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A Novel Multi-Attribute Decision-Making Approach for Improvisational Emergency Supplier Selection: Partial Ordinal Priority Approach

January 2024
SSRN Electronic Journal

January 2024

DOI:10.2139/ssrn.4708945

Authors:

Renlong Wang

Chinese Academy of Sciences

Xueyan Shao

Chinese Academy of Sciences

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A Novel Multi-Attribute Decision-Making Approach for

Improvisational Emergency Supplier Selection: Partial Ordinal

Priority Approach

Renlong Wanga, Rui Shenb, Shutian Cuic, Xueyan Shaod, Hong Chia,d, Mingang Gaod,∗

aSchool of Emergency Management Science and Engineering, University of Chinese Academy of

Sciences, Beijing, 100049, China

bSchool of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049, China

cSchool of Economics and Management, Northwest A&F University, Yangling, 712100, China

dInstitutes of Science and Development, Chinese Academy of Sciences, Beijing, 100190, China

Abstract

The frequency of disaster occurrences highlights the importance of scientiﬁcally and logically

selecting emergency suppliers to ensure the timely distribution of relief provisions to aﬀected

regions. The abrupt, unpredictable, and intricate nature of disasters presents formidable

challenges to the improvisational emergency supplier selection (IESS). It demands swift

decision-making within constrained time frames amidst information scarcity, necessitating

coordinating various stakeholders and identifying potential Pareto optimal solutions. There-

fore, this study proposes a novel multi-attribute decision-making (MADM) approach, the

Partial Ordinal Priority Approach (POPA), tailored to address the challenges inherent in

IESS. POPA utilizes easily accessible and stable ranking data, encompassing expert pref-

erence information, as model input. This study derives the decision weight optimization

model and the adversarial Hasse diagram of the partial-order cumulative transformation set

based on ranking preference information. POPA can simultaneously determine the weights

of experts, criteria, and alternatives, generating the adversarial Hasse diagram. This dia-

gram streamlines the redundant dominance structure among alternatives and furnishes infor-

mation on Pareto optimal alternatives, suboptimal alternatives, and alternative clustering

details. To validate the eﬀectiveness of POPA, a case study on IESS for the Zhengzhou

mega-rainstorm disaster is conducted with sensitivity and comparative analysis. Overall,

POPA facilitates swift and stable decision-making while considering potential Pareto op-

timal solutions amidst time constraints, high information uncertainty, and involvement of

multiple stakeholders.

Keywords: Improvisational emergency supplier selection (IESS), Partial Ordinal Priority

Approach (POPA), Multi-attribute decision-making (MADM), Partial-order relationship,

Adversarial Hasse diagram

∗Corresponding author

Email addresses: 13127073530@163.com (Renlong Wang), shenrui0913@163.com (Rui Shen),

C19061092@163.com (Shutian Cui), xyshao@casisd.cn (Xueyan Shao), chihong@casisd.cn (Hong Chi),

mggao@casisd.cn (Mingang Gao)

January 29, 2024

Electronic copy available at: https://ssrn.com/abstract=4708945

1. Introduction

In today’s high-risk society, disasters such as earthquakes, ﬂoods, terrorist attacks, and

pandemics pose severe threats to human life, property, and social stability (Wang et al.,

2022). These disasters often exhibit characteristics of suddenness, uncertainty, and com-

plexity, making accurate prediction and control diﬃcult (Song et al.,2024). When disasters

strike, they often result in infrastructure damage, transportation disruptions, communica-

tion breakdowns, and shortages of emergency supplies, causing signiﬁcant loss of life and

property (Zhang et al.,2022). For instance, during the outbreak of the COVID-19 pan-

demic in Wuhan, Hubei province, the unique nature and rapid spread of the virus severely

impacted the medical system, leading to a shortage of medical resources in the region. The

surge of infected patients overwhelmed hospitals, and there was a scarcity of personal pro-

tective equipment like medical masks and protective suits, putting frontline medical workers

at a high risk of infection. They even had to reuse protective equipment, further increasing

the risk of transmission. Two months after the COVID-19 outbreak, the country allocated

40 million medical masks, 5 million sets of protective suits, and 5,000 sets of infrared ther-

mometers to meet the urgent needs of healthcare personnel in Hubei province (People’sDaily,

2020). During disaster response and emergencies, the stability and reliability of the sup-

ply chain are crucial to ensure the timely delivery of supplies to aﬀected areas (Liu et al.,

2022a). Swift, stable, and suﬃcient emergency supplies can shorten response times, enhance

rescue eﬀectiveness, ensure personnel safety, and reduce disaster losses. Therefore, the scien-

tiﬁc and rational selection of emergency suppliers, a typical multi-attribute decision-making

(MADM) problem, is essential to emergency decision-making (Zhang et al.,2022;Liao et al.,

2020;Mahmoudi and Javed,2022). However, in the face of extreme disasters, there is often

a need for improvisational emergency supplier selection (IESS), when pre-disaster emer-

gency supplier selection often fails to meet demand. Unlike traditional MADM problems,

the characteristics of disasters impose new requirements on IESS: decisions need to be made

rapidly in a limited time, with high information uncertainty and the involvement of multiple

stakeholders (Su et al.,2022;Pamucar et al.,2022;Li et al.,2022a).

Over the past decade, MADM technique has emerged as a crucial instrument for address-

ing IESS, capable of addressing inherent conﬂicting objectives, diverse data, and signiﬁcant

uncertainty within IESS. At present, the predominant approach to tackling the problem of

IESS relies on MADM methods such as GRA (Zhang et al.,2022), TOPSIS (Afrasiabi et al.,

2022;Ge et al.,2020), VIKOR (Zhu and Wang,2023;Zhang et al.,2023) (Zhu and Wang,

2023;Zhang et al.,2023), TODIM (Liu et al.,2022a;Su et al.,2022;Wang et al.,2023),

BWM (Tavakoli et al.,2023;Song et al.,2024), DEMATEL (G¨okler and Boran,2023;Wu

and Liao,2024). Speciﬁcally, due to the limited availability and challenging acquisition of

objective data in the decision-making process of IESS, current research predominantly relies

on subjective decision data from experts. The primary forms include evaluation values (Ge

et al.,2020;Sureeyatanapas et al.,2018), semantic values (Li et al.,2022b;Pamucar et al.,

2020;Wang et al.,2023), and pairwise comparison values (Wang et al.,2023;Yang et al.,

2020). Furthermore, some research extends existing MADM methods using grey system

theory (Yang et al.,2020;Zhang et al.,2022), fuzzy set theory (Ge et al.,2020;Qin and

Liu,2019), rough set theory (Rong and Yu,2024;Jiang et al.,2020;Sun et al.,2020) to ad-

dress the high uncertainty and substantial ambiguity in the decision-making process of IESS.

When aggregating opinions and preferences from multiple stakeholders, most studies em-

ploy average-based techniques, such as weighted averages, fuzzy averages, and mean square

Electronic copy available at: https://ssrn.com/abstract=4708945

deviation methods (Liu et al.,2022a;Ning et al.,2022;Zulqarnain et al.,2021). Only a few

studies utilize social network analysis to aggregate expert opinions from the perspective of

consensus and divergence within the expert group (Liu et al.,2022b).

Through the literature review, four critical limitations in current research have been

identiﬁed: (1) Most existing MADM methods in IESS only provide a total-order ranking of

alternatives, lacking the capability to identify potential Pareto optimal solutions. However,

in the practical application of IESS, it is crucial to enhance decision-making transparency and

reliability by eﬀectively identifying and prioritizing Pareto optimal alternatives (Grierson,

2008). (2) Most existing MADM methods in IESS overly rely on subjective opinions from

experts in the form of evaluative values and pairwise comparison values (Afrasiabi et al.,

2022). Nevertheless, stakeholders from diﬀerent backgrounds demonstrate diﬀerences in

professional knowledge and the extent of available information. This variability can lead

to notable discrepancies in expert judgments, and obtaining these opinions may involve

considerable time consumption. (3) Most existing MADM methods in IESS heavily rely on

algebraic logic for integrating viewpoints of multiple stakeholders (Ataei et al.,2020). This

reliance may lead to neglecting the valuable insights and preferences of experts, resulting

in decision outcomes deviating from the actual circumstances. Consequently, it adversely

aﬀects the accuracy and eﬀectiveness of the IESS process. (4) Most existing MADM methods

in IESS often integrate techniques of data standardization, expert opinion aggregation, and

pre-acquisition of weight information based on classical MADM methods. Nevertheless,

the above approach unavoidably ampliﬁes the complexity and error probability of MADM

models.

To overcome the above limitations in IESS, this study endeavors to introduce a MADM

approach (1) utilizing more available and stable decision data as IESS inputs to the MADM

model; (2) dispensing with the necessity for integrating data standardization, expert opinion

aggregation, or pre-weight acquisition approaches; and (3) expeditiously generating decision

outcomes for experts, criteria, and alternatives in MADM, while considering Pareto op-

timality condition. Therefore, this study proposes the Partial Ordinal Priority Approach

(POPA) for solving the IESS problem. Speciﬁcally, the proposed approach is based on

the Ordinal Priority Approach (OPA) with the ranking data of the criteria, experts, and

alternatives as model inputs. Based on the preference information embedded in ranking

data, this study derives a decision weight optimization model and partial-order cumulative

transformation to generate the most simpliﬁed dominance structure of alternatives (adver-

sarial Hasse diagram). The proposed approach can simultaneously determine the weight

of experts, criteria, and alternatives and generate dominance structure with information on

Pareto optimal alternatives, sub-optimal alternatives, and clustering details. Ultimately, the

computed weight outcomes are integrated with the adversarial Hasse diagram to address the

IESS while considering the Pareto optimality condition and expert preference information.

The remaining parts of this paper are organized as follows: Section 2 conducts a literature

review of the MADM approach in IESS. Section 3 outlines the criteria for IESS. In Section

4, the research method (POPA) is presented. Section 5 employs the IESS process for the

Zhengzhou mega-rainstorm disaster as a case study to demonstrate and validate POPA.

Lastly, Section 6 presents the conclusions and outlines future directions.

Electronic copy available at: https://ssrn.com/abstract=4708945

2. Literature Review

When confronted with natural disasters or other emergencies, organizations and govern-

ments require swift access to emergency supplies to cater to people’s fundamental needs and

facilitate disaster relief operations (Zhang et al.,2022). The process of selecting emergency

suppliers assumes a pivotal role, as their product quality, service responsiveness, and supply

chain reliability directly impact the eﬃcacy of rescue operations and the well-being of those

aﬀected (Chen et al.,2020). The selection of emergency suppliers can be conceptualized as

a classical MADM problem (Liao et al.,2020;Mahmoudi and Javed,2022;Mousavi et al.,

2020;Shakeel et al.,2020). In current research concerning emergency supplier selection,

scholars typically execute it in three steps, as illustrated in Fig.1(Liu et al.,2022a). The

initial step involves gathering decision-making data concerning evaluation criteria, experts,

and alternatives, encompassing historical records, statistical data, or expert opinions. The

subsequent step entails determining the weights assigned to the criteria and experts. Ul-

timately, rational MADM methods are utilized to ascertain the comprehensive evaluation

value for ranking the alternatives. However, it is noteworthy that the selection of emergency

suppliers mainly focuses on the emergency preparedness phase, and the decision-making

process is typically unrestricted by severe time pressures (Li et al.,2022b;Liu et al.,2022a).

