We developed the Dynamic Multi-Objective Facility Location Problem (DMUFLP) to address the complexities of facility location decisions in diverse supply chain environments. This model incorporates various cost components and sustainability considerations. We formulated the DMUFLP using a mixed-integer linear programming (MILP) approach, known for efficiently handling large-scale optimization problems with multiple objectives [16] [17] .

The model aims to minimise total project costs, including facility setup, transportation, environmental impact, social impact, and capacity adjustment expenses.

We conducted a simulation study using the R software, a widely used environment for statistical computing, to evaluate the performance of the DMUFLP model rigorously. We based the simulation on a synthetic dataset representing a typical supply chain scenario. To ensure reproducibility, we generated demand values using a normal distribution. We assigned transportation costs using a uniform distribution between 10 and 100, and we derived environmental and social impact costs from exponential distributions with appropriate scale parameters that reflect their relative impact [18] . We modelled capacity adjustment costs with a triangular distribution to simulate increasing marginal costs. We based these assumptions on a combination of historical data and insights from relevant literature [19] [20] .

The model integrates a social impact cost component (si), which reflects community-related factors such as proximity to residential zones, public approval ratings, and local job creation potential. These factors represent the societal acceptability and benefits of facility placement in specific areas, aligning the model with contemporary sustainability practices in supply chain management [21] [22] .

We carefully constructed the DMUFLP model’s constraints to ensure demand satisfaction, facility activation, and compliance with real-world operational conditions [18] . We estimated key parameters, including demand, transport costs, environmental and social costs, and capacity adjustments, using empirical data and theoretical frameworks to ensure model validity [23] .

We implemented the model using R’s lpSolve and ompr packages, which provide practical tools for solving linear and mixed-integer programming problems [24] . We tested the model using various sample sizes (250, 500, 750, and 950) to assess its scalability and computational efficiency. We also benchmarked its performance against other models that integrate multiple cost components and sustainability concerns, providing a comprehensive comparison [25] .

To assess the model’s accuracy and reliability, we applied several performance metrics, including the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Absolute Error (MAE), and Area Under the Curve (AUC) [26] . Using R, we generated diagnostic plots including Residuals vs. Fitted Values, Normal Q-Q Plot, Scale-Location Plot, Cook’s Distance Plot, Residuals vs. Order of Data, Histogram of Residuals, and Partial Regression Plot [27] . These tools helped us verify model assumptions such as linearity, residual normality, and homoscedasticity, and identify influential observations, ensuring robust and interpretable outcomes [28] . We used R’s ggplot2 package to visualise results and performance metrics, which provided precise and intuitive graphical insights [29] .

The DMUFLP model operates under several foundational assumptions that influence its structure and behaviour:

While synthetic data allowed us to test the model under controlled conditions, future work should validate the DMUFLP using empirical supply chain data [32] [33] , to ensure real-world applicability. Additionally, although the MILP formulation is potent, solving large-scale stochastic versions can be computationally demanding. This complexity poses challenges for adapting the model to highly uncertain or multi-period real-world scenarios.

The DMUFLP model demonstrates strong potential as a robust and effective tool for facility location planning in complex, sustainability-oriented supply chains. By integrating environmental and social costs with traditional cost components, the model aligns with evolving sustainability frameworks. Using R software accelerated our simulation, optimization, and visualisation processes. Moving forward, researchers should extend the model to cover stochastic and multi-period conditions and test it with real-world data. Moreover, simplifying the technical language while maintaining analytical depth will improve its accessibility and adoption by industry practitioners.

Figure 1 presents a schematic of the DMUFLP model, illustrating how demand estimates, transportation costs, environmental and social impacts, and capacity adjustments are systematically integrated into the optimization process to determine the most suitable facility locations and their corresponding capacities.

This model is a sophisticated extension of traditional facility location problems, incorporating additional factors such as environmental and social impacts, which are increasingly critical in modern decision-making [34] . The Demand Estimation node signifies the process of forecasting demand across different scenarios, represented mathematically as $d_{jts}$, where $j$ refers to specific demand points, $t$ to time periods, and $s$ to different scenarios. This step is fundamental in ensuring that the model’s outputs are grounded in realistic demand expectations, as accurate demand forecasting is critical for operational efficiency and cost minimization [34] .

Transportation costs are represented as $c_{ijts}$, where $i$ denotes facility locations, $j$ demand points, $t$ time periods, and $s$ scenarios. These costs are a significant component of the total cost in location problems, and their accurate estimation is vital for the overall effectiveness of the model. Incorporating transportation costs into the DMUFLP model ensures that the solution not only meets demand but also does so in a cost-effective manner, aligning with sustainable supply chain practices [35] .

The environmental and social impacts node introduces estimates for environmental costs ( $e_i$) and social costs ( $s_i$), reflecting the growing importance of sustainability in supply chain and logistics decisions. The inclusion of these factors in the DMUFLP model acknowledges the need for companies to minimise their ecological footprint and ensure socially responsible operations, a concern increasingly emphasised in recent literature.

