Journal of Global Optimization

https://doi.org/10.1007/s10898-025-01509-1

AI-based predictive analytics for enhancing data-driven

supply chain optimization

Funda Iseri1,4 · Halil Iseri2,4 · Natasha J. Chrisandina1,4 · Eleftherios Iakovou2,3,4 ·

Efstratios N. Pistikopoulos1,4

Received: 1 October 2024 / Accepted: 3 June 2025

© The Author(s) 2025

Abstract
Data analytics and machine learning are emerging as leading technologies to develop
next-generation data-driven decision-making tools in supply chain management. Predictive
analytics play an important role in providing deep insights to lower uncertainty and boost
overall efficiency in terms of demand fulfillment, inventory management, and resource allo-
cation, thereby enhancing informed decision-making. By leveraging historical data, efficient
forecasting can further be developed, guiding supply chain design and operational decisions.
In this work, we focus on reverse logistics in support of development of closed-loop sup-
ply chains for photovoltaic panels (PV), and employ advanced statistical and deep learning
forecasting techniques for predicting demand and commodity prices. The selected predic-
tion models are integrated into the supply chain optimization model, and the impact on the
whole system performance is investigated. The proposed methodology streamlines opera-
tions, reduces costs, and allows for quick adjustments to shifting market dynamics. This
work underscores the transformative potential and competitive advantage of AI-employed
data-driven analytics in ensuring sustainable and resilient supply chains within the circular
economy, particularly for critical materials in PV recycling.

Keywords AI-driven demand planning · deep learning · closed-loop PV supply chain ·

circular economy · reverse logistics · supply chain optimization · supply chain resilience

1 Introduction

The integration of data-driven approaches such as big data analytics (BDA), machine learn-
ing (ML), and artificial intelligence (AI) has received increased attention as technologies
continues to improve. Supply chain management (SCM) is one significant field where BDA
B Efstratios N. Pistikopoulos
stratos@tamu.edu
1 Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX,
USA
2 Department of Multidisciplinary Engineering, Texas A&M University, College Station, TX, USA
3 Department of Engineering Technology and Industrial Distribution, Texas A&M University,
College Station, TX, USA
4 Texas A&M Energy Institute, Texas A&M University, College Station, TX, USA

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 Journal of Global Optimization

has been extensively analyzed and shown to offer added value regarding risk assessment and
predictive analytics to improve performance. These new approaches can help improve the
seamless data and information exchange by effective and accurate data transfer among supply
chain stakeholders, thereby increasing the overall supply chain performance. Furthermore,
data-driven analytics can also help in designing resilient and robust supply chains, which will
have accurate prediction models to guide well-informed decision-making and fast recovery
from disruptions [1, 2]. Similarly, it is widely accepted that the integration of AI and Industry
4.0 can further enhance supply chain responsiveness and flexibility [3].
Supply chain analytics can be categorized into three main types: descriptive, predictive,
and prescriptive. Descriptive techniques aim to explain or reconstruct the system and to pro-
vide situational awareness. Real-time data exchange and information flow related to location
and quantity of products in the supply chain enhance delivery schedules, order fulfillment,
handling of emergencies, changing transportation modes, and other tactical decisions [4].
On the other hand, predictive techniques harness historical data to develop foresight to pre-
dict demand and price patterns, as well as other future events that could affect operations
[4]. This approach mainly utilizes big data analytics mixed with AI techniques to generate
predictions. Various statistical analysis methods such as moving averages and exponential
smoothing, and deep learning (DL) models like neural networks, long-short-term-memory
(LSTM), and others are often used to analyze data structures and predict future trends and
behaviors. In addition to predictive analysis, ML-based data mining methods like cluster-
ing analyses are also leveraged to investigate market trends and classification of demand
or products. Prescriptive techniques use both descriptive and predictive analytics to support
mathematical optimization models of the supply chain at hand [4]. These techniques allow
for the development of possible strategies and action plans regarding design, operational,
logistics, and transportation decisions in the supply chain. Examples of where prescriptive
analytics might be applied include production planning, task selection, location selection,
resource allocation, cost minimization and profit maximization [1]. Integrating these strate-
gies into comprehensive decision-making is essential to enhance supply chain performance,
efficiency, responsiveness, and resilience. Building on this foundation, the use of big data ana-
lytics advances the prediction of future trends, providing valuable decision-making insights
into various aspects of SCM.
In line with the growing use of AI in supply chains, there is a growing interest for the use of
neural networks as surrogate models to optimize objective functions in complex supply chain
problems. [5] investigate the impact of neural networks as being effective surrogate models to
approximate non-linear objective functions in optimization contexts. [6] examine production
planning in hybrid manufacturing and remanufacturing systems including both new produc-
tion and remanufacturing of end-of-life products. With the proposed scenario-based mixed
integer linear programming (MILP) model, the demand for both new and remanufactured
products are analyzed, further emphasizing how AI-driven surrogate models can support
optimization in these complex systems. The application of deep learning (DL) models is also
increasingly common in the supply chain domain. [7] present deep learning (DL) to predict
shipment risks and improve supply chain resilience during the COVID-19 pandemic. Their
study proposes a hybrid model, temporal convolutional network, which is able to accurately
forecast the risk of of shipment to a particular destination under COVID-19 restrictions,
improving supply chain resilience. Moreover, [8] presents a hybrid ML model for demand
forecasting, and investigates how the use of these demand forecasting models can improve a
steel manufacturer’s supply chain performance.
Beyond traditional supply chains, data-driven decision-support frameworks are making
significant impacts in infrastructure management. [9] integrate track condition predictions

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Journal of Global Optimization

