# Suitability of a Consensual Fuzzy Inference System to Evaluate Suppliers of Strategic Products

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Terminology Matters

#### 2.2. Main Methodologies and Factors Used in SSE Models

## 3. Supplier Evaluation Model for Strategic Products Purchasing

#### Knowledge about Strategic Product Bid Evaluation

## 4. Inference Methods Applied in the Evaluation Model

#### 4.1. Factor Weighting Method

#### 4.2. Fuzzy Inference System (FIS)

#### 4.2.1. Variables Domain Partitioning Using 2-Tuplas and Central Symmetry

- (i)
- Definition of an ordered set {Si} of linguistic labels of preference (see Table 5).
- (ii)
- (iii)
- Preference assessment {Si} from the experts for each proposed structure (see Table 6).
- (iv)
- In each structure, aggregate all the expert estimations (e.g., through the “Extended Arithmetic Mean (EAM)”, taking as values the order of labels in the scale {Si} [87]. Example: EAM (Struc_1) = (0 × 0 + 1 × 2 + 2 × 2)/4 = 1.5).
- (v)
- By means of a symbolic translation process based on the interval [−0.5, 0.5), development of the 2-tuples related to each structure (each 2-tuple should identify the original preference label nearest to the calculated EAM and its closeness (to left or right) See Figure 5.
- (vi)
- After identifying two-tuples of all structures, the one representing the highest preference according to their lexicographic order will be chosen as optimal (in this case, “Struc_1”).
- (vii)
- To obtain the cores of the internal fuzzy labels, an agreement should be reached on the value given by the experts, e.g., through modal value (mode). In this case, after revealing the cores of the external fuzzy labels, the following mode values were obtained: Low (L = 6), Medium (M = 7) and High (H = 8).

#### 4.2.2. Knowledge Elicitation for Rule Bases

#### 4.2.3. Performance of the FIS

#### 4.3. Case Analysis Results and Discussion

- (1)–(5): Input variable values.
- (4’): Normalized values of “Price” (4) based on the interpolation function shown in Figure 11.
- (6W)/(6F): Bid scores by WM and FIS methods.
- (7): Variation rate (%) between the two methods.
- (8W)/(8F): Final evaluation by the two proposed methods.

_{pro}= 0.3 W

_{pri}= 0.1 W

_{del}= 0.6 for bid scoring subsystem; W

_{apri}= 0.6 W

_{Bid}= 0.4 for final evaluation subsystem in new suppliers (and W

_{post}= 0.8 W

_{Bid}= 0.2 in historical ones).

- (i)
- FIS provides less linearity, penalizing suppliers (02) and (06) for their low “Delivery” rates and suppliers (07), (08) and (09) for not exceeding the lower limit value in “Eval_post”.
- (ii)
- With FIS, it is not necessary to standardize the domain range of the factor “Price”.
- (iii)
- The inserted knowledge in the bid subsystem rule-base concords with the features of strategic products given by experts (priority variable: “Delivery”, followed by “Product” and lastly “Price”). Similarly, the agreed inserted knowledge in the final subsystem permits to adequately restrict the global valuation of strategic products suppliers, preventing their acceptance when “Bid” and “Eval_Pri”/”Eval_Post” factors present deficient ratings (see suppliers (03) and (07)/(08)/(09) in Table 10 and Table 11).

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 9.**(

**a**) Bid inference maps for strategic products based on Product/Price variables. Constant value of Delivery: Low (left) and High (right); (

**b**) Bid inference maps for strategic products based on Delivery/Price variables. Constant value of Product: Low (left) and High (right).

**Table 1.**Most significant factors used in suppliers’ selection and evaluation (SSE) models. Adapted from [66].

Variable Mentioned in Literature | Frequency of Mention | % | Name Assigned in the Proposed Model |
---|---|---|---|