Nevertheless, disaster events often entail complexity and uncertainty, leading to the pos-

sibility that pre-selected emergency suppliers may not adequately meet the requirements

for disaster response (Pamucar et al.,2022). In such instances, improvisational selection of

emergency suppliers becomes essential, exhibiting characteristics distinct from conventional

emergency supplier selection. Speciﬁcally, IESS must eﬃciently acquire and aggregate ex-

pert opinions, considering preferences information, to achieve stable decision outcomes in

situations characterized by intense time pressure, poor quality or diﬃcult accessibility of

decision data, and the involvement of multiple stakeholders.

Figure 1: Typical steps of the MADM for IESS

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In the present IESS research, widely utilized MADM methods encompass prominent

techniques such as GRA (Zhang et al.,2022), TOPSIS (Afrasiabi et al.,2022;Ge et al.,2020),

VIKOR (Zhu and Wang,2023;Zhang et al.,2023), TODIM (Liu et al.,2022a;Su et al.,

2022;Wang et al.,2023), BWM (Tavakoli et al.,2023;Song et al.,2024), MARCOS (Rong

and Yu,2024), MACBETH (Pamucar et al.,2020,2022), DEMATEL (G¨okler and Boran,

2023;Wu and Liao,2024), and others. Considering the challenges associated with acquiring

data in emergency situations, only a few studies incorporate objective decision data, while

the majority rely on subjective decision data derived from expert opinions due to poor

data quality. Subjective decision data takes the form of evaluation values (Ge et al.,2020;

Sureeyatanapas et al.,2018;Wang and Cai,2017), semantic values (Li et al.,2022b;Shakeel

et al.,2020;Su et al.,2022;Wang et al.,2023), and pairwise comparison values (Akter

et al.,2022;Liu et al.,2022a;Wang et al.,2023;Yang et al.,2020). Given the uncertainty,

ambiguity of crises, and limitations in expert experience and available information, the

subjective decision data provided by experts also carries a level of uncertainty and ambiguity.

Consequently, some studies apply grey system theory (Pamucar et al.,2020;Yang et al.,2020;

Zhang et al.,2022), fuzzy theory (Liu et al.,2023;Pamucar et al.,2022;Ge et al.,2020;Qin

and Liu,2019), and rough set theory (Rong and Yu,2024;Pamucar et al.,2022;Afrasiabi

et al.,2022) to enhance subjective decision data, addressing inherent uncertainties and

ambiguities. Regarding expert opinion aggregation, most studies mainly rely on algebraic

logic-based aggregation techniques, such as weighted averages and geometric means (Ataei

et al.,2020;Liu et al.,2022a;Ning et al.,2022;Pamucar et al.,2022;Zulqarnain et al.,

2021). Only a few studies explore the application of expert consensus and trust networks

in the aggregation of expert opinions in IESS (Liu et al.,2022b). Additionally, to account

for risk preferences of decision-makers arising in emergency scenarios, numerous current

studies integrate prospect theory (Zhang et al.,2023;Liu et al.,2019), cumulative prospect

theory (Liao et al.,2020), and regret theory (Liu et al.,2023) into the IESS decision-making

process. For example, in addressing the critical challenge of emergency medical supplier

selection during the COVID-19 pandemic, Liu et al. (2022a) employed a comprehensive

approach. They integrated expert semantic values transformed into interval Type-2 fuzzy

sets (IT2FSs) as inputs and incorporated an extended IT2FSs assessment method along with

an original ISM-BWM-Cosine Similarity-Max Deviation Method (IBCSMDM) to account

for psychological factors and bidirectional inﬂuence relationships. Su et al. (2022) utilized

probabilistic linguistic values provided by experts as input data and employed a TODIM

approach, augmented with prospect theory, for assessing suppliers, tackling the complexities

of MADM in the intricate and dynamic environment. Moreover, Furthermore, in addressing

the critical issue of medical supplier selection during the COVID-19 pandemic, Pamucar

et al. (2020) proposed a novel decision-making approach called Measuring Attractiveness

through a Categorical-Based Evaluation Technique (MACBETH), which incorporates fuzzy

rough numbers to eﬀectively manage the inherent high uncertainty associated with the

selection process. Zhang et al. (2022) proposed a novel approach, the Spherical Fuzzy Grey

Relational Analysis based on Cumulative Prospect Theory (SF-CPT-GRA), which integrates

GRA with spherical fuzzy sets and incorporates decision-makers’ risk preferences for eﬀective

emergency supplier selection.

The essence of the above MADM process lies in projecting the supplier’s performance

or utility evaluated by experts on various criteria into single comprehensive evaluation cri-

teria for supplier ranking. However, results obtained through the above approach often

lack stability and fail to accurately identify potential Pareto-optimal solutions in the IESS

Electronic copy available at: https://ssrn.com/abstract=4708945

(Kacprzyk et al.,2022). The practice has shown that considering Pareto-optimal solutions

in decision analysis can eﬀectively enhance the transparency and stability of decision-making

while also providing insights for decision portfolios (Grierson,2008). In addition, despite

the prevailing tendency in most studies to rely on expert opinions, which are more readily

available compared to objective decision data, the evaluation values, semantic values, and

pairwise comparison values also require the support of substantial professional knowledge

and information (Afrasiabi et al.,2022). Experts need to provide speciﬁc estimates for alter-

natives, but this process can be cumbersome and time-consuming, especially when obtaining

pairwise comparison values. Moreover, the involvement of multiple stakeholders results in

signiﬁcant variations in decision data. However, the current expert opinion aggregation

methods based on algebraic logic often struggle to objectively reﬂect experts’ actual views

and preferences in the above context (Ataei et al.,2020). Such disparities may lead to a

disconnect between decision outcomes and actual circumstances, thereby impacting the ac-

curacy and eﬀectiveness of supplier selection. Furthermore, the existing IESS methodology,

grounded in the traditional MADM process, incorporates a range of methods involving data

standardization, expert opinion aggregation, and pre-acquisition of weight information. Un-

doubtedly, this adds complexity to the model and increases the likelihood of errors. Notably,

the dominance relationships among alternatives (i.e., ranking data) are more accessible for

experts to establish and more stable and reliable for decision-makers (Wang et al.,2021).

This superiority arises as experts only need to assess which one is better without explicitly

determining the degree of diﬀerence between alternatives. Consequently, employing ranking

data as decision data for IESS is a promising approach.

Hence, this research endeavors to present an innovative MADM approach for IESS with

uncertainty and intense time pressure. The proposed approach leverages more reliable and

easily accessible ranking data as input, eliminating the necessity for data standardization,

expert opinion aggregation, and pre-weight acquisition techniques. Additionally, it considers

preference information among expert opinions and potential Pareto optimal solutions.

3. Evaluation Criteria of IESS

This section discusses the evaluation criteria for IESS from the perspective of supplier

emergency response capability and emergency supply capacity, as outlined in Table 1.

Table 1: Evaluation criteria for IESS

Perspective Criteria

Supplier emergency response capability

Emergency response speed (C1)

Emergency delivery reliability (C2)

Emergency geographic coverage (C3)

Operation sustainability (C4)

Collaborative experience and credibility (C5)

Emergency supply capacity

Emergency suppliy availability(C6)

Emergency supply quality (C7)

Emergency supply cost-eﬀectiveness (C8)

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3.1. Supplier Emergency Response Capability

Disasters entail instability, immediacy, and intricacy, potentially causing disruptions in

the supply chain, logistics delays, and inventory deﬁcits. In such situations, the pivotal

factor becomes the suppliers’ emergency response capability. Broadly, emergency response

capability denotes to the suppliers’ ability to swiftly respond to and adjust their supply

chain, ensuring production and delivery capacity when faced with disasters (Pamucar et al.,

2022). Through literature analysis, this study classiﬁes emergency response capability into

ﬁve core criteria: response speed (C1), delivery reliability (C2), geographic coverage (C3),

sustainability (C4), and collaborative experience and credibility (C5). Response speed per-

tains to the supplier’s timeliness in executing emergency response measures (Li et al.,2022a;

Zhang et al.,2022). Emergency delivery reliability encompasses the precision and depend-

ability of suppliers in delivering emergency supplies during crises (Afrasiabi et al.,2022;

Pamucar et al.,2020). A high level of reliability in emergency supplier deliveries ensures the

rapid and accurate distribution of supplies, thereby mitigating losses and impacts arising

from disasters. Emergency geographic coverage refers to the extent to which a supplier’s

emergency supplies can reach aﬀected areas during crises (Mohamadi and Yaghoubi,2017).

Expansive geographic coverage enhances a supplier’s ability to promptly deliver emergency

supplies and assistance to disaster-stricken regions, thereby minimizing the repercussions of

disasters. Sustainability of emergency supplier pertains to their capacity to consistently de-

liver reliable services and products amidst crises, incorporating sustainable practices across

economic, social, and environmental dimensions (Kannan et al.,2020). The collaborative

experience and credibility of emergency suppliers refers to the expertise and trustworthi-

ness demonstrated by the supplier in past collaborations when dealing with crises (Li et al.,

2022a;Pamucar et al.,2022). Emergency suppliers, drawing from their past cooperative

experiences, may be better equipped to handle unforeseen events due to the nature and

scale of challenges they may have previously encountered. Moreover, emergency suppliers

usually collaborate with multiple organizations, exhibiting excellent coordination and team-

work skills. They are familiar with the communication and collaboration processes with

all parties involved, enabling them to cooperate closely with other key stakeholders, thus

forming a uniﬁed force to respond to unexpected events.

3.2. Emergency Supply Capacity

Emergency supplies capacity primarily focuses on the inherent attributes of emergency

supplies that emergency suppliers can provide (Zhang et al.,2022). This capacity can be

broadly classiﬁed into quality, availability, and cost (Li et al.,2022a;Pamucar et al.,2022).

Supply quality encompasses the overall quality and durability, which are critical considera-

tions given the challenging environmental conditions and extensive usage during emergencies

(Afrasiabi et al.,2022). High-quality supplies ensure reliability and stability in urgent sit-

uations, while low-quality supplies may deteriorate swiftly, hindering or jeopardizing the

rescue process and adding further challenges and risks for the aﬀected population. Supply

availability pertains to the timely provision and acquisition of supplies during disasters or

emergencies (Ge et al.,2020). Reserving an appropriate quantity of supplies ensures swift

assistance to aﬀected areas, providing necessary support during critical moments. Supply

cost is associated with procurement and usage expenses (Liu et al.,2022a). Exorbitant

costs may limit the acquisition and reserves of supplies, impacting the scale and scope of

rescue operations. Identifying cost-eﬀective supplies or implementing preemptive storage

and procurement measures helps alleviate cost pressures during emergency relief eﬀorts.

Electronic copy available at: https://ssrn.com/abstract=4708945

4. The Proposed Partial Ordinal Priority Approach

Partial Ordinal Priority Approach (POPA) builds upon the Ordinal Priority Approach

(OPA) proposed by Ataei et al. (2020), integrating the partial-order theory and graph theory,

thereby representing a partial-order extension of OPA. POPA can be divided into three

parts: (1) weight optimization based on ranking preference information, (2) partial-order

cumulative transformation, and (3) adversarial Hasse diagram generation. Consequently,

POPA yields results regarding the weights of experts, criteria, and alternatives, along with

an adversarial Hasse diagram that illustrates the hierarchical dominance structure among

alternatives. This section primarily concentrates on elucidating the principle of POPA.

