the ranking of 
the lathe machines based on the total net flows. 

 
Figure 2. Plot of total net flows versus lathe machine type 

 
From Figure 2, Lathe 2 is the best alternative because it 

has the highest total net flows of 0.325, followed by Lathe 5, 
which has total net flows of 0.03. Next is Lathe 1, which has a 
total net flow of -0.0188, followed by Lathe 3, having a total 
net flow of -0.0975, and finally, Lathe 4, which is the worst 
ranking alternative, having a total net flow of -0.2388. This 
provides the ranking of the various lathe machines under 
consideration based on power, price, complexity, and weight 
criteria with linear preference functions. 

 
 
 
 
 
 
 

 Lathe 1 Lathe 2 Lathe 3 Lathe 4 Lathe 5 

Lathe 1 0 0 0.3 0.5 0.5 

Lathe 2 0.25 0 0.55 0.5 0.5 

Lathe 3 0.25 0 0 0.46 0.5 

Lathe 4 0.375 0 0.25 0 0.38 

Lathe 5 0.5 0.5 0.5 0.5 0 

Lathes 
Total Positive 

Flows 

Total 

Negative 

Flows 

Total Net 

Flows 

Lathe 1 0.325 0.34375 -0.0188 

Lathe 2 0.45 0.125 0.325 

Lathe 3 0.3025 0.4 -0.0975 

Lathe 4 0.25125 0.49 -0.2388 

Lathe 5 0.5 0.47 0.03 



OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

20 

 

4. Conclusion 

During the procurement of machines for manufacturing, 

managers are faced with the problem of selecting the best 

machine among similar alternatives. The machine selection 

problem is complex because decisions usually have to be 

made based on more than one criterion. This study utilizes 

PROMETHEE to select the best lathe machine from a group of 

five similar alternatives, namely Lathe 1, Lathe 2, Lathe 3, 

Lathe 4, and Lathe 5. The machines were evaluated based on 

criteria such as power, price, complexity, and weight, having 

preference weights of 0.25, 0.3, 0.25, and 0.2, respectively. 

The results indicated that Lathe 2 is the best alternative 

because it has the highest total net flows of 0.325, followed by 

Lathe 5, which has total net flows of 0.03. Next is Lathe 1, 

which has a total net flow of -0.0188, followed by Lathe 3, 

having a total net flow of -0.0975, and finally, Lathe 4, which 

is the worst ranking alternative, having a total net flow of -

0.2388. The study provides a procedure for selecting the best 

alternative among a group of similar alternatives using 

PROMETHEE. For further research, other multi-criteria 

decision analysis methods and method combinations can be 

utilized to select the best machine among a group of machines 

during procurement. Also, the performance of each 

alternative can be investigated considering a situation where 

the preference functions of each criterion is not linear. 

Ethical issue 

The author is aware of and complies with best practices in 
publication ethics, specifically with regard to authorship 
(avoidance of guest authorship), dual submission, 
manipulation of figures, competing interests, and compliance 
with policies on research ethics. The author adheres to 
publication requirements that the submitted work is original 
and has not been published elsewhere. 

Data availability statement 
The manuscript contains all the data. However, more data will 

be available upon request from the authors. 

Conflict of interest 

The author declares no potential conflict of interest. 

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OK. Wofuru-Nyenke /Future Sustainability                                                                       November 2024| Volume 02 | Issue 04 | Pages 15-21 

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DOI:10.1002/9781118644898 

 

 

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