with tactical maintenance planning and operational scheduling, enabling more informed and
effective maintenance interventions. [10] present a data-driven approach to optimize dynamic
multiple commodity supply chains with relocatable modular production units, addressing
complex supply chain problems that dynamically adjust to changes in demand, price, and
availability. Regarding the energy supply chain, especially for design and planning decisions,
there is increasing interest in ML-based prediction models. [11] utilize a multi-criteria opti-
mization model to minimize costs and emissions. To reduce uncertainties, a recurrent neural
network approach is implemented to predict energy supply and demand for a hybrid renew-
able energy system coupled with reverse osmosis desalination. [12] propose a ML-based
approach to optimize shale gas supply chain network, where a forecasting model is imple-
mented to predict local natural gas demand and used to identify re-fracture candidates. [13]
present a ML framework to select a subset of sub-problems to design of large-scale energy
systems and illustrates how ML helps to reduce the computational burden. In a related study,
[14] explore the use of neural networks to enhance optimization by approximating objec-
tive functions more efficiently, reducing computational costs in large-scale supply chain
networks. [15] investigate the strategic power sector transition in Mexico, optimizing the
selection of technologies while minimizing cost, emission, and water consumption based on
the obtained demand profile via a deep learning model. [16] employ Agent-Based Modeling
(ABM) to generate different energy consumption scenarios to predict energy demand; these
generated scenarios are then used in a Mixed Integer Linear Programming (MILP) optimiza-
tion model to optimize system design. [17] utilize an Artificial Neural Network (ANN) model
to investigate energy price boundaries, and implement a multi-stage planning along with the
uncertainties in energy load growth and price fluctuations. Similarly, [18] proposes a multi-
stage investment planning framework that quantifies uncertainty and variability in energy
demand, incorporating data-driven scenario generation to support flexible decision-making
under uncertainty.
Table 1 provides a taxonomy of the state-of-the-art literature on the different ways data
analytics and AI-employed models have been used in supply chain management. Building
on the insights gained from the literature, it is clear that there is a growing interest towards
advanced data analytics and ML-based models, and their integration into supply chain models
to reduce uncertainty and increase accuracy and resilience. The growing interest in this area
emphasizes the need for additional investigations into the quantifiable benefits this integration
could offer for decision-makers.
In this study, we present a comprehensive approach for integrating predictive analytics
techniques into supply chain management by focusing on different statistical and DL-based
forecasting models to generate predictions for both demand and commodity prices, further
investigating the collective impact of these forecasting models on the closed-loop supply
chain of PV waste management. We not only select the most accurate predictive models for
demand and commodity price but also rigorously investigate the impact of forecasting on
various aspects of supply chain performance, including total cost, inventory management,
and demand fulfillment. By statistically analyzing the quality of forecasting models and their
predictions, we demonstrate the importance of validating forecasts models to ensure accuracy.
To incorporate the forecasts generated into SCM, a multi-period mixed-integer linear (MILP)
programming model is built to optimize the closed-loop supply chain for PV panels, where the
predictions are used to guide medium-term tactical decisions. This enables us to statistically
validate how the accuracy of forecasts enhances supply chain performance. Furthermore,
this study contributes to the field by introducing a novel application of AI-driven forecasting
models for demand and commodity prices specifically tailored to the complexities of PV
recycling and integrating these predictions into a comprehensive optimization framework.

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 Table 1 Data Analytics, AI Applications in Supply Chain
Publication BDA AI Decision-Making Supplier-Selection Forecasting Key-Performance (KPIs)

[19]  
[20]  

123
[21]   
[22]   
[23]  
[24]   
[25]  
[26]    
[27]   
[28]   
[29]  
[30]  
[31]  
[32]  
[33]    
[34]   
[35]  
[36]  
[37]  
[38] 
[39]  
[40]  
[41] 
[42]  
[43]  
[44]  
[45]  
Journal of Global Optimization

[46]    
Journal of Global Optimization

We aim to underscore the critical role that AI-based predictive analytics and strategic decision-
making play in developing sustainable and competitive supply chain solutions.
The remainder of the paper is structured as follows. In Section 2, the specific proper-
ties and complexities involved in the closed-loop supply chain for PV panels are discussed
to motivate the problem and the need for accurate forecasts for demand planning and sup-
ply chain management. In Section 3, we discuss our proposed methodological framework,
encompassing six steps. Various forecasting methods are assessed by their ability to describe
and predict the dynamics of PV demand and commodity prices. Metrics used to compare
supply chain performance with and without forecast integration are further discussed. In
Section 4, a numerical case study is presented to illustrate this approach on an indicative PV
supply chain. Section 5 outlines the performance of various forecasting models in predicting
PV demand and commodity prices, as well as how forecast models affect the supply chain
performance metrics. Finally, Section 6 provides concluding remarks along with relevant
managerial insights and further research directions.

2 Distinct challenges in PV supply chains

The comprehensive literature review and taxonomy presented in Section 1 highlights the
utility of prescriptive and predictive models in developing well-informed decision-making
tools for maximizing supply chain performance, especially through the integration of data
analytics and AI-based techniques. The strong connection between demand planning and sup-
ply chain efficiency in particular, underscores the need for AI-employed approaches which
leverage advanced data analytics for efficient demand planning to navigate the complexi-
ties of today’s dynamic market environment to develop cost-competitively resilient supply
chains. With effective forecasting of demand and price, organizations can better utilize their
resources, minimize waste, and take prompt actions to market fluctuations.
Next-generation data-driven approaches will be instrumental in emerging and challenging
supply chains of material circularity that need immediate attention such as the reverse supply
chain networks of end-of-life (EoL) PV panels [47, 48]. Solar PV technology has become a
key renewable energy source in the ongoing energy transition supported by industrial policy
in the US and the EU. Long-term projections from the International Renewable Energy
Agency (IRENA) suggest that global PV capacity will reach 1,630 GW by 2030 and 4,500
GW by 2050 [49]. The rapid growth of solar PV investments for the energy transition brings
along other problems that need immediate attention, such as how to manage the handling
of end-of-life panels and recirculate the valuable materials they possess back into critical
supply chains.
PV panels have a limited lifespan (20–30 years), resulting in significant waste as they
reach the end of their operational life. This is further compounded by material losses and
damaged items during the packaging and logistics stages of the end-to-end supply chain.
Despite the surging demand, the current recycling ratio is only around 10%, and the majority
of PV waste materials end up in landfills [50]. This underscores the urgent need for closed-
loop supply chain networks that can effectively recover and reintroduce valuable resources
into the economies.
PV panels contain valuable materials essential for critical supply chains such as silver,
silicon, copper, aluminum, and other rare-earth materials which are highly critical for the
production of high-value goods such as batteries, EV vehicles, semiconductors, and new PV
panels. Figure 1 shows the distribution of PV waste materials by mass and value, highlighting

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Fig. 1 Distribution of materials of PV waste by mass and value, respectively Data source [53] .

the fact that while high-value components such as silver and other rare-earth metals represent
a smaller portion of a panel by mass compared to aluminum and glass, they have significant
economic potential due to their use in various industries. Indicatively as of 2024, the demand
for silver remains strong, consistently outrunning supply in recent years. This is largely due
to growing industrial demand, driven in large part by the solar-panel manufacturing sector,
where silver plays an integral role. From 2019 to 2023, solar-sector demand for silver surged
by 158%, and it is expected to increase by an additional 20% this year [51]. The growing
reliance on rare-earth elements, essential to a low-carbon transition, presents a vital challenge
to global sustainability due to the impact of mining for these elements, further providing an
incentive to recover rare-earth elements from waste materials [52]. The potential for value
creation through raw material recovery from PV recycling is estimated to be around $450
million by 2030 and $15 billion by 2050 [49].
The contrast between the growing market for solar energy and the inadequate recycling
infrastructure emphasizes the strong need for innovative solutions for reverse logistics to
support the development of closed-loop supply chains (CLSCs) for PV panels. To navigate
the inherent complexities and overcome the challenges of this emerging field, there is a
strong demand for robust, flexible, and AI-employed decision-making tools. These tools
should also incorporate the concept of "radical optionality" [54], enabling the evaluation
and navigation of multiple scenarios to manage the uncertainties that are ever-present in the
recycling landscape. Key decision-making options within these tools should include:

• Retaining valuable materials such as silver, silver, a critical commodity for multiple
supply chains,
• Selling materials on the spot market,
• Utilizing materials in the manufacturing of additional PV panels.

