# Comparative Analysis of the Simple WISP and Some Prominent MCDM Methods: A Python Approach

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Simple Weighted Sum Product Method

_{ij}denotes a performance or rating of alternative i concerning criterion j, m denotes number of alternatives, and n denotes number of criteria.

#### 2.2. The Cosine Similarity Measure

## 3. Comparison of the WISP Method and Some Efficient MCDM Methods

_{j}= {0.25, 0.25, 0.25, 0.25}.

_{1}were generated using Python for in-range loops, while ratings of alternatives A

_{2}to A

_{5}were generated using the numpy.random.randint (1, 6) function and numpy.random.seed (0).

_{1}–C

_{4}of the alternative A

_{1}, is shown graphically in Figure 2 and Figure 3.

_{1}–C

_{4}of the alternative A

_{1}, ${C}_{i}\in \left[1,5\right]$, which were formed using the Python for in range (1, 6) loops.

_{1}–C

_{4}, ${C}_{1,3}\in \left(1,3,5\right)$, and ${C}_{2,4}\in \left[1,5\right],$ where the ratings of criteria C

_{2}and C

_{4}were formed using the Python for in range (1, 6, 2) loops.

**The first analysis**. In the first of the five conducted analyses, the correlation between ranking orders of alternatives was obtained by using the WISP method and ranking orders of alternatives obtained by applying TOPSIS, SAW, ARAS, WASPAS, and CoCoSo methods. The correlation was examined based on four “randomly selected” initial decision-making matrices.

**The second analysis**. In the second of the five conducted analyses, the correlation of the ranks of alternative A

_{1}concerning the ranks of the same alternative obtained by applying the selected MCDM methods was examined.

_{1}was performed for five cases, with a different number of variations of the values of criteria C

_{1}–C

_{4}, where different number of variations was realized by using different combinations of range (1,6) and range (1,6,2) function in the Python for in loop. In each of the five cases, for each MCDM method used, a vector containing the rank of alternative A

_{1}was formed.

_{1}between WISP–WASPAS, WISP–SAW, and WISP–ARAS methods. There is a high correlation that also exists between WISP–TOPSIS and WISP–CoCoSo methods, but it is lower compared to the above-mentioned.

**The third analysis**. In the third analysis, the correlation between the best-ranked alternatives obtained by applying several MCDM methods was examined. As in the previous analysis, the correlation was performed for five cases with a different numbers of variations.

**The fourth analysis**. The fourth analysis was conducted to determine the similarity between the ranking orders of alternatives obtained by applying WISP and mentioned MCDM methods. As in previous cases, the analysis was performed on five cases with different numbers of variations. The obtained results of this analysis are shown in Table 12.

**The fifth analysis**. Unlike previous analyses, in this analysis, the values of alternative A

_{2}are also varied, as in the case of alternative A

_{1}. In this analysis, the similarity of the first-ranked method obtained using WISP and the above-mentioned MCDM methods was checked. The results of the calculation, i.e., the similarity of the obtained ranks calculated using the cosine similarity measure are shown in Table 13.

## 4. A Numerical Illustration

- ―
- Constructional parameters;
- ―
- Economical parameters;
- ―
- Technical parameters.

_{4}is the most acceptable in the case of applying both methods.

_{7}from 0.12 to 0.054, with a corresponding increase of weights of other criteria in order to meet the following constraint, or more precisely by applying the weighting vector w

_{j}= {0.075, 0.075, 0.075, 0.150, 0.215, 0.086, 0.054, 0.134, 0.054, 0.081}, all alternatives gave the same ranking order of alternatives, as it is shown in Table 23.

_{7}caused that all MCDM methods gave the same ranking order of alternatives, i.e., that the alternative A

_{3}is the most acceptable by applying all considered MCDM methods. In this case, criterion C

_{7}was chosen because of its significant weight. Similar analyses can be performed with increasing or decreasing weights of other criteria with higher weights, or with groups of criteria with lower weights.

