# Evaluating Industry 4.0 Manufacturing Configurations: An Entropy-Based Grey Relational Analysis Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methodology and the Problem on Hand

_{11k}is the measure of performance criterion 1, and weightage is assigned to performance criterion 1 (expressed as W

_{1k}). The structure of the derived decision matrix for market scenario k, for all alternative manufacturing configurations (i ϵ (1 to m)), is as presented in Table 1 below.

## 3. Adopted Approach and Its Application

_{mnk}is the measure of a performance criterion n for an alternative m, for a given market scenario k (refer to Table 1). Refer to Figure 2 for the adopted approach. The steps involved in the adopted approach are presented in the following subsections.

#### 3.1. Estimate and Normalize Signal to Noise Ratios

_{ijk}for the jth performance criterion and the ith alternative for the kth market scenario is expressed as follows:

_{ij}is the measure of the performance criterion j for an alternative i, for a given market scenario k. After normalization of all measured performance criteria values for each alternative configuration for a market scenario k, the equation for this is expressed as follows:

_{ijk}) for all six market scenarios (refer to Table 2) are displayed as follows:

#### 3.2. Estimate Grey Relational Coefficients and Grey Relational Grade

_{ijk}’ are further processed to get relational coefficients (i.e., ξ

_{ijk}) for the jth performance characteristic and the ith alternative for the kth scenario, and are expressed as follows:

_{ij}is the measure of the performance criterion j for an alternative I and corresponding market scenario k, and the equation for this is expressed as follows:

_{ijk}) for all six market scenarios (refer Table 2) are displayed as follows:

_{ik}). ϒ

_{ik}is used for ith alternative corresponding to the kth scenario over an n number of performance criteria. The ϒ

_{ik}, which is estimated using ξ

_{ijk}, is a relational coefficient, and W

_{jk}is the performance criterion j with a certain weightage for a given market scenario k. Researchers [46,47] initially proposed a different criterion weighting methods and later developed these methods into a useful statistical tool for analysis. In the absence of a performance criterion’s subjective weights, Shannon’s entropy and principal component analysis [49,50] are among the approaches used for obtaining a performance criterion’s weights for a multi-criteria decision-making scenario. These two techniques were developed as effective analytical tools for the optimization of multi-criteria measures because they are based on statistical approaches, free from subjective judgment, and use original information. The various steps involved in Shannon’s entropy weighting and principal component weighting are described and compared below.

#### 3.3. Performance Criterion’s Weight Estimation

_{ijk}is used to calculate W

_{jk}. The criterion entropy weight value for a given market scenario k is represented as E

_{jk}and calculated by using Equation (5) below.

_{ik}are estimated for ith alternative for a given kth market scenario over an n number of performance criteria (refer to Section 3.4 for details).

_{mk}is formulated and presented in Equation (11) that follows. The square of principal components gives the performance criterion weight W

_{jk}for a given market scenario k.

_{ik}, are estimated for ith alternative for a given kth market scenario over an n number of performance criteria (refer to Section 3.4 below).

#### 3.4. Ranking of Alternative Configurations for Industry 4.0

_{ik}, are estimated using Equation (13). The highest ϒ

_{ik}provides the best alternative choice for the kth market scenario.

_{ijk}is performed to obtain the value for a given market scenario, and the corresponding alternative relational grade values. The calculated relational grade values and ranks are based on entropy-based criterion weights (refer to Equations (14) and (15)).

_{jk}with relational coefficients ξ

_{ijk}is performed to obtain relational grade values, and alternatives are ranked in a descending order (refer to Equations (16) and (17)).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Performance Criterion (j) → Alternative (i) ↓ | 1 | 2 | … | n |
---|---|---|---|---|

1 | X_{11k} | X_{12k} | … | X_{1nk} |

2 | X_{21k} | X_{22k} | … | X_{2nk} |

. | . | . | … | . |

m | X_{m1k} | X_{m2k} | … | X_{mnk} |

Criterion weight → | W_{1k} | W_{2k} | … | W_{nk} |

**Table 2.**The alternative manufacturing configurations (six different market scenarios) and their evaluation based on five performance criteria.

