A Non-Parametric Approach-Based Trade-Off between Food System Efficiency and Robustness

Abstract

Balancing the efficiency and robustness of food systems is a well-known trade-off process. Over-optimization of efficiency may lead to excessive use of resources. On the other hand, the under-optimization of efficiency may lead to excessive waste of resources. This paper presents a novelly structured approach that integrates two well-suited non-parametric methods for analyzing and balancing the interconnection between the efficiency and robustness of food systems. This approach, which consists of three main steps, provides a theoretical framework and offers practical solutions. First, slacks-based data envelopment analysis (SBM-DEA) is utilized to analyze the efficiency of five food robustness dimensions. Second, the weighted efficiency of these dimensions is computed and analyzed to provide insight into their influence on food system efficiency. Finally, two search methods were developed to identify improving efficiency and robustness opportunities. The outcomes of these methods were analyzed and validated using data from 37 countries, with robustness dimension weights determined via the Analytic Hierarchy Process (AHP). While the first improvement method provided valuable insights, the second method proved more effective in identifying the sources of inefficiency of robustness dimensions.

1. Introduction

The trade-off between the efficiency and robustness of food systems is essential to reduce waste and ensure they resilient enough to withstand shocks and disruptions. Achieving a better balance between efficiency and robustness depends on the level of consistency between the constraints and characteristics of the food systems, such as standards and regulations, food strategy and vision, diversification of resources, and technological innovation. Several managerial approaches to achieving this balance have been explored over recent years, such as cost–benefit analysis, demand forecasting, and resource optimization. Managerial approaches used in food systems share a core set of principles that facilitate the adoption of methods and performance comparison across all industry scales. The selection of a method mainly depends on the specific goal and objectives of the industry, which may vary from improving productivity, enhancing customer satisfaction, improving food security and resilience, etc. While these approaches are widely known for their ability to provide robust solutions, their applicability and effectiveness may be subject to serious challenges relating to data structure analysis, data accessibility, data dynamicity, and estimating relative weights.

Analyzing data structures is crucial for understanding the intricate relationships impacting food system efficiency and robustness. Improper data analysis can yield misleading results that fail to represent a food system’s true performance accurately. For example, neglecting the impacts of climate change, such as water scarcity and crop diversity, might superficially enhance efficiency but compromise long-term robustness. Key methods include outlier detection, data flow, and correlation analysis [1].

Data accessibility ensures the acquisition of adequate datasets for analysis. Incomplete or inappropriate data can lead to high uncertainty and flawed insights. For instance, an incomplete dataset on electricity consumption might understate this factor’s significant impact on food system performance. Engaging various stakeholders is essential for validating relevant factors and dimensions. Reliable sources and databases at local and global levels adopt robust data collection and analysis methods.

Data dynamicity, i.e., constantly changing data, is typical in food systems. Static data fail to provide a comprehensive or forward-looking perspective. Dynamic factors, such as urban population growth and food demand, necessitate methods like time series analysis and system dynamic models to balance efficiency and robustness over time effectively.

Composite indicators are widely utilized to provide a comprehensive overview of food system performance. Composite indicators assess a system’s efficiency or robustness by combining several indicators into a single index, such as the efficiency ratio, robustness, and cost efficiency. However, the relative weights assigned to the indicators (or factors) become essential to a composite indicator’s effectiveness. In such a case, weighting methods that involve stakeholders are more recommended. Individuals and stakeholders with enough knowledge and experience concerning food systems can significantly contribute to assigning relative weights reflecting the importance of indicators (or factors) to a food system’s capacity. This process should be conducted by meeting individuals with enough knowledge and experience regarding food systems. Due to the variety of principles and procedures of existing weighting methods, Section 2 will briefly introduce some of the most known of these methods.

However, overcoming these challenges has become critical to developing more robust methods to capture the relationship between food system efficiency and robustness and provide stakeholders and decision-makers with opportunities to balance these two performance measures. Various parametric methods, such as Stochastic Frontier Analysis (SFA), the Cobb–Douglas Production Function, etc., are used to calculate food system efficiency [2,3]. Non-parametric methods, such as Data Envelopment Analysis (DEA), are highly advantageous for assessing food system efficiency due to their flexibility and robustness. Unlike parametric methods, non-parametric approaches do not require specific assumptions about the functional form of the data, making them better suited to handle the complex, non-linear relationships inherent in food systems [4]. These methods are remarkably robust to outliers and measurement errors, common in agricultural and food production data [5]. Furthermore, non-parametric methods can effectively manage multiple inputs and outputs without needing large sample sizes, making them practical for the varied and often limited datasets typical in food system studies [6]. Hence, this study proposes a non-parametric method to calculate food system efficiency. Concurrently, we will investigate strategies for balancing food system efficiency with robustness.

The significance of the proposed approach mainly lies in its ability to overcome challenges affecting the applications of several existing managerially-based approaches. Below is a list of the distinctive advantages of the proposed approach:

• Utilizing slacks-based data envelopment analysis (SBM-DEA), a well-known non-parametric (distribution-free) method, allows incorporating various indicators (or factors) without imposing distributional assumptions. These methods are also valuable for their robustness to the impacts of outliers and extreme values on the credibility of the findings. These methods are also advantageous due to their ability to handle the influence of outliers and extremes on the credibility of the findings and conclusions.
• Utilizing the input–output-based SBM-DEA model is essential to provide stakeholders with opportunities to incorporate input and output indicators to reveal the resources available and how efficiently these resources are used to achieve the desired food system outcomes.
• Utilizing the Analytic Hierarchy Process (AHP) promotes the incorporation of stakeholders and practitioner expertise. This collaborative approach allows them to assign relative weights to different indicators (or factors) and determine the low-performing robustness dimensions. This participatory feature would enhance the applicability of this approach such that it could be easily extended to and repeatedly used at different food system levels. An indicator’s weight may vary at each level.

The subsequent parts of this paper are summarized below. Section 2 presents a detailed literature review of food system efficiency and robustness and existing weighting methods used in the literature. Section 3 explains the methodology used in this study. Section 4 defines the concepts used in this study, i.e., the Analytic Hierarchy Process (AHP), and the Slacks-based measure Data Envelopment Analysis (SBM-DEA). Section 5 introduces the proposed approach through an illustrative example using 37 food-producing countries. Section 6 discusses the results and their application. Finally, Section 7 reports the conclusions of this study and provides recommendations.

2. Literature Review

Balancing food system efficiency and robustness is crucial for ensuring sustainable and secure food production. The concept of robustness in agriculture involves this system’s ability to withstand disturbances and fluctuations, as highlighted in various research papers [7,8,9]. While efficiency is essential for productivity and profitability [10], solely focusing on efficiency without considering robustness can lead to vulnerabilities in the food system [11]. Wang et al. emphasize the importance of efficiency, profitability, sustainability, and fairness for ensuring the robustness of a food system, proposing the Food System Robustness Index (FSRI) model to evaluate and optimize food systems [8]. The computation of the FSRI involves the utilization of the AHP and correlation analysis. De Goede et al. discuss the concept of robustness in agricultural systems, highlighting the need to navigate vulnerability and stability to achieve optimal robustness states that can cope with environmental fluctuations [9]. Additionally, Napel et al. advocate for using an adaptation model in agriculture, focusing on ameliorating the consequences of disturbances rather than solely aiming for control to enhance sustainability [11].

