A proper CNC machine selection problem is an important issue for manufacturing companies under competitive market conditions. The selection of an improper machine tool can cause many problems such as production capabilities and productivity indicators considering time and money industrially and practically. In this paper, a comprehensive solution approach is presented for the CNC machine tool selection problem according to the determined criteria. Seven main and thirteen sub-criteria were determined for the evaluation of the seven alternatives. To purify the selection process from subjectivity, instead of a single decision-maker, the opinions of six different experts on the importance of the criteria were taken and evaluated using the Best-Worst method. According to the evaluations, the order of importance of the main criteria has been determined as cost, productivity, flexibility, and dimensions. After the weighting of the criteria, three different ranking methods (GRA, COPRAS, and MULTIMOORA) were preferred due to the high investment costs of the selected alternatives. The findings obtained by solving the problem of selection of the CNC machine are close to those obtained by past researchers. As a result, using the suggested methodology, effective alternative decision-making solutions are obtained.
Companies need to have many plans related to marketing, financing, and production in today’s competitive markets. On the other hand, companies, based on these strategies, have to take a series of decisions, especially at the stage of establishment and when making growth decisions. One of these decisions is the selection of machines and equipment to be used in manufacturing. Identifying the appropriate machine or equipment from among the alternatives available is also a very important decision which, in the long run, affects the efficiency of the production system. The use of suitable machinery improves the manufacturing process, ensures the effective use of manpower, increases productivity, and enhances the versatility of the system (Dağdeviren,
Generally, computer numerical control (CNC) machines, which can be used with high precision to perform repetitive, challenging, and unsafe production jobs, are considered cost-effective equipment (Athawale and Chakraborty,
The scope of this paper, which is based on these needs, is to select a proper machine tool using Best-Worst weighted GRA, COPRAS, and MULTIMOORA methods. These methods are used to determine the order of priority with managerial insights and implications. However, this paper tries to answer the following questions:
What are the criteria of the most used features in the CNC machine tool selection process?
Which alternative CNC machine tool may be more suitable under variable weighted uncertainties?
How different weights of expert opinions will affect this selection problem on the Best-Worst methodology?
The rest of this study is organized as follows: a related literature review is given in Section
For several years, machine tool selection has been an important decision problem for manufacturing firms. The primary explanation for this is that there are several issues with the selection of an inappropriate machine that affects overall efficiency and production capabilities in the long run (Taha and Rostam,
Since there is more than one criterion, Multi-Criteria Decision Making (MCDM) methods are widely used in the solution of the Machine Tool Selection (MTS) problem. Several options and criteria are evaluated in these studies to decide the best alternative. It is considered as the most suitable option for the decision-maker who, after rating the alternatives, gets the highest score (Ayağ and Özdemir,
Due to uncertainties in the decision-maker’s decisions, a fuzzy AHP instead of traditional AHP was used for the evaluation and justification of an advanced production system (Ayag and Ozdemir, 2006) with developing a software (Durán and Aguilo,
Moreover, in order to measure the level of benefit provided by using fuzzy numbers in multi-criteria decision models, Yurdakul and Ic (
Detailed literature review.
Application area | Method | Uncertainty | ||||||
Source | Machine selection | Tool selection | Technology selection | MCDM Method | Integrated method | Others | Crisp | Fuzzy |
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Methods such as SAW (Patel
In this study, a new solution approach is proposed where criteria weights are determined by the Best-Worst method, and rankings are determined by considering with GRA, COPRAS, and MULTIMOORA methods. Within the scope of the study, a new solution approach in which weighting and ranking methods are used together has been tried to be put forward. The methods used are powerful methods that have not been used before in the machine tool selection problem and their effectiveness has been shown in previous studies in the literature and this study.
In the machine tool market, there are hundreds of CNC machine alternatives. In the first step, machine tool alternatives that can satisfy the company’s needs should be identified. In the second stage, the defined alternatives are evaluated using any decision model. When comparing various machine tools, decision-makers use a set of criteria. These criteria are generally related to the technological features of the machine, but they also include criteria such as productivity, flexibility, cost, maintenance, and service. Ayağ and Özdemir (
In this paper, the Best-Worst method is applied for determining the criteria weights using the mean of the expert opinions via taking advantage of pairwise comparison from best to worst. This method has been preferred for reasons such as making less and more consistent comparisons, being able to be used with other methods to be used for sorting, and not having to deal with fractional numbers. On the other hand, the choice of CNC machine tool is one of the decision problems that require a very high investment. For this reason, alternatives and decisions can be compared by using more than one method rather than a single method for ranking the alternatives. As for the choice of alternatives, GRA is selected with reference series, COPRAS is also selected to evaluate the performance of each alternative, taking into account the contradictory situations, and MULTIMOORA is preferred to apply dominance solution in terms of the subordinate ranking methods for this study. These alternative selection methods are used with the determined criteria weights from the Best-Worst method. Consequently, the whole solution procedure is designed for the proper decision-making process on the CNC selection research problem.
The method proposed by Rezaei (
Step 4: A binary comparison is made between the other criteria and the worst criterion, again using the scale 1–9, and the OW vector (
Step 5: Optimal weights (
Here, the status
Then the expressions here are converted into the mathematical model shown below:
With the solving of the model, the value of the optimal weights is obtained that is the criterion that shows how consistent the evaluations are. If this value is close to zero, it means that a consistent evaluation has been made.
