# Environmentally Friendly Supplier Selection Using Prospect Theory

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## Abstract

**:**

## 1. Introduction

## 2. Supplier Selection Decision Model

- Cost index:
- 1.1.
- Product price (I1): the cost directly related to obtaining the corresponding supplier’s product.
- 1.2.
- Product failure ratio (I2): the product’s failure rate when environmentally friendly factors are considered. For the environmentally friendly supplier selection problem, the environmental performance index (EPI) is one method to quantify and numerically mark a supplier’s environmental performance. If a product has environmentally friendly attributes, it can be accepted as a satisfactory product; this directly connects the product failure ratio to an index related to the environment.
- 1.3.
- Asset liability ratio (I3): the proportion of a company’s assets that are being financed with debt, rather than equity. This ratio is used to determine the financial risk of a business.
- 1.4.
- Supply cycle (I4): the speed at which the supplier provides the ordered products.
- 1.5.
- Research and development (R & D) capital investment ratio (I5): the ratio of R & D investment divided by the revenues.

- Profitability index:
- 2.1.
- The Quality of the Staff (I6): staff quality indicates employee quality.
- 2.2.
- Product market share (I7): percentage of a market (defined in terms of either units or revenue) accounted for by a specific entity.
- 2.3.
- Delivery accuracy (I8): an important operational performance measure that combines non-financial elements, namely time and quality.
- 2.4.
- Return on net assets (I9): a measure of financial performance, calculated as net income divided by fixed assets and net working capital.
- 2.5.
- Order Completion Rate (I10): percent of orders shipped containing all items of a sales order. It is calculated as the number of annual sales orders delivered as a complete shipment according to the sales order (with no items on back order), divided by the total number of annual sales orders.

#### 2.1. Calculation of Gain and Loss

#### 2.2. The Calculation of the Foreground Value and the Ranking of the Schemes

- (1)
- Based on (4)–(19): Establish risk return matrix ${G}_{t}={[{G}_{ij}^{t}]}_{m\times n}$ and risk loss matrix ${L}_{t}={[{L}_{ij}^{t}]}_{m\times n}$
- (2)
- Based on (20)–(24): Establish decision matrix $V={[{V}_{ij}]}_{m\times n}$
- (3)
- Based on (25) and (26): Establish a standardized decision-making matrix
- (4)
- Based on (27): Calculate the overall outlook for each vendor ${U}_{i}$, and based on the ${U}_{i}$ value, determine the size of all suppliers using a row sort.

## 3. Example Analysis

_{2}and S

_{4}. This demonstrates the validity of the comprehensive prospect value calculation method using supplier prospect theory, prospect theory based on fuzzy research, and the impact of psychological factors on real decision making. Compared with other methods, prospect theory has clear advantages. Factors are consistent when comparing model and decision-maker conclusions with the actual situation; the model answers specific requirements related to the practical application of the decision process; the model combines theoretical guidance with good practical significance; and the calculated results are accurate. These factors contributed to the better fit of the model to the actual values. In summary, the traditional maximum utility theory cannot reflect the actual decision-making behavior of the decision makers. In the actual decision-making process, the decision makers focus on the gains and losses. The prospect theory fully takes into account this demand, and considers the attitude of the decision maker towards the gains and losses, while the computational complexity can be significantly reduced by using the programming language. These advantages make the application of prospect theory effective and easy.

## 4. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Variable | Meaning |
---|---|

$S=\{{S}_{1},{S}_{2},\mathrm{...},{S}_{i},\mathrm{...},{S}_{m}\}$ | A collection of m supplier alternatives |

$m$ | Number of suppliers |

$I=\{{I}_{1},{I}_{2},\mathrm{...},{I}_{i},\mathrm{...},{I}_{n}\}$ | A set of n indicators |

$N=\{1,2,\mathrm{...},n\}$ | Subscripted collections of index |

${N}_{1},{N}_{2}\mathrm{and}{N}_{1}\cup {N}_{2}=N,{N}_{1}\cap {N}_{2}=\varnothing $ | Subscripted collections of profit index and cost index |

