# Reframing Demand Forecasting: A Two-Fold Approach for Lumpy and Intermittent Demand

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Decoupling the demand forecasting problem into two separate problems: classification (demand occurrence) and regression (demand quantity estimation);
- Using four measurements to assess demand forecast performance: (i) the area under the receiver operating characteristic curve (AUC ROC) (Bradley [23]) to assess demand occurrence, (ii) two variations of the mean absolute scaled error (MASE) (Hyndman et al. [24]) to assess demand quantity forecasts, and (iii) stock-keeping-oriented prediction error cost (SPEC), proposed by Martin et al. [25] as an inventory metric;
- A new demand classification schema based on the existing literature and our research findings.

_{adj}model described by Hasni et al. [32]. We measured their performance using classification, regression, and inventory metrics. We also developed a compound model that outperforms the ones listed above.

## 2. Related Work

#### 2.1. Demand Characterization

#### 2.2. Forecasting Sparse Demand

#### 2.3. Demand Forecasting Models

#### Forecasting Features

#### 2.4. Metrics

## 3. Reframing Demand Forecasting

#### 3.1. A Classification of Existing Demand Forecasting Models for Lumpy and Intermittent Demand

**Type I**: uses a single model to predict the expected demand size for a given time step.**Type II**: uses aggregation to remove demand intermittency and benefit from regular time series models to forecast demand.**Type III**: uses separate models to estimate whether demand will take place at a given point in time and the expected demand size.**Type IV**: uses separate models to estimate the demand interval and demand size.

#### 3.2. Demand Characterization and Forecasting Models

^{2}, is accepted as a measure of whether a collection of observed demand is smooth, lumpy, intermittent, or erratic (Lowas III et al. [87]); it remains relevant for statistical methods. Nevertheless, we consider its relevance to blur with respect to different machine learning models. Its importance may be rendered irrelevant for global machine learning classification models that predict demand occurrence. By considering multiple items simultaneously, global models perceive a higher density of demand events and less irregularity than models developed with data regarding a single demand item. Simultaneously, the model can learn underlying patterns, which may be related to specific behaviors (e.g., deliveries that take place only on certain days). It is important to note that event scarcity usually results in imbalanced classification datasets, posing an additional challenge.

#### 3.3. Metrics

_{1}= α

_{2}= 0.5 since we have no empirical data that would support weighting α asymmetrically.

_{I}and MASE

_{II}). Following the criteria in Wallström et al. [42], we compute MASE

_{I}for the time series that results from ignoring zero-demand values. By doing so, we assess how well the regression model performs against a näive forecast, assuming a perfect demand occurrence prediction. On the other hand, MASE

_{II}is computed for the time series that considers all points where demand either took place or is predicted according to the classification model. By doing so, we measure the impact of demand event occurrence misclassification on the demand size forecast. When the model predicting demand occurrence has perfect performance, (i) should equal (ii). Finally, we compute SPEC for the whole time series (considering zero and non-zero demand occurrences). This way, the metric measures the overall forecast impact on inventory, weighting stock-keeping, and opportunity costs.

## 4. Methodology

#### 4.1. Business Understanding

#### 4.2. Data Understanding

#### 4.3. Data Preparation, Feature Creation, and Modeling

**MC+RAND**: a hybrid model proposed by Willemain et al. [29]. Demand occurrence is estimated as a Markov process, while demand sizes are randomly sampled from previous occurrences.**NN+SES**: a hybrid model proposed by Nasiri Pour et al. [28]. Considers a NN model (see Figure 2) to forecast demand occurrence; demand size is computed by exponential smoothing over non-zero demand quantities in past periods. We used the following parameters for the NN: a maximum of 300 iterations, a constant learning rate of 0.01, and a hyperbolic tangent activation. Given that no description was given on whether scaling was applied to the dataset prior to training the network, we explored two models: without feature scaling (NN_{NS}+SES) and with feature scaling (NN_{WS}+SES).**ADIDA**forecasting method, proposed by Nikolopoulos et al. [30], which removes intermittence through aggregation and then disaggregates the forecast back to the original aggregation level.**ELM**: an ELM model as proposed by Lolli et al. [31]. We initialized the model with the following parameters: 15 hidden units, ReLU activation, a regularization factor of 0.1, and normal weight initialization. We trained two models: ELM(C1) (two models, trained per demand type) and ELM(C2) (global model, considering all the demand types).**VZ**: a method proposed by Hasni et al. [32], considering only positive demands when the predicted lead-time demand was equal to the forecasting horizon considered._{adj}

