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A Fuzzy Multi-objective Supplier Selection Model in Green Supply Chain
Network: Case in Palm Oil Industry
Parapat Gultom1, Rizky Habibi2, Esther Sorta Mauli Nababan3, Ismail Husein4
1
Department of Mathematics, Universitas Sumatera Utara, Medan, Indonesia, parapat@usu.ac.id
2
Institut Akademi Informatika dan Komputer Medicom, Medan. Indonesia,
pakhabibi@gmail.com
3
Department of Mathematics, Universitas Sumatera Utara, Medan, Indonesia, esther@usu.ac.id
4
Department of Mathematics, Universitas Islam Negeri Sumatera Utara, Medan, Indonesia,
husein_ismail@uinsu.ac.id
Abstract
Supplier selection plays a crucial role in purchasing management, as it has a significant impact
on supply chain performance. Building strong and strategic relationships with suppliers can
enhance overall business performance. To ensure the best supplier is chosen, businesses need to
employ various selection criteria. Selecting the right supplier not only reduces purchasing costs
but also improves the quality of the final product and enhances the company's competitiveness,
leading to increased customer satisfaction. The problem of supplier selection is complex, and
recent works in this field have emphasized the importance of using a highly demanded approach.
This paper introduces a novel multi-objective model that considers demand allocation,
greenhouse gas emissions, and the quality and service level of suppliers in a fuzzy environment.
Few studies have explored models that incorporate all these four objective functions, making this
research unique. The proposed model is transformed into a single objective form using the
Zimmermann fuzzy approach based on the proposed fuzzy model. Numerical experiments are
conducted to validate the effectiveness of the proposed model.
Keywords: Supplier selection, Green supply chain, Fuzzy multi-objective, Zimmermann fuzzy
approach.
1 Introduction
Selecting the right supplier has a significant impact on reducing operational costs and improving
product quality, while making the wrong choice can lead to financial and operational issues [1]. The
importance of choosing the right provider has been emphasized in production network frameworks
and extensively discussed in the literature, driven by experts and scholars in recent years. Supplier
selection offers various advantages in terms of reducing costs in purchasing raw materials and
minimizing lead times for regulated products. It also contributes to enhancing product quality and
increasing competitiveness for companies [2]. However, supplier selection is not solely the
responsibility of the purchasing division; it is a complex multi-objective optimization problem with
conflicting objectives and limited constraints [3].

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The selection of suppliers falls under the realm of Multiple Criteria Decision Making (MCDM) and is
influenced by factors such as the environment, final products, and economic conditions of the
producer [4]. Dickson identified 23 criteria that purchasing managers consider when making supplier
choices, revealing the existence of multiple objective functions beyond cost minimization, such as
maximizing quality and minimizing delivery time. These complexities transform supplier selection
into a challenging multi-objective optimization problem [6]. Simultaneously considering different
objective functions in supplier selection becomes difficult due to conflicts between goals [7].
Achieving a tradeoff between the main criteria of supplier selection becomes the primary objective.
MCDM techniques provide effective tools for evaluating options and satisfactorily weighting criteria
based on purchaser conditions [8]. However, unpredictable events and vague criteria phrases like
"very top quality" or "too low price" introduce further limitations and make it challenging to handle
ambiguities using deterministic models. To address these difficulties, uncertainty tools such as fuzzy
or stochastic programming approaches are necessary to design supplier selection networks that align
better with real-world applications [4]. The fuzzy sets theory has been employed by various authors as
a well-known tool for handling uncertainty in this domain [4-9].
This study makes significant contributions to the field of supplier selection, including factors such as
product purchase price, shipping costs, storage, and interest rates in the final product cost.
Additionally, it formulates the problem using fuzzy theory to enhance practicality. The study also aims
to maximize supplier service levels and considers the environmental impact of greenhouse gas
emissions from CPO mills. These considerations expand the dimensions of the supplier selection
problem, leading to more accurate selections of final suppliers. The main limitations of the proposed
model encompass acceptable quality levels, minimum order allocation, storage, supplier capacity, and
trust. Based on the existing literature, no similar study integrates all these aspects simultaneously.