$w=({w}_{1},{w}_{2},\mathrm{...},{w}_{n}),{w}_{j}\ge 0\mathrm{and}{\displaystyle \sum _{j=1}^{n}{w}_{j}=1}$ | Weight vector of index |

Y | The index of natural state |

$A=\{{A}_{1},{A}_{2},\mathrm{...},{A}_{t},\mathrm{...},{A}_{y}\}$ | Natural state set |

$Y=\left\{1,2,\cdots ,y\right\}$ | A collection of natural states of y |

${p}_{t},{p}_{t}\ge 0and{\displaystyle \sum _{t=1}^{y}{p}_{t}=1}$ | Probability of occurrence of state ${A}_{t}$ |

$E=({e}_{1},{e}_{2},\mathrm{...},{e}_{n})$ | The desired vector of the index |

${e}_{j}=({e}_{j}^{1},{e}_{j}^{2},\mathrm{...},{e}_{j}^{y})$ | The expectation of decision makers for attribute ${I}_{j}$ |

${e}_{j}^{t}$ | The status of ${A}_{t}$ decision makers to target the expectations of ${I}_{j}$ |

$X={\left[{x}_{ij}^{t}\right]}_{m\times n}$ | Risk decision matrix |

${x}_{ij}^{t}$ | The status of ${A}_{t}$, supplier ${S}_{i}$ aims at decision results of ${I}_{j}$ |

${x}_{ij}^{t1}$,${x}_{ij}^{tz}$ | Lower and upper bounds of interval numbers |

${a}_{ij}^{t},{b}_{ij}^{t},{c}_{ij}^{t},{d}_{ij}^{t},{a}_{ij1}^{t},{b}_{ij}^{t},{c}_{ij}^{t},{d}_{ij1}^{t}$ | Value of fuzzy number |

${I}^{K}=\{{I}_{1},{I}_{2},\mathrm{...},{I}_{{J}_{1}}\}$ | Index value is the index set of clear number |

${I}^{L}=\{{I}_{{J}_{1}+1},{I}_{{J}_{1}+2},\mathrm{...},{I}_{{J}_{2}}\}$ | Index value is the index set of interval number |

${I}^{F}=\{{I}_{{J}_{2}+1},{I}_{{J}_{2}+2},\mathrm{...},{I}_{n}\}$ | Index value is the index set of intuitionistic trapezoidal fuzzy number |

${N}^{K}=\{1,2,\mathrm{...},{J}_{1}\}$ | Subscripted collections of index subset ${I}^{K}$ |

${N}^{L}=\{{J}_{1}+1,{J}_{1}+2,\mathrm{...},{J}_{2}\}$ | Subscripted collections of index subset ${I}^{L}$ |

${N}^{F}=\{{J}_{2}+1,{J}_{2}+2,\mathrm{...},n\}$ | Subscripted collections of index subset ${I}^{F}$ |

Index | State | State | Supplier | Mean Vector | ||||
---|---|---|---|---|---|---|---|---|

