Introduction

Due to their superior predictive performance, complex machine learning and deep neural network-based models have received high attention and are widely exploited in the business domain (Bawack et al., 2022; Janiesch et al., 2021; Cliff et al., 2011) along with other fields including image processing (Jiao and Zhao, 2019), health (Panesar, 2019; Bartoletti, 2019) and bioinformatics (Cao et al., 2020; Li et al., 2019). The tasks of those technologies range across different application areas including supply chain management, credit risk prediction (Moscato et al., 2021; Bussmann et al., 2021), detection of fraud credit card transaction (Carcillo et al., 2021; Randhawa et al., 2018) and marketing campaigns in retail banking (Ładyyżyński et al., 2019).

Generally, artificial intelligence (AI) techniques employ a huge size of training data for making predictions. While there is a huge interest in such predictions in various business domains (Ribeiro et al., 2016), one of the major problems of complex machine learning models is that they are very difficult to understand (Abedin et al., 2022; Adadi et al., 2018; Thiebes et al., 2021). Several Methods using induced ordered weighted averaging (IOWA) adaptive neuro-fuzzy inference system (ANFIS) can deal with multidimensional data to predict the quality of service and hence it help stakeholders in the decision-making process (Hussain et al., 2022a, b, 2021). As decisions often depend on a huge number of model parameters (Alvarez-Melis and Jaakkola, 2017), machine learning and deep learning techniques are like black-boxes or magic boxes to the general users (and often even for developers). The higher the accuracy of a complex machine learning model, the more opaque the models tend to become (Ribeiro et al., 2016). This opaqueness leads to a situation where users might question the predictions, because they are unable to understand the underlying decision making processes (i.e. the reasons for why a maybe counter-intuitive recommendation has been given) (Arya et al., 2019).

User acceptance is generally one of the main barriers for the success of technologies in companies. As AI-based recommendations can potentially have a huge impact on operational as well as strategic decisions in companies, it seems to be beneficial if users or consumers of AI models could better understand why those recommendations have been made (Meske et al., 2022). Apart from increasing trust in AI-recommendations, having factual explanations of a certain decision would also help users to learn about the field of application (for instance gaining a better understanding in the importance or non-importance of certain factors for business decisions) (Förster et al., 2020). In addition, according to the general data protection regulation (GDPR) by the European Union, EU citizens have the right to receive explanations about AI-based decisions, for instance if an AI recommendation affects credit worthiness or insurance rates (Meske et al., 2022; Došilović et al., 2018).

In this research, we propose a novel explainable predictive model for product backorder prediction. A backorder is a situation where customers can order a product even though that particular product is out of stock at the time when the order is placed (Hajek and Abedin, 2020; Ntakolia et al., 2021). Basically, its an order to a future inventory, going along with contingencies as time of delivery can vary and is not definitely known. Backorders are especially common for items that are highly popular. While for some items such as the latest flagship Apple iPhone, such events are quite common, it can be very unpredictable for other types of products. When retail companies order high amounts of products based on backorders, they risk their reputation if they are unable to keep the expected delivery dates. Another risk is that customers can cancel their orders because they don’t want to wait any longer or found another retailer where the product is in stock, leaving the company with excess products in their inventory. Here, predictive models can help to tackle these challenge by predicting the probability whether a certain product will be backordered or not, giving companies more time to plan and supporting them in their inventory management. In related works, researchers have proposed complex machine learning based methods to predict future product backorders. The predictive models include the application of support vector machine, XBoost, ensemble classifier and deep neural networks (Islam and Amin, 2020; Hajek and Abedin, 2020; Li, 2017; Shajalal et al., 2021).