The proposed study introduces a new multi-objective supplier selection problem that incorporates
various parameters such as quality, vehicle capacity, product price, order amount, supplier capacity,
and environmental impact when purchasing CPO from suppliers. Fuzzy theory is used to estimate
demand, and the dependent variables in the proposed methodology include the intended suppliers and
purchase quantities. Taking all these factors into account, a new multi-objective optimization model is
proposed. In summary, the main contributions of this paper can be summarized as follows: proposing
a new multi-objective model for supplier selection considering greenhouse gas emissions in a fuzzy
environment, applying the Zimmermann fuzzy methodology to handle natural uncertainties, and
providing an overview of the research structure, including a review of related works, the development
of a modeling approach, application of the proposed fuzzy approach with numerical experiments, and
concluding remarks with future directions in Section 5.
2 Literature review
The supplier selection literature extensively employs various mathematical programming approaches
and computational techniques, including heuristics and metaheuristics, to tackle the complexities of
supplier selection models [10, 11]. The process of selecting suppliers becomes intricate due to the
consideration of multiple criteria [1]. Three crucial factors for proper supplier selection are standard
quality, timely delivery, and performance history [5]. The primary evaluation criteria involve the total

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cost and quality of the final product [12]. The current trend in supplier selection involves adopting a
multi-objective optimization model using uncertainty tools and incorporating additional factors such
as discount rates, order quantities, acceptable trust, refunds, possible replacements, installment
purchases, and quality controls [13].
Given the conflicting goals among criteria in supplier selection, the application of multi-objective
programming techniques proves valuable [1]. Aissaoui et al. [14] review the inclusion of an Internet-
based environment and the consideration of the entire procurement process as key criteria for selecting
final suppliers, aiming to identify the best supplier group and allocate orders accordingly. Chai et al.
[15] explore decision-making techniques from three perspectives: Multiple Criteria Decision Making
(MCDM), mathematical programming approaches, and artificial intelligence techniques, including
popular heuristics and metaheuristics.
Lee et al. [16] propose a mixed integer programming model to address the allocation of orders across
multiple suppliers over multiple periods and products, while considering additional price discounts.
Their main objective is to minimize the total cost, including ordering, holding, purchasing, and
shipping costs, without allowing inventory shortages. Arikan [17] employs a fuzzy linear multi-
objective programming model to select suppliers with multiple sources, aiming to minimize total
costs, maximize service quality, and maximize on-time delivery. Ghadimi and Heavey [18] evaluate
sustainability criteria for supplier selection in the medical device industry, introducing an Efficient
Fuzzy Inference System (EFIS) to calculate registered data based on sub-criteria. Li et al. [19] propose
a fuzzy inhomogeneous multi-attribute decision-making approach to solve the outsourcing supplier
selection problem, aiming to optimize the total network cost. Azadnia et al. [20] introduce a
coordinated method that combines rule-based weighted fuzzy and fuzzy analytic hierarchy process
within a multi-objective programming approach to model supplier selection and continuous order
allocation. Wan et al. [21] propose an intuitionistic fuzzy linear programming approach to optimize a
two-stage logistic network. Torabi et al. [22] develop a bi-objective scenario-based mixed-integer
programming model that considers supplier reserves. Mazdeh et al. [23] consider lot size in the
supplier selection framework and propose a single-solution heuristic to solve the model for large-scale
problems.
Nourmohamadi Shalke et al. [24] present a sustainable supplier selection strategy that incorporates
quantity discounts for the first time. Cheraghalipour and Farsad [25] develop a bi-objective supplier
selection and order allocation model considering quantity discounts. The primary objective is
minimizing total cost, while the secondary objective is reducing environmental emissions. Based on
our literature review, no existing research simultaneously addresses the four objective functions of
minimizing costs, minimizing environmental impact, maximizing product quality influenced by the
quality of raw materials from suppliers, and maximizing the level of trust in suppliers, as reviewed in
this study. Cost minimization is achieved by considering the purchase price, shipping costs with
different vehicle types and capacities, and disassembly costs, all of which impact the final product
cost.

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3 Research methodology
In this research, a fuzzy multi-objective mathematical model is created to address the challenge of
selecting CPO mill suppliers within the GSCN. The model incorporates four objective functions:
minimizing economic costs, minimizing greenhouse gas emissions, maximizing supplier quality, and
maximizing the level of service provided. It is important to note that the model operates within a fuzzy
environment, taking into account uncertainties and ambiguities.
The study outlines the key aspects of the problem and provides a detailed description of the problem
formulation and the mathematical model employed. Supplier selection is a complex task, and finding a
single supplier that fulfills all criteria optimally is often difficult. Different suppliers may excel in
different areas such as cost, environmental impact, product quality, and service level. The model takes
into consideration the interplay between these criteria and the capabilities of the suppliers, enabling a
comprehensive evaluation.