_{8}from 0.125 to 0.022, i.e., by applying the weighting vector w

_{j}= {0.078, 0.078, 0.078, 0.157, 0.224, 0.089, 0.134, 0.022, 0.056, 0.084}, the alternative A

_{4}will become most appropriate, except by applying the CoCoSo method, as shown in Table 24.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Correlation of utility alternatives achieved by applying different MCDM methods formed based on 225 variations.

C_{1} | C_{2} | C_{3} | C_{4} | |
---|---|---|---|---|

A_{1} | 1 | 5 | 1 | 3 |

A_{2} | 5 | 1 | 4 | 4 |

A_{3} | 4 | 2 | 4 | 3 |

A_{4} | 5 | 1 | 1 | 5 |

A_{5} | 3 | 2 | 1 | 2 |

WISP | TOPSIS | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|

u_{i} | S_{i} | S_{i} | Q_{i} | Q_{i} | k_{i} | |

A_{1} | 0.908 | 0.650 | 0.717 | 0.744 | 0.660 | 2.470 |

A_{2} | 0.690 | 0.350 | 0.488 | 0.459 | 0.443 | 1.389 |

A_{3} | 0.735 | 0.384 | 0.529 | 0.518 | 0.505 | 1.907 |

A_{4} | 0.799 | 0.469 | 0.650 | 0.604 | 0.591 | 1.762 |

A_{5} | 1.000 | 0.555 | 0.750 | 0.722 | 0.725 | 2.823 |

WISP | TOPSIS | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|

A_{1} | 2 | 1 | 2 | 1 | 2 | 2 |

A_{2} | 5 | 5 | 5 | 5 | 5 | 5 |

A_{3} | 4 | 4 | 4 | 4 | 4 | 3 |

A_{4} | 3 | 3 | 3 | 3 | 3 | 4 |

A_{5} | 1 | 2 | 1 | 2 | 1 | 1 |

Cosine similarity | 0.982 | 1.000 | 0.982 | 1.000 | 0.982 |

C_{1} | C_{2} | C_{3} | C_{4} | |
---|---|---|---|---|

A_{1} | 3 | 3 | 3 | 1 |

A_{2} | 5 | 1 | 4 | 4 |

A_{3} | 4 | 2 | 4 | 3 |

A_{4} | 5 | 1 | 1 | 5 |

A_{5} | 3 | 2 | 1 | 2 |

C_{1} | C_{2} | C_{3} | C_{4} | |
---|---|---|---|---|

A_{1} | 5 | 1 | 3 | 5 |

A_{2} | 5 | 1 | 4 | 4 |

A_{3} | 4 | 2 | 4 | 3 |

A_{4} | 5 | 1 | 1 | 5 |

A_{5} | 3 | 2 | 1 | 2 |

C_{1} | C_{2} | C_{3} | C_{4} | |
---|---|---|---|---|

A_{1} | 5 | 5 | 5 | 3 |

A_{2} | 5 | 1 | 4 | 4 |

A_{3} | 4 | 2 | 4 | 3 |

A_{4} | 5 | 1 | 1 | 5 |

A_{5} | 3 | 2 | 1 | 2 |

WISP | TOPSIS | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|

A_{1} | 2 | 1 | 1 | 1 | 1 | 1 |

A_{2} | 5 | 5 | 5 | 5 | 5 | 5 |

A_{3} | 4 | 4 | 4 | 4 | 4 | 3 |

A_{4} | 3 | 3 | 3 | 3 | 3 | 4 |

A_{5} | 1 | 2 | 2 | 2 | 2 | 2 |

Cosine similarity | 0.982 | 0.982 | 0.982 | 0.982 | 0.964 |

WISP | TOPSIS | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|

A_{1} | 4 | 4 | 5 | 5 | 4 | 4 |

A_{2} | 5 | 5 | 4 | 4 | 5 | 4 |

A_{3} | 3 | 3 | 3 | 3 | 3 | 2 |

A_{4} | 2 | 2 | 2 | 2 | 2 | 3 |

