# Supply Chain Demand Forecasting and Price Optimisation Models with Substitution Effect

^{1}

^{2}

^{*}

^{th}Birthday of Prof. Adil M. Bagirov)

## Abstract

**:**

## 1. Introduction

- investigating demand prediction models for multiple products with substitution effect and proposing a new demand prediction procedure;
- utilising data-driven machine learning approaches in demand forecasting;
- studying price optimisation models for the retailer that reflects all price changes of products and constructing a novel price optimisation procedure;
- applying the proposed demand prediction and price optimisation procedures to real-world transaction data.

## 2. Data Set

#### 2.1. Data Description

#### 2.2. Data Preprocessing

#### 2.3. Exploratory Data Analysis (EDA)

## 3. Demand Prediction Models

#### 3.1. Existing Models

#### 3.2. Selection of Variables

#### 3.3. Validation of Models

#### 3.4. Evaluation and Comparison of Models

#### 3.5. The Proposed Demand Prediction Procedure

- 1.
- Start modelling for the first product on the product list p (p = 1–12);
- 2.
- Generate input data including key features, price of all products, time lags, and sales of the target product from the database;
- 3.
- Split features and target variable;
- 4.
- Select the individual demand prediction model from the predefined model set m (m = 1–14);
- 5.
- Select the hyperparameter option combination from the predefined parameter set h (h is different depends on the model);
- 6.
- Increase the rolling forecasting windows from 52 weeks to [the total number of weeks of each product − test weeks];
- 7.
- Perform the demand prediction model training with selected demand prediction model, hyperparameter option and rolling forecasting window obtained in steps 4–6;
- 8.
- Calculate and store RMSPE score from the model validation process using (9);
- 9.
- After finishing the rolling forecasting loop, calculate and store the average RMSPE score of all the rolling forecasting for selected hyperparameter option;
- 10.
- After finishing the hyperparameter option loop, store the best hyperparameter option with the lowest RMSPE score for selected demand prediction model;
- 11.
- After finishing the demand prediction model loop, store the best demand prediction model with the best hyperparameter option for selected product;
- 12.
- After finishing the product loop, print the best demand prediction model for all the products.

## 4. Price Optimisation Models

#### 4.1. Basic Model

- 1.
- Demand prediction and model training:
- Train the selected best demand prediction model for individual product in the product category;
- Identify the demand prediction error of each model through validation process;
- Identify the impact of variables on sales by producing the feature importance score;

- 2.
- Generate new input data:
- Generate new input data to be entered into the trained demand prediction model, which has all the price combinations and key features (${K}^{N}$ combinations for each product);

- 3.
- Demand forecasting:
- Perform the demand forecasting using the trained demand prediction model and new input data;
- Produce demand forecast results which are predicted sales for each combination;

- 4.
- Price optimisation:
- Perform the price optimisation with the basic price optimisation model and demand forecast results;
- Produce the optimal solution that maximises the profit on the prediction week.

#### 4.2. Model 1

#### 4.3. Model 2

#### 4.4. The Proposed Price Optimisation Procedure

- 1.
- Start modelling for the first product on the product list p (p = 1–12);
- 2.
- Import the best demand prediction model and the best hyperparameter option for selected product;
- 3.
- Train the best demand prediction model with the best hyperparameter option using the previous 104 weeks of the prediction week;
- 4.
- Store the feature importance scores and rankings if the best demand prediction model is machine learning model;
- 5.
- Calculate and store the RMSPE score using the formula (9);
- 6.
- Generate price combination options ${K}^{N}$ (K is the number of products in the product category and N is the number of discount rate options) in the loop;
- 7.
- 8.
- After finishing the price combination options loop, perform the demand forecasting for the selected product using the selected best demand prediction model and generated new input data within the loop;
- 9.
- Store the demand forecast results which are the predicted sales for each price combination;
- 10.
- After finishing the product loop, perform the price optimisation with the price optimisation model: Constraint (10) and generated demand forecast results;
- 11.
- Produce the final optimal discount rate for all the products in the product category that maximise the sales revenue on the prediction week.