An essential approach of the grey system theory (GST) used in the decision-making process and measuring the changes of similarities and differences between its factors over time is called Grey Relational Analysis (GRA) (Feng and Wang,
In the method, when the decision-maker has no information, that is, when the information is black, the greyness of a process is done. In most decision problems with insufficient and/or incomplete information, the GRA method is used to select, rank, and evaluate (Chan and Tong,
After the normalization process, all values take values between 0–1. A decision alternative (
In the following step, the relationship between the desired and actual experimental data is determined by calculating the grey relational coefficient from the absolute difference matrix. Grey coefficients (
In the last step, the grey relational degree is calculated by taking the average of the grey relational coefficients and the ranking is performed according to this value. Grey relational degrees (
The COPRAS method developed by Zavadskas
The COPRAS method assumes a direct and proportional dependence of the degree of importance and utility of decision options on a system of criteria that adequately defines the alternatives and the values and weights of the criteria. Determining the importance, priority order, and degree of use of alternatives is carried out in five stages (Kaklauskas
The sum of the dimensionless weighted index values is equal to “
The larger the value of
The Multi-Objective Optimization Based on Ratio Analysis (MOORA) method proposed by Brauers and Zavadskas (
MULTIMOORA is mostly used as a multi-criteria decision-making technique in fields such as industry, economy, environment, health services, and information technologies as practical applications. In this section, we first explain the MULTIMOORA method in terms of the subordinate ranking methods. The first step also involves generating a decision matrix and weight vector, as seen below, with
Also, on the MCDM problems, the ratings of alternatives may have different dimensions generally, so, the normalized ratings should be required and for this, Van Delft and Nijkamp normalization approach is used in MULTIMOORA application considering the most robust choice and proving by Brauers
In certain cases, the triple subordinate methods are also known as the ratio, complete multiplicative, and reference point forms, and they are used to solve the exits problem. The ratio method should be used as a completely compensatory model if the problem has any independent criteria. The ratio system is computed by Eq. (
The reference point approach, on the other hand, is a conservative method for measuring and comparing the ratio system and complete multiplicative form with Eqs. (
Eq. (
The best alternative found by the Reference Point Approach has the least benefit (
Although Brauers and Zavadskas (
The maximum utility alternative is the best alternative based on the Full Multiplicative Form, and the sequence of this technique is obtained by equation (
Using these subordinate ranks, we also should decide the final ranking of the alternatives in the final phase. The aggregating multiple subordinate rankings are presented by Brauers and Zavadskas (
The basic framework of the proposed method.
One of the most important decisions in the design and construction of a competitive manufacturing environment is the selection of the appropriate machine tools. This chapter contains the application of the proposed method to solve the machine tool selection problem. The basic framework of the methods proposed within the scope of the study and detailed in Section
According to the consumer specifications, the appropriate machine should be selected from the existing database. At the beginning of the research, 4 main and 13 sub-criteria were determined to be used in the solution of the problem, taking into account the literature research and expert opinions. Dimensions (C_{1}), Flexibility (C_{2}), Productivity (C_{3}), and Cost (C_{4}) criteria, whose sub-criteria are shown in Table
The determined weights can be used with equal weight or they can be weighted differently according to the needs of the company. The importance of the criteria was determined by using the Best-Worst method, details of which are given in Section
The mainand sub-criteria.
Main criteria | Sub-criteria | Objective | Unit |
Dimensions (C_{1}) | Table load (C_{11}) | Max | kg s |
Main travel (C_{12}) | Max | mm | |
Table size (C_{13}) | Max | m^{2} | |
Machine weight (C_{14}) | Min | kg s | |
Flexibility (C_{2}) | Spindle rate (C_{21}) | Max | rpm |
Spindle power (C_{22}) | Max | kw | |
Max. tool weight (C_{23}) | Max | kg s | |
Productivity (C_{3}) | Feed rate (axis |
Max | mm/min s |
Tool magazine capacity (C_{32}) | Max | set | |
Cutting feed rate (C_{33}) | Max | mm/min s | |
Cost (C_{4}) | Procurement price (C_{41}) | Min | $ |
Operation cost (C_{42}) | Min | $ | |
Maintenance cost (C_{43}) | Min | $ |
BO vectors for main criteria.
Experts no. | Best | Dimensions (C_{1}) | Flexibility (C_{2}) | Productivity (C_{3}) | Cost (C_{4}) |
Experts 1 | Cost (C_{4}) | 6 | 4 | 2 | 1 |
Experts 2 | Cost (C_{4}) | 5 | 3 | 2 | 1 |
Experts 3 | Cost (C_{4}) | 7 | 4 | 2 | 1 |
Experts 4 | Cost (C_{4}) | 6 | 5 | 3 | 1 |
Experts 5 | Cost (C_{4}) | 6 | 2 | 4 | 1 |
Experts 6 | Cost (C_{4}) | 6 | 2 | 4 | 1 |
OW vectors for main criteria.
