Supply Chain Elastic Strain

Abstract

The introduction of the concepts of shear elastic strain (${\epsilon }^{-}$) and tensile elastic strain (${\epsilon }^{+}$) is a catalyst for new horizons of research into supply chain elasticity. Functional formulas encompassing the metrics of ${\epsilon }^{-}$ and ${\epsilon }^{+}$, their critical point, maximum strain value, and similar parameters are established through rigorous mathematical derivations. The supply chain elasticity of agricultural commodities, including grains, apples, and wheat, are assessed by utilizing the derived formulas. The results show that the metrics of supply chain elastic strain serve as direct metrics of measuring the supply chain’s anti-interference capability, and they also facilitate an objective assessment of the supply chain’s safety and stability. The formula is succinctly derived, and it yields objective outcomes with general applicability, particularly suited for research and application for supply chain elasticity.

1. Introduction

The key indicators for measuring the security and stability of supply chains are supply chain elasticity and supply chain resilience. Supply chain elasticity, an inherent characteristic of supply chains, explains how the supply chain undergoes deformation when subjected to external forces or interference. Supply chain resilience, on the other hand, measures the capacity of supply chains to recover after interference, which is a multivariate function that encompasses recovery time and state of restoration.

The solution of supply chain elasticity is critical for the objective evaluation of supply chains. Currently, the main evaluation method for supply chain elasticity involves constructing a comprehensive evaluation system; direct function-based approaches to solving supply chain elasticity remain unexplored. This paper combines theories of material elasticity and economic elasticity to deduce that supply chain elasticity is a function of supply chain elastic stress and supply chain elastic strain. It conducted in-depth research on supply chain elastic strain, deriving models to solve shear elastic strain and tensile elastic strain in the supply chain. Finally, numerical experiments were applied to solve the shear and tensile elasticity of plant-based product supply chains in 134 countries, providing a visual representation of the stability of global supply chain elasticity for these products.

2. Literature Review

The Obama administration pioneered the concept of supply chain elasticity in 2012, positioning supply chain security as a cornerstone of national economic and security imperatives. This stance galvanized global attention in regard to the significance of supply chain elasticity. China delineated a fresh directive aimed at “safeguarding industrial and supply chain stability in 2020”. Following suit, the Biden administration released its “100-day review” in 2021, signaling a commitment to addressing supply chain disruptions. The UK’s Royal Institute of International Affairs advocated for an intensified international collaboration to buttress national economic security in 2022, emphasizing the importance of robust global supply chain elasticity. Over the past decade, supply chain elasticity has emerged as a focal point in academic and operational circles. Research in this domain has predominantly spanned four key arenas: evaluative metrics for elasticity, elasticity of supply–demand within supply chains, strategic paradigms to enhance elasticity, and the development of standards pertinent to supply chain elasticity.

Research on evaluating supply chain resilience includes studies on computational methods for supply chain resilience [1,2,3] and the construction of supply chain resilience calculation models and evaluation frameworks from different perspectives to assess post-disruption and recovery states [4,5]. For instance, developing an evaluation index system for supply chain resilience from an industry perspective, which utilizes reliable quantitative metrics to assess and enhance the elasticity of industry-specific supply chains, focusing on automotive manufacturing and its associated semiconductor supply chain [6], perishable agricultural commodities alongside their associated cold chain logistics [7,8], prefabricated construction methodologies [9], and the overarching industrial manufacturing sector [10]. Apart from this, an evaluation framework has been developed as well for supply chain network elasticity from the network-centric perspective, to evaluate supply chain network elasticity based on connectivity, node density, and node critical complexity, and to assess supply chain network elasticity based on network efficiency within industrial zones [11,12]. In addition, a supply chain elasticity evaluation framework was established from the key elements of the supply chain process, such as logistics, production processes, transportation modalities, and warehousing and operational management, to enhance supply chain elasticity [13,14,15,16]. For instance, formulating internal and external elasticity of enterprise from the perspective of organizational management in sectors like construction and maritime enterprises, aiming to boost the adaptability of their supply chains to market demands [17,18]. In addition to this, prioritizing environmental sustainability, the Green Elasticity evaluation system was formulated for supply chains, offering a strategic guide for environmentally conscious supplier selection and promoting sustainable supply chain practices [19,20,21,22].

The development of elasticity strategies for supply chains to fortify defenses against external interference and enhance recovery mechanisms from interruptions is one of the paramount objectives. Previous studies, such as tailoring industry-specific elasticity strategies for the industries of petrochemicals, retail, tire manufacturing, and clotting, based on their characteristics, have emphasized improving interference resistance and long-term sustainability [23,24,25,26]. Elasticity strategies have been customized to reduce uncertainties and reinforce stability based on the features of each stage of the supply chain, from supplier selection to product distribution and network design [27,28,29,30]. Strategies centered on elasticity recovery have been designed in response to supply chain interruption. These encompass the design of models tailored for supply chain re-establishment, the erection of closed-loop supply chains fortified against external disruptions, and the enhancement of performance in the problematic segments of the chain, aiming to elevate the supply chain’s recuperative capacity and hasten recovery timelines [31,32,33]. Developing supply chain strategies based on complementary resilience [34], along with developing supply chain resilience strategies from strategic and operational perspectives, to mitigate or alleviate supply chain risks [35] also underscores a significant research trajectory in supply chain elasticity strategy.