#### 4.5. Evaluation and Comparison of Models

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Nagle, T.T.; Müller, G. The Strategy and Tactics of Pricing: A Guide to Growing More Profitably, 6th ed.; Routledge: Oxfordshire, UK, 2018. [Google Scholar] [CrossRef]
- Cressman, G.E. Value-based pricing: A state-of-the-art review. In Handbook of Business-to-Business Marketing; Edward Elgar Publishing: Cheltenham, UK, 2012. [Google Scholar]
- Cohen, M.C.; Leung, N.H.Z.; Panchamgam, K.; Perakis, G.; Smith, A. The Impact of Linear Optimization on Promotion Planning. Oper. Res.
**2017**, 65, 446–468. [Google Scholar] [CrossRef] - Talluri, K.T.; Van Ryzin, G. The Theory and Practice of Revenue Management, 1st ed.; International Series in Operations Research & Management Science, 68; Kluwer Academic Publishers: Boston, MA, USA, 2004. [Google Scholar] [CrossRef]
- Ferreira, K.J.; Lee, B.H.A.; Simchi-Levi, D. Analytics for an Online Retailer: Demand Forecasting and Price Optimization. Manuf. Serv. Oper. Manag.
**2016**, 18, 69–88. [Google Scholar] [CrossRef] - Caro, F.; Gallien, J. Clearance Pricing Optimization for a Fast-Fashion Retailer. Oper. Res.
**2012**, 60, 1404–1422. [Google Scholar] [CrossRef] - Cosgun, Ö.; Kula, U.; Kahraman, C. Markdown optimization for an apparel retailer under cross-price and initial inventory effects. Knowl.-Based Syst.
**2017**, 120, 186–197. [Google Scholar] [CrossRef] - Felgate, M.; Fearne, A. Analyzing the Impact of Supermarket Promotions: A Case Study Using Tesco Clubcard Data in the UK. In The Sustainable Global Marketplace: Proceedings of the 2011 Academy of Marketing Science (AMS) Annual Conference; Developments in Marketing Science: Proceedings of the Academy of Marketing Science; Springer International Publishing: Cham, Switzerland, 2014; pp. 471–475. [Google Scholar] [CrossRef]
- Cohen, M.C.; Perakis, G. Optimizing Promotions for Multiple Items in Supermarkets; Springer Series in Supply Chain Management; Springer International Publishing: Cham, Switzerland, 2019; pp. 71–97. [Google Scholar] [CrossRef]
- Ma, S.; Fildes, R. A retail store SKU promotions optimization model for category multi-period profit maximization. Eur. J. Oper. Res.
**2017**, 260, 680–692. [Google Scholar] [CrossRef] - Fildes, R.; Goodwin, P.; Önkal, D. Use and misuse of information in supply chain forecasting of promotion effects. Int. J. Forecast.
**2019**, 35, 144–156. [Google Scholar] [CrossRef] - Ailawadi, K.L.; Harlam, B.A.; Cesar, J.; Trounce, D. Promotion Profitability for a Retailer: The Role of Promotion, Brand, Category, and Store Characteristics. J. Mark. Res.
**2006**, 43, 518–535. [Google Scholar] [CrossRef] - Kamakura, W.A.; Kang, W. Chain-wide and store-level analysis for cross-category management. J. Retail.
**2007**, 83, 159–170. [Google Scholar] [CrossRef] - Leung, A.; McGregor, M.; Chesney, J. Income and Substitution Effects: Graphical Analysis for Intermediate Microeconomics. J. Econ. Educ.
**2014**, 14, 97–107. [Google Scholar] - Gelper, S.; Wilms, I.; Croux, C. Identifying Demand Effects in a Large Network of Product Categories. J. Retail.
**2016**, 92, 25–39. [Google Scholar] [CrossRef] - Leeflang, P.S.H.; Parreño-Selva, J. Cross-category demand effects of price promotions. J. Acad. Mark. Sci.
**2011**, 40, 572–586. [Google Scholar] [CrossRef] [PubMed] - Thomassen, Ø.; Smith, H.; Seiler, S.; Schiraldi, P. Multi-Category Competition and Market Power: A Model of Supermarket Pricing. Am. Econ. Rev.
**2017**, 107, 2308–2351. [Google Scholar] [CrossRef] - Karray, S.; Martín-Herrán, G.; Zaccour, G. Pricing of demand-related products: Can ignoring cross-category effect be a smart choice? Int. J. Prod. Econ.