Experts No. | Worst | Dimensions (C_{1}) | Flexibility (C_{2}) | Productivity (C_{3}) | Cost (C_{4}) |
Experts 1 | Dimensions (C_{1}) | 1 | 2 | 4 | 6 |
Experts 2 | Dimensions (C_{1}) | 1 | 2 | 3 | 5 |
Experts 3 | Dimensions (C_{1}) | 1 | 2 | 3 | 7 |
Experts 4 | Dimensions (C_{1}) | 1 | 2 | 2 | 6 |
Experts 5 | Dimensions (C_{1}) | 1 | 3 | 2 | 6 |
Experts 6 | Dimensions (C_{1}) | 1 | 4 | 2 | 6 |
The weights of all the main and sub-criteria are shown in Table
After determining the decision alternatives and criteria weights, the ranking process was started with the GRA, COPRAS, and MULTIMOORA methods. The following section explains the details of the sorting process with the aforementioned methods.
The weights of the main criteria.
Criteria | Weights of criteria | Mean | |||||
Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6 | ||
Dimensions (C_{1}) | 0.0784 | 0.0923 | 0.0784 | 0.0879 | 0.0811 | 0.0709 | 0.0815 |
Flexibility (C_{2}) | 0.1373 | 0.1692 | 0.1373 | 0.1255 | 0.2703 | 0.2196 | 0.1765 |
Productivity (C_{3}) | 0.2745 | 0.2538 | 0.2549 | 0.2092 | 0.1351 | 0.1318 | 0.2099 |
Cost (C_{4}) | 0.5098 | 0.4846 | 0.5294 | 0.5774 | 0.5135 | 0.5777 | 0.5321 |
Ksi | 0.0392 | 0.0231 | 0.0196 | 0.0502 | 0.0270 | 0.0811 | 0.0400 |
The final weights of criteria.
Main criteria | The weight of main criteria | Sub-criteria | The weight of sub-criteria | Final weights |
Dimensions (C_{1}) | 0.0815 | Table load (C_{11}) | 0.197 | 0.016 |
Main travel (C_{12}) | 0.558 | 0.046 | ||
Table size (C_{13}) | 0.153 | 0.012 | ||
Machine weight (C_{14}) | 0.092 | 0.007 | ||
Flexibility (C_{2}) | 0.1765 | Spindle rate (C_{21}) | 0.222 | 0.039 |
Spindle power (C_{22}) | 0.591 | 0.104 | ||
Max. tool weight (C_{23}) | 0.187 | 0.033 | ||
Productivity (C_{3}) | 0.2099 | Feed rate (C_{31}) | 0.159 | 0.033 |
Tool magazine capacity (C_{32}) | 0.581 | 0.122 | ||
Cutting feed rate (C_{33}) | 0.261 | 0.055 | ||
Cost (C_{4}) | 0.5321 | Procurement price (C_{41}) | 0.719 | 0.382 |
Operation cost (C_{42}) | 0.169 | 0.090 | ||
Maintanance cost (C_{43}) | 0.113 | 0.060 |
The data of the alternatives (decision matrix).
C_{1} | C_{2} | C_{3} | C_{4} | ||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |
Units | kg s | mm | m^{2} | kg s | rpm | kw | kg s | mm/min s | set | mm/min s | $ | $ | $ |
Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min |
Alt.∖Weights | 0.016 | 0.046 | 0.012 | 0.007 | 0.039 | 0.104 | 0.033 | 0.033 | 0.122 | 0.055 | 0.382 | 0.090 | 0.060 |
400 | 687 | 0.465 | 5800 | 8000 | 7.5 | 7 | 28.0 | 24 | 10.0 | 206250 | 25800 | 5150 | |
400 | 720 | 0.500 | 6000 | 8000 | 5.5 | 6 | 26.7 | 24 | 12.0 | 262500 | 32000 | 5800 | |
800 | 710 | 0.720 | 8000 | 15000 | 7.5 | 10 | 27.9 | 32 | 1.2 | 318750 | 39800 | 7000 | |
300 | 600 | 0.550 | 3300 | 10000 | 10.1 | 3 | 52.0 | 21 | 30.0 | 335750 | 41000 | 8000 | |
1600 | 953 | 0.975 | 11000 | 8000 | 11 | 6 | 18.3 | 24 | 5.0 | 412500 | 51500 | 10300 | |
250 | 567 | 0.336 | 3800 | 12000 | 5.5 | 6 | 48.0 | 25 | 15.0 | 262500 | 31250 | 5470 | |
3000 | 980 | 1.000 | 12500 | 8000 | 11 | 15 | 19.3 | 20 | 10.0 | 487500 | 58000 | 11000 | |
Min | 250 | 566.67 | 0.336 | 3300 | 8000 | 5.5 | 3 | 18.333 | 20 | 1.2 | 206250 | 25800 | 5150 |
Max | 3000 | 980 | 1.000 | 12500 | 15000 | 11 | 15 | 52 | 32 | 30 | 487500 | 58000 | 11000 |
The method consists of three basic steps: normalization, grey relational coefficient calculation, and grey relational degree calculation. In the first step, the data of the alternatives are transformed into comparison sequences by normalizing the criteria according to the benefits, cost, and optimality of the criteria. Normalized versions of the data presented in Table
The normalized decision matrix.