Market Scenario: k | Alternative: i | Performance Criteria: j | ||||
---|---|---|---|---|---|---|

Performance Criterion 1 | Performance Criterion 2 | Performance Criterion 3 | Performance Criterion 4 | Performance Criterion 5 | ||

Market Scenario: 1 Market Demand: High Product Variety: Low | Alternative: 1 | X_{111} = 87.72 | X_{121} = 236.25 | X_{131} = 325.38 | X_{141} = 27.11 | X_{151} = 545.04 |

Alternative: 2 | X_{211} = 84.60 | X_{221} = 416.82 | X_{231} = 211.72 | X_{241} = 103.85 | X_{251} = 374.15 | |

Alternative: 3 | X_{311} = 86.21 | X_{321} = 566.62 | X_{331} = 156.73 | X_{341} = 194.53 | X_{351} = 521.10 | |

Alternative: 4 | X_{411} = 65.01 | X_{421} = 509.64 | X_{431} = 186.90 | X_{441} = 161.59 | X_{451} = 469.28 | |

Alternative: 5 | X_{511} = 61.94 | X_{521} = 445.11 | X_{531} = 208.25 | X_{541} = 117.40 | X_{551} = 391.36 | |

Market Scenario: 2 Market Demand: Low Product Variety: Low | Alternative: 1 | X_{112} = 64.79 | X_{122} = 183.38 | X_{132}= 376.58 | X_{142} = 15.36 | X_{152} = 406.30 |

Alternative: 2 | X_{212} = 70.24 | X_{222} = 274.68 | X_{232}= 304.50 | X_{242} = 45.70 | X_{252} = 239.77 | |

Alternative: 3 | X_{312} = 77.06 | X_{322} = 360.64 | X_{332}= 254.39 | X_{342} = 78.30 | X_{352} = 315.98 | |

Alternative: 4 | X_{412} = 63.26 | X_{422} = 331.99 | X_{432}= 272.72 | X_{442} = 59.80 | X_{452} = 297.91 | |

Alternative: 5 | X_{512} = 60.32 | X_{522} = 300.54 | X_{532} = 287.77 | X_{542} = 42.39 | X_{552} = 266.17 | |

Market Scenario: 3 Market Demand: High Product Variety: Medium | Alternative: 1 | X_{113} = 92.93 | X_{123} = 692.41 | X_{133} = 157.22 | X_{143} = 301.51 | X_{153} = 1690.42 |

Alternative: 2 | X_{213} = 89.31 | X_{223} = 1132.45 | X_{233} = 116.65 | X_{243} = 711.53 | X_{253} = 1084.61 | |

Alternative: 3 | X_{313} = 90.14 | X_{323}= 1534.96 | X_{333} = 81.53 | X_{343} = 718.11 | X_{353} = 1297.11 | |

Alternative: 4 | X_{413} = 69.84 | X_{423}= 1160.99 | X_{433} = 115.04 | X_{443} = 727.81 | X_{453} = 1113.32 | |

Alternative: 5 | X_{513} = 71.04 | X_{523} = 1059.74 | X_{533} = 117.77 | X_{543} = 628.13 | X_{553} = 963.02 | |

Market Scenario: 4 Market Demand: Low Product Variety: Medium | Alternative: 1 | X_{114}= 80.21 | X_{124}= 266.73 | X_{134} = 314.28 | X_{144} = 25.08 | X_{154} = 620.56 |

Alternative: 2 | X_{214} = 81.78 | X_{224} = 464.44 | X_{234} = 215.22 | X_{244} = 136.70 | X_{254} = 432.62 | |

Alternative: 3 | X_{314} = 83.60 | X_{324} = 574.05 | X_{334} = 183.76 | X_{344} = 207.47 | X_{354} = 528.20 | |

Alternative: 4 | X_{414} = 67.45 | X_{424} = 495.62 | X_{434} = 217.26 | X_{444} = 156.69 | X_{454} = 462.08 | |

Alternative: 5 | X_{514} = 64.93 | X_{524} = 449.30 | X_{534} = 230.45 | X_{544} = 122.52 | X_{554} = 395.75 | |