To showcase the effectiveness of the proposed model, numerical examples are presented. These
examples illustrate how the model can be applied in real-world scenarios and demonstrate its ability to
make informed supplier selection decisions. By considering multiple objectives and operating in a
fuzzy environment, the model provides a valuable tool for organizations within the GSCN to optimize
their supplier selection process. It enables them to strike a balance between economic costs,
environmental sustainability, supplier quality, and service level.
3.1 Problem description and mathematical model
Supplier selection involves considering a wide range of criteria, and this poses a significant challenge.
It is important to acknowledge that it is unlikely for a single supplier to fulfill all selection criteria
perfectly. In most cases, while one supplier may excel in one criterion, another supplier may excel in a
different criterion. For instance, an organization focused on producing high-quality products may
prioritize product quality over price. Additionally, such organizations may value prompt response
times and flexibility in production, indicating the importance of a supplier's service level.
To effectively analyze suppliers, there needs to be an interactive assessment that takes into account the
selection criteria, environmental impact, and the suppliers' capabilities in a comprehensive manner. It
is essential to understand that no supplier will meet all criteria perfectly, but by considering the
interactions between various factors, a well-rounded evaluation can be conducted. By assessing the
needs and benefits associated with different selection criteria and acknowledging the trade-offs
between them, organizations can make informed decisions when selecting suppliers.
3.1.1 Assumptions
This research introduces a novel approach through a multi-objective mathematical model that
considers shipping and warehousing costs, interest rates, supplier selection, and order allocation in the
supply chain simultaneously. The model is formulated using a fuzzy multi-objective programming
approach. The following assumptions and model characteristics are proposed based on this new
development:
• The model focuses on a single type of product sourced from multiple suppliers.

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• Various types of vehicles with different capacities and costs are taken into account.
• The model incorporates both variable and fixed shipping costs.
• Triangular fuzzy numbers are utilized within the model to handle uncertainty.
• The model assumes no shortages or delays from any supplier are allowed.
• Similar to previous studies, each supplier is assumed to use only one type of transportation.
3.1.2 Sets
𝑖 : Suppliers;
𝑗 : Types of vehicles.
3.1.3 Parameters
𝑎𝑖 : Purchase cost per metric ton (MT) of CPO from Supplier i;
𝑏𝑖𝑗 : Fixed shipping costs from Supplier i in one delivery using vehicle j;
𝑐 : Cost of delay in supplying per MT of CPO;
𝑑 : Disassembly cost per MT of CPO;
𝐷 : The amount of demand CPO;
𝐸𝑖 : Greenhouse gas emissions per 1 metric ton CPO purchasing from Supplier i;
𝑓 : Fees charged per 1 metric ton by the system in the case of purchasing from green
suppliers;
𝑔𝑖 : Percentage of total volume of CPO purchased from Supplier i that are supplied
with delays;
𝑂𝑖 : Greenhouse gas emissions in Supplier i;
𝑄𝑖 : Percentage of quality level of Supplier i;
𝑆𝑖 : Percentage of service level of Supplier i;
𝐿𝑖 : The maximum acceptable level of greenhouse gas emissions by Supplier i
according to environmental indicators;
𝑐𝑎𝑝𝑖 : CPO capacity production at Supplier i (in metric ton);
𝑐𝑎𝑝𝑗 : Capacity of Vehicle j (in metric ton);
𝐶𝑖 : Total purchase cost per metric ton (MT) of CPO from Supplier i;
𝐸𝑖 : Percentage of greenhouse gas emissions per 1 metric ton CPO of Supplier i;
3.1.4 Variables
𝑥𝑖 : Number of per metric ton CPO purchased from Supplier i;
𝑦𝑖 : 1 if Supplier i is selected to purchase, 0 otherwise;
𝑤𝑖 : 1 if total greenhouse gas emissions from Supplier i still acceptable level, 0
otherwise.
3.1.5 The multiobjective supplier selection mathematical model
The considered multi-objective mathematical formulation given as a typical linear model for supplier
selection problems is 𝑚𝑖𝑛 𝑍1, 𝑚𝑖𝑛 𝑍2, 𝑚𝑎𝑥 𝑍3, and 𝑚𝑎𝑥 𝑍4.