C_{1} | C_{2} | C_{3} | C_{4} | ||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |
Alt.∖Ref. serie | 1.000 | 1.000 | 1.000 | 0.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.000 | 0.000 | 0.000 |
A_{1} | 0.055 | 0.290 | 0.194 | 0.728 | 0.000 | 0.364 | 0.333 | 0.287 | 0.333 | 0.306 | 1.000 | 1.000 | 1.000 |
0.055 | 0.371 | 0.247 | 0.707 | 0.000 | 0.000 | 0.250 | 0.248 | 0.333 | 0.375 | 0.800 | 0.807 | 0.889 | |
0.200 | 0.347 | 0.578 | 0.489 | 1.000 | 0.364 | 0.583 | 0.284 | 1.000 | 0.000 | 0.600 | 0.565 | 0.684 | |
0.018 | 0.081 | 0.322 | 1.000 | 0.286 | 0.836 | 0.000 | 1.000 | 0.083 | 1.000 | 0.540 | 0.528 | 0.513 | |
0.491 | 0.935 | 0.962 | 0.163 | 0.000 | 1.000 | 0.250 | 0.000 | 0.333 | 0.132 | 0.267 | 0.202 | 0.120 | |
0.000 | 0.000 | 0.000 | 0.946 | 0.571 | 0.000 | 0.250 | 0.881 | 0.417 | 0.479 | 0.800 | 0.831 | 0.945 | |
1.000 | 1.000 | 1.000 | 0.000 | 0.000 | 1.000 | 1.000 | 0.030 | 0.000 | 0.306 | 0.000 | 0.000 | 0.000 |
After the normalization process, the absolute value table is created by using the equation shown in equation (
The absolute value table.
C_{1} | C_{2} | C_{3} | C_{4} | ||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |
Alt.∖Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min |
0.945 | 0.710 | 0.806 | 0.728 | 1.000 | 0.636 | 0.667 | 0.713 | 0.667 | 0.694 | 1.000 | 1.000 | 1.000 | |
0.945 | 0.629 | 0.753 | 0.707 | 1.000 | 1.000 | 0.750 | 0.752 | 0.667 | 0.625 | 0.800 | 0.807 | 0.889 | |
0.800 | 0.653 | 0.422 | 0.489 | 0.000 | 0.636 | 0.417 | 0.716 | 0.000 | 1.000 | 0.600 | 0.565 | 0.684 | |
0.982 | 0.919 | 0.678 | 1.000 | 0.714 | 0.164 | 1.000 | 0.000 | 0.917 | 0.000 | 0.540 | 0.528 | 0.513 | |
0.509 | 0.065 | 0.038 | 0.163 | 1.000 | 0.000 | 0.750 | 1.000 | 0.667 | 0.868 | 0.267 | 0.202 | 0.120 | |
1.000 | 1.000 | 1.000 | 0.946 | 0.429 | 1.000 | 0.750 | 0.119 | 0.583 | 0.521 | 0.800 | 0.831 | 0.945 | |
0.000 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | 0.970 | 1.000 | 0.694 | 0.000 | 0.000 | 0.000 |
Grey coefficients (
The absolute value table.
C_{1} | C_{2} | C_{3} | C_{4} | Weighted rank | |||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |||
Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min | ||
Weights | 0.016 | 0.046 | 0.012 | 0.007 | 0.039 | 0.104 | 0.033 | 0.033 | 0.122 | 0.055 | 0.382 | 0.090 | 0.060 | ||
0.346 | 0.413 | 0.383 | 0.407 | 0.333 | 0.440 | 0.429 | 0.412 | 0.429 | 0.419 | 0.333 | 0.333 | 0.333 | 0.0286 | 7 | |
0.346 | 0.443 | 0.399 | 0.414 | 0.333 | 0.333 | 0.400 | 0.399 | 0.429 | 0.444 | 0.385 | 0.382 | 0.360 | 0.0298 | 6 | |
0.385 | 0.434 | 0.542 | 0.505 | 1.000 | 0.440 | 0.545 | 0.411 | 1.000 | 0.333 | 0.455 | 0.469 | 0.422 | 0.0411 | 3 | |
0.337 | 0.352 | 0.425 | 0.333 | 0.412 | 0.753 | 0.333 | 1.000 | 0.353 | 1.000 | 0.481 | 0.486 | 0.494 | 0.0402 | 4 | |
0.495 | 0.886 | 0.930 | 0.754 | 0.333 | 1.000 | 0.400 | 0.333 | 0.429 | 0.365 | 0.652 | 0.712 | 0.807 | 0.0493 | 2 | |
0.333 | 0.333 | 0.333 | 0.346 | 0.538 | 0.333 | 0.400 | 0.808 | 0.462 | 0.490 | 0.385 | 0.376 | 0.346 | 0.0314 | 5 | |
1.000 | 1.000 | 1.000 | 1.000 | 0.333 | 1.000 | 1.000 | 0.340 | 0.333 | 0.419 | 1.000 | 1.000 | 1.000 | 0.0645 | 1 |
The second method used to sort the alternatives is the COPRAS method. This method starts with the formation of the weighted decision matrix with the help of Eq. (
The weighted normalized decision matrix for the COPRAS method.