This work represents an initial effort towards developing AI-empowered demand plan-
ning embedded into a decision-making framework to strengthen cost-competitiveness. We
explore the integration of statistical and DL-based forecasting models into a supply chain
optimization model, focusing on predicting both demand and commodity prices. By inte-
grating these decision processes, our approach provides a flexible and adaptive framework
that enhances supply chain resilience and adaptability. This dynamic approach empowers
decision-makers to assess various scenarios and make optimal choices in a highly competi-
tive and volatile market. While the current model is an initial prototype, it demonstrates the
significant potential for leveraging advanced data analytics to empower decision-makers to
assess various scenarios and make more informed choices with agility and foresight.

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Journal of Global Optimization

3 Methodological framework

Our methodological framework aims to build forecasting models for a broad set of commodity
prices such as silver, aluminium, copper, glass, polymer, and demand for PV panels, each
with unique characteristics and challenges to forecast accurately. A flowchart for the proposed
methodology is represented in Figure 2.

3.1 Data collection and preprocessing

The complexity of the forecasting process is increased by the vast number of distinct time
series involved, each with its own unique patterns, trends, and cycles. All the data sources
for demand and price are obtained from open-access sources, and are reported on a monthly
basis.
The data preprocessing is initiated by splitting the data into 80% for training and 20% for
testing. To manage variability, mathematical transformations like logarithms or Box-Cox are
applied as needed. The data is then processed to check for stationarity in mean and variance
using Seasonal and Trend decomposition using Loess (STL) decomposition and differencing
to remove trends and seasonal effects [55]. The choice between additive and multiplicative
decomposition is checked and applied by analyzing seasonal fluctuations that vary with data
levels. Stationarity analysis is crucial for statistical predictive model implementations to
ensure that statistical properties of the time series remain consistent over time. Differencing
and transformations stabilize the mean and variance, with unit root tests determining the
necessary order of differencing.

3.2 Selection of prediction models and implementation

A well-designed methodology is essential for producing reliable and insightful predictions.

The success of any forecasting process depends on the careful selection of models that are
tailored to the characteristics of the data being analyzed. This research prioritizes a diverse
set of forecasting models, selected for their proven effectiveness and strong validation in
the literature [56]. To this effect, we incorporate two deep learning approaches - Multi-
layer Perceptron (MLP) and Long Short-Term Memory (LSTM) networks - alongside a
statistical method. This combination enhances predictive accuracy and allows for a deeper
understanding of complex data patterns.
MLP is a deep neural network that processes input signals through multiple hidden layers
to capture complex, non-linear relationships in the data. An MLP network can be expressed
mathematically as follows:

q
y = g( wiO . f (x T .wiI + bih ) + b O ), (1)
i=1

where w I represents the input weight matrix and w O is the output weight matrix. Similarly,
b O is the bias vector of the output layer and bh is the bias vector of the hidden layer, while
f is the activation function for the hidden layer and g is the activation function of the output
layer [57]. MLP models are effective on supervised problems indicating that the correlation
between input and output are trained by a data-set. In the training step, the weights and
biases are modified to minimize error values. The weight and bias adjustments are applied

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with back-propagation. The number of hidden layers, the number of neurons in each layer,
and the weights between neurons affect the performance of MLP.
LSTM, distinct from traditional feed-forward networks, incorporates a memory cell that
retains information over time making it especially effective for time series forecasting by
capturing long-term dependencies. The information processing process of LSTM [58] can
be expressed mathematically by the following equations:

i t = σ (Wl xt + Hl h t−1 + bl ) (2)

f t = σ (W f x f + H f h t−1 + b f ) (3)
ot = σ (Wo xt + Ho h t−1 + bo ) (4)
c̃t = tanh(Wc xt + Hc h t−1 + bc ) (5)
ct = f t ∗ ct−1 + i t ∗ c̃t (6)
h t = ot ∗ tanh(ct ) (7)
1
σ (x) = (8)
1 + e−x
e x − e−x
tanh(x) = x , (9)
e + e−x

where i t is the input gate, f t is the forget gate, and ot is the output gate, σ refers to the
sigmoid function with a range of 0 to 1, the tanh function is applied to determine the value
between -1 and 1. Also, c̃t is the candidate value for the states of the memory cell at time t.
Similarly, ct is the state of the current memory cell at time t and finally, h t is the output value
filtered by the output gate. On the right side of the equations, x t is the input to the memory
cell at time, t, weight matrices are Wi , W f , Wo , Wc , Hi , H f , Ho , Hc , and bias vectors are
bi , b f , bo and bc , respectively.
Alongside these, our research employs ARIMA (AutoRegressive Integrated Moving Aver-
age), a robust statistical model known for its effectiveness in modeling time series data.
ARIMA combines autoregressive and moving average components, with differencing to
handle non-stationarity, making it particularly useful for data with consistent trends. When
seasonality is detected, the Seasonal ARIMA (SARIMA) model is applied to incorporate
these patterns; otherwise, standard ARIMA is used.
The ARIMA model can be expressed mathematically as follows:


p 
q
xt = c + φi xt−i + t + θi t−i , (10)
i=1 i=0

where xt is the stationary variable, c is constant, φi , are auto-correlation coefficients and t

are the residuals. The θi terms are the weights applied to the current and prior values of a
stochastic term in the time-series.
The model assumes that the residuals follow a Gaussian white noise process, with a mean
of zero and a constant variance. The parameters φi and θi must be non-zero for the model to
be meaningful. This framework allows the ARIMA model to effectively capture and forecast
time series patterns by considering both past values and past errors [55]. By incorporating
seasonal terms on the general ARIMA model, a Seasonal ARIMA (SARIMA) model can be
expressed as ARIMA(p,d,q)(P,D,Q)m, where (p,d,q) represents non-seasonal and (P,D,Q)m
accounts for seasonal aspects. To analyze potential seasonal effects on the forecasting results,
both ARIMA and SARIMA methods are utilized.

