# QAmplifyNet: Pushing the Boundaries of Supply Chain Backorder Prediction
Using Interpretable Hybrid Quantum–Classical Neural Network

Md Abrar Jahin  Department of Industrial Engineering and Management, Khulna
University of Engineering and Technology (KUET), Khulna 9203, Bangladesh  Md
Sakib Hossain Shovon  Department of Computer Science, American International
University-Bangladesh (AIUB), Dhaka 1229, Bangladesh  Md. Saiful Islam
Department of Industrial Engineering and Management, Khulna University of
Engineering and Technology (KUET), Khulna 9203, Bangladesh  Jungpil Shin
Department of Computer Science and Engineering, The University of Aizu,
Aizuwakamatsu 965-8580, Japan  Corresponding author: Jungpil Shin (e-mail:
jpshin@u-aizu.ac.jp).  M. F. Mridha  Department of Computer Science, American
International University-Bangladesh (AIUB), Dhaka 1229, Bangladesh  Yuichi
Okuyama  Department of Computer Science and Engineering, The University of
Aizu, Aizuwakamatsu 965-8580, Japan

###### Abstract

Supply chain management relies on accurate backorder prediction for optimizing
inventory control, reducing costs, and enhancing customer satisfaction.
Traditional machine-learning models struggle with large-scale datasets and
complex relationships. This research introduces a novel methodological
framework for supply chain backorder prediction, addressing the challenge of
collecting large real-world datasets with 90% accuracy. Our proposed model
demonstrates remarkable accuracy in predicting backorders on short and
imbalanced datasets. We capture intricate patterns and dependencies by
leveraging quantum-inspired techniques within the quantum-classical neural
network QAmplifyNet. Experimental evaluations on a benchmark dataset establish
QAmplifyNet’s superiority over eight classical models, three classically
stacked quantum ensembles, five quantum neural networks, and a deep
reinforcement learning model. Its ability to handle short, imbalanced datasets
makes it ideal for supply chain management. We evaluate seven preprocessing
techniques, selecting the best one based on Logistic Regression’s performance
on each preprocessed dataset. The model’s interpretability is enhanced using
Explainable Artificial Intelligence techniques. Practical implications include
improved inventory control, reduced backorders, and enhanced operational
efficiency. QAmplifyNet also achieved the highest F1-score of 94% for
predicting "Not Backorder" and 75% for predicting "Backorder," outperforming
all other models. It also exhibited the highest AUC-ROC score of 79.85%,
further validating its superior predictive capabilities. QAmplifyNet
seamlessly integrates into real-world supply chain management systems,
empowering proactive decision-making and efficient resource allocation. Future
work involves exploring additional quantum-inspired techniques, expanding the
dataset, and investigating other supply chain applications. This research
unlocks the potential of quantum computing in supply chain optimization and
paves the way for further exploration of quantum-inspired machine learning
models in supply chain management. Our framework and QAmplifyNet model offer a
breakthrough approach to supply chain backorder prediction, offering superior
performance and opening new avenues for leveraging quantum-inspired techniques
in supply chain management.

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## Introduction

Supply chain management (SCM) plays a critical role in ensuring the smooth
flow of goods and services from manufacturers to end consumers. In this
context, accurate prediction of backorders, which refers to unfulfilled
customer orders due to temporary stockouts, is of paramount importance. Supply
chain backorder prediction (SCBP) enables proactive inventory management,
efficient resource allocation, and enhanced customer satisfaction. It assists
in mitigating the negative impacts of stockouts, such as lost sales, decreased
customer loyalty, and disrupted production schedules. Predicting backorders
for products in the future is challenging, mainly because the demand for a
particular product can fluctuate unexpectedly. To develop an accurate
predictive model, it is crucial to have an adequate amount of training data
derived from the inventory tracking system. This data allows the model to
learn the patterns that indicate whether a product will likely be backordered.
However, a significant challenge in building such a model is the inherent
imbalance in the dataset. The number of samples where a product is backordered
is much lower than those where products are not backordered. This class
imbalance creates a skewed dataset, which can negatively impact the model’s
performance.

There needs to be more research available on product backordering,
specifically addressing the challenges of class imbalance [1, 2]. However,
extensive work has been conducted in the past to optimize inventory
management. Inventory managers encounter various challenges when faced with
material shortages, which can result in complete backlogs or lost orders.
Previous literature has categorized material backordering as fixed, partial,
or time-weighted backorders [3]. Customers’ willingness to wait for a
replenished stock depends on factors such as supplier reputation, recency of
the backorder placement, and waiting time. Some customers may be patient and
wait, while others may seek alternative options due to impatience. In such
cases, the supplier experiences sales loss and missed revenue opportunities,
leading to customer dissatisfaction and potential doubts about the supplier’s
inventory management capabilities.

Traditional prediction models, predominantly based on classical machine
learning (CML) algorithms, have been widely utilized for backorder prediction.
However, these models face several challenges when dealing with large-scale
datasets typically encountered in supply chain (SC) applications. Traditional
models often need help handling these datasets’ complexity and dimensionality,
leading to suboptimal performance and limited scalability. Moreover, the
ability to capture intricate patterns and dependencies within the data is
crucial for accurate prediction, which remains a challenge for conventional
approaches. Despite the widespread use of CML models, tuning millions of
hyperparameters during training CML models like DNNs requires significant
computing power. The fast-rising data volume required for training,
particularly in the post-Moore’s Law era, exceeds the limit of semiconductor
production technology, which limits the field’s advancement. On the other
hand, quantum computing (QC) has proven to be more effective at solving issues
that are insurmountable for conventional computers, such as factoring big
numbers and doing unstructured database searches [4]. Nevertheless, because of
the noise produced by the quantum gates and the absence of quantum error
correction on Noisy Intermediate Scale Quantum (NISQ) devices, QC with
substantial circuit depth faces significant difficulties. So, creating quantum
algorithms with a reasonable level of noise-resistant circuit depth would be
of fundamental relevance. The performance of CML models is now being
outperformed by quantum machine learning (QML), which is based on variational
quantum circuits [5]. The vastly decreased number of model parameters is one
of the key advantages of variational quantum models over their classical
counterparts. As a result, variational quantum models reduce the overfitting
issues related to CML. Moreover, under some circumstances, they may learn more
quickly or attain better test accuracy compared to their conventional
counterparts [6]. The variational quantum model plays a vital role as the
quantum component of a modern QML architecture, with the circuit parameters
being updated by a classical computer [7].

The emergence of quantum-inspired techniques has opened up new avenues for
addressing the limitations of CMLs [8]. These techniques, inspired by QC
principles, leverage the inherent parallelism and quantum-inspired
optimization algorithms to enhance predictive capabilities. QML models exhibit
promising potential in handling large-scale datasets, capturing complex
patterns, and improving prediction accuracy in various domains. SCBP can
benefit from enhanced model performance, improved accuracy, and more efficient
resource allocation by harnessing the power of quantum-inspired techniques.
The utilization of QML algorithms can enable the identification of intricate
relationships between variables, facilitating more accurate prediction of
backorder occurrences. Consequently, these techniques have the potential to
optimize SCM, minimize stockouts, reduce costs, and enhance customer
satisfaction.

In the field of inventory management research, numerous studies have delved
into enhancing forecasting and decision-making regarding backorders. A
multitude of these studies have explored various mathematical and algorithmic
approaches, such as Markov decision processes, reinforcement learning (RL),
fuzzy models, and ML techniques. However, a conspicuous research gap exists
when it comes to effectively handling short and imbalanced datasets, a common
real-world scenario where acquiring extensive data is often impractical. While
ensemble forecasting models have demonstrated superiority over other methods,
their computational efficiency becomes a limiting factor, particularly with
large warehouse datasets, thus restricting their practical applicability. In
light of the limitations of traditional ML models and the potential advantages
offered by quantum-inspired techniques, this research aims to develop a novel
Hybrid Quantum-Classical Keras Neural Network (NN) tailored for SCBP to bridge
this research gap. The proposed model combines the flexibility and
interpretability of Keras NNs with quantum-inspired optimization algorithms to
overcome the limitations of classical approaches. By integrating quantum-
inspired techniques into the prediction process, we anticipate achieving
improved accuracy, robustness, and scalability in SCBP.

The novelty of this research lies in the application and benchmarking of QML
techniques against the classical ML techniques in the field of SCBP for short
and imbalanced datasets. Extensive literature review and analysis confirm that
this study represents the first known instance of QML implementation in the
context of SCBP. By introducing QML to this domain, our objective is to unlock
new possibilities and harness the potential advantages that quantum-inspired
techniques can offer to the field of SCM. This research contributes to the
field of SCM by exploring the potential of QML techniques for accurate and
efficient backorder prediction. A novel hybrid Quantum-Classical Neural
Network (Q-CNN) was developed as part of this study, combining the strengths
of parallel-processed NN computing and quantum physics. Hybrid classical-
quantum computing is a computational paradigm that combines classical
infrastructure with quantum computers to address specific problem domains. In
this approach, classical computers play a crucial role in pre-processing data,
controlling quantum machines during computation, and post-processing the
results obtained from quantum computations. By harnessing quantum phenomena
such as entanglement and superposition, quantum computers possess the ability
to perform parallel processing in a manner unprecedented by classical
computers [9]. By leveraging the strengths of both classical and quantum
computing, hybrid systems enable the utilization of quantum resources while
utilizing classical algorithms and techniques to enhance overall computational
performance. This synergistic combination allows for the efficient utilization
of quantum resources and the effective integration of classical and quantum
computing capabilities to tackle complex problems. The hybrid algorithms
employed in this study outperformed their classical counterparts by leveraging
quantum and classical computing capabilities.

In light of these considerations, this research provides a novel and thorough
methodology for anticipating inventory backorders. The goal is to maximize
profits while minimizing costs related to product backorders, maintaining good
relationships with suppliers and customers, and preventing sales from being
lost. Customers and businesses alike can profit from precise projections of
future backorders for individual products with the help of a well-developed
predictive model. A current topic of research is the simplification of quantum
algorithms for usage with NISQ computers [10]. Quantum algorithms that scale
well may be efficiently executed on computers that use photons,
superconductors, or trapped ions [13, 11, 12]. Particularly exciting is QML
because of its compatibility with current NISQ designs [14, 15]. Predicting
product backorders, for example, requires access to massive amounts of data,
which is a strength of traditional ML algorithms. For this reason, this
research introduces a novel Q-CNN model that can deal with data imbalances
even when trained on a small dataset. NISQ devices are effective in running
shallow-depth algorithms requiring a few qubits [10]. Given the specific
difficulties and prerequisites of product backordering prediction, it becomes
sensitive to take advantage of QML run on NISQ devices by means of the SCBP
dataset. Classification in inspection tests for small-size datasets was made
possible by searching for a quantum advantage on the classifier [16].