## 5. Experiments and Results

_{1}and α

_{2}equal to 0.5. We summarize our experiments in Table 5. In Table 6, we summarize the results obtained for our own models, while in Table 7, we compare the best-performing of our models against the models described in the scientific literature and described above (in Section 4.3).

_{I}and MASE

_{II}values and even larger differences regarding the SPEC metric for most models. The only exceptions were the ELM(C1) and ELM(C2) models, which achieved better MASE metrics but not competitive AUC ROC or SPEC values. These results confirm the importance of considering the demand forecasting of irregular demand patterns as two separate problems (demand occurrence and demand size), each with its features and optimized against its own set of metrics. The improvements in classification scores substantially impacted demand size and inventory metrics.

_{I}and MASE

_{II}, while MFV had the best median SPEC score and remained competitive on MASE

_{I}and MASE

_{II}values. The Näive variant achieved the second-best SPEC score while remaining competitive on MASE values.

## 6. Conclusions

_{I}and MASE

_{II}for regression, and SPEC to assess the impact on inventory). We also developed a novel model that outperformed not only seven models described in the literature for lumpy and intermittent demand but also the state-of-the-art self-improving mechanism known as ADIDA. Considering the problem separation mentioned above, we propose a new demand classification schema based on the approach described herein that provides a good demand forecast. We consider two types of demands: ’R’ for demands where only regression is required and ’C+R’ where classification and regression models estimate demand occurrence and size, respectively.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A-MAPE | Alternative mean absolute percentage error |

ADI | Average demand interval |

ADIDA | Aggregate–disaggregate intermittent demand approach |

AUC ROC | Area under the curve of the receiver operating characteristic |

CV | Coefficient of variation |

ELM | Extreme learning machine |

FSN | Fast-slow-non moving |

GMAE | Geometric mean absolute error |

GMAMAE | Geometric mean (across series) of the arithmetic mean (across time) of the absolute errors |

GMRAE | Geometric mean relative absolute error |

MAD | Mean absolute deviation |

MADn | Mean absolute deviation over non-zero occurrences |

MAE | Mean absolute error |

MAPE | Mean absolute percentage error |

MAR | Mean absolute error |

MASE | Mean absolute scaled error |

MdAE | Median absolute error |

MdRAE | Median relative absolute error |

ME | Mean error |

MFV | Most frequent value |

mGMRAE | Mean-based geometric mean relative absolute error |

MLP | Multilayer perceptron |

mMAE | Mean-based mean absolute error |

mMAPE | Mean-based mean absolute percentage error |

mMdAE | Mean-based median absolute error |

mMSE | Mean-based mean squared error |

mPB | Mean-based percentage of times better |

MSE | Mean squared error |

MSEn | Mean squared error over non-zero occurrences |

MSR | Mean squared rate |

NN | Neural network |

PB | Percentage of times better |

PIS | Periods in stock |

RGRMSE | Relative geometric root mean squared error |

RMSE | Root mean squared error |

RMSSE | Root mean squared scaled error |

sAPIS | Scaled absolute periods in stock |

SBA | Syntetos–Boylan approximation |

SES | Simple exponential smoothing |

SKU | Stock-keeping unit |

sMAPE | Symmetric mean absolute percentage error |

SPEC | Stock-keeping-oriented prediction error cost |

TSB | Teunter, Syntetos, and Babai |

VED | Vital–essential–desirable |

WRMSSE | Weighted root mean squared scaled error |

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**Figure 1.**Demand pattern classification. (

**A**) depicts different demand patterns, while (

**B**) shows the classification proposed by Syntetos et al. [5] based on empirical findings.

**Figure 2.**MLP for the hybrid approach proposed by Nasiri Pour et al. [28]. The inputs to the model are the demand size at the end of the preceding period, the number of periods between the last two demand occurrences, the number of periods between the target period and the last demand occurrence, and the number of periods between the target period and the first immediately preceding zero-demand period.