Probability | S_{1} | S_{2} | S_{3} | S_{4} | S_{5} | |||

I_{1} | A_{1} | 0.2 | 15 | 16 | 16 | 17 | 14 | 15 |

A_{2} | 0.2 | 13 | 15 | 14 | 16 | 13 | ||

A_{3} | 0.2 | 12 | 14 | 11 | 15 | 12 | ||

A_{4} | 0.2 | 11 | 13 | 10 | 13 | 11 | ||

A_{5} | 0.2 | 10 | 12 | 9 | 9 | 10 | ||

I_{2} | A1 | 0.2 | 0.95 | 0.93 | 0.95 | 0.93 | 0.90 | 0.90 |

A2 | 0.2 | 0.93 | 0.90 | 0.80 | 0.91 | 0.88 | ||

A_{3} | 0.2 | 0.90 | 0.85 | 0.75 | 0.90 | 0.85 | ||

A_{4} | 0.2 | 0.88 | 0.82 | 0.72 | 0.85 | 0.83 | ||

A_{5} | 0.2 | 0.85 | 0.80 | 0.70 | 0.80 | 0.80 | ||

I_{3} | A_{1} | 0.2 | 0.55 | 0.60 | 0.50 | 0.55 | 0.50 | 0.60 |

A_{2} | 0.2 | 0.58 | 0.62 | 0.60 | 0.58 | 0.57 | ||

A_{3} | 0.2 | 0.60 | 0.65 | 0.65 | 0.60 | 0.60 | ||

A_{4} | 0.2 | 0.62 | 0.67 | 0.66 | 0.65 | 0.62 | ||

A_{5} | 0.2 | 0.65 | 0.70 | 0.68 | 0.70 | 0.65 | ||

I_{4} | A_{1} | 0.2 | 15 | 16 | 14 | 15 | 16 | 15 |

A_{2} | 0.2 | 18 | 18 | 20 | 16 | 18 | ||

A_{3} | 0.2 | 20 | 22 | 21 | 18 | 20 | ||

A_{4} | 0.2 | 21 | 24 | 22 | 19 | 21 | ||

A_{5} | 0.2 | 22 | 25 | 23 | 20 | 22 | ||

I_{5} | A_{1} | 0.2 | 0.25 | 0.25 | 0.20 | 0.30 | 0.25 | 0.20 |

A_{2} | 0.2 | 0.23 | 0.22 | 0.18 | 0.22 | 0.22 | ||

A_{3} | 0.2 | 0.20 | 0.15 | 0.15 | 0.20 | 0.20 | ||

A_{4} | 0.2 | 0.15 | 0.12 | 0.14 | 0.15 | 0.18 | ||

A_{5} | 0.2 | 0.10 | 0.10 | 0.12 | 0.10 | 0.15 | ||

I_{6} | A_{1} | 0.2 | ([5,6,7,8]; 0.7,0.3) | ([6,7,8,9]; 0.8,0.1) | ([4,6,7,8]; 0.6,0.3) | ([6,7,8,9]; 0.8,0.2) | ([6,7,8,9]; 0.8,0.1) | 6 |

A_{2} | 0.2 | ([4,5,6,7]; 0.7,0.2) | ([5,7,8,9]; 0.8,0.2) | ([3,4,7,8]; 0.6,0.4) | ([5,6,8,9]; 0.8,0.2) | ([5,6,7,8]; 0.8,0.2) | ||

A_{3} | 0.2 | ([3,4,5,6]; 0.6,0.2) | ([4,5,6,7]; 0.8,0.2) | ([3,4,5,6]; 0.6, 0.4) | ([5,6,7,8]; 0.8,0.2) | ([4,5,7,8]; 0.8,0.2) | ||

A_{4} | 0.2 | ([2,4,5,6]; 0.6,0.3) | ([3,4,5,6]; 0.8,0.2) | ([2,3,4,5]; 0.6,0.3) | ([4,5,6,7]; 0.8,0.2) | ([4,5,6,7]; 0.8,0.2) | ||

A_{5} | 0.2 | ([2,3,5,6]; 0.6,0.3) | ([2,4,6,7]; 0.8,0.2) | ([1,3,4,5]; 0.6,0.3) | ([3,4,6,7]; 0.8,0.2) | ([3,5,6,7]; 0.8,0.2) | ||