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Table 2 Hyperparameter tuning through grid search

Model Hyperparameters Tuned

ARIMA / SARIMA p: Order of the autoregressive part, [1−−5];

d: Degree of first differencing, d = 1;
q: Order of the moving average part, [1−−4];
P: Order of seasonal autoregressive part, [1−−5];
D: Degree of seasonal differencing, [0, 1];
Q: Order of seasonal moving average part, [1−−4];
s: Seasonal period, s = 12
MLP & LSTM I nputs: Number of prior inputs to use as model input, [h, 2h];
N odes: Number of nodes in hidden layer, [32, 64, 128];
Dr opout.rate: [0.1];
Lear ning.rate: [1e-4, 1e-6];
E pochs: Number of training epochs, [250, 500];
Batch.si ze: Number of samples per mini-batch, [8, 16, 32];
Di f f er ences: Applied differencing to inputs, [0, 1, h];
Standar di zation: [True, False]

Due to the extensive number of hyperparameters to be determined, both deep learning and
ARIMA/SARIMA models are evaluated separately using grid search to identify the optimal
configuration for each approach. Grid search is a systematic tuning method that exhaustively
explores a predefined set of hyperparameter combinations, selecting the set that yields the
best performance based on validation metrics. The detailed hyperparameter settings explored
during grid search for each model can be found in Table 2. This strategy leverages the strengths
of each method: deep learning models excel in capturing complex patterns, while ARIMA
and SARIMA offer robust statistical modeling for linear and seasonal trends. By comparing
the performance of these models, the study aims to identify the best forecasting method,
ultimately providing accurate and reliable predictions that support informed decision-making
in complex environments.

3.3 Evaluation metrics for prediction models

In order to select the best performed prediction models, we employ widely-applied accuracy
metrics, which are the root mean squared error (RMSE) and the mean absolute error (MAE).
MAE is often preferred for its straightforward interpretation and ease of calculation [55]. The
fundamental difference between MAE and RMSE lies in their error calculation methods.
MAE determines the absolute error loss, providing a direct measure of forecast accuracy,
whereas RMSE uses a squared error loss function, giving more weight to larger errors. This
makes RMSE particularly effective at detecting significant errors. Additionally, in datasets
with infrequent outliers, RMSE may outperform MAE due to its increased sensitivity to
extreme values. Based on these accuracy metrics, the optimized hyperparameters for each
model were selected through grid search, and the corresponding values are presented in the
Appendix B for each commodity price and demand.

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3.4 Integration of prediction models to supply chain optimization

The closed-loop supply chain model is designed as a deterministic multi-period model which
represents the entire process from the collection of end-of-life PV panels to the production
of new panels using both recycled materials and additional virgin material from external
suppliers. At recycling centers, PV wastes are recycled into precious commodities including
silver, aluminum, copper, glass, and polymer. The Full Recovery End-of-Life Photovoltaic
(FRELP) process technology is assumed for the recycling process, with conversion rates
taken from [59]. At manufacturing centers, the required materials to produce new PV panels
[53] are obtained from both recycled centers and external suppliers. The model objective is
to reduce total expenses while meeting demand, controlling inventory levels, and choosing
external material suppliers (supply chain sourcing). To ensure accuracy and relevance, real-
world U.S. market pricing for PV panel demand and commodity prices are obtained from
publicly available data.
Once the supply chain model is established, the next important step is integrating the
most accurate predictive models. Among SARIMA, MLP, and LSTM models, based on the
obtained error values, the model having the highest accuracy (minimum loss) is selected
and integrated to the optimization model. These forecasts are then used to inform key deci-
sion variables within the optimization framework, such as material flows, inventory levels,
and external supplier selection. By embedding the forecasts into the supply chain model,
predictive analytics are translated into critical parameters that drive overall performance.

3.5 Supply chain key performance indicators

In this step, the key performance metrics for evaluating the supply chain are defined. These
metrics focus on cost minimization, inventory management, demand fulfillment, and the
efficient selection of external suppliers. These metrics are essential for assessing how well
the integrated forecasting and optimization model meets the overall objectives of the supply
chain.
The Supply Chain Performance Metric (SCPM) is defined as a weighted sum of several
key components:

SC P M = w1 · T Cnorm + w2 · I Tnorm + w3 · S L norm ,

Where:

– w1 , w2 , and w3 are weights assigned to each component, reflecting their relative impor-
tance in the overall performance assessment. In this study, the assigned weights have no
effect on the Analysis of Variance (ANOVA) analysis so they have equal weights.
– T Cnorm represents the normalized total cost, ensuring that cost efficiency is a central
focus.
– I Tnorm stands for the normalized inventory turnover, a key metric for evaluating how
effectively inventory is managed. The Inventory Turnover metric is calculated as:
Cost of Goods Sold
Inventory Turnover =
Value of Average Inventory
This metric provides insight into how efficiently the inventory is being used to generate
sales.

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Journal of Global Optimization

– S L norm denotes the normalized service level, which captures the supply chain’s ability to
fulfill orders on time. Thus, it measures the supply chain’s reliability in meeting customer
demand promptly. The Service Level is calculated as:
Number of Orders Delivered on Time
Service Level =
Total Number of Orders
Together, these metrics provide a comprehensive evaluation framework, allowing for a multi-
objective assessment of how well the supply chain meets its goals in terms of cost, efficiency,
and customer satisfaction. By normalizing and weighting these components, the SCPM pro-
vides a balanced and flexible approach to performance evaluation, ensuring that the supply
chain operates optimally across all critical areas.