Over the last decade, several weighting methods have been developed and examined to compute composite indices. Using expert opinions or stakeholder inputs to assign weights to food system indicators (or factors) is common in these methods (see [12,13]). Selecting individuals with enough years of experience is essential to avoid the influence of personal biases and subjective opinions (see [14]). Despite this approach being extensively criticized in the literature; it is still applied in various contexts due to its computational simplicity and applicability.

Using analytical approaches for weight estimation is another promising research trend. These approaches can be used separately or in combination to understand a food system’s performance comprehensively. Several techniques commonly utilized in the literature include the Analytic Network Process (ANP), the Analytical Hierarchy Approach (AHP), and the Technique for Order of Preference by Similarity of Ideal Solution (TOPSIS). Although these methods can be helpful for the involvement of stakeholders, it is essential to consider that they are time-consuming and labor-intensive, especially with a high-dimensional space of indicators (see [15]).

Several weighting methods utilizing statistical principles such as principal component analysis (PCA), factor analysis (FA), and multiple regression analysis (MRA) have been examined and validated to generate relative weights in different research contexts; for more information, see [13,16,17,18]. However, these methods differ from the expert-based weighting methods in their heavy dependence on generating weights from the data. Hence, any violation of the statistical assumptions of the weighting method may significantly affect the accuracy of the estimated weights. PCA is considered more favorable than other statistical models [17] because it relies on actual data variability [14,18]. PCA assumes that the indicators are linearly related and of equal importance.

The MRA methods have also received notable attention recently; for more information, see [1,19,20,21,22,23]. The MRA methods are essential in measuring the efficiency and robustness of food systems when food system performance is modeled using both input and output indicators [24]. The MRA methods help reveal the relationships between input and output indicators and hence facilitate the establishment of the relative weights for each indicator [12]. The last decade has witnessed a notable trend of extending the applications of advanced MRA-based-weighting methods, such as ridge regression, least absolute shrinkage, selection operator (lasso), and elastic-net regression [25,26]. The accuracy of the weights generated by these methods remains an active area of research, particularly in contexts like sustainability and eco-efficiency assessment [24].

However, despite the various weighting methods available, no one can fit all research contexts. One common feature among these methods is that their effectiveness is heavily influenced by the success of involving individuals and stakeholders from the relevant industry with appropriate knowledge and experience.

3. Methodology

This paper introduces a non-parametric-based approach to provide decision-makers and stakeholders with a realistic data-driven tool to effectively balance the trade-off between food system efficiency and robustness while enhancing overall resilience. The approach is structured in a way that includes three modules (see Figure 1). Each module aims to achieve specific objectives, as described below:

Module 1 (Data Integration and Management): This module involves procedures to be followed to develop the I-O impact matrix. This module mainly involves three steps. These are (1) selecting the input-output indicators, (2) developing the AAEEG impact matrix (availability, affordability, economic, environmental, and governance), and (3) weighing the AAEEG dimensions (availability, affordability, economic, environmental, and governance) using the AHP. The fourth step, although not a primary step, checks the adequacy of the impact matrix to ensure the consistency of the data. Some other steps might be added to this module to ensure the consistency and credibility of the impact matrix, such as outlier detection, data imputation, and data normalization.

Module 2 (Optimizing SBM-DEA Model): This module involves discussing the outcomes of the SBM-DEA model. The practitioner should examine the efficiency scores of all Decision-Making Units (DMUs) and categorize the DMUs into two groups: efficient and inefficient DMUs. Comparing the scores of the DMUs across the five dimensions (AAEEG) can help provide a comprehensive insight into the performance of the underlying DMUs. A composite index of the technical efficiency across various robustness dimensions is calculated. The weights for the dimensions are estimated using the AHP.

Module 3 (Investigating Improvement Plans): This module describes the procedures for exploring opportunities for balancing food system efficiency and robustness. The composite index constructed in the previous module is then analyzed using defined criteria to pinpoint the indicators that necessitate intervention to improve their efficiency and, consequently, the overall robustness of the underlying dimension(s). Two methods have been proposed in this study. The roles of stakeholders and practitioners in this module are to determine the extent of improvement required for low-performing indicators (or dimensions), considering this study’s specific goals and objectives.

4. Methods and Techniques

This section contains the methods and concepts used in this research. Two concepts were mainly used, and they are detailed below.

4.1. Slack-Measure-Based DEA (SBM-DEA)

The DEA is a non-parametric data-driven approach that helps assess the effectiveness of decision-making units (DMUs) [27]. A DMU is an entity or system that converts inputs into outputs. The DEA ranks the DMUs based on their efficiency scores. Several DEA models have been proposed and evaluated. The slack-based DEA (SBM-DEA) model is the most widely known among these models. The SBM-DEA is an advanced DEA model using the slack term ($s^{-}\ge 0\mathrm{and}{s}^{+}\ge 0$, $s^{-}\in$ ${\mathbb{R}}^m$ and $s^{+}\in$ ${\mathbb{R}}^s$) to identify ineffective DMUs and highlight the degree to which these DMUs can adjust their inputs or outputs to reach the point where a DMU can decrease its inputs or increase outputs without impacting its efficiency score [27,28,29]. The SBM-DEA model compares a DMU’s inputs and outputs to those of its peers and determines how much the unit falls short of achieving the same output level with the same input level. The SBM-DEA approach helps identify an inefficient DMU and benchmark its efficiency performance versus its counterpart DMUs.

Let $\mathbb{Z}$_(n,(m+s)) be the input–output matrix of n DMUs; then, the input and output score vectors of the $j$th DMU_j ($j$ = 1, …,$n$) are denoted as X = $x_{ij}(i=1,\dots ,$m$)\in {\mathbb{R}}^{m\times n}$ and Y = $y_{rj}(r=1,\dots ,$s$)\in {\mathbb{R}}^{s\times n}$, respectively. The special structure of the $\mathbb{Z}$_(n,(m+s)) matrix is as follows:

		Input Scores					Output Scores
DMU1	$x_{11}$	$x_{21}$	$\dots$	$x_{(m-1)1}$	$x_{m1}$	$y_{11}$	$y_{21}$	$\dots$	$y_{\left(s-1\right)1}$	$y_{s1}$
DMU2	$x_{12}$	$x_{22}$	$\dots$	$x_{\left(m-1\right)2}$	$x_{m2}$	$y_{12}$	$y_{22}$	$\dots$	$y_{\left(s-1\right)2}$	$y_{s2}$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
DMUn−1	$x_{1\left(n-1\right)}$	$x_{2(n-1)}$	$\dots$	$x_{\left(m-1\right)(n-1)}$	$x_{m(n-1)}$	$y_{1(n-1)}$	$y_{2(n-1)}$	$\dots$	$y_{\left(s-1\right)(n-1)}$	$y_{s(n-1)}$
DMUn	$x_{1n}$	$x_{2n}$	$\dots$	$x_{\left(m-1\right)n}$	$x_{mn}$	$y_{1n}$	$y_{2n}$	$\dots$	$y_{\left(s-1\right)n}$	$y_{sn}$