The realm of supply-and-demand elasticity in supply chains boasts substantial research contributions. For agricultural products such as dairy, coffee, and wheat, enhancing the supply–demand price elasticity of their respective supply chains can be achieved through devising appropriate policies, modifying the structure of the supply chain, adapting to retail market preferences, and tapping into consumption potentials. Such efforts can mitigate or eliminate the impacts of price and market fluctuations on these supply chains [36,37,38]. For the industries of grain, coffee, and forestry, elevating the entire industry’s supply price elasticity and demand price elasticity can be realized by enhancing the sustainability and value addition of agricultural and forestry products [39,40]. For energy sectors, especially electricity and liquefied gas, marked outcomes in scaling up the overarching supply–demand elasticity are evident through endeavors like distributed energy storage and swift demand feedback mechanisms [41,42]. Studies pinpoint that fuel taxation is a cardinal determinant affecting the elasticity of energy-driven supply chains [43]. In other realms, fortifying the capabilities of the distribution channel can augment supply-and-demand elasticities for industrial domains [44]. Creating distribution systems aligned with production scheduling and stockpile management can boost enterprise’s supply chain elasticity [45]. Meticulously crafting robust import–export supply–demand frameworks and curtailing exorbitant non-tariff hindrances have emerged as keystones for refining the elasticity in global trading supply chains [46]. Establishing stable import–export supply–demand networks and eliminating excessive non-tariff barriers are paramount for enhancing the supply–demand elasticity of global import–export trade supply chains [47]. Conversely, the influence of supply chain supply–demand elasticity on external forces on the supply chain is significant, with an inherent elasticity equilibrium potential existing within the supply chain [48,49].

Notable research outcomes have highlighted diversified facets of supply chain elasticity evaluation. For instance, criteria for elasticity assessment that emphasize supplier selection and order distribution have been proposed [50,51,52]. In scenarios following supply chain interruptions, emergent resilient optimization models and feedback control frameworks have been formulated [53,54]; the relationship between demand elasticity and supply chain stability has been studied [55]; and the optimal matching of elasticity and fragility in cross-border e-commerce supply chains has been analyzed, aspiring to achieve enhanced supply chain elasticity [56].

Drawing conclusions from the above, extensive research has been conducted on the evaluation of supply chain elasticity. However, an in-depth literature review indicated a gap in studies offering a generalizable approach to the evaluation of elasticity. This paper endeavors to delve into the study of supply chain share strain and reveal the intrinsic nature of supply chain elasticity. The objective is to evaluate the ability of supply chains to withstand external disruption by employing a generalized research approach, which offers innovation perspectives for future research on supply chain elasticity.

3. Supply Chain Elasticity, Elastic Strain

There are two forms of elasticity in objects: one is shear elasticity, triggered when the object deforms due to applied stress, and the other is tensile elasticity (Young’s modulus), induced when the object deforms from tensile forces. Similarly, supply chain elasticity is bifurcated into two types. The first type relates to underloaded conditions of the supply chain, which result in reduced profits, and the anti-interference capability of the supply chain showing a positive trend with regard to load reduction, potentially leading to supply chain breakage, termed as shear strain on the supply chain. The other type pertains to scenarios of supply chain overload, consequent increase in costs, and the anti-interference capability of the supply chain moving in an inverse direction with load variations, potentially leading to chain breakage, termed as tensile strain on the supply chain (Young’s modulus).

Supply chain elasticity is quantified by the supply chain elastic modulus, which refers to stress divided by strain under unidirectional stress conditions. The formula is defined as: E = $\frac{\sigma }{\epsilon }$, where $\sigma$ stands for stress and $\epsilon$ for strain. The “stress” in the supply chain refers to the external forces applied to it, which are interference imposed by the external environment. The elasticity “strain” of the supply chain refers to the economic phenomenon whereby the supply chain’s ability to resist interference changes when subjected to external forces. Supply chain elastic strain can be classified into two categories: shear elastic strain and tensile elastic strain.

4. Supply Chain Shear Elastic Strain (ε⁻) and the Crital Point (${\mathit{\epsilon }}_{\mathbf{0}}^{-}$)

Disregarding supply chain constraints on product supply and market demand, as well as price fluctuations, the shear elasticity strain of a supply chain refers to the ratio of the reduction in supply chain revenue or flow when the supply chain operates below full capacity due to external interference, relative to its profitability or profit under normal full-capacity operations, denoted as ${\epsilon }^{-}$. Based on this definition, we assume that the calculation formula for supply chain shear elasticity strain is as follows:

$${\epsilon }^{-}=\frac{\Delta I^{-}}{I}$$

(1)

where I is the sales revenue when the supply chain is operating at full capacity, I > 0, and where $\Delta I^{-}$ is the reduction in sales revenue, $\Delta I^{-}\ge$ 0.

Ignoring price fluctuations, the formula for shear strain in supply chain elasticity is

$${\epsilon }^{-}=\frac{\Delta Q^{-}}{Q}$$

(2)

where Q is the flow rate of the supply chain when operating at full capacity, Q > 0; $\Delta Q^{-}$ is the reduction in supply chain flow, $\Delta Q^{-}\ge$ 0; $\Delta Q^{-}$ represents the amount of shear elasticity deformation that occurs in the supply chain; and $\Delta Q^{-}$/Q represents the degree of shear elasticity deformation occurring in the supply chain, specifically indicating the ratio of shear elasticity deformation, termed as shear elasticity strain in the supply chain.