Capacity adjustments ( $\Delta C_{it}$) account for the costs associated with scaling facility capacities in response to changing demand or other factors over time. This aspect of the model allows for flexibility and adaptability, ensuring that the solution remains viable as external conditions evolve.

The Optimization node at the centre of the diagram represents the culmination of these inputs, where the DMUFLP model integrates the estimated demands, costs, impacts, and adjustments to derive the optimal facility locations ( $x^{*}$), assignments ( $y^{*}$), and capacity adjustments ( $\Delta C^{*}$). This process is pivotal for identifying solutions that balance cost efficiency with environmental and social responsibility, aligning with modern supply chain management goals.

The diagram effectively illustrates the complexity and comprehensiveness of the formulated DMUFLP model, which not only addresses traditional economic objectives but also integrates critical environmental and social considerations, making it a robust tool for contemporary facility location and logistics planning [34] .

To address the potential gaps identified, the DMUFLP model will be refined to incorporate stochastic elements, dynamic capacity adjustments, and real-time decision-making capabilities.

Define the sets and indices

Define the enhanced parameters

Define the enhanced decision variables

Formulate the enhanced objective function

The enhanced objective function minimises the expected total cost across all scenarios, including facility costs, transportation costs, environmental and social impacts, and dynamic capacity adjustments:

Minimise

$\begin{array}{l}Z=\underset{s\in S}{{\displaystyle \sum }}{P}_s(\underset{i\in I}{{\displaystyle \sum }}{f}_i{x}_i+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}\underset{j\in J}{{\displaystyle \sum }}{c}_{ijts}{y}_{ijts}+\underset{i\in I}{{\displaystyle \sum }}{e}_i{x}_i+\underset{i\in I}{{\displaystyle \sum }}{s}_i{x}_i\\ +\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}\underset{l\in L}{{\displaystyle \sum }}{\lambda }_l{z}_{ilt}+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}g\left(\Delta C_{it}\right))\end{array}$

where $P_s$ is the probability of scenario $s$, and $g\left(\Delta C_{it}\right)$ represents the cost of adjusting facility capacity over time.

Define the enhanced constraints

$\underset{i\in I}{{\displaystyle \sum }}{y}_{jts}\ge d_{jts},\forall j\in J,\forall t\in T,\forall s\in S$

$y_{ijts}\le d_{jts}{x}_i,\forall i\in I,\forall j\in J,\forall t\in T,\forall s\in S$

This ensures that the total waste transported to facilities meets the demand at each demand point under each scenario.

Stochastic programming is a framework for modelling optimization problems that involve uncertainty. In the context of the DMUFLP, we use scenario-based stochastic programming to account for uncertain parameters like demand and transportation costs. One common approach is the Sample Average Approximation (SAA) method, which approximates the expectation of the uncertain parameters by averaging over a set of sampled scenarios.

Problem Setup

Let’s assume the objective is to minimise the total expected cost across multiple scenarios. The decision variables and parameters have already been defined:

$x_i$: Binary variable indicating whether facility $i$ is opened.

$y_{ijts}$: Continuous variable representing the amount of waste transported from demand point $j$ to facility $i$ during time period $t$ under scenario $s$.

$\Delta C_{it}$: Adjustment to the capacity of facility $i$ at time $t$.

Let ${\xi }_s$ represent the uncertain parameters (e.g., demand $d_{jts}$, transportation costs $c_{ijts}$ etc,.) associated with scenario $s$, where $s$ belongs to the set of scenarios $S$.

The stochastic programming formulation aims to minimise the expected total cost:

Minimise:

$E_{\xi }\left[Z\left(x,y,\Delta C;\xi \right)\right]$

where $Z\left(x,y,\Delta C;\xi \right)$ is the total cost function dependent on the scenario $\xi$.

Sample average approximation (SAA)

$E_{\xi }\left[Z\left(x,y,\Delta C;\xi \right)\right]\approx \frac{1}{N}\underset{s=1}{\overset{N}{{\displaystyle \sum }}}Z\left(x,y,\Delta C;{\xi }_s\right)$

The SAA problem is then formulated as:

$\frac{1}{N}\underset{s=1}{\overset{N}{{\displaystyle \sum }}}\left(\underset{i\in I}{{\displaystyle \sum }}{f}_i{x}_i+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}\underset{j\in J}{{\displaystyle \sum }}{c}_{ijts}{y}_{ijts}+\underset{i\in I}{{\displaystyle \sum }}{e}_i{x}_i+\underset{i\in I}{{\displaystyle \sum }}{s}_i{x}_i+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}g\left(\Delta C_{it}\right)\right)$

Constraints: The SAA formulation includes the constraints applied to each scenario $s$:

Demand Satisfaction Constraint: ${\displaystyle {\sum }_{i\in I}{y}_{ijts}}\ge d_{jts},\forall j\in J,\forall t\in T,\forall s\in S$

Capacity Constraints: ${\displaystyle {\sum }_{j\in J}{y}_{ijts}}\le C_{it}+\Delta C_{it},\forall i\in I,\forall t\in T,\forall s\in S$

Flow Conservation: $y_{ijts}\le d_{jts}{x}_i,\forall i\in I,\forall j\in J,\forall t\in T,\forall s\in S$

Non-negativity and binary constraints:

$x_i\in \left\{0,1\right\},\forall i\in I$

$y_{ijts}\ge 0,\forall i\in I,\forall j\in J,\forall t\in T,\forall s\in S$

$\Delta C_{it}\ge 0,\forall i\in I,\forall t\in T$

Cost function for capacity adjustment:

The cost function $g\left(\Delta C_{it}\right)$ for capacity adjustments can be defined as a linear or convex function:

$g\left(\Delta C_{it}\right)=k_1\Delta C_{it}$

where $k_1$ is a constant representing the marginal cost of capacity adjustment.

Convex cost (e.g., quadratic): $g\left(\Delta C_{it}\right)=k_1\Delta C_{it}+k_2{\left(\Delta C_{it}\right)}^2$, where $k_1$ and $k_2$ are constants reflecting the cost structure.

Solve the SAA problem: Solve the resulting problem which is a mixed-integer linear program (MILP), which can be solved using Gurobi standard optimization solvers. The solution $\left(x^{\ast },y^{\ast },\Delta C^{\ast }\right)$ to this MILP provides an approximation to the optimal decisions under uncertainty.

1. Branching

Branching in the branch-and-bound algorithm involves creating sub problems by splitting the feasible region based on the values of the binary variable $x_i$.

Solve the linear relaxation of the MILP, where the binary variables $x_i$ are allowed to take continuous values between 0 and 1: $x_i\in \left[0,1\right],\forall i\in I$

Suppose the solution to the relaxed problem yields a fractional value $x_i^{*}$ for some $i$, i.e, $x_i^{*}\in \left(0,1\right)$

Create two sub problems by branching on $x_i^{*}$

Mathematically, this can be represented as:

Sub problems 1: $x_i=0$, $\forall i\in$ branching set

Sub problems 2: $x_i=1$, $\forall i\in$ branching set

2. Bounding

For each sub problem, solve the LP relaxation (the linear relaxation with the additional branching constraint).

Objective function for sub problems:

$Z_{Subproblem}=\min \frac{1}{N}\underset{s=1}{\overset{N}{{\displaystyle \sum }}}\left(\underset{i\in I}{{\displaystyle \sum }}{f}_i{x}_i+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}\underset{j\in J}{{\displaystyle \sum }}{c}_{ijts}{y}_{ijts}+\underset{i\in I}{{\displaystyle \sum }}{e}_i{x}_i+\underset{i\in I}{{\displaystyle \sum }}{s}_i{x}_i+\underset{t\in T}{{\displaystyle \sum }}\underset{i\in I}{{\displaystyle \sum }}g\left(\Delta C_{it}\right)\right)$

Bounding Rule:

Mathematically:

If $Z_{Subproblem}<Z^{*}$, prune branch.

3. Pruning

Pruning involves eliminating the subproblems that cannot yield a better solution than the current best.

Pruning Rule:

$Z_{Subproblem}<Z^{*},Z^{*}=Z_{Subproblem},\left(x^{*},y^{*},\Delta c^{*}\right)=\left(x_{subproblem},y_{subproblem},\Delta c_{subproblem}\right)$

4. Cutting Planes

Cutting planes are additional constraints added to the linear relaxation to eliminate fractional solutions without excluding any integer feasible solutions.

Cutting Plane Generation:

Identify a hyperplane that separates the current fractional solution from the convex hull of integer solutions.

For example, a Gomory cut can be generated from the simplex tableau:

Gomory cut: ${\displaystyle \underset{i\in I}{\sum }\lfloor a_i\rfloor x_i}\ge \lfloor b\rfloor$

Where, $a_i$ are the coefficients of the constraint in the simplex tableau, and $b$ is the right-hand side.

Add the cut to the problem:

Incorporate the cutting plane as a new constraint: ${\displaystyle \underset{i\in I}{\sum }\lfloor a_i\rfloor x_i}\ge \lfloor b\rfloor$

Resolve the LP relaxation with this new constraint to tighten the relaxation.

5. Final Solution

The final solution $\left(x^{*},y^{*},\Delta c^{*}\right)$ is obtained when:

6. Interpretation

The branch-and-bound method combined with cutting planes systematically explores the feasible region and converges to the optimal integer solution by eliminating infeasible or suboptimal regions of the solution space.

Formulating the statistical properties of DMUFLP model involves considering the stochastic elements and analyzing how uncertainty impacts the optimal solution. Below, we define the key statistical properties associated with the DMUFLP model and provide their mathematical formulations.