However, the mere prediction of future backorders only solves part of the problem. Suppose you are responsible for a particular inventory management system at a retail company. When you are notified that the AI model decided that a particular product is going to be backordered in the near future, what will you do? Would you increase the inventory level (i.e. obtaining more products in advance)? Would you change any policy (negotiating with suppliers about faster transit times, lead times etc)? If you increase the inventory level, how many products would you order, assuming that some would surely be cancelled? For taking these decisions, you would need to understand the reasons for the prediction. Hence, our approach tries to provide insights into the factors that contribute to a certain prediction, helping users to adapt their strategies accordingly. Our paper contributes in the following ways:

  • We proposed a new CNN-based model for backorder prediction. Since backorders are rare events in inventory management systems, it is a challenging task to identify them. Their rarity leads to an extremely imbalanced distribution within datasets. Often, the percentage of the backodered samples is less than 0.01% (specifically 0.007%, de Santis et al. (2017)). To address this data imbalance, we incorporated an adaptive synthetic oversampling (ADASYN) technique that generates synthetic samples for a minority class. The results, based on diverse experimental settings and comparison with existing known related works, illustrated that our method achieved better prediction performance achieving a new state-of-the-art methods performance in terms of standard evaluation metrics.

  • To provide an overall insight of the predictive model’s decision-making priorities, we investigate the impact of different attributes of an order in the predictive models. We introduce an XAI technique, namely SHAP (Shapely additive explanations), that can interpret and/or explain the predictive model to identify the most important attributes of the decision making. Hence the stakeholders are enabled to better understand the model’s decision-making priorities and consider that when they have to work with such technologies.

  • By explaining specific predictions, our method can answer why a particular product will be backordered or not. Every order has different feature’s values which are considered to make predictions. Therefore, we trained a local interpretable surrogate model employing LIME (local interpretable model-agnostic explanations), and present explanations for an individual prediction to answer the question “why has this specific decision been made?” Hence, stakeholders can not only assess the models’ priorities in general, but also analyse singular decisions to better understand them.

The organization of the rest of this paper is as follows: Section 2 summarises related works on predicting product backorders. We present a brief discussion about different XAI terminologies in Section 3. In Section 4, we present our method for predicting future product backorders and the explanation generation techniques. The predictive performance of our proposed CNN-based method and performance comparison with classical machine learning classifiers and known related works are presented in detail in Section 5. The decisions of complex machine learning and deep learning models are explained through different types of explanations both for models’ priorities as well as specific predictions in Section 6. Finally, Section 7 concludes our proposed methods and findings of this study by discussing the prospects of introducing XAI technology in the business domain.

Related work

This section presents the discussion of related research on backorder prediction and explainable artificial intelligence in supply chain management. Existing works proposed different models to predict plausible future backorders in inventory management systems. Based on the types of techniques used, the predictive models can broadly be classified into two categories: i) Classical machine learning classifiers and ii) Deep learning-based predictive models. In the former category, the classifiers include support vector machine (Hajek and Abedin, 2020), gradient boosting (Ntakolia et al., 2021; de Santis et al., 2017), decision trees, and random forests (Islam and Amin, 2020). The deep learning-based models employed recurrent neural networks (RNN) (Li, 2017), deep auto-encoders (Saraogi et al., 2021), as well as deep neural networks (DNN) (Shajalal et al., 2021).

Islam and Amin (2020) proposed a method to predict future backorders by applying distributed random forest and gradient boosting classifiers. They introduced a ranged-based approach to cope with the numerous types of real-time data. However, they did not include some features of the samples such as features related to inventory level, previous sales, future sale forecasting, and lead time. A profit-maximizing function based approach is introduced by Hajek and Abedin (2020).

Table 1 The summary of existing study on product backorder prediction
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They aligned their profit maximization function with classical machine learning classifiers. The performance of their methods demonstrated how much profit can be increased by predicting future backorders. An explainable classical machine learning-based method is proposed by Ntakolia et al. (2021). Their method applied several classifiers such as random forest, XGBoost, SVM, etc. They also applied shapely additive values to present the global explanations to interpret the models. Similarly, de Santis et al. (2017) also used different classical classifiers. The performance of deep learning approaches is comparatively better than the classical classifiers. Shajalal et al. (2021) proposed a deep neural network (DNN) based backorder prediction model. Inspired by the success of deep learning classifiers, Li (2017), Saraogi et al. (2021) and Lawal and Akintola (2021) applied deep auto-encoder, a recurrent neural network-based classification models.