**2020**, 223, 107512. [Google Scholar] [CrossRef] - Kok, A.G.; Fisher, M.L. Demand Estimation and Assortment Optimization Under Substitution: Methodology and Application. Oper. Res.
**2007**, 55, 1001–1021. [Google Scholar] [CrossRef] - Vaidyanathan, R. Retail Demand Management: Forecasting, Assortment Planning and Pricing. Doctoral Dissertation, University of Pennsylvania, Philadelphia, PA, USA, 2011. [Google Scholar]
- Mahajan, S.; van Ryzin, G. Stocking Retail Assortments Under Dynamic Consumer Substitution. Oper. Res.
**2001**, 49, 334–351. [Google Scholar] [CrossRef] - Smith, S.A.; Agrawal, N. Management of Multi-Item Retail Inventory Systems with Demand Substitution. Oper. Res.
**2000**, 48, 50–64. [Google Scholar] [CrossRef] - Vulcano, G.; Van Ryzin, G.; Ratliff, R. Estimating Primary Demand for Substitutable Products from Sales Transaction Data. Oper. Res.
**2012**, 60, 313–334. [Google Scholar] [CrossRef] - Islam, S.; Amin, S.H.; Wardley, L.J. Machine learning and optimization models for supplier selection and order allocation planning. Int. J. Prod. Econ.
**2021**, 242, 108315. [Google Scholar] [CrossRef] - Makridakis, S.; Spiliotis, E.; Assimakopoulos, V. The M5 competition: Background, organization, and implementation. Int. J. Forecast.
**2021**, 38, 1325–1336. [Google Scholar] [CrossRef] - Chawla, A.; Singh, A.; Lamba, A.; Gangwani, N.; Soni, U. Demand Forecasting Using Artificial Neural Networks—A Case Study of American Retail Corporation. In Applications of Artificial Intelligence Techniques in Engineering; Advances in Intelligent Systems and Computing; Springer: Singapore, 2018; pp. 79–89. [Google Scholar] [CrossRef]
- Punia, S.; Nikolopoulos, K.; Singh, S.P.; Madaan, J.K.; Litsiou, K. Deep learning with long short-term memory networks and random forests for demand forecasting in multi-channel retail. Int. J. Prod. Res.
**2020**, 58, 4964–4979. [Google Scholar] [CrossRef] - Vairagade, N.; Logofatu, D.; Leon, F.; Muharemi, F. Demand Forecasting Using Random Forest and Artificial Neural Network for Supply Chain Management. In Computational Collective Intelligence; Lecture Notes in Computer Science; Springer International Publishing: Cham, Switzerland, 2019; pp. 328–339. [Google Scholar] [CrossRef]
- Akçay, Y.; Natarajan, H.P.; Xu, S.H. Joint Dynamic Pricing of Multiple Perishable Products Under Consumer Choice. Manag. Sci.
**2010**, 56, 1345–1361. [Google Scholar] [CrossRef] - Bolton, R.N. The relationship between market characteristics and promotional price elasticities. Mark. Sci.
**1989**, 8, 153–169. [Google Scholar] [CrossRef] - Wooldridge, J.M. Introductory Econometrics: A Modern Approach; Cengage Learning: Boston, MA, USA, 2015. [Google Scholar]
- Meng, Y.; Brennan, A.; Purshouse, R.; Hill-McManus, D.; Angus, C.; Holmes, J.; Meier, P.S. Estimation of own and cross price elasticities of alcohol demand in the UK—A pseudo-panel approach using the Living Costs and Food Survey 2001–2009. J. Health Econ.
**2014**, 34, 96–103. [Google Scholar] [CrossRef] - Pappenberger, F.; Ramos, M.H.; Cloke, H.L.; Wetterhall, F.; Alfieri, L.; Bogner, K.; Mueller, A.; Salamon, P. How do I know if my forecasts are better? Using benchmarks in hydrological ensemble prediction. J. Hydrol.
**2015**, 522, 697–713. [Google Scholar] [CrossRef] - Gilliland, M.; Sglavo, U. Worst practices in business forecasting. Analytics
**2010**, 12, 17. [Google Scholar] - Anđelić, O.; Rakićević, Z. Time-series analysis application in demand forecasting: A case study of FMCG. In Proceedings of the XVII International Symposium, Belgrade, Serbia, 7–9 September 2020; p. 10. [Google Scholar]
- Kohli, S.; Godwin, G.T.; Urolagin, S. Sales prediction using linear and KNN regression. In Advances in Machine Learning and Computational Intelligence: Proceedings of ICMLCI 2019; Springer: Singapore, 2021; pp. 321–329. [Google Scholar]