3.6 Supply chain performance evaluation under multiple scenarios

Finally, the constructed supply chain model is used to generate various scenarios to evaluate
supply chain performance under different conditions. This step allows for the assessment of
how changes in demand, commodity prices, or other factors impact the overall efficiency
and cost-effectiveness of the supply chain. The results from these scenarios provide valuable
insights into the robustness and adaptability of the supply chain model.
In this study, scenarios are generated by varying the recycling ratio within the supply
chain. Specifically, the proportion of materials used in the manufacturing of new PV panels
that need to be sourced from recycling facilities is adjusted to different levels from 0% to
100%, resulting in a total of 15 different scenarios. For each scenario, the SCPM is calculated
to evaluate the impact of these changes. These scenarios are then investigated using three
different data sets for demand and price as key variables in the optimization model:
• monthly actual/historical values,
• values predicted by the best-performing forecasting models,
• simple average values.
By calculating the SCPM for each of these scenarios, we can compare the performance
under different conditions.
To analyze the impact of forecasting accuracy on the supply chain performance, the
ANOVA (analysis of variance) method is implemented. ANOVA is a statistical technique
used to determine whether there are significant differences between the means of different
groups. In this context, it helps to assess whether the differences in SCPM across scenarios
using actual, forecasted, and simple average data are statistically significant. To construct
and test the hypotheses, we define the following:
– Null Hypothesis (H0): Forecasting, particularly the most accurate ones, does not signif-
icantly impact the supply chain performance metric (SCPM). It can be expressed as:
– H0: μSCPMactual = μSCPMfcast = μSCPMs.avg ,
where null hypothesis indicates that the mean SCPM values across actual, forecasted,
and simple average scenarios are equal.
– Alternative Hypothesis (H1): Forecasting significantly impacts the supply chain perfor-
mance metric. Formally, these hypotheses are expressed as:
– H1: At least one of the means is different.
To test the hypotheses, the ANOVA method is applied which provides critical insights
into the significance of forecasting in supply chain performance. If the null hypothesis is

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 Journal of Global Optimization

Fig. 2 Methodological framework.

rejected, it indicates that forecasting has a statistically significant impact on the SCPM,
thereby validating the importance of incorporating advanced forecasting techniques into
supply chain management practices.

4 Illustrative case study

Closed-loop PV supply chain

We consider a central planner who is tasked with optimizing a closed-loop supply chain
for PV panel recycling and manufacturing. This system spans multiple locations and centers,
represented by different sets, and operates over a series of planning periods, T . The EoL PV
panels are collected in PV collection centers before they are sent to recycling centers. In the
recycling centers, the EoL PV panels are recycled with processes selected from the existing
literature and practice in order to extract and re-gain the valuable materials within the pan-
els such as aluminum, copper, silicon etc. These recycled materials are then transported to

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Journal of Global Optimization

Fig. 3 Representation of closed-loop supply chain for PV.

Material Collection Centers, which supply the PV manufacturers with the materials needed
in the production of new PV panels. As the efficiency of the recycling processes is not 100%,
it is assumed that any additional materials required to produce PV panels are acquired from
the commodities market; i.e. external suppliers. The overall aim of the model is to mini-
mize total cost of the supply chain, including costs of production, transportation, additional
material procurement from the external market, storage, and penalties with respect to unmet
demand. The optimization problem involves several key sets that define the components and
relationships within the supply chain:
Centers I represent PV Waste Collection Centers where EoL PV panels are initially col-
lected and stored. Centers J indicate Recycling Centers, in which PV waste are recycled. The
recycled materials are then stored at Material Collection Centers K , which also act as storage
areas for purchased materials from external suppliers. The set of materials that are required
to produce new PV panels are referred to as L. In the final stage, new PV panels are man-
ufactured in PV Manufacturing Centers, N . Based on the given demand, the required type
and amount of material are transported to these centers. In order to satisfy the material need
to produce new PV panels, we employ External Suppliers E which represent corresponding
set of external suppliers for each material in set of L. As an example, Suppliersilver pro-
vides additional need of silver from external supplier. The planning period of the model T ,
represents the planning horizon, which has multiple periods. In this case study, the strategic
planning period is employed with 12 months.
The planner must make strategic decisions about the allocation of materials, including how
much material to move from Centers I to Centers J , how to distribute recycled materials
from Centers K to Centers N , which external suppliers to engage for additional resources,
and how much material to store at Centers K . The aim of the model is to minimize total costs;
constituting of production, transportation, additional material procurement, and storage costs,
along with penalties due to unmet demand, while ensuring that inventory levels are optimal
and that demand for new PV panels is met throughout the planning period. At Center J , the
FRELP [59] process is selected as the process technology for recycling, and at Centers N ,

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new PV panels are manufactured based on the conversion rates obtained from [53]. While
the model is set up, the additional steps can be categorized into 3 parts as:

(a) Using actual/historical demand and commodity prices

(b) Selecting the best forecasting model among SARIMA, MLP and LSTM
(c) Using a statistical approach as simple average.

The aforementioned steps are integrated into the optimization model to analyze the impact
of input data accuracy on total cost, inventory management and demand satisfaction. To
observe the effect of the predictive analysis on the overall supply chain, different scenarios
are generated by varying the overall recycling ratio in the supply chain model, from 0% to
100%. A total of 15 scenarios are generated and the overall performance is analyzed for each.
Details of the Mixed Integer Linear Programming (MILP) model used in this analysis are
provided in Section A. The MILP model is conceptually described below:

Capital Expenses
+Fixed and Variable Operational Expenses
+Transportation Cost
minimize T otalCost{ +Material Cost
+Demand Penalty
+EoL PV cost
−Revenue of new PVs
subject to: Demand fulfillment
Inventory management
h(x, y) = 0 Supplier selection
g(x, y) ≤ 0 Capacity utilization
Material/mass balances
Transportation constraints,

where the goal is to find values of the operational (x ∈ Rn ) and strategic (y ∈ Y = {0, 1}m )
decision variables, subject to the set of equality (h(x, y) = 0) and inequality (g(x, y) ≤ 0)
constraints, and the Total Cost is minimized for the whole supply chain ecosystem.

5 Results and discussions

Table 3 summarizes the selected best forecasting models, among SARIMA, ML and LSTM,
and obtained error metrics. The forecasting results represents the seven time series including
PV demand and different commodity prices. For all forecasting models to have the optimum
structure, grid search is applied to define their optimal hyperparameters. The results clearly
show that MLP is the most frequently utilized forecasting method, being selected for three
different commodity products. It is followed by LSTM and SARIMA, each selected for two
commodity products. These findings emphasize that neural network-based models, such as
MLP and LSTM, perform better in handling complex datasets thus making them ideal for
capturing intricate trends and fluctuations. On the other hand, traditional statistical models
like SARIMA are more effective for datasets that are relatively stationary and have simpler
structures. The accuracy and performance of each model highly depend on the specific char-
acteristics of the dataset and its optimized hyperparameters. It is also important to note that
the findings further underscore that deep learning based models outperform statistical meth-
ods when the dataset displays increased complexity, like non-linearity or non-stationarity.