**Figure 3.**Demand categorization schemas. On the right, (

**A**) corresponds to the influential categorization developed by Syntetos et al. [5]. On the left, (

**B**) proposes a new schema that only considers demand occurrence, dividing demand into two groups. ‘R’ denotes regular demand occurrence, where demand size can be predicted with a regression model. ‘C+R’ denotes irregular demand occurrence that requires a model to predict demand occurrence and a model to predict demand size.

**Figure 4.**Two-fold machine learning approach to demand forecasting. (

**A**) shows a basic architecture for demand forecasting when reframing demand forecasting as classification and regression problems. (

**B**) shows a flowchart with steps followed to create the demand forecasting models and issue demand forecasts.

**Figure 5.**The ADI-CV

^{2}categorizations for the time series based on the classification proposed by Syntetos et al. [5].

**Figure 6.**We computed predictions for two forecasting horizons, i.e., 14 (

**A**) and 56 (

**B**) days, to test the sensitivity of predictions regarding demand occurrence and demand size to the forecasting horizon.

**Table 1.**Metrics identified in main related works we reviewed on the topic of lumpy and intermittent demand.

Metric | [54] | [79] | [55] | [6] | [80] | [28] | [66] | [81] | [53] | [82] | [16] | [39] | [83] | [42] | [71] | [73] | [74] | [75] | [76] | [77] | [24] | [25] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A-MAPE | X | X | X | |||||||||||||||||||

GMAE | X | |||||||||||||||||||||

GMAMAE | X | |||||||||||||||||||||

GMRAE | X | |||||||||||||||||||||

MAD | X | X | X | X | X | X | ||||||||||||||||

MAE | X | X | X | |||||||||||||||||||

MAPE | X | X | X | X | X | X | X | |||||||||||||||

MAR | X | |||||||||||||||||||||

MASE | X | X | X | X | ||||||||||||||||||

MdAE | X | |||||||||||||||||||||

MdRAE | X | X | ||||||||||||||||||||

ME | X | |||||||||||||||||||||

mGMRAE | X | |||||||||||||||||||||

mMAE | X | |||||||||||||||||||||

mMAPE | X | |||||||||||||||||||||

mMdAE | X | |||||||||||||||||||||

mMSE | X | |||||||||||||||||||||

mPB | X | |||||||||||||||||||||

MSE | X | X | X | X | X | X | X | |||||||||||||||

MSR | X | |||||||||||||||||||||

MSEn | X | |||||||||||||||||||||

MADn | X | |||||||||||||||||||||

PB | X | X | X | X | ||||||||||||||||||

PIS | X | X | ||||||||||||||||||||

RGRMSE | X | X | X | |||||||||||||||||||

RMSSE | X | |||||||||||||||||||||

RMSE | X | X | X | |||||||||||||||||||

sAPIS | X | |||||||||||||||||||||

sMAPE | X | X | X | X | ||||||||||||||||||

SPEC | X | X | ||||||||||||||||||||

Theil’s U statistic | X | |||||||||||||||||||||

WRMSSE | X |

**Table 2.**Model types identified in main related works we reviewed on the topic of lumpy and intermittent demand.

Model Type | Related Work |
---|---|

I | [16,26,31,44,45,46,49,50,51,52,53,54,55] |

II | [8,30,58,59,60,61,63] |

III | [28,29,64] |

IV | [32,65] |

Metric | Mean | Std | Min | 25% | 50% | 75% | Max |
---|---|---|---|---|---|---|---|

ADI | 86.00 | 87.26 | 1.97 | 11.86 | 37.29 | 156.60 | 261.00 |

CV^{2} | 1.44 | 1.04 | 0.50 | 0.70 | 1.10 | 1.90 | 4.83 |

Metric | Mean | Std | Min | 25% | 50% | 75% | Max |
---|---|---|---|---|---|---|---|

ADI | 56.72 | 70.58 | 1.41 | 9.79 | 25.26 | 71.18 | 261.00 |

CV^{2} | 0.09 | 0.11 | 0.00 | 0.02 | 0.05 | 0.13 | 0.48 |

**Table 5.**Description of the reference models we evaluated for demand occurrence forecasting and demand size estimation.