Quality | 68 | 16.7% | Q_System |

Delivery | 64 | 15.7% | Delivery |

Price/Cost | 63 | 15.4% | Price |

Manufacturing capability | 39 | 9.6% | Structure |

Service | 35 | 8.6% | Response |

Management | 25 | 6.1% | Structure/Product |

Technology | 25 | 6.1% | Structure/Product |

Research & Development | 24 | 5.9% | Structure/Product |

Finance | 23 | 5.6% | Economic |

Flexibility | 18 | 4.4% | Quality/Response |

Reputation | 15 | 3.7% | Structure |

Relationship | 3 | 0.7% | n.a. |

Risk | 3 | 0.7% | n.a. |

Safety & Environment | 3 | 0.7% | n.a. |

2010—Adapted from [66] | |||||

Single | papers | % | Hybrid | papers | % |

DEA | 14 | 35.0% | AHP+ others | 16 | 39.0% |

MP | 9 | 22.5% | Fuzzy+ others | 9 | 22.0% |

AHP | 7 | 17.5% | MP+ others | 8 | 19.5% |

CBR | 7 | 17.5% | DEA+ others | 5 | 12.2% |

ANP | 3 | 7.5% | SMART+ others | 3 | 7.3% |

2017—Adapted from [1] | |||||

Single | papers | % | Hybrid | papers | % |

AHP | 29 | 36.7% | AHP+ others | 56 | 42.4% |

TOPSIS | 28 | 35.4% | TOPSIS+ others | 47 | 35.6% |

VIKOR | 10 | 12.7% | ANP+ others | 21 | 15.9% |

ANP | 7 | 8.9% | VIKOR+ others | 8 | 6.1% |

DEMATEL | 5 | 6.3% |

Variable Names | Description | Data Source | Variable Score Calculus | Range and Interpretation | |
---|---|---|---|---|---|

Input Variables | [Product] | Measures the adaptability of the product to the requirements of the company: Multi-functionality, ease of storage, proper packaging, shelf life, guarantee ... | Questionnaire designed “ad hoc” | Proportion of points obtained in the survey respect to the total possible points × 10 | Range: 0–10 <6: Deficient ~8: Acceptable >9: Optimal |

[Price] | Rate of change in the price of the supplier’s bid with respect to the minimum bid (taking into account collateral aspects such as the payment period, the discount or taxes) | Provider bids. | $\frac{pric{e}_{i}-pric{e}_{min}}{pric{e}_{min}}$ | Range: 0–1 >0.50: Deficient ~0.25: Acceptable <0.05: Optimal | |

[Delivery] | Measures the adaptability to the transportation company used by the supplier and the delivery conditions, the adaptation of the batch size, frequency and delivery schedules, handling units, identification and labelling, returns at no cost ... | Questionnaire design “ad hoc” | Proportion of points obtained in the survey respect to the total possible points × 10 | Range: 0–10 <6: Deficient ~8: Acceptable >9: Optimal | |

Output Variable | [BID] | Score of the Provider’s bid. | Values awarded [Product], [Price] and [Delivery] | Weighting Method (WM) or Fuzzy Inference System (FIS) | Range: 0–10 <6: Deficient ~8: Acceptabl e >9: Optimal |

Variable Names | Description | Data Source | Variable Score Calculus | Range and Interpretation | |
---|---|---|---|---|---|

Input Variables | [Eval_Pri] or [Eval_Post] | “A Priori” or “A posteriori” score of the supplier | List of Approved Suppliers or Historical behavior reports | Previous evaluation process | Range: 0–10 <6: Not approved ~8: Acceptable >9: Optimal |

[BID] | Score of the supplier’s bid | Assigned values to [Product], [Price] and [Delivery] | (WM) or (FIS) | Range: 0–10 <6: Deficient ~8: Acceptable >9: Optimal | |

Output Variable | [SCORE_New] or [SCORE_Hist] | Score of a new supplier or Score of a previous supplier | Assigned values to [Eval_Pri] or [Eval_Post], and [BID]—from previous subsystem | (WM) or (FIS) | Range: 0–10 <6: Deficient ~8: Acceptable >9: Optimal |

Label | Concept | TrFNs | 2-Tuples | |
---|---|---|---|---|

s0 | D | Disagreement | (0.0 0.0 0.3 0.5) | (D, 0) |

s1 | P | Partial Agreement | (0.3 0.5 0.5 0.7) | (P, 0) |

s2 | T | Total Agreement | (0.5 0.7 1.0 1.0) | (T, 0) |

**Table 6.**Preference assessment from the experts in four proposed structures to define the cores of the partition.

Struc_1 | Struc_2 | Struc_3 | Struc_4 | |
---|---|---|---|---|

Exp1 | T | P | - | - |

Exp2 | T | - | - | P |

Exp3 | P | T | - | - |

Exp4 | P | - | T | - |

EAM | 1.5 | 0.75 | 0.5 | 0.25 |

2-tuples | (T, −0.5) | (P, −0.25) | (P, −0.5) | (D, 0.25) |

**Table 7.**Rating on linguistic preferences by experts in five alternative labels to define the output label of a rule.

- | VL | L | M | H | VH |
---|---|---|---|---|---|

Exp1 | - | P | T | - | - |

Exp2 | - | - | P | T | P |

Exp3 | P | T | P | - | - |

Exp4 | - | P | T | P | - |

EAM | 0.25 | 1.00 | 1.5 | 0.75 | 0.25 |

2-tuples | (D, 0.25) | (P, 0.00) | (T, −0.5) | (P, −0.25) | (D, 0.25) |

Delivery | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Insuf | Accep | Ideal | Insuf | Accep | Ideal | Insuf | Accep | Ideal | |||