Hypothesis: When supply chain shortages lead to a net profit of zero, there are merchants (or enterprises) within the supply chain that also face zero or even negative net profits. At this point, merchants (or enterprises) choose to interrupt production and operations due to lack of profitability, exiting the supply chain. The shear elastic strain serves as the critical threshold for the supply chain’s shear elasticity, also representing the maximum shear elastic strain of the supply chain, denoted as ${\epsilon }_0^{-}$. The corresponding supply chain flow (sales volume) represents the minimum shear elastic load of the supply chain, denoted as $Q_0^{-}$, and the change in supply chain flow (or sales volume) is termed the maximum shear elasticity strain of the supply chain, symbolized by $\Delta Q_0^{-}$.

When the supply chain operates at full capacity, P represents the product selling price, Q represents the supply chain product flow (sales volume), F represents fixed costs (such as rent, property taxes, etc.), $\alpha$ represents the gross profit margin (the ratio of unit product gross profit to sales price), $\beta$ represents the net profit margin, and $\alpha$ > $\beta$. The net profit of the supply chain is

$$PQ\beta =PQ\alpha -F$$

further delineated in

$$F=PQ(\alpha -\beta )$$

(3)

When the supply chain is disrupted by external factors, resulting in a decrease in sales volume $\Delta Q^{-}$, the net profit Y of the supply chain decreases with an increase in $\Delta Q^{-}$, expressed by the following formula:

$$Y=P(Q-\Delta Q^{-})\alpha -F$$

(4)

F represents fixed costs, which remain constant. Substitute (3) into (4):

$$Y=P(Q-\Delta Q^{-})\alpha -PQ(\alpha -\beta )$$

Upon simplification, it yields

$$Y=PQ\beta -P\Delta Q^{-}\alpha$$

(5)

Based on these assumptions, the net profit is zero. Thus, when $Y=0$, the supply chain experiences interruption, leading to the shear elasticity deformation reaching $\Delta Q^{-}$ = $\Delta Q_0^{-}$, expressed by the following formula:

When $\Delta Q^{-}$ = $\Delta Q_0^{-}$,

$$PQ\beta -P\Delta Q^{-}\alpha =0$$

(6)

Upon simplification, this yields

$$\Delta Q^{-}=\Delta Q_0^{-}=\frac{Q\beta }{\alpha }$$

(7)

At this point, the shear elasticity strain of the supply chain also reaches its maximum, expressed by the following formula:

$${\epsilon }_0^{-}=\frac{\Delta Q^{-}}{Q}=\frac{\beta }{\alpha }$$

(8)

Findings: When $\Delta Q^{-}$ = $\Delta Q_0^{-}$, the supply chain has a net profit of 0, and the shear elastic strain of the supply chain has reached its maximum. The shear elastic strain of the supply chain ${\epsilon }^{-}$ = ${\epsilon }_0^{-}$ = $\beta /\alpha$.

According to the definition of the supply chain’s shear elastic minimum load $Q_0^{-}$ and Formulas (7) and (8), the solution formula can be expressed as

$$\Delta Q_0^{-}=Q-\Delta Q_0^{-}=Q(1-{\epsilon }_0^{-})$$

(9)

All the variables derived above satisfy non-negative constraints.

5. Supply Chain Tensile Strain (${\mathit{\epsilon }}^{+}$) and the Critical Point (${\mathit{\epsilon }}_{\mathbf{0}}^{+}$)

Disregarding the constraints on product supply and market demand in the supply chain, as well as factors such as price fluctuations, the tensile elasticity strain (or Young’s modulus strain) of a supply chain refers to the ratio of increased supply chain flow due to external interference causing the supply chain to operate beyond full capacity, denoted as ${\epsilon }^{+}$. At this point, supply chain costs increase and profits decrease. Based on this definition, the formula of supply chain tensile elastic strain is defined as follows:

$${\epsilon }^{+}=\frac{\Delta Q^{+}}{Q}$$

(10)

where Q represents the flow rate of the supply chain at full capacity, $Q>0$; $\Delta Q^{+}$ represents the difference between the flow rate of the supply chain under overload conditions and at full capacity, $\Delta Q^{+}\ge$ 0; $\Delta Q^{+}$ represents the amount of tensile deformation in the supply chain elasticity; and $\frac{\Delta Q^{-}}{Q}$ signifies the extent of tensile strain, which is the ratio of occurrence, referred to as the tensile elastic strain of the supply chain.

Hypothesis: When the supply chain is overloaded, leading to increased costs that result in a net profit of zero, there are merchants (or enterprises) within the supply chain that also face zero or even negative net profits. Consequently, these merchants (or enterprises) may cease their operations and withdraw from the supply chain, due to its unprofitability. The tensile elastic strain at the point of this state is termed as the critical point of tensile elastic strain or the maximum amount of tensile elastic strain, symbolized by ${\epsilon }_0^{+}$. The corresponding sales revenue (or sales volume) in this situation is referred to as the maximum load of tensile in the supply chain elasticity, symbolized by $Q_0^{+}$. The change in sales revenue (or sales volume) is recognized as the maximum tensile amount in the supply chain elasticity, symbolized by $\Delta Q_0^{+}$.