The open-access Kaggle dataset used in this research was gathered from an
8-week inventory management system [1]. Unfortunately, as shown in Figure 1,
the dataset needs to be balanced because the number of backordered items is
disproportionately high (137:1). Figure 3 shows a dataset heatmap, indicating
that high feature correlations are required, increasing the difficulty of
working with the dataset. The issue is made more complicated by the fact that
any prediction model will need help dealing with imbalanced datasets [17].
Gradient boosting model (GBM), random forest (RF), and logistic regression
(LR) are only some of the traditional ML methods that have been presented for
similar jobs in the past [1, 2]. It has also been common practice to use
undersampling and oversampling strategies to rectify grossly unbalanced
business statistics [19, 18].

(a) Before Removal of Null Values

(b) After Removal of Null Values

Figure 1: Barplots showing the distribution of null values in the dataset (a)
before and (b) after removal. The top of each bar shows the number of samples
present in each feature.

This research presents an innovative approach for SCBP that incorporates
effective preprocessing methods, resulting in a novel quantum-classical ML-
based prediction model. There are various steps in the methodological
flowchart. We first preprocess each SKU’s features using seven possible
combinations of methods. We benchmark each preprocessed dataset by applying
the LR model and then select the most effective preprocessing technique based
on its accuracy. The selected preprocessing tasks involve converting
categorical features into numerical features, handling missing values, log
transforming numerical features, normalizing feature values within a specific
range, and dropping redundant numerical features using variation inflation
factor (VIF) treatment. In this classification problem, there are
substantially fewer positive samples (backordered) than negative samples (non-
backordered). Consequently, we address the issue of class imbalance by
employing an undersampling technique called NearMiss. We choose undersampling
instead of oversampling because QML models struggle to train on large datasets
compared to CML models. Furthermore, we utilize principal component analysis
(PCA) to extract four input principal components from the preprocessed
dataset. These components capture the most significant features for
prediction.

Finally, we propose our hybrid Q-CNN model, named QAmplifyNet, which
incorporates key aspects of the architecture in its mnemonic name. The "Q"
signifies the utilization of QC principles, highlighting the model’s quantum
component. "Amplify" represents the concept of amplifying information through
the model’s layers. Lastly, "Net" refers to the NN nature of the model,
incorporating both classical and QML components. For the performance
evaluation of our model, we compare it against eight commonly used CML models,
one deep RL model, five quantum NNs, and three quantum-classical stacked
ensembles. Through this comprehensive comparison, we aim to demonstrate the
superiority and robustness of our proposed QAmplifyNet model for SCBP on short
datasets. Despite the excellent accuracy of CML models on this complete
dataset, our proposed QAmplifyNet model holds significant value. It showcases
remarkable performance on short, imbalanced data, which is a common challenge
in SC inventory management. Additionally, the application of QML in this
domain represents a pioneering effort, making it the first-ever QML
application in the SC inventory management field.

Using a benchmark SCBP dataset titled "Can you predict product backorder?", we
run tests utilizing the proposed model. The experimental findings demonstrate
the higher performance of our technique in SCBP, as evaluated by accuracy and
area under the receiver operating characteristic (ROC) curve. Moreover, we
compare our models to well-known classification models and come to the
conclusion that our strategy performs noticeably better than other comparable
models.

Figure 2: Methodological framework illustrating (a) data sources, (b) data
collection and splitting, (c) data preprocessing, and (d) proposed Q-CNN model
development for SCBP.

In summary, this paper makes eight-fold contributions:

  1. 1.

This study represents the pioneering application of QC in the SCM domain.

  2. 2.

We introduce a novel theoretical framework for predicting inventory
backorders.

  3. 3.

We present a comprehensive data preprocessing technique that combines log
transformation, normalization, VIF treatment, and NearMiss undersampling to
address the imbalanced nature of the dataset in the rare SCM domain.

  4. 4.

We propose a hybrid quantum-classical Keras NN-based technique for forecasting
product backorders, enhancing the suppliers’ overall efficiency.

  5. 5.

We demonstrate that the hybrid Q-CNN model overcomes the challenge of limited
availability of large SCM datasets by showcasing its enhanced performance
compared to CML and QML models on short datasets with few features.

  6. 6.

We enhance the interpretability of the proposed model by implementing
Explainable Artificial Intelligence (XAI) methods, specifically SHAP and LIME.

  7. 7.

Our novel methodology significantly improves prediction accuracy, reducing
misclassification errors, especially false positives and false negatives, and
ultimately increasing enterprise profitability.

  8. 8.

Lastly, we discuss how the proposed methodology can be applied homogeneously
to other supervised binary classification domains, such as predicting credit
card defaults.

Here’s how the paper breaks down: the current literature on SCBP, CML models,
quantum-inspired models, and RL-based techniques are reviewed in Section
“Related Work". It draws attention to the unanswered questions that our
proposed model seeks to answer. Section “Methods" introduces the proposed
hybrid Q-CNN-based backorder prediction system to address class imbalance on
the short dataset. It describes the selected preprocessing steps followed by
the architectures and working principles of the models used in this paper. In
Section “Results", we use the experimental data and robustness tests to
evaluate, compare, and verify the effectiveness of the proposed model. In
Section “Discussion", we conclude and make comparisons between the proposed
model and alternative methods. Possible real-world SCM implementations of the
suggested method are also discussed. The report finishes with a summary of the
main findings and the main contributions in Section “Conclusion and Future
Work". Some future research directions in SCBP employing quantum-inspired
approaches are also presented.

## Related Work

In the field of scientific research on inventory management, various studies
have been conducted to improve forecasting and decision-making related to
backorders. [20] proposed a solution based on Markov decision processes to
define inventory and backorder strategies. They treated production system
yield as a stochastic process. [21] examined a stock inventory system
incorporating periodic reviews and a partial backorder forecast. They
developed a framework considering the distribution of demand and its factors
to assess uncertainty in the inventory system. [22] analyzed estimating errors
and derived an inventory model’s predicted lead time demand distribution. This
distribution could be used to optimize inventory management. [23] determined
ordering policies in inventory systems using RL. They viewed SCM as a multi-
agent system and utilized the Q-learning technique to solve the model. [24]
combined the N-retailer problem and overall cost considerations to develop an
objective function for ordering, storing, and backordering in a single
inventory. They optimized three decisions jointly: lot sizing, routing and
distribution, and replenishment.

[25] developed an integrated multi-item manufacturing and routing problem
model considering constrained warehousing and manufacturing capabilities. For
optimal lot-sizing, routing and distribution, and replenishment, they factored
in a penalty cost for backordered products. [26] highlighted the significance
of accurately calculating backorder levels to enhance fill rates and reduce
total costs. [27] considered partial backordering and discounts on backordered
products in a production-inventory system to maximize net present value. [28]
emphasized the importance of incorporating backorder decisions and costs into
an ideal inventory policy, noting that previous models often overlooked SCBP.
[29] introduced fuzzy number-based optimization models to account for
uncertain demand and lead times, outperforming conventional methods. [30]
presented a fuzzy model that included human reasoning with backorders, while
[31] and [32] constructed economic ordering quantity models with various
factors such as special sale pricing, poor quality, partial backordering, and
quantity discounts. [33] proposed an integrated inventory model that optimized
multiple decisions simultaneously, including lead time, lot size, number of
shipments, and safety factor. In a fuzzy condition, [34] analyzed a warehouse
model incorporating backorders using fuzzy numbers and a graded mean
integration model. [35] optimized spare component allocation decisions for
serviceable standalone systems with dependent backorders. [36] devised an
approach to forecasting order for line-replaceable unit components with
backorders, highlighting the need to consider the dynamic features of these
factors. [37] proposed a framework to reduce overall costs and anticipated
risk costs of backorder replenishment plans using a Bayesian belief network.
[38] investigated a dynamic rationing scheme that considered demand dynamics,
while [39] used a Markov decision support system to determine the best
rationing levels across all categories of demand. [40, 41] developed non-
parametric and parametric prediction models, such as kernel density and GARCH
algorithms, to predict safety stock and reduce long lead times. [1] proposed
ensemble-based machine learning algorithms, GBM and RF paired with
undersampling, for SCBP. [2] discussed the benefits and limitations of
ensemble prediction methods and undersampling in dealing with noisy data and
improving prediction accuracy. [42] improved SCBP with the use of the
Conditional Wasserstein Generative Adversarial Network (CWGAN) model along
with Randomized Undersampling (RUS). Initially, the majority of the non-
backorder samples were reduced using RUS. Second, CWGAN was used as a
technique for oversampling to provide superior backorder samples. Ultimately,
RF was implemented to predict backorders. The class imbalance problem was
successfully addressed by [43] densely linked DNNs, which combined SMOTE and
randomized undersampling. The experimental outcomes indicate better prediction
performance and predicted profit on a thorough product backordering dataset,
proving the proposed model’s superiority over existing ML approaches.

In handling noisy data and minimizing overfitting, ensemble forecasting models
have shown superiority to non-parametric and parametric forecasting methods.
However, their computational efficiency becomes a limitation when analyzing
large warehouse datasets in real time, limiting their practical utility. On
the other hand, undersampling techniques can enhance computational
performance, but they may also exclude potentially valuable training data and
compromise prediction accuracy. To address these challenges, we propose a
hybrid Q-CNN applied to a short backorder dataset; our preprocessing approach
involves several steps. Firstly, we apply a log transform to the data,
followed by standard scaling to normalize the features. We also address
multicollinearity issues by implementing Variable Inflation Factor (VIF)
treatment. With the training dataset being unbalanced, we employ the NearMiss
undersampling method, which involves deliberately reducing the majority of
class occurrences.

The choice of a hybrid Q-CNN for analyzing the short dataset is driven by its
unique advantages, primarily rooted in the challenging nature of SCBP with
limited data. In real-world SCM, acquiring extensive datasets is a formidable
task due to the hesitance of companies to disclose their internal data.
Collecting vast data is often infeasible, considering the significant time and
effort required. Given these constraints, our approach focuses on developing
QAmplifyNet, a model specifically designed to predict backorders accurately,
even when trained on relatively short datasets. We balanced and shortened the
preprocessed dataset using the NearMiss algorithm after selecting the optimal
preprocessing technique to accomplish this. Despite the inherent challenges
associated with short datasets, our hybrid Q-CNN demonstrates remarkable
predictive accuracy. By offering these insights, we aim to elucidate the
unique advantages of our approach when dealing with the inherent limitations
of short SCBP datasets. Combining classical and QC techniques, this approach
harnesses the power of quantum algorithms for specific tasks while leveraging
the robustness and versatility of CML frameworks like Keras. Exploiting
quantum principles like superposition and entanglement through the use of
quantum-inspired algorithms inside a classical NN framework can result in
efficient and more accurate calculations [44]. Compared to purely classical or
purely quantum models, the hybrid Q-CNN is anticipated to outperform in
several aspects. Firstly, the combination of classical and quantum techniques
enables more powerful computations, leading to increased accuracy in backorder
predictions. The utilization of quantum-inspired algorithms within the
classical framework allows for more efficient exploration of the solution
space and better identification of patterns and trends in the data. The hybrid
approach offers practical advantages over pure quantum models. Quantum
computers are still in the early stages of development, and their availability
and scalability may pose limitations in real-world applications. CML models
often grapple with scalability issues when handling extensive datasets,
resulting in computational bottlenecks that hinder their practicality. The
intricate relationships and dynamic nature of SC data constrain their real-
time prediction capabilities. These limitations become particularly evident
when striving for timely SCBPs. Our research highlights the imperative need
for advanced solutions by quantum-inspired techniques, such as our QAmplifyNet
model, to transcend these constraints, ultimately enhancing the accuracy and
efficiency of real-time predictions while addressing the scalability
challenges inherent to pure classical or pure quantum models.