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𝑀𝑖𝑛 𝑍1 = ∑ ∑ 𝑥𝑖 (𝑎𝑖 + ⌈
𝑥𝑖
𝑐𝑎𝑝𝑗
⌉ 𝑏𝑖𝑗 + 𝑐𝑔𝑖 + 𝑑 + (𝑓𝑤𝑖𝑦𝑖))
𝑗
𝑖
(1)
If 𝐶𝑖 is the total cost, where:
𝐶𝑖 = 𝑎𝑖 + ⌈
𝑥𝑖
𝑐𝑎𝑝𝑗
⌉ 𝑏𝑖𝑗 + 𝑐𝑔𝑖 + 𝑑 + (𝑓𝑤𝑖𝑦𝑖) ∀𝑖, 𝑗
Then the first objective function (1) can be written in the form:
𝑀𝑖𝑛 𝑍1 = ∑ 𝐶𝑖𝑥𝑖
𝑖
(2)
𝑀𝑖𝑛 𝑍2 = ∑ 𝐸𝑖𝑥𝑖
𝑖
(3)
𝑀𝑎𝑥 𝑍3 = ∑ 𝐻𝑖𝑥𝑖
𝑖
(4)
𝑀𝑎𝑥 𝑍4 = ∑ 𝐾𝑖𝑥𝑖
𝑖
(5)
s.t.
∑ 𝑥𝑖
𝑖
≥ 𝐷 (6)
𝑥𝑖 ≤ 𝑐𝑎𝑝𝑖 ∀𝑖 (7)
𝑤𝑖𝑂𝑖 ≤ 𝐿𝑖 ∀𝑖 (8)
𝑥𝑖 ≥ 0 ∀𝑖 (9)
𝑦𝑖, 𝑤𝑖 ∈ {0,1} ∀𝑖 (10)
To address the problem of selecting CPO mill suppliers, four objective functions are formulated: net
price, greenhouse gas emissions, quality, and service. The first objective function aims to minimize the
total monetary cost, including the cost of purchasing CPO, shipping costs, disassembly costs, penalty
costs for delays, and costs associated with selecting environmentally friendly suppliers. The second
objective function focuses on minimizing greenhouse gas emissions from the suppliers. The third
objective function is to maximize the quality of the purchased CPO, as it directly impacts customer
satisfaction. The fourth objective function aims to maximize the level of trust in the suppliers,
emphasizing their trustworthiness and loyalty.

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Several constraints are considered in the model. The demand constraint ensures that the demand for
CPO is satisfied. The order quantity from each supplier should not exceed its capacity, as indicated by
the constraint set. The constraint related to environmental friendliness determines whether a supplier
meets the environmentally friendly criteria. Negative orders are prohibited, as specified by a
constraint, and binary variables are described by another constraint. In real cases, decision-makers
often lack precise and complete information regarding the decision criteria and constraints. The
variables C_i, E_i, H_i, K_i, and D represent either crisp or fuzzy values, accounting for the
uncertainty in the data.
For supplier selection problems the collected data does not behave crisply and they are typically fuzzy
in nature. A fuzzy multiobjective model is developed to deal with the problem. Let X be a universe of
discourse, A is a fuzzy subset of X if, for all x∈X, there is a number μ_A (x)∈[0,1] assigned to
represent the membership of x to A, and μ_A (x) is called the membership function of A. α_cut
represent the (crisp) set of elements that belong to the fuzzy set A for which the degree of its
membership function exceeds the level α: A_α=[x∈X| μ_A (x)≥α]. A fuzzy decision is defined in an
analogy to non-fuzzy environments ‘‘as the selection of activities which simultaneously satisfy
objective functions and constraints’’. In fuzzy set theory the intersection of sets normally corresponds
to the logical ‘‘and’’. The ‘‘decision’’ in a fuzzy environment can therefore be viewed as the
intersection of fuzzy constraints and fuzzy objective functions [26]. The fuzzy decision can be divided
into two categories, symmetric and asymmetric fuzzy decision-making. In a symmetrical fuzzy
decision there is no difference between the weight of objectives and constraints while in the
asymmetrical multi-objective fuzzy decision, the objectives and constraints are not equally important
and have different weights [26,27,28].
Constructing either the symmetrical or the asymmetrical model depends upon the selection of
operators. For fuzzy decision-making, the selection of appropriate operators is very important.