When the supply chain is overloaded, some goods fail to enter or enter late into the supply chain, leading to a higher product destruction rate in the supply chain than during full-capacity operation. For clarity, this paper categorizes all rates of goods destruction exceeding those of full-capacity operation as destruction of goods in the overloaded portion of the supply chain, with all fixed costs categorized under the full-capacity portion. The destruction rate of goods in the overloaded portion is denoted as $\gamma (\gamma \ge 0)$. At this point, the loss due to the supply chain operating in overload is $P\Delta Q^{+}(1-\alpha )\gamma -P\Delta Q^{+}(1-\gamma )\alpha$, simplified to $P\Delta Q^{+}(\gamma -\alpha )$. The net profit Y of the supply chain is expressed in function form, as follows:

$$Y=PQ\beta -P\Delta Q^{+}(\gamma -\alpha )$$

further delineated in

$$Y=PQ\beta +p\Delta Q^{+}(\alpha -\gamma )$$

(11)

The profit of the supply chain needs to be analyzed under three conditions according to the relationship between $\alpha$ and $\gamma$:

(1)

$\gamma <\alpha$

$$Y=PQ\beta +p\Delta Q^{+}(\alpha -\gamma )>PQ\beta$$

(12)

At condition $\gamma <\alpha$, the net profit of the supply chain increases with flow and there is no upper limit to the supply chain flow. Therefore, the maximum load of supply chain tensile elasticity has no limit, $Q_0^{+}=+\infty$. The corresponding maximum tensile elasticity of the supply chain also has no upper limit, $\Delta Q_0^{+}=+\infty$.

(2)

$\gamma =\alpha$

$$Y=PQ\beta +p\Delta Q^{+}(\alpha -\gamma )=PQ\beta$$

(13)

At condition $\gamma =\alpha$, the net profit of the supply chain does not change with an increase in load and there is no upper limit to the supply chain load. In this situation, the maximum tensile load of the supply chain elasticity has no upper limit, $Q_0^{+}=+\infty$. The corresponding maximum tensile elasticity of the supply chain also has no upper limit, $\Delta Q_0^{+}=+\infty$.

(3)

$\gamma >\alpha$

$$P\Delta Q^{+}(\alpha -\gamma )<0$$

further delineated in

$$Y=PQ\beta +p\Delta Q^{+}(\alpha -\gamma )<PQ\beta$$

(14)

When $\gamma >\alpha$, the supply chain’s net profit Y decreases with the increase of the overloaded flow $\Delta Q^{+}$. When the supply chain’s net profit is 0, the overloaded flow $\Delta Q^{+}$ reaches its maximum, denoted as $\Delta Q^{+}=\Delta Q_0^{+}$. The expression in formula is

$$PQ\beta +p\Delta Q^{+}(\alpha -\gamma )=0$$

Upon simplification, it is

$$\Delta Q^{+}=\Delta Q_0^{+}=\frac{Q\beta }{(\gamma -\alpha )}$$

(15)

That is, the maximum tensile strain of the supply chain elasticity, expressed as a function, is

$${\epsilon }_0^{+}=\frac{\Delta Q_0^{+}}{Q}=\frac{\beta }{(\gamma -\alpha )}$$

(16)

Findings: When $\gamma >\alpha$, and $\Delta Q^{+}=\Delta Q_0^{+}$, the supply chain’s net profit is 0, and the supply chain’s tensile elastic strain reaches its maximum, denoted as ${\epsilon }^{+}={\epsilon }_0^{+}=\frac{\beta }{(\gamma -\alpha )}$. At this point, according to Formula (16), the formula for determining the maximum extension $\Delta Q_0^{+}$ of the supply chain—and based on the definition of the maximum load $Q_0^{+}$ of the supply chain’s tensile elasticity, combined with Formulas (15) and (16)—is as follows:

$$Q_0^{+}=Q+\Delta Q_0^{+}=Q(1+{\epsilon }_0^{+})$$

(17)

6. Experiment on the Elasticity of Product Supply Chain

6.1. Selection of Experimental Subjects and Parameter Settings

6.1.1. Selection of Experimental Subjects and Data Description

Based on the HS4 customs classification, this study selected the second category of goods, which are closely related to daily life, as the research subject. Specifically, it analyzed the elasticity strain of the product supply chain. The second category includes plant products, such as live plants, vegetables, fruits, nuts, coffee, tea, grains, flour, gums, plant fluids, and other plant products.

The import and export data were collected from 10 to 20 publicly listed companies in each of 134 countries. Based on this data, we experimentally validated the model for calculating the elasticity strain of product supply chains.

6.1.2. Experimental Parameter Settings

Production Output (I): The calculation of elastic strain was based on the trade volume of import and export products, with the maximum trade volume in the past five years serving as the full-load capacity of the product supply chain. Gross profit margin ($\alpha$), net profit margin ($\beta$): these values were determined by compiling data on the gross and net profit margins of the top 10–20 listed companies involved in the import and export of the product in each country over the past five years.