The hybrid model can leverage existing computational resources and
infrastructures by integrating CML frameworks like Keras, making it more
accessible and practical for implementation in real-world SC environments.
This integration allows for more accurate predictions, improved decision-
making, and better inventory control, making it a promising approach for
addressing the challenges of backorder management in real-world contexts.

Our study focused on analyzing a short and imbalanced dataset obtained by
undersampling a larger dataset. We aimed to benchmark our proposed hybrid
Q-CNN against CML, QML, and RL models. In working with a short and imbalanced
dataset, our hybrid model showcased its strength and outperformed the CML,
QML, and RL models. It is essential to emphasize that our hybrid model’s
superior performance on this particular short and imbalanced dataset
highlights its effectiveness in addressing the specific challenges associated
with such data characteristics. This milestone underscores the practicality
and utility of the hybrid Q-CNN in real-world scenarios where acquiring large
datasets may be difficult, yet accurate predictions are crucial. Our findings
have implications for domains with similar short and imbalanced datasets. The
success of our proposed model indicates its practicality and usefulness in
situations where obtaining extensive datasets is challenging, but accurate
predictions are of paramount importance.

## Methods

### Data Collection

We used a benchmarking dataset called “Can you predict product backorder?"
obtained from the Kaggle data repository to conduct extensive experiments on
our proposed hybrid Q–CNN–based prediction model. The data was gathered from
the Kaggle repository. There are many orders for various products included in
the dataset. A total of 22 features characterize the eight-week trajectory of
each order, and a target binary feature denotes if the corresponding product
is a backorder or not. Table 1 summarizes the features.

One significant challenge in working with this dataset is the inherent class
imbalance related to product backorders. Notably, backorder instances are
relatively uncommon in real-world SC scenarios. In this dataset, out of a
total of 1,929,935 orders, only 13,981 orders (approximately 0.72%) are
classified as delayed or experiencing a backorder. Conversely, a vast majority
of 1,915,954 orders (about 99.28%) are categorized as negative cases,
indicating that the products were not subject to backorders. Figure 4 shows
the dataset’s class distribution. This substantial class imbalance poses a
notable challenge for predictive modeling. With an imbalance ratio of
approximately 1:137, the dataset is heavily skewed toward the negative class,
making it highly unbalanced. This imbalance can significantly impact the
performance of ML models, as they may tend to favor the majority class,
resulting in suboptimal predictive accuracy for the minority class.

To address this challenge and ensure a rigorous evaluation of our proposed
model, we adopted a stratified k-fold cross-validation approach with five
splits, ensuring that the class distribution is maintained in each fold’s
training and testing sets. Additionally, we chose not to shuffle the dataset
during this process, preserving the integrity of the class distribution.

Figure 3: Spearman correlation heatmap to analyze the relationship between the
target feature ’went_on_backorder’ and 14 preprocessed input features of the
SCBP dataset. Figure 4: Class distribution of the imbalanced dataset used in
our study.

Features | Notation | Description
---|---|---
sku (Stock Keeping Unit) | x0 | A distinctive identifier for each instance in the dataset.
national_inv | x1 | Current level of inventory of the product.
lead time | x2 | Time taken for a shipment to be delivered from its starting point to the final destination.
in_transit_qty | x3 | This quantity is calculated based on the most recent picking slip or the cumulative
|  | quantity, and it represents the amount of product currently in transit.
forecast_3_months, forecast_6_months, forecast_9_months | x4, x5, x6 | Sales forecasts for the product over the following three, six, and nine months, respectively.
sales_1_month, Sales_3_month, sales_6_month, sales_9_month | x7, x8, x9, x10 | Sales quantity of the product in the last one, three, six, and nine months, respectively.
min_bank | x11 | Minimum recommended stocking level for the product.
potential_issue | x12 | Any problems or issues associated with the product or its parts.
pieces_past_due | x13 | Quantity of overdue parts for the product, if any.
perf_6_months_avg, perf_12_months_avg | x14, x15 | Average performance of the product over the past six months and twelve months, respectively.
local_bo_qty | x16 | Amount of stock orders that are currently overdue.
ppap_risk, deck_risk, stop_auto_buy, oe_constraint, rev_stop | x17 – x21 | Binary flags (yes or no) associated with specific risks
|  | or constraints related to the products.
went_to_backorder | y | Target variable by which the status of the product’s backorder is indicated.

Table 1: Descriptions of the dataset features of a particular product order.

### Data Preprocessing

Data preprocessing is a crucial step in enhancing the performance of ML
models. Our preprocessing approach initially focused on identifying and
addressing irrelevant data points. For instance, we observed that variables
like perf_6_month_avg and perf_12_month_avg contained negative values deemed
inconsistent and removed. We encountered features that included the symbol ’?’
indicating missing values, which were also eliminated. Furthermore, we
transformed categorical features into binary numerical representations to
facilitate analysis, separating them from the original dataset for subsequent
processing and analysis. For instance, the value of certain features
containing either ‘yes’ or ‘no’ were converted into binary 1 and 0,
respectively.

We tried seven different combinations of preprocessing steps and tested LR on
each preprocessed data to evaluate and choose the best preprocessing step for
our model development. Seven alternative techniques are as follows:

  1. 1.

IFLOF: We removed anomalies from the dataset using the Isolation Forest Local
Outlier Factor (IFLOF), which is a method that combines the Isolation Forest
algorithm with the Local Outlier Factor algorithm. IFLOF identifies outliers
by constructing an ensemble of isolation trees and measuring the local outlier
factor for each data point. It provides a measure of abnormality for each
instance in the dataset.

  2. 2.

IFLOF+VIF: This preprocessing step combines IFLOF outlier detection with the
VIF. VIF is a measure of multicollinearity, which assesses the correlation
between predictor variables in a regression model. Applying IFLOF to identify
outliers and then using VIF to identify highly correlated variables helps
address outlier detection and multicollinearity issues.

  3. 3.

IQR+VIF: We applied the Interquartile Range (IQR), a statistical dispersion
measure, to identify outliers and then applied the VIF to detect and remove
multicollinearity.

  4. 4.

VIF: We only applied VIF in this method without using any log transformation,
standard scaling, or anomaly detection algorithm.

  5. 5.

No log transform+VIF: This preprocessing step involves applying VIF to the
dataset without performing a log transform on the variables. This method
allows for the detection of multicollinearity without the influence of log
transformations.

  6. 6.

RobScaler+VIF: In this alternative, we tried RobScaler, a method used for
robust feature scaling to the dataset, before using VIF to detect
multicollinearity. RobScaler is particularly useful when dealing with data
that contains outliers, as it scales the features by removing the median and
scaling them according to the interquartile range.

  7. 7.

Log transform+StandardScaler+VIF: This proposed preprocessing step involves
three stages. First, a log transform is applied to the variables, which can
help normalize skewed data and handle nonlinear relationships. We removed the
infinity values from the resulting data. Then, the StandardScaler is used for
feature scaling, which ensures that all variables have a mean and standard
deviation of 0 and 1, respectively. Finally, the VIF with threshold = 5 is
applied to detect multicollinearity in the transformed and scaled dataset.
This method aims to handle skewness, standardize the data, and identify
multicollinearity simultaneously. We selected a subset of 14 features (x1–x4,
x10–x13, x15, x16, x17–x20) from the dataset in this technique. We made this
selection by excluding the remaining features due to the existence of
multicollinearity among them.

To choose the best preprocessing method, we undersampled the dataset to make
it balanced, maintaining the majority-to-minority ratio of 3:1 using the
NearMiss algorithm. Compared to the other approaches, Log
transform+StandardScaler+VIF produces the best ROC-AUC of 66% by the LR model
(see Table 2). LR was chosen specifically to evaluate the performance of
different preprocessing methods because it is a widely used and well-
established classification algorithm. By applying various preprocessing
methods and evaluating their effects on LR’s performance, we gain valuable
insights into which techniques are most effective in improving the predictive
capabilities of LR.

After selecting the best preprocessing technique, we finally balanced and
shortened the preprocessed dataset using the NearMiss algorithm, which was
further fed as the input data for all the models used in this study. The input
training data has 1000 samples having a 1:1 majority-to-minority class ratio.
The test data was intentionally made imbalanced using the undersampling
majority-to-minority ratio of 3:1, having 267 samples, among which 67 went
backorder and the rest did not.

Performance Metrics | Preprocessing Techniques
---|---
IFLOF | IFLOF+VIF | IQR+VIF | VIF | No log transform+VIF | RobScaler+VIF | Log transform+StandardScaler+VIF (Proposed)
Not Backorder (0) | Precision | 99% | 99% | 99% | 100% | 100% | 99% | 100%
Recall | 17% | 19% | 21% | 40% | 58% | 100% | 59%
F1-score | 29% | 31% | 35% | 58% | 73% | 100% | 74%
Backorder (1) | Precision | 1% | 1% | 1% | 1% | 1% | 0% | 1%
Recall | 84% | 82% | 79% | 88% | 62% | 0% | 72%
F1-score | 1% | 1% | 2% | 2% | 2% | 0% | 2%
Accuracy | 17% | 19% | 22% | 41% | 58% | 99% | 60%
ROC AUC | 50% | 50% | 50% | 64% | 60% | 50% | 66%
Micro average precision | 50% | 50% | 50% | 50% | 50% | 50% | 50%
Micro average recall | 50% | 50% | 50% | 50% | 60% | 50% | 66%

Table 2: Performance evaluation of LR model on the undersampled data with
different preprocessing techniques compared in this study

### Classical Models

We implemented 8 CML models using the scikit-learn [45] library, which
provides a comprehensive set of tools for ML tasks in Python. Additionally,
the parallel-computing library Dask [46] was utilized to enhance the
efficiency and scalability of these algorithms. It enabled the distribution
and execution of computations across multiple processors or machines, allowing
for faster processing. We performed hyperparameter tuning using GridSearchCV
with a 3-fold cross-validation to identify the optimal hyperparameters for
each CML, as shown in Table 3.