Backorders are not a common scenario in inventory management systems. In turn, the number of non-backordered items is much larger than the backordered ones. Hence, real-time data collected from any inventory system will be strongly imbalanced, leading to challenges in predicting future backorders on that basis. In this particular task (Li, 2017), the ratio between majority (non-backordered) and minority (backordered) samples is 100:0.007. In the case of an imbalanced dataset, the classifiers might learn the pattern with potential bias. That is why different under-sampling, oversampling, and class weight-based approaches are common to balance the dataset and bias (Hajek and Abedin, 2020). Randomly duplicating the minority samples or randomly discarding the majority samples has also been applied to balance the dataset (Chawla et al., 2002). But randomly duplicating the minority samples will increase redundant samples and hence the model might be biased. Therefore, generating synthetic minority samples based on the Euclidean distance is a popular approach to balance the dataset. This method is called SMOTE (Synthetic Minority Over-sampling Technique) (Chawla et al., 2002). The combination of SMOTE and random under-sampling has been applied by Hajek and Abedin (2020) and Shajalal et al. (n.d., 2021). Li (2017) applied different balancing techniques including SMOTE, ADASYN (Adaptive Synthetic Sampling) (He et al., 2008) and random under-sampling. Bagging (Błaszczyński and Stefanowski, 2015) is also applied for the same purpose by (de Santis et al., 2017). Table 1 summarises the existing methods for predicting product backorders.

Table 2 Existing research gaps in explainable product backorder prediction and our steps to fulfil the research gaps
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However, to the best of our knowledge, none of the studies applied XAI to interpret their machine learning model except Ntakolia et al. (2021).

In our paper, we propose a convolutional neural network framework-based model that outperformed different classifiers including classical and deep learning-based models in backorder prediction. Ntakolia et al. (2021) interpreted only classical models mainly with global explanations. Our method integrated explainable artificial intelligence that generates global explanations for the classical and deep learning-based prediction model. Though the global interpretation is useful to illustrate the general mechanisms and behavior of the model, it can not explain a particular prediction. We introduced a model applying shapely additive explanation (Lundberg and Lee, 2017) and local interpretable model-agnostic explanation (Ribeiro et al., 2016) to interpret the overall model and local specific decisions.

To clearly illustrate the research gap in the existing literature and our research focus, we present a comparative analysis in Table 2.

XAI terminology

In this paper, we employed two XAI techniques, namely shapely additive explanations (SHAP) and local interpretable model-agnostic explanations (LIME). Here, we present the background and working principle of these two techniques.

SHapely Additive exPlanation (SHAP)

Lundberg and Lee Lundberg and Lee (2017) first proposed a unified approach to explain and interpret the prediction of machine learning models. The explanations basically illustrate the contributions (positive and negative importance or influence) of different features for the predicted decision of a particular sample x. The overall feature importance of different features of the whole model can also be interpreted as global explanations. In that case, the importance score resembles the weight of features as in the linear model. The SHAP values represent the importance of the features. The explanation of every single prediction can be seen as a vector of shap values. The same representation is used to interpret the overall model. For a given instance x, the explanation using SHAP can be defined as

$$\begin{aligned} g(z') = \phi _0 + \sum \limits _{i=1}^{M} \phi _jz'_{j}, \end{aligned}$$
(1)

where g is denoted as the explanation model. The vector for simplified features, known as the coalition vector is represented by \(z'\) (\(z' \in \{0,1\}^M\)). The 1 represents that features’ values are the same as the original instances and vice-versa. The attribution of particular features j of the instance x is denoted by \(\phi _j\) which is a real number. The higher the value of \(\phi _j\), the more important the feature j. The \(\phi _j\) is computed based on Shapely values (Nowak and Radzik, 1994), a game-theoretic approach that identifies and detects the contribution of all players in a collaborative game. The collaborative game with multiple players is analogue to the prediction of the instance having multiple features. In turn, applying this game-theoretic approach we can examine the contribution of each feature to a particular decision. For a given feature vector \(x'\) and a predictive model f, the computation is done as follows:

$$\begin{aligned} \phi _i(f,x') = \sum \limits _{z'\subseteq \{ x'_1, x'_2,...,x'_n \} \setminus \{x'_i\}} \frac{(|z'|)!\,\, (M-|z'|-1)!}{M!} \cdot [f(z' \cup x'_i) - f(z')], \end{aligned}$$
(2)

The subset of the features employed by the model is denoted as \(z'\). \(x'\) is the vector with features values to be explained and can be defined as \([f(z' \cup x'_i) - f(z')]\) and M is the number of features. The prediction by the model f is denoted by \(f(z')\). Moreover, SHAP values are computed by a standard game-theoretical approach and utilized Shapely values to have a unified interpretable model with fast computation. More mathematical and technical details for SHAP can be found in the study published by Lundberg and Lee (2017) as well as in Nowak and Radzik (1994).