Loss Rate ($\gamma$): This refers to the loss or depreciation rate of products that fail to enter the supply chain in a timely manner after a certain period. It was used as the loss rate for the product.

6.2. Experiment on Elastic Strain of Product Supply Chain

6.2.1. Numerical Experiments

The maximum shear strain value of the supply chain can directly reflect the stability and safety of the product supply chain’s shear elasticity. For plant products, according to the formula ${\epsilon }_0^{-}=\beta /\alpha$, the maximum shear strain values of the supply chain for 134 countries were calculated. The results are shown in

Table 1. Results of the maximum shear strain calculation (${\epsilon }_0^{-}$) in the supply chain of plant products.

C/R	${\mathit{M}}_{\mathit{SES}}$	C/R	${\mathit{M}}_{\mathit{SES}}$	C/R	${\mathit{M}}_{\mathit{SES}}$	C/R	${\mathit{M}}_{\mathit{SES}}$	C/R	${\mathit{M}}_{\mathit{SES}}$
ANG	0.6736	TUN	0.6737	NGR	0.6739	CYP	0.7161	SWE	0.7176
BRN	0.2499	UGA	0.6736	OMN	0.8482	CZE	0.7169	SUI	0.7161
BEN	0.6736	ZAM	0.6737	PAK	0.8526	COD	0.6736	UKR	0.7161
BOT	0.6736	ZIM	0.6734	PHI	0.8526	DEN	0.7486	CAN	0.7478
BFA	0.6735	AFG	0.8526	KSA	0.6738	FIN	0.7164	CRC	0.7481
CMR	0.6736	ARM	0.8537	SIN	0.8528	FRA	0.7186	DOM	0.7473
CIV	0.6736	AZE	0.8527	KOR	0.8531	GER	0.7167	ESA	0.7474
EGY	0.5407	BAN	0.8528	SRI	0.8528	GRE	0.7165	EST	0.7161
ETH	0.6735	MYA	0.8529	TPE	0.8531	HUN	0.7164	GUA	0.7474
GAB	0.6738	CAM	0.8528	TJK	0.8533	ISL	0.7160	HON	0.7473
GHA	0.6735	CHN	0.2500	THA	0.8527	IRL	0.7166	MEX	0.7483
IRQ	0.8528	GEO	0.8539	TUR	0.8528	ITA	0.5767	NCA	0.7474
JAM	0.6736	HKG	0.8529	TKM	0.8529	LTU	0.7160	PAN	0.7475
KEN	0.7825	IND	0.8535	UAE	0.8551	MLT	0.7162	USA	0.3333
LBY	0.6735	INA	0.5699	UZB	0.8528	MDA	0.7179	AUS	0.9921
MAD	0.6737	IRI	0.8539	VIE	0.8527	NED	0.7173	NZL	0.9921
MTN	0.6737	ISR	0.8527	YEM	0.8526	MKD	0.7161	ARG	0.5127
MRI	0.6803	JPN	0.7173	ALB	0.7161	NOR	0.7189	BOL	0.5128
MAR	0.6736	JOR	0.9183	ALG	0.7162	POL	0.7164	BRA	0.5135
MOZ	0.6736	KAZ	0.8540	AUT	0.6968	POR	0.7165	CHI	0.5322
NIG	0.6736	KUW	0.8533	BLR	0.7161	QAT	0.7172	COL	0.5184
COG	0.6736	KGZ	0.8528	BEL	0.7167	ROM	0.7165	ECU	0.5128
SEN	0.6738	LAO	0.8526	BER	0.7473	RUS	0.7181	PAR	0.5137
RSA	0.6740	LIB	0.8527	BIH	0.7160	RSB	0.7161	PER	0.5128
SUD	0.6735	LUX	0.7164	GBR	0.7159	SVK	0.7165	URU	0.5144
TAN	0.6737	MAS	0.8527	BUL	0.9129	SLO	0.7165	VEN	0.5127
TOG	0.6736	MGL	0.8527	CRO	0.7164	ESP	0.7178

C/R: Abbreviation for country/region; M_SES: the maximum supply chain shear elastic strain.

.

In order to reflect the shear elasticity of plant products’ supply chains across different countries more accurately, this paper categorized the maximum strain values into five classes from low to high: Class 1 (Very Weak), Class 2 (Weak), Class 3 (Moderate), Class 4 (Strong), and Class 5 (Very Strong). For algorithm selection, to eliminate the influence of subjective factors and classify more realistically and objectively, this paper adopted the k-means clustering algorithm in unsupervised machine learning, using the differences between strain values as the basis for clustering. The classification ranges and results are shown in

Table 2. Classification of shear elastic strain in plant-based product supply chains.