Product | Insuf | VL | VL | L | L | M | M | H | H | VH | BID |

Accep | VL | VL | L | L | M | M | H | H | VH | ||

Ideal | VL | VL | L | L | M | M | H | H | VH | ||

Insuf | Accep | Ideal | |||||||||

Price | |||||||||||

VL = Very Low/L = Low/M = Medium/H = High/VH = Very High |

Eval_Pri or Eval_Post | ||||||
---|---|---|---|---|---|---|

VL | L | M | H | |||

BID | VL | L | L | L | L | |

L | L | L | M | H | Score_New | |

M | L | M | H | H | or | |

H | L | M | H | VH | Score_Hist | |

VH | L | H | VH | VH |

[1] | [2] | [3] | [4] | [4’] | [5] | [6W] | [6F] | [7] | [8W] | [8F] | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

New/Hist | Eval_Pri | Eval_Post | Product | Price | Price’ | Delivery | Bid (WM) | Bid (FIS) | Variation Rate (%) | Final Eval (WM) | Final Eval (FIS) | |||

New | Hist | New | Hist | |||||||||||

SUPP 01 | New | 7.17 | 6.0 | 0.03 | 9.4 | 8.7 | 8.0 | 6.7 | 15.83% | 7.5 | 4.3 | |||

SUPP 02 | New | 6.13 | 7.5 | 0.54 | 4.6 | 7.8 | 7.4 | 4.8 | 35.05% | 6.6 | 2.7 | |||

SUPP 03 | New | 8.30 | 3.2 | 0.12 | 8.3 | 2.8 | 3.5 | 2.8 | 19.31% | 6.4 | 2.7 | |||

SUPP 04 | New | 7.87 | 7.0 | 0.75 | 2.5 | 9.0 | 7.8 | 6.5 | 16.13% | 7.8 | 5.7 | |||

SUPP 05 | Hist | 6.66 | 9.0 | 0.04 | 9.2 | 7.4 | 8.1 | 7.7 | 4.47% | 6.9 | 6.2 | |||

SUPP 06 | Hist | 8.08 | 6.5 | 0.33 | 6.4 | 6.1 | 6.2 | 2.9 | 53.57% | 7.7 | 2.7 | |||

SUPP 07 | Hist | 5.28 | 7.5 | 0.19 | 7.6 | 9.5 | 8.7 | 8.1 | 7.00% | 6.0 | 5.0 | |||

SUPP 08 | Hist | 5.91 | 8.5 | 0.03 | 9.4 | 7.0 | 7.7 | 7.0 | 8.97% | 6.3 | 5.0 | |||

SUPP 09 | Hist | 5.51 | 7.0 | 0.75 | 2.5 | 9.0 | 7.8 | 6.5 | 16.13% | 6.0 | 5.0 | |||

SUPP 10 | Hist | 7.32 | 3.2 | 0.12 | 8.3 | 2.8 | 3.5 | 2.8 | 19.31% | 6.6 | 2.7 |

Bid Subsystem | Final Evaluation Subsystem | |||
---|---|---|---|---|

Behaviour WM | Behaviour FIS | Chosen Method | Score | |

SUPP 01 | √ | √√ | FIS | 4.3 |

SUPP 02 | X | √ | FIS | 2.,7 |

SUPP 03 | √ | √√ | FIS | 2.7 |

SUPP 04 | √ | √√ | FIS | 5.7 |

SUPP 05 | √ | √ | WM/FIS | 6.9/6.2 |

SUPP 06 | X | √ | FIS | 2.7 |

SUPP 07 | √ | √ | FIS | 5.0 |

SUPP 08 | √ | √ | FIS | 5.0 |

SUPP 09 | √ | √√ | FIS | 5.0 |

SUPP 10 | √ | √√ | FIS | 2.7 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Garcia, N.; Puente, J.; Fernandez, I.; Priore, P.
Suitability of a Consensual Fuzzy Inference System to Evaluate Suppliers of Strategic Products. *Symmetry* **2018**, *10*, 22.
https://doi.org/10.3390/sym10010022

**AMA Style**

Garcia N, Puente J, Fernandez I, Priore P.
Suitability of a Consensual Fuzzy Inference System to Evaluate Suppliers of Strategic Products. *Symmetry*. 2018; 10(1):22.
https://doi.org/10.3390/sym10010022

**Chicago/Turabian Style**

Garcia, Nazario, Javier Puente, Isabel Fernandez, and Paolo Priore.
2018. "Suitability of a Consensual Fuzzy Inference System to Evaluate Suppliers of Strategic Products" *Symmetry* 10, no. 1: 22.
https://doi.org/10.3390/sym10010022