Models | Best hyperparameters
---|---
Catboost | boost_from_average = False
boosting_type = ’Plain’
border_count = 254
depth = 20
devices = ’0:1’
early_stopping_rounds = 500
eval_metric = ’AUC’
feature_border_type = ’GreedyLogSum’
grow_policy = ’Lossguide’
leaf_estimation_backtracking = ’Any Improvement’
learning_rate = 0.5
loss_function = ’Logloss’
max_leaves = 100
model_size_reg = 0.5
posterior_sampling = False
random_seed = 786
random_strength = 1
rsm = 1
scale_pos_weight = 3
score_function = ’cosine’
sparse_features_conflict_fraction = 0
task_type = ’GPU’
LGBM | colsample_bytree = 1.0
learning_rate = 0.01
max_depth = 5
n_estimators = 500
subsample = 0.8
RF | bootstrap = True
criterion = ’gini’
max_depth = 30
max_features = ’auto’
min_samples_leaf = 2
min_samples_split = 2
n_estimators = 200
XGBoost | learning_rate = 0.3
max_depth = 20
min_child_weight = 2
n_estimators = 100
scale_pos_weight = 0.5
colsample_bytree = 1
KNN | algorithm = ’ball_tree’
n_neighbors = 4
SVM | kernel = ’rbf’
C = 0.9
DT | min_samples_leaf = 6
criterion = ’gini’
max_depth = 3
Table 3: Best hyperparameters selected by GridSearchCV for the CML models

The CML models used include Categorical Boosting (Catboost), Light Gradient
Boosting Machine (LGBM), Random Forest (RF), Extreme Gradient Boosting
(XGBoost), Artificial Neural Network (ANN), K-Nearest Neighbors (KNN), Support
Vector Machines (SVM), and Decision Tree (DT). The classical ANN architecture
employed in this study consists of an input layer with 14 neurons, followed by
two dense layers with 14 and 10 neurons, respectively.

### Stacked Ensemble Models

Using qiskit [47] and qiskit_machine_learning modules, we explore the
following classically-stacked quantum ensemble algorithm:

  1. 1.

The base classifiers are trained using the provided training data.

  2. 2.

The trained base classifiers are then used to make predictions on both train
and test datasets.

  3. 3.

The output labels generated by the base classifiers on the training and
testing data are appended as additional features to the original training and
testing datasets.

  4. 4.

Next, the meta-classifier is trained using the updated train data, and its
performance is evaluated on the updated testing data to obtain the final
prediction values.

#### QSVM+LGBM+LR

We initialize two base classifiers, namely Quantum Support Vector Machine
(QSVM) and LGBM. For the QSVM classifier, we utilize the ZZFeatureMap to
calculate the kernel matrix. The computation of the kernel matrix is performed
using the following equation:

$K(\vec{x_{i}},\vec{x_{j}})=K_{ij}=|\langle\phi^{\dagger}(\vec{x_{j}})|\phi(\vec{x_{i}})\rangle|^{2}$
(1)

where $x_{i},x_{j}\in X$ (training dataset), and $\phi$ represents the feature
map. To simulate the results of the quantum computer, we employ the state
vector simulator, which can be substituted with a backend for hardware
results. We consider two base classifiers, with the second one being LGBM. For
the ensemble construction, we utilize LR as the meta-classifier, which
combines the predictions of the two base classifiers.

#### VQC+QSVM

We used the ZZFeatureMap to define the feature map for the Variational Quantum
Classifier (VQC) as a base classifier and QSVM as a meta-classifier. The input
data was mapped to a higher-dimensional quantum space using this feature map.
For the VQC, we chose the TwoLocal ansatz, which involved the use of $R_{Y}$
(Equation 6) and $R_{Z}$ (Equation 7) gates for the parameterized rotations
and the $CZ$ (Equation 2) gate for entanglement.

$CZ=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\\\ \end{bmatrix}$
(2)

This ansatz was repeated for two iterations. Then COBYLA optimizer and a
QuantumInstance with the statevector_simulator backend were configured. The
QSVM’s kernel was initialized using the QuantumKernel, which employed the
chosen feature map and a QuantumInstance with the statevector_simulator
backend.

#### VQC+LGBM

We utilized the previously mentioned initialization techniques for both the
VQC and LGBM models, employing them as the base and meta classifiers. These
models were then integrated into a stacking ensemble framework, where the
predictions of the base classifiers were combined and used as features for the
meta-classifier LGBM.

### Quantum Neural Network (QNN) Models

We used Pennylane [48] dependencies for developing the QNN models. Pennylane
is employed to simulate quantum circuits and conduct quantum experiments,
facilitating the development of QC programs.

#### MERA-VQC

Our scheme has an ansatz based on a tensor network named Multi-scale
Entanglement Renormalization Ansatz (MERA). With only 16 variables designed,
the amplitude embedded one layer for each tensor network. We initialized the
device as the QC backend with 4 qubits using Pennylane’s QML library [48]. The
entanglement structure for the MERA circuit was implemented using CNOT gates
between the qubits 0 and 1, and two rotation gates, $R_{Y}$, were applied to
each qubit using the specified weights. The number of block wires, parameters
per block, and the total number of blocks parameterized the MERA quantum
circuit. Then a quantum circuit was implemented using the defined MERA
structure to process the training data. We defined a VQC classifier that
utilized the previously constructed quantum circuit. The classifier took
weights, bias, and classical data as inputs and produced predictions based on
the output of the circuit. We implemented a square loss function to measure
the difference between the predicted and true labels and an accuracy function
to assess the model’s performance. We defined a cost function that was used to
optimize the weights and bias parameters to enable the model to learn from the
training dataset by quantifying the overall loss between the predictions and
the true labels.

#### RY-CNOT-VQC

RY-CNOT-VQC 6-layered classifier highlights the use of $R_{Y}$ and CNOT gates
in the circuit structure, providing more detailed information about the
model’s architecture. We employed a 2-qubit simulator to translate classical
vectors into quantum state amplitudes. The circuit was encoded using the
method described by [49]. Also, following the work of [50], we had to break
down controlled Y-axis rotations into simpler circuits. The quantum state
preparation process was defined using quantum gates such as $R_{Y}$ (rotation
around the y-axis), controlled-NOT (CNOT), and $Pauli-X$ gates. The primary
quantum circuit incorporates the state preparation process, applying multiple
rotations layers based on the given weights. Then a function applies rotation
gates on qubits 0 and 1 and performs a CNOT operation between them. The
quantum circuit was evaluated on a test input by applying the state
preparation process and estimating the expectation value of the $Pauli-Z$
operator on qubit 0.

#### Classical NN+Encoder+QNN

As suggested in [51], this hybrid model is made up of a classical NN, an
encoder circuit, and a QNN. There are two qumodes that make up the quantum
circuit. Each vector entry was used as the parameter of available quantum
gates to encode classical data into quantum states. Two 10-neuron hidden
layers, each with an ’ELU’ activation function and a 14-neuron output layer,
comprise the classical NN. Then, 14 entries of the classical NN’s output
vectors are sent into squeezer, interferometers, displacement gates, and Kerr
gates as input parameters. Kerr gates, Interferometer-1, interferometer-2,
squeezers, and displacement gates were employed in the QNN’s four-step
sequence. Using the $Pauli-X$ gate’s $\langle\phi_{k}|X|\phi_{k}\rangle$
expectation value, for the final state $|\phi_{k}\rangle$ of each qumode, a
two-element vector [$\langle\phi_{0}|X|\phi_{0}\rangle$,
$\langle\phi_{1}|X|\phi_{1}\rangle$] was constructed. The ROC value of this
model is 71.09%, and the closest threshold to optimal ROC is 54%.

### Deep Reinforcement Learning Model

We used TensorFlow 2.3+ [52] and TF Agents 0.6+ [53] to implement Double Deep
Q-Network (DDQN) [54]. By treating the classification problem as an Imbalanced
Classification Markov Decision Process, DDQN predicts that the episode will
end when the agent misclassifies a sample from the minority-class but not a
majority-class sample. The training process involved 100,000 episodes, and a
replay memory was used with a length matching the number of warmup steps.
Mini-batch training was performed with a batch size of 32, and the Q-network
was updated using 2,000 steps of data collected during each episode. The
policy was updated every 500 steps, and a soft update strategy was employed
with a blending factor of 1 to update the target Q-network every 800 steps.
The model architecture consisted of three dense layers with 256 units and ReLU
activation, followed by dropout layers with a rate of 0.2. The final layer
directly outputted the Q-values. Adam optimization was applied with a learning
rate of 0.00025, and future rewards were not discounted. The exploration rate
decayed from 1.0 to min_epsilon over $\frac{1}{10}th$ of the total episodes,
and the minimum and final chance of choosing a random action was set to 0.5.

### Proposed QAmplifyNet Model

The provided Figure 2 presents an overview of our proposed methodological
framework. The first phase in the framework is gathering baseline information,
which may include supplier efficiency, lead times, inventory levels, and
product sales. Information on sales, supplier efficiency, inventory levels,
and lead times for suppliers is gathered from a wide variety of data sources.
These data are then combined and grouped into weekly time intervals for
orders. The dataset is subsequently divided into training and testing sets.
The collected data undergoes preprocessing using our suggested ‘Log
transformation+Standard Scaling+VIF treatment’ method to address the common
anomalies found in manufacturing industrial sensor data. This involves
eliminating inconsistent data points, managing null values, and scaling and
normalizing the data within a specified range. We applied PCA on both the
train and test datasets to prepare the input for our 2-qubit Amplitude
Encoder, resulting in 4 features. This dimensionality choice aligns with the
model’s requirements, as it operates on $\log_{2}{4}$, which yields a
2-dimensional classical data input. The aggregated data is then prepared for
predictive analytics, employing a hybrid Q-CNN named QAmplifyNet as the core
component of the proposed framework. The classical layers process the input
data, while the quantum layer performs quantum computations on the encoded
data. This comprehensive framework enables us to effectively leverage the
collected data and utilize the hybrid model for analysis and prediction
purposes.

Figure 5: Model architecture of QAmplifyNet model.

In our implementation, we leveraged the capabilities of PennyLane [48] to
convert QNodes into Keras layers. This integration allowed us to combine these
quantum layers with a diverse set of classical layers available in Keras,
enabling the creation of genuinely hybrid models. Figure 5 explains the
proposed architecture of QAmplifyNet, which consists of a Keras Sequential
model consisting of an input layer, three classical hidden layers, one quantum
layer, and a classical output layer. Here is an explanation of each layer:

  1. 1.

Input Layer: The input layer accepts inputs from 4 PC features and comprises 4
neurons.

  2. 2.

Hidden Layer 1: This dense layer has 512 units with a ReLU activation
function. It receives input from the input layer (with a dimension of 4) and
is set as non-trainable, serving the purpose of embedding the input data.