Table 2provides the involved indexes, parameters, sets, variables, and partial-order theory

symbols.

Table 2: Notation deﬁnition of the proposed approach

Type Notation Deﬁnition

Index

iIndex of alternatives (1,...,i,...,m)

jIndex of criteria (1, . . . , j, . . . , n)

kIndex of experts (1, . . . , k, . . . , p)

Parameter

rekRanking of the experts k

rcjk Ranking of the criteria junder the preference of the expert k

raijk

Ranking of the alternative ifor the criteria junder the

preference of the expert k

Set

ASet of alternatives ∀i∈A

CSet of criteria ∀j∈C

ESet of expert ∀k∈E

Variable ZObjective function

Wra

ijk

Weight of the alternative ifor the criteria jwith the ranking of

raijk under the preference of the expert k

Partial-order

theory symbol

(A, ⪯P OC T ) Partial-order cumulative transformation set (POCTS)

P RP OC T POCTS in binary matrix form

A+

x,P OC T Upper set of the alternative xof POCTS

A−

x,P OC T Lower set of the alternative xof POCTS

A=

x,P OC T Incomparable set of the alternative xof POCTS

GSP O CT General skeleton matrix of POCTS

4.1. Preliminaries

Deﬁnition 1. Partial-Order Relation

Let Rbe a binary relation on a set X, denoted as R⊆X×X(Ris a subset of the

Cartesian product of X). R is deﬁned as a partial-order relation on set X, denoted as ⪯, if

it satisﬁes the following properties:

(1) Self-reversibility: Ris self-reversible if xRx for ∀x∈X.

(2)Transmissibility: Ris transmissible if xRy, yRz ⇒xRz for ∀x, y, z ∈X.

(3)Antisymmetry: Ris antisymmetric if xRy, yRx ⇒x=yfor ∀x, y ∈X.

Deﬁnition 2. Total-Order Relation

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A partial-order relation Ron a set Xis deﬁned as total-order relation on the set Xif it

satisﬁes the strong completeness (i.e., xRy ∨yRx for ∀x, y ∈X)

The partial-order relation and total-order relation on the alternative set Aoriginating

from the utility of the criteria set Care called partial-order set (denoted as (A, ⪯C)) and

total-order set (denoted as (A, ≤C)), respectively. Considering evaluations from various

experts, (A, ⪯C|E) signiﬁes the partial-order set, while the total-order set is expressed in a

similar manner. It is worth noting that partial-order relation is a more “ﬂexible” relation

compared to total-order relation. This ﬂexibility allows for instances where the compared

alternatives are either equivalent or not comparable.

Deﬁnition 3. Lower Set and Upper Set of Partial-Order Set

Given the partial-order set (A, ⪯C), for ∀x∈A,A−

x,C ={y|y⪯Cx, y ∈A}is deﬁned as

the lower set of xon the partial-order set (A, ⪯C), and A+

x,C ={y|x⪯Cy, y ∈A}is deﬁned

as the upper set of xon the partial-order set (A, ⪯C).

Property 1. Given the partial-order set (A, ⪯C), for ∀x, y ∈A, there exists: (1) x∈

A−

y,C ⇔y∈A+

x,C ; (2) x⪯Cy⇔A−

x,C ⊆A−

y,C .

Deﬁnition 4. Order-Preserving Mapping of Partial-Order Set

Let (A, ⪯C1)and (B, ⪯C2)be the partial-order set, and function f:A→Bis the

mapping. If x⪯C1y⇒f(x)⪯C2f(y)holds for ∀x, y, ∈A, then function fis deﬁned as

order-preserving mapping of the partial-order set (A, ⪯C1).

Deﬁnition 5. Inclusion Relation in Partial-Order Set

Let A−

x,C1and B−

x,C2be the lower set of xon the partial-order set (A, ⪯C1)and (B, ⪯C2),

respectively. If A−

x,C1⊆B−

x,C2holds for ∀x∈A, then it is deﬁned that the partial-order set

(A, ⪯C1)is a subset of the partial-order set (B, ⪯C2), denoted as (A, ⪯C1)⊆(B, ⪯C2).

Theorem 1. Suppose that the partial-order set (A, ⪯C1)⊆(B, ⪯C2)and A=B. If x⪯C1y

holds for ∀x, y ∈A, then there exists x⪯C2y.

Proof of Theorem 1.Since ∀x, y ∈A, then ∀x, y ∈B. Given (A, ⪯C1)⊆(B, ⪯C2),

by Deﬁnition 5, it follows that A−

x,C1⊆B−

x,C2, A−

y,C1⊆B−

y,C2for x, y ∈A. By Property 1,

x⪯C1yimplies that x∈A−

y. And since A−

y,C1⊆B−

y,C2, it follows that x∈B−

y,C2such that

x⪯C2y.

Theorem 1provides an insight that when two partial-order sets, sharing the same al-

ternatives yet diﬀering in criteria, exhibit an inclusive relation, the evaluation outcomes

are order-preserving in the context of MADM. In addition, according to Property 1of the

partial-order set, it can be inferred that if two alternatives are comparable, no matter how

the weights of experts and criteria change, the partial-order relation between the two alter-

natives remains unchanged. The above insight will serve as the basis for constructing the

partial-order set in POPA that incorporates the information of criteria weight.

4.2. Weight Optimization Based on Ranking Preference Information

In MADM, a pivotal concern is accurately determining the weights assigned to experts,

criteria, and alternatives while considering the experts’ preference. The weight calculation

procedure in POPA is mainly based on the original OPA model with ranking data as input.

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The proposed approach formulates a weight optimization model from partial-order theory,

facilitating the integration of expert preference information without requiring averaging and

data standardization techniques.

When employing POPA, the decision-maker initiates the process by assigning the ranking

rekto each expert. Subsequently, each expert should independently provide the ranking for

each criteria rcjk ,and the ranking for each alternative under each criteria raijk. Suppose

that Ara

ijk is the alternative iwith the ranking raij k of the criteria junder the evaluation

of the expert k. The above ranking provided by the experts independently can reﬂect the

experts’ preferences. Then, the partial-order relation among alternatives under each expert

and criteria can be presented as Eq.(1).

Ara=m

ijk ⪯C|EAra=m−1

ijk ⪯C|E· · · ⪯C|EAra=r+1

ijk ⪯C|EAra=r

ijk ⪯C|E· · · ⪯C|EAra=1

ijk ∀i, j, k (1)

Let Wra

ijk be the assigned weight or utility of the alternative iunder the evaluation of the

expert kwith the ranking raij k of the criteria j. There exists a corollary Wra=r+1

ijk ≤Wra=r

ijk

such that Ara=r+1

ijk ⪯C|EAra=r

ijk holds for ∀i, j, k (i.e., for the same criteria and expert, the

weight of alternative with the ranking raijk =ris greater than or equal to the weight of the

alternative with the ranking raijk =r+ 1). Then, the following statement holds:

Ara=m

ijk ⪯C|EAra=m−1

ijk ⪯C|E· · · ⪯C|EAra=r+1

ijk ⪯C|EAra=r

ijk ⪯C|E· · · ⪯C|EAra=1

ijk ≜

Wra=m

ijk ≤Wra=m−1

ijk ≤ · ·· ≤ Wra=r+1

ijk ⪯Wra=r

ijk ≤ · ·· ≤ Wra=1

ijk ∀i, j, k (2)

The weight disparities among the consecutive rankings of the alternatives stated in Eq.(2)

can be segregated into mautonomous inequalities, as shown in Eq.(3).

Wra=1

ijk −Wra=2

ijk ≥0

Wra=2

ijk −Wra=3

ijk ≥0

. . .

Wra=r

ijk −Wra=r+1

ijk ≥0

. . .

Wra=m−1

ijk −Wra=m

ijk ≥0

(3)

To evaluate the impact of ranking preference information of experts on the weight of the

alternatives, POPA integrates rek,rcj k , and raij k into the assessment of weight disparities

of alternatives with consecutive rankings. Thus, this study then multiplies both sides of the

inequality (Eq.(3)) by the above ranking parameter, as illustrated in Eq.(4).

rekrcj k raijk(Wra=r

ijk −Wra=r+1

ijk )≥0∀i, j, k (4)

Eq.(1)-Eq.(4) present a logic for analyzing the weight or utility disparities of alterna-

tives with consecutive rankings assigned by experts with preference information. This above

process can also be extended to analyze the weight disparities of experts and criteria. It is

notable that decision-makers are willing to seek decision weight computations that mirror

the expert’s preferences and exhibit maximum discrimination. Thus, this study formulates

a multi-objective optimization model for decision weight optimization based on the rank-

ing preference information, as shown in Eq.(5). The optimization objectives encompass

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maximizing the weights of last-ranked alternatives and the weight discrepancies between

alternatives with consecutive rankings.

max{rekrcj k raijk(Wra=r

ijk −Wra=r+1

ijk ), rekrcjk raijk Wra=m

ijk } ∀i, j, k

s.t.

i=1

j=1

k=1

Wra

ijk = 1

Wra

ijk ≥0

(5)

Where the variable Wra

ijk is the decision variable of the optimization model and other

parameters are consistent with the deﬁnition speciﬁed in Table 2.

Then, maximum the minimization of the objective function:

maxmin{rekrcj k raijk(Wra=r

ijk −Wra=r+1

ijk ), rekrcjk raijk Wra=m

ijk } ∀i, j, k

s.t.

i=1

j=1

k=1

Wra

ijk = 1

Wra

ijk ≥0

(6)

The optimization model in max-min form can be further transformed into a linear pro-

gramming problem by variable substitution:

Z= min{rekrcj k raijk(Wra=r

ijk −Wra=r+1

ijk ), rekrcjk raijk Wra=m

ijk } ∀i, j, k (7)

Substituting Eq.(7) into Eq.(6) yields a linear optimization model for MADM weights

based on ranking preferences, as shown in Proposition 1.

Proposition 1. Decision Weight Optimization Model Based on Ranking Preference Infor-

mation

Given the ranking of experts rek, the ranking of criteria rcjk and alternatives under each

criteria raijk independently given by the experts, the decision weight optimization model

based on ranking preference information of experts is formulated as Eq.(8).

maxZ

s.t.

Z≤rekrcj k raijk(Wra=r

ijk −Wra=r+1

ijk )∀i, j, k

Z≤rekrcj k raijkWra=m

ijk ∀i, j, k

i=1

j=1

k=1

Wra

ijk = 1

Wra

ijk ≥0

(8)

By resolving Eq.(8), the weight of the alternative iwith the ranking raijk of the criteria j

under the evaluation of the expert kcan be determined. To ensure the active participation of

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experts in the decision-making process, it is recommended to set a lower bound of Wra

ijk ≥η,

where η=1

8×m×n×p.

Eq.(9) provides the formulations for computing the weight of experts, criteria, and al-

ternatives.

j=1

k=1

Wra

ijk ∀i

i=1

k=1

Wra

ijk ∀j

i=1

j=1

Wra

ijk ∀k

(9)

Through the above derivation, this study obtains an optimization model for determin-

ing decision weights while considering experts’ ranking preference information. Nevertheless,

the computed alternative weights merely project numerous criteria onto a single comprehen-

sive criteria (Yue and Yao,2023). Consequently, the outcomes lack stability in addressing

situations involving Pareto optimal solutions within the alternatives, which is the common

problem with the most MADM methods (Cao et al.,2023). To address this limitation,

the outcomes of weight optimization based on ranking preference information will undergo a

partial-order cumulative transformation in POPA, which incorporates information of criteria

weight.