Market Scenario: 5 Market Demand: Low Product Variety: High | Alternative: 1 | X_{115} = 96.78 | X_{125} = 1162.72 | X_{135} = 73.79 | X_{145} = 700.89 | X_{155} = 2871.57 |

Alternative: 2 | X_{215} = 88.44 | X_{225} = 1676.26 | X_{235} = 58.29 | X_{245} = 1208.01 | X_{255} = 1543.52 | |

Alternative: 3 | X_{315} = 89.64 | X_{325} = 1993.89 | X_{335} = 44.00 | X_{345} = 1511.55 | X_{355} = 1807.63 | |

Alternative: 4 | X_{415} = 66.93 | X_{425} = 1746.08 | X_{435} = 56.78 | X_{445} = 1266.71 | X_{455} = 1641.52 | |

Alternative: 5 | X_{515} = 67.31 | X_{525} = 1621.92 | X_{535} = 56.14 | X_{545} = 1140.46 | X_{555} = 1394.66 | |

Market Scenario: 6 Market Demand: High Product Variety: High | Alternative: 1 | X_{116} = 93.93 | X_{126} = 711.83 | X_{136} = 123.62 | X_{146} = 300.14 | X_{156} = 1728.41 |

Alternative: 2 | X_{216} = 84.71 | X_{226} = 1056.70 | X_{236} = 85.51 | X_{246} = 617.61 | X_{256} = 1008.13 | |

Alternative: 3 | X_{316} = 87.28 | X_{326} = 1287.48 | X_{336} = 64.56 | X_{346} = 824.13 | X_{356} = 1221.62 | |

Alternative: 4 | X_{416} = 65.59 | X_{426} = 1149.35 | X_{436} = 91.50 | X_{446} = 704.92 | X_{456} = 1085.18 | |

Alternative: 5 | X_{516} = 61.68 | X_{526} = 1061.38 | X_{536} = 88.10 | X_{546} = 612.57 | X_{556} = 931.75 |

Set of Weights | Rankings of Alternative Configurations for Each Scenarios | |||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Scenario:1 ^{#} | Scenario:2 | Scenario:3 | Scenario:4 | Scenario:5 | Scenario:6 | |||||||||||||||||||||||||

Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | Alternative: 1 | Alternative: 2 | Alternative: 3 | Alternative: 4 | Alternative: 5 | |

Set 1: equal weights for all performance measures | 2 | 4 | 1 | 3 | 5 | 3 | 4 | 1 | 2 | 5 | 3 | 2 | 1 | 4 | 5 | 2 | 4 | 1 | 3 | 5 | 2 | 3 | 1 | 4 | 5 | 2 | 4 | 1 | 3 | 5 |

Set 2: entropy weight | 2 | 4 | 1 | 3 | 5 | 3 | 4 | 1 | 2 | 5 | 3 | 2 | 1 | 4 | 5 | 3 | 2 | 1 | 4 | 5 | 2 | 3 | 1 | 4 | 5 | 2 | 3 | 1 | 4 | 5 |

Set 3: principal component weight | 2 | 4 | 1 | 3 | 5 | 3 | 4 | 1 | 2 | 5 | 3 | 2 | 1 | 4 | 5 | 2 | 4 | 1 | 3 | 5 | 2 | 3 | 1 | 4 | 5 | 2 | 4 | 1 | 3 | 5 |

^{#}Six different market scenarios (refer to Table 2).

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**MDPI and ACS Style**

Rehman, A.U.; AlFaify, A.Y.
Evaluating Industry 4.0 Manufacturing Configurations: An Entropy-Based Grey Relational Analysis Approach. *Processes* **2023**, *11*, 3151.
https://doi.org/10.3390/pr11113151

**AMA Style**

Rehman AU, AlFaify AY.
Evaluating Industry 4.0 Manufacturing Configurations: An Entropy-Based Grey Relational Analysis Approach. *Processes*. 2023; 11(11):3151.
https://doi.org/10.3390/pr11113151

**Chicago/Turabian Style**

Rehman, Ateekh Ur, and Abdullah Yahia AlFaify.
2023. "Evaluating Industry 4.0 Manufacturing Configurations: An Entropy-Based Grey Relational Analysis Approach" *Processes* 11, no. 11: 3151.
https://doi.org/10.3390/pr11113151