  3. 3.

Hidden Layer 2: The second dense layer has 256 units with a ReLU activation
function. It receives input from the previous layer and is non-trainable.

  4. 4.

Hidden Layer 3: This dense layer has 4 units with a ReLU activation function.
It takes input from the previous layer and is non-trainable. It passes
4-dimensional outputs to the next quantum layer.

  5. 5.

QNN KerasLayer: The next layer incorporates a 2-qubit quantum node (QNode) and
weight shapes. It represents the quantum part of the hybrid model and takes
input from the previous dense layer, which receives these four-dimensional
classical data as inputs and converts them into four amplitudes, representing
a quantum state of two qubits.

  6. 6.

Output Layer: The final output probabilities are generated via a softmax
activation function in the output dense layer. There are just two possible
classes that need to be classified; hence this layer only has 2 units. The
softmax activation function can be characterized as follows:

$\sigma(\vec{\theta})_{i}=\frac{e^{\theta_{i}}}{\sum_{j=1}^{K}e^{\theta_{j}}}$
(3)

$\displaystyle Where,$ $\displaystyle\sigma$ $\displaystyle=\text{softmax
function}$ $\displaystyle\vec{\theta}$ $\displaystyle=\text{input vector}$
$\displaystyle e^{\theta_{i}}$ $\displaystyle=\text{standard exponential
function for input vector}$ $\displaystyle K$ $\displaystyle=\text{class count
in multi-classifier}$

Using a learning rate of 0.01 and a loss function of ’binary_crossentropy,’ we
employed the Adam optimizer.

In the QAmplifyNet mode, we have implemented distinct classical and quantum
parts that work together to form the overall architecture. Let’s delve into
the details of each part:

#### Classical Part

The classical part of the model primarily consists of classical layers that
operate on classical data. In our specific implementation, we have used
classical dense layers with various activation functions (e.g., ReLU) and
configurations. These classical layers process the input data using classical
computations, performing operations like linear transformations and nonlinear
activations. Our model has three classical dense layers: Dense Layer 1, Dense
Layer 2, and Dense Layer 3. These layers receive inputs from the previous
layer and are set as non-trainable, as indicated by the ‘trainable=False‘
parameter. The classical part culminates in Dense Layer 4, which has two units
and employs the Softmax activation function for generating the final output
probabilities.

#### Quantum Part

The quantum part of the model is integrated into the classical part using the
‘qml.qnn.KerasLayer’ from PennyLane [48]. This part includes the QNode, which
represents the quantum circuit, and weight shapes that define the structure of
the quantum operations. In our implementation, the QNode is defined, which
consists of quantum operations from PennyLane’s templates of ‘Amplitude
Embedding’ (AE) and ‘Strongly Entangling Layers’ (SEL).

Classical data items must be embedded as quantum states on qubits for
processing by a quantum computer due to the quantum nature of the computer’s
operation. In the circuit, the state preparation component, AE, is responsible
for encoding classical data onto the two data qubits. The key advantage of AE
is its ability to handle significantly large amounts of information with a
relatively small number of qubits. With amplitude encoding, the number of
amplitudes available is practically limitless, allowing for encoding a
significant amount of data. Notably, the number of qubits required for
encoding a given number of features follows a logarithmic relationship
$(log_{2}(n))$, meaning that as the number of data features increases, only a
logarithmic increase in the number of qubits is needed. This scalability
enables encoding a vast amount of information with each additional qubit,
making amplitude encoding a powerful approach for handling complex datasets in
QC. The AE is composed of a parameterized quantum circuit comprising an
embedding circuit and a variational circuit (see Figure 6). The embedding
circuit incorporates an Amplitude Encoder, which is designed to encode a
maximum of $2^{n}$ data features into the amplitudes of a quantum state
consisting of $n$ qubits. Alternatively, a vector containing $N$ number of
features can be encoded using [$\log_{2}n$] qubits. The amplitudes of a
quantum state $|\phi_{x}\rangle$ with $n$ qubits can be thought of as a
representation of a normalized classical datapoint $x$ with $N$ dimensions, as

$|\phi_{x}\rangle=\sum_{i=1}^{N}x_{i}|i\rangle$ (4)

In the given equation, where $N$ is equal to $2^{n}$, $x_{i}$ represents the
$i$-th element of the variable $x$, and $|i\rangle$ refers to the $i$-th state
in the computational basis. Nevertheless, $x_{i}$ can be a float or integer.
The $x$ vector must be normalized according to the definition.

$\sum_{i=1}|x_{i}|^{2}=1$ (5)

Figure 6: Quantum circuit representation of QAmplifyNet, featuring two qubits
labeled "0" and "1". The circuit comprises variational layers utilizing the
SEL approach for two qubits with a depth of one layer. Initial blue lines
depict the embedding of features into the quantum state’s amplitudes. Two
$R_{Y}$-gates (see Equation 6) introduce $\frac{\pi}{2}$ rotations to both
qubits. Subsequently, two U3 rotation gates involving $R_{Z}$, $R_{Y}$, and
$R_{Z}$ (see Equation 7) single-qubit rotations are optimized during training.
Blue CNOT (see Equation 8) entangling gates connect qubits 0 and 1,
reinforcing their entanglement in a circular topology. The measurement layer
includes two $Pauli-Z$ operators (Graphics generated using Pennylane-Qiskit).

If the number of features to encode is not a power of 2, the remaining
amplitudes can be filled with constant values. The AE technique transforms the
4 features obtained from the classical component into the amplitudes of a
quantum state with 2 qubits.

$R_{Y}(\phi)=\begin{bmatrix}\cos(\phi/2)&-\sin(\phi/2)\\\
\sin(\phi/2)&\cos(\phi/2)\end{bmatrix}$ (6)

$R_{Z}(\psi)=\begin{bmatrix}e^{-i\psi/2}&0\\\ 0&e^{i\psi/2}\end{bmatrix}$ (7)

$CNOT=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{bmatrix}$ (8)

In the variational stage, the number of SEL, $L$ is variable. SEL consists of
generic trainable rotational gates $Rot(\alpha_{i},\beta_{i},\gamma_{i})$
implemented on qubits 0 and 1, and then a set of CNOT gates are used to
connect adjacent qubit pairs, with the last qubit being regarded as a neighbor
of the first. The number of SEL for an $n$-qubit circuit can be modified to
tune the complexity of this circuit. The model has precisely
$3\times{n}\times{L}$ number of trainable parameters. The SEL utilizes a
circuit-centric approach in its design. In this approach, each individual
qubit, denoted as $G$, is represented by a $2\times 2$ unitary matrix, as
shown in Equation 9, where $\theta,\phi,\psi\in[0,\pi]$.

$G(\theta,\phi,\psi)=\begin{bmatrix}e^{i\phi}\cos(\theta)&e^{i\psi}\sin(\theta)\\\
-e^{-i\psi}\sin(\theta)&e^{-i\phi}\cos(\theta)\end{bmatrix}$ (9)

Due to the lack of support for the "reversible" differentiation method in SEL,
PennyLane [48] automatically chooses the most suitable differentiation method
available. The state of the two qubits can be measured using the $Pauli-Z$
operator. Upon measurement, the qubits will collapse to a specific state. The
matrix representation of the $Pauli-Z$ operator is illustrated in Equation
(10).

$\sigma_{z}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}$ (10)

The measurement of the first qubit’s $Pauli-Z$ operator is denoted as
$\langle\sigma_{z}^{0}\rangle\in[-1,+1]$. This expectation value is
subsequently utilized to determine the probabilities involved
$P_{notbackorder}$ and $P_{backorder}$ of being "not backorder" or "backorder"
state, respectively:

$P_{notbackorder}=\frac{1}{2}(\langle\sigma_{z}^{0}\rangle+1)$ (11)
$P_{backorder}=\frac{1}{2}(1-\langle\sigma_{z}^{0}\rangle)=1-P_{notbackorder}$
(12)

These quantum operations help encode the input features into quantum states
and perform quantum computations. The QNode calculates the $Pauli-Z$
operators’ expectation values for each quantum circuit qubit. The QNode’s
output is then input for the subsequent classical layers.

#### Combining Classical and Quantum Parts

The classical and quantum parts of the model are seamlessly integrated within
the ‘Sequential’ framework of Keras. The classical layers process the data up
to a certain point, and then the output is fed into the quantum layer
(KerasLayer), which incorporates the QNode. The challenge lies in the
necessity of the 4-neuron dense layer preceding the 2-qubit quantum layer, as
it serves to pass the essential 4-dimensional input features to the quantum
layer, rendering its removal or replacement unfeasible. The Adam optimizer is
utilized to train the parameters of the model, which include the weights and
biases. The model is optimized during training based on the binary cross-
entropy loss function. The training steps involve iteratively updating the
parameters to improve the model’s performance and accuracy. We also enabled
the ’EarlyStopping’ mechanism during the training process, ensuring that the
model stops when the desired metric stops improving, which helped prevent
overfitting and saved training time. We ensured that the model’s primary goal
was to generalize effectively to new, unseen data, which was confirmed during
testing. Although the validation loss remained higher, the model achieved a
good balance between fitting the training data and generalizing to new
instances. The use of ’EarlyStopping’ added robustness to our convergence
analysis, preventing overfitting during training. The training procedure took
place on a Kaggle kernel environment equipped with 2 CPU cores, 13 GB RAM, and
2 Nvidia Tesla T4 GPUs, each of 15 GB. The model parameters were carefully
selected through multiple trial runs to optimize accuracy. The training
process concluded after 18 epochs using a batch size of 5.

The loss curve of Figure 8 indicates the model’s ability to minimize errors
during the training and validation phases. Loss curves aid in evaluating the
model’s learning progress, generalization capability, and potential for
effective predictions on new instances.

Figure 7: Confusion matrix and classification reports (from left to right) for
QAmplifyNet model. Figure 8: From left to right, the curves represent the ROC
and Loss evolution vs. Epochs of the QAmplifyNet model. These curves provide
insights into the model’s ability to classify backorder instances accurately
and its overall predictive performance as training progresses.

## Results

### Evaluation Metrics

In order to assess the effectiveness of the predictive models employed in this
study, various performance metrics have been utilized. In the context of SCBP,
True Positive (TP) refers to the number of correctly classified instances of
backorder occurrences, while True Negative (TN) represents the number of
correctly classified instances of non-backorder occurrences. False Positive
(FP) indicates the number of backorder instances mistakenly classified, and
False Negative (FN) signifies the number of misclassified non-backorder
instances. Both FP and FN are significant as higher FN results in missed
opportunities with potential customers, leading to increased opportunity
costs. On the other hand, higher FP leads to increased inventory holding costs
and a greater risk of product obsolescence due to the long-term accumulation
of unnecessary inventory. Here are the definitions and equations for the
performance metrics used to evaluate our models:

#### Accuracy

Accuracy is a metric that evaluates the overall accuracy of predictions by
determining the ratio of correct predictions to the total number of
predictions made.