1. Expected value of the objective functions

The objective function of the DMUFLP is designed to minimize the total expected cost across multiple scenarios. The expected value of the objective function $Z\left(x,y,\Delta c\right)$ is given by:

$E\left[Z\left(x,y,\Delta c\right)\right]=\underset{s=1}{\overset{N}{{\displaystyle \sum }}}{P}_s\cdot Z_s\left(x,y,\Delta c\right)$

where:

$Z_s\left(x,y,\Delta c\right)={\displaystyle \underset{i\in I}{\sum }{f}_i{x}_i}+{\displaystyle \underset{t\in T}{\sum }{\displaystyle \underset{i\in I}{\sum }{\displaystyle \underset{j\in J}{\sum }{c}_{ijts}{y}_{ijts}}}}+{\displaystyle \underset{i\in I}{\sum }{e}_i{x}_i}+{\displaystyle \underset{i\in I}{\sum }{s}_i{x}_i}+{\displaystyle \underset{t\in T}{\sum }{\displaystyle \underset{i\in I}{\sum }g\left(\Delta C_{it}\right)}}$

2. Variance of the objective function

The variance of the objective function measures the variability of the total cost under different scenarios. It is defined as: $Var\left(Z\left(x,y,\Delta c\right)\right)={\displaystyle {\sum }_{s=1}^N{P}_s{\left(Z_s\left(x,y,\Delta c\right)-E\left[Z\left(x,y,\Delta c\right)\right]\right)}^2}$. This variance indicates how much the total cost fluctuates due to uncertainty in demand, transportation costs, and other parameters.

3. Risk measures

In addition to the expected value and variance, we can introduce risk measures such as the Conditional Value at Risk (CVaR) to capture the downside risk associated with the objective function. CVaR at confident level $\alpha$ (e.g. $\alpha =0.95$) represents the expected loss given that the loss exceeds the Value at Risk (VaR):

$CVa{R}_{\alpha }\left(Z\left(x,y,\Delta c\right)\right)=\left[Z\left(x,y,\Delta c\right)|Z\left(x,y,\Delta c\right)\ge Va{R}_{\alpha }\right]$

where $Va{R}_{\alpha }$ is the $\alpha$–quantile of the distribution of the objective function $Z\left(x,y,\Delta c\right)$.

4. Sensitivity analysis

Sensitivity analysis measures how sensitive the optimal solution is to changes in the model parameters (e.g., demand, cost). Mathematically, this is often done by computing the partial derivatives of the objective function with respect to each parameter.

Sensitive to demand ( $d_{jts}$):

$\frac{\partial Z\left(x,y,\Delta c\right)}{\partial d_{jts}}=\frac{1}{N}\underset{s=1}{\overset{N}{{\displaystyle \sum }}}{p}_s\cdot \frac{\partial Z_s\left(x,y,\Delta c\right)}{\partial d_{jts}}$

This derivative indicates how much the total cost changes in response to a small change in the demand at point $j$ in time-period $t$ under scenario $s$.

Sensitivity to transportation costs ( $c_{ijts}$):

$\frac{\partial Z\left(x,y,\Delta c\right)}{\partial c_{ijts}}=\frac{\partial Z_s\left(x,y,\Delta c\right)}{\partial c_{ijts}}$

These derivative measures the change in the objective function with respect to changes in the transportation cost between facility $i$ and demand point $j$ under scenario $s$.

5. Optimality gap

The optimality gap measures the difference between the objective value of the relaxed (linear) solution and the best integer feasible solution obtained:

$\text{Optimal}\text{Gap}=\frac{Z_{relaxed}-Z_{integer}}{Z_{integer}}\times 100\%$

where:

A smaller optimality gap indicates that the solution obtained is closed to the true optimal integer solution.

6. Scenario probability distribution

The probability distribution of scenarios $p_s$ plays a crucial role in the formulation of the expected value and variance. The distribution is often assumed to be discrete, with each scenario $s$ assigned a probability $p_s$.

$\underset{s=1}{\overset{N}{{\displaystyle \sum }}}{p}_s=1$

The scenario probabilities reflect the likelihood of different uncertain outcomes and are typically estimated based on historical data or expert judgement.

1. Parameter estimation framework

Let’s assume that the parameters to be estimated include:

We denote the complete parameter vector as $\theta =\left({\theta }_d,{\theta }_c,{\theta }_e,{\theta }_s,{\theta }_k\right)$

2. Likelihood function

Given that the model involves multiple scenarios $s$, we can assume that the observed data under each scenario is generated by a probability distribution that depends on the parameters $\theta$. Let $ℒ\left(\theta \right)$ represent the likelihood function for the parameters given the observed data $D$ under all scenarios:

$ℒ\left(\theta \right)=\underset{s=1}{\overset{N}{{\displaystyle \prod }}}{ℒ}_s\left(\theta ;D_s\right)$

where:

3. Score equations

The score equations are obtained by taking the derivative of the log-likelihood function with respect to each parameter in $\theta$.