- Kalla, R.; Murikinjeri, S.; Abbaiah, R. An improved demand forecasting with limited historical sales data. In Proceedings of the 2020 International Conference on Computer Communication and Informatics (ICCCI), Coimbatore, India, 22–24 January 2020; pp. 1–5. [Google Scholar]
- Nikolopoulos, K.I.; Babai, M.Z.; Bozos, K. Forecasting supply chain sporadic demand with nearest neighbor approaches. Int. J. Prod. Econ.
**2016**, 177, 139–148. [Google Scholar] [CrossRef] - Bala, P.K. Decision tree based demand forecasts for improving inventory performance. In Proceedings of the 2010 IEEE International Conference on Industrial Engineering and Engineering Management, Macao, China, 7–10 December 2010; pp. 1926–1930. [Google Scholar]
- Kotsiantis, S.B. Decision trees: A recent overview. Artif. Intell. Rev.
**2013**, 39, 261–283. [Google Scholar] [CrossRef] - Liu, Y.; Wang, Y.; Zhang, J. New machine learning algorithm: Random forest. In Proceedings of the Information Computing and Applications: Third International Conference, ICICA 2012, Chengde, China, 14–16 September 2012; Proceedings 3. Springer: Berlin/Heidelberg, Germany, 2012; pp. 246–252. [Google Scholar]
- Mert, B.; Eskiocak, D.İ.; Oğul, İ.Ü.; Aslan, M.K.; Karalar, E. Determining Annual and Monthly Sales Targets for Stores and Product Categories in FMCG Retail. In Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal-Proceedings of the INFUS 2022 Conference; Springer: Cham, Switzerland, 2022; Volume 2, pp. 524–530. [Google Scholar]
- Geurts, P.; Ernst, D.; Wehenkel, L. Extremely randomized trees. Mach. Learn.
**2006**, 63, 3–42. [Google Scholar] [CrossRef] - Friedman, J.H. Greedy function approximation: A gradient boosting machine. Ann. Stat.
**2001**, 29, 1189–1232. [Google Scholar] [CrossRef] - Friedman, J.H. Stochastic gradient boosting. Comput. Stat. Data Anal.
**2002**, 38, 367–378. [Google Scholar] [CrossRef] - Henzel, J.; Sikora, M. Gradient boosting application in forecasting of performance indicators values for measuring the efficiency of promotions in FMCG retail. In Proceedings of the 2020 15th Conference on Computer Science and Information Systems (FedCSIS), Sofia, Bulgaria, 6–9 September 2020; pp. 59–68. [Google Scholar]
- Antipov, E.A.; Pokryshevskaya, E.B. Interpretable machine learning for demand modeling with high-dimensional data using Gradient Boosting Machines and Shapley values. J. Revenue Pricing Manag.
**2020**, 19, 355–364. [Google Scholar] [CrossRef] - Chen, T.; He, T.; Benesty, M.; Khotilovich, V.; Tang, Y.; Cho, H.; Chen, K.; Mitchell, R.; Cano, I.; Zhou, T.; et al. Xgboost: Extreme gradient boosting. R Package Version 0.4-2
**2015**, 1, 1–4. [Google Scholar] - Ke, G.; Meng, Q.; Finley, T.; Wang, T.; Chen, W.; Ma, W.; Ye, Q.; Liu, T.Y. Lightgbm: A highly efficient gradient boosting decision tree. Adv. Neural Inf. Process. Syst.
**2017**, 30, 1–9. [Google Scholar] - Guryanov, A. Histogram-based algorithm for building gradient boosting ensembles of piecewise linear decision trees. In Proceedings of the Analysis of Images, Social Networks and Texts: 8th International Conference, AIST 2019, Kazan, Russia, 17–19 July 2019; Revised Selected Papers 8. Springer: Cham, Switzerland, 2019; pp. 39–50. [Google Scholar]