This paper uses the measure of efficiency ($\tau )$ using $s^{-}\ge 0\mathrm{and}{s}^{+}\ge 0$, $s^{-}\in$ ${\mathbb{R}}^m$ and $s^{+}\in$ ${\mathbb{R}}^s$, which are referred to as slacks and represent input surplus and output shortage, respectively [29,30].

$$\tau ={\displaystyle \frac{1-\frac{1}{m}{\sum }_{i=1}^m{s}_i^{-}/x_{io}}{1+\frac{1}{s}{\sum }_{r=1}^s{s}_r^{+}/y_{ro}}}$$

(1)

where 0 <$\tau$ $\le$ 1. According to [29], the DMU can be described using Equations (2) and (3) below:

$$\lambda \mathrm{X}+s^{-}=x_0$$

(2)

$$\lambda \mathrm{Y}-{\mathrm{s}}^{+}={\mathrm{y}}_0$$

(3)

where $\lambda \ge$0 is a positive vector ${\mathbb{R}}^j$, $\mathrm{X}>0$, and $\mathrm{Y}>0$. Considering Equations (1)–(3), the SBM is defined as below [29]:

$$\mathrm{Minimize}\tau ={\displaystyle \frac{1-\frac{1}{m}{\sum }_{i=1}^m{s}_i^{-}/x_{io}}{1+\frac{1}{s}{\sum }_{r=1}^s{s}_r^{+}/y_{ro}}}$$

(4)

The efficiency score $\tau$ for each DMU indicates the extent to which it can achieve the best possible performance for each performance measure, subject to its input and output requirements while considering any slack in the input and output variables. However, according to [29], a DMU ($x_0$,$y_0$) is considered an efficient DMU when it fully utilizes its resources to achieve its desired outputs. For further reading and information regarding the slack measure properties, see [29,30,31,32,33].

4.2. Analytic Hierarchy Process (AHP)

The AHP is a structured decision-making approach developed by Saaty [34,35] to prioritize alternatives based on multiple criteria and sub-criteria. The AHP is widely used for weighing and prioritizing criteria in decision-making processes. The AHP uses a hierarchical structure to represent a problem and allows decision-makers to compare and prioritize different criteria and alternatives based on their relative importance (see [36,37,38]). Figure 2 illustrates the typical hierarchal structure of the AHP, with $n$ decision criteria and $m$ generated alternative solutions.

The consistency ratio (CR) is an essential tool in the AHP used to evaluate the consistency of pairwise comparisons made by decision-makers, ensuring their judgments are reliable and consistent. The CR is computed by dividing the consistency index (CI) by the random consistency index (RI); for more information, see Equation (5). Ideally, the is resulting ratio should be less than 0.1 for the comparison to be considered consistent. When CR ≤ 0.1, the judgment matrix has acceptable consistency; otherwise, the decision-maker may need to revise their judgments [39].

$$\mathrm{CR}={\displaystyle \frac{\mathrm{Consistency}\mathrm{Index}}{\mathrm{Random}\mathrm{Consistency}\mathrm{Ratio}}}={\displaystyle \frac{\mathrm{CI}}{\mathrm{RI}}}$$

(5)

The CI is calculated using Equation (6), where $\gamma$_max is the maximum eigenvalue obtained by multiplying the total sums vector by the weighted values vector for paired comparisons and $p$ represents the number of decision criteria being compared.

$$\mathrm{CI}={\displaystyle \frac{{\gamma }_{\max }-p}{p-1}}$$

(6)

When CI = 0, the judgment matrix has completion consistency. As the CI increases, the degree of consistency decreases. The RI is a reference value based on the size of the number of criteria being compared; for more information, see [40]. The RI values for matrices of different sizes are pre-defined and can be found in tables; for instance, see [34].

5. Illustrative Example

This section provides a practical illustration to demonstrate the operational and mathematical procedures of the four modules shown in Figure 1.

5.1. Selecting Robustness Indicators

The effectiveness of the proposed approach heavily relies on carefully selecting the appropriate inputs and outputs to measure the efficiency of food robustness dimensions. Therefore, two selection approaches were followed in this study. First, several previous research works conducted concerning the same context were reviewed, and the most relevant inputs and outputs indicators were selected. Second, several experts were consulted to assess and validate the indicators’ relevance to the efficiency of food robustness dimensions.

This example uses a set of 30 indicators (25 inputs and 5 outputs) of 37 food-producing countries.

Table 1. Robustness dimensions and indicators.

Dimensions	Indicators	Source	Symbol	Units
Availability	1. Diversity of domestic production: food calories	FAO	OAV1	Kcal (1)
	2. Diversity of import sources: food calories	FAO	IAV1	Kcal
	3. Diversity of stocks: food calories	FAO	IAV2	Kcal
	4. Availability of electricity access	World Bank	IAV3	% of population
	5. Land area equipped for irrigation	FAO	IAV4	103 Ha (2)
	6. Availability of irrigation facility land	FAO	IAV5	103 Ha
Accessibility	7. Cereal yield	World Bank	OAC1	Kg/Ha
	8. Adult literacy	World Bank	IAC1	%
	9. Unit labor cost	OECD	IAC2	EUR
	10. Mobile phone subscribers	World Bank	IAC3	Per 100 people
	11. Total number of paved and unpaved roads	Eurostat	IAC4	Km
	12. Short-term household credit	OECD	IAC5	EUR Bn.
	13. Access to electricity (rural areas)	World Bank	IAC6	% of rural population
Economic	14. Consumer price index	World Bank	OEC1	(2010 = 100)
	15. Affordability of daily calorie intake	FAO	IEC1	%
	16. Personal disposable income	Eurostat	IEC2	EUR
	17. Expenditure on food per capita	World Bank	IEC3	EUR 103
	18. Ratio of urban population	World Bank	IEC4	%
	19. Energy imports	EUROPA	IEC5	%
	20. Household annual disposable income	OECD	IEC6	USD 1000
Environmental	21. Air quality index	World Air Quality Report	OEV1	µg/m3
	22. Renewable freshwater resources	Eurostat	IEV1	106 m3
	23. Energy used in agriculture	FAOSTAT	IEV2	TJ (3)
	24. Purchased electricity consumption	FAOSTAT	IEV3	TJ
	25. Internal renewable water resources	World Bank	IEV4	Bn. m3
	26. Annual rate of net forest change	FAOSTAT	IEV5	%
	27. Water used for drinking	EUROPA	IEV6	liters per capita
Governance	28. Transportation infrastructure value	OECD	OGN1	EUR
	29. Protection of intellectual property	EUROPA	IGN1	BoP (4), USD
	30. Government spending	OECD	IGN2	% of GDP (5)

⁽¹⁾ Kcal = Kilocalories; ⁽²⁾ Ha = Hectare; ⁽³⁾ TJ = Terajoule; ⁽⁴⁾ BoP = Balance of Payment; ⁽⁵⁾ GDP = Gross Domestic Product.

reports these indicators along with their symbols and measurement units. This study employs data from various sources, including the Food and Agriculture Organization (FAO), the World Bank, the Organization for Economic Co-operation and Development (OECD), Eurostat, and global economic data statistics. The selected indicators were distributed over five robustness dimensions (AAEEG) based on managerial and computational considerations. The selected input-output indicators are aligned with the 17 sustainable development goals (SDGs). This study provides governmental policymakers with data-driven advice for establishing a regulatory environment, investing in infrastructure and research, and allocating resources to support the development of robust and sustainable food systems.

