# Demand Forecasting Approaches Based on Associated Relationships for Multiple Products

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Forecasting Model and Evaluation

#### 2.1. VAR and DFM

#### 2.2. The Forecasting Approach Based on Correlation

- Scheme (I): Only select the product having the highest correlation with the object from the proper variable subset as a predictor added in final model.
- Scheme (II): Extract the first principle component of all elements in the proper variable subset as a predictor added in the final model.

#### 2.3. The Forecasting Approach Based on Granger Causality

- Scheme (I): Only select the product having lowest p-value of Granger test with the object from the proper variable subset as a predictor added in final model.
- Scheme (II): Extract the first principle component of all elements in the proper variable subset as a predictor added in the final model.

#### 2.4. The Forecast Accuracy Measures

## 3. Data Description

#### 3.1. Data and Pretreatment

#### 3.2. Correlation Analysis

#### 3.3. Granger Causality Analysis

## 4. Empirical Analysis

#### 4.1. Experimental Setup

#### 4.2. Results and Analysis

#### 4.2.1. Forecasting Accuracy Analysis

#### 4.2.2. Inventory Performance Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Inventory ratios for approaches with different critical values satisfying various service levels. (

**a**) is for the approach based on correlation and (

**b**) is for the approach based on Granger causality.

Family | Category | Sub-Category | Product | Total Purchase | Average Purchase of Subcategory ^{1} | Average Purchase of Category ^{1} |
---|---|---|---|---|---|---|

Primary equipment | AC circuit breaker | 1 | 1 | 2.65 | 2.65 | 2.65 |

AC transformer | 2 | 18 | 102.30 | 51.15 | 5.68 | |

AC disconnector | 2 | 2 | 2.37 | 1.19 | 1.19 | |

Switch cabinet | 3 | 9 | 71.25 | 23.75 | 7.92 | |

High-voltage fuse | 1 | 1 | 4.01 | 4.01 | 4.01 | |

Lightning arrestor | 1 | 1 | 1.70 | 1.70 | 1.70 | |

Load switch | 1 | 3 | 11.86 | 11.86 | 3.95 | |

Equipment material | Tower pole | 2 | 13 | 112.90 | 56.45 | 8.68 |

Wire & ground wire | 4 | 40 | 140.43 | 35.11 | 3.51 | |

Cable | 3 | 49 | 272.65 | 90.88 | 5.56 | |

Insulator | 6 | 11 | 11.29 | 1.88 | 1.03 | |

Metal fittings | 24 | 97 | 16.75 | 0.70 | 0.17 | |

Cable accessory | 5 | 83 | 11.01 | 2.20 | 0.13 | |

Optical cable accessory | 2 | 5 | 0.16 | 0.08 | 0.03 | |

Optical cable | 2 | 2 | 0.87 | 0.44 | 0.44 |

^{1}Units of average purchase: Million yuan per month.

Category | Cluster | Degree Centrality | Betweeness Centrality | Eigenvector Centrality | PageRank Centrality | Clustering Coefficient |
---|---|---|---|---|---|---|

AC transformer | G1 | 6 | 0.833 | 0.125 | 1.119 | 0.800 |

AC disconnector | G1 | 4 | 0.000 | 0.100 | 0.787 | 1.000 |

High-voltage fuse | G1 | 5 | 0.250 | 0.111 | 0.952 | 0.900 |

Insulator | G1 | 6 | 0.833 | 0.125 | 1.119 | 0.800 |

Metal fittings | G2 | 7 | 3.000 | 0.143 | 1.299 | 0.667 |

Tower pole | G2 | 6 | 1.417 | 0.125 | 1.299 | 0.733 |

Ground wire | G2 | 5 | 0.667 | 0.111 | 0.964 | 0.800 |

lightning arrester | G2 | 3 | 0.000 | 0.091 | 0.631 | 1.000 |

Category | G | In-Degree | Out-Degree | Betweenes Centrality | Closeness Centrality | Eigenvector Centrality | PageRank Centrality | Clustering Coefficient |
---|---|---|---|---|---|---|---|---|