4.3. Partial-Order Cumulative Transformation

The partial-order cumulative transformation of alternative weights in POPA primarily

aims to construct a partial-order set incorporating the information of criteria weight, which

is more ﬂexible than the partial-order set originating from the strict Pareto optimality

condition over each criteria, and more robust than the total-order set based on the single

projected comprehensive criteria computed by decision weight optimization model. The

core of this process lies in ensuring the newly constructed partial-order set has the property

of order-preserving. The ﬁrst step is to compute alternative weights under the criteria, as

depicted in Eq.(10).

WAC

ij =

k=1

Wra

ijk ∀i, j (10)

Denote (A, ≤SP C C ) as the total-order set originating from WA(i.e., single projected

comprehensive criteria), and (A, ⪯AC ) as the partial-order set based on WAC originating

from the strict Pareto optimality condition. Given the strict Pareto optimality-based partial-

order relation of alternatives, it refers explicitly to the partial-order relation formed by

directly examining the unprocessed weights of alternatives on various criteria.

Deﬁnition 6. Partial-Order Cumulative Transformation Set

Suppose that the criteria in WAC are arranged in descending ranking of the calculated

criteria weight. The partial-order cumulative transformation weight can be calculated by

Eq.(11). Then, (A, ⪯P OCT )is deﬁned as the partial-order cumulative transformation set

(POCTS) of (A, ⪯AC ).

WP OC T

ij =

j=1

WAC

ij l∈[n], and ∀i, j (11)

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The partial-order cumulative transformation weight can be expressed as Eq.(12) in matrix

form, where His the upper triangular matrix. The above procedure represents a linear

mapping of cumulative transformations applied to WAC, incorporating weight information.

WP OC T =WAC H=





WAC

11 WAC

11 +WAC

12 · · · WAC

11 +WAC

12 +· · · +WAC

WAC

21 WAC

21 +WAC

22 · · · WAC

21 +WAC

22 +· · · +WAC

.....

WAC

m1WAC

m1+WAC

m2· · · WAC

m1+WAC

m2+· · · +WAC







(12)

Comparing the magnitudes of the row vectors of WPO CT yields the POCTS in binary

matrix form among the alternatives P RP OCT = [prP OC T

xy ]m×m. Where prP OC T

xy can be deter-

mined by Eq.(13).

prP OC T

xy =(1,if WP OC T

xj ≥WP OC T

yj ,∀j

0,otherwise ∀x, y ∈A(13)

Subsequently, the order-preserving property of the POCTS is veriﬁed by Theorem 2and

Theorem 3.

Theorem 2. Suppose that (A, ⪯P OC T )is the partial-order cumulative transformation set of

(A, ⪯AC ), then it follows that (A, ⪯AC )⊆(A, ⪯P O CT ).

Proof of Theorem 2.Let A−

x,AC ={t|t⪯AC x, t ∈A}and A−

x,P OC T ={t|t⪯P OC T x, t ∈

A}denote the lower set of x∈Aon (A, ⪯AC) and (A, ⪯P OC T ), respectively. For ∀t∈A−

x,AC ,

there exists t⪯AC x⇔WAC

tj ≤WAC

xj ,∀j. It follows that Pl

j=1 WAC

tj ≤Pl

j=1 WAC

xj , l ∈[n],

such that t∈A−

x,P OC T , which implies that A−

x,AC ⊆A−

x,P OC T . By Deﬁnition 5, (A, ⪯AC )⊆

(A, ⪯P OC T ) holds.

Theorem 3. Suppose that (A, ⪯P OC T )is the partial-order cumulative transformation set of

(A, ⪯AC ). For ∀x, y ∈A, when x⪯Cyexists, WP OCT

xj ≤WP OC T

yj holds.

Proof of Theorem 3.Prove by mathematical induction. Consider WAC , and its ele-

ments WAC

ij can be decomposed into WC

j·VAC

ij , where WC

jsigniﬁes the criteria weight

computed in POPA, and VAC

ij can be interpreted as an unweighted utility with respect to

the criteria j. Given x⪯Cy, it follows that WAC

xj ≤WAC

yj ⇔VAC

xj ≤VAC

yj such that

(WAC

y1−WAC

x1)+(WAC

y2−WAC

x2) + · · · + (WAC

yn −WAC

xn )≥0⇔

1(VAC

y1−VAC

x1) + WC

2(VAC

y2−VAC

x2) + · · · +WC

n(VAC

yn −VAC

xn )≥0(14)

When r= 2, there exists WC

1≥WC

2and VAC

y1≥VAC

x1such that

1(VAC

y1−VAC

x1)≥WC

2(VAC

y1−VAC

x1) (15)

It follows that WC

1(VAC

y1−VAC

x1) + WC

2(VAC

y2−VAC

x2)≥

2(VAC

y1−VAC

x1) + WC

2(VAC

y2−VAC

x2) =

2(VAC

y1−VAC

x1+VAC

y2−VAC

x2)≥0

(16)

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When r=l, there exists {WC

l−1≥WC

l,∀j∈l}and {VAC

yj ≥VAC

xj ,∀j∈l}such that

1(VAC

y1−VAC

x1) + · · · +WC

l(VAC

yl −VAC

xl )≥

l(VAC

y1−VAC

x1+· · · +VAC

yl −VAC

xl )≥0(17)

Thus, when r=n, there exists:

1(VAC

y1−VAC

x1) + · · · +WC

n(VAC

yn −VAC

xn )≥

n(VAC

y1−VAC

x1+· · · +VAC

yn −VAC

xn )(18)

By the given premise WC

1(VAC

y1−VAC

x1) + WC

2(VAC

y2−VAC

x2) + · · · +WC

n(VAC

yn −VAC

xn )≥0,

it follows that WC

1(VAC

y1−VAC

x1) + · · · +WC

n(VAC

yn −VAC

xn )≥

n(VAC

y1−VAC

x1+· · · +VAC

yn −VAC

xn )≥0(19)

Thus, WP OCT

xj ≤WP OC T

yj holds.

Theorem 2and Theorem 3prove the order-preserving property of the newly constructed

POCTS incorporating information of criteria weight. It is noteworthy that the last column

element in WP OCT equals to the WAcomputed by decision weight optimization model.

Then, by Deﬁnition 2of total-order relation, it follows that (A, ⪯AC )⊆(A, ⪯P OC T )⊆

(A, ≤SP C C ). It is further demonstrated by theoretical derivation that the relationship be-

tween the constructed POCTS and the original partial-order set based on the strict Pareto

optimality condition over each criteria and the total-order set based on the single projected

comprehensive criteria. This implies that, theoretically, the partial-order set constructed

based on the partial-order cumulative transformation (A, ⪯POC T ) makes a trade-oﬀ between

the total-order relation based on comprehensive evaluation weights and the partial-order

relation based on the strict Pareto optimality condition, thereby forming a more robust

partial-order relation. While the partial-order relation of the strict Pareto optimality condi-

tion (A, ⪯AC ) emphasizes the strict dominance relation between alternatives on each criteria,

whereas the total-order relation (A, ≤SP C C ) is of a stronger property in which each pair of

alternatives is comparable. From the perspective of MADM practice, the POCTS signiﬁes

that if alternatives face a disadvantage in a relatively important criteria, they may still

be deemed viable as long as succeeding criteria can compensate for the deﬁciency in that

speciﬁc criteria. This ensures a more stable and dependable outcome for managers when

selecting optimal alternatives.

However, based on the transmissibility of the partial-order relation, it is evident that

redundant information exists in the generated POCTS binary matrix. It means that the

redundant dominance relations can be streamlined according to the transmissibility of the

partial-order relations to generate a more concise dominance structure among alternatives.

This structure will provide visual support to decision-makers for optimal alternative selec-

tion.

4.4. Adversarial Hasse Diagram Generation

In order to streamline the reduction information in the parital-order relation, this study

proposes the adversarial Hasse diagram of POPA, drawing inspiration from the Hasse dia-

gram (Wu et al.,2021) and the adversarial interpretative structural modeling (AISM) (Su

et al.,2023). Compared to the conventional Hasse diagram, the adversarial Hasse diagram

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not only identiﬁes the most simpliﬁed dominance relations but also extracts hierarchical

dominance structures of alternatives based on the non-dominant ascending and descending

rules. These dual-direction extraction rules empower decision-makers with a more compre-

hensive perspective for decision-making. Notably, the dominance structure of alternatives

oﬀers intuitive insights into Pareto-optimal alternatives, and clustering hierarchy informa-

tion of alternatives.

Proposition 2. Dominant Hierarchy Extraction of Partial-Order Cumulative Transforma-

tion Set in Binary Matrix Form

By Deﬁnition 3, the lower set, upper set, and incomparable set of x∈Aof the POCTS

in binary matrix form can be determined by Eq.(20).

A−

x,P OC T ={y|prP OC T

xy = 1, y ∈A}

A+

x,P OC T ={y|prP OC T

yx = 1, y ∈A}

A=

x,P OC T =A−A−

x,P OC T −A+

x,P OC T

(20)

Regarding dominant hierarchy extraction based on non-dominant ascending rules, if the

condition (A−

x,P OC T ∪ {x})∩(A+

x,P OC T ∪ {x}) = (A−

x,P OC T ∪ {x}), x ∈Ais satisﬁed for y∈A,

then position yat the top layer. Subsequently, eliminate the rows and columns in P RP O CT

corresponding to y. Iterate this process, positioning y∈Afrom the highest to the lowest

layer, until all elements in P RP O CT are eliminated.

Regarding dominant hierarchy extraction based on non-dominant descending rules, if the

condition (A−

x,P OC T ∪ {x})∩(A+

x,P OC T ∪ {x}) = (A+

x,P OC T ∪ {x}), x ∈Ais satisﬁed for

y∈A, then position yat the bottom layer. Subsequently, eliminate the rows and columns in

P RP OC T corresponding to y. Iterate this process, positioning y∈Afrom the lowest to the

highest layer, until all elements in P RP O CT are eliminated.

Following Proposition 2, the dominant hierarchy is extracted, and then P RP OCT is sub-

jected to edge contraction to eliminate redundant dominance relation information. Following

Eq.(21) for edge contraction, the general skeleton matrix (GS) is determined.

GSP O CT = (P RP OCT +I)′−((P RP O CT )′)2−I(21)

Where, Iis unit matrix, and all operators involved are Boolean operations.

Ultimately, the adversarial Hasse diagram can be derived by substituting the general

skeleton matrix into the extracted dominant hierarchy, with its property presented as follows.

Property 2. Given the adversarial Hasse diagram of the partial-order cumulative transfor-

mation set (A, ⪯P OC T ), the following holds:

(1) The top-layer alternatives, identiﬁed through non-dominant ascending and descending

rules, represent Pareto optimal alternatives.

(2) The count of dominant hierarchies, determined by non-dominant ascending and de-

scending rules, is uniform, with at least one consistent alternative for each layer.

(3) The dominance relation within the general skeleton matrix maintains transitivity.

(4) The alternatives within the same layer is incomparable.

