# A New Integrated Multi-Criteria Decision Making and Multi-Objective Programming Model for Sustainable Supplier Selection and Order Allocation

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Sustainable Supplier Selection

#### 2.2. Supplier Selection and Order Allocation

## 3. Preliminaries

**Definition 1**

**[18].**

**Definition 2**

**[18].**

**Definition 3**

**[18].**

**Definition 4**

**[18].**

- (1)
- ${h}_{{S}_{{O}_{1}}}\otimes {h}_{{S}_{{O}_{1}}}={F}^{-1}\left({\displaystyle \underset{{\eta}_{1}\in F\left({h}_{{S}_{{O}_{1}}}\right),{\eta}_{2}\in F\left({h}_{{S}_{{O}_{2}}}\right)}{\cup}\left\{{\eta}_{1}{\eta}_{2}\right\}}\right);$
- (2)
- ${\left({h}_{{S}_{O}}\right)}^{\lambda}={F}^{-1}\left({\displaystyle \underset{\eta \in F\left({h}_{{S}_{O}}\right)}{\cup}\left\{{\eta}^{\lambda}\right\}}\right).$

**Definition 5**

**[18].**

**Definition 6**

**[44].**

**Definition**

**7.**

## 4. The Developed SSS&OA Model

_{g}, where ${h}_{{S}_{{O}_{ij}}}^{g}=\left\{{s}_{{\varphi}_{ij}<{o}_{{\phi}_{ij}}>}^{g}|{s}_{{\varphi}_{ij}<{o}_{{\phi}_{ij}}>}^{g}\in {S}_{O}\right\}$ is a DHHLE denoting the evaluation for supplier ${A}_{i}$ with respect to ${C}_{j}$ based on the linguistic term sets $S=\left\{{s}_{t}|t=-\tau ,\dots ,-1,0,1,\dots ,\tau \right\}$ and $O=\left\{{o}_{k}|k=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$. The decision makers’ weights are assumed as ${\nu}_{g}\left(g=1,2,\dots ,q\right)$ with $\sum _{g=1}^{q}{\nu}_{g}}=1$ and $0<{\nu}_{g}<1$. Next, the detailed procedure of the developed SSS&OA model is described.

**Stage 1.**Determine criteria weights by the PIPRCIA method.

**Step 1.1.**Rank the selection criteria in descending order.

**Step 1.2.**Determine the comparative importance of criteria.

**Step 1.3.**Compute the relative weights of selection criteria.

_{j}(j = 1,2,…,n) is calculated through the following formula:

**Step 1.4.**Calculate the normalized weights of selection criteria.

**Stage 2.**Select qualified sustainable suppliers by the DFT.

**Step 2.1.**Aggregate the decision makers’ evaluations.

**Step 2.2.**Calculated the distance matrix $\dot{D}$ between sustainable suppliers.

**Step 2.3.**Determine the feedback matrix $\dot{F}$.

**Step 2.4.**Determine the contract matrix $\dot{C}$.

**Step 2.5.**Obtain the ranking of sustainable suppliers.

_{i}at the time t and can be calculated by

**Stage 3.**Allocate order sizes to the qualified sustainable suppliers.

**Step 3.1.**Define related indexes and parameters.

**Indexes**

- p: Index of products p, $p=1,2,\dots ,\lambda $.
- i: Index of supplier i, $i=1,2,\dots ,m$.
- r: Index of discount intervals, $r=1,2,\dots ,R$.

**Parameters:**

- ${O}_{pi}$: Ordering cost of product p offered by green supplier.
- ${P}_{pir}$: Purchase price of product p offered by green supplier i in discount interval r
- ${B}_{pir}$: Lower quantity bound of the discount interval r in product p provided by supplier i.
- ${T}_{pi}$: Unit transportation cost of product p offered by green supplier i.
- ${H}_{p}$: Unit holding cost of product p.
- ${D}_{p}$: Demand of product p.
- ${C}_{pi}$: Capacity of pth product for ith green supplier.
- ${Q}_{pi}$: Defective rate of product p offered by green supplier i.
- ${Q}_{p}$: Maximum defective rate of product p can be accepted.
- ${L}_{pi}$: Delay rate of supplier p in product i.
- ${L}_{p}$: Maximum acceptable delay rate of product p.
- ${P}_{i}^{}$: Priority value of sustainable supplier i obtained by the DFT.

**Decision variables:**

- ${X}_{pir}$: Order size of product p purchased from sustainable supplier i at discount interval r.
- ${Y}_{pir}$: Binary variable (=1) if product p is offered by sustainable supplier i at discount interval r, 0 otherwise.