Model Task | Model Type | ID | Models | Data |
---|---|---|---|---|

Classification | Global, per demand type (lumpy or intermittent) | C1 | CatBoost | Demand occurrence features |

Global, over all instances (lumpy and intermittent) | C2 | CatBoost | Demand occurrence features | |

Regression | Local, one per each time series | R1 | Naive SES MA(3) MFV RAND | Past non-zero demand sizes |

Global, per demand type (lumpy or intermittent) | R2 | LightGBM | Demand size features | |

Global, over all instances (lumpy and intermittent) | R3 | LightGBM | Demand size features |

**Table 6.**Overall results obtained with the models we proposed, for both forecasting horizons. Best results are bolded, second-best results are displayed in italics.

Model | Regression | Forecasting Horizon: 14 Days | Forecasting Horizon: 56 Days | ||||||
---|---|---|---|---|---|---|---|---|---|

AUC ROC ↑ | MASE_{I} ↓ | MASE_{II} ↓ | SPEC_{median} ↓ | AUC ROC ↑ | MASE_{I} ↓ | MASE_{II} ↓ | SPEC_{median} ↓ | ||

C1R1 | Naive | 0.9408 | 0.5764 | 1.1084 | 121.9809 | 0.9409 | 0.4861 | 1.1084 | 121.9809 |

MA(3) | 0.9408 | 0.5482 | 1.0544 | 281.2004 | 0.9409 | 0.4624 | 1.0544 | 281.2004 | |

SES | 0.9408 | 0.5229 | 1.0058 | 1210.7634 | 0.9409 | 0.4410 | 1.0058 | 1210.7634 | |

MFV | 0.9408 | 0.5437 | 1.0457 | 141.4351 | 0.9409 | 0.4585 | 1.0457 | 141.4351 | |

RAND | 0.9408 | 0.7552 | 1.4524 | 2011.5744 | 0.9409 | 0.6353 | 1.4485 | 2675.2844 | |

C1R2 | ML | 0.9408 | 0.5813 | 1.1183 | 46,422.3206 | 0.9409 | 0.4917 | 1.1215 | 46,162.9523 |

C1R3 | ML | 0.9408 | 0.5758 | 1.1250 | 48,807.4847 | 0.9409 | 0.4613 | 1.1274 | 48,279.0973 |

C2R1 | Naive | 0.9700 | 0.5271 | 1.0460 | 110.3435 | 0.9700 | 0.4445 | 1.0460 | 110.3435 |

MA(3) | 0.9700 | 0.4906 | 0.9736 | 245.1851 | 0.9700 | 0.4137 | 0.9736 | 245.1851 | |

SES | 0.9700 | 0.4611 | 0.9152 | 1343.2328 | 0.9700 | 0.3888 | 0.9152 | 1343.2328 | |

MFV | 0.9700 | 0.4760 | 0.9448 | 94.8092 | 0.9700 | 0.4014 | 0.9448 | 94.8092 | |

RAND | 0.9700 | 0.7267 | 1.4422 | 2034.5172 | 0.9700 | 0.5975 | 1.4059 | 2938.0534 | |

C2R2 | ML | 0.9700 | 0.5194 | 1.0310 | 44,255.2309 | 0.9700 | 0.4398 | 1.0352 | 44,101.6088 |

C2R3 | ML | 0.9700 | 0.5295 | 1.0510 | 39,918.1584 | 0.9700 | 0.4479 | 1.0542 | 40219.3378 |

**Table 7.**Comparison of methods found in related works and three of the best models we created: C2R1-SES, C2R1-MFV, and C2R1-Näive. Our three models achieved the highest AUC ROC and showed competitive MASE

_{I}and MASE

_{II}values. C2R1-MFV achieved the best overall performance on the SPEC metric, while C2R1-Näive achieved the second-best result. Best results are bolded, and second-best results are presented in italics.