Zimmermann [29] classified eight important criteria that may be helpful for selecting the appropriate
operators in fuzzy decisions. The multiobjective linear formulation of numerical example is presented
as min Z_1, min Z_2, max Z_3, and max Z_4 (2-5). Let n as a number of objective functions,
n=1,2,3,4. We solve problems (2-5) by using fuzzy linear programming as has been done by
Zimmermann [29]. We formulate fuzzy linear programming by uniting each objective function Z_n
to be the maximum Z_n^+ and minimum Z_n^- value by solving:
𝑍1
−
= 𝑀𝑖𝑛 𝑍1
𝑍2
−
= 𝑀𝑖𝑛 𝑍2
𝑍3
+
= 𝑀𝑎𝑥 𝑍3
𝑍4
+
= 𝑀𝑎𝑥 𝑍4
(11)
𝑍1
−
, 𝑍2
−
, 𝑍3
+
, and 𝑍4
+
are obtained through solving the multiobjective problem as a single objective
using, each time, only one objective. Since for every objective function 𝑍𝑛, its value changes
linearly from 𝑍𝑛
−
to 𝑍𝑛
+
, it may be considered as a fuzzy number with the linear membership
function 𝜇𝑍𝑛(𝑥) as shown in Fig. 1.

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Fig.1. Objective function as fuzzy number: (a) & (b) for minimizing objective function (negative
objective) and (c) & (d) for maximizing objective function (positive objective).
Assuming that membership functions, based on preference or satisfaction are linear the linear
membership for minimization objective function and maximization objective function are given as
follows:
𝜇𝑍1
(𝑥) = {
1
𝑍1
+
− 𝑍1(𝑥) 𝑍1
+
− 𝑍1
−
⁄
0
𝑓𝑜𝑟 𝑍1(𝑥) ≤ 𝑍1
−
,
𝑓𝑜𝑟 𝑍1
−
≤ 𝑍1(𝑥) ≤ 𝑍1
+
,
𝑓𝑜𝑟 𝑍1(𝑥) ≥ 𝑍1
+
.
𝜇𝑍2
(𝑥) = {
1
𝑍2
+
− 𝑍2(𝑥) 𝑍2
+
− 𝑍2
−
⁄
0
𝑓𝑜𝑟 𝑍2(𝑥) ≤ 𝑍2
−
,
𝑓𝑜𝑟 𝑍2
−
≤ 𝑍2(𝑥) ≤ 𝑍2
+
,
𝑓𝑜𝑟 𝑍2(𝑥) ≥ 𝑍2
+
.
𝜇𝑍3
(𝑥) = {
1
𝑍3(𝑥) − 𝑍3
−
𝑍3
+
− 𝑍3
−
⁄
0
𝑓𝑜𝑟 𝑍3(𝑥) ≥ 𝑍3
+
,
𝑓𝑜𝑟 𝑍3
−
≤ 𝑍3(𝑥) ≤ 𝑍3
+
,
𝑓𝑜𝑟 𝑍3(𝑥) ≤ 𝑍3
−
.
(12)
𝑍1
−
𝑍1
+
1
(a)
1
𝜇𝑍1(𝑥) 𝜇𝑍2(𝑥)
(b)
𝑍2
+
𝑍2
−
1
𝑍3
−
𝑍3
+
𝜇𝑍3
(𝑥)
1
(c) (d)
𝑍4
+
𝑍4
−
𝜇𝑍4
(𝑥)

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𝜇𝑍4
(𝑥) = {
1
𝑍4(𝑥) − 𝑍4
−
𝑍4
+
− 𝑍4
−
⁄
0
𝑓𝑜𝑟 𝑍4(𝑥) ≥ 𝑍4
+
,
𝑓𝑜𝑟 𝑍4
−
≤ 𝑍4(𝑥) ≤ 𝑍4
+
,
𝑓𝑜𝑟 𝑍4(𝑥) ≤ 𝑍4
−
.
The linear membership function for the fuzzy constraints is given as:
𝜇𝐷
1
(𝑥) = {
1
𝐷 − 𝐷−
𝐷+
− 𝐷−
⁄
0
𝑓𝑜𝑟 𝐷 ≥ 𝐷+
,
𝑓𝑜𝑟 𝐷−
≤ 𝐷 ≤ 𝐷+
,
𝑓𝑜𝑟 𝐷 ≤ 𝐷−
.