**Step 3.2.**Construct an MOLP model considering multiple objectives.

**Step 3.3:**Determine optimum order quantities of the selected suppliers.

## 5. Case Study

#### 5.1. Illustration of the Proposed Model

_{1}is shown in Table 3.

**Stage 1.**Determine criteria weights by the PIPRCIA method.

**Step 1.1.**Based on their estimated importance to supplier sustainable performance, the 10 criteria are ranked in descending order as: ${C}_{1}\succ {C}_{5}\succ {C}_{3}\succ {C}_{2}\succ {C}_{4}\succ {C}_{6}\succ {C}_{8}\succ {C}_{10}\succ {C}_{9}\succ {C}_{7}$.

**Step 1.2.**Through Equation (11), the comparative importance evaluation information of criteria is aggregated and the result is given in Table 2.

**Step 1.3.**Using Equation (12), the relative weights of criteria ${w}_{j}\left(j=1,2,\dots ,10\right)$ are computed as shown in Table 4.

**Step 1.4.**According to Equation (13), the normalized weights of criteria ${\overline{w}}_{j}\left(j=1,2,\dots ,10\right)$ are calculated and displayed in Table 4.

**Stage 2.**Select qualified sustainable suppliers by the DFT.

**Step 2.1.**By Equation (14), the collective DHHL evaluation matrix $H={\left[{h}_{{S}_{{O}_{ij}}}\right]}_{5\times 10}$ is established as shown in Table 5.

**Step 2.2**. Via Equation (15), the distance matrix of the five sustainable suppliers $\dot{D}$ is computed as follows:

**Step 2.3**. Let $\phi =0.1$ and $\delta =20$, and by Equation (16), the feedback matrix $\dot{F}$ is determined as:

**Step 2.4.**Based on Equation (17), the contract matrix $\dot{C}$ is obtained as follows:

**Step 2.5.**Using Equation (19) and let t = 1000, the final preference vector of five sustainable suppliers is obtained as: $P=\left(1490,-910,-460,1770,-1880\right)$. As a result, the five sustainable suppliers are ranked as: ${A}_{4}\succ {A}_{1}\succ {A}_{3}\succ {A}_{2}\succ {A}_{5}$.

**Stage 3.**Allocate order sizes to the qualified sustainable suppliers.

**Steps 3.1–3.2.**Based on the ranking of sustainable suppliers, the suppliers A

_{4}and A

_{1}are selected to participate in the following order allocation process. The supplier data used in the MOLP model are given in Table 6. The product demand, maximum defective rate, and maximum delay rate are given as:$D=12500,Q=3.0\%,$ $L=2.8\%$, respectively. In order to maintain a friendly relationship with the suppliers, it is assumed that the two suppliers can obtain an order quantity of at least 10% of the total demand.

**Step 3.3.**According to decision makers’ opinions, the relative importance of the four objective functions are given as: ${\omega}_{1}=0.3,{\omega}_{2}=0.2,{\omega}_{3}=0.2$ and ${\omega}_{4}=0.3$. By solving the following single objective programming model, the order allocation results of the two selected sustainable suppliers are presented in Table 7.

_{1}is 5000 units with a unit price of 26.5. At the same time, A

_{4}received the order of 7500 units at a unit price of 26. For the first objective function, the minimum purchase cost is 327,500.

#### 5.2. Comparison and Discussion

_{5}). Moreover, the orders for three of the five sustainable suppliers by the hesitant fuzzy DFT (i.e., A

_{2}, A

_{3}, and A

_{5}) and the fuzzy GRA (i.e., A

_{1}, A

_{4}, and A

_{5}) are consistent with those obtained by the developed model. Therefore, the effectiveness of the proposed SSS&OA approach is validated.

_{1}has a higher priority in comparison with A

_{4}, and is the best option for the considered sustainable supplier selection problem. However, A

_{4}is assumed to be the best choice, which is more important than A

_{1}according to our proposed model. For the two sustainable suppliers, giving a higher priority to A

_{4}is also verified by the other two methods. Similarly, the result of the fuzzy GRA method suggests that A

_{4}has a higher priority compared with A

_{3}. But, A

_{3}is better than A

_{2,}as indicted by our proposed model, the hesitant fuzzy DFT and the intuitionistic fuzzy VIKOR methods. Actually, A

_{4}is the best sustainable supplier since it is given high evaluation values on C

_{3}, C

_{4}, C

_{8}, and C

_{9}while A

_{1}obtains low evaluation on these criteria. A

_{3}is better than A

_{2}because it is evaluated with higher values than A

_{2}on important criteria (e.g., C

_{1}, C

_{6}and C

_{8}).