Local interpretable model-agnostic explanation (LIME)

LIME mainly provides model-agnostic explanations based on local surrogate models. Ribeiro et al. (2016) first introduced this approach for training a local surrogate model instead of a global model for providing explanations for a particular prediction. LIME employed a new local dataset containing the permuted samples with corresponding predictions to train the local interpretable surrogate model. This surrogate model is then used to explain individual predictions. The model is considered as an approximation of the original complex, black-box predictive model. The computation of the surrogate model can be defined as follows:

$$\begin{aligned} \xi (x) = \underset{g \in \mathbf {G}}{\mathrm {arg \, \min }} \,\, \textit{L}(f,g,\pi _{x'}) + \Omega (g) \end{aligned}$$
(3)

The explanation model for a particular instance x and the explanation family are represented by g and G, respectively. The original model is denoted by f and \(\textit{L}\) is the loss function. The complexity of the model can be defined by \(\Omega (g)\). LIME is useful to explain specific decision predicted by the model (i.e., local prediction).

Explainable product backorder prediction

The overview of our proposed explainable product backorder prediction framework is depicted in Fig. 1. We first apply preprocessing step to handle the missing values, converting qualitative variables into quantitative ones and normalizing the values in a similar range. Next, we apply our proposed convolutional neural network-based backorder prediction model to classify the product. Finally, we introduce explainable AI techniques to explain both global, model agnostic aspects as well as individual decisions with the intent to make the inventory manager understand better why his or her backorder prediction system acts as it does.

Fig. 1
figure 1

Proposed explainable backorder prediction approach

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Preprocessing and feature analysis

In our dataset, each particular sample has 21 different features/attributes including current inventory, lead time, forecasting for a different time, sales performance, different risk flags. The details of the dataset are presented in Section 5.1.1. The value of different features is varied widely among binary, quantitative, qualitative, and categorical. In this step, all the feature values are transformed into a real number. The missing values are handled by filling them in with the median of other samples’ values. A normalization technique is then applied to convert each feature value into a certain range [0,1]. Here, we applied the most widely recognized MinMax normalization technique.

However, a dataset having highly correlated features is not suitable for applying classification methods. We investigate to see whether any high correlated features are available, exploiting the Pearson correlation coefficient measure for this purpose. According to the findings, we observe that there are no features with a high correlation (\(\rho >.80\)). Hence, the dataset should now be suitable for our purpose of backorder predictions.

figure a

Algorithm 1 ADASYN: Adaptive Synthetic Oversampling

Handling class imbalance with ADASYN

As we noted earlier, a product backorder scenario is a rare event that leads the dataset to be extremely imbalanced. Therefore, we employed one of the efficient synthetic oversampling methods, ADASYN (Adaptive Synthetic Oversampling) (He et al., 2008) to balance the dataset. Considering the difficulty level of learning, ADASYN generates synthetic minority class examples utilizing the weighted distribution. ADASYN focused on generating more synthetic minority class examples for those minority samples that are harder to classify. Given a training dataset, \(D_{train}\) with N number of samples where each sample is denoted as x, y, the vector x is represented by a K dimensional vector containing different attributes of an ordered product and y is the binary value that indicates the label (0 for non-backordered and 1 for backordered one).

Let \(m_{\min }\) and \(m_{maj}\) be the number of examples of minority class and majority class, respectively such that \(m_{\min } + m_{maj} = N\), and in this backorder prediction task \(m_{\min }<< m_{maj}\). ADASYN oversampling techniques generate synthetic minority class examples to balance the dataset according the algorithm illustrated in Algorithm 1.