Category	Range of Shear Elastic Strain	Countries/Regions
First (Very Weak)	0.2499–0.5767	United States, Italy, China, Uruguay, Colombia, Brazil, Indonesia, Argentina, Egypt, Chile, Paraguay, Peru, Ecuador, Venezuela, Bolivia, Bahrain
Second (Weak)	0.6734–0.6968	Austria, Saudi Arabia, Tunisia, South Africa, Morocco, Senegal, Ethiopia, Zambia, Benin, Togo, Botswana, Angola, Zimbabwe, Madagascar, Uganda, Côte d’Ivoire, Tanzania, Mozambique, Republic of the Congo, Libya, Ghana, Mauritania, Gabon, Niger, Cameroon, Burkina Faso, Sudan, Nigeria, Democratic Republic of the Congo, Mauritius, Jamaica
Third (Moderate)	0.7160–0.7472	Japan, Switzerland, Germany, Czech Republic, Slovenia, Sweden, United Kingdom, Finland, France, Hungary, Slovakia, Ireland, Romania, Belgium, Poland, Netherlands, Lithuania, Spain, Belarus, Croatia, Portugal, Serbia, Bosnia and Herzegovina, Norway, Greece, Ukraine, North Macedonia, Russia, Moldova, Albania, Qatar, Algeria, Cyprus, Estonia, Iceland, Malta, Luxembourg
Fourth (Strong)	0.7473–0.8482	Denmark, Mexico, Canada, Panama, Costa Rica, El Salvador, Dominican Republic, Guatemala, Oman, Kenya, Honduras, Nicaragua, Bermuda
Fifth (Very Strong)	0.8526–0.9921	Taiwan (China), South Korea, Singapore, Israel, Hong Kong, Malaysia, Thailand, Turkey, Bulgaria, Philippines, India, New Zealand, Vietnam, Lebanon, United Arab Emirates, Kuwait, Jordan, Georgia, Armenia, Sri Lanka, Kyrgyzstan, Iran, Uzbekistan, Pakistan, Laos, Cambodia, Australia, Bangladesh, Kazakhstan, Azerbaijan, Myanmar, Tajikistan, Turkmenistan, Afghanistan, Yemen, Mongolia, Iraq

.

Based on the experimental results, the heatmap of shear elastic strain in the plant-based product supply chain is as shown in Figure 1.

6.2.2. Analysis and Discussion of Experimental Results

(1)

From an overall perspective, the shear elasticity strain of this product category’s supply chain is relatively strong. In most countries, the shear strain elasticity values are above 0.5767. This suggests that even with a significant decrease in the supply of plant products, for any reason, the supply chain can maintain basic stability. This also indicates that this product category generally enjoys a high profit margin and significant added value. Observing the supply chain, the stability of the backend supply chain for this product category tends to be better than that of the frontend supply chain. In summary, the shear elasticity of the supply chain for this category of product is relatively stable, and increasing the product supply can enhance the overall revenue of the supply chain for these products.

(2)

The shear elasticity strain of plant product supply chains in China, Indonesia, the United States, Italy, Egypt, and South American countries appears relatively weak. This indicates that these countries and regions generally experience lower net profit margins in this product category’s supply chains, particularly in China and the United States, both around 0.3. Given that these countries are major producers of this product category, a decrease in export volumes significantly impacts the stability of their supply chains and heightens the risk of disruptions. Therefore, these countries and regions need to enhance the added value of these products to improve net profit margins and strengthen the shear elasticity strain of their supply chains, thereby mitigating the risk of disruptions.

(3)

Asian countries (regions)—except for China, Malaysia, and Indonesia—and Oceania countries exhibit robust shear elasticity strains in their plant product supply chains, all surpassing 0.85. These nations predominantly serve as consumers of this product category, indicating generally higher net profit margins in their supply chains for these products. A reduction in the import volumes of these products has minimal impact on the stability of their supply chains, highlighting their overall resilience and stability in handling fluctuations.

6.3. Experiment on Tensile Elastic Strain of Product Supply Chain

6.3.1. Numerical Experiments

The maximum strain value of supply chain tensile elasticity can directly indicate the stability and safety of the tensile elasticity of the product supply chain. For plant products, according to the formula ${\epsilon }_0^{+}=\beta /(\gamma -\alpha )$, the maximum tensile elastic strain values for the supply chains in 134 countries were calculated. The results are shown in

Table 3. Results of the maximum tensile strain calculation (${\epsilon }_0^{+}$) in the supply chain of plant products.