- (1)
- By applying the DHHLTSs, the hesitant evaluation information from each decision maker can be expressed more accurately and comprehensively. Thus, the propped model can reduce information distortion and improve the accuracy of evaluation in the sustainable supplier selection process.
- (2)
- Using the DFT, the decision behaviors of decision makers can be depicted preferably. Further, the preference values of the proposed model are obtained with a dynamic process, which overcomes the disadvantages of the previous methods that just rely on the information processing at a certain time.
- (3)
- After determining the ranking of sustainable suppliers, the order sizes for the selected suppliers can be determined according to their preference values based on the constructed MOLP model considering quantity discount.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Stević, Ž.; Pamučar, D.; Puška, A.; Chatterjee, P. Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to COmpromise solution (MARCOS). Comput. Ind. Eng.
**2020**, 140, 106231. [Google Scholar] [CrossRef] - Liu, H.C.; Quan, M.Y.; Li, Z.; Wang, Z.L. A new integrated MCDM model for sustainable supplier selection under interval-valued intuitionistic uncertain linguistic environment. Inf. Sci.
**2019**, 486, 254–270. [Google Scholar] [CrossRef] - Meksavang, P.; Shi, H.; Lin, S.M.; Liu, H.C. An extended picture fuzzy VIKOR approach for sustainable supplier management and its application in the beef industry. Symmetry
**2019**, 11, 468. [Google Scholar] [CrossRef] [Green Version] - Quan, M.; Wang, Z.; Liu, H.; Shi, H. A hybrid MCDM approach for large group green supplier selection with uncertain linguistic information. IEEE Access
**2018**, 6, 50372–50383. [Google Scholar] [CrossRef] - Lo, H.W.; Liou, J.J.H.; Wang, H.S.; Tsai, Y.S. An integrated model for solving problems in green supplier selection and order allocation. J. Clean. Prod.
**2018**, 190, 339–352. [Google Scholar] [CrossRef] - Liu, Y.; Jin, L.; Zhu, F. A multi-criteria group decision making model for green supplier selection under the ordered weighted hesitant fuzzy environment. Symmetry
**2019**, 11, 17. [Google Scholar] [CrossRef] [Green Version] - Wang, J.; Gao, H.; Wei, G.; Wei, Y. Methods for multiple-attribute group decision making with q-rung interval-valued orthopair fuzzy information and their applications to the selection of green suppliers. Symmetry
**2019**, 11, 56. [Google Scholar] [CrossRef] [Green Version] - Shi, H.; Quan, M.Y.; Liu, H.C.; Duan, C.Y. A novel integrated approach for green supplier selection with interval-valued intuitionistic uncertain linguistic information: A case study in the agri-food industry. Sustainability
**2018**, 10, 733. [Google Scholar] [CrossRef] [Green Version] - Wang, K.Q.; Liu, H.C.; Liu, L.; Huang, J. Green supplier evaluation and selection using cloud model theory and the QUALIFLEX method. Sustainability
**2017**, 9, 688. [Google Scholar] [CrossRef] [Green Version] - Alegoz, M.; Yapicioglu, H. Supplier selection and order allocation decisions under quantity discount and fast service options. Sustain. Prod. Consum.
**2019**, 18, 179–189. [Google Scholar] [CrossRef] - Duan, C.Y.; Liu, H.C.; Zhang, L.J.; Shi, H. An extended alternative queuing method with linguistic Z-numbers and its application for green supplier selection and order allocation. Int. J. Fuzzy Syst.
**2019**, 21, 2510–2523. [Google Scholar] [CrossRef] - Esmaeili-Najafabadi, E.; Fallah Nezhad, M.S.; Pourmohammadi, H.; Honarvar, M.; Vahdatzad, M.A. A joint supplier selection and order allocation model with disruption risks in centralized supply chain. Comput. Ind. Eng.
**2019**, 127, 734–748. [Google Scholar] [CrossRef] - Govindan, K.; Sivakumar, R. Green supplier selection and order allocation in a low-carbon paper industry: Integrated multi-criteria heterogeneous decision-making and multi-objective linear programming approaches. Ann. Oper. Res.