**Figure 1.**Univariate analysis: Histogram and Kernel Density Estimators (KDE) of sales units for different products.

**Figure 6.**Regression Analysis: Own Product Price Elasticity—Log-Log Regression (Price vs. Sales Units).

Type | Models |
---|---|

Benchmarking | Simple Average |

Simple Median | |

Naïve Bayes | |

Seasonal Naïve | |

Time Series | Auto Regressive Integrated Moving Averages (ARIMA) |

Machine Learning | Multivariate Linear Regression (MLR) |

K-Neighbours | |

Decision Tree | |

Random Forest | |

Extremely Randomised Tree | |

Gradient Boosting Machine (GBM) | |

XGBoost | |

Light GBM | |

Histogram-based GBM |

Features | Description |
---|---|

LOG_PRICE | Log Prices of different brand products within product category |

DISCOUNT | Discount rate of product |

D_RATE | Discount rate (0 = 0%, 1 = 0–10%, 2 = 10–20%, …, 5 = 40–50%) |

DISPLAY | Display promotion (0 = No, 1 = Yes) |

FEATURE | Feature promotion (0 = No, 1 = Yes) |

DISCOUNT_TYPE2 | Discount types (0 = No Discount, 1 = Unsupported Discount, 2 = Supported Discount) |

PREVIOUS_DISCOUNT | Discount rate of previous periods |

DISCOUNT_WEEKS | Consecutive discount weeks |

NO_DISCOUNT_WEEKS | Consecutive nondiscount weeks |

MONTH | Month |

WEEK | Week |

t-1 | Time lag 1 |

t-2 | Time lag 2 |

t-3 | Time lag 3 |

Product | Model | Best_Parameters | Best_Score |
---|---|---|---|

P1 | GradientBoosting | {‘max_depth’:3, ‘n_estimators’:100, ‘subsample’: 0.7} | 0.1534 |

P2 | ExtraTrees | {‘max_depth’:3, ‘n_estimators’:500} | 0.2243 |

P3 | GradientBoosting | {‘max_depth’:3, ‘n_estimators’:100, ‘subsample’: 0.8} | 0.0763 |

P4 | ExtraTrees | {‘max_depth’:5, ‘n_estimators’:100} | 0.0639 |

P5 | ExtraTrees | {‘max_depth’:3, ‘n_estimators’:500} | 0.0528 |

P6 | ExtraTrees | {‘max_depth’:3, ‘n_estimators’:1000} | 0.0660 |

P7 | ExtraTrees | {‘max_depth’:5, ‘n_estimators’:100} | 0.0763 |

P8 | ExtraTrees | {‘max_depth’:4, ‘n_estimators’:100} | 0.0896 |

P9 | RandomForest | {‘max_depth’:5, ‘n_estimators’:100} | 0.1200 |

P10 | GradientBoosting | {‘max_depth’:4, ‘n_estimators’:100, ‘subsample’: 0.8} | 0.2084 |

P11 | RandomForest | {‘max_depth’:3, ‘n_estimators’:500} | 0.1034 |

P12 | ExtraTrees | {‘max_depth’:4, ‘n_estimators’:500} | 0.0813 |

Parameters | Values |
---|---|

Set of products: N | 12 products in cold cereal category |

(P1–P12) | |

Set of discount sates: K | 3 discount rate options |

(0%, 25%, 50%) | |

Products’ base price | base price of each product in prediction week |

($1.98, $1.98, $2.44, $3.04, $4.79, $2.8, $3.53, $3.25, $3.32, $3.12, $3.12, $2.99) | |

Products’ cost | cost of each product in prediction week |

($0.59, $0.61, $0.73, $0.91, $1.44, $0.84, $1.06, $0.98, $0.99, $0.94, $0.94, $0.90) | |

Price combinations | combinations of discount rate of all products within the product category: ${X}_{k1,k2,\dots ,kn}$ |

$N\times {K}^{N}=12\times {3}^{12}=12\times 531,441=6,377,292$ combinations |

Product | Final Optimal Solution |
---|---|

P1 | 0% |

P2 | 25% |

P3 | 0% |

P4 | 0% |

P5 | 50% |

P6 | 0% |

P7 | 0% |

P8 | 0% |

P9 | 25% |

P10 | 0% |

P11 | 0% |

P12 | 25% |

Optimal profit | $2456.777 |

No. of Products N | No. of Discount Rates K | No. of Combinations $\mathit{N}\times \left({\mathit{K}}^{\mathit{N}}\right)$ | Reduced Search Space ^{1} | Execution Time (Demand Prediction) | Execution Time (Price Optimisation) |
---|---|---|---|---|---|