C_{1} | C_{2} | C_{3} | C_{4} | ||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |
Alt.∖Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min |
0.0010 | 0.0060 | 0.0013 | 0.0009 | 0.0046 | 0.0135 | 0.0044 | 0.0042 | 0.0172 | 0.0066 | 0.0345 | 0.0083 | 0.0059 | |
0.0010 | 0.0063 | 0.0014 | 0.0009 | 0.0046 | 0.0099 | 0.0037 | 0.0040 | 0.0172 | 0.0079 | 0.0439 | 0.0103 | 0.0066 | |
0.0019 | 0.0062 | 0.0020 | 0.0012 | 0.0085 | 0.0135 | 0.0062 | 0.0042 | 0.0229 | 0.0008 | 0.0533 | 0.0128 | 0.0080 | |
0.0007 | 0.0052 | 0.0015 | 0.0005 | 0.0057 | 0.0181 | 0.0019 | 0.0079 | 0.0151 | 0.0197 | 0.0562 | 0.0132 | 0.0091 | |
0.0038 | 0.0083 | 0.0027 | 0.0016 | 0.0046 | 0.0197 | 0.0037 | 0.0028 | 0.0172 | 0.0033 | 0.0690 | 0.0165 | 0.0117 | |
0.0006 | 0.0049 | 0.0009 | 0.0006 | 0.0068 | 0.0099 | 0.0037 | 0.0073 | 0.0179 | 0.0099 | 0.0439 | 0.0100 | 0.0062 | |
0.0071 | 0.0086 | 0.0027 | 0.0019 | 0.0046 | 0.0197 | 0.0094 | 0.0029 | 0.0143 | 0.0066 | 0.0816 | 0.0186 | 0.0125 |
After calculating the normalized decision matrix, the sum of the criteria values to be minimized for each alternative (
Calculations of the COPRAS method.
Alternatives | Order of alternatives | ||||||||
1.059 | 0.395 | 0.395 | 4.000 | 1.000 | 5.258 | 1.8195 | 86 | 4 | |
1.038 | 0.458 | 0.862 | 1.6936 | 80 | 7 | ||||
1.278 | 0.573 | 0.689 | 1.8020 | 85 | 5 | ||||
1.376 | 0.511 | 0.774 | 1.9647 | 93 | 2 | ||||
1.337 | 0.778 | 0.508 | 1.7236 | 82 | 6 | ||||
1.147 | 0.406 | 0.974 | 1.8877 | 90 | 3 | ||||
1.766 | 0.878 | 0.451 | 2.1089 | 100 | 1 |
The first step of the MULTIMOORA method also includes creating a decision matrix and weight vector with
Normalized decision matrix.
Alternatives | C_{1} | C_{2} | C_{3} | C_{4} | |||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |
Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min |
Weights | 0.0160 | 0.0455 | 0.0125 | 0.0075 | 0.0393 | 0.1042 | 0.0330 | 0.0333 | 0.1219 | 0.0547 | 0.3825 | 0.0897 | 0.0599 |
0.1120 | 0.3410 | 0.2540 | 0.2780 | 0.2970 | 0.3300 | 0.3160 | 0.3130 | 0.3700 | 0.2590 | 0.2300 | 0.2360 | 0.2480 | |
0.1120 | 0.3580 | 0.2730 | 0.2870 | 0.2970 | 0.2420 | 0.2710 | 0.2980 | 0.3700 | 0.3100 | 0.2930 | 0.2930 | 0.2800 | |
0.2250 | 0.3530 | 0.3930 | 0.3830 | 0.5570 | 0.3300 | 0.4510 | 0.3120 | 0.4930 | 0.0310 | 0.3560 | 0.3640 | 0.3380 | |
0.0840 | 0.2980 | 0.3000 | 0.1580 | 0.3710 | 0.4440 | 0.1350 | 0.5820 | 0.3230 | 0.7760 | 0.3750 | 0.3750 | 0.3860 | |
0.4490 | 0.4740 | 0.5330 | 0.5270 | 0.2970 | 0.4840 | 0.2710 | 0.2050 | 0.3700 | 0.1290 | 0.4600 | 0.4710 | 0.4970 | |
0.0700 | 0.2820 | 0.1840 | 0.1820 | 0.4460 | 0.2420 | 0.2710 | 0.5370 | 0.3850 | 0.3880 | 0.2930 | 0.2860 | 0.2640 | |
0.8430 | 0.4870 | 0.5460 | 0.5980 | 0.2970 | 0.4840 | 0.6770 | 0.2160 | 0.3080 | 0.2590 | 0.5440 | 0.5310 | 0.5300 |
After the normalized decision matrix is created, the alternative ranking is determined according to the decreasing order of the calculated
Calculations of the ratio system.
C_{1} | C_{2} | C_{3} | C_{4} | Order | |||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |||
Alt.∖Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min | ||
0.002 | 0.016 | 0.003 | 0.002 | 0.012 | 0.034 | 0.010 | 0.010 | 0.045 | 0.014 | 0.088 | 0.021 | 0.015 | 0.020 | 1 | |
0.002 | 0.016 | 0.003 | 0.002 | 0.012 | 0.025 | 0.009 | 0.010 | 0.045 | 0.017 | 0.112 | 0.026 | 0.017 | −0.018 | 4 | |
0.004 | 0.016 | 0.005 | 0.003 | 0.022 | 0.034 | 0.015 | 0.010 | 0.060 | 0.002 | 0.136 | 0.033 | 0.020 | −0.024 | 5 | |
0.001 | 0.014 | 0.004 | 0.001 | 0.015 | 0.046 | 0.004 | 0.019 | 0.039 | 0.042 | 0.143 | 0.034 | 0.023 | −0.016 | 3 | |
0.007 | 0.022 | 0.007 | 0.004 | 0.012 | 0.050 | 0.009 | 0.007 | 0.045 | 0.007 | 0.176 | 0.042 | 0.030 | −0.087 | 6 | |
0.001 | 0.013 | 0.002 | 0.001 | 0.017 | 0.025 | 0.009 | 0.018 | 0.047 | 0.021 | 0.112 | 0.026 | 0.016 | −0.001 | ||
0.014 | 0.022 | 0.007 | 0.004 | 0.012 | 0.050 | 0.022 | 0.007 | 0.038 | 0.014 | 0.208 | 0.048 | 0.032 | −0.106 |
In the Reference Point Approach (RPA), which is a conservative method, first of all, the absolute difference (distance) between the
In the Full Multiplicative Form (FMF), the multiplication values of the criteria in the normalized decision matrix, which are in the direction of maximization, are divided by the multiplication value of the criteria to be minimized, and “
At the last stage, the rankings found as a result of the calculations above have been converted into a single line with the theory of dominance. The final ranks were determined by taking the average of the rankings. The final rankings determined by applying the theory of dominance in the MULTIMOORA method are given in Table
Calculations of the reference point approach.