It first calculates the degree of imbalance d and then, depending on the tolerated imbalance ratio, computes the number G that denotes the number of synthetic minority class examples needed to be generated. Here \(\beta \in [0,1]\) indicates the desired bleaching ratio, \(\beta = 1\) indicated that the dataset will be fully balanced. For each minority example \(x_i\), ADASYN then calculate the ratio \(r_i\) applying K-nearest neighbors with Euclidean distance, where \(\Delta _i\) is the number of nearest neighbors of \(x_i\). Using the normalized ratio \(\hat{r_i}\), then it computes the number of synthetic examples for each minority examples \(x_i\). Finally it generates the synthetic minority class examples applying the distance vector and the random number \(\lambda\).

Convolutional neural network-based prediction model

Inspired by the success of the convolutional neural network (CNN)-based models in computer vision, natural language processing and other classification tasks, we proposed a 1-dimensional CNN classifier to predict product backorder in advance. The structure of our proposed CNN-based predictive model is illustrated in Fig. 2.

Fig. 2
figure 2

Structure of our proposed convolutional neural network-based backorder prediction model

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Our CNN-based predictive model has two convolutional hidden layers with batch normalization, max-pooling, and dropout layers. To extract unique and low-level features, the max-pooling layers are exploited. In addition, max-pooling makes the computation faster by reducing the dimension and parameters (Wu and Gu, 2015). Moreover, it reduces the variance. Then we utilized one flattened layer followed by three dense layers with dropout layers. To overcome the over-fitting problem, dropout layers are applied to randomly drop some neurons in the training process for regularization (Kingma et al., 2015; Srivastava, 2013). The parameters and activation functions in different layers of convolutional neural networks are summarized in Table 3. In the convolutional layers and all hidden dense layers, we employed the Relu (Ramachandran et al., 2017) activation function. Finally, Sigmoid (Ramachandran et al., 2017) activation function is applied in the output layer.

Table 3 The summary of different layers with parameters and activation functions
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Experiments and evaluation

Dataset collection and evaluation metrics

This section presents the details of dataset that is leveraged to conduct experiments using our proposed method. We also present a brief discussion about the evaluation metrics considered to measure and validate the performance.

Dataset

We carried out a wide range of experiments to validate the performance of our methods on a publicly available benchmark dataset called “Can you Predict Product BackorderFootnote 1”The dataset has an 8 weeks inventory of historical data. The brief statistical summary of the dataset is depicted in Table 4.

Table 4 Brief statistical summary of the dataset
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Fig. 3
figure 3

Distribution of backordered and non-backordered samples

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The numerical figures in Table 4 illustrate that the number of backordered (positive) samples is much lower than the number of non-backordered (negative) samples. Hence, the ratio (1:137) indicates that this dataset is an extremely imbalanced one. For a better understanding of why this is a challenging problem, we illustrated the distribution of backordered (positive) and non-backordered (negative) samples using a doughnut chart in Fig. 3. There are 22 features for each sample and the attributes/features include current inventory, transit time, quantity, forecasting, and different risk flag. The list of features with a brief description is depicted in Table 5.

Table 5 Description of different features/attributes of a particular ordert
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Evaluation metrics

Generally, the performance of any classification method is measured based on the common evaluation metrics including accuracy, precision, recall and f\(_{1}\)-score. The confusion metrix is used to compute those metrics. However, the backorder prediction dataset is extremely class imbalanced, and the above mentioned evaluation metrics are not enough to validate the performance of any classifier on a imbalanced dataset. Therefore, we employed accuracy, AUC (Area Under the Curve) and ROC (Receiver Operating Characteristics) curves to measure and visualize the performance of our proposed backorder prediction method. The accuracy score is calculated by using the measures from confusion metrics as follows:

$$\begin{aligned} Acc = \frac{tp + tn}{tp + fp + fn + tn}, \end{aligned}$$
(4)

where tp, fp, fn and tn denote the number of classified samples as true positive, false positive, false negative and true negative, respectively.

AUC is one of the most efficient metrics to measure the performance of any classification model on imbalanced data. The AUC is calculated as follows:

$$\begin{aligned} AUC = \frac{1 + P - F}{2}, \end{aligned}$$
(5)

where P is the precision of the classifier and F is the false positive rate. The details of the these metrics can be found elsewhere in the published study by Chawla et.al (2002) and de Santis et al. (2017).