𝜇𝐷
2
(𝑥) = {
1
𝐷+
− 𝐷 𝐷+
− 𝐷−
⁄
0
𝑓𝑜𝑟 𝐷 ≥ 𝐷+
,
𝑓𝑜𝑟 𝐷−
≤ 𝐷 ≤ 𝐷+
,
𝑓𝑜𝑟 𝐷 ≤ 𝐷−
.
(13)
The first operator discussed is the max-min operator, which Zimmermann [27, 29] utilized for
handling fuzzy multiobjective problems. This operator aims to identify the worst-case scenario by
maximizing the minimum values among the objectives.
Next, the convex operator, also known as the weighted additive operator, is introduced. This operator
allows decision-makers to assign different weights to each criterion, reflecting their relative
importance. By assigning appropriate weights, decision-makers can effectively balance the impact of
various criteria in the decision-making process.
In fuzzy programming modeling, following Zimmermann's approach, a fuzzy solution is obtained by
finding the intersection of all fuzzy sets associated with fuzzy objectives and fuzzy constraints. This
intersection represents the common elements that satisfy both the fuzzy objectives and constraints. By
considering the intersection of these fuzzy sets, a comprehensive fuzzy solution is derived, taking into
account all the fuzzy objectives and h fuzzy constraints in the problem formulation.
𝜇𝐴(𝑥) = {{⋂ 𝜇𝑍𝑛
(𝑥)
𝑛
} ⋂ {⋂ 𝜇𝐷
𝑚(𝑥)
𝑚
}} (14)
In order to find optimal solution (𝑥∗) in the above fuzzy model, it is equivalent to solving the
following crisp model [26]:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝜆𝑛 (15)
s.t.:
𝜆1 ≤ 𝜇𝑍1
(𝑥)
𝜆2 ≤ 𝜇𝑍2
(𝑥)
𝜆3 ≤ 𝜇𝑍3
(𝑥)
𝜆4 ≤ 𝜇𝑍4
(𝑥)
𝛾1 ≤ 𝜇𝐷
1
(𝑥)
𝛾2 ≤ 𝜇𝐷
2
(𝑥)
(16)

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In the proposed fuzzy solution, the membership functions μ_A(x), μ_(Z_n)(x) (for n=1,2,3,4), and
μ_D^m(x) (for m=1,2) represent the membership levels of the solution, the four objective functions,
and the constraints (specifically, the demand market), respectively.
The symmetry between constraints and objective functions in a fuzzy environment is emphasized,
meaning that there is no distinction between fuzzy goals and fuzzy constraints in this definition of the
fuzzy decision [26]. However, in certain supplier selection problems, it is necessary to consider
situations where fuzzy goals and fuzzy constraints have varying levels of importance to the decision-
maker (DM) and other stakeholders. To address this, the weighted additive model is employed, which
allows for the incorporation of unequal importance among objectives and constraints.
The weighted additive model is commonly used in multi-objective optimization problems, where a
single utility function is used to express the overall preference of the DM and determine the relative
importance of different criteria [30]. In this model, the membership functions of the fuzzy objectives
are multiplied by their corresponding weights and then summed together, resulting in a linear
weighted utility function. This approach is aligned with the convex fuzzy model proposed by Bellman
and Zadeh [31], Sakawa [28], and the weighted additive model introduced by Tiwari et al. [32].
𝜇𝐴(𝑥) = ∑ 𝑢𝑛𝜇𝑍𝑛
(𝑥)
𝑛
+ ∑ 𝑣𝑚𝜇𝐷
𝑚
(𝑥)
𝑚
,
∑ 𝑢𝑛
𝑛
+ ∑ 𝑣𝑚
𝑚
= 1, 𝑢𝑛, 𝑣𝑚 ≥ 0,
(17)
where 𝑢𝑛 and 𝑣𝑚 are the weighting coefficients that present the relative importance among the
fuzzy goal and fuzzy constraints. The following crisp single objective programming is equivalent
to the above fuzzy model:
max ∑ 𝑢𝑛𝜆𝑛
𝑛
+ ∑ 𝑣𝑚
𝑚
𝛾𝑚
(18)
s.t.:
𝜆1 ≤ 𝜇𝑍1
(𝑥)
𝜆2 ≤ 𝜇𝑍2
(𝑥)
𝜆3 ≤ 𝜇𝑍3
(𝑥)
𝜆4 ≤ 𝜇𝑍4
(𝑥)
𝛾1 ≤ 𝜇𝐷
1
(𝑥)
𝛾2 ≤ 𝜇𝐷
2
(𝑥)
𝜆𝑛, 𝛾𝑚 ∈ [1,0] 𝑛 = 1,2,3,4; 𝑚 = 1,2;
∑ 𝑢𝑛
𝑛
+ ∑ 𝑣𝑚
𝑚
= 1, 𝑢𝑛, 𝑣𝑚 ≥ 0,
(19)

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𝑥𝑖 ≥ 0 ∀𝑖
3.2 Solution approach
The last few years have seen a lot of interest in applying fuzzy sets to model various real-world
applications [33-37]. In the previous section, the solution approach has been initiated by introducing
the formulation of the proposed multi-objective fuzzy model for the supplier selection problem. After
that, for decision making problems, adjustments were made to several operators based on the
Zimmermann method [26]. Finally, a numerical experimental example is provided using the
hypothesized data.