Through the above derivation, this study formulates POPA and clariﬁes the motivations

and underlying logic behind each part. In general, POPA utilizes more stable and easily ob-

tainable ranking data that mirrors expert preference information, serving as inputs for the

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model. The decision weight optimization model based on ranking preference information

is initially derived to determine the weights of experts, criteria, and alternatives. POPA

then proposes a partial-order cumulative transformation to handle the inability to address

potential Pareto optimal problems in ranking the alternatives through single projected com-

prehensive criteria, thus constructing a more robust partial-order relation. Ultimately, by

generating the adversarial Hasse diagram, POPA streamlines redundant information within

the dominance structure, concurrently presenting Pareto optimal alternatives and the sim-

pliﬁed dominance structure with clustering hierarchy information of alternatives. These

sequential steps aim to ensure the eﬀectiveness and reliability of the model while addressing

potential Pareto optimality scenarios, thereby enhancing the practical guidance and decision

support eﬀectiveness of the ﬁnal decision outcomes.

The steps for implementing POPA are outlined as follows:

Step 1: Determining the experts, criteria, and alternatives.

Step 2: Ranking the experts and acquiring the ranking of criteria and alternatives under

each criteria independently provided by each expert.

Step 3: Computing the weight of experts, criteria, and alternatives based on the ranking

preference information as expressed in Eq.(8) and Eq.(9).

Step 4: Performing the partial-order cumulative transformation and determining the

POCTS in binary matrix form in accordance with Eq.(10), Eq.(11), and Eq.(12).

Step 5: Generating the adversarial Hasse diagram following the process of dominant

hierarchy extraction and edge contraction as expressed in Proposition 2and Eq.(21).

5. Case study and Discussion

5.1. Case Description and Data Collection

This study utilizes the IESS associated with the 7.20 mega-rainstorm disaster in Zhengzhou,

China, as a case study to illustrate and validate the proposed POPA. The unprecedented

intensity and geographic extent of the torrential rain in Zhengzhou shattered historical prece-

dents, surpassing established ﬂood control and drainage capacities, resulting in widespread

waterlogging and inundation (Peng and Zhang,2022). The exigency of the situation neces-

sitates a substantial inﬂux of medical supplies, sustenance, essential provisions, and other

relief materials-outstripping the provisioning capacity of the pre-identiﬁed suppliers during

the disaster preparedness phase. Confronted with this overwhelming catastrophe, it is neces-

sary to carry out IESS to ensure the seamless advancement of rescue protocols. Considering

the tight time constraints, highly uncertain information, and the involvement of multiple

stakeholders, the scenario of IESS for the Zhengzhou mega-rainstorm disaster emerges as a

representative application scenario for POPA.

Fifteen emergency suppliers, designated as A1 to A15, are at the disposal for selection

within the stricken region. These suppliers encompass a spectrum of characteristics, includ-

ing location, responsiveness, and supply capacity, each exhibiting distinct variations. Certain

entities prioritize a stable supply chain and swift responsiveness, albeit at a cost premium

beyond the norm. Conversely, other entities boast advantageous geographical placement

and eﬃcient traﬃc connectivity, expediting the provision of ﬂood relief in times of crisis.

However, these advantages might be counterbalanced by uncertainties surrounding the qual-

ity and durability of the provided rescue materials. These distinguishing characteristics will

serve as benchmarks aiding experts in ranking alternatives. Following this, ﬁve decision-

makers from various departments act as representatives for stakeholders in the selection of

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emergency suppliers. This group includes representatives from the Zhengzhou Emergency

Management Department, the Zhengzhou Civil Aﬀairs Department, and the Zhengzhou Mu-

nicipal Health Commission. The experts have been prioritized based on their authority in

emergency response decision-making, with E5 having the highest rank, followed by E2, E1,

E3, and E4 in descending order (E5 >E2 >E1 >E3 >E4). These decision-makers evaluate

the essential requirements for specifying emergency suppliers and provide rankings for both

sub-criteria and suppliers, as detailed in Table A.1.

5.2. Emergency Supplier Selection Results and Analysis

Figure 2displays the weights of emergency suppliers, sub-criteria with the computed

weight presented in Table B.1. Regarding the expert weight outcomes, Expert E5, possess-

ing the utmost authority, is assigned 0.4380, followed by Expert E2 with 0.2190. Subse-

quently, Experts E1, E3, and E4 trail behind with weights of 0.1460, 0.1095, and 0.0876,

respectively. Notably, the results of expert weights reveal a distinct trend of diminishing

marginal eﬀects. In the context of criteria, the most critical factor is the supplier response

speed (C1), carrying a weight of 0.2444. Closely followings are supplier collaborative experi-

ence and credibility(C5), supplier delivery reliability (C2), and supplier geographic coverage

(C3), with weights of 0.2007, 0.1481, and 0.1329, respectively. Conversely, the weighs for

the supply availability (C6), the supply quality (C7), the supply cost-eﬀectiveness (C8),

and supplier sustainability (C4) are relatively lower, with weights of 0.0931, 0.0694, 0.0602,

and 0.0512, respectively. From the criteria weight results of the case study, it is evident

that during the improvisational selection of emergency suppliers by critical stakeholders,

the primary focus lies on the emergency response capability of suppliers rather than the

attributes of the supplies they can provide. Notably, despite the increasing emphasis on the

sustainability of humanitarian operations in alignment with UN Sustainable Development

Goals, stakeholders in the case study still regard the sustainability of emergency suppliers

as a relatively insigniﬁcant criteria. Based on the computed weights of alternatives, the top

ﬁve alternatives are ranked as follows: A8, A3, A7, A5, and A13. Speciﬁcally, A8 exhibits

the most signiﬁcant weight, reaching 0.1021, followed by A3, with a weight of 0.0907. In

contrast, the weights for A7 and A5 are similar, standing at 0.0829 and 0.0803, respectively.

The weight for A13 is 0.072.

However, based on the discussion in the literature review and methodology, the above

ranking of alternatives is derived by projecting the evaluation criteria of IESS onto a com-

prehensive criteria. This outcome raises concerns regarding the stability of decision-making,

posing challenges in identifying potential Pareto optimal scenarios. Therefore, for a more

comprehensive decision-making insight into the Pareto optimal alternatives and the domi-

nant structure among alternatives, it is imperative to undertake further partial-order cumu-

lative transformation, especially within the context of IESS. Table 3presents the result of

partial-order cumulative transformation.

Through dominant hierarchy extraction and edge contraction operations, the partial-

order cumulative transformation set in binary matrix form is streamlined to eliminate re-

dundant information and further generate the adversarial Hasse diagram of emregency sup-

pliers, as depicted in Figure 3. The adversarial Hasse diagram distinctly illustrates the IESS

information regarding the dominance structure, Pareto-optimal alternatives, and hierarchi-

cal clustering details. The dashed blocks within the adversarial Hasse diagram denote the

alternatives with altered hierarchy.

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Figure 2: Computed weights of experts, sub-criteria and suppliers of the case IESS

As exempliﬁed in Table 4, the dominant hierarchy of the case IESS unfolds over ﬁve

layers, elucidating the alternatives with unchanged hierarchy and altered hierarchy inherent

within each layer. The structure originating from non-dominant descending rules positions

A3 at Layer 1, succeeded by A5 and A8 at Layer 2, and A2, A10, A11, A1, and A13 at

Layer 3. Layer 4 encompasses A4, A6, A12, and A14, while Layer 5, the bottom layer,

contains A9, A15, and A7. Within the adversarial Hasse diagram based on non-dominant

descending rules, the Pareto optimal alternatives for the case IESS emerges as A3, which

excels in pivotal dimensions of supplier response speed (C1) and consistently performs ex-

ceptionally across other criteria, encompassing supplier delivery reliability (C2) and supply

quality (C7). A proximate alternative of note lies in Layer 2, represented by A5 and A8.

Notably, under the transitive conditions of the adversarial Haase diagram, A5 serves as a

sub-optimal alternative for A3. Regarding structure based on non-dominant ascending rules,

the dominance hierarchy of A8 has transitioned from Layer 2 to Layer 1, whereas A7 now

occupies Layer 2 rather than Layer 5. This reconﬁguration positions A8 and A3 in Layer

1, each with inherent merits and limitations. A comparison between A8 and A3 reveals

that, in terms of the supplier collaborative experience and credibility(C5) and the supply

cost-eﬀectiveness (C8), A8 takes precedence. Furthermore, A5 and A7 are sub-optimal alter-

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Table 3: The results of partial-order cumulative transformation of the case IESS

WP OC T

ij1WP OC T

ij2WP OC T

ij3WP OC T

ij4WP OC T

ij5WP OC T

ij6WP OC T

ij7WP OC T

ij8

A1 0.0211 0.0301 0.0407 0.0454 0.0531 0.0563 0.0607 0.0653

A2 0.0119 0.0290 0.0351 0.0496 0.0521 0.0556 0.0608 0.0627

A3 0.0342 0.0435 0.0624 0.0722 0.0785 0.0860 0.0887 0.0907

A4 0.0142 0.0173 0.0241 0.0341 0.0418 0.0461 0.0490 0.0555

A5 0.0292 0.0409 0.0539 0.0613 0.0684 0.0735 0.0763 0.0803

A6 0.0121 0.0154 0.0249 0.0330 0.0462 0.0518 0.0538 0.0559

A7 0.0063 0.0337 0.0554 0.0645 0.0721 0.0770 0.0790 0.0829

A8 0.0267 0.0678 0.0719 0.0770 0.0832 0.0862 0.0954 0.1021

A9 0.0080 0.0161 0.0272 0.0363 0.0422 0.0487 0.0516 0.0530

A10 0.0177 0.0318 0.0362 0.0474 0.0509 0.0545 0.0584 0.0597

A11 0.0161 0.0221 0.0302 0.0438 0.0541 0.0602 0.0658 0.0696

A12 0.0112 0.0247 0.0350 0.0449 0.0473 0.0523 0.0560 0.0605

A13 0.0114 0.0338 0.0446 0.0553 0.0620 0.0672 0.0694 0.0720

A14 0.0133 0.0244 0.0321 0.0382 0.0417 0.0444 0.0478 0.0512

A15 0.0111 0.0144 0.0195 0.0230 0.0256 0.0289 0.0358 0.0385

Table 4: Hierarchical clustering information of the case IESS

Alternatives with

unchanged hierarchy

Alternatives with altered

hierarchy (ascending rules)

Alternatives with altered

hierarchy (descending rules)

Layer 1 A3 — A8

Layer 2 A5 A8 A7

Layer 3 A1, A2, A10, A11, A13 — —

Layer 4 A4, A6, A12, A14 — —

Layer 5 A9, A15 A7 —

natives to A3 and A8. Comparing A5 with A7, the latter excels in the supplier collaborative

experience and credibility(C5), supplier delivery reliability (C2), and supplier geographic

coverage (C3). Leveraging their respective strengths, A7 demonstrates suitability for large-

scale disaster scenarios that require a stable resource supply, whereas A1 is better suited to

resource supply scenarios with extreme time pressures.

Therefore, the IESS is not solely determined by the highest weights of alternatives.

Projecting multiple criteria onto a comprehensive criteria for the IESS may result in losing

valuable decision insights. Instead, this decision-making process necessitates considering

the speciﬁc capability advantages of each emergency supplier and tailoring the selection to

diﬀerent scenarios. As illustrated by the case study, POPA can provide information on the

alternative dominance structure, Pareto-optimal and sub-optimal alternatives, which will aid

decision-makers in formulating more transparent and robust decisions. While, this type of

information is not possible in some of MADM methods like TOPSIS, VIKOR, and TODIM.