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Table 3 The best prediction models for PV demand and commodity prices along with error metrics
PV demand & commodity prices Selected Model MAE RMSE

PV Demand, GW MLP 1.43 2.15

Silver, $/kg LSTM 34.78 43.49
Silicon, $/kg SARIMA 2.33 2.71
Polymer, $/kg SARIMA 0.304 0.33
Aluminum, $/kg LSTM 0.113 0.17
Copper, $/kg MLP 0.303 0.44
Glass, $/kg MLP 0.72 0.73

Table 4 Post Hoc Test Result: Group 1 Group 2 Mean Diff. p-adj Reject
Tukey HSD
SCPM_actual SCPM_s.avg 0.221 0.0132 True
SCPM_actual SCPM_fcast 0.025 0.9498 False

Consequently, it is essential to evaluate a range of models, as the best-performing model can

vary significantly based on the underlying data structure.
After making the decision on the best forecasting models, the next step involves replacing
the historical PV demand and commodity prices with the forecasted values and analyzing
how these predictions affect total cost, inventory management and demand fulfillment in the
whole supply chain. To achieve this, the SCPM metric is derived for each generated scenario
for capturing the effect of prediction accuracy in supply chain optimization.
Figure 3 illustrates the impact of prediction performance on SCPM values across different
scenarios. The left-most (green) box-plot represents the baseline scenario utilizing historical
data for PV demand and commodity prices, while the middle box-plot shows the SCPM
values generated with the most accurate forecasting models identified in Table 3. The right-
most (blue) box-plot represents the SCPM values derived by using a simple statistical simple
average data to assess the effect of accuracy of forecast precision. The results show that
SCPM values derived from historical and accurately forecasted data exhibit similar medians
and variability, indicating close alignment between these two approaches. Notably, the SCPM
values obtained with accurate forecasts demonstrate slightly lower variance, suggesting that
the supply chain operates more stably and predictably, with reduced exposure to fluctuations.
Lower variance in SCPM values reflects a lower risk profile, enabling the supply chain to
maintain consistent performance in cost, inventory, and demand management. In contrast,
the SCPM values based on the simple average data display significantly greater variability,
highlighting increased unpredictability and a higher level of operational risk. Such high
variance signals instability and implies greater difficulty in anticipating and responding to
changing market conditions. Overall, accurate forecasting plays a critical role in reducing
uncertainty and variability, directly contributing to enhanced, reliable, and stable supply chain
performance metrics
To quantify the impact of forecast accuracy on SCPM values, the Tukey’s Honest Signifi-
cant Difference (HSD) test is conducted to determine whether there is a statistically significant
difference between each pair of groups [60]. The obtained results in Table 4 show that there
is no statistically significant difference between SC P Mactual and SC P Mfcast groups, as indi-
cated by a p-value higher than 0.05, which confirms that with a 95% confidence level we

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Fig. 4 Historical data vs the best forecasting models for demand and commodity prices.

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Fig. 5 SCPM values for the comparison.

fail to reject the null hypothesis. This further shows that the SCPM values derived from the
actual and forecasted data are quite similar. On the other hand, a significant difference is evi-
dent between SC P Mactual and SC P Ms.avg groups, as shown by a p-value of 0.0132, which
is less than 0.05. This leads us to reject the null hypothesis, demonstrating that the SCPM
values from the simple average method differ significantly from the actual values, further
underscoring the importance of using accurate forecasting methods in maintaining reliable
SCPM values.

6 Conclusions

In this study, statistical and DL-based forecasting approaches are implemented to predict PV
demand and commodity prices such as silver, aluminum, copper, silicon, polymer, and float
glass which are not only vital for PV ecosystem but also for other strategically important
feed-forward supply chains. These materials are essential components in industries such as
electronics, electric vehicles, and battery manufacturing, which face similar challenges in
material availability, price volatility, and sustainability constraints. Therefore, the proposed
forecasting-optimization framework has the potential to be adapted to various circular supply
chain structures beyond PV recycling.
To assess the impact of forecasting accuracy on supply chain performance, SCPM values
are calculated across a set of generated scenarios. In these scenarios, the actual historical input
data for PV demand and commodity prices is systematically replaced with two alternative
datasets: (i) predictions from the best-performing forecasting models, and (ii) simple average
values. This substitution has resulted in three distinct SCPM groups, each representing a
different data input strategy. An analysis of variance (ANOVA), followed by a post-hoc
HSD test, is conducted to compare differences between these groups. The results indicate no
statistically significant difference between the SCPM values obtained using actual data and
those derived from accurate forecasts, reinforcing the reliability of the predictive models. In

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contrast, the group based on simple average inputs has performed significantly worse, both in
mean and variance, underscoring the critical importance of advanced forecasting techniques
in capturing market dynamics and supporting effective supply chain management.
Our work emphasizes the role of predictive analytics in reducing uncertainty and improv-
ing supply chain decisions. By integrating forecasts into optimization frameworks, we
demonstrate how these models can enhance the overall efficiency, resilience and sustainabil-
ity of supply chains, particularly in industries like renewable energy. To further harness the
abilities of AI-based tools in the PV supply chain, these tools can be utilized in areas beyond
demand planning such as for material identification, sorting, and processing to optimize recy-
cling precision and efficiency. While the current study highlights the potential of AI-driven
forecasting and optimization in circular supply chains, it is important to acknowledge certain
limitations. These include potential challenges in data availability, the computational burden
of training and tuning complex models, and simplifying assumptions used during model-
ing. Furthermore, despite achieving high forecasting accuracy, deep learning models such as
MLP and LSTM are often considered "black-box" models due to their limited interpretabil-
ity. Understanding the influence of individual input variables is crucial for building trust and
transparency in forecasting outcomes. Future work could address these challenges by incor-
porating real-time data streams, uncertainty-aware modeling, and hybrid AI-optimization
approaches, along with model interpretability techniques such as feature importance anal-
ysis. This would not only enhance transparency and explainability but also support more
informed and accountable decision-making in supply chain planning frameworks. Addition-
ally, developing models which employ disruption analysis and strategic planning capabilities
could add significant value towards a cost-effective and sustainable circular supply chain
ecosystems.

A Nomenclature and model formulation

The nomenclature and model formulation for the case study in Section 4 are outlined below.
Objective Function
The objective of the model is to minimize the total cost, which includes capital expenditures
(CapEx), operating expenditures (OpEx), transportation costs, material costs from external
suppliers, storage costs, contracting costs, penalties for unmet demand, waste panel costs,
and revenue from selling new PV panels.