Complete formulations of supplier selection problems to the fuzzy multiobjective are stated in the
following steps:
Step 1: Construct the supplier selection model according to the criteria and constraints of the
buyer and suppliers.
Step 2: Solve the multiobjective supplier selection problem as a single-objective supplier
selection problem using each time only one objective. This value is the best value for
this objective as other objectives are absent.
Step 3: Determine the corresponding values for every objective at each solution derived.
Step 4: For each objective function find a lower bound and an upper bound corresponding to
the set of solutions for each objective. Let 𝑍𝑛
−
and 𝑍𝑛
+
denote the lower bound and
upper bound for the 𝑛-th objective (𝑍𝑛) from (11).
Step 5: For the objective functions and fuzzy constraints find the membership function
according to (12–13).
Step 6: Based on fuzzy convex decision-making, formulate the equivalent crisp model of the
fuzzy optimization problem according to (18–19).
Step 7: Find the optimal solution vector 𝑥∗
, where 𝑥∗
is the efficient solution of the original
multiobjective supplier selection problem with the decision-makers preferences.
4 Numerical Experimentation
In the context of introducing a new product to the market, the management of three suppliers is
required. The criteria for supplier selection include net price, quality, and service. Additionally, the
capacity constraints of the suppliers need to be taken into account. However, the exact values of these
criteria and constraints are uncertain. The table provided, Table 1, presents the de-fuzzified values of
cost, emissions, quality, service level, and supplier constraints. On the other hand, Table 2 shows the
fuzzy number representing the predicted demand, which is approximately 1200.
The multi-objective linear formulation of the numerical example aims to minimize Z_1, Z_2, and
maximize Z_3 and Z_4. These objective functions correspond to different aspects of the supplier
selection problem, such as minimizing cost and emissions while maximizing quality and service. 𝑍1 =
3𝑥1 + 2𝑥2 + 5𝑥3 + 4𝑥4
𝑍2 = 40𝑥1 + 30𝑥2 + 10𝑥3 + 20𝑥4
𝑍3 = 0,8𝑥1 + 0,85𝑥2 + 0,9𝑥3 + 0,95𝑥4

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𝑍4 = 0,9𝑥1 + 0,7𝑥2 + 0,7𝑥3 + 0,8𝑥4
s.t.:
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 1200
𝑥1 ≤ 300
𝑥2 ≤ 400
𝑥3 ≤ 500
𝑥4 ≤ 600
𝑥𝑖 ≥ 0 ∀𝑖
Table 1. Suppliers quantitative information
Cost Emission Quality (%) Service (%) Capacity
Supplier 1 3 4 80 90 300
Supplier 2 2 3 85 70 400
Supplier 3 5 1 90 70 500
Supplier 4 4 2 95 80 600
The linear membership function is used for fuzzifying the objective functions and demand constraint
for the above problem according to steps 1–4. The data set for the values of the lower bounds and
upper bounds of the objective functions and a fuzzy number for the demand are given in Table 2.