**2016**, 238, 243–276. [Google Scholar] [CrossRef] - dos Santos, B.M.; Godoy, L.P.; Campos, L.M.S. Performance evaluation of green suppliers using entropy-TOPSIS-F. J. Clean. Prod.
**2019**, 207, 498–509. [Google Scholar] [CrossRef] - Memari, A.; Dargi, A.; Akbari Jokar, M.R.; Ahmad, R.; Abdul Rahim, A.R. Sustainable supplier selection: A multi-criteria intuitionistic fuzzy TOPSIS method. J. Manuf. Syst.
**2019**, 50, 9–24. [Google Scholar] [CrossRef] - Li, J.; Fang, H.; Song, W. Sustainable supplier selection based on SSCM practices: A rough cloud TOPSIS approach. J. Clean. Prod.
**2019**, 222, 606–621. [Google Scholar] [CrossRef] - Liu, P.; Gao, H.; Ma, J. Novel green supplier selection method by combining quality function deployment with partitioned Bonferroni mean operator in interval type-2 fuzzy environment. Inf. Sci.
**2019**, 490, 292–316. [Google Scholar] [CrossRef] - Gou, X.; Liao, H.; Xu, Z.; Herrera, F. Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: A case of study to evaluate the implementation status of haze controlling measures. Inf. Fusion
**2017**, 38, 22–34. [Google Scholar] [CrossRef] - Wang, X.; Gou, X.; Xu, Z. Assessment of traffic congestion with ORESTE method under double hierarchy hesitant fuzzy linguistic environment. Appl. Soft Comput.
**2020**, 86, 105864. [Google Scholar] [CrossRef] - Duan, C.Y.; Chen, X.Q.; Shi, H.; Liu, H.C. A new model for failure mode and effects analysis based on k-means clustering within hesitant linguistic environment. IEEE Trans. Eng. Manag.
**2019**. [Google Scholar] [CrossRef] - Krishankumar, R.; Subrajaa, L.S.; Ravichandran, K.S.; Kar, S.; Saeid, A.B. A framework for multi-attribute group decision-making using double hierarchy hesitant fuzzy linguistic term set. Int. J. Fuzzy Syst.
**2019**, 21, 1130–1143. [Google Scholar] [CrossRef] - Liu, N.; He, Y.; Xu, Z. Evaluate public-private-partnership’s advancement using double hierarchy hesitant fuzzy linguistic PROMETHEE with subjective and objective information from stakeholder perspective. Technol. Econ. Dev. Econ.
**2019**, 25, 386–420. [Google Scholar] [CrossRef] [Green Version] - Montserrat-Adell, J.; Xu, Z.; Gou, X.; Agell, N. Free double hierarchy hesitant fuzzy linguistic term sets: An application on ranking alternatives in GDM. Inf. Fusion
**2019**, 47, 45–59. [Google Scholar] [CrossRef] - Bai, C.; Kusi-Sarpong, S.; Badri Ahmadi, H.; Sarkis, J. Social sustainable supplier evaluation and selection: A group decision-support approach. Int. J. Prod. Res.
**2019**, 57, 7046–7067. [Google Scholar] [CrossRef] - Liang, Y.; Liu, J.; Qin, J.; Tu, Y. An improved multi-granularity interval 2-tuple TODIM approach and its application to green supplier selection. Int. J. Fuzzy Syst.
**2019**, 21, 129–144. [Google Scholar] [CrossRef] - Phochanikorn, P.; Tan, C. An integrated multi-criteria decision-making model based on prospect theory for green supplier selection under uncertain environment: A case study of the Thailand palm oil products industry. Sustainability
**2019**, 11, 1872. [Google Scholar] [CrossRef] [Green Version] - Wu, Q.; Zhou, L.; Chen, Y.; Chen, H. An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods. Inf. Sci.
**2019**, 502, 394–417. [Google Scholar] [CrossRef] - Busemeyer, J.R.; Townsend, J.T. Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment. Psychol. Rev.
**1993**, 100, 432–459. [Google Scholar] [CrossRef] - Lee, S.; Son, Y.J.; Jin, J. Decision field theory extensions for behavior modeling in dynamic environment using Bayesian belief network. Inf. Sci.
**2008**, 178, 2297–2314. [Google Scholar] [CrossRef] - Lee, S.; Son, Y.J. Extended decision field theory with social-learning for long-term decision-making processes in social networks. Inf. Sci.
**2020**, 512, 1293–1307. [Google Scholar] [CrossRef] - Song, C.; Zhang, Y.; Xu, Z.; Hao, Z.; Wang, X. Route selection of the Arctic northwest passage based on hesitant fuzzy decision field theory. IEEE Access
**2019**, 7, 19979–19989. [Google Scholar] [CrossRef] - Hao, Z.; Xu, Z.; Zhao, H.; Zhang, R. Novel intuitionistic fuzzy decision making models in the framework of decision field theory. Inf. Fusion
**2017**, 33, 57–70. [Google Scholar] [CrossRef] - Abad, A.G.; Jin, J.; Son, Y.J. Estimation of expected human attention weights based on a decision field theory model. Inf. Sci.