Prediction performance

We conducted a wide range of experiments with multiple settings to illustrate the performance of our backorder prediction methods. Since our major goal in this study is to introduce the explainability of in backorder prediction, we first applied classical machine learning and a deep neural network-based classification approach. Then, we exploit XAI technologies (SHAP and LIME) to explain the model’s priorities and individual prediction. Classifiers from classical machine learning including decision tree, support vector machine, gradient boosting, etc., were applied. All experimental settings can be broadly classified into three different types based on the chosen dataset balancing strategy, classical ML, and deep learning. In all experimental setups, we applied two different dataset balancing techniques ADASYIN and SMOTE (Chawla et.al, 2002). Based on the predictive models, we report the experimental results in two categories, classical and deep classifiers.

Table 6 Performance of classical machine learning models in terms of accuracy and AUC
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The prediction performance of all experimental setups using classical machine learning is presented in Table 6. The results conclude that the ASASYN balancing technique is more efficient and achieved higher accuracy as well as AUC than SMOTE in most of the experimental setups. In turn, we can conclude that for the backorder prediction task, our introduced ADASYN balancing technique would be a better choice to implement any real-time backorder prediction system. Among all five different classical machine learning models, the gradient boosting (XGBoost) classifier achieved higher accuracy. On the other hand, in terms of the most effective and important evaluation metric, AUC, support vector machine performs better than other models. In addition, other classification models including decision tree, SVM, and KNN also achieved effective performance in backorder prediction except for Gaussian Naive Bayes.

Table 7 Performance of CNN-based models in terms of accuracy and AUC
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The experimental setup for deep learning techniques can be classified based on the parameters. The experiments are conducted by training CNN-based models with different settings. Two types of CNN models are applied. One has max-pooling layers and the other does not. The models were trained using two different epoch sizes which are 50 and 100. The performance of all experimental settings is presented in Table 7.

From the results, we can see that the convolutional neural network-based model with max-pooling layer (MxCNN_100 and MxCNN_50) performed better among other experimental settings. It can also be concluded that our introduced ADASYN data balancing technique achieved efficient performance in both evaluation metrics. We added dropout layers that randomly drop some neurons in the training process for regularisation to overcome the over-fitting problem. To illustrate the necessity of dropout layers in CNN-based models, we carried out experiments with and without dropout layers. The performance based on the evaluation metrics concludes that dropout layers overcome the over-fitting problem. The MxCNN model without dropout layers achieved accuracy and AUC in the training data of 0.9081 and 0.9651, respectively. On the other hand, for testing data, the performance is lower in terms of both metrics. Without dropout layers, the performance of the model on test data based on accuracy and AUC are 0.8792 and 0.9411, respectively. Though the performance difference (2.89% in accuracy and 2.4% in AUC) between the training and testing data is not that big but still it has over-fitting. The method with dropout layers got the training accuracy and AUC of 0.8843 and 0.9499, respectively. For test data, the performance is quite consistent with accuracy and AUC of 0.8903 and 0.9489, respectively. Thus, we can say that the inclusion of dropout layers overcomes the over-fitting problem and eventually increases the performance.

Fig. 4
figure 4

Performance of convolutional neural network based predictive model in terms of Receiver Operating Characteristic curve (ROC curve). The X-axis and Y-axis indicate the false positive and true positive percentage, respectively

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As compared to the performance of classical machine learning models reported in Table 6, the prediction power of our proposed CNN-based approach is way higher than the performance of ML methods. Although classical machine learning classifiers achieved higher accuracy, our CNN based model achieved a huge improvement in predicting future backorder prediction in terms of AUC, which is a more important metric to judge the performance of a classifier on data imbalance, because higher accuracy alone might not guarantee the predictive power of a classifier in case of extreme data imbalance. The performance of our method is also depicted by the Receiver Operating Characteristic curve (ROC curve) in Fig. 4. The curve illustrated the performance of our predictive model as compared to a random classifier. The area within the green curve shows the higher AUC achieved in predicting product backorders.