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Figure 3: Adversarial Hasse diagram of the case IESS

5.3. Discussion

5.3.1. Validation of POPA Results Based on Sensitivity Analysis

Performing sensitivity analysis on input data or parameters is a vital numerical analysis

technique to evaluate the eﬃcacy of MADM methods. The core of POPA involves ranking

data with expert preferences, where the rankings of criteria and alternatives under criteria,

independently provided by experts, can aptly reﬂect expert preference information. Ana-

lyzing the sensitivity of rankings provided by experts involves perturbing their preference

information, thus creating numerous new expert scenarios. Nevertheless, this method lacks

a distinct benchmark, impeding thorough analysis. Therefore, this paper conducts a sensi-

tivity analysis on expert rankings supplied by decision-makers to ensure logical coherence.

Speciﬁcally, this study conducts a complete permutation of rankings for ﬁve experts with 120

experiments. The weights assigned to experts, criteria, and alternatives computed by POPA

are aggregated. Concurrently, the frequency of instances where each alternative emerges as

the Pareto optimal solution is examined within the adversarial Hasse diagram. Figure 4and

Table 5depict box plots and descriptive statistics of the computed weight outcomes.

In the descriptive statistical analysis of expert weights, the average values obtained

by the ﬁve experts all demonstrate a consistent feature, standing at 0.2. Expert weights’

maximum and minimum values are 0.4380 and 0.0876, respectively. This consistency is

also evident across all other indicators. This outcome aligns with intuitive observations

from implementing a whole permutation experimental design. Each expert’s frequency of

occurrence at various ranking positions is equal, resulting in the uniformity of descriptive

statistical results among the experts. Regarding the mean values of criterion weights, the

most signiﬁcant is C1, with a weight of 0.2821, followed by C3, with a weight of 0.1717.

Subsequently, C5, C2, and C6 closely follow, with weights of 0.1295, 0.1214, and 0.1134,

respectively, exhibiting a relatively similar trend. In contrast, the weights of C8 and C4

are the lowest, at 0.0541 and 0.0517, respectively. The Skewness results of the criteria

indicate that, except for C1 exhibiting left Skewness, the remaining criteria demonstrate

right Skewness. Meanwhile, the Kurtosis results for the criteria reveal that all criteria

exhibit negative Kurtosis, implying that the criteria weights concentrate around the mean,

with relatively fewer data points in the tails. Notably, the coeﬃcients of variation for both

C3 and C5 surpass the designated threshold (0.15), while the coeﬃcients for other criteria

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Figure 4: Box plots of the computed weight outcomes

remain below the threshold. Speciﬁcally, the coeﬃcient of variation for C5 reaches 0.2956,

indicating a pronounced dispersion trend. Furthermore, an examination of Figure 4reveals

a distinct portion of C5 weights reaching 0.2033. This is primarily due to the ranking of C5

provided by expert E5, which is 1, contrasting with rankings of 4, 6, 5, and 7 from other

experts. The divergent nature of these evaluations results in abnormal ﬂuctuations in the

weight of C5. Upon analyzing the mean weights of alternatives, it is evident that the top four

ranked alternatives are A5, A8, A3, and A11, with weights of 0.0881, 0.0868, 0.0841, and

0.0806, respectively. Notably, the maximum weight among the alternatives is associated with

A8, reaching 0.1039, while A8 exhibits a relatively high degree of dispersion. Additionally,

the coeﬃcients of variation for the weights of the alternatives do not exceed the threshold

(0.15). Furthermore, both the standard deviation and variance are smaller than that of the

criteria, indicating a more excellent stability of the alternatives compared to the criteria.

Furthermore, it is observed that A1, A2, A5, A7, A8, A10, A11, A12, and A13 all display

a negative skewness, indicating a right-skewed distribution. This implies that the weight

distribution of alternatives with negative Skewness extends more gradually to the right,

with the possibility of some relatively large values concentrated overall around the mean.

In contrast, the remaining alternatives exhibit characteristics of a left-skewed distribution

in their negative Skewness. Through statistical analysis of criteria and alternative weights,

it can be deduced that the weight outcomes of POPA are reasonably stable.

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Table 5: Descriptive statistics of the computed weight outcomes

Mean Max Min Standard Deviation Variance Skewness Kurtosis Coeﬃcient of Variation

E1 0.2000 0.4380 0.0876 0.1276 0.0163 1.1019 -0.3233 0.6380

E2 0.2000 0.4380 0.0876 0.1276 0.0163 1.1019 -0.3233 0.6380

E3 0.2000 0.4380 0.0876 0.1276 0.0163 1.1019 -0.3233 0.6380

E4 0.2000 0.4380 0.0876 0.1276 0.0163 1.1019 -0.3233 0.6380

E5 0.2000 0.4380 0.0876 0.1276 0.0163 1.1019 -0.3233 0.6380

C1 0.2821 0.3263 0.2202 0.0341 0.0012 -0.3725 -1.3224 0.1209

C2 0.1214 0.1488 0.0980 0.0164 0.0003 0.2634 -1.4249 0.1353

C3 0.1717 0.2375 0.1307 0.0331 0.0011 0.8466 -0.6099 0.1927

C4 0.0517 0.0551 0.0490 0.0018 0.0000 0.4307 -0.9298 0.0348

C5 0.1295 0.2033 0.0895 0.0383 0.0015 1.0689 -0.3561 0.2956

C6 0.1135 0.1394 0.0877 0.0144 0.0002 0.0569 -0.8765 0.1267

C7 0.0760 0.0909 0.0680 0.0076 0.0001 1.0011 -0.4367 0.0998

C8 0.0541 0.0605 0.0501 0.0032 0.0000 0.8783 -0.5356 0.0599

A1 0.0683 0.0856 0.0559 0.0093 0.0001 0.4050 -1.2665 0.1362

A2 0.0555 0.0639 0.0485 0.0045 0.0000 0.2496 -1.0751 0.0813

A3 0.0841 0.0929 0.0686 0.0080 0.0001 -1.0532 -0.3691 0.0952

A4 0.0623 0.0696 0.0545 0.0042 0.0000 -0.1595 -0.7932 0.0672

A5 0.0881 0.0967 0.0798 0.0047 0.0000 0.1111 -0.7965 0.0536

A6 0.0676 0.0779 0.0540 0.0070 0.0000 -0.5076 -0.8752 0.1035

A7 0.0694 0.0842 0.0598 0.0076 0.0001 0.9465 -0.4578 0.1089

A8 0.0868 0.1039 0.0715 0.0091 0.0001 0.2538 -0.9885 0.1046

A9 0.0577 0.0647 0.0494 0.0047 0.0000 -0.2852 -1.3624 0.0807

A10 0.0639 0.0734 0.0554 0.0053 0.0000 0.2339 -1.3156 0.0826

A11 0.0806 0.0992 0.0687 0.0095 0.0001 0.9569 -0.4535 0.1178

A12 0.0541 0.0607 0.0488 0.0034 0.0000 0.4575 -1.0154 0.0636

A13 0.0702 0.0816 0.0602 0.0062 0.0000 0.2458 -1.2380 0.0876

A14 0.0524 0.0581 0.0461 0.0033 0.0000 -0.2151 -1.0294 0.0638

A15 0.0391 0.0428 0.0351 0.0021 0.0000 -0.2569 -0.8675 0.0545

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Table 6provides a detailed depiction of the frequencies at which various alternatives

emerge as Pareto-optima solutions. It is evident from the observations that A3, A5, A8,

A11, and A1 consistently stand out with frequencies exceeding 10 in multiple experiments.

Notably, A3, A5, and A8 frequently manifest as Pareto-optimal solutions in experiments ex-

ceeding 50 instances. The disclosed information indicates that, under speciﬁc circumstances,

the above alternatives all have the potential to become Pareto optimal solutions. This ﬁnd-

ing contrasts signiﬁcantly with the weight-based total-order ranking results, focusing solely

on the alternative with the highest weight. Such diﬀerences suggest the importance of con-

sidering the potential superiority of alternatives in diﬀerent contexts during decision-making

rather than relying solely on their relative positions in a total-order based on weights. How-

ever, the weight-based total-order ranking results exhibit a certain degree of alignment with

the alternatives expected to be Pareto optimal. This demonstrates the reliability and ratio-

nality of the adversarial Hasse diagram based on partial-order cumulative transformation in

identifying Pareto optimality.

Table 6: Frequency of attaining Pareto optimal solutions in adversarial Hasse diagram

Frequency in non-dominant

ascending structure

Frequency in non-dominant

descending structure Discrepancy

A1 13 9 4

A2 0 0 0

A3 99 81 18

A4 0 0 0

A5 82 79 3

A6 0 0 0

A7 0 0 0

A8 51 24 27

A9 0 0 0

A10 3 2 1

A11 33 1 32

A12 0 0 0

A13 6 0 6

A14 6 0 6

A15 0 0 0

5.3.2. Validation of POPA results Based on Comparative Analysis

This study undertakes a comparative analysis within the IESS context of the Zhengzhou

mega-rainstorm disaster to validate POPA. Speciﬁcally, it contrasts the alternative ranking

outcomes of POPA with those derived from conventional MADM methods, encompassing

TOPSIS, VIKOR, TODIM, and RSR. Typically, the above MADM methods employ decision

matrices with objective values or evaluation scores, necessitating the prior acquisition of

criterion weights. Consequently, aligning the ranking outcomes of MADM methods based

on the decision matrix with those of POPA facilitates a more eﬃcient demonstration of the

merits and validity of POPA. It is worth noting that ELECTRE is not selected due to the

additional requirement of setting subjective threshold parameters during its implementation.

Regarding the input of the selected MADM methods, the criteria weights produced by POPA

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are employed for the criteria weight input, with the data presented in Table B.1 serving as

the decision matrix. The computed alternative ranking results are shown in Table 7.

Table 7: Alternative ranking results of multiple MADM methods

POPA weight

ranking

POPA average

dominant hierarchy TOPSIS VIKOR TODIM RSR

7 3 8 7 6 11

8 3 7 8 8 8

2 1 1 1 2 10

12 4 10 10 11 12

4 2 3 3 3 2

11 4 9 11 10 14

3 3.5 4 4 13 4

1 1.5 2 2 1 1

13 5 13 13 14 9

10 3 11 12 4 7

6 3 5 5 5 6

9 4 12 9 9 5

5 3 6 6 7 3

14 4 14 14 12 13

15 5 15 15 15 15

This paper employs the Spearman correlation coeﬃcient to assess the correlation among

the ordinal sequences of alternative ranking results, as shown in Eq.(22).

ρ= 1 −6Pn

i=1 d2

n(n2−1) (22)

Where nis the number of alternative and d2

iis the diﬀerence squared between two

rankings. As ρapproaches 1, it indicates a pronounced positive correlation between the

two rankings; whereas when the ρapproaches -1, it signiﬁes a signiﬁcant inverse correlation

between the two rankings. When ρapproaches 0, it suggests a lack of evident correlation

between the two rankings. Table 8shows the Spearman correlation coeﬃcients of alternative

rankings of multiple MADM methods.