The availability dimension consists of five input indicators and one output indicator used to measure the ability of a food system to provide a sufficient and reliable food supply to society. The availability dimension comprises six inputs and one output representing food production, distribution, infrastructure, and logistics. The accessibility dimension consists of six input indicators and one output indicator measuring the capability of individuals to access safe and sufficient food. The economic dimension consists of six input indicators and one output indicator used to measure the economic outputs and externalities of a food system. The environmental dimension consists of six input indicators and one output indicator used to measure the impacts of food system activities on the surrounding natural environment. Finally, the governance dimension consists of two input indicators and one output indicator describing the interactions between food production, processing, and consumption concerning the different drivers of food systems.

5.2. Developing Impact Matrix

The existing data for the $\mathbb{Z}$ matrix were transformed into a common scale using the feature-scaling method. This step is important to ensure the indicators are uniformly represented and comparable across different dimensions. The normalized scores of the input and output indicators can be computed using Equations (7) and (8), respectively:

$$x_{ij}^N=a_0+{\displaystyle \frac{\left(x_{ijk}-{\mathrm{M}\mathrm{i}\mathrm{n}}_i\right)\left(a_1-a_0\right)}{\left({\mathrm{M}\mathrm{a}\mathrm{x}}_i-{\mathrm{M}\mathrm{i}\mathrm{n}}_i\right)}};i=\mathrm{1,2},\dots ..,m,j=\mathrm{1,2},\dots ,n$$

(7)

$$y_{rj}^N=b_0+{\displaystyle \frac{\left(y_{rj}-{\mathrm{M}\mathrm{i}\mathrm{n}}_r\right)\left(b_1-b_0\right)}{\left({\mathrm{M}\mathrm{a}\mathrm{x}}_r-{\mathrm{M}\mathrm{i}\mathrm{n}}_r\right)}};r=\mathrm{1,2},\dots ..,s,j=\mathrm{1,2},\dots ,n$$

(8)

where (${\mathrm{M}\mathrm{i}\mathrm{n}}_i$, ${\mathrm{M}\mathrm{a}\mathrm{x}}_i$) and (${\mathrm{M}\mathrm{i}\mathrm{n}}_r$, ${\mathrm{M}\mathrm{a}\mathrm{x}}_r$) represent the maximum and minimum scores of the i^th input and r^th output indicators. The pairs ($a_0,a_1)$ and ($b_0,b_1)$ are predetermined constants for the range of $x_{ij}$ and $y_{rj}$, respectively. The two pairs of constants were set to their customary values (0, 1), and then Equations (7) and (8), given above, were used to generate the normalized pairs $x_{rj}^N(i=1,\dots ,m)$ and $y_{rj}^N(r=1,\dots ,s)$ of the $\mathbb{Z}$ matrix. The new version of the $\mathbb{Z}$ matrix will be referred to as ${\mathbb{Z}}_k^N$.

5.3. Weighting AAEEG Dimensions

This paper uses the AHP method to generate weights for the robustness dimensions. The initial step of the AHP is the development of a pair-wise comparison matrix. This comparison is critical to the weight scores generated by the AHP. There are two different approaches, in practice, to conducting this comparison. These are indicator-based or dimension-based pair-wise approaches. The indicator-based approach follows a simple mechanism in which the stakeholders and experts compare all possible pairs of indicators and assign scores representing the relative importance of each indicator relative to the other. The dimension-based approach follows the same mechanism, but the pairs are formed from the dimensions rather than the indicators. The stakeholders and experts compare pairs of dimensions and assign a score representing the importance of each dimension relative to the other.

For the convenience of experts and stakeholders, this study follows the dimension-based approach. First, the relative importance of the five dimensions from the governance expert’s perspective is determined to develop the pairwise comparison matrix. Then, a group of experts and stakeholders in the relevant field provide their ratings using a scale from 1/9 to 9, where 9 means “absolutely more importance” and 1/9 means “absolutely low importance” [41,42].

Table 2. Governance-expert-based AHP pair-wise comparison matrix.

	Accessibility	Availability	Economic	Environmental	Governance
Accessibility	1.000	1.000	0.143	0.333	0.200
Availability	1.000	1.000	0.143	0.333	0.200
Economic	7.000	7.000	1.000	5.000	0.333
Environmental	3.000	3.000	0.200	1.000	0.3333
Governance	5.000	5.000	3.000	3.000	1.000

shows the results of the dimension-based pairwise comparison.

The relative weights (${\omega }_k$) of each of the AAEEG dimensions were estimated and are reported in

Table 3. Governance-expert-based AHP weights.

Dimension	Dimension Weight
Accessibility	5.20%
Availability	5.20%
Economic	33.90%
Environmental	12.10%
Governance	43.60%

. The results show that the governance dimension (43.60%) leads in the list of dimensions, followed by the economic dimension (33.90). The results also show that the rest of the dimensions (availability, affordability, and environmental) significantly differ from these two dimensions.

Following this, we computed the CI ratio using Equation (6) to check the consistency of the pairwise comparison. In calculating the weighted sum value of each dimension, ${\omega }_k$ (k = 1, 2, …, 5) represents the weight of the kth dimension (Table 3), and $C_{ij}$ represents the priority index of the ith pair of dimensions (

Table 4. The weighted sum values matrix.

	Accessibility	Availability	Economic	Environmental	Governance	${\mathit{W}\mathit{S}\mathit{V}}_{\mathit{k}}$
Accessibility	0.052	0.052	0.048	0.040	0.087	0.14
Availability	0.052	0.052	0.048	0.040	0.087	0.14
Economic	0.364	0.364	0.339	0.605	0.145	6.89
Environmental	0.156	0.156	0.068	0.121	0.145	0.91
Governance	0.260	0.260	1.017	0.363	0.436	7.41

).

Then, the weighted sum value (${WSV}_k$) of the kth dimension was calculated using Equation (9).

$${WSV}_k=\sum _{i=1}^5\sum _{j=1}^5{C}_{ij}\times {\omega }_k\mathrm{for}\mathrm{all}\mathrm{values}\mathrm{of}k$$

(9)

The maximum eigenvalue ($\gamma$_max) was then calculated as follows:

$${\gamma }_{max}={\left(\sum _{i=1}^5{\displaystyle \frac{{WSV}_j}{C_{ij}}}\right)}^{-n}={\left(26.82\right)}^{-5}=5.36$$

(10)

The CI value was calculated using Equation (6), as follows:

$$\mathrm{CI}={\displaystyle \frac{{\gamma }_{\max }-p}{p-1}}={\displaystyle \frac{5.36-5}{5-1}}=0.09$$

(11)

Finally, we calculated the CR as follows:

$$\mathrm{CR}={\displaystyle \frac{\mathrm{CI}}{\mathrm{RI}}}={\displaystyle \frac{0.09}{1.12}}=0.081$$

(12)

A CR value of 1.12 value is often used at n = 5. Since the calculated CR of 0.08 is less than 0.1, we can conclude that the pair-wise comparison matrix is adequate.