Performance comparison with state-of-the-art methods

The performance comparison of our proposed backorder prediction model with existing state-of-the-art methods is presented in Table 8. We directly reported the performance in terms of accuracy and AUC from existing published papers. Some existing works reported the performance only in terms of AUC but did not use accuracy and some others did the opposite. The blank cells (i.e., “-”) in the table indicate that the performance based on particular evaluation metric is not reported in the published paper. According to the performance of different state-of-the-art methods reported in the table, we can see that our CNN-based predictive model outperformed the known related works in terms of both evaluation metrics except for one method by Shajalal et al. (2021). In turn, our methods significantly outperformed all methods based on accuracy. Shajalal et al. (2021) applied a deep neural network with SMOTE oversampling technique. They applied four different variants of their methods utilising oversampling and under sampling techniques. Compared to the performance of those methods, our model got the best performance except one. Though one of their methods achieved a higher performance in terms of AUC, the performance difference with our method is subtle. In addition, their method lacks global interpretability and local explainability.

Table 8 Performance comparison of our method with known related work on the same dataset in terms of accuracy and AUC
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Islam and Amin (2020) applied a distributed random forest (DRF) and gradient boosting machine (GBM) classifier to model product backorder. The performance of their models is struggling compared to the CNN-based model in terms of both evaluation metrics. Another noticeable concern in the performance of their method is substantial over-fitting. Numerically, their training accuracy of 0.9835 is way higher than the testing accuracy of 0.8436. Another work by Hajek and Abedin (2020) applied classical machine learning classifiers to model product backorder prediction. From their applied ML classifiers, random forest (RF) achieved the best AUC, which is still lower than our method. Note that they did not report the accuracy in the paper. Similar to Shajalal et al. (2021); Ntakolia et al. (2021) proposed a multi-layer perceptron (MLP) based neural network (NN) for modelling product backorder. But their performance is still much lower than ours in terms of both evaluation metrics. We think adding ADASYN oversampling technique overcome the data imbalance problem better. With this, our convolutional neural network-based predictive model capture the product backorder more efficiently as compared to other state-of-the-art methods. Having this comparative analysis, we can conclude that our method has got a new state-of-the-art performance in predicting product backorder in the inventory management system.

Explaining backorder prediction model

This section presents the explainability of our introduced XAI techniques to interpret and/or explain the overall model and particular decisions of the proposed backorder prediction model. We first present the global model agnostic explanations generated to interpret the overall model’s prediction priorities. Then the local explanations for a certain prediction are presented to provide the overall insight to understand a certain product will be going to be backordered or not.

Explaining overall model’s priority

To interpret and explain the overall model, we exploit Shapely Additive values (Shap Values) that highlight the overall features’ contributions in predicting the model’s decisions. The feature contributions for the best performing model are depicted in Fig. 5 and 6.

Fig. 5
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Global interpretation of the features’ contributions of backorder prediction model as summary plot

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Fig. 6
figure 6

Global interpretation of the features’ contributions of backorder prediction model as bar chart

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We can see both figures indicate the top ten most important features that the prediction model has given higher importance, such as current inventory, transit quantity, lead time, performance in the last 12 and 6 months, and sales. We can conclude that these are the top 10 most important features on which the model depends more to predict whether a product is going to be backordered or not.

Explaining individual predictions

The features described in the previous figures (Fig. 5 and 6) have overall high importance in the predictive models. But it is expected that every sample (order) is different and unique in terms of features’ values. Therefore, the importance and contributions of different features also will be different for particular order. To identify the most contributing features of each order, we employed local interpretable model agnostic explanation (LIME) to explain individual predictions. Using LIME, we trained a surrogate model with a portion of training data that mimics the performance and decision-making priorities of the proposed backorder prediction model. The explanation using LIME is depicted in Figs. 7 and 8. These are the explanations for two individual backordered samples.

Fig. 7
figure 7

Local explanations of an individual prediction using LIME

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Fig. 8
figure 8

Local explanations of an individual prediction using LIME

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The labels of these two products are 1 (backordered) and the model also predicted the same. The features in the right side marked by Yellow color pushed the predictive model to classify as backordered, while Blue colored features in the left side did the opposite. From Fig. 7, we can see that the probability for being classified as backordered and non-backordered is 66% and 34%, respectively. The figure also indicates that the most important features (features in the right side) that lead to the prediction as backordered are local_bo_qty, current_inventory, and sales in the last 1 and 3 months and a risk flag. On the other hand, features (features in the left side) like lead time, in_transit_quantity, performance in the last 6 and 12 months, etc. try to push the model to predict a product as non-backordered. However, for another backordered sample, we can see that the list of contributed features for the backorder decision is different than the previous one. In Fig. 8, features such as lead time contributed the most to pushing the model to decide as a backordered one, which was the opposite for the previous sample.