Model | Forecasting Horizon: 14 days | Forecasting Horizon: 56 days | ||||||
---|---|---|---|---|---|---|---|---|

AUC ROC ↑ | MASE_{I} ↓ | MASE_{II} ↓ | SPEC_{median} ↓ | AUC ROC↑ | MASE_{I} ↓ | MASE_{II} ↓ | SPEC_{median} ↓ | |

Croston [16] | 0.5000 | 1.5997 | 96.2732 | 182,590,225.2842 | 0.5000 | 1.4769 | 1.4769 | 173,846,999.3033 |

SBA [26] | 0.5000 | 1.5196 | 91.4593 | 173,457,612.1803 | 0.5000 | 1.4047 | 88.2762 | 165,151,599.2596 |

TSB [27] | 0.5337 | 1.0340 | 42.8525 | 82,848,841.2377 | 0.5448 | 0.7491 | 11.9425 | 2,0024,649.5191 |

MC+RAND [29] | 0.5000 | 0.8616 | 1.3786 | 199.3702 | 0.5000 | 0.8619 | 1.3786 | 199.3702 |

NN [28]_{WS}+SES | 0.5000 | 0.3872 | 0.8588 | 18,804,861.1954 | 0.5000 | 0.3825 | 0.8588 | 18,240,504.9768 |

NN [28]_{NS}+SES | 0.5000 | 0.3834 | 0.8588 | 18,804,861.1954 | 0.5000 | 0.3825 | 0.8588 | 18,240,504.9768 |

ELM(C1) [31] | 0.6931 | 1.1020 | 0.0004 | 1593.4317 | 0.7024 | 0.3087 | 0.0004 | 1566.1803 |

ELM(C2) [31] | 0.6955 | 0.1034 | 0.0004 | 1605.4686 | 0.6967 | 0.0317 | 0.0004 | 1581.5055 |

VZ [32]_{adj} | 0.5000 | 0.9998 | 1.9864 | 23,883.8074 | 0.5000 | 0.9922 | 1.9864 | 23,883.8074 |

C2R1-Naive | 0.9700 | 0.5271 | 1.0460 | 110.3435 | 0.9700 | 0.4445 | 1.0460 | 110.3435 |

C2R1-SES | 0.9700 | 0.4611 | 0.9152 | 1343.2328 | 0.9700 | 0.3888 | 0.9152 | 1343.2328 |

C2R1-MFV | 0.9700 | 0.4760 | 0.9448 | 94.8092 | 0.9700 | 0.4014 | 0.9448 | 94.8092 |

**Table 8.**Comparison of AUC ROC for lumpy and intermittent demand, obtained from the C1 and C2 models. Using all data for a single classification model to predict demand occurrence showed improvements for both groups and time horizons. The largest improvement was observed for lumpy demand, with an improvement greater than 0.17.

Model | Forecasting Horizon: 14 Days | Forecasting Horizon: 56 Days | ||
---|---|---|---|---|

AUC ROC_{lumpy} ↑ | AUC ROC_{intermittent} ↑ | AUC ROC_{lumpy} ↑ | AUC ROC_{intermittent} ↑ | |

C1 | 0.7368 | 0.9666 | 0.7379 | 0.9666 |

C2 | 0.9097 | 0.9776 | 0.9097 | 0.9776 |

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## Share and Cite

**MDPI and ACS Style**

Rožanec, J.M.; Fortuna, B.; Mladenić, D.
Reframing Demand Forecasting: A Two-Fold Approach for Lumpy and Intermittent Demand. *Sustainability* **2022**, *14*, 9295.
https://doi.org/10.3390/su14159295

**AMA Style**

Rožanec JM, Fortuna B, Mladenić D.
Reframing Demand Forecasting: A Two-Fold Approach for Lumpy and Intermittent Demand. *Sustainability*. 2022; 14(15):9295.
https://doi.org/10.3390/su14159295

**Chicago/Turabian Style**

Rožanec, Jože Martin, Blaž Fortuna, and Dunja Mladenić.
2022. "Reframing Demand Forecasting: A Two-Fold Approach for Lumpy and Intermittent Demand" *Sustainability* 14, no. 15: 9295.
https://doi.org/10.3390/su14159295