The results show that the alternative ranking derived from POPA weights displays a

noteworthy positive association with TOPSIS and RSR rankings, with Spearman correla-

tion coeﬃcients of 0.9536 and 0.9750, respectively. In contrast, a moderate correlation is

observed with VIKOR and TODIM rankings, featuring corresponding Spearman correlation

coeﬃcients of 0.7321 and 0.7500, respectively. The dominant hierarchy of POPA reveals a

notable Spearman correlation coeﬃcient of 0.9315 with VIKOR, signiﬁcantly surpassing cor-

relations with VIKOR from other methods. Furthermore, the dominant hierarchy of POPA

shows a robust correlation with TOPSIS and RSR, with corresponding Spearman correlation

coeﬃcients of 0.8654 and 0.8452. However, the correlation with TODIM is comparatively

lower, at only 0.5806. Notably, the correlation levels of all other methods with TODIM rank-

ings are generally low, ranging between [0.5429, 0.6786]. In summary, the ranking outcomes

derived from POPA reveal substantial correlations with other methods, except for some

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Table 8: Spearman correlation coeﬃcients of alternative ranking results of multiple MADM methods

POPA weight

ranking

POPA average

dominant hierarchy TOPSIS VIKOR TODIM RSR

POPA weight

ranking 1.0000 0.8599 0.9536 0.7321 0.7500 0.9750

POPA average

dominant hierarchy 0.8599 1.0000 0.8654 0.9315 0.5806 0.8452

TOPSIS 0.9536 0.8654 1.0000 0.7250 0.6179 0.9714

VIKOR 0.7321 0.9315 0.7250 1.0000 0.5429 0.7214

TODIM 0.7500 0.5806 0.6179 0.5429 1.0000 0.6786

RSR 0.9750 0.8452 0.9714 0.7214 0.6786 1.0000

variations with TODIM. Based on the presented ﬁndings, POPA demonstrates reasonable

performance compared to other traditional MADM methods, relying on ordinal data that is

easily accessible and without the need for prior acquisition of additional weight information.

5.3.3. Advantage Analysis of POPA over Other MADM Methods

Currently, classical MADM methods can be broadly categorized into three main groups:

(1) weighting methods, exempliﬁed by entropy-related methods; (2) ranking methods, in-

cluding TOPSIS, VIKOR, ELECTRE, PROMETHEE, and Hasse diagram; and (3) compre-

hensive methods, such as AHP, BWM, and OPA. Table 9presents a comparative analysis

between POPA and various other MADM methods across diﬀerent decision-making char-

acteristics. The table reveals that POPA transforms MADM issues into a mathematical

optimization model capable of concurrently determining weights for experts, criteria, and

alternatives while identifying Pareto optimal solutions. Furthermore, POPA employs rank-

ing data as model inputs, eliminating the need for pairwise comparison data and decision

matrix, where obtaining pairwise comparison data and decision matrix has been proven

challenging and time-consuming. Additionally, there is no requirement for standardizing

decision data or aggregating expert opinions in POPA.

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Table 9: Comparison of POPA with other MADM methods

AHP/

ANP Entropy TOPSIS TODIM VIKOR ELECTRE PROMETHEE BWM Hasse

diagram OPA POPA

Weighting of criteria ✓ ✓ ✓ ✓ ✓

Weighting of experts ✓ ✓ ✓

Weighting of alternatives ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Utilizing decision matrix ✓ ✓ ✓ ✓ ✓ ✓ ✓

Utilizing pairwise

comparison data ✓ ✓ ✓

Utilizing ranking data ✓ ✓

No translation of

qualitative

into quantitative

variables required

✓ ✓

No data standardization

required ✓ ✓ ✓ ✓ ✓

No expert opinion

aggregation required ✓ ✓ ✓

No eﬀect of positive and

negative ideal solutions

on results

✓ ✓ ✓ ✓ ✓ ✓

Formulating the problem

as a mathematical

optimization model

✓ ✓ ✓

Ability to undertake

group decision-making ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Ability to address Pareto

optimal solutions ✓ ✓

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6. Conclusion

In the contemporary high-risk society, frequent extreme climate disasters, terrorist at-

tacks, and major infectious diseases introduce notable abruptness and uncertainty, present-

ing substantial threats to human welfare, economic stability, and the security of individuals’

lives and assets. In such scenarios, the rapid surge in demand for emergency supplies often

exceeds the existing supply from emergency suppliers. Decision-makers must judiciously and

promptly navigate this uncertain environment to make improvisational selections of suitable

emergency suppliers. Presently, the MADM within IESS frequently encounters a series of

formidable challenges, manifested as follows: (1) insuﬃcient, imprecise, and time-consuming

acquisition of decision data; (2) the necessity to coordinate and integrate decision opinions

and preferences from multiple stakeholders; (3) the identiﬁcation of potential Pareto optimal

solutions during the decision-making process.

Therefore, this study presents POPA for addressing the challenges that IESS encounters.

Speciﬁcally, POPA builds upon the OPA with the more accessible and reliable rankings of

experts, criteria, and alternatives as the model inputs. The study develops a decision weight

optimization model capable of simultaneously obtaining the weights of experts, criteria,

and alternatives, considering expert ranking preferences. To enhance decision stability and

eﬀectively identify potential Pareto optimal solutions, this study derives the partial-order

cumulative transformation of decision weights and the corresponding partial-order cumula-

tive transformation set of alternatives. Subsequently, an adversarial Hasse diagram of the

partial-order cumulative transformation set is introduced. This not only streamlines the re-

dundant dominance structure among alternatives but also furnishes information on Pareto

optimal alternatives, suboptimal alternatives, and alternative clustering details. Utilizing

the IESS problem during the Zhengzhou heavy rain disaster as a case study, this paper

presents an illustrative demonstration of POPA and further validates its eﬃcacy through

sensitivity and comparative analysis.

The fundamental contribution of this study lies in the introduction of POPA to address

the IESS. Methodologically, this study incorporates the partial-order theory into OPA for

the ﬁrst time, achieving a partial-order extension of OPA. Speciﬁcally, POPA utilizes eas-

ily accessible and stable ranking data from experts as input, making it more suitable for

decisions that are characterized by a lack of suﬃcient and accurate data. POPA facilitates

the simultaneous determination of weights for experts, criteria, and alternatives, consid-

ering expert preference information without needing data standardization, expert opinion

aggregation, or pre-acquisition of criteria weights. Additionally, the partial-order cumulative

transformation and adversarial Hasse diagram generation of POPA, supported by theoreti-

cal derivation, can eﬀectively identify potential Pareto optimal solutions and facilitate more

robust decision-making. Practically, this study utilizes POPA by integrating representative

criteria system to address IESS, oﬀering insights and guidance for authentic decision-making

processes. Furthermore, POPA is not limited to IESS but is also applicable to any decision

scenario exhibiting similar characteristics.

Finally, it is essential to emphasize that the conclusions and ﬁndings of this study are

based on limited case scenarios. Consequently, there is a need for further exploration of

practical applications of POPA to validate its eﬀectiveness. Notably, the implementation of

IESS based on POPA relies on the assumption of criteria independence. Therefore, future

studies could reﬁne POPA, considering the interplay of criteria for a more accurate reﬂection

of real-world situations. Lastly, this study does not encompass the impact of decision-

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maker’s risk preferences on decision outcomes. Accordingly, future research is expected to

incorporate the risk preferences of decision-makers into the model extension of POPA.

Author Contributions

Renlong Wang: Writing - original draft, Conceptualization, Methodology, Validation,

Software. Rui Shen: Data curation, Validation, Visualization, Formal analysis. Shutian

Cui: Validation, Visualization, Formal analysis. Xueyan Shao: Writing – review & edit-

ing, Validation, Investigation. Hong Chi: Supervision, Investigation. Mingang Gao :

Supervision, Writing – review & editing, Funding acquisition, Investigation.

Funding

This research was funded by the National Natural Science Foundation of China (Grant

number: 72134004).

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal

relationships that could have appeared to inﬂuence the work reported in this paper.

Data Availability

Data will be made available on request.

Appendix A.

Table A.1: Ranking of criteria and alternatives under criteria provided by experts of the case IESS

Expert ID Supplier ID SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8

E1 — 15274368

E2 — 12486357

E3 — 14385267

E4 — 25167438

E5 — 32471865

A1 1 14 10 8 3 6 8 15

A2 9 10 6 10 15 11 11 5

A3 2 15 5 15 6 15 10 1

A4 10 7 4 11 9 5 3 9

A5 3 4 15 1 4 7 7 13

A6 4 13 3 12 7 8 1 6

A7 8 3 14 2 5 3 14 8

A8 7 9 12 3 10 12 4 4

A9 11 2 11 5 1 2 9 10

A10 6 12 1 14 11 13 2 3

A11 12 1 13 4 2 1 12 11

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Table A.1: Ranking of criteria and alternatives under criteria provided by experts of the case IESS

Expert ID Supplier ID SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8

A12 14 6 2 13 14 14 13 2

A13 5 11 7 9 8 4 15 12

A14 13 8 8 6 12 9 5 7

A15 15 5 9 7 13 10 6 14

A1 13 1 4 11 3 1 15 10

A2 4 15 11 3 7 13 10 3

A3 1 14 10 9 12 14 9 6

A4 767465511

A5 2 9 12 5 5 8 11 7

A6 11 4 6 6 11 2 3 12

A7 15 2 3 12 10 9 4 15

A8 3 7 14 2 1 4 12 4

A9 14 3 2 15 9 6 1 14

A10 6 13 13 7 14 11 14 2

A11 12 5 1 14 15 3 2 13

A12 5 12 15 1 2 12 13 8

A13 10 8 5 13 8 7 8 5

A14 8 10 9 8 4 10 6 9

A15 9 11 8 10 13 15 7 1

A1 7 9 14 1 7 13 15 6

A2 15 6 4 11 12 12 2 5

A3 2 14 15 2 2 3 10 12

A4 14 2 1 15 4 2 14 14

A5 1 13 13 7 3 14 9 1

A6 13 5 3 12 5 1 1 15

A7 6 10 12 6 11 10 12 4

A8 3 15 11 3 1 4 11 13

A9 12 1 2 14 10 9 3 7

A10 5 11 10 8 13 11 13 3

A11 10 3 5 4 9 5 4 10

A12 8 7 8 13 15 15 8 2

A13 11 4 6 5 6 6 5 8

A14 4 12 9 9 14 8 7 11

A15 9 8 7 10 8 7 6 9

A1 11 6 8 14 15 14 5 4

A2 6 10 7 13 9 6 4 10

A3 15 5 12 8 4 7 10 7

A4 7 9 6 12 14 5 3 11

A5 10 7 3 5 3 13 2 3

A6 3 4 13 4 13 1 15 15

A7 12 1 9 9 2 2 9 14

A8 9 11 4 15 8 8 6 6

A9 5 8 11 6 10 12 11 5

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Table A.1: Ranking of criteria and alternatives under criteria provided by experts of the case IESS

Expert ID Supplier ID SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8

A10 2 14 5 7 6 9 7 8

A11 13 3 1 10 5 3 1 12

A12 8 12 10 1 11 10 14 2

A13 14 2 2 11 1 4 8 13

A14 1 13 15 2 12 15 13 1

A15 4 15 14 3 7 11 12 9

A1 4 14 15 3 10 15 6 4

A2 14 7 1 14 4 10 11 5

A3 10 1 2 12 9 2 1 15

A4 3 12 11 1 15 14 12 6

A5 13 3 6 10 8 1 5 14

A6 9 8 10 11 14 4 15 9

A7 15 2 3 5 2 3 4 10

A8 2 13 12 2 1 9 10 1

A9 6 11 7 13 11 13 7 7

A10 8 9 8 15 5 5 8 13

A11 1 15 13 4 12 12 13 2

A12 11 4 4 9 6 6 2 12

A13 7 5 9 6 3 7 3 11

A14 12 6 5 7 7 8 14 8

A15 5 10 14 8 13 11 9 3

Appendix B.