I_{7} | A_{1} | 0.2 | 0.25 | 0.25 | 0.26 | 0.26 | 0.25 | 0.22 |

A_{2} | 0.2 | 0.22 | 0.23 | 0.25 | 0.25 | 0.23 | ||

A_{3} | 0.2 | 0.20 | 0.22 | 0.20 | 0.23 | 0.22 | ||

A_{4} | 0.2 | 0.19 | 0.21 | 0.19 | 0.20 | 0.19 | ||

A_{5} | 0.2 | 0.18 | 0.20 | 0.18 | 0.18 | 0.18 | ||

I_{8} | A_{1} | 0.2 | 0.90 | 0.95 | 0.95 | 0.95 | 0.90 | 0.90 |

A_{2} | 0.2 | 0.88 | 0.90 | 0.92 | 0.92 | 0.88 | ||

A_{3} | 0.2 | 0.85 | 0.85 | 0.90 | 0.90 | 0.85 | ||

A_{4} | 0.2 | 0.82 | 0.83 | 0.85 | 0.88 | 0.83 | ||

A_{5} | 0.2 | 0.80 | 0.80 | 0.80 | 0.85 | 0.80 | ||

I_{9} | A_{1} | 0.2 | 0.15 | 0.16 | 0.15 | 0.14 | 0.15 | 0.13 |

A_{2} | 0.2 | 0.13 | 0.15 | 0.14 | 0.13 | 0.14 | ||

A_{3} | 0.2 | 0.12 | 0.14 | 0.13 | 0.12 | 0.12 | ||

A_{4} | 0.2 | 0.11 | 0.13 | 0.12 | 0.11 | 0.11 | ||

A_{5} | 0.2 | 0.10 | 0.12 | 0.11 | 0.10 | 0.10 | ||

I_{10} | A_{1} | 0.2 | 0.95 | 0.96 | 0.95 | 0.96 | 0.95 | 0.93 |

A_{2} | 0.2 | 0.93 | 0.95 | 0.90 | 0.95 | 0.92 | ||

A_{3} | 0.2 | 0.90 | 0.90 | 0.88 | 0.90 | 0.90 | ||

A_{4} | 0.2 | 0.88 | 0.85 | 0.86 | 0.89 | 0.89 | ||

A_{5} | 0.2 | 0.85 | 0.80 | 0.85 | 0.88 | 0.88 |

Index | State | Supplier | ||||
---|---|---|---|---|---|---|

S_{1} | S_{2} | S_{3} | S_{4} | S_{5} | ||

I_{1} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 |

A_{2} | 2.00 | 0.00 | 1.00 | 0.00 | 2.00 | |

A_{3} | 3.00 | 1.00 | 4.00 | 0.00 | 3.00 | |

A_{4} | 4.00 | 2.00 | 5.00 | 2.00 | 4.00 | |

A_{5} | 5.00 | 3.00 | 6.00 | 6.00 | 5.00 | |

I_{2} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.10 | 0.00 | 0.02 | |

A_{3} | 0.00 | 0.05 | 0.15 | 0.00 | 0.05 | |

A_{4} | 0.02 | 0.08 | 0.18 | 0.05 | 0.07 | |

A_{5} | 0.05 | 0.10 | 0.20 | 0.10 | 0.10 | |

I_{3} | A_{1} | 0.05 | 0.00 | 0.10 | 0.05 | 0.10 |

A_{2} | 0.02 | 0.00 | 0.00 | 0.02 | 0.03 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{4} | A_{1} | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{5} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.02 | 0.00 | 0.00 | |

A_{3} | 0.00 | 0.05 | 0.05 | 0.00 | 0.00 | |

A_{4} | 0.05 | 0.08 | 0.06 | 0.05 | 0.02 | |

A_{5} | 0.10 | 0.10 | 0.08 | 0.10 | 0.05 | |

I_{6} | A_{1} | 4.00 | 12.83 | 4.00 | 12.83 | 12.83 |

A_{2} | 0.17 | 12.83 | 4.00 | 15.83 | 4.00 | |

A_{3} | 0.00 | 0.17 | 0.00 | 4.00 | 4.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.17 | 0.17 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.17 | 0.17 | |

I_{7} | A_{1} | 0.03 | 0.03 | 0.04 | 0.04 | 0.03 |

A_{2} | 0.00 | 0.01 | 0.03 | 0.03 | 0.01 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{8} | A_{1} | 0.00 | 0.05 | 0.05 | 0.05 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.02 | 0.02 | 0.00 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{9} | A_{1} | 0.02 | 0.03 | 0.02 | 0.01 | 0.02 |

A_{2} | 0.00 | 0.02 | 0.01 | 0.00 | 0.01 | |

A_{3} | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{10} | A_{1} | 0.02 | 0.03 | 0.02 | 0.03 | 0.02 |

A_{2} | 0.00 | 0.02 | 0.00 | 0.02 | 0.00 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Index | State | Supplier | ||||
---|---|---|---|---|---|---|

S_{1} | S_{2} | S_{3} | S_{4} | S_{5} | ||

I_{1} | A_{1} | 0.00 | −1.00 | −1.00 | −2.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.00 | −1.00 | 0.00 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{2} | A_{1} | −0.05 | −0.03 | −0.05 | −0.03 | 0.00 |