Switch cabinet | G1 | 4 | 4 | 52.600 | 0.056 | 0.120 | 1.848 | 0.179 |

load switch | G1 | 0 | 3 | 2.667 | 0.038 | 0.056 | 0.768 | 0.333 |

Optical cable | G1 | 2 | 1 | 1.000 | 0.036 | 0.042 | 0.808 | 0.333 |

Optical cable accessory | G1 | 0 | 2 | 0.000 | 0.034 | 0.032 | 0.575 | 0.500 |

AC transfer | G2 | 2 | 1 | 0.400 | 0.034 | 0.038 | 0.543 | 0.000 |

High-voltage fuse | G2 | 6 | 5 | 28.967 | 0.053 | 0.119 | 1.578 | 0.167 |

Tower pole | G2 | 7 | 1 | 31.067 | 0.053 | 0.119 | 1.583 | 0.190 |

Cable accessory | G2 | 1 | 5 | 10.900 | 0.042 | 0.074 | 1.187 | 0.050 |

AC disconnector | G2 | 2 | 3 | 3.467 | 0.045 | 0.086 | 0.932 | 0.417 |

Insulator | G2 | 2 | 2 | 1.467 | 0.037 | 0.059 | 0.738 | 0.167 |

Metal fittings | G3 | 3 | 3 | 19.500 | 0.050 | 0.106 | 1.376 | 0.300 |

AC circuit breaker | G3 | 1 | 1 | 0.000 | 0.037 | 0.045 | 0.541 | 0.500 |

Lightning arrester | G3 | 0 | 2 | 0.833 | 0.034 | 0.037 | 0.549 | 0.000 |

Wire &ground wire | G3 | 4 | 1 | 7.133 | 0.040 | 0.067 | 0.972 | 0.083 |

Index | Level | SES ^{1} | AR | VAR | DFM | CI (0.513) | CII (0.513) | GI (0.1) | GII (0.1) |
---|---|---|---|---|---|---|---|---|---|

MASE | Product | 0.7837 | 0.7895 | - | 0.7712 | 0.8102 | 0.5806 | 0.7001 | 0.6953 |

Subcategory | 0.7376 | 0.7228 | - | 0.6820 | 0.7425 | 0.5593 | 0.6749 | 0.6843 | |

Category | 0.7266 | 0.6692 | 0.7985 | 0.7436 | 0.6813 | 0.5412 | 0.6283 | 0.6448 | |

Family | 0.8107 | 0.6697 | 0.6933 | - | 0.6693 | 0.6693 | 0.6749 | 0.6749 | |

GMRASE | Product | 1 | 1.1006 | - | 1.1102 | 1.1068 | 0.7918 | 1.0478 | 1.0004 |

Subcategory | 1 | 1.0310 | - | 1.0031 | 1.0560 | 0.7979 | 1.0012 | 1.0151 | |

Category | 1 | 0.9207 | 1.1981 | 1.1029 | 0.9337 | 0.7479 | 0.8687 | 0.8846 | |

Family | 1 | 0.8332 | 0.8514 | - | 0.8289 | 0.8289 | 0.8387 | 0.8387 |

^{1}SES is the baseline model when calculate relative errors.

**Table 5.**Forecasting errors of the approaches based on correlation under different critical conditions.

Index | Level | CI | CII | CI | CII | CI | CII | CI | CII |
---|---|---|---|---|---|---|---|---|---|

(0.6) | (0.7) | (0.8) | (0.9) | ||||||

MASE | Product | 0.8095 | 0.6216 | 0.7991 | 0.6926 | 0.7962 | 0.7589 | 0.7933 | 0.7784 |

Subcategory | 0.7405 | 0.5721 | 0.7400 | 0.5861 | 0.7386 | 0.6112 | 0.7373 | 0.6633 | |

Category | 0.6776 | 0.5346 | 0.6797 | 0.5523 | 0.6651 | 0.5770 | 0.6692 | 0.6692 | |

Family | 0.6693 | 0.6693 | 0.6693 | 0.6693 | 0.6697 | 0.6697 | 0.6697 | 0.6697 | |

GRMASE | Product | 1.1065 | 0.8516 | 1.0900 | 0.9325 | 1.0990 | 1.0184 | 1.1035 | 1.0760 |

Subcategory | 1.0528 | 0.8392 | 1.0519 | 0.8403 | 1.0546 | 0.8950 | 1.0499 | 0.9485 | |

Category | 0.9306 | 0.7421 | 0.9353 | 0.7789 | 0.9143 | 0.8094 | 0.9207 | 0.9207 | |