A_{5} | 1 | 1 | 1 | 1 | 1 | 1 |

Cosine similarity | 1.000 | 0.982 | 0.982 | 1.000 | 0.974 |

WISP | TOPSIS | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|

A_{1} | 2 | 1 | 2 | 2 | 2 | 1 |

A_{2} | 5 | 5 | 5 | 5 | 5 | 4 |

A_{3} | 4 | 4 | 4 | 4 | 4 | 2 |

A_{4} | 3 | 3 | 3 | 3 | 3 | 5 |

A_{5} | 1 | 2 | 1 | 1 | 1 | 3 |

Cosine similarity | 0.982 | 1.000 | 1.000 | 1.000 | 0.873 |

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Max | Min | Mean | |
---|---|---|---|---|---|---|---|---|

WISP–TOPSIS | 0.970 | 0.968 | 0.966 | 0.967 | 0.967 | 0.970 | 0.966 | 0.968 |

WISP–SAW | 0.988 | 0.990 | 0.990 | 0.988 | 0.988 | 0.990 | 0.988 | 0.989 |

WISP–ARAS | 0.982 | 0.986 | 0.986 | 0.985 | 0.984 | 0.986 | 0.982 | 0.985 |

WISP–WASPAS | 0.990 | 0.992 | 0.992 | 0.992 | 0.992 | 0.992 | 0.990 | 0.992 |

WISP–CoCoSo | 0.957 | 0.968 | 0.970 | 0.967 | 0.969 | 0.970 | 0.957 | 0.966 |

Variations: | 81 | 135 | 225 | 375 | 625 |

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Max | Min | Mean | |
---|---|---|---|---|---|---|---|---|

WISP–TOPSIS | 0.928 | 0.925 | 0.919 | 0.925 | 0.922 | 0.928 | 0.919 | 0.924 |

WISP–SAW | 0.960 | 0.971 | 0.979 | 0.976 | 0.977 | 0.979 | 0.960 | 0.973 |

WISP–ARAS | 0.950 | 0.965 | 0.972 | 0.969 | 0.971 | 0.972 | 0.950 | 0.965 |

WISP–WASPAS | 0.965 | 0.976 | 0.985 | 0.985 | 0.987 | 0.987 | 0.965 | 0.980 |

WISP–CoCoSo | 0.939 | 0.931 | 0.940 | 0.934 | 0.931 | 0.940 | 0.931 | 0.935 |

Variations: | 81 | 135 | 225 | 375 | 625 |

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Max | Min | Mean | |
---|---|---|---|---|---|---|---|---|

WISP–TOPSIS | 0.988 | 0.988 | 0.987 | 0.988 | 0.988 | 0.988 | 0.987 | 0.988 |

WISP–SAW | 0.993 | 0.994 | 0.995 | 0.995 | 0.995 | 0.995 | 0.993 | 0.994 |

WISP–ARAS | 0.992 | 0.994 | 0.994 | 0.994 | 0.994 | 0.994 | 0.992 | 0.994 |

WISP–WASPAS | 0.994 | 0.995 | 0.996 | 0.996 | 0.996 | 0.996 | 0.994 | 0.995 |

WISP–CoCoSo | 0.944 | 0.944 | 0.953 | 0.952 | 0.952 | 0.953 | 0.944 | 0.949 |

Variations: | 81 | 135 | 225 | 375 | 625 |

Cosine Similarity Measure | |
---|---|

WISP–TOPSIS | 0.948 |

WISP–SAW | 0.985 |

WISP–ARAS | 0.987 |

WISP–WASPAS | 0.998 |

WISP–CoCoSo | 0.926 |

Variations: | 50,625 |

Criteria | Criteria Names | Category | Opt. | Criteria Weights |
---|---|---|---|---|

C_{1} | The size and the shape of the machine | Constructional parameters | max | 0.070 |