A_{2} | −0.03 | 0.00 | 0.00 | −0.01 | 0.00 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{3} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | −0.02 | 0.00 | 0.00 | 0.00 | |

A_{3} | 0.00 | −0.05 | −0.05 | 0.00 | 0.00 | |

A_{4} | −0.02 | −0.07 | −0.06 | −0.05 | −0.02 | |

A_{5} | −0.05 | −0.10 | −0.08 | −0.10 | −0.05 | |

I_{4} | A_{1} | 0.00 | −1.00 | 0.00 | 0.00 | −1.00 |

A_{2} | −3.00 | −3.00 | −5.00 | −1.00 | −3.00 | |

A_{3} | −5.00 | −7.00 | −6.00 | −3.00 | −5.00 | |

A_{4} | −6.00 | −9.00 | −7.00 | −4.00 | −6.00 | |

A_{5} | −7.00 | −10.00 | −8.00 | −5.00 | −7.00 | |

I_{5} | A_{1} | −0.05 | −0.05 | 0.00 | −0.10 | −0.05 |

A_{2} | −0.03 | −0.02 | 0.00 | −0.02 | −0.02 | |

A_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

I_{6} | A_{1} | −0.17 | 0.00 | −1.33 | 0.00 | 0.00 |

A_{2} | −3.33 | −0.17 | −10.50 | −0.17 | −0.17 | |

A_{3} | −8.17 | −3.33 | −8.17 | −0.17 | −3.33 | |

A_{4} | −24.33 | −8.17 | −11.17 | −3.33 | −3.33 | |

A_{5} | −34.67 | −24.33 | −16.83 | −10.50 | −6.00 | |

I_{7} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{3} | −0.02 | 0.00 | −0.02 | 0.00 | 0.00 | |

A_{4} | −0.03 | −0.01 | −0.03 | −0.02 | −0.03 | |

A_{5} | −0.04 | −0.02 | −0.04 | −0.04 | −0.04 | |

I_{8} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | −0.02 | 0.00 | 0.00 | 0.00 | −0.02 | |

A_{3} | −0.05 | −0.05 | 0.00 | 0.00 | −0.05 | |

A_{4} | −0.08 | −0.07 | −0.05 | −0.02 | −0.07 | |

A_{5} | −0.10 | −0.10 | −0.10 | −0.05 | −0.10 | |

I_{9} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

A_{3} | −0.01 | 0.00 | 0.00 | −0.01 | −0.01 | |

A_{4} | −0.02 | 0.00 | −0.01 | −0.02 | −0.02 | |

A_{5} | −0.03 | −0.01 | −0.02 | −0.03 | −0.03 | |

I_{10} | A_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

A_{2} | 0.00 | 0.00 | −0.03 | 0.00 | −0.01 | |

A_{3} | −0.03 | −0.03 | −0.05 | −0.03 | −0.03 | |

A_{4} | −0.05 | −0.08 | −0.07 | −0.04 | −0.04 | |

A_{5} | −0.08 | −0.13 | −0.08 | −0.05 | −0.05 |

Method | Result |
---|---|

This paper’s method | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

Yao [38] | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

Krohling [39] | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

Mahdavi [40] | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{2}\succ {S}_{4}$ |

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## Share and Cite

**MDPI and ACS Style**

Song, W.; Chen, Z.; Wang, X.; Wang, Q.; Shi, C.; Zhao, W.
Environmentally Friendly Supplier Selection Using Prospect Theory. *Sustainability* **2017**, *9*, 377.
https://doi.org/10.3390/su9030377

**AMA Style**

Song W, Chen Z, Wang X, Wang Q, Shi C, Zhao W.
Environmentally Friendly Supplier Selection Using Prospect Theory. *Sustainability*. 2017; 9(3):377.
https://doi.org/10.3390/su9030377

**Chicago/Turabian Style**

Song, Wei, Zhiya Chen, Xuping Wang, Qian Wang, Chenghua Shi, and Wei Zhao.
2017. "Environmentally Friendly Supplier Selection Using Prospect Theory" *Sustainability* 9, no. 3: 377.
https://doi.org/10.3390/su9030377