**2014**, 278, 520–534. [Google Scholar] [CrossRef] - Qin, H.; Guan, H.; Wu, Y.J. Analysis of park-and-ride decision behavior based on decision field theory. Transp. Res. Part F Traffic Psychol. Behav.
**2013**, 18, 199–212. [Google Scholar] [CrossRef] - Keshavarz Ghorabaee, M.; Amiri, M.; Salehi Sadaghiani, J.; Hassani Goodarzi, G. Multiple criteria group decision-making for supplier selection based on COPRAS method with interval type-2 fuzzy sets. Int. J. Adv. Manuf. Technol.
**2014**, 75, 1115–1130. [Google Scholar] [CrossRef] - You, X.Y.; You, J.X.; Liu, H.C.; Zhen, L. Group multi-criteria supplier selection using an extended VIKOR method with interval 2-tuple linguistic information. Expert Syst. Appl.
**2015**, 42, 1906–1916. [Google Scholar] [CrossRef] - Kannan, D.; Jabbour, A.B.L.d.S.; Jabbour, C.J.C. Selecting green suppliers based on GSCM practices: Using fuzzy TOPSIS applied to a Brazilian electronics company. Eur. J. Oper. Res.
**2014**, 233, 432–447. [Google Scholar] [CrossRef] - Xu, X.G.; Shi, H.; Zhang, L.J.; Liu, H.C. Green supplier evaluation and selection with an extended MABAC method under the heterogeneous information environment. Sustainability
**2019**, 11, 6616. [Google Scholar] [CrossRef] [Green Version] - Lu, Z.; Sun, X.; Wang, Y.; Xu, C. Green supplier selection in straw biomass industry based on cloud model and possibility degree. J. Clean. Prod.
**2019**, 209, 995–1005. [Google Scholar] [CrossRef] - Mohammed, A.; Setchi, R.; Filip, M.; Harris, I.; Li, X. An integrated methodology for a sustainable two-stage supplier selection and order allocation problem. J. Clean. Prod.
**2018**, 192, 99–114. [Google Scholar] [CrossRef] [Green Version] - Gören, H.G. A decision framework for sustainable supplier selection and order allocation with lost sales. J. Clean. Prod.
**2018**, 183, 1156–1169. [Google Scholar] [CrossRef] - Cheraghalipour, A.; Farsad, S. A bi-objective sustainable supplier selection and order allocation considering quantity discounts under disruption risks: A case study in plastic industry. Comput. Ind. Eng.
**2018**, 118, 237–250. [Google Scholar] [CrossRef] - Moheb-Alizadeh, H.; Handfield, R. Sustainable supplier selection and order allocation: A novel multi-objective programming model with a hybrid solution approach. Comput. Ind. Eng.
**2019**, 129, 192–209. [Google Scholar] [CrossRef] - Gou, X.; Xu, Z.; Liao, H.; Herrera, F. Multiple criteria decision making based on distance and similarity measures under double hierarchy hesitant fuzzy linguistic environment. Comput. Ind. Eng.
**2018**, 126, 516–530. [Google Scholar] [CrossRef] - Stanujkic, D.; Zavadskas, E.K.; Karabasevic, D.; Smarandache, F.; Turskis, Z. The use of the pivot pairwise relative criteria importance assessment method for determining the weights of criteria. Rom. J. Econ. Forecast.
**2017**, 20, 116–133. [Google Scholar] - Stević, Ž.; Stjepanović, Ž.; Božičković, Z.; Das, D.K.; Stanujkić, D. Assessment of conditions for implementing information technology in a warehouse system: A novel fuzzy PIPRECIA method. Symmetry
**2018**, 10, 586. [Google Scholar] [CrossRef] [Green Version] - Berkowitsch, N.A.J.; Scheibehenne, B.; Rieskamp, J. Rigorously testing multialternative decision field theory against random utility models. J. Exp. Psychol. Gen.
**2014**, 143, 1331–1348. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.; You, X.Y.; Liu, H.C.; Wu, S.M. An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection. Symmetry
**2017**, 9, 169. [Google Scholar] [CrossRef] - Banaeian, N.; Mobli, H.; Fahimnia, B.; Nielsen, I.E.; Omid, M. Green supplier selection using fuzzy group decision making methods: A case study from the agri-food industry. Comput. Oper. Res.
**2018**, 89, 337–347. [Google Scholar] [CrossRef]