C_{2} | The volume or the capacity of the machine | min | 0.070 | |

C_{3} | The construction of agitation and aeration system | max | 0.070 | |

C_{4} | The number of the machines | max | 0.140 | |

C_{5} | Investments | Economical parameters | min | 0.200 |

C_{6} | Terms of payment and maintenance | max | 0.080 | |

C_{7} | Operating costs | min | 0.120 | |

C_{8} | Warranty period | Technical parameters | max | 0.125 |

C_{9} | Delivery time | min | 0.050 | |

C_{10} | Maintenance conditions | max | 0.075 |

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | |
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | 3 | 7 | 4 | 4 | 6 | 4 | 6 | 8 | 5 | 8 |

A_{2} | 4 | 6 | 5 | 5 | 5 | 5 | 5 | 8 | 6 | 9 |

A_{3} | 6 | 4 | 5 | 6 | 4 | 5 | 5 | 9 | 7 | 9 |

A_{4} | 5 | 6 | 6 | 5 | 3 | 6 | 4 | 7 | 8 | 9 |

A_{5} | 2 | 8 | 3 | 4 | 6 | 3 | 6 | 7 | 7 | 8 |

TOPSIS | VIKOR | |||
---|---|---|---|---|

S_{i} | Rank | Q_{i} | Rank | |

A_{1} | 0.20 | 4 | 0.88 | 4 |

A_{2} | 0.45 | 3 | 0.41 | 3 |

A_{3} | 0.72 | 2 | 0.00 | 1 |

A_{4} | 0.74 | 1 | 0.28 | 2 |

A_{5} | 0.04 | 5 | 1.00 | 5 |

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | |
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | 0.50 | 0.88 | 0.67 | 0.67 | 1.00 | 0.67 | 1.00 | 0.89 | 0.63 | 0.89 |

A_{2} | 0.67 | 0.75 | 0.83 | 0.83 | 0.83 | 0.83 | 0.83 | 0.89 | 0.75 | 1.00 |

A_{3} | 1.00 | 0.50 | 0.83 | 1.00 | 0.67 | 0.83 | 0.83 | 1.00 | 0.88 | 1.00 |

A_{4} | 0.83 | 0.75 | 1.00 | 0.83 | 0.50 | 1.00 | 0.67 | 0.78 | 1.00 | 1.00 |

A_{5} | 0.33 | 1.00 | 0.50 | 0.67 | 1.00 | 0.50 | 1.00 | 0.78 | 0.88 | 0.89 |

${\mathit{u}}_{\mathit{i}}^{\mathit{s}\mathit{d}}$ | ${\mathit{u}}_{\mathit{i}}^{\mathit{p}\mathit{d}}$ | ${\mathit{u}}_{\mathit{i}}^{\mathit{s}\mathit{r}}$ | ${\mathit{u}}_{\mathit{i}}^{\mathit{p}\mathit{r}}$ | |
---|---|---|---|---|

A_{1} | −0.01 | −0.00005 | 0.98 | 0.0013 |

A_{2} | 0.12 | −0.00003 | 1.33 | 0.0054 |

A_{3} | 0.22 | −0.00002 | 1.71 | 0.0175 |

A_{4} | 0.21 | −0.00002 | 1.76 | 0.0132 |

A_{5} | −0.08 | −0.00007 | 0.82 | 0.0003 |

max | 0.22 | −0.00002 | 1.76 | 0.0175 |

${\overline{\mathit{u}}}_{\mathit{i}}^{\mathit{s}\mathit{d}}$ | ${\overline{\mathit{u}}}_{\mathit{i}}^{\mathit{p}\mathit{d}}$ | ${\overline{\mathit{u}}}_{\mathit{i}}^{\mathit{s}\mathit{r}}$ | ${\overline{\mathit{u}}}_{\mathit{i}}^{\mathit{p}\mathit{r}}$ | |
---|---|---|---|---|