Fig. 9
figure 9

Local explanations of an individual prediction using force plot

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Fig. 10
figure 10

Local explanations of an individual prediction using force plot

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Fig. 11
figure 11

Local explanations of an individual prediction using waterfall plot

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Fig. 12
figure 12

Local explanations of an individual prediction using waterfall plot

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To explain the local individual predictions more transparently, we applied shap values to plot the explanation as a force plot. Figures 9 and 10 illustrate the explanations for two different backordered samples. In both figures, the predictions from the models, referred to as base values are 0.67 and 0.75, respectively for both samples. The closer the value is to 1, the more the prediction leans toward backordered, while the closer to 0, the decision will then predicted as non-backordered. The red marked features contributed to increase the base value that help to decide the sample as a backordered one, and blue marked features did the opposite. The features having more impact on the base value remain closer to the boundary. For example, the two most-contributing features that push the model to decide the samples as backordered are current inventory and per_6_months for the first force plot (Fig. 9). For another example, the features with the most impact are current inventory and the forecasting for 9 months (Fig. 10). The explanations for the same two samples’ decisions are also presented using the waterfall plot in Figs. 11 and 12. The red marked features contributed to predicting the sample backordered and the blue colored try to push the classifier to predict the sample as non-backordered. Here the number and the span indicate the level of contributions of the features towards the decision.

With the help of our approach, stakeholders without in-depth knowledge of how backorder prediction systems work can have a better understanding both in terms of how the models generally factor in different types of data for making their decisions, as well as be enabled to analyse concrete decisions (that might seem counter intuititve or risky) in more depth than is possible with existing approaches. By applying such visualisations in practice, stakeholders would thus be enabled to enact suggestions from AI based systems more competently, and adapt their business strategies and decisions accordingly. This has the potential to both improve the usefulness, as well as also the willingness to adopt such systems in practice. While our introduced explainabilty has still to be evaluated with users, its is not trivial to implement such systems in practice. In this regard, our paper contributes a demonstration of the applicability, and shows how such techniques can be implemented in a way that provides value to other stakeholders than developers of machine learning systems, to whose such applications are currently targeted.

Conclusion & future directions

This paper proposed a novel CNN-based model for product backorder prediction in an inventory management system and introduced global and local explainability that can explain the overall model decision-making priorities and answer the “why” question regarding any specific prediction. First, we proposed a novel convolutional neural network-based prediction model incorporating ADASYN oversampling technique to address data imbalance problem. The performance carrying out diverse experiment setups concluded that our proposed CNN-based backorder prediction model achieved a new state-of-the art result in product backorder prediction. In addition, the performance comparison with some known related methods demonstrated that our methods outperformed others in terms of multiple evaluation metrics. Secondly, our model is not only able to predict the backordered item but also can explain the reasons why the model predicts that a product is going to be backordered. For doing so, we utilised existing successful XAI techniques, SHAP and LIME, to explain the overall predictive model and individual decisions. Using global explanations, the stakeholder, and inventory managers can have an idea and understanding of how the overall model is making the decision. On the other hand, they can explicitly know and analyse why a certain product has a high chance to be backordered in the future, leveraging the explanations for their business decisions. Hence, they can identify which attributes have the most impact on a particular decision, and then react by adapting controllable attributes (i.e. current inventory, lead time, etc.). Therefore, even when our approach still needs to be evaluated in practice, we believe these explanations can help the stakeholders to make their decision and minimize future losses. Most importantly, these explanations can increase the trust, transparency, and accountability of the AI-based predictive models in business problems, thus helping to overcome limitations of existing approaches that are more like black boxes for the users. While our study demonstrated the applicability of XAI techniques in the business domain on the concrete example of backorder predictions, there are multiple application areas such as customer churn prediction, customer behavior prediction, credit-worthiness assessment, fraud detection etc. where our explainable predictive model can be applied.

In the future, we plan to develop a collaborative interface to represent the explanations so that people can understand the decision-making more efficiently. We are also planning to introduce counterfactual explanations to provide a clear understanding about what are the possible actions she needs to take into account in the future.