Family | 0.8289 | 0.8289 | 0.8289 | 0.8289 | 0.8332 | 0.8332 | 0.8332 | 0.8332 |

**Table 6.**Forecasting errors of the approaches based on Granger causality under different critical conditions.

Index | Level | GI | GII | GI | GII |
---|---|---|---|---|---|

(0.05) | (0.01) | ||||

Product | 0.7001 | 0.6602 | 0.7007 | 0.6287 | |

MASE | Subcategory | 0.6749 | 0.6607 | 0.6841 | 0.6727 |

Category | 0.6360 | 0.6572 | 0.6336 | 0.6251 | |

Family | 0.6697 | 0.6697 | 0.6697 | 0.6697 | |

GRMASE | Product | 1.0478 | 0.9601 | 1.0483 | 0.9159 |

Subcategory | 1.0012 | 0.9851 | 1.0070 | 0.9768 | |

Category | 0.8814 | 0.9059 | 0.8802 | 0.8686 | |

Family | 0.8332 | 0.8332 | 0.8332 | 0.8332 |

Model | ASE | RASE | |||||
---|---|---|---|---|---|---|---|

Product | Subcategory | Category | Product | Subcategory | Category | ||

AR | 0.849 | −1.189 | −3.301 | 5.090 | 0.842 | −3.521 | |

VAR | - | - | 1.624 | - | - | 1.780 | |

DFM | −1.392 | −2.692 | 0.398 | 3.020 | 0.051 | 1.154 | |

CI | (0.513) | 2.875 | 0.330 | −2.672 | 6.194 | 1.564 | −2.788 |

(0.6) | 2.871 | 0.195 | −2.601 | 6.147 | 1.471 | −2.833 | |

(0.7) | 2.067 | 0.173 | −2.500 | 5.644 | 1.458 | −2.738 | |

(0.8) | 1.725 | 0.071 | −3.337 | 5.580 | 1.404 | −3.582 | |

(0.9) | 1.395 | −0.021 | −3.301 | 5.317 | 1.352 | −3.521 | |

CII | (0.513) | −14.119 | −6.281 | −4.799 | −11.726 | −4.730 | −5.256 |

(0.6) | −11.002 | −5.640 | −4.570 | −7.693 | −3.033 | −4.991 | |

(0.7) | −7.353 | −5.008 | −3.832 | −3.996 | −3.271 | −4.139 | |

(0.8) | −2.675 | −4.055 | −3.221 | 1.126 | −1.915 | −3.600 | |

(0.9) | −0.643 | −2.664 | −3.301 | 3.777 | −1.016 | −3.521 | |

GI | (0.1) | −7.493 | −4.178 | −3.603 | 1.405 | 0.023 | −4.114 |

(0.05) | −7.493 | −4.178 | −3.274 | 1.405 | 0.023 | −3.806 | |

(0.01) | −7.437 | −3.621 | −3.450 | 1.422 | 0.136 | −4.060 | |

GII | (0.1) | −8.197 | −3.260 | −3.325 | 0.016 | 0.202 | −3.547 |

(0.05) | −10.745 | −4.404 | −2.282 | −1.478 | −0.220 | −2.564 | |

(0.01) | −11.906 | −3.846 | −3.264 | −3.566 | −0.489 | −3.623 | |

Sample size | 335 | 59 | 15 | 335 | 59 | 15 | |

p = 0.1 | 1.284 | 1.296 | 1.341 | 1.284 | 1.296 | 1.341 | |

p = 0.05 | 1.649 | 1.671 | 1.753 | 1.649 | 1.671 | 1.753 | |

p = 0.01 | 2.338 | 2.391 | 2.602 | 2.338 | 2.391 | 2.602 |

**Table 8.**Total inventory costs for different approaches considered satisfying various service levels.

GMRASE | Stock Cost Parameters ^{1} | SES | AR | DFM | CI (0.513) | CII (0.513) | GI (0.1) | GII (0.1) |
---|---|---|---|---|---|---|---|---|

0.9 | a = 0.4, b = 0.4 | 10.96 | 10.70 | 10.33 | 11.02 | 8.87 | 10.14 | 10.22 |

a = 0.4, b = 0.6 | 11.01 | 10.76 | 10.38 | 11.07 | 8.94 | 10.19 | 10.28 | |

a = 0.4, b = 0.8 | 11.06 | 10.81 | 10.43 | 11.13 | 9.01 | 10.25 | 10.33 | |

0.93 | a = 0.4, b = 0.4 | 11.83 | 11.56 | 11.21 | 11.87 | 9.67 | 11.01 | 11.10 |

a = 0.4, b = 0.6 | 11.86 | 11.59 | 11.24 | 11.90 | 9.71 | 11.04 | 11.13 | |

a = 0.4, b = 0.8 | 11.88 | 11.61 | 11.26 | 11.93 | 9.74 | 11.07 | 11.16 | |

0.95 | a = 0.4, b = 0.4 | 12.29 | 12.02 | 11.69 | 12.33 | 10.12 | 11.48 | 11.57 |

a = 0.4, b = 0.6 | 12.31 | 12.04 | 11.70 | 12.35 | 10.14 | 11.50 | 11.59 | |

a = 0.4, b = 0.8 | 12.32 | 12.06 | 11.71 | 12.37 | 10.17 | 11.52 | 11.60 | |

0.97 | a = 0.4, b = 0.4 | 12.77 | 12.51 | 12.18 | 12.82 | 10.60 | 11.97 | 12.06 |

a = 0.4, b = 0.6 | 12.78 | 12.52 | 12.19 | 12.83 | 10.62 | 11.98 | 12.07 | |

a = 0.4, b = 0.8 | 12.79 | 12.53 | 12.20 | 12.84 | 10.63 | 12.00 | 12.08 | |

0.99 | a = 0.4, b = 0.4 | 13.57 | 13.31 | 12.98 | 13.61 | 11.39 | 12.76 | 12.85 |

a = 0.4, b = 0.6 | 13.58 | 13.32 | 12.98 | 13.62 | 11.40 | 12.77 | 12.86 | |

a = 0.4, b = 0.8 | 13.58 | 13.33 | 12.99 | 13.63 | 11.41 | 12.78 | 12.87 |

^{1}Units of total inventory costs: Million.