5.4. Analyzing the Impact Matrix

This section evaluates the adequacy of the impact matrix to ensure data consistency. Five impact matrices (${\mathbb{Z}}_k;k=\mathrm{1,2},\dots 5)$ were developed in this study, and each matrix has a size of 37 $\times$ ($m_k$ + $s_k$), where $n$ = 37 is the total number of DMUs, and ($m_k$ + $s_k$) is the sum of input and output indicators of the kth dimension. The terms $x_{ijk}$ and $y_{rjk}$ represent the impact of the i^th input indicator and the r^th output indicator of the j^th DMU of the kth dimension, where$i$= 1, 2, …, $m_k$, $r$ = 1, 2, …, $s_k$, and $k$ = 1, 2, …, 5.

Ensuring an adequate sample size is crucial, and it depends on various factors, including the number of DMUs, inputs, and outputs. In the literature, two rules have been extensively discussed with respect to assessing the adequacy of a sample size, as proposed by Kumar et al. [43]. The first rule states that the sample size should be greater than or equal to the product of the inputs and outputs (i.e., $n\ge m\times s$). The second rule, according to Kumar et al. [43], suggests that the number of DMUs should be at least three times the sum of inputs and outputs (i.e., $n\ge 3\left(m+s\right)$).

Table 5. Evaluation of the adequacy of the impact matrices (n = 37).

$\mathit{k}$	Dimension	${\mathit{m}}_{\mathit{k}}$	${\mathit{s}}_{\mathit{k}}$	Rule 1	Rule 2	$(\mathbf{n}\ge {\mathit{m}}_{\mathit{k}}\times {\mathit{s}}_{\mathit{k}})$	$\mathbf{n}\ge {3(\mathit{m}}_{\mathit{k}}+{\mathit{s}}_{\mathit{k}})$	Result
1	Availability	5	1	5	18	Yes	Yes	Pass
2	Accessibility	6	1	6	21	Yes	Yes	Pass
3	Economic	6	1	6	21	Yes	Yes	Pass
4	Environmental	6	1	6	21	Yes	Yes	Pass
5	Governance	2	1	2	9	Yes	Yes	Pass

reveals the adequacy of the impact matrix of the robustness dimensions involved in this study.

5.5. SBM-DEA Output Analysis

The SBM-DEA model was utilized to estimate each dimension’s efficiency ($T_E$) and slack (S_E) scores. The scores were collected to form the efficiency, $\mathbb{E}$_{((37,1), k)}, and slack, $\mathbb{S}$_{(37,1, k)}, lists. The $T_E$ scores measure the capacity of each DMU to utilize resources to produce outputs relative to the other DMUs. The $T_E$ scores assume a value from 0 to 1, where 1 indicates an efficient DMU, and 0 indicates an inefficient DMU. The S_E scores measure how each DMU can reduce inputs or increase output to enhance the efficiency relative to the other DMUs. To help decision-makers focus their efforts on a subset of DMUs that require urgent intervention, we used the quartile method to classify the inefficient DMUs into two categories (inefficient and moderately efficient). We define inefficient DMUs as countries for which $T_E$< Q₁, while moderately efficient DMUs are those DMUs for Q₁ < $T_E$< Q₃.

Table 6. $T_E$ Statistics regarding the availability dimension.

Statistic	$\mathbb{E}$((37,1), 1)	${\mathbf{Q}}_1<{\mathit{T}}_{\mathit{E}}$ < Q3	${\mathbf{T}}_{\mathbf{E}}\le$ Q1
No. of DMUs	37	16	9
Minimum ${\mathrm{T}}_{\mathrm{E}}$	0.425	0.513	0.425
Maximum ${\mathrm{T}}_{\mathrm{E}}$	1.000	0.998	0.511
1st Quartile (${\mathrm{Q}}_1$)	0.513	0.556	0.448
3rd Quartile (${\mathrm{Q}}_3$)	1.000	0.680	0.468
Average (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}1}})$	0.718	0.649	0.465
Standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}1}}$)	0.228	0.131	0.029

reports the statistics on the $T_E$ scores with respect to the availability dimension. The statistics show that the availability dimension has average efficiency (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}}})$ equal to 0.718 and a standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}}}$) equal to 0.228. The results also reveal that 12 countries achieved the maximum efficiency score. These countries are usually referred to as Frontiers. Other DMUs, that is, the majority (67.4%) of the DMUs, achieved efficiency scores ranging from low efficiency (9 DMUs) to moderate efficiency (16 DMUs). This percentage highlights that most countries are not effectively converting resources into sufficient food. The minimum score of the ${\mathrm{T}}_{\mathrm{E}}$ achieved in this dimension is 42.5%, meaning that at least one DMU could not utilize (1 − 0.425)% = 57.47% of the available resources. Some factors that might have affected the ${\mathrm{T}}_{\mathrm{E}}$ scores are the food system typology, technological advancement, and climate change. However, the corresponding decision-makers can still improve food availability, especially for the inefficient DMUs, by examining several input-based strategies, such as food waste management, sustainable agriculture, and diversifying food resources.

Table 7. $T_E$ statistics of the accessibility dimension.

Statistic	$\mathbb{E}$((37,1), 1)	${\mathbf{Q}}_1<{\mathit{T}}_{\mathit{E}}$ < Q3	${\mathbf{T}}_{\mathbf{E}}\le$ Q1
No. of DMUs	37	14	10
Minimum ${\mathrm{T}}_{\mathrm{E}}$	0.457	0.569	0.457
Maximum ${\mathrm{T}}_{\mathrm{E}}$	1.000	1.000	0.556
1st Quartile (${\mathrm{Q}}_1$)	0.556	0.634	0.499
3rd Quartile (${\mathrm{Q}}_3$)	1.000	0.761	0.529
Average (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}1}})$	0.769	0.739	0.511
Standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}1}}$)	0.217	0.156	0.030

reports the statistics regarding the $T_E$ scores in the availability dimension. The statistics show that the accessibility dimension’s average (${\mu }_{T_{E2}}$ = 0.769) slightly outperforms the availability dimension’s average (${\mu }_{T_{E1}}$ = 0.718), meaning that the corresponding countries are more efficient in utilizing resources relevant to food accessibility than food availability. The number of countries that achieved maximum efficiency in this dimension is 13. Table 7 shows that 27.70% of the DMUs are classified as inefficient and 37.78% are classified as moderate-efficient DMUs. The minimum score of the $T_E$ achieved in the accessibility dimension is 45.7%, meaning that at least one DMU could not utilize (1 − 0.457)% = 54.3% of its accessibility resources, a value slightly less than the ratio in the availability dimension. Even though these two dimensions show almost similar efficiency scores, the decision-makers may require different improvement strategies due to the distinct nature of the context of these dimensions. Food availability focuses more on the food supply, such as food production, local and national food stocks, and agricultural land areas. Food accessibility is more focused on the ability of individuals and society to obtain enough food, relating to factors such as household income, food prices, and food market diversity.