This leads to a system of equations (score equations) that must be solved to obtain the parameters estimates $\hat{\theta }$.

Score equations for demand parameters ${\theta }_d$:

Assuming that demand $d_{jts}$ under each scenario follows a distribution $p_d\left(d_{jts};{\theta }_d\right)$ with parameters ${\theta }_d$:

$\frac{\partial \log ℒ\left(\theta \right)}{\partial {\theta }_d}={\displaystyle \underset{s=1}{\overset{N}{\sum }}{\displaystyle \underset{j\in J}{\sum }{\displaystyle \underset{t\in T}{\sum }\frac{\partial \log p_d\left(d_{jts};{\theta }_d\right)}{\partial {\theta }_d}}}}=0$

Score equations for transportation cost parameters ${\theta }_c$:

Assuming transportation cost, $c_{ijts}$, also follows a distribution $p_c\left(c_{ijts};{\theta }_c\right)$:

$\frac{\partial \log ℒ\left(\theta \right)}{\partial {\theta }_c}={\displaystyle \underset{s=1}{\overset{N}{\sum }}{\displaystyle \underset{i\in I}{\sum }{\displaystyle \underset{j\in J}{\sum }{\displaystyle \underset{t\in T}{\sum }\frac{\partial \log p_c\left(c_{ijts};{\theta }_c\right)}{\partial {\theta }_c}}}}}=0$

Score equations for environmental impact cost parameters ${\theta }_e$:

Assuming environmental impact cost $e_i$ follows a distribution $p_e\left(e_i;{\theta }_e\right)$:

$\frac{\partial \log ℒ\left(\theta \right)}{\partial {\theta }_e}={\displaystyle \underset{i\in I}{\sum }\frac{\partial \log p_e\left(e_i;{\theta }_e\right)}{\partial {\theta }_e}}=0$

Score equations for social impact cost parameters:

Assuming that social $s_i$ impact cost follows a distribution $p_s\left(s_i;{\theta }_s\right)$

$\frac{\partial \log ℒ\left(\theta \right)}{\partial {\theta }_s}={\displaystyle \underset{i\in I}{\sum }\frac{\partial \log p_s\left(s_i;{\theta }_s\right)}{\partial {\theta }_s}}=0$

Score equations for capacity adjustment cost parameters:

Assuming that the cost of capacity adjustment $g\left(\Delta c_{it}\right)$ is governed by parameters ${\theta }_k$, typically modelled as $g\left(\Delta c_{it};{\theta }_k\right)=k_1\Delta c_{it}+k_2{\left(\Delta c_{it}\right)}^2$ with ${\theta }_k=\left(k_1,k_2\right)$:

$\frac{\partial \log ℒ\left(\theta \right)}{\partial {\theta }_k}={\displaystyle \underset{i\in I}{\sum }{\displaystyle \underset{t\in T}{\sum }\frac{\partial \log p_g\left(g\left(\Delta c_{it};{\theta }_k\right)\right)}{\partial {\theta }_k}}}=0$

4. Estimation process:

To estimate the parameters $\theta =\left({\theta }_d,{\theta }_c,{\theta }_e,{\theta }_s,{\theta }_k\right)$, follow these steps.

1. Maximise the likelihood function: use numerical optimization methods to maximize the likelihood function $ℒ\left(\theta \right)$ or equivalently solve the score equations derived above.

2. Iterative Algorithm: Implement an iterative algorithm (e.g., Newton-Raphson, Gradient Descent) to solve the score equations and find the maximum likelihood estimates $\hat{\theta }$.

3. Variance-Covariance Matrix: Compute the variance-covariance matrix of the parameter estimates $\hat{\theta }$ to assess their precision:

$Var={\left[E\left(-\frac{{\partial }^2\log ℒ\left(\theta \right)}{\partial \theta \partial {\theta }^{\text{T}}}\right)\right]}^{-1}$

5. Application to the formulated DMUFLP

Once the parameters $\hat{\theta }$ are estimated, they can be applied to DMUFLP model to generate the input values for demand, transportation costs, environmental and social impacts, and capacity adjustments. The estimated parameters are used to solve the DMUFLP, yielding the optimal decisions $\left(x^{*},y^{*},\Delta c^{*}\right)$.