Table 5 Sets
Notation Description

I Set of PV Waste Collection Centers

J Set of Recycling Facilities
K Set of Material Sorting and Storage Centers
L Set of Recovered Materials (glass, polymer, silver, copper, aluminum, silicon)
N Set of PV Manufacturing Plants
E Set of External Material Suppliers
T Set of Time Periods
Z Set of Transportation Modes

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Table 6 Decision Variables

Notation Description
coll
qi,t Quantity of PV waste collected at site i during period t
trans
qi, Quantity of PV waste transported from i to j at time t
j,t
q rec
j,k,l,t Quantity of recovered material l from j sent to k at time t
dist
qk,n,l,t Quantity of material l from k to plant n at time t
prod
qn,t Quantity of PV panels produced at plant n at time t
ext
qe,k,l,t Quantity of material l bought from external supplier e for k at t
sk,l,t Inventory of material l at center k at time t
ut ≥ 0 Unmet demand at time t
be,l ∈ {0, 1} Binary: 1 if supplier e is selected for material l
z i, j,t ∈ {0, 1} Binary: 1 if transport mode is selected among centers at time t

Table 7 Parameters
Notation Description

citrans
j Transport cost per unit from center i to center j
ctrans
jkl Transport cost from center j to center k for material l
trans
ckln Transport cost from center k to center n for material l
cap
ci, j,k,n Capital expenditure per unit capacity at centers i, j, k, n
op op op op
ci , c j , ck , cn Operational expenditure per unit for centers i, j, k, n
p pv Unit selling price of PV panel
cwaste Cost per unit of PV waste
ηlrec Recycling yield for material l
ηlmanu Manufacturing yield for material l
Cimax
j Max transport capacity of the selected transport mode
stor
ck,l,t Storage cost of material l at k at time t
contr
ce,l Contracting cost for supplier e and material l
cpenalty Penalty for unmet demand
sup
cape,l,t Max capacity of supplier e for l at t
demandt New PV demand at time t
pe,k,l,t Unit price of material l from supplier e to center k at time t

Minimize: CapEx + OpEx + Transportation Cost + Material Cost + Storage Cost +

Contracting Cost + Unmet Demand Penalty + Waste Panel Cost − Revenue from new PVs
min Total Cost =
   cap
ci, j,k,n · Capi, j,k,n
i∈I j∈J k∈K n∈N
 op
   op
+ coll
qi,t · ci + j,k,l,t · c j
q rec
i∈I t∈T j∈J k∈K l∈L t∈T

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   op
 prod op
+ dist
qk,n,l,t · ck + qn,t · cn
k∈K n∈N l∈L t∈T n∈N t∈T
   
+ qi,trans
j,t · citrans
j + j,k,l,t · c jkl
q rec trans

i∈I j∈J t∈T j∈J k∈K l∈L t∈T

  
+ dist
qk,n,l,t · ckln
trans

k∈K n∈N l∈L t∈T

    
+ ext
qe,k,l,t · pe,k,l,t + stor
ck,l,t · sk,l,t
e∈E k∈K l∈L t∈T k∈K l∈L t∈T

+ contr
ce,l · be,l
e∈E l∈L

+ cpenalty · u t
t∈T

+ cwaste · qi,t
coll

i∈I t∈T
 prod
− p pv · qn,t
n∈N t∈T

Constraints
• Facility Capacity Constraints:

j,t ≤ Capi,t ,
qi,trans ∀i ∈ I , t ∈ T
coll

j∈J
coll
qi,t ≤ Capi,t
coll
, ∀i ∈ I , t ∈ T

q j,k,l,t ≤ Cap rec
rec
j,t , ∀ j ∈ J, t ∈ T
k∈K l∈L

j,t ≤ Cap j,t ,
qi,trans ∀ j ∈ J, t ∈ T
rec

i∈I

dist
qk,n,l,t ≤ Capk,t
stor
, ∀k ∈ K , t ∈ T
n∈N l∈L
 
j,k,l,t +
q rec ≤ Capk,t , ∀k ∈ K , t ∈ T
ext stor
qe,k,l,t
j∈J e∈E
 prod
qn,t ≤ Capn,t
manu
, ∀n ∈ N
t∈T
 sup
ext
qe,k,l,t ≤ cape,l,t · be,l , ∀e ∈ E, l ∈ L, t ∈ T
k∈K

• Material Balances:

j,t ≤ qi,t ,
qi,trans ∀i ∈ I , t ∈ T
coll

j∈J
 
j,k,l,t ≤ ηl ·
q rec j,t , ∀ j ∈ J , l ∈ L, t ∈ T
rec
qi,trans
k∈K i∈I
  
j,k,l,t +
q rec ≤ , ∀k ∈ K , l ∈ L, t ∈ T
ext dist
qe,k,l,t qk,n,l,t
j∈J e∈E n∈N

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prod

qn,t · ηlmanu ≤ dist
qk,n,l,t , ∀n ∈ N , l ∈ L, t ∈ T
k∈K

• Demand Satisfaction:
 prod
qn,t ≥ 0.8 · demandt , ∀t ∈ T
n∈N

 prod
qn,t + u t = demandt , ∀t ∈ T
n∈N

• Inventory Balance:
 
dist
qk,n,l,t ≤ sk,l,t−1 + j,k,l,t +
q rec ext
qe,k,l,t , ∀k ∈ K , l ∈ L, t ∈ T
j∈J e∈E
  
sk,l,t = sk,l,t−1 + j,k,l,t +
q rec ext
qe,k,l,t − dist
qk,n,l,t , ∀k ∈ K , l ∈ L, t ∈ T
j∈J e∈E n∈N

• Transport Mode Capacity:


j,t ≤ Ci, j · z i, j,t , ∀i ∈ I , j ∈ J , t ∈ T
qi,trans max

i∈I j∈J

• Supplier Activation:
ext
qe,k,l,t ≤ M · be,l , ∀e ∈ E, ∀k ∈ K , ∀l ∈ L, ∀t ∈ T
M, Big-M, represents maximum possible purchase quantity of material l from supplier
e to center k at time t.

B Key hyperparameters

This section provides the detailed information for the the obtained key hyperparameters for
each commodity price and demand.