Dimension | Criteria | Definition |
---|---|---|

Supplier performance | Product quality (C_{1}) | Ensure the quality of products in accordance with ISO 19000, QS9000 and other relevant requirements and specifications |

Green manufacturing (C_{2}) | Committed to the production of clean and environmentally friendly products. | |

Service flexibility (C_{3}) | Products need to meet customer requirements and can ensure on-time delivery when orders are changed | |

Environmental protection | Environmental performance (C_{4}) | The ability of environmental protection and to observe environmental supervision for products and reduce waste as much as possible. |

Innovation ability (C_{5}) | Innovative product design to ensure the product detachable, recyclable and sustainable | |

Green logistic (C_{6}) | The ability to reduce transportation cost and pollution through logistics planning | |

Supplier risk | Labor intensive (C_{7}) | The extent to which a supplier relies on labor in productive activities |

Financial stability (C_{8}) | The financial status and financial stability of supplier | |

Supplier reputation (C_{9}) | The reputation of supplier in the industry, as well as past cooperation experience | |

Information safety (C_{10}) | Suppliers’ ability to protect product information |

Criteria | Decision Makers | |||||
---|---|---|---|---|---|---|

DM_{1} | DM_{2} | DM_{3} | DM_{4} | DM_{5} | Aggregated Vector | |

(C_{1}, C_{3}) | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>},{{s}^{\prime}}_{2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{1}>},{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{2.94}>},{{s}^{\prime}}_{2<{{o}^{\prime}}_{0.35},{{o}^{\prime}}_{0.70},{{o}^{\prime}}_{1.13}>}\right\}$ |

(C_{3}, C_{5}) | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>},{{s}^{\prime}}_{2<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-2}>},{{s}^{\prime}}_{0<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{2<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{1.10},{{o}^{\prime}}_{2.40}>},{{s}^{\prime}}_{1<{{o}^{\prime}}_{0.88}>}\right\}$ |

(C_{5}, C_{2}) | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-2}>},{{s}^{\prime}}_{0<{{o}^{\prime}}_{1}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{-1.29},{{o}^{\prime}}_{0.26}>}\right\}$ |

(C_{2}, C_{4}) | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-2}>},{{s}^{\prime}}_{0<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{-2.04},{{o}^{\prime}}_{0.13}>}\right\}$ |

(C_{4}, C_{6}) | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{3}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-2}>},{{s}^{\prime}}_{0},{{s}^{\prime}}_{1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{-1.23},{{o}^{\prime}}_{-0.08},{{o}^{\prime}}_{0.05}>}\right\}$ |

(C_{6}, C_{8}) | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{0.09}>}\right\}$ |

(C_{8}, C_{10}) | $\left\{{{s}^{\prime}}_{2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>},{{s}^{\prime}}_{2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0.63},{{o}^{\prime}}_{1.02}>}\right\}$ |

(C_{10}, C_{9}) | $\left\{{{s}^{\prime}}_{-1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{-2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{-1.86}>}\right\}$ |

(C_{9}, C_{7}) | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{1}>},{{s}^{\prime}}_{1<{{o}^{\prime}}_{-2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{2<{{o}^{\prime}}_{-1}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2}>}\right\}$ | $\left\{{{s}^{\prime}}_{1<{{o}^{\prime}}_{0}>}\right\}$ | $\left\{{{s}^{\prime}}_{0<{{o}^{\prime}}_{2.57}>}\right\}$ |

Suppliers | Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

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

${A}_{1}$ | $\left\{{s}_{2<{o}_{1}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>},{s}_{2<{o}_{-1}>}\right\}$ | $\left\{{s}_{0<{o}_{2}>}\right\}$ | $\left\{{s}_{-1<{o}_{0}>}\right\}$ | $\left\{{s}_{3<{o}_{-2}>}\right\}$ | $\left\{{s}_{1<{o}_{2}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>}\right\}$ | $\left\{{s}_{-2<{o}_{1}>}\right\}$ | $\left\{{s}_{-2<{o}_{1}>},{s}_{-1<{o}_{2}>}\right\}$ | $\left\{{s}_{-1<{o}_{2}>}\right\}$ |

${A}_{2}$ | $\left\{{s}_{-1<{o}_{0}>}\right\}$ | $\left\{{s}_{0<{o}_{-1}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>},{s}_{2<{o}_{-1}>}\right\}$ | $\left\{{s}_{2<{o}_{2}>}\right\}$ | $\left\{{s}_{2<{o}_{-1}>}\right\}$ | $\left\{{s}_{-1<{o}_{2}>}\right\}$ | $\left\{{s}_{2<{o}_{1}>}\right\}$ | $\left\{{s}_{0<{o}_{-1}>}\right\}$ | $\left\{{s}_{0<{o}_{0}>},{s}_{1<{o}_{-1}>}\right\}$ | $\left\{{s}_{-2<{o}_{0}>}\right\}$ |