$\text{{Accuracy}}=\frac{{TN+TP}}{{TN+TP+FN+FP}}$ (13)

#### Precision

Precision is a metric that evaluates the accuracy of positive predictions made
by a model. It represents the proportion of correctly identified positive
instances out of all instances that were predicted as positive. This metric
helps assess the model’s capability to minimize false positive predictions,
providing insights into its precision and reliability in identifying positive
cases accurately.

$\text{{Precision}}=\frac{{TP}}{{FP+TP}}$ (14)

#### Recall (True Positive Rate)

Recall is a metric that quantifies the model’s effectiveness in correctly
identifying positive instances among all the actual positive instances. It
provides insight into how well the model can detect and capture the positive
cases in the dataset.

$\text{{Recall}}=\frac{{TP}}{{TP+FN}}$ (15)

#### F1-measure

The F1-measure is a metric that combines precision and recall into a single
value, giving equal importance to both. It serves as a balanced measure that
is particularly beneficial when dealing with imbalanced datasets. By
considering both precision and recall, the F1 measure provides a comprehensive
evaluation of a model’s performance, considering both the ability to correctly
identify positive instances (precision) and capture all positive instances
(recall). This makes it a valuable metric in uneven class distribution
scenarios like the SCBP dataset, as it offers a balanced assessment of the
model’s effectiveness.

$\text{{F1-measure}}=2\times\frac{{\text{{Precision}}\times\text{{Recall}}}}{{\text{{Precision}}+\text{{Recall}}}}$
(16)

#### Specificity (True Negative Rate)

Specificity is a metric that quantifies the accuracy of a model in correctly
identifying negative instances among all the actual negative instances. It
provides insight into the model’s capability to detect and classify negative
instances accurately.

$\text{{Specificity}}=\frac{{TN}}{{TN+FP}}$ (17)

#### Gmean

The Gmean is a metric represented by the equation 18 that aims to achieve a
balance between maximizing the TP and TN. It takes into account both TP and TN
while minimizing the adverse effects caused by imbalanced class distributions.
It is crucial to acknowledge that the Gmean metric does not offer insights
into the specific contributions made by each class towards the overall index.
Consequently, various combinations of TN and TP can result in identical Gmean
values.

$\text{{Gmean}}=\sqrt{{\text{{TP}}\times\text{{TN}}}}$ (18)

#### IBA

IBA is a measure to estimate the performance of binary classifiers on
imbalanced datasets using the following equation:

$\text{{IBA}}=\text{(Gmean)}^{2}\times(1+\text{Dominance})$ (19)

Here, $Dominance$ refers to the absolute difference between TP and TN, which
is utilized to gauge the relationship between these two measures. By
substituting $Dominance$ and $Gmean$ into the equation, we can gain valuable
insights into how the IBA balances the trade-off between $Dominance$ and the
$Gmean$.

#### AUC-ROC index

The AUC is a metric that evaluates the overall performance of a model by
considering its ability to differentiate between positive and negative
instances at various classification thresholds. It is represented graphically
as a ROC curve. The AUC serves as an indicator of the model’s discriminative
power and its capacity to classify different instances accurately.

These performance metrics are relevant to the SCBP problem as they provide
insights into the model’s accuracy, precision, recall, and ability to handle
imbalanced datasets. They help assess the model’s effectiveness in correctly
identifying backorder and non-backorder instances.

### Results and Analysis

The results presented in Table 4 demonstrate the performance comparison of
different algorithms for the task at hand. Our proposed QAmplifyNet algorithm
achieves the highest accuracy score of 90%, outperforming all the other models
used in this study. Among the QNN models, MERA 4-layered, Classical
NN+Encoder+QNN, and RY-CNOT 6-layered exhibit respectable accuracy scores of
78%, 77%, and 75%, respectively. Nevertheless, when choosing an ML algorithm,
it is vital to take into account factors beyond accuracy as the sole
criterion. Considerations such as the algorithm’s ability to generalize well
to unseen data, its interpretability in providing understandable insights, and
its computational efficiency should also be taken into consideration.

In our scenario, we have two classes: ’0’ represents "Not Backorder" and ’1’
represents "Backorder." While different models demonstrate better performance
in either precision or recall, it is essential to consider both measures by
assessing the F1-score. QAmplifyNet achieves the best macro-average F1-score
of 84%, with 94% for predicting class 0 and 75% for predicting class 1. Given
the imbalanced nature of the dataset, we employed the ‘imblearn’ module from
scikit-learn to gain insights into specificity, Gmean, and IBA values.
QAmplifyNet yields the highest Gmean (77%) and IBA scores (62% for class 0 and
57% for class 1), outperforming the other models. Furthermore, QAmplifyNet
achieved an AUC-ROC value of 79.85%, indicating that the model exhibits
stronger discriminatory power than the other models. The AUC-ROC analysis
allows us to assess the model’s overall ability to rank instances correctly
and provides insights into their predictive capabilities.

QAmplifyNet achieves the highest macro-average precision and recall scores of
94% and 80%, respectively. Regarding class 0, QAmplifyNet achieves a precision
of 88% (see Figure 7), indicating that 88% of instances predicted as class 0
are correctly classified. The recall of 100% signifies that the model
successfully identifies all true class 0 instances. The specificity of 60%
suggests that the model accurately identifies 60% of the true class 1
instances as class 1. Concerning class 1, the precision of 100% reveals that
all instances predicted as class 1 are classified correctly. Nevertheless,
with a recall of 60%, it signifies that the model only manages to identify 60%
of the actual instances belonging to class 1. On the other hand, a specificity
of 100% implies that all instances belonging to class 0 are accurately
classified as class 0. QAmplifyNet demonstrated significant outperformance
compared to the other models, achieving a macro-average specificity of 80%.
This indicates that QAmplifyNet excelled in correctly identifying the negative
instances, surpassing the performance of the other models in terms of
distinguishing non-backorder cases accurately.

Notably, QAmplifyNet consistently demonstrates superior performance across all
evaluation metrics: accuracy, AUC-ROC, precision, recall, F1-score,
specificity, Gmean, and IBA (macro-average 59.50%). In contrast, other models
exhibit inconsistent performance across some of the metrics.

The comparison of confusion matrix components, namely TP, TN, FN, and FP, for
various models in SCBP is depicted in the provided Figure 9. Upon analyzing
the results, it becomes evident that QAmplifyNet outperforms other models in
terms of predictive performance. QAmplifyNet achieved a TP rate of 14.98% and
a TN rate of 74.91%, demonstrating its ability to classify positive and
negative instances accurately. Significantly, it achieved a notable 0% FP
rate, signifying a complete absence of incorrect predictions labeling non-
backorder instances as backorders. Furthermore, QAmplifyNet exhibited a
relatively low FN rate of 10.11%, implying a minimal number of missed positive
predictions. In contrast, several models displayed higher FP rates,
erroneously identifying actual non-backorders as backorders. Similarly, other
models demonstrated higher FN rates, misclassifying backorder cases. This
achievement is particularly significant given the imbalanced nature of the
SCBP problem. For instance, the MERA 4-Layered and RY-CNOT 6-Layered models
achieved 0% FP rates but at the expense of higher FN rates of 22.10% and
25.09%, respectively. Additionally, their TP rates (3% and 0%) were lower than
that of QAmplifyNet.

Figure 9: Bar plot illustrating the TP, TN, FP, and FN rates of various models
used in this paper for SCBP. The obtained values are derived from the
confusion matrices of each model, offering valuable information regarding
their ability to accurately classify instances as either positive or negative.

Comparatively, the CML models exhibited significantly higher average FP rates
(47.99%) and relatively lower average TN rates (26.92%). Similarly, the
Stacked Ensemble models demonstrated FP rates of 53.18% and TN rates of
21.72%. Classical NN+Encoder+QNN had a 15.73% higher FP rate and a 15.73%
lower TN rate compared to QAmplifyNet. It is worth noting that DDQN achieved a
high TP rate of 24.34% but at the cost of a substantial FP rate of 50.56%.
Conversely, QAmplifyNet achieved a competitive TN rate while maintaining a
significantly lower FP rate of 0%, underscoring its robustness in minimizing
false positive predictions.

The comparison of QAmplifyNet with other models highlights its superiority in
achieving a balanced trade-off between TP, TN, FP, and FN rates, resulting in
more accurate SCBP with minimal FP and FN predictions. This substantiates
QAmplifyNet’s potential for enhancing the reliability and robustness of SCBP
systems.

Model Category | Models | Evaluation Metrics
---|---|---
Precision | Recall | F1-score | Specificity | Gmean | IBA | Accuracy | ROC–AUC
CML Models | Catboost | 0 | 90% | 33% | 48% | 90% | 54% | 27% | 47% | 73.83%
1 | 31% | 90% | 46% | 33% | 54% | 31%
DT | 0 | 88% | 33% | 47% | 87% | 53% | 27% | 46% | 68.07%
1 | 30% | 87% | 45% | 33% | 53% | 30%
KNN | 0 | 81% | 39% | 53% | 73% | 53% | 28% | 48% | 55.03%
1 | 29% | 73% | 41% | 39% | 53% | 29%
LGBM | 0 | 88% | 33% | 47% | 87% | 53% | 27% | 46% | 75.23%
1 | 30% | 87% | 45% | 33% | 53% | 30%
RF | 0 | 88% | 33% | 47% | 87% | 53% | 27% | 46% | 73.96%
1 | 30% | 87% | 45% | 33% | 53% | 30%
SVM | 0 | 84% | 39% | 53% | 78% | 55% | 29% | 49% | 63.74%
1 | 30% | 78% | 43% | 39% | 55% | 31%
XGBoost | 0 | 88% | 33% | 47% | 87% | 53% | 27% | 46% | 71.90%
1 | 30% | 87% | 45% | 33% | 53% | 30%
3 Dense Layered NN | 0 | 87% | 47% | 61% | 79% | 61% | 36% | 55% | 72.52%
1 | 33% | 79% | 47% | 47% | 61% | 38%
Stacked Ensemble Models | QSVM+LGBM+LR | 0 | 91% | 29% | 44% | 91% | 51% | 25% | 45% | 70.00%
1 | 30% | 91% | 45% | 29% | 51% | 28%
VQC+QSVM | 0 | 94% | 29% | 44% | 94% | 52% | 25% | 45% | 62.00%
1 | 31% | 94% | 46% | 29% | 52% | 29%
VQC+LGBM | 0 | 91% | 29% | 44% | 91% | 44% | 25% | 45% | 60.00%
1 | 30% | 91% | 45% | 29% | 45% | 28%
QNN Models | MERA 1-Layered | 0 | 83% | 47% | 60% | 72% | 58% | 33% | 53% | 59.32%
1 | 31% | 72% | 43% | 47% | 58% | 35%
MERA 2-Layered | 0 | 83% | 29% | 43% | 82% | 49% | 23% | 42% | 55.54%
1 | 28% | 82% | 42% | 29% | 49% | 25%
MERA 4-Layered | 0 | 77% | 100% | 87% | 12% | 35% | 13% | 78% | 55.97%
1 | 100% | 12% | 21% | 100% | 35% | 11%
RY-CNOT 6-Layered | 0 | 75% | 100% | 86% | 0% | 0% | 0% | 75% | 50.00%
1 | 0% | 0% | 0% | 100% | 0% | 0%
Classical NN+Encoder+QNN | 0 | 80% | 89% | 84% | 79% | 75% | – | 77% | 71.09%
1 | 71% | 53% | 61% | 79% | 75% | –
Deep RL Model | DDQN | 0 | 88% | 33% | 47% | 87% | 53% | 27% | 46% | 47.58%
1 | 30% | 87% | 45% | 33% | 53% | 30%
Proposed | QAmplifyNet | 0 | 88% | 100% | 94% | 60% | 77% | 62% | 90% | 79.85%
1 | 100% | 60% | 75% | 100% | 77% | 57%