C/R	${\mathit{M}}_{\mathit{TES}}$	C/R	${\mathit{M}}_{\mathit{TES}}$	C/R	${\mathit{M}}_{\mathit{TES}}$	C/R	${\mathit{M}}_{\mathit{TES}}$	C/R	${\mathit{M}}_{\mathit{TES}}$
ANG	0.7306	TUN	0.2019	NGR	0.2242	CYP	0.0708	SWE	0.0948
BRN	0.1780	UGA	0.1742	OMN	0.1429	CZE	0.1061	SUI	0.0664
BEN	0.2573	ZAM	0.7886	PAK	0.3671	COD	0.2268	UKR	0.0714
BOT	0.2887	ZIM	0.2025	PHI	0.3677	DEN	0.1367	CAN	0.2482
BFA	0.6662	AFG	0.2751	KSA	0.2005	FIN	0.0981	CRC	0.2091
CMR	0.1982	ARM	0.6030	SIN	0.2794	FRA	0.1097	DOM	0.2191
CIV	0.2176	AZE	0.4020	KOR	0.3615	GER	0.0836	ESA	0.4385
EGY	0.7338	BAN	0.2877	SRI	0.3552	GRE	0.0818	EST	0.1292
ETH	0.2251	MYA	0.6461	TPE	0.2977	HUN	0.0776	GUA	0.2308
GAB	0.0955	CAM	1.5396	TJK	1.5398	ISL	0.1229	HON	0.2703
GHA	0.2119	CHN	0.2332	THA	0.3900	IRL	0.0722	MEX	0.245
IRQ	0.1653	GEO	0.3471	TUR	0.4090	ITA	0.0500	NCA	0.3083
JAM	0.2337	HKG	0.2986	TKM	0.2955	LTU	0.0935	PAN	0.1679
KEN	0.2005	IND	0.4395	UAE	0.2857	MLT	0.1165	USA	0.221
LBY	1.0478	INA	0.5998	UZB	0.4300	MDA	0.0970	AUS	0.4786
MAD	0.2619	IRI	0.1944	VIE	0.3962	NED	0.0754	NZL	0.2147
MTN	0.8993	ISR	0.3320	YEM	0.1992	MKD	0.0819	ARG	0.5586
MRI	0.0501	JPN	0.0743	ALB	0.1056	NOR	0.0683	BOL	1.5396
MAR	0.2413	JOR	0.2082	ALG	0.0796	POL	0.0997	BRA	0.9379
MOZ	0.3250	KAZ	0.9694	AUT	0.0539	POR	0.0897	CHI	0.1382
NIG	0.1561	KUW	0.4116	BLR	0.0504	QAT	0.0696	COL	1.4103
COG	0.1976	KGZ	1.1217	BEL	0.0821	ROM	0.0998	ECU	0.3646
SEN	0.1604	LAO	0.4138	BER	0.0950	RUS	0.0592	PAR	0.2415
RSA	0.2514	LIB	0.2540	BIH	0.0626	RSB	0.0666	PER	0.3813
SUD	0.2780	LUX	0.0862	GBR	0.0562	SVK	0.1159	URU	1.4008
TAN	0.1806	MAS	0.3882	BUL	0.1205	SLO	0.0816	VEN	1.5396
TOG	0.5262	MGL	0.4344	CRO	0.1009	ESP	0.0732

C/R: Abbreviation for country/region; $M_{TES}$: the maximum supply chain tensile elastic strain.

.

In line with the numerical experiment for the shear elastic strain of the supply chain, the maximum tensile strain values of supply chain elasticity were divided into five categories from low to high: Category 1 (Very Weak), Category 2 (Weak), Category 3 (Moderate), Category 4 (Strong), and Category 5 (Very Strong). The k-means clustering algorithm, a method of unsupervised machine learning, was used for classification. The classification ranges and results are shown in

Table 4. Classification of tensile elastic strain in the supply chain of plant-based products.

Category	Range of Tensile Elastic Strain	Countries/Regions
First (Very Weak)	0.05–0.0803	Japan, Switzerland, Slovenia, Austria, United Kingdom, Italy, Hungary, Ireland, Belgium, Netherlands, Spain, Belarus, Serbia, Bosnia and Herzegovina, Norway, Ukraine, Russia, Qatar, Algeria, Cyprus, Mauritius
Second (Weak)	0.0806–0.1557	Germany, Czechia (Czech Republic), Sweden, Finland, France, Slovakia, Romania, Poland, Denmark, Lithuania, Croatia, Portugal, Bulgaria, Greece, North Macedonia, Moldova, Albania, Oman, Chile, Gabon, Niger, Estonia, Bermuda, Iceland, Malta, Luxembourg
Third (Moderate)	0.1603–0.2835	Singapore, United States, China, Mexico, Saudi Arabia, Canada, Panama, Costa Rica, Tunisia, New Zealand, Lebanon, United Arab Emirates, Jordan, Dominican Republic, South Africa, Guatemala, Paraguay, Iran, Kenya, Honduras, Morocco, Senegal, Ethiopia, Benin, Zimbabwe, Madagascar, Uganda, Côte d’Ivoire, Afghanistan, Tanzania, Yemen, Republic of the Congo, Ghana, Cameroon, Sudan, Nigeria, Iraq, Democratic Republic of the Congo, Bahrain, Jamaica
Fourth (Strong)	0.2874–0.4384	Taiwan (China), China (depending on the context), South Korea, Israel, Hong Kong, Malaysia, Thailand, Turkey, Philippines, India, Vietnam, El Salvador, Kuwait, Georgia, Sri Lanka, Uzbekistan, Pakistan, Laos, Bangladesh, Azerbaijan, Peru, Nicaragua, Ecuador, Turkmenistan, Botswana, Mozambique, Mongolia
Fifth (Very Strong)	0.4786–1.5394	Uruguay, Colombia, Brazil, Indonesia, Argentina, Egypt, Armenia, Kyrgyzstan, Cambodia, Australia, Kazakhstan, Zambia, Myanmar, Tajikistan, Togo, Venezuela, Angola, Bolivia, Libya, Mauritania, Burkina Faso

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The heatmap of tensile elastic strain for plant-based products in various countries is shown in Figure 2:

6.3.2. Analysis and Discussion of Experimental Results

(1)

On the whole, the tensile elastic strain of the supply chain for this product category is relatively weak. In most countries, the maximum tensile elastic strain values are below 0.3, with an average value of 0.32. This reflects a high spoilage rate and low storage value, typical of seasonal consumer products. The supply chain’s capacity to handle overload services for these products is weak, resulting in low tensile elasticity stability. Enhancing storage properties through secondary or deep processing can improve the stability of the product supply chain’s tensile elasticity.