${A}_{3}$ | $\left\{{s}_{1<{o}_{-1}>}\right\}$ | $\left\{{s}_{2<{o}_{-1}>}\right\}$ | $\left\{{s}_{0<{o}_{1}>}\right\}$ | $\left\{{s}_{-1<{o}_{0}>}\right\}$ | $\left\{{s}_{-1<{o}_{1}>}\right\}$ | $\left\{{s}_{1<{o}_{2}>}\right\}$ | $\left\{{s}_{0<{o}_{0}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>},{s}_{2<{o}_{-2}>}\right\}$ | $\left\{{s}_{1<{o}_{2}>}\right\}$ | $\left\{{s}_{2<{o}_{-1}>}\right\}$ |

${A}_{4}$ | $\left\{{s}_{2<{o}_{0}>}\right\}$ | $\left\{{s}_{2<{o}_{0}>}\right\}$ | $\left\{{s}_{1<{o}_{1}>},{s}_{2<{o}_{0}>}\right\}$ | $\left\{{s}_{3<{o}_{-1}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>}\right\}$ | $\left\{{s}_{0<{o}_{0}>}\right\}$ | $\left\{{s}_{1<{o}_{-2}>}\right\}$ | $\left\{{s}_{2<{o}_{2}>}\right\}$ | $\left\{{s}_{2<{o}_{0}>}\right\}$ | $\left\{{s}_{2<{o}_{-1}>}\right\}$ |

${A}_{5}$ | $\left\{{s}_{0<{o}_{-1}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>}\right\}$ | $\left\{{s}_{1<{o}_{0}>}\right\}$ | $\left\{{s}_{1<{o}_{2}>}\right\}$ | $\left\{{s}_{-2<{o}_{0}>}\right\}$ | $\left\{{s}_{0<{o}_{1}>}\right\}$ | $\left\{{s}_{-1<{o}_{-1}>},{s}_{0<{o}_{0}>}\right\}$ | $\left\{{s}_{0<{o}_{1}>}\right\}$ | $\left\{{s}_{-1<{o}_{2}>}\right\}$ | $\left\{{s}_{0<{o}_{-1}>}\right\}$ |

Criteria | Relative Weights | Normalized Weights |
---|---|---|

C_{1} | 3.71 | 0.19 |

C_{5} | 2.62 | 0.13 |

C_{3} | 1.9 | 0.1 |

C_{2} | 1.74 | 0.09 |

C_{4} | 1.95 | 0.1 |

C_{6} | 2.04 | 0.1 |

C_{8} | 2.02 | 0.1 |

C_{10} | 1.16 | 0.06 |

C_{9} | 1.4 | 0.07 |

C_{7} | 1 | 0.05 |

Suppliers | Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

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

${A}_{1}$ | $\left\{{s}_{2<{o}_{0.91,}{o}_{1.20}>}\right\}$ | $\left\{{s}_{1<{o}_{1.44,}{o}_{2.07}>}\right\}$ | $\left\{{s}_{1<{o}_{1.28}>}\right\}$ | $\left\{{s}_{1<{o}_{2.99}>},{s}_{2<{o}_{0.27}>}\right\}$ | $\left\{{s}_{2<{o}_{0.86}>}\right\}$ | $\left\{{s}_{1<{o}_{0.70,}{o}_{1.17}>}\right\}$ | $\left\{{s}_{0<{o}_{0.10}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.53}>}\right\}$ | $\left\{{s}_{0<{o}_{-2.57,}{o}_{-1.08}>}\right\}$ | $\left\{{s}_{0<{o}_{1.32}>}\right\}$ |

${A}_{2}$ | $\left\{{s}_{-1<{o}_{-0.74,}{o}_{-0.18}>}\right\}$ | $\left\{{s}_{0<{o}_{-0.47}>}\right\}$ | $\left\{{s}_{0<{o}_{2.99}>},{s}_{1<{o}_{0.56}>}\right\}$ | $\left\{{s}_{2<{o}_{0.47}>}\right\}$ | $\left\{{s}_{1<{o}_{2.40}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.11}>},{s}_{0<{o}_{-2.31}>}\right\}$ | $\left\{{s}_{2<{o}_{0.74}>}\right\}$ | $\left\{{s}_{0<{o}_{0.56}>}\right\}$ | $\left\{{s}_{0<{o}_{0.70,}{o}_{-0.19}>}\right\}$ | $\left\{{s}_{0<{o}_{0.82}>}\right\}$ |