Table 4: Performance comparisons of the models used in this study against
QAmpliNet on short SCBP dataset

### XAI Interpretation using LIME and SHAP

To gain insights into the interpretability of QAmplifyNet, we applied two
popular XAI techniques: Local Interpretable Model-agnostic Explanations (LIME)
and SHapley Additive exPlanations (SHAP) in Python programming language. By
employing these methods, we were able to gain insights into the model’s
predictions and provide explanations by identifying the specific contributions
of individual features.

#### LIME

LIME is a local interpretability method that provides explanations for
individual predictions by approximating the model’s behavior around specific
instances. By introducing perturbation into the input data and tracking how
the hybrid model’s predictions changed, we were able to utilize LIME to
provide potential explanations for the model’s behavior. This process allowed
us to identify the most significant features and understand their influence on
the model’s decision-making. LIME achieves this by generating local surrogate
models around a particular instance of interest. These surrogate models are
simpler and more interpretable, such as linear or decision tree models, and
capture the local behavior of the complex model. LIME examines the
significance and influence of individual aspects on the model’s decision-
making process by perturbation of the input features and evaluating the
ensuing changes in predictions. The equation of LIME can be expressed as
follows:

$\gamma(x)=\arg\min_{h\in H}L(f,h,\pi_{x})+\lambda(h)$ (20)

Here, the loss function $L$ quantifies the similarity between the original,
sophisticated model $f$ and the interpretable model $h$. The family of
interpretable models is denoted by $H$, while $\pi_{x}$ represents the
closeness of the instances being evaluated to a specific instance $x$. The
term $(h)$ indicates the significance or importance assigned to the model $h$,
which can involve additional weighting or importance factors. LIME aims to
find the interpretable model $h$ that minimizes the loss function and
adequately captures the complex model $f$ behavior while considering the
proximity and criticality aspects.

The family of interpretable models is denoted by $H$, while $\pi_{x}$
represents the closeness of the instances being evaluated to a specific
instance $x$. The term $(h)$ indicates the significance or importance assigned
to the model $h$, which can involve additional weighting or importance
factors. LimeTabularExplainer is a specific implementation of LIME designed
for tabular data. It leverages the idea of perturbation by generating
perturbed instances around the instance of interest. LimeTabularExplainer
constructs a local surrogate model by fitting a weighted linear model to these
perturbed instances. The weights assigned to each perturbed instance reflect
their similarity to the original instance, and the model’s predictions on
these instances are used to approximate feature importance. Figure 10(a) shows
the LIME-based feature importance bar plot showcasing the explanations for a
specific instance’s prediction. The plot visualizes individual features’
contributions towards classifying the instance into "No Backorder" or
"Backorder" categories. Figure 10(b) depicts the LIME-generated explanation
plot for another instance, depicting the feature importance and their
contributions to the prediction. The features ’PC1’, ’PC2’, ’PC3’, and ’PC4’
are considered, and the predicted probabilities are obtained using the model.

(a) LIME explanation for a single instance including predicted probabilities
of each class

(b) LimeTabularExplainer for a single instane

Figure 10: The figure comprises two subfigures illustrating the LIME
explanations for different instances in the classification task. (a) displays
a bar plot depicting the feature importance for a specific instance, while (b)
exhibits the LIME-generated explanation plot for another instance. Both (a)
and (b) highlight the contributions of the features ’PC1’, ’PC2’, ’PC3’, and
’PC4’ towards the predictions, providing insights into the classification
process of the model.

#### SHAP

SHAP is another popular XAI technique that provides global interpretability by
attributing the model’s predictions to individual features across the entire
dataset. Therefore, the contribution of each feature in the prediction was
computed and visualized using the SHAP Python library. By utilizing SHAP
values from cooperative game theory, SHAP quantifies each feature’s influence
on a prediction. They provide a quantitative assessment of how much each
feature contributes to the overall prediction, denoted as $\phi_{j}(x)$, which
is defined as follows:

$\phi_{j}(x)=\frac{1}{m!}\sum_{s\subseteq\\{x_{1},x_{2},...,x_{m}\\}\setminus\\{x_{j}\\}}\frac{|s|!(m-|s|-1)!}{m!}\times\left(\text{val}(s\cup\\{x_{j}\\})-\text{val}(s)\right)$
(21)

In Equation 21, $\phi_{j}(x)$ represents the Shapley value for the feature
$x_{j}$, where $x_{j}$ denotes a specific feature value. The feature subset of
the model is denoted as $s$. The parameter $m$ represents the total number of
features in the model. The term $val(s)$ represents the projection of feature
values in the set $s$. This equation calculates the Shapley value by summing
over all possible subsets $s$, considering their cardinality and the
difference between the valuation of the subset including $x_{j}$ and the
valuation of the subset excluding $x_{j}$. The division by $m!$ and the
factorials account for the different permutations and combinations of the
subsets.

(a) SHAP value visualization for a single prediction

(b) Decision plot of misclassified instances using SHAP values

Figure 11: (a) SHAP force plot on the selected instance using the
KernelExplainer and (b) SHAP decision plot showing the features’ contributions
to the misclassifications in the QAmplifyNet model.

We applied SHAP to our hybrid model to understand the importance and influence
of different features in determining the model’s output. SHAP values explain
how the model’s expected or base output, denoted by $E[f(x)]$, transitions to
the actual output, denoted as $f$ when the specific features $(x)$ are not
known. These values quantify each feature’s contribution to the prediction and
indicate the pattern of connection between the features and the target
variable $y$. When the SHAP value of a feature is close to -1 or +1, it
substantially impacts the prediction of that data point. On the other hand, a
SHAP value close to 0 for a feature indicates less importance in making the
prediction. Figure 11(a) displays the impact of each PC on the prediction for
the specific test instance.

SHAP decision plots provide insights into how the model reaches its decisions,
especially when numerous significant features are at play. Each line
corresponds to a prediction in these plots and interacts with the x-axis at
the predicted value. The color of the line reflects the prediction’s value,
creating a visual spectrum. Moving from the bottom to the top, SHAP values for
each feature are added to the model’s baseline, effectively revealing the
individual contributions of these features to the overall prediction. These
plots are invaluable for understanding the inner workings of complex models by
highlighting feature importance and showcasing the impact of each feature on
the final prediction. Figure 11(b) shows how each feature contributes to
shifting the model’s output from the expected value.

### Statistical Test

We conducted a rigorous statistical analysis with a significance level
($\alpha$) set at 5% to assess the performance of the QAmplifyNet model
against 17 ML models employed in our study. Our research investigated the
following two hypotheses:

  * •

Null Hypothesis (H0): There is no significant difference in accuracy and ROC-
AUC between the QAmplifyNet and the tested models.

  * •

Alternative Hypothesis (Ha): There is a significant difference in accuracy and
ROC-AUC between the QAmplifyNet and the tested models.

We conducted a ten-fold cross-validated paired t-test, a well-established
method for comparing classification model performance. The dataset was
randomly split into ten subsets, with shuffling for statistical robustness.
Then, for each cross-validation fold, we train the models on the training
subset and evaluate their accuracy on the corresponding test subset. Table 5
displays the results, highlighting significant differences in accuracy and
ROC-AUC scores between the QAmplifyNet model and its counterparts. These
results reject H0 in all 17 cases, indicating that the observed differences in
accuracy and ROC-AUC scores are statistically significant. When the p-value is
less than 5%, it signifies that these differences are unlikely to have
occurred by chance, providing robust evidence of the QAmplifyNet model’s
superior performance. The consistently low p-values underscore our confidence
in the model’s capabilities.

Model | t-statistic (Accuracy) | p-value (Accuracy) | t-statistic (ROC-AUC) | p-value (ROC-AUC)
---|---|---|---|---
Catboost | 71.81148356 | 9.95E-14 | 17.63726588 | 2.74E-08
DT | 63.94133809 | 2.82E-13 | 17.72554207 | 2.63E-08
KNN | 57.87545827 | 6.91E-13 | 73.09019106 | 8.49E-14
LGBM | 63.01465217 | 3.22E-13 | 14.14082681 | 1.88E-07
RF | 68.07699279 | 1.61E-13 | 14.08814707 | 1.94E-07
SVM | 65.40333765 | 2.31E-13 | 34.80494872 | 6.59E-11
XGBoost | 71.80695901 | 9.96E-14 | 24.47760546 | 1.52E-09
3 Dense Layered NN | 51.25034807 | 2.06E-12 | 11.31959904 | 1.26E-06
QSVM+LGBM+LR | 102.2786348 | 4.14E-15 | 26.03690965 | 8.77E-10
VQC+QSVM | 116.0140734 | 1.33E-15 | 34.87689033 | 6.47E-11
VQC+LGBM | 78.4710231 | 4.49E-14 | 29.79826722 | 2.64E-10
MERA 1-Layered | 68.41589025 | 1.54E-13 | 40.62670723 | 1.65E-11
MERA 2-Layered | 66.83965127 | 1.90E-13 | 48.78815222 | 3.20E-12
MERA 4-Layered | 30.86211809 | 1.93E-10 | 53.36665826 | 1.43E-12
RY-CNOT 6-Layered | 23.77391947 | 1.97E-09 | 38.83465142 | 2.47E-11
Classical NN+Encoder+QNN | 30.96578613 | 1.87E-10 | 15.72792266 | 7.47E-08
DDQN | 62.70623552 | 3.37E-13 | 61.82682204 | 3.82E-13

Table 5: Ten-fold Cross-Validated Paired t-Test for Model Performance
Comparison

## Discussions

SCM is a complex and critical process that relies heavily on accurate
prediction of backorders to optimize inventory control, reduce costs, and
ensure customer experience. This research introduced a groundbreaking hybrid
Q-CNN model called QAmplifyNet for SCBP, which integrates quantum-inspired
techniques into the conventional ML framework. The discussion aims to
comprehensively analyze the proposed model’s benefits, limitations, practical
implications, and potential applications. The integration of quantum-inspired
techniques in the proposed model offers several advantages over classical and
hybrid models. Firstly, the utilization of quantum-inspired algorithms enables
the model to grasp intricate data patterns and interdependencies, which is
crucial for accurate SCBP. The parallelism inherent in QC allows for more
efficient solution space exploration, leading to improved prediction accuracy.
QAmplifyNet benefits from the flexibility and interpretability of the Keras NN
framework. Combining quantum-inspired optimization algorithms with Keras’s
well-established architecture enhances the model’s overall performance and
interpretability. Our proposed model demonstrates robustness in handling
short, imbalanced datasets commonly encountered in SCM. By employing a
combination of preprocessing techniques, undersampling, and principal
component analysis, the model effectively addresses the challenges posed by
limited data availability and class imbalance.