Table 8. $T_E$ Statistics on the economic dimension.

Statistic	$\mathbb{E}$((37,1), 1)	${\mathbf{Q}}_1<{\mathit{T}}_{\mathit{E}}$ < Q3	${\mathbf{T}}_{\mathbf{E}}\le$ Q1
No. of DMUs	37	9	21
Minimum ${\mathrm{T}}_{\mathrm{E}}$	0.424	0.424	0.478
Maximum ${\mathrm{T}}_{\mathrm{E}}$	1.000	0.470	1.000
1st Quartile (${\mathrm{Q}}_1$)	0.478	0.428	0.589
3rd Quartile (${\mathrm{Q}}_3$)	1.000	0.459	0.800
Average (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}1}})$	0.697	0.447	0.704
Standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}1}}$)	0.218	0.018	0.158

reports the statistics regarding the T_E scores in the economic dimension. The number of inefficient DMUs in this dimension is 21. This explains the low T_E average score (${\mu }_{T_{E3}}$ = 0.637) in the economic dimension compared with the scores in the previous two dimensions. However, 81.08% of the countries were observed to be inefficient to moderately efficient. This percentage highlights the poor performance of most countries in this dimension. The minimum score of the $T_E$, in the economic dimension, 48.8% means that at least one DMU could not effectively utilize (1 − 0.424)% = 57.6% of its resources, a value less than the ratio in the two previous dimensions. Considering the high priority of this dimension (33.90%), as seen in Table 3, we can surely say that the economic dimension requires more attention from decision-makers than the previous two dimensions.

Table 9. $T_E$ Statistics regarding the environmental dimension.

Statistic	$\mathbb{E}$((37,1), 1)	${\mathbf{Q}}_1<{\mathit{T}}_{\mathit{E}}$ < Q3	${\mathbf{T}}_{\mathbf{E}}\le$ Q1
No. of DMUs	37	10	17
Minimum ${\mathrm{T}}_{\mathrm{E}}$	0.075	0.075	0.165
Maximum ${\mathrm{T}}_{\mathrm{E}}$	1.000	0.161	1.000
1st Quartile (${\mathrm{Q}}_1$)	0.161	0.100	0.185
3rd Quartile (${\mathrm{Q}}_3$)	1.000	0.151	0.435
Average (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}1}})$	0.475	0.123	0.373
Standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}1}}$)	0.385	0.031	0.270

reports the statistics on the T_E scores in the environmental dimension. The average T_E score in this dimension is 0.475. This value is the lowest among all the previous dimensions. There are 17 DMUs in the inefficient category and 10 DMUs in the moderately efficient category, meaning that 72.97% of the countries failed to convert their environmental resources into outputs efficiently. These numbers stress the need for these countries to adopt ecological practices to enhance food system efficiency and robustness. The minimum score of the $T_E$ achieved in the environmental dimension is 7.5%. This is a very low score. However, investigating the factors beyond this low-efficiency performance requires the collaboration of food authorities and stakeholders at the country level. Some factors that deserve to be considered are inefficient farming practices, improper food system typologies, and lack of food technology advancement.

Table 10. $T_E$ Statistics regarding the governance dimension.

Statistic	$\mathbb{E}$((37,1), 1)	${\mathbf{Q}}_1<{\mathit{T}}_{\mathit{E}}$ < Q3	${\mathbf{T}}_{\mathbf{E}}\le$ Q1
No. of DMUs	37	9	19
Minimum ${\mathrm{T}}_{\mathrm{E}}$	0.365	0.365	0.457
Maximum ${\mathrm{T}}_{\mathrm{E}}$	1.000	0.448	1.000
1st Quartile (${\mathrm{Q}}_1$)	0.457	0.394	0.496
3rd Quartile (${\mathrm{Q}}_3$)	1.000	0.435	0.652
Average (${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{E}1}})$	0.659	0.414	0.613
Standard deviation (${\mathsf{\sigma }}_{T_{\mathrm{E}1}}$)	0.247	0.030	0.176

reports the statistics regarding the T_E scores in the governance dimension. According to the AHP results, the dimension of governance is the most important, with a priority of 43.60%. The governance role is crucial to developing food system strategies, promoting research and development, and addressing the implications of a food system’s regulations. Despite the highest importance of the governance dimension for food system robustness, the average of its $T_E$ scores (${\mu }_{T_{E5}}$ = 0.650) is less than the availability and accessibility dimensions, having the lowest priority. This finding emphasizes the need to find practical solutions to enhance the utilization of governance inputs and maintain appropriate food robustness and resilience levels.

5.6. Estimating Weighted Efficiency Score

This section further analyzes the efficiency scores of the AAEEG robustness dimensions. Five impact matrices (${\mathbb{Z}}_k;k=\mathrm{1,2},\dots 5)$ were developed in this study. Each matrix has a size of 37 $\times$ ($m_k$ + $s_k$), where $n$ = 37 is the total number of DMUs, and ($m_k$ + $s_k$) is the sum of the input and output indicators of the kth dimension. The terms $x_{ijk}$ and $y_{rjk}$ represent the impact of the i^th input indicator and the r^th output indicator of the j^th DMU of the kth dimension, where $i$ = 1, 2, …, $m_k$, $r$ = 1, 2, …, $s_k$, and $k$ = 1, 2, …, 5.

The weighted-SBM-DEA optimization model was then applied for all sub-matrices separately to calculate the efficiency of the DMUs. The weight of each input ($I_i^k$) and output ($O_i^k$) indicator in the $k$th dimensions were distributed using Equation (13) below:

$$I_i^k={\displaystyle \frac{1}{m_k}}{\mathrm{and}O}_i^k={\displaystyle \frac{1}{s_k}},i=\mathrm{1,2},\dots ,k$$

(13)

To determine the aggregated efficiency performance of each dimension, we first calculate the weighted sum of the T_E in each dimension as follows:

$${WSI}_i^k={\omega }_k{T_E}_i^k;i=\left(\mathrm{1,2},\dots ,n\right);k=\left(\mathrm{1,2},\dots ,5\right)$$

(14)

where ${\omega }_k$ is the weight of the kth dimension calculated using the AHP. The aggregated efficiency of the $k$th dimension ($A^k)$ can be computed as follows:

$$A^k={\displaystyle \frac{1}{n}}\sum _{i=1}^n{WSI}_i^k;i=\left(\mathrm{1,2},\dots ,n\right);k=\left(\mathrm{1,2},\dots ,5\right)$$

(15)

Figure 3 shows the distribution of the ${WSI}_i^k$ and $A^k$ for the robustness distributions. The results show that the availability dimension is the lowest ($A^k$ = 0.037) among the other dimensions. Economic and governance are the best dimensions. These results can be explained by the high weights of the economic and governance dimensions and the low weights of the accessibility and availability dimensions.