${A}_{3}$ | $\left\{{s}_{0<{o}_{-1.26}>}\right\}$ | $\left\{{s}_{1<{o}_{1.16}>}\right\}$ | $\left\{{s}_{0<{o}_{2.90}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.48}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.23}>}\right\}$ | $\left\{{s}_{1<{o}_{0.20,}{o}_{0.33}>}\right\}$ | $\left\{{s}_{0<{o}_{-2.21}>}\right\}$ | $\left\{{s}_{1<{o}_{0.74,}{o}_{1.05}>}\right\}$ | $\left\{{s}_{1<{o}_{1.80}>}\right\}$ | $\left\{{s}_{0<{o}_{2.21}>}\right\}$ |

${A}_{4}$ | $\left\{{s}_{1<{o}_{2.92}>}\right\}$ | $\left\{{s}_{1<{o}_{1.11}>}\right\}$ | $\left\{{s}_{1<{o}_{2.77}>},{s}_{2<{o}_{0.42}>}\right\}$ | $\left\{{s}_{2<{o}_{0.69}>}\right\}$ | $\left\{{s}_{1<{o}_{0.67}>}\right\}$ | $\left\{{s}_{0<{o}_{0.67}>}\right\}$ | $\left\{{s}_{1<{o}_{0.22}>}\right\}$ | $\left\{{s}_{1<{o}_{2.89}>},{s}_{2<{o}_{0.27}>}\right\}$ | $\left\{{s}_{1<{o}_{2.67}>}\right\}$ | $\left\{{s}_{0<{o}_{2.41}>}\right\}$ |

${A}_{5}$ | $\left\{{s}_{0<{o}_{-2.99}>}\right\}$ | $\left\{{s}_{0<{o}_{0.54}>}\right\}$ | $\left\{{s}_{0<{o}_{1.79}>}\right\}$ | $\left\{{s}_{1<{o}_{0.99}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.63}>}\right\}$ | $\left\{{s}_{0<{o}_{-1.50}>}\right\}$ | $\left\{{s}_{0<{o}_{-2.23,}{o}_{-0.92}>}\right\}$ | $\left\{{s}_{0<{o}_{1.93}>}\right\}$ | $\left\{{s}_{-1<{o}_{-0.47}>}\right\}$ | $\left\{{s}_{0<{o}_{0.94}>}\right\}$ |

Suppliers | Raw Material Data | ||||
---|---|---|---|---|---|

r | ${\mathit{P}}_{1\mathit{i}\mathit{r}}$ | ${\mathit{L}}_{1\mathit{i}}$ | ${\mathit{C}}_{1\mathit{i}}$ | ${\mathit{Q}}_{1\mathit{i}}\left(\mathit{\%}\right)$ | |

A_{1} | 0–5000 | 28.5 | 2.2 | 8500 | 2.3 |

$\ge 5000$ | 26.5 | ||||

A_{4} | 0–6500 | 28 | 3 | 10,000 | 1.8 |

$\ge 6500$ | 26 |

Suppliers | Order Sizes |
---|---|

A_{1} | ${X}_{111}=0;\text{}{X}_{112}=5000;$ |

A_{4} | ${X}_{141}=0;\text{}{X}_{142}=7500;$ |

Objective function result | |

Z_{1} | 327 500 |

Z_{2} | 250 |

Z_{3} | 335 |

Z_{4} | 20 725 000 |

© 2020 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**

You, S.-Y.; Zhang, L.-J.; Xu, X.-G.; Liu, H.-C.
A New Integrated Multi-Criteria Decision Making and Multi-Objective Programming Model for Sustainable Supplier Selection and Order Allocation. *Symmetry* **2020**, *12*, 302.
https://doi.org/10.3390/sym12020302

**AMA Style**

You S-Y, Zhang L-J, Xu X-G, Liu H-C.
A New Integrated Multi-Criteria Decision Making and Multi-Objective Programming Model for Sustainable Supplier Selection and Order Allocation. *Symmetry*. 2020; 12(2):302.
https://doi.org/10.3390/sym12020302

**Chicago/Turabian Style**

You, Shan-Yong, Li-Jun Zhang, Xue-Guo Xu, and Hu-Chen Liu.
2020. "A New Integrated Multi-Criteria Decision Making and Multi-Objective Programming Model for Sustainable Supplier Selection and Order Allocation" *Symmetry* 12, no. 2: 302.
https://doi.org/10.3390/sym12020302