Table 2 The data set for membership functions
μ0 μ1 μ0
𝑍1 (net cost) - 3700 5200
𝑍2 (emission) - 2000 3400
𝑍3 (quality level) 1030 1105 -
𝑍4 (Service level) 870 960 -
Demand 1100 1200 1300
The fuzzy multiobjective formulation for the example problem aims to minimize the total monetary
cost and greenhouse gas emissions, while maximizing the total quality and service level of the
purchased CPO. The membership functions for the four objective functions and the demand constraint
are provided. The formulation can be represented as follows:
Find 𝑥𝑇
= (𝑥1, 𝑥2, 𝑥3, 𝑥4)
to satisfy:
𝑍
̃1 = 3𝑥1 + 2𝑥2 + 5𝑥3 + 4𝑥4 ≤
̃ 𝑍1
0

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𝑍
̃2 = 4𝑥1 + 3𝑥2 + 𝑥3 + 2𝑥4 ≤
̃ 𝑍2
0
𝑍
̃3 = 0,8𝑥1 + 0,85𝑥2 + 0,9𝑥3 + 0,95𝑥4 ≥
̃ 𝑍3
0
𝑍
̃4 = 0,9𝑥1 + 0,7𝑥2 + 0,7𝑥3 + 0,8𝑥4 ≥
̃ 𝑍4
0
s.t.:
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 =
̃ 1200
𝑥1 ≤ 300
𝑥2 ≤ 400
𝑥3 ≤ 500
𝑥4 ≤ 600
𝑥𝑖 ≥ 0 ∀𝑖
𝑤𝑗 (𝑗 = 1, 2, 3, 4) and 𝛽1 are the weights associated with the jth objective and demand constraint.
In this example, the assumed decision-makers relative importance or weights of the fuzzy goals
are given as: 𝑤1 = 0,2; 𝑤2 = 0,35; 𝑤3 = 0,15; 𝑤4 = 0,1; and the weight of the fuzzy constraint
is 𝛽1 = 0,2.
Based on the convex fuzzy decision-making (18–19) and the weights which are given by
decision-makers, the crisp single objective formulation for the numerical example is as follows
(step 6):
max 0,2𝜆1 + 0,35𝜆2 + 0,15𝜆3 + 0,1𝜆4
s.t.:
𝜆1 ≤
5200 − (3𝑥1 + 2𝑥2 + 5𝑥3 + 4𝑥4)
1500
𝜆2 ≤
2400 − (4𝑥1 + 3𝑥2 + 𝑥3 + 2𝑥4)
400
𝜆3 ≤
(0,8𝑥1 + 0,85𝑥2 + 0,9𝑥3 + 0,95𝑥4) − 1030
75
𝜆4 ≤
(0,9𝑥1 + 0,7𝑥2 + 0,7𝑥3 + 0,8𝑥4) − 870
90
𝛾1 ≤
1300 − (𝑥1 + 𝑥2 + 𝑥3 + 𝑥4)
100

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𝛾1 ≤
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4) − 1150
50
𝑥1 ≤ 300
𝑥2 ≤ 400
𝑥3 ≤ 500
𝑥4 ≤ 600
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0
The linear programming software QM is used to solve this problem. The optimal solution for the
above formulation is obtained as follows:
𝑥1 = 0, 𝑥2 = 100, 𝑥3 = 500, 𝑥4 = 600
𝑍1 = 5100, 𝑍2 = 2000, 𝑍3 = 1105, 𝑍4 = 900
Corresponding to decision-makers preferences (0.2, 0.35, 0.15, 0.1), in this solution, 600 (maximum
capacity) items are assigned to be purchased from supplier 4, because of the highest quality level of
supplier 4 performances on the quality criterion. The remaining items are split between supplier 2 and
supplier1. The membership function values are obtained as follows.
5 Conclusion
Supplier selection is a critical aspect of supply chain management in today's highly competitive global
market. Companies recognize the importance of choosing the right suppliers, as it directly impacts cost
savings, product quality, and overall service performance. However, supplier selection is a complex
decision-making process involving multiple criteria, both qualitative and quantitative, with potential
conflicting objectives. This research aims to address this challenge by proposing a novel multi-
objective model for supplier selection, considering factors such as environmental impact, quality level,
and service level.
The proposed model incorporates a range of criteria, including cost minimization, environmental
impact consideration, and product quality maximization. It takes into account constraints related to
supplier capacity, demand, shipping costs, and potential delays. To achieve flexibility and handle the
inherent uncertainty in the decision-making process, a fuzzy approach is adopted.
The Zimmermann fuzzy approach is applied to solve the model, leveraging its capability to handle
multi-criteria decision-making problems. Numerical experiments are conducted using solving
software, taking advantage of the single-objective nature of Zimmermann's fuzzy approach.
By utilizing this multi-objective fuzzy model, companies can make informed decisions in supplier
selection, balancing various criteria and achieving optimal outcomes in terms of cost, environmental
impact, and product quality.

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