5.7. Finding Improvement Strategies and Applying Improvements

On further analyzing the weighted efficiency scores of the DMUs and AAEEG dimensions, two approaches are suggested in this paper to identify opportunities for efficiency improvement. These are (1) an inefficient-dimension-based approach and (2) a criterion-based approach. The first approach initially identifies the dimensions with the lowest $A^k$; then, the inefficient DMUs in this dimension are selected for efficiency improvement. The main role of stakeholders and experts in this approach focuses on prioritizing and validating the set of resource indicators (inputs or outputs) in each DMU’s inefficient subset of DMUs. The second approach uses a specific criterion to identify the inefficient subset of DMUs across all the dimensions, for instance, the DMUs with the highest frequency of being inefficient among all the dimensions. The slack scores of the indicators of the selected DMUs are then analyzed to reveal better opportunities for efficiency improvement.

5.7.1. Approach One: Inefficient-Dimension-Based Improvement

This approach follows a hierarchal sequence in identifying the opportunity for improvement. The dimension with the minimum sum of ${WSI}_i^k$ or $A^k$ is first identified. Once it is determined, the practitioner starts to investigate the DMU(s)’s performance in the corresponding dimension and determines $i$^th DMU with a minimum (${WSI}_i^k$). The final step involves defining the improvement strategy. However, the involvement of industry and government representatives is essential to the outcome of this step.

Three distinct approaches can be applied to enhance the efficiency performance of DMUs within this system. The first is the input-oriented approach, which focuses on reducing the input while keeping the output at its current level. The second is the output-oriented approach, which focuses on increasing the output while maintaining the same level of input. The third is the mix-oriented approach, which combines the two previous approaches. The approach selected from these three orientations might be based on the intervention of decision-makers and stakeholders.

In this example, the input-oriented approach is used. The initial analysis of the ${WSI}_i^k$ or $A^k$ scores (see Figure 3) showed that the availability dimension is the least efficient (${WSI}_i^1=$ 1.381, and $A^1$ = 0.037). The DMUs within the availability dimension were also analyzed, and three DMUs—Italy, Portugal, and the Netherlands—were identified as having better opportunities to promote their efficiency scores in the availability dimension.

A correlation analysis was conducted to understand how indicators are related to each other so that we could target them to optimize robustness efficiency. From the correlation analysis of the availability dimensions (see

Table 11. Correlation measurements between all pairs of indicators in the availability dimension.

	OAV1	IAV1	IAV2	IAV3	IAV4	IAV5
OAV1	1	−0.444	0.152	−0.788	0.343	0.19
IAV1	−0.444	1	0.064	0.131	0.043	0.104
IAV2	−0.152	0.064	1	0.221	0.537	0.092
IAV3	−0.788	0.131	0.221	1	0.413	−0.171
IAV4	−0.343	0.043	0.537	0.413	1	0.028
IAV5	0.19	0.104	0.092	−0.171	0.028	1

), it was inferred that there is a heavy positive correlation between the indicators “Land area equipped for irrigation” and “Diversity of food in stocks,” with a coefficient of 0.53. This positive correlation means that when one indicator increases, the other increases. Similarly, the indicator “Land area equipped for irrigation” also positively correlates with “Diversity of food in stocks” with a coefficient of 0.41. A negative correlation can also be found between the indicators. The indicator “Diversity of domestic production” is negatively correlated with two indicators, with one being “Availability of electricity access” and the other being “Diversity of import sources”, with correlation coefficients of −0.78 and −0.44, respectively.

As mentioned, selecting indicators should be completed in coordination with stakeholders to incorporate their demands and expertise to optimize the aggregated score. However, the investigations found that the “Land area equipped for irrigation” or IAV4 is the more valuable indicator for enhancing food system efficiency and robustness. Note that regardless of which improvement approach is followed, the reduction in inputs or increase in outputs should not exceed the slack scores of the targeted indicators.

Table 12. Suggested improvement plans.

DMU	Indicator	Slack Reduction, %	Saved, Ha
Italy	IAV4	70	4831
The Netherlands		70	704.83
Portugal		40	380.61

shows the suggested improvement plans. Several strategies for enhancing robustness using the unutilized land area can be suggested and discussed with decision-makers. Some of these strategies include promoting crop diversification, implementing sustainable food production, and increasing livestock to meet demand. Since the DMUs used in this study are countries, we can say Italy has saved the most land compared with the others.

The SBM-DEA was applied to determine the ${WSI}_i^1$ and $A^1$; for more information, see Figure 4. The results show a slight improvement in both the ${WSI}_i^1$ and $A^1$ of the availability dimension. Despite this relatively small improvement in efficiency performance, considering the cost of land area equipped for irrigation, one should expect a significant cost reduction, especially in the European region.

However, the effectiveness of approach one may be influenced by several factors, such as the number of DMUs involved in the improvement, the weight of the chosen dimension, and the input or output reduction percentage. With this in mind, another approach is suggested in this study, which is mainly to overcome these factors’ impact on the improvement plan’s effectiveness.

5.7.2. Approach Two: A Criterion-Based Improvement

This approach uses a specific criterion to identify the inefficient DMUs across all dimensions. Its search range is more comprehensive than that of the first approach. However, one possible challenge of using this criterion is that the number of DMUs might be significantly high. To overcome the high-dimensionality issue, this study uses a combined criterion; for more information, see Equation (16).

$${DMU}_{in}=\left\{T_E<0.2and{DMU}_{in}^f\ge 2\right\}$$

(16)

where ${DMU}_{in}$ is the subset of inefficient DMUs, and ${DMU}_{in}^f$ is the number of times that a DMU is classified as efficient across all dimensions. Unlike approach one, the two main advantages of this criterion are its ability to control the number of DMUs for improvement and its coverage of all the robustness dimensions.

The results showed that seven DMUs revealed scores meeting the Equation (16) criterion. An improvement plan with equal slack reduction is suggested to examine the impact of the second approach.

Table 13. Suggested improvement plan based on approach two.

DMU	Dimension	Indicator	Slack Reduction, %
Iceland	Economic	IEC3	50
Luxembourg	Environment	IEV1	50
	Governance	IGV1	50
Macedonia	Environment	IEV1	50
Montenegro	Governance	IGV1	50
	Economic	IEC2	50
Romania	Economic	IEC2	50
Spain	Accessibility	IAC4	50
	Availability	IAV4	50
Sweden	Governance	IGT1	50

shows the targeted dimensions and indicators.

Table 13 shows the improvement plan. The target inputs of all the DMUs involved in this plan are 50% of their original values. Other percentages of slack reduction might be used, considering that the maximum limit of the target reduction is specified by the slack scores of each indicator.

Figure 5 illustrates the distribution of the robustness dimension’s efficiency after the improvement using the second approach. The economic, environmental, and governance-weighted averages have increased marginally. The availability dimension has shown efficiency performance similar to that yielded by approach one.