9 | 2 | 4608 | 3024 | 15.4 s | 0.163 s |

9 | 3 | 177,147 | 102,816 | 1 m 16 s | 21.5 s |

9 | 4 | 2,359,296 | 918,540 | 9 m 31 s | 1 m 14 s |

9 | 5 | 17,578,125 | 4,548,096 | 48 m 18 s | 6 m 47 s |

12 | 2 | 49,152 | 26,532 | 1 m 14 s | 3.87 s |

12 | 3 | 6,377,292 | 1,108,800 | 13 m 39 s | 7 m 14 s |

12 | 4 | 201,326,592 | 11,274,924 | 1 h 58 m 46 s | 25 m 12 s |

12 | 5 | 2,929,687,500 | NA ^{2} | NA | NA |

^{1}Reduction in price combinations from applying constraints to the basic model.

^{2}Not applicable due to the lack of computer resources.

No. of Products N | No. of Discount Rates K | No. of Combinations $\mathit{N}\times \left({\mathit{K}}^{\mathit{N}}\right)$ | Reduced Search Space ^{1} | Execution Time (Demand Prediction) | Execution Time (Price Optimisation) |
---|---|---|---|---|---|

9 | 2 | 4608 | 1710 | 15.1 s | 0.232 s |

9 | 3 | 177,147 | 50,544 | 56.5 s | 5.86 s |

9 | 4 | 2,359,296 | 413,343 | 5 m 21 s | 26.2 s |

9 | 5 | 17,578,125 | 1,928,448 | 21 m 57 s | 1 m 11 s |

12 | 2 | 49,152 | 10,692 | 31.5 s | 0.469 s |

12 | 3 | 6,377,292 | 513,204 | 7 m 1 s | 3 m 58 s |

12 | 4 | 201,326,592 | 5,275,044 | 1 h 10 m 47 s | 12 m 13 s |

12 | 5 | 2,929,687,500 | NA ^{2} | NA | NA |

^{1}Reduction in price combinations from applying constraints to Model 1.

^{2}Not applicable due to the lack of computer resources.

No. of Products N | No. of Discount Rates K | No. of Combinations $\mathit{N}\times \left({\mathit{K}}^{\mathit{N}}\right)$ | Reduced Search Space ^{1} | Execution Time (Demand Prediction) | Execution Time (Price Optimisation) |
---|---|---|---|---|---|

9 | 2 | 4608 | 90 | 13.4 s | 0.105 s |

9 | 3 | 177,147 | 720 | 14.7 s | 0.120 s |

9 | 4 | 2,359,296 | 2430 | 20.4 s | 0.137 s |

9 | 5 | 17,578,125 | 5760 | 56.4 s | 0.199 s |

12 | 2 | 49,152 | 240 | 25.9 s | 0.219 s |

12 | 3 | 6,377,292 | 3840 | 48.6 s | 0.254 s |

12 | 4 | 201,326,592 | 19,440 | 11 m 13 s | 0.711 s |

12 | 5 | 2,929,687,500 | 61,440 | 2 h 31 m 33 s | 1.38 s |

^{1}Reduction in price combinations from applying constraints to Model 2.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, K.H.; Abdollahian, M.; Schreider, S.; Taheri, S.
Supply Chain Demand Forecasting and Price Optimisation Models with Substitution Effect. *Mathematics* **2023**, *11*, 2502.
https://doi.org/10.3390/math11112502

**AMA Style**

Lee KH, Abdollahian M, Schreider S, Taheri S.
Supply Chain Demand Forecasting and Price Optimisation Models with Substitution Effect. *Mathematics*. 2023; 11(11):2502.
https://doi.org/10.3390/math11112502

**Chicago/Turabian Style**

Lee, Keun Hee, Mali Abdollahian, Sergei Schreider, and Sona Taheri.
2023. "Supply Chain Demand Forecasting and Price Optimisation Models with Substitution Effect" *Mathematics* 11, no. 11: 2502.
https://doi.org/10.3390/math11112502