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Table 8 Best forecasting methods and configurations

Demand & Commodity Products Selected Method Best Configuration

PV Demand MLP # of indices tested = 108, # of inputs = 13, # of nodes

= 64, dropout = 0.1, learn_rate = 0.0001, # of epochs =
250, # of batches = 16, differences taken = 1, data scaled
= False, future horizon = 13
Silver LSTM # of indices tested = 24, # of inputs = 13, # of nodes = 64,
dropout = 0.1, learn_rate = 0.0001, # of epochs = 250, #
of batches = 8, differences taken = 1, data scaled = True,
future horizon = 13
Silicon SARIMA p = 0, d = 0, q = 1, P = 0, D = 0, Q = 0, seasonal
period s = 12, future horizon = 12
Polymer SARIMA p = 0, d = 0, q = 0, P = 0, D = 1, Q = 0, seasonal
period s = 12, future horizon = 12
Aluminum LSTM # of indices tested = 12, # of inputs = 13, # of nodes = 32,
dropout = 0.1, learn_rate = 0.0001, # of epochs = 250, #
of batches = 16, differences taken = 1, data scaled = True,
future horizon = 13
Copper MLP # of indices tested = 24, # of inputs = 13, # of nodes = 32,
dropout = 0.1, learn_rate = 0.0001, # of epochs = 250, #
of batches = 16, differences taken = 1, data scaled = True,
future horizon = 13
Glass MLP # of indices tested = 12, # of inputs = 13, # of nodes = 32,
dropout = 0.1, learn_rate = 0.0001, # of epochs = 250, #
of batches = 8, differences taken = 1, data scaled = True,
future horizon = 13

C Forecast-driven objective approximation and convexity assurance

C.1 Forecast-based continuous formulation

We isolate the forecast-dependent terms in the total cost:

J = Jstatic + Jdemand (d(t)) + Jprice ( pSi (t)) (i)

where:

• Jstatic : CapEx, OpEx, transportation, storage, contracting, and waste-related terms

T
• Jdemand (d(t)) = 0 cpenalty · u(t) dt with u(t) = max{0, d(t) − qprod (t)}
T
• Jprice ( pSi (t)) = 0 qext,Si (t) · pSi (t) dt

Using forecasted values d̂(t) and p̂Si (t), the approximation becomes:

 T  
Jˆ = Jstatic + cpenalty · û(t) + qext,Si (t) · p̂Si (t) dt (ii)
0

where û(t) = max{0, d̂(t) − qprod (t)}.

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C.2 Discrete-time approximation and MILP structure

To operationalize the forecast-based continuous formulation, we discretize the planning hori-

zon into finite time periods t ∈ T = {1, 2, . . . , N } and define functional forms for demand
and silicon price at each t.
Demand Function:

demandt = αt 2 + βt + γ , ∀t ∈ T

Silicon Price Function (Piecewise Time-Dependent):


at 2 + bt + c, if t ≤ T thresh
p̃e,k,Si,t = ∀e ∈ E, k ∈ K , t ∈ T
et + f , if t > T thresh

Substituting the functions of demand and price into the forecast-based cost approximation
(in Section C.1), Equation (ii) becomes:
 
Jˆ ≈ Jstatic + cpenalty · û t + qext,Si,t · p̃e,k,Si,t (iii)
t∈T

The objective function can be revised by substituting the approximated demand and price
functions as follows:


Minimize Total Cost = ... + cpenalty · u t
t∈T
  
+ ext
qe,k,l,t · pe,k,l,t
e∈E k∈K l∈L\{Si} t∈T

+ ext
qe,k,Si,t · p̃e,k,Si,t
e∈E k∈K t∈T

Revised Constraints:
 prod
qn,t + u t = αt 2 + βt + γ , ∀t ∈ T
n∈N
 prod
qn,t ≥ 0.8 · (αt 2 + βt + γ ), ∀t ∈ T
n∈N
u t ≥ 0, ∀t ∈ T

Table 9 Additional Variables and Parameters

Notation Description

α, β, γ Coefficients for the quadratic demand function demandt = αt 2 + βt + γ

T thresh Threshold time period after which silicon price becomes linear
a, b, c Coefficients of the quadratic price function for silicon (when t ≤ T thresh )
e, f Coefficients of the linear decreasing price function (when t > T thresh )
p̃e,k,Si,t Time-dependent piecewise price function for silicon

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C.3 Approximation error and sensitivity to forecasts

Define the forecast errors:

εd (t) = d(t) − d̂(t), ε p (t) = pSi (t) − p̂Si (t)
Assuming f (x; d, p) is Lipschitz continuous with constants L d , L p , we have:

| f (x; d, p) − f (x; d̂, p̂)| ≤ L d · |εd (t)| + L p · |ε p (t)|

 T

| Jˆ − J | ≤ L d · |εd (t)| + L p · |ε p (t)| dt
0
This bound quantifies the impact of forecast errors on the objective.

C.4 Convexity and global optimality

The incorporation of time-dependent demand and piecewise-defined silicon prices does not
alter the fundamental structure of the optimization model. Both functions, forecasted demand
and forecasted silicon prices, are exogenous parameters. They are defined solely as functions
of time and are independent of the model’s decision variables. As such, they enter the objective
function and constraints as known parameters.
Among the cost components, only the material cost associated with silicon and the penalty
for unmet demand are directly influenced by these predicted quantities. All other terms such as
capital expenditures, operational costs, transportation, storage, are structurally independent
of forecast uncertainty and remain linear in the decision variables.
Moreover, the forecast-dependent terms are integrated into the model through linear
expressions. The unmet demand variable u t is a continuous decision variable, and the time-
varying silicon price p̃e,k,Si,t is embedded into the material cost as a deterministic coefficient.
Consequently, no nonlinear decision-dependent relationships are introduced by these fore-
casts.
The complete model formulation maintains the following characteristics:

• All continuous decision variables appear linearly or affinely in the objective function and
constraints.
• All constraints, including capacity limits, material balances, supplier availability, and
demand fulfillment, are defined as linear equalities or inequalities.
• Forecasted input (demand and price) are entirely exogenous and do not induce nonlin-
earity with respect to the decision variables.

Conclusion: Given these properties, the problem is formally classified as a Mixed-Integer

Linear Program (MILP). Despite the presence of non-linear time-dependent expressions in
the form of quadratic or piecewise-linear functions, the model retains linearity in the deci-
sion space, and hence convexity with respect to continuous decision variables is preserved.
Solvers such as Gurobi are guaranteed to identify globally optimal solutions, ensuring both
tractability and mathematical rigor. Furthermore, the forecast approximation framework pro-
vides a bounded error envelope that enables systematic evaluation of the impact of prediction
quality on the overall optimization outcome.
Acknowledgements The authors express their gratitude for the financial support from Texas A&M University
and the Texas A&M Energy Institute. This work was partially supported by the National Science Founda-
tion Convergence Accelerator Program under the Grant titled “Securing Critical Material Supply Chains by

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Journal of Global Optimization

Enabling Photovoltaic Circularity (SOLAR)” with Grant No 49100423C0005, and by the National Institutes
of Health, with grant number P42ES027704, (Texas A&M University Superfund Research Center).

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
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