C_{1} | C_{2} | C_{3} | C_{4} | Order | |||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |||
Alt.∖Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min | ||
0.012 | 0.007 | 0.004 | 0.002 | 0.010 | 0.016 | 0.012 | 0.009 | 0.015 | 0.028 | 0.120 | 0.026 | 0.017 | 0.1201 | 7 | |
0.012 | 0.006 | 0.003 | 0.002 | 0.010 | 0.025 | 0.013 | 0.009 | 0.015 | 0.025 | 0.096 | 0.021 | 0.015 | 0.0961 | 5-6 | |
0.010 | 0.006 | 0.002 | 0.002 | 0.000 | 0.016 | 0.007 | 0.009 | 0.000 | 0.041 | 0.072 | 0.015 | 0.012 | 0.0720 | 4 | |
0.012 | 0.009 | 0.003 | 0.003 | 0.007 | 0.004 | 0.018 | 0.000 | 0.021 | 0.000 | 0.065 | 0.014 | 0.009 | 0.0648 | 3 | |
0.006 | 0.001 | 0.000 | 0.001 | 0.010 | 0.000 | 0.013 | 0.013 | 0.015 | 0.035 | 0.032 | 0.005 | 0.002 | 0.0353 | 2 | |
0.012 | 0.009 | 0.005 | 0.003 | 0.004 | 0.025 | 0.013 | 0.001 | 0.013 | 0.021 | 0.096 | 0.022 | 0.016 | 0.0961 | 5-6 | |
0.000 | 0.000 | 0.000 | 0.000 | 0.010 | 0.000 | 0.000 | 0.012 | 0.023 | 0.028 | 0.000 | 0.000 | 0.000 | 0.0283 | 1 | |
0.014 | 0.022 | 0.007 | 0.004 | 0.022 | 0.050 | 0.022 | 0.019 | 0.060 | 0.042 | 0.208 | 0.048 | 0.032 |
Calculations of the full multiplicative form.
C_{1} | C_{2} | C_{3} | C_{4} | Order | |||||||||||
C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | |||
Alt.∖Goal | Max | Max | Max | Min | Max | Max | Max | Max | Max | Max | Min | Min | Min | ||
0.966 | 0.952 | 0.983 | 0.990 | 0.953 | 0.891 | 0.963 | 0.962 | 0.886 | 0.929 | 0.570 | 0.879 | 0.920 | 1.28132 | 1 | |
0.966 | 0.954 | 0.984 | 0.991 | 0.953 | 0.862 | 0.958 | 0.961 | 0.886 | 0.938 | 0.625 | 0.896 | 0.927 | 1.10843 | 3 | |
0.976 | 0.954 | 0.988 | 0.993 | 0.977 | 0.891 | 0.974 | 0.962 | 0.917 | 0.827 | 0.674 | 0.913 | 0.937 | 0.99529 | 5 | |
0.961 | 0.946 | 0.985 | 0.986 | 0.962 | 0.919 | 0.936 | 0.982 | 0.871 | 0.986 | 0.687 | 0.916 | 0.945 | 1.06753 | 4 | |
0.987 | 0.967 | 0.992 | 0.995 | 0.953 | 0.927 | 0.958 | 0.949 | 0.886 | 0.894 | 0.743 | 0.935 | 0.959 | 0.90819 | 6 | |
0.958 | 0.944 | 0.979 | 0.987 | 0.969 | 0.862 | 0.958 | 0.979 | 0.890 | 0.950 | 0.625 | 0.894 | 0.923 | 1.15174 | 2 | |
0.997 | 0.968 | 0.992 | 0.996 | 0.953 | 0.927 | 0.987 | 0.950 | 0.866 | 0.929 | 0.792 | 0.945 | 0.963 | 0.89015 | 7 |
Calculations of the full multiplicative form.
Alternatives | RS | RPA | FMF | Mean | Final order |
A_{1} | 1 | 7 | 1 | 3.00 | 1 |
A_{2} | 4 | 6 | 3 | 4.33 | 4 |
A_{3} | 5 | 4 | 5 | 4.67 | 5 |
A_{4} | 3 | 3 | 4 | 3.33 | 3 |
A_{5} | 6 | 2 | 6 | 4.67 | 6 |
A_{6} | 2 | 5 | 2 | 3.00 | 2 |
A_{7} | 7 | 1 | 7 | 5.00 | 7 |
As a result, the different sequences shown in Table
Rankings obtained by different methods.