While QAmplifyNet offers numerous advantages, it is important to acknowledge
its limitations. One potential limitation is the computational complexity
associated with quantum-inspired techniques. QC is still in its nascent
stages, and the current hardware limitations, such as noise and limited qubit
connectivity, can hinder the scalability and practical implementation of
quantum algorithms. Therefore, the proposed model may face challenges when
scaling up to larger datasets or real-time applications. Additionally, the
training and optimization of quantum-inspired models require specialized
knowledge and expertise. The integration of quantum and classical components
in the proposed model adds complexity, requiring researchers and practitioners
to have a strong understanding of both QC principles and traditional ML
techniques.

Accurate SCBP has significant practical implications for various aspects of
SCM. By leveraging the proposed model, organizations can optimize inventory
control, reduce backorders, and enhance customer satisfaction. The ability to
predict backorders enables proactive management of inventory levels,
minimizing stockouts, and ensuring the availability of products to meet
customer demands. This, in turn, leads to improved customer loyalty and
increased revenue opportunities. The accurate prediction of backorders allows
for more efficient resource allocation. Organizations can optimize their
production schedules, procurement processes, and transportation logistics
based on predicted demand, leading to cost savings and improved operational
efficiency. Additionally, accurate SCBP facilitates better supplier
communication and coordination, ensuring timely replenishment and minimizing
delays.

Our proposed model can be seamlessly integrated into real-world SCM systems.
Organizations can enhance their decision-making processes and automate SCBP by
incorporating the QAmplifyNet into existing inventory management software.
This integration enables real-time monitoring of inventory levels, proactive
order fulfillment, and efficient allocation of resources. The model can also
be used to identify potential bottlenecks or vulnerabilities in the SC,
allowing organizations to implement preventive measures and improve overall SC
resilience. QAmplifyNet has the potential for broader applications beyond SCM.
The hybrid nature of this model makes it adaptable to other supervised binary
classification tasks, such as credit card default prediction or fraud
detection, where imbalanced datasets and limited feature sets are common
challenges.

The comparative analysis sheds light on the strengths and weaknesses of each
model, which has direct implications for SCBP. QAmplifyNet emerges as the top-
performing model, consistently demonstrating strong performance across
multiple evaluation metrics, including accuracy, F1-score, specificity, Gmean,
IBA, and AUC-ROC. Its ability to achieve high accuracy and F1-scores indicates
its effectiveness in correctly predicting positive and negative instances,
which is crucial for efficient SCM. The superior performance of QAmplifyNet in
various metrics implies that it can effectively minimize false positives and
false negatives, addressing the challenge of imbalanced data in SCBP. This is
particularly noteworthy given the significant impact of FPs and FNs on
inventory management and customer satisfaction. Businesses may improve
customer satisfaction, reduce the likelihood of disruptions, and maximize
inventory efficiency by quickly and correctly detecting instances at risk of
backorders.

However, it is essential to consider the practical implications beyond model
performance metrics. Factors such as generalizability, interpretability, and
computational efficiency are critical for real-world implementation.
QAmplifyNet exhibits strong generalization capabilities, as evidenced by its
robust performance on the validation dataset. Its incorporation of
amplification techniques ensures scalability and computational efficiency,
enabling timely predictions for large-scale SC operations. Interpretability is
also a crucial factor in SC decision-making. While QAmplifyNet performs
exceptionally well in terms of accuracy and other metrics, its black-box
nature may limit the understanding of how and why specific predictions are
made. To address this, we presented the interpretability of QAmplifyNet using
SHAP and LIME.

## Conclusions and Future Works

In this research, our primary contribution lies in the development of
QAmplifyNet, a novel hybrid Q-CNN model designed explicitly for backorder
prediction in the SC domain. By harnessing the power of quantum-inspired
techniques within the well-established Keras framework, we aimed to
significantly enhance the accuracy of backorder prediction. Furthermore, we
proposed a comprehensive methodological framework encompassing various stages,
including data source identification, data collection, data splitting, data
preprocessing, and implementing the QAmplifyNet model. To ensure the optimal
performance of our model, we thoroughly explored seven different preprocessing
alternatives and meticulously evaluated their effectiveness by assessing the
performance of LR on each preprocessed dataset. This rigorous evaluation
process allowed us to select the most suitable preprocessing technique for our
specific application. Through extensive experiments and evaluations on a short
SCBP dataset, we compared the performance of QAmplifyNet with eight
traditional CML models, three classically stacked quantum ensemble models,
five QNN models, and one deep RL model. Our findings clearly demonstrate the
exceptional backorder prediction accuracy achieved by QAmplifyNet, surpassing
all other models in terms of accuracy with an impressive 90% accuracy rate.
Notably, QAmplifyNet also achieved the highest F1-score of 94% for predicting
"Not Backorder" and 75% for predicting "Backorder," outperforming all other
models. Additionally, QAmplifyNet exhibited the highest AUC-ROC score of
79.85%, further validating its superior predictive capabilities. By seamlessly
integrating quantum-inspired techniques into our model, we successfully
captured complex patterns and dependencies within the data, leading to
significant improvements in prediction accuracy.

The significance of the proposed model lies in its ability to optimize
inventory control, reduce backorders, and enhance overall SCM. Accurate SCBP
enables proactive decision-making, efficient resource allocation, and improved
customer satisfaction. By integrating the QAmplifyNet into real-world SC
systems, organizations can achieve cost savings, increased revenue
opportunities, and improved operational efficiency. By implementing XAI
techniques, specifically SHAP and LIME, we could successfully enhance the
interpretability of the proposed model. Understanding the model’s decision-
making process was greatly aided by these XAI techniques, shedding light on
the significance and contribution of different features in predicting
backorders. By leveraging SHAP and LIME, we were able to gain a deeper
understanding of how the model arrived at its predictions and identify the key
factors influencing those predictions.

Our pioneering research in quantum-backed backorder prediction presents
intriguing future prospects while acknowledging certain technological
limitations. We exclusively employed AE as part of our quantum feature
mapping. However, the potential of Angle Encoding and Basis Encoding remains
untapped, awaiting exploration in subsequent research. Due to constraints in
current quantum hardware, our QAmplifyNet model was crafted with only 2
qubits. Yet, as QC technology progresses, the utilization of circuits with a
larger qubit capacity, such as 4 or more, holds great promise for tackling
more complex computational tasks. Our study featured VQC with a single layer,
but the potential for increased depth to capture more nuanced data
dependencies beckons; furthermore, by diversifying the types of quantum layers
used beyond the SEL, the model’s performance can be fine-tuned with the
incorporation of Basic Entangler Layers, Continuous-Variable Neural Net
Layers, and Random Layers. Additionally, our research primarily integrated VQC
with fully connected dense layers. However, there is considerable potential in
coupling quantum models with a broader array of neural network architectures,
including Long Short-Term Memory, Gated Recurrent Unit, and Convolutional NN.
These endeavors are intricately linked to the ongoing evolution of QC
capabilities. They will play a vital role in the continued development of QML
for diverse and complex applications.

Deploying QAmplifyNet in real-world scenarios presents several potential
pitfalls and challenges that must be considered. It’s important to acknowledge
the current limitations in quantum hardware. Quantum computers are still in
their nascent stages, and access to large-scale, fault-tolerant quantum
devices is limited. This can restrict the deployment of quantum models in
real-world applications, especially when handling extensive datasets or
requiring real-time processing. However, with the ongoing advancements in
quantum technology, these hardware limitations may gradually diminish. Deeper
circuits may demand more computational resources and time, potentially
limiting their practicality in real-time applications. Striking a balance
between circuit depth and model performance is essential. The adoption of
quantum technology can be associated with substantial costs, including
hardware, software, and expertise. Assessing the cost-effectiveness of
deploying quantum models in real-world scenarios is crucial, particularly for
organizations with budget constraints. As QC matures and becomes more
accessible, many of these challenges may be addressed, opening up new
opportunities for leveraging QML in diverse and complex scenarios.

There are several promising avenues for future work in this field. Firstly,
further improvements can be made to the proposed model by exploring additional
quantum-inspired techniques and algorithms. As the field of QC continues to
advance, more efficient quantum hardware and algorithms are expected to become
available, which could enhance the performance and scalability of the model.
Expanding the dataset used for training and evaluation could further improve
the model’s accuracy and generalizability. The model’s ability to capture a
wider variety of patterns and trends might benefit from the incorporation of a
more extensive and diversified dataset. Furthermore, the potential for
applying the proposed model in other domains of SCM, such as demand
forecasting or inventory optimization, warrants exploration. The versatility
of QML models opens up opportunities for their application in various aspects
of SC operations.

In addition to introducing a novel strategy for SCBP by making use of the
hybrid Q-CNN, this study is notable for being the first use of QML in the
field of SCM. The results stress the necessity of quantum-inspired methods to
enhance prediction accuracy and optimize SCM. Future research has the
potential to change the field of SCM and stimulate breakthroughs in QML models
by continuing to improve the model, expanding the dataset, and exploring other
quantum-inspired approaches.

## Data availability

This research used the scikit-learn package for CML trials [45]:
https://scikit-learn.org/stable/. This is a link to the readily accessible
SCBP dataset: "Can you predict product backorder?".

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## Acknowledgements

We gratefully acknowledge the support of the Competitive Research Fund of The
University of Aizu, Japan, for funding this research.

## Author contributions statement

M.A.J.: Conceptualization, Methodology, Data curation, Writing - Original
Draft Preparation, Software, Visualization, Investigation. M.S.H.S.:
Methodology, Writing - Original Draft Preparation. M.S.I.: Supervision,
Writing - Reviewing and Editing. J.S.: Funding acquisition, Writing -
Reviewing and Editing. M.F.M.: Writing - Reviewing and Editing. Y.O.: Writing
- Reviewing and Editing.

## Competing interests

The author declares no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to M.A.J.