(2)

The tensile elastic strain of plant product supply chains in Japan, European regions, and countries like Algeria is relatively weak. This indicates limited potential for enhancing the circulation and supply chain capabilities of these product categories in these countries and regions. Specifically, countries such as Russia, the United Kingdom, Belarus, and Italy have tensile elastic strains around 0.05, highlighting poor overall service elasticity of their supply chains. Keeping the overload service of these supply chains below 5% is essential to mitigate a high probability of disruptions. Other countries and regions at this level also face significant impacts on the stability of plant product supply chains, due to overload services, thereby significantly increasing the risk of chain disruptions. Therefore, these countries and regions need to enhance the storage capacity and improve the tensile elastic strain of plant product supply chains to enhance their ability to handle overload services.

(3)

In Southeast Asia, South America, Oceania, and a few countries in Africa, the tensile elastic strain of plant product supply chains is relatively strong. This indicates that these regions generally possess superior overload service capabilities compared to others. Most of these countries are primary producers of plant products, with relatively strong storage service capacities within their supply chains. This enables a greater influx of products into the supply chain, with minimal destabilizing effects from overloaded products. Consequently, these countries and regions should encourage higher production of such products to bolster the overall profitability of their plant product supply chains.

7. Conclusions and Discussion

7.1. Conclusions

By deducing and analyzing the formula for supply chain elastic strain, the following conclusions are drawn:

(1)

Theoretical Conclusion

The amount of the supply chain’s shear elastic strain critical point ${\epsilon }_0^{-}$ has a direct correlation with its maximum shear amount $I_0^{-}$: the larger the former, the higher the latter, and the lower its minimum shear elastic load; whereafter, the supply chain’s elasticity to external interferences, such as declining client base or market downturns, becomes stronger, resulting in enhanced safety and stability. The reverse proposition is equally valid. The amount of the supply chain’s tensile elastic strain critical point ${\epsilon }_0^{+}(\gamma >\alpha )$ has a direct correlation with its maximum tensile amount $\Delta I_0^{+}$: the larger the former, the greater the latter, and the higher its maximum tensile elastic load; whereafter, the supply chain’s elasticity to external interferences, such as abrupt increases in customer demand and sharp market upticks, with a heightened capability for overload operations, leads to superior supply chain stability. The opposite holds true under contrasting conditions. As the supply chain adheres to Condition $\gamma \le \alpha$, the critical point of its tensile elastic strain is ${\epsilon }_0^{+}\in +\infty$, the maximum amount of tensile elastic strain is $\Delta I_0^{+}\in +\infty$, and its maximum tensile elastic load is $I_0^{+}\in +\infty$. Under such circumstances, abrupt increases in customer demand or sharp market upticks do not perturb the supply chain, highlighting its steadfast stability. Inherently, items with diminished damage rates present better overcapacity operational stability and safety within their supply chains than their higher-damage-rate counterparts.

(2)

Practical Implications

For seasonal products, the shear elastic strain values of the product supply chain are generally high, indicating higher profit margins. The reduction in product volume entering the supply chain has minimal impact on its stability, and the risk of chain disruptions is also relatively low.

For products with high storage demands, the tensile elastic strain in producing countries is typically higher than in consuming countries, indicating that producing nations possess stronger storage service capabilities. Additionally, their product supply chains exhibit stronger overload service capabilities compared to consuming countries.

7.2. Discussion Advantages and Limitations

This paper has derived the functional formula for supply chain elastic strain of $\epsilon$, encompassing critical points for both shear elastic strain ${\epsilon }_0^{-}$ and tensile elastic strain ${\epsilon }_0^{+}$. Prior to this, research in this domain predominantly assessed supply chain elasticity by constructing evaluation systems. Evidently, the findings of this paper have introduced a novel perspective to the evaluation of supply chain elasticity. Our computational approach simplifies the process, producing quantified results that more intuitively reflect the relative merits of supply chain elasticity and possess greater generalizability.

The study of supply chain elastic strain in this paper was conducted under the assumption that there were no constraints related to product supply, market demand, and fluctuations in pricing. However, practical scenarios have limited product supplies, capped market demands, and price variations that align with shifts in market dynamics. Thus, future investigations into supply chain elasticity should build upon this research while incorporating factors such as supply, demand, and price fluctuations.

The study of supply chain elastic strain in this paper was based on a singular supply chain scenario. However, supply chains inherently exhibit a networked structure with distinct network characteristics. Hence, subsequent studies on supply chain elasticity should build upon the foundation of this research while incorporating the network features inherent to supply chains. The study in this paper was primarily based on customs import and export data, which led to certain discrepancies in measuring supply chain elastic strain. Consequently, when calculating the elastic strain of product supply chains, it is essential to use more comprehensive and accurate data. This approach will yield more precise results and higher reliability, especially for companies that require detailed information to ensure the accuracy of the calculation results.

The study of this paper offers a methodology for calculating supply chain elastic strain; however, further investigation is needed to determine how to apply the results and establish standards. Additionally, this study explores supply chain elastic strain by considering the supply chain as a unified entity consisting of a networked structure of multiple enterprises. Therefore, further investigations into supply chain elastic strain could analyze the impact of network characteristics based on this study.