Alternatives | GRA | COPRAS | MULTIMOORA |
A_{1} | 7 | 1 | 1 |
A_{2} | 6 | 4 | 4 |
A_{3} | 3 | 5 | 5 |
A_{4} | 4 | 3 | 3 |
A_{5} | 2 | 7 | 6 |
A_{6} | 5 | 2 | 2 |
A_{7} | 1 | 6 | 7 |
The final rankings obtained by different methods.
The abundance of machine alternatives, the difficulty in accessing reliable information, and the lack of experts in evaluating machine features make machine tool selection a difficult and important problem. In addition, it is known that an unsuitable machine selection adversely affects the efficiency, sensitivity, and flexibility of the entire production system. When all these situations are taken into consideration, it is seen that the right information should be made by the right people and using the appropriate methods for the selection of a proper machine tool. When the studies in the literature are examined, many different methods provide different solutions. In this context, many studies have been conducted in which the uncertainty situation, as well as deterministic methods, are taken into account. The important thing here is to make the right decision by evaluating the opinions of more than one expert working in the production environment with different methods, rather than a single method and a single expert’s opinion.
In this paper, a new framework is proposed to examine the performance of different methods using the same criteria weights for a suitable machine tool (CNC machine) selection problem. Weights of the criteria determined by BWM were used for weighting decision matrices for the sorting methods in this new framework. Using seven alternatives, four main, and thirteen sub-criteria in the problem, machine alternatives were evaluated with GRA, COPRAS, and MULTIMOORA methods. To create a reliable final ranking in the MULTIMOORA method, the theory of dominance was used and the final rankings were determined by averaging the different rank values. As a result, it is aimed to increase the reliability of the final solution with this new approach including BWM as the criteria weighting method. In the evaluations made for the main criteria, it has been seen that the cost of the machine tool is the most important criterion, as in the studies of similar criteria in the literature (Arslan
The proposed solution procedure is well-designed for the research problem. The CNC machine selection problem is also studied in many pieces of research. In this way, the selected seven alternatives, four main, and thirteen sub-criteria can also be accepted as the main research limitation. On the other hand, the obtained results are shown effective and robust decisions for the problem using comprehensive methods as the main advantage. In future studies, it may be considered that fuzzy logic-based methods can be used for the solution in cases where decision-makers express the importance levels of the criteria with linguistic variables. The evaluation of the expert opinions can be considered by intuitionistic approaches on MCDM methodologies.
The weight calculation of sub-criteria of dimensions (C_{1}) main criteria.
BO vectors for sub-criteria | Experts No. | Best | Table load (C_{11}) | Main travel (C_{12}) | Table size (C_{13}) | Machine weight (C_{14}) |
Experts 1 | Main travel (C_{12}) | 4 | 1 | 2 | 6 | |
Experts 2 | Main travel (C_{12}) | 3 | 1 | 4 | 7 | |
Experts 3 | Main travel (C_{12}) | 4 | 1 | 5 | 8 | |
Experts 4 | Main travel (C_{12}) | 3 | 1 | 5 | 7 | |
Experts 5 | Main travel (C_{12}) | 2 | 1 | 6 | 4 | |
Experts 6 | Main travel (C_{12}) | 3 | 1 | 5 | 8 |
OW vectors for sub-criteria | Experts No. | Worst | Table load (C_{11}) | Main travel (C_{12}) | Table size (C_{13}) | Machine weight (C_{14}) |
Experts 1 | Machine weight (C_{14}) | 2 | 6 | 3 | 1 | |
Experts 2 | Machine weight (C_{14}) | 2 | 3 | 2 | 1 | |
Experts 3 | Machine weight (C_{14}) | 2 | 8 | 2 | 1 | |
Experts 4 | Machine weight (C_{14}) | 3 | 1 | 5 | 7 | |
Experts 5 | Table size (C_{13}) | 2 | 6 | 1 | 2 | |
Experts 6 | Machine weight (C_{14}) | 3 | 8 | 2 | 1 |
The weights of sub-criteria | Sub-criteria | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6 | Mean |
Table load (C_{11}) | 0.135 | 0.231 | 0.163 | 0.208 | 0.238 | 0.206 | 0.197 | |
Main travel (C_{12}) | 0.514 | 0.496 | 0.630 | 0.589 | 0.524 | 0.598 | 0.558 | |
Table size (C_{13}) | 0.270 | 0.174 | 0.130 | 0.125 | 0.095 | 0.124 | 0.153 | |
Machine weight (C_{14}) | 0.081 | 0.099 | 0.076 | 0.079 | 0.143 | 0.072 | 0.092 | |
Ksi | 0.027 | 0.198 | 0.022 | 0.034 | 0.048 | 0.021 | 0.058 |
The weight calculation of sub-criteria of flexibility (C_{2}) main criteria.