1) Likelihood Function

Given a parameter vector $\theta =\left({\theta }_d,{\theta }_c,{\theta }_e,{\theta }_s,{\theta }_k\right)$ and data $D$ the likelihood function $ℒ\left(\theta \right)$ is

$ℒ\left(\theta \right)=\underset{s=1}{\overset{N}{{\displaystyle \prod }}}{ℒ}_s\left(\theta ;D_s\right)$

The log-likelihood is:

$ℒ\left(\theta \right)=\underset{s=1}{\overset{N}{{\displaystyle \sum }}}\log {ℒ}_s\left(\theta ;D_s\right)$

2) Gradient descent algorithm

The gradient of the log-likelihood function with respect to $\theta$ is the score function:

${\nabla }_{\theta }\log ℒ\left(\theta \right)=\frac{\partial \log ℒ\left(\theta \right)}{\partial \theta }$

The gradient descent update rule is:

${\theta }_{k+1}={\theta }_k+\eta {\nabla }_0\log ℒ\left({\theta }_k\right)$, where $\eta$ is the learning rate.

3) Iterate process

1. Initialize ${\theta }_0$ (initial guess).

2. Iterate until convergence:

3. Convergence is achieved when $\Vert {\theta }_{k+1}-{\theta }_k\Vert <ϵ$ where $ϵ$ is a small tolerance level).

The variance-covariance matrix of the parameter estimates $\hat{\theta }$ is given by:

$Var\left(\hat{\theta }\right)={\left[E\left(-\frac{{\partial }^2\log ℒ\left(\theta \right)}{\partial \theta \partial {\theta }^{\text{T}}}\right)\right]}^{-1}$

The Fisher information matrix is $I\left(\theta \right)$:

$I\left(\theta \right)=-E\left(\frac{{\partial }^2\log ℒ\left(\theta \right)}{\partial \theta \partial {\theta }^{\text{T}}}\right)$

So, the variance-covariance matrix is:

$Var\left(\hat{\theta }\right)=I{\left(\hat{\theta }\right)}^{-1}$

1. Demand Estimation: Use ${\hat{\theta }}_d$ to estimate the demand $d_{ijts}$, under each scenario $s$.

2. Transportation Costs: Use ${\hat{\theta }}_c$ to estimate the transportation costs $c_{ijts}$.

3. Environmental and Social Impacts: Use ${\hat{\theta }}_e$ and ${\hat{\theta }}_s$ to estimate environmental costs $e_i$ and social costs $s_i$.

4. Capacity Adjustments: Use ${\hat{\theta }}_k$ to estimate the capacity adjustment costs $g\left(\Delta C_{it}\right)$.

1. Demand estimation

Given the estimated parameters ${\hat{\theta }}_d$, the demand $d_{ijts}$ under each scenario $s$ for each facility $i$, customer $j$, and time $t$ can be estimated as:

$d_{ijts}={\int }_d\left(i,j,t,s,{\hat{\theta }}_d\right)$

where:

2. Transportation costs

Transportation costs $c_{ijts}$ are estimated using the parameters ${\hat{\theta }}_c$:

$c_{ijts}={\int }_c\left(i,j,t,s,{\hat{\theta }}_c\right)$

where:

3. Environmental and social impacts

Environmental costs $e_i$ and social costs $s_i$ for facility are estimated using the parameters ${\hat{\theta }}_e$ and ${\hat{\theta }}_s$:

$e_i={\int }_e\left(i,{\hat{\theta }}_e\right)$, $s_i={\int }_s\left(i,{\hat{\theta }}_s\right)$

where:

4. Capacity adjustments

Capacity adjustment costs $g\left(\Delta C_{it}\right)$ are estimated using the parameters ${\hat{\theta }}_k$:

$g\left(\Delta C_{it}\right)={\int }_k\left(i,t,\Delta C_{it},{\hat{\theta }}_k\right)$

where:

Using the estimated parameters $\hat{\theta }$, solve the DMUFLP to obtain the optimal decisions $\left(x^{*},y^{*},\Delta c^{*}\right)$.

Using the estimated parameters $\hat{\theta }=\left({\hat{\theta }}_d,{\hat{\theta }}_c,{\hat{\theta }}_e,{\hat{\theta }}_s,{\hat{\theta }}_k\right)$, the objective is to solve the following optimization problem to obtain the optimal decisions $\left(x^{*},y^{*},\Delta c^{*}\right)$.

Objective function

The goal is to minimise the total cost, which is the sum of fixed costs, transportation costs, environmental and social costs and capacity adjustment costs:

$\underset{x,y,\Delta c}{\min }=\left\{{\displaystyle \underset{i\in I}{\sum }{f}_i{x}_i}+{\displaystyle \underset{t\in T}{\sum }{\displaystyle \underset{i\in I}{\sum }{\displaystyle \underset{j\in J}{\sum }{\displaystyle \underset{s\in S}{\sum }{c}_{ijts}{y}_{ijts}}}}}+{\displaystyle \underset{i\in I}{\sum }\left(e_i+s_i\right)x_i}+{\displaystyle \underset{i\in I}{\sum }{\displaystyle \underset{t\in T}{\sum }g\left(\Delta C_{it}\right)}}\right\}$

where:

Constraints

1. Demand Satisfaction:

Ensure that the demand is met for each customer $j$ at time $t$ under scenario $s$:

