OK. Wofuru-Nyenke /Future Sustainability November 2024| Volume 02 | Issue 04 | Pages 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 https://doi.org/10.55670/fpll.fusus.2.4.3 November 2024| Volume 02 | Issue 04 | Pages 15-21 Journal homepage: https://fupubco.com/fusus ISSN 2995-0473 mailto:ovundah.wofuru-nyenke@ust.edu.ng https://doi.org/10.55670/fpll.fusus.2.4.3 https://fupubco.com/fusus 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 18 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 19 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. 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