Experts No. | Best | Spindle rate (C_{21}) | Spindle power (C_{22}) | Max. tool weight (C_{23}) | |
Experts 1 | Spindle power (C_{22}) | 2 | 1 | 4 | |
Experts 2 | Spindle power (C_{22}) | 3 | 1 | 5 | |
Experts 3 | Spindle power (C_{22}) | 3 | 1 | 2 | |
Experts 4 | Spindle power (C_{22}) | 4 | 1 | 2 | |
Experts 5 | Spindle power (C_{22}) | 3 | 1 | 2 | |
Experts 6 | Spindle power (C_{22}) | 3 | 1 | 6 |
OW vectors for sub-criteria | Experts No. | Worst | Spindle rate (C_{21}) | Spindle power (C_{22}) | Max. Tool Weight (C_{23}) |
Experts 1 | Max. tool weight (C_{23}) | 2 | 4 | 1 | |
Experts 2 | Max. tool weight (C_{23}) | 2 | 5 | 1 | |
Experts 3 | Spindle rate (C_{21}) | 1 | 3 | 2 | |
Experts 4 | Spindle rate (C_{21}) | 1 | 4 | 2 | |
Experts 5 | Spindle rate (C_{21}) | 1 | 3 | 2 | |
Experts 6 | Max. tool weight (C_{23}) | 2 | 6 | 1 |
The weights of sub-criteria | Sub-criteria | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6 | Mean |
Spindle rate (C_{21}) | 0.286 | 0.225 | 0.167 | 0.143 | 0.292 | 0.222 | 0.222 | |
Spindle power (C_{22}) | 0.571 | 0.650 | 0.542 | 0.571 | 0.542 | 0.667 | 0.591 | |
Max. tool weight (C_{23}) | 0.143 | 0.125 | 0.292 | 0.286 | 0.167 | 0.111 | 0.187 | |
Ksi | 0.000 | 0.025 | 0.042 | 0.000 | 0.042 | 0.000 | 0.0181 |
The weight calculation of sub-criteria of productivity (C
BO vectors for sub-criteria | Experts No. | Best | Feed rate (C_{31}) | Tool magazine capacity (C_{23}) | Cutting feed rate (C_{33}) |
Experts 1 | Tool magazine capacity (C_{23}) | 5 | 1 | 3 | |
Experts 2 | Tool magazine capacity (C_{23}) | 4 | 1 | 2 | |
Experts 3 | Tool magazine capacity (C_{23}) | 2 | 1 | 3 | |
Experts 4 | Tool magazine capacity (C_{23}) | 2 | 1 | 2 | |
Experts 5 | Tool magazine capacity (C_{23}) | 5 | 1 | 3 | |
Experts 6 | Tool magazine capacity (C_{23}) | 4 | 1 | 2 |
OW vectors for sub-criteria | Experts No. | Worst | Feed rate (C_{31}) | Tool magazine capacity (C_{23}) | Cutting feed rate (C_{33}) |
Experts 1 | Feed rate (C_{31}) | 1 | 5 | 2 | |
Experts 2 | Feed rate (C_{31}) | 1 | 4 | 2 | |
Experts 3 | Cutting feed rate (C_{33}) | 2 | 3 | 1 | |
Experts 4 | Feed (C_{31}) | 1 | 2 | 1 | |
Experts 5 | Feed rate (C_{31}) | 1 | 5 | 2 | |
Experts 6 | Feed rate (C_{31}) | 1 | 4 | 2 |
The weights of sub-criteria | Sub-criteria | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6 | Mean |
Feed rate (C_{31}) | 0.125 | 0.143 | 0.167 | 0.250 | 0.125 | 0.143 | 0.159 | |
Tool magazine capacity (C_{32}) | 0.650 | 0.571 | 0.542 | 0.500 | 0.650 | 0.571 | 0.581 | |
Cutting feed rate (C_{33}) | 0.225 | 0.286 | 0.292 | 0.250 | 0.225 | 0.286 | 0.261 | |
Ksi | 0.025 | 0.000 | 0.042 | 0.000 | 0.025 | 0.000 | 0.015 |
The weight calculation of sub-criteria of cost (C
BO vectors for sub-criteria | Experts No. | Best | Procurement price (C_{41}) | Operation cost (C_{42}) | Maintenance cost (C_{43}) |
Experts 1 | Procurement price (C_{41}) | 1 | 5 | 8 | |
Experts 2 | Procurement price (C_{41}) | 1 | 4 | 7 | |
Experts 3 | Procurement price (C_{41}) | 1 | 6 | 4 | |
Experts 4 | Procurement price (C_{41}) | 1 | 3 | 5 | |
Experts 5 | Procurement price (C_{41}) | 1 | 4 | 9 | |
Experts 6 | Procurement price (C_{41}) | 1 | 5 | 8 |
OW vectors for sub-criteria | Experts No. | Worst | Procurement price (C_{41}) | Operation cost (C_{42}) | Maintenance cost (C_{43}) |
Experts 1 | Maintenance cost (C_{43}) | 8 | 2 | 1 | |
Experts 2 | Maintenance cost (C_{43}) | 7 | 2 | 1 | |
Experts 3 | Operation cost (C_{42}) | 6 | 1 | 2 | |
Experts 4 | Maintenance cost (C_{43}) | 5 | 2 | 1 | |
Experts 5 | Maintenance cost (C_{43}) | 9 | 2 | 1 | |
Experts 6 | Maintenance cost (C_{43}) | 8 | 2 | 1 |
The weights of sub-criteria | Sub-criteria | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6 | Mean |
Procurement price (C_{41}) | 0.753 | 0.717 | 0.704 | 0.650 | 0.736 | 0.753 | 0.719 | |
Operation cost (C_{42}) | 0.156 | 0.183 | 0.111 | 0.225 | 0.181 | 0.156 | 0.169 | |
Maintanance cost (C_{43}) | 0.091 | 0.100 | 0.185 | 0.125 | 0.083 | 0.091 | 0.113 | |
Ksi | 0.026 | 0.017 | 0.037 | 0.025 | 0.014 | 0.026 | 0.024 |