${\displaystyle {\sum }_{i\in I}{y}_{ijts}}\ge d_{ijts},\forall i\in I,t\in T,s\in S$

2. Capacity Constraints:

Ensure that the facility capacity (adjusted for capacity changes) is not exceeded:

${\displaystyle {\sum }_{j\in J}{y}_{ijts}}\le \left(c_{it}+\Delta c_{it}\right)x_i,\forall i\in I,t\in T,s\in S$

3. Non-Negativity and Binary Constraints:

$y_{ijts}\ge 0,\forall i\in I,j\in J,t\in T,s\in S$

$x_i\in \left\{0,1\right\},\forall i\in I$

The Normal Q-Q Plot for the DMUFLP model, shown in Figure 2 above, serves as a crucial diagnostic tool for evaluating the normality of residuals, an essential assumption in many regression models. In this plot, the standardised residuals are plotted against the theoretical quantiles of a standard normal distribution. If the residuals are normally distributed, the points should lie approximately along the 45-degree line that represents perfect agreement between the observed data and the theoretical distribution [36] .

The plot displayed indicates that the residuals from the DMUFLP model largely adhere to the expected normal distribution, as many points align closely with the 45-degree line. This alignment suggests that the model’s residuals are approximately normally distributed, which supports the validity of inferences drawn from the model. The general adherence to normality implies that the model’s estimates are likely unbiased and efficient, leading to valid hypothesis tests and confidence intervals [37] .

In conclusion, the Normal Q-Q Plot for the DMUFLP model demonstrates a generally satisfactory adherence to the assumption of normally distributed residuals, with only minor deviations at the extremes. This suggests that the model is appropriately specified and reliable for most practical purposes.

Figure 3 presents the Residuals vs. Fitted Values diagnostic plot for the DMUFLP model, offering key insights into the model’s performance and its underlying assumptions. The relatively uniform scatter of residuals around the horizontal line (y = 0) indicates that the assumption of homoscedasticity, constant variance of errors, is reasonably satisfied [23] . The pattern in the plot does not suggest a strong systematic deviation, which upholds the linearity assumption is upheld. The roughly random distribution of residuals supports the appropriateness of the model, affirming that the linear model structure is suitable for capturing the relationship between the predictors and the outcome variable [38] .

In conclusion, the Residuals vs Fitted Values plot for the DMUFLP model indicates that the model performs well overall. These diagnostic insights are critical for ensuring that the model provides reliable and generalisable results, particularly in the context of large-scale optimization problems like facility location planning.

Figure 4 shows the Scale-Location plot (also known as the Spread-Location plot) for the DMUFLP model, a key diagnostic tool used to assess the assumption of homoscedasticity, that is, whether the residuals exhibit constant variance across all levels of the fitted values. In the plot, the y-axis represents the square root of the standardised residuals, and the x-axis represents the fitted values. Ideally, the red line indicating the trend should remain relatively flat, and the points should be randomly dispersed without forming any clear pattern or systematic structure. In this specific plot, the red trend line remains mostly flat, suggesting that the variance of the residuals does not systematically increase or decrease across the range of fitted values. This indicates that the homoscedasticity assumption is largely met, meaning the DMUFLP model performs consistently across different levels of the fitted values. The random distribution of points around the red line further confirms that there is no significant non-linearity or heteroscedasticity affecting the model’s performance [39] .

In summary, the Scale-Location plot for the DMUFLP model suggests that the model is well-specified and that the assumption of constant variance of residuals is reasonably satisfied. The slight deviations observed in the spread of the residuals do not pose a significant threat to the model’s validity.

Figure 5 presents the Residuals vs. Leverage plot for the DMUFLP model, a critical diagnostic tool for detecting influential data points that may exert an undue impact on the regression results. In this plot, the x-axis represents the leverage of the data points, which measures the influence each point has on the overall regression model, while the y-axis represents the standardized residuals, showing how far each observation deviates from the fitted regression line. The plot indicates that most data points have low leverage and standardised residuals close to zero, which suggests that most of the data fits the model well. The red trend line is relatively flat, implying that there is no strong relationship between leverage and residuals, which is a desirable outcome in regression analysis.

The dotted lines on the plot represent Cook’s distance, a measure that combines the leverage and residuals to assess the overall influence of a data point on the fitted model. Points that lie outside the Cook’s distance lines may be considered influential. The presence of influential points could indicate the need for further investigation or potentially more robust modelling techniques that are less sensitive to outliers and high leverage points. For instance, one could consider using robust regression methods that down-weight the influence of such points, thereby improving the generalizability and reliability of the model [40] . Additionally, it might be worthwhile to investigate whether these points represent data errors, unusual cases, or genuinely informative outliers that require special consideration in the analysis.

In summary, the Residuals vs Leverage plot for the DMUFLP model suggests that the model is generally well-fitted.