The results above show the outperformance of the second criterion compared with the criterion used in approach one. Despite this, it should be noted that setting the separation limit might affect the second approach’s performance and applicability. A deep understanding of how the separation limit may affect the selection of the indicators involved in the improvement plan is crucial to the second approach. However, one way to avoid this challenge is by involving experts and stakeholders with the appropriate knowledge of the analyzed food system’s characteristics. Another way is by modifying the criterion such that only the DMUs most frequently appearing as inefficient DMUs’ are selected.

6. Discussion

Analyzing the efficiency scores of 37 food-producing countries across five dimensions—availability, accessibility, economic, environmental, and governance—presents compelling insights with far-reaching implications for policymakers and stakeholders involved in food security. The results reveal a spectrum of performance levels across these dimensions, delineating areas of excellence and opportunities for enhancement, allowing food systems’ robustness to be bolstered.

In the availability dimension, characterized by the stability of food supply, the average efficiency score was notable, amounting to 0.718. However, with most countries demonstrating efficiency levels ranging from low to moderate, it becomes evident that resource utilization for ensuring a stable food supply remains a challenge for many nations. Strategies such as sustainable agriculture, effective food waste management, and diversifying food resources emerge as crucial focal points for improvement in this dimension.

Moving on to the accessibility dimension, which pertains to the ease of obtaining food, the average efficiency score slightly surpassed that of availability, standing at 0.769. Despite this, approximately 27.70% of the investigated countries fell into the low-efficiency category, indicating substantial room for progress in ensuring equitable access to food. Socioeconomic factors like household income, food prices, and market diversity emerge as critical determinants in this dimension.

The average efficiency score in the economic dimension, which evaluates food systems’ affordability and financial aspects, was lower, amounting to 0.637. This dimension exhibited a high prevalence of low to moderate efficiency, emphasizing the urgent need for targeted economic policies and interventions to bolster food system performance. Initiatives to improve consumer affordability, increase disposable income, and reduce food expenditures are imperative in addressing economic inefficiencies.

The environmental dimension, gauging the sustainability of resource use and environmental impacts, presented the lowest average efficiency score, amounting to 0.475. With a significant proportion of countries failing to convert environmental resources into food outputs efficiently, there is a pressing need for sustainable environmental practices and policies. Factors such as sustainable farming practices, efficient water use, and renewable energy adoption emerge as pivotal in improving ecological efficiency and mitigating negative impacts on food systems.

Lastly, the governance dimension, deemed most critical, with a weight of 43.60%, exhibited an average efficiency score lower than the scores for availability and accessibility, standing at 0.650. Governance is pivotal in devising food system strategies, promoting research, and ensuring regulatory compliance. Strengthening institutional frameworks, enhancing infrastructure, and fostering stakeholder collaboration are essential for enhancing governance efficiency and overall food system resilience.

The study’s findings offer several practical applications that can significantly enhance food system robustness. In terms of policy development, governments can leverage these insights to design and implement targeted policies addressing specific inefficiencies within each dimension. For instance, economic policies might focus on reducing food costs and increasing disposable incomes, while environmental policies could prioritize sustainable agricultural practices and resource conservation.

Efficient resource allocation, based on the identified strengths and weaknesses, is crucial for optimizing food system performance. Infrastructure, technology, and education investments should be prioritized in dimensions where countries exhibit low-efficiency scores. This strategic allocation of resources can substantially improve the overall robustness of food systems.

Collaboration among government agencies, private sector entities, and international organizations is essential for addressing multidimensional food system challenges. Engaging stakeholders in the decision-making process can enhance the effectiveness of implemented strategies and ensure sustainable outcomes. Such collaboration fosters a holistic approach to problem-solving and promotes sharing best practices and innovations.

Finally, it is critical to establish robust monitoring and evaluation frameworks to assess food system performance regularly. Continuous data collection and analysis will provide timely insights, enabling proactive adjustments to policies and interventions. Stakeholders can adapt strategies by identifying emerging issues early, ensuring food systems’ long-term resilience and sustainability.

7. Conclusions

This study proposes an analytical approach based on SBM-DEA to balance the efficiency and robustness of food systems. Two methods were proposed in this study to investigate opportunities to improve the efficiency of the robustness dimensions. The applicability and operational procedures of the proposed approach have been evaluated using a dataset representing the impact of five robustness dimensions. The proposed approach has revealed an acceptable level of applicability in identifying the inefficient robustness dimensions and providing decision-makers with relevant measures required for establishing plans to improve the efficiency of the underlying dimensions. These measures help provide management with data-based insight into the most effective practices for evaluating and developing resource allocation plans and identifying deficiencies in food systems.

Two aspects make the proposed approach applicable to high-dimensional spaces of food robustness dimensions. First, this approach adopts the grouping principle, in which all related indicators are listed in the same robustness dimension. Second, the SBM-DEA model can comprehend the interactions within or between robustness dimensions and provide outcomes that help management or stakeholders examine the effectiveness of any potential intervention, even when the number of dimensions is significant. In contrast to regression methods, the SBM-DEA is a non-parametric approach that does not rely on any statistical assumptions about a dataset. This attribute enhances the proposed approach’s usability and applicability, making it more suitable for many dataset types.

The first approach to identifying opportunities for improvement has revealed some notable findings that should be considered in real-life practice. It has been noted that the first approach is not recommended when there are many dimensions. This approach focuses on one robustness dimension at a time. Despite this approach being practically straightforward, stakeholders might recommend it for food robustness improvement, which might not be characterized by one dimension. In addition, this approach may not deliver the expected outcomes when the selected dimension has a low relative weight. The second approach has shown more desirable features. This approach has two setting parameters, which stakeholders can use to examine multiple improvement plans. In addition, this approach has a wide search range for low-efficiency dimensions, as it is not limited to one dimension, as is the case for the first approach. The inefficient dimensions may include DMUs from multiple food dimensions, meeting the practitioner’s and stakeholders’ expectations. In this study, the second approach selected inefficient DMUs across all robustness dimensions, resulting in more significant efficiency improvement and resource-saving than the first approach.

This study used data only from 37 countries for the analysis. Expanding the scope of analysis to include more countries and regions could provide a more comprehensive picture of global food system efficiencies. Researchers can uncover unique challenges and opportunities specific to each context by conducting comparative studies across different economic and climatic zones, enriching the findings. The AHP was used to weigh the dimensions in this study. Although it is the preferred choice, it has limitations, such as subjective bias, scalability, and inconsistency issues, especially when dealing with many indicators [44,45]. The authors suggest using other data-based weighting methods, such as principal component analysis, regression analysis, etc. This study utilizes SBM-DEA for efficiency analysis [43] to mitigate the risk of yielding inaccurate outcomes. The proposed approach relied on stakeholders and experts to group the robustness dimension. However, although this approach has practical advantages, the authors recommend other methods, such as k-means clustering, k-neighborhood clustering, and Gaussian mixtures models, for future research. Finally, evaluating the long-term impacts of implemented policies and interventions on food system robustness is essential for identifying best practices and areas for continuous improvement. Longitudinal studies that track progress over time can provide a deeper understanding of how various strategies affect the sustainability of food systems.