Table B.1: Computed weight outcomes of the case IESS

Expert ID Supplier ID C1 C2 C3 C4 C5 C6 C7 C8

A1 0.0119 0.0001 0.0009 0.0004 0.0016 0.0012 0.0004 0.0000

A2 0.0021 0.0004 0.0019 0.0003 0.0001 0.0005 0.0002 0.0006

A3 0.0083 0.0000 0.0022 0.0000 0.0009 0.0001 0.0003 0.0015

A4 0.0018 0.0006 0.0027 0.0002 0.0005 0.0015 0.0011 0.0003

A5 0.0065 0.0011 0.0001 0.0017 0.0013 0.0010 0.0005 0.0001

A6 0.0053 0.0002 0.0033 0.0002 0.0008 0.0009 0.0020 0.0005

A7 0.0026 0.0013 0.0002 0.0012 0.0011 0.0022 0.0001 0.0003

A8 0.0031 0.0004 0.0005 0.0009 0.0004 0.0004 0.0009 0.0007

A9 0.0014 0.0017 0.0007 0.0006 0.0030 0.0028 0.0004 0.0002

A10 0.0037 0.0002 0.0059 0.0001 0.0003 0.0003 0.0014 0.0008

A11 0.0011 0.0024 0.0004 0.0008 0.0021 0.0040 0.0002 0.0002

A12 0.0005 0.0007 0.0042 0.0001 0.0001 0.0002 0.0001 0.0010

A13 0.0044 0.0003 0.0016 0.0003 0.0006 0.0018 0.0000 0.0001

A14 0.0008 0.0005 0.0013 0.0005 0.0003 0.0007 0.0007 0.0004

A15 0.0002 0.0009 0.0011 0.0004 0.0002 0.0006 0.0006 0.0001

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Table B.1: Computed weight outcomes of the case IESS

Expert ID Supplier ID C1 C2 C3 C4 C5 C6 C7 C8

A1 0.0012 0.0089 0.0020 0.0003 0.0016 0.0059 0.0001 0.0004

A2 0.0080 0.0002 0.0005 0.0012 0.0008 0.0004 0.0005 0.0014

A3 0.0178 0.0004 0.0007 0.0004 0.0003 0.0002 0.0006 0.0008

A4 0.0047 0.0028 0.0012 0.0010 0.0009 0.0022 0.0013 0.0003

A5 0.0125 0.0016 0.0004 0.0008 0.0011 0.0013 0.0004 0.0007

A6 0.0021 0.0040 0.0014 0.0007 0.0003 0.0042 0.0020 0.0002

A7 0.0004 0.0062 0.0024 0.0002 0.0004 0.0011 0.0016 0.0001

A8 0.0098 0.0023 0.0002 0.0016 0.0030 0.0027 0.0003 0.0011

A9 0.0007 0.0049 0.0031 0.0000 0.0005 0.0019 0.0036 0.0001

A10 0.0056 0.0006 0.0003 0.0006 0.0001 0.0007 0.0001 0.0018

A11 0.0016 0.0033 0.0045 0.0001 0.0001 0.0033 0.0025 0.0002

A12 0.0066 0.0008 0.0001 0.0022 0.0021 0.0005 0.0002 0.0006

A13 0.0026 0.0019 0.0017 0.0001 0.0006 0.0016 0.0008 0.0009

A14 0.0039 0.0013 0.0008 0.0005 0.0013 0.0009 0.0011 0.0005

A15 0.0032 0.0010 0.0010 0.0003 0.0002 0.0001 0.0009 0.0025

A1 0.0023 0.0004 0.0001 0.0011 0.0005 0.0003 0.0000 0.0004

A2 0.0002 0.0007 0.0013 0.0001 0.0002 0.0004 0.0010 0.0005

A3 0.0062 0.0001 0.0001 0.0008 0.0012 0.0024 0.0002 0.0001

A4 0.0004 0.0016 0.0030 0.0000 0.0008 0.0031 0.0001 0.0001

A5 0.0089 0.0001 0.0002 0.0003 0.0010 0.0002 0.0003 0.0013

A6 0.0006 0.0008 0.0016 0.0001 0.0007 0.0045 0.0015 0.0000

A7 0.0028 0.0003 0.0003 0.0003 0.0002 0.0007 0.0001 0.0006

A8 0.0049 0.0000 0.0003 0.0006 0.0018 0.0020 0.0002 0.0001

A9 0.0008 0.0022 0.0021 0.0000 0.0003 0.0008 0.0008 0.0003

A10 0.0033 0.0003 0.0004 0.0002 0.0001 0.0005 0.0001 0.0007

A11 0.0013 0.0012 0.0011 0.0005 0.0003 0.0017 0.0007 0.0002

A12 0.0019 0.0006 0.0006 0.0001 0.0000 0.0001 0.0003 0.0009

A13 0.0010 0.0010 0.0009 0.0004 0.0006 0.0014 0.0006 0.0003

A14 0.0040 0.0002 0.0005 0.0002 0.0001 0.0010 0.0004 0.0001

A15 0.0016 0.0005 0.0008 0.0002 0.0004 0.0012 0.0005 0.0002

A1 0.0004 0.0004 0.0016 0.0000 0.0000 0.0001 0.0009 0.0004

A2 0.0011 0.0002 0.0019 0.0001 0.0002 0.0006 0.0011 0.0001

A3 0.0001 0.0005 0.0006 0.0003 0.0005 0.0005 0.0004 0.0002

A4 0.0009 0.0003 0.0022 0.0001 0.0000 0.0007 0.0013 0.0001

A5 0.0005 0.0004 0.0039 0.0004 0.0006 0.0001 0.0017 0.0005

A6 0.0020 0.0006 0.0005 0.0005 0.0001 0.0018 0.0000 0.0000

A7 0.0003 0.0014 0.0013 0.0002 0.0007 0.0012 0.0004 0.0000

A8 0.0006 0.0002 0.0032 0.0000 0.0002 0.0004 0.0007 0.0003

A9 0.0013 0.0003 0.0008 0.0004 0.0002 0.0002 0.0003 0.0003

A10 0.0025 0.0001 0.0027 0.0003 0.0003 0.0003 0.0006 0.0002

A11 0.0002 0.0008 0.0071 0.0002 0.0004 0.0010 0.0024 0.0001

A12 0.0008 0.0001 0.0011 0.0012 0.0001 0.0003 0.0001 0.0006

A13 0.0001 0.0010 0.0050 0.0001 0.0010 0.0008 0.0005 0.0001

Electronic copy available at: https://ssrn.com/abstract=4708945

Table B.1: Computed weight outcomes of the case IESS

Expert ID Supplier ID C1 C2 C3 C4 C5 C6 C7 C8

A14 0.0036 0.0001 0.0001 0.0008 0.0001 0.0000 0.0002 0.0009

A15 0.0016 0.0000 0.0003 0.0007 0.0003 0.0002 0.0002 0.0002

A1 0.0053 0.0007 0.0002 0.0028 0.0053 0.0001 0.0019 0.0032

A2 0.0005 0.0047 0.0089 0.0002 0.0160 0.0007 0.0007 0.0027

A3 0.0018 0.0178 0.0062 0.0005 0.0064 0.0031 0.0059 0.0001

A4 0.0065 0.0016 0.0010 0.0051 0.0007 0.0002 0.0005 0.0022

A5 0.0008 0.0098 0.0028 0.0008 0.0078 0.0045 0.0022 0.0003

A6 0.0021 0.0039 0.0013 0.0006 0.0015 0.0020 0.0001 0.0013

A7 0.0002 0.0125 0.0049 0.0019 0.0249 0.0024 0.0027 0.0011

A8 0.0083 0.0012 0.0008 0.0036 0.0356 0.0008 0.0009 0.0071

A9 0.0037 0.0021 0.0023 0.0003 0.0042 0.0003 0.0016 0.0019

A10 0.0026 0.0032 0.0019 0.0001 0.0133 0.0017 0.0013 0.0005

A11 0.0119 0.0004 0.0006 0.0023 0.0032 0.0004 0.0004 0.0050

A12 0.0014 0.0080 0.0040 0.0009 0.0111 0.0014 0.0042 0.0006

A13 0.0031 0.0066 0.0016 0.0016 0.0195 0.0012 0.0033 0.0008

A14 0.0011 0.0056 0.0033 0.0013 0.0093 0.0010 0.0002 0.0016

A15 0.0044 0.0026 0.0004 0.0011 0.0023 0.0005 0.0011 0.0039

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The supplier selection problem is one of the most important issues in supply chain management. So, many papers have investigated the mentioned problem. However, the related literature shows that researchers had less attention to the sustainability and resilience aspects based on the customer preferences in supplier selection problem. To cover this gap, this research tries to investigate the customer-based sustainable-resilient supplier selection problem. In this way, a Markovian-based fuzzy decision-making method is proposed. At the outset, the customer preferences are evaluated using a combination of the quality function deployment and the Markov transition matrix. Then, by combining the transition matrix and the fuzzy best–worst method, the weights of the indicators are calculated. Finally, the decision matrix is formed and the performance of suppliers is measured based on the multiplication of the decision matrix and vector of sub-criteria weights. Regarding the recent pandemic disruption (COVID-19), the importance of online marketplaces is highlighted more than the past. Hence, this study considers an online marketplace as a case study. Results show that in a pandemic situation, the preferences of customers when they cannot go shopping normally will change after a while. Based on the Markov steady state, these changes are from the priority of price, availability, and performance in initial time to serviceability, reliability, and availability in the future. Finally, based on the FBWM results, from the customer point of view, the top five sub-criteria for sustainable-resilient supplier selection include cost, quality, delivery, responsiveness, and service. So, based on these priorities, the case study potential suppliers are prioritized, respectively.

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A novel resilient and sustainable supplier selection model based on D-AHP and DEMATEL methods

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Purpose In the supply chain disruption risk, the issue of supplier evaluation and selection is solved by an extended VIKOR method based on regret theory. Design/methodology/approach Considering the influence of irrational emotions of decision makers, an evaluation model is designed by the regret theory and VIKOR method, which makes the decision-making process closer to reality. Findings The paper has some innovations in the evaluation index system and evaluation model construction. The method has good stability under the risk of supply chain interruption. Originality/value The mixed evaluation information is used to describe the attributes, and the evaluation index system is constructed by the combined method of the social network analysis method and the literature research method to ensure the accuracy and accuracy of the extracted attributes. The issue of supplier evaluation and selection is solved by an extended VIKOR method based on regret theory.

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In the AHP method, it is essential to control the uncertainty in obtaining weight. The paper aims to reduce the uncertainty introduced when using the AHP method, so as to improve the stability of the weight. Firstly, the paper ranks the indicators according to their importance, and the lowest lower limits of the importance ratio of two adjacent indicators are obtained, which can replace the accurate values of the indicator comparison in the AHP method and significantly reduce the introduced uncertainty. Secondly, the schemes are compared under infinite groups of weights, avoiding the blindness and instability of the comparison results obtained by only one group of weights. At the same time, the schemes’ comprehensive comparison results are displayed in the partial order Hasse diagram, which can clearly identify whether the comparison between scheme pairs is stable and evaluate the stability of the weight. Finally, an example is given to show the specific operation process of the proposed method, and simulation is used to verify the method’s superiority.

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