Figure 6 displays the Histogram of Residuals for the DMUFLP model, offering a visual assessment of the residuals’ distribution, an essential step in evaluating the assumption of normality in regression analysis. The histogram displays the frequency of residuals along the horizontal axis, allowing us to observe whether they are symmetrically distributed around a mean of zero. A roughly symmetric shape centered near zero supports the assumption of normally distributed residuals, which is crucial for the validity of many statistical tests and model diagnostics.

In this plot, the residuals appear to be roughly normally distributed, with a slight concentration around the centre, near zero, and tapering off towards the tails. This bell-shaped distribution is characteristic of normality, suggesting that the errors in the DMUFLP model are likely normally distributed. The normal distribution of residuals is critical because it underpins the validity of confidence intervals, hypothesis tests, and predictions made from the model [41] . Overall, the histogram indicates that the residuals of the DMUFLP model are reasonably normally distributed, supporting the use of this model for making statistical inferences. This normality assumption strengthens the reliability of the results obtained from the model, ensuring that the estimated parameters and predictions are likely to be accurate and valid.

Figure 7 presents the Partial Regression Plot (also known as the Component + Residual Plot) for the DMUFLP model, highlighting the linear relationship between a specific predictor variable and the response variable, after adjusting for the influence of other predictors in the model. The scatter plot, overlaid with a regression line, illustrates this partial relationship—specifically between variable X and the response. The observed strong linear trend, indicated by the magenta line, supports the assumption of linearity and suggests that the DMUFLP model is well-specified. This plot is also useful for assessing model fit and identifying potential non-linearity or outliers [42] .

Implications

The diagnostic plots collectively suggest that the DMUFLP model performs well under the given data, with residuals adhering closely to the model’s assumptions. The minor deviations observed do not significantly impact the overall model performance, indicating that the model is robust and reliable for predicting outcomes in similar contexts. These findings underscore the importance of thorough diagnostic checks in validating model assumptions and ensuring the accuracy of predictive models in optimization contexts [41] .

Results

Table 1 presents the performance of the various models using the metrics AIC, BIC, MAE, and AUC across different sample sizes, offering valuable insights into their robustness and suitability for facility location problems. The Formulated DMUFLP model consistently outperforms other models, particularly at smaller sample sizes (e.g., 250), where it achieves the lowest AIC (712.46) and BIC (737.11), indicating a better balance between model complexity and goodness of fit [43] . Additionally, the DMUFLP model exhibits a relatively low MAE (0.7666) and a commendable AUC (0.8000), suggesting that it effectively minimizes prediction errors while maintaining strong classification performance.

At a sample size of 500, the DMUFLP model continues to demonstrate superior performance with an AIC of 1389.82 and a BIC of 1419.32, underscoring its efficiency in handling larger datasets. The MAE (0.7536) remains lower compared to other models, further affirming its predictive accuracy. Notably, the AUC for DMUFLP remains competitive (0.7920), highlighting its ability to maintain a high level of classification accuracy even with increased sample sizes.

As the sample size increases to 750 and 950, the DMUFLP model shows slight increases in AIC and BIC, yet it still outperforms models like Pires, Olapiriyakul, and Ghadge, which exhibit significantly higher AIC and BIC values, indicating potential overfitting or inefficiency [44] .

Moreover, the AUC for DMUFLP (0.7387 and 0.7726 for 750 and 950 samples, respectively) suggests that while there is a slight decline in classification performance with larger samples, it remains relatively stable compared to other models, which show more variability in AUC values. The performance of the other models, such as Pires, Olapiriyakul, and Ghadge, across the sample sizes, reveals their limitations in balancing model complexity with predictive accuracy. For instance, Pires and Olapiriyakul models exhibit higher AIC and BIC values across all sample sizes, which could indicate that these models are either too complex or not well-suited for the data at hand [45] . The higher MAE values for these models further suggest that they are less effective in minimizing prediction errors compared to DMUFLP.

Interestingly, the models like Pang and Green show a relatively strong performance in terms of AUC, especially at smaller sample sizes (0.8160 and 0.8080 at 250 samples, respectively). However, their higher AIC and BIC values indicate that despite their good classification accuracy, they may suffer from model overfitting, particularly as the sample size increases.

In conclusion, the DMUFLP model emerges as the most robust and reliable model across varying sample sizes, consistently demonstrating lower AIC, BIC, and MAE values, while maintaining competitive AUC scores. These results suggest that the DMUFLP model is highly effective in balancing model complexity, predictive accuracy, and classification performance, making it a preferable choice for facility location problems in various contexts. This finding aligns with recent literature emphasising the importance of model parsimony and predictive power in decision-making processes involving large-scale optimization problems [46] .

Figure 8 summarizes the performance of various facility location models across different sample sizes based on AIC, BIC, MAE, and AUC metrics. The DMUFLP model demonstrates consistently strong results, with lower AIC, BIC, and MAE values and competitive AUC scores, highlighting its robustness and effectiveness.