A_{1} | 0.812 | 0.99997 | 0.719 | 0.984 |

A_{2} | 0.914 | 0.99999 | 0.844 | 0.988 |

A_{3} | 1.000 | 1.00000 | 0.983 | 1.000 |

A_{4} | 0.993 | 1.00000 | 1.000 | 0.996 |

A_{5} | 0.754 | 0.99995 | 0.659 | 0.983 |

${\mathit{u}}_{\mathit{i}}$ | Rank | |
---|---|---|

A_{1} | 0.879 | 4 |

A_{2} | 0.937 | 3 |

A_{3} | 0.996 | 2 |

A_{4} | 0.997 | 1 |

A_{5} | 0.849 | 5 |

WISP | TOPSIS | VIKOR | SAW | ARAS | WASPAS | CoCoSo | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

u_{i} | Rank | S_{i} | Rank | Q_{i} | Rank | S_{i} | Rank | Q_{i} | Rank | Q_{i} | Rank | k_{i} | Rank | |

A_{1} | 0.879 | 4 | 0.20 | 4 | 0.88 | 4 | 0.68 | 4 | 0.66 | 4 | 0.67 | 4 | 7.31 | 4 |

A_{2} | 0.937 | 3 | 0.45 | 3 | 0.41 | 3 | 0.78 | 3 | 0.77 | 3 | 0.77 | 3 | 16.68 | 3 |

A_{3} | 0.996 | 2 | 0.72 | 2 | 0.00 | 1 | 0.89 | 2 | 0.88 | 2 | 0.88 | 2 | 22.23 | 1 |

A_{4} | 0.997 | 1 | 0.74 | 1 | 0.28 | 2 | 0.90 | 1 | 0.90 | 1 | 0.89 | 1 | 19.30 | 2 |

A_{5} | 0.849 | 5 | 0.04 | 5 | 1.00 | 5 | 0.61 | 5 | 0.60 | 5 | 0.60 | 5 | 0.88 | 5 |

WISP | TOPSIS | VIKOR | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|---|

Rank | Rank | Rank | Rank | Rank | Rank | Rank | |

A_{1} | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

A_{2} | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

A_{3} | 2 | 2 | 1 | 2 | 2 | 2 | 1 |

A_{4} | 1 | 1 | 2 | 1 | 1 | 1 | 2 |

A_{5} | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

Wisp | TOPSIS | VIKOR | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|---|

Rank | Rank | Rank | Rank | Rank | Rank | Rank | |

A_{1} | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

A_{2} | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

A_{3} | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

A_{4} | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

A_{5} | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

WISP | TOPSIS | VIKOR | SAW | ARAS | WASPAS | CoCoSo | |
---|---|---|---|---|---|---|---|

Rank | Rank | Rank | Rank | Rank | Rank | Rank | |

A_{1} | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

A_{2} | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

A_{3} | 2 | 2 | 2 | 2 | 2 | 2 | 1 |

A_{4} | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

A_{5} | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

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**MDPI and ACS Style**

Stanujkić, D.; Karabašević, D.; Popović, G.; Zavadskas, E.K.; Saračević, M.; Stanimirović, P.S.; Ulutaş, A.; Katsikis, V.N.; Meidute-Kavaliauskiene, I. Comparative Analysis of the Simple WISP and Some Prominent MCDM Methods: A Python Approach. *Axioms* **2021**, *10*, 347.
https://doi.org/10.3390/axioms10040347

**AMA Style**

Stanujkić D, Karabašević D, Popović G, Zavadskas EK, Saračević M, Stanimirović PS, Ulutaş A, Katsikis VN, Meidute-Kavaliauskiene I. Comparative Analysis of the Simple WISP and Some Prominent MCDM Methods: A Python Approach. *Axioms*. 2021; 10(4):347.
https://doi.org/10.3390/axioms10040347

**Chicago/Turabian Style**

Stanujkić, Dragiša, Darjan Karabašević, Gabrijela Popović, Edmundas Kazimieras Zavadskas, Muzafer Saračević, Predrag S. Stanimirović, Alptekin Ulutaş, Vasilios N. Katsikis, and Ieva Meidute-Kavaliauskiene. 2021. "Comparative Analysis of the Simple WISP and Some Prominent MCDM Methods: A Python Approach" *Axioms* 10, no. 4: 347.
https://doi.org/10.3390/axioms10040347