**Table 9.**Total inventory costs for the approach based on correlation with different critical values.

Target Service Level | Stock Cost Parameters ^{1} | CI | CII | CI | CII | CI | CII | CI | CII |
---|---|---|---|---|---|---|---|---|---|

(0.6) | (0.7) | (0.8) | (0.9) | ||||||

0.9 | a = 0.4, b = 0.4 | 10.99 | 9.11 | 10.88 | 9.53 | 10.84 | 10.57 | 10.72 | 10.68 |

a = 0.4, b = 0.6 | 11.04 | 9.18 | 10.93 | 9.59 | 10.90 | 10.62 | 10.77 | 10.74 | |

a = 0.4, b = 0.8 | 11.10 | 9.24 | 10.99 | 9.65 | 10.95 | 10.67 | 10.82 | 10.79 | |

0.93 | a = 0.4, b = 0.4 | 11.85 | 9.94 | 11.73 | 10.40 | 11.70 | 11.43 | 11.58 | 11.54 |

a = 0.4, b = 0.6 | 11.87 | 9.97 | 11.76 | 10.43 | 11.72 | 11.46 | 11.60 | 11.57 | |

a = 0.4, b = 0.8 | 11.90 | 10.01 | 11.79 | 10.45 | 11.75 | 11.48 | 11.63 | 11.59 | |

0.95 | a = 0.4, b = 0.4 | 12.30 | 10.40 | 12.19 | 10.86 | 12.15 | 11.89 | 12.04 | 12.00 |

a = 0.4, b = 0.6 | 12.32 | 10.42 | 12.21 | 10.88 | 12.17 | 11.91 | 12.05 | 12.02 | |

a = 0.4, b = 0.8 | 12.34 | 10.44 | 12.23 | 10.90 | 12.19 | 11.92 | 12.07 | 12.03 | |

0.97 | a = 0.4, b = 0.4 | 12.79 | 10.88 | 12.68 | 11.35 | 12.64 | 12.38 | 12.52 | 12.49 |

a = 0.4, b = 0.6 | 12.81 | 10.90 | 12.69 | 11.36 | 12.66 | 12.39 | 12.54 | 12.50 | |

a = 0.4, b = 0.8 | 12.82 | 10.92 | 12.71 | 11.38 | 12.67 | 12.40 | 12.55 | 12.51 | |

0.99 | a = 0.4, b = 0.4 | 13.59 | 11.68 | 13.47 | 12.14 | 13.44 | 13.17 | 13.32 | 13.29 |

a = 0.4, b = 0.6 | 13.60 | 11.69 | 13.48 | 12.15 | 13.45 | 13.18 | 13.33 | 13.30 | |

a = 0.4, b = 0.8 | 13.61 | 11.70 | 13.49 | 12.16 | 13.46 | 13.19 | 13.34 | 13.30 |

^{1}Units of total inventory costs: Million.

**Table 10.**Total inventory costs for the approach based on Granger causality with different critical values.

Target Service Level | Stock Cost Parameters ^{1} | GI | GII | GI | GII |
---|---|---|---|---|---|

(0.05) | (0.01) | ||||

0.9 | a = 0.4, b = 0.4 | 10.14 | 10.03 | 10.14 | 9.52 |

a = 0.4, b = 0.6 | 10.19 | 10.09 | 10.20 | 9.57 | |

a = 0.4, b = 0.8 | 10.25 | 10.14 | 10.25 | 9.63 | |

0.93 | a = 0.4, b = 0.4 | 11.01 | 10.91 | 11.02 | 10.39 |

a = 0.4, b = 0.6 | 11.04 | 10.93 | 11.05 | 10.41 | |

a = 0.4, b = 0.8 | 11.07 | 10.96 | 11.07 | 10.44 | |

0.95 | a = 0.4, b = 0.4 | 11.48 | 11.38 | 11.48 | 10.86 |

a = 0.4, b = 0.6 | 11.50 | 11.39 | 11.50 | 10.87 | |

a = 0.4, b = 0.8 | 11.52 | 11.41 | 11.52 | 10.89 | |

0.97 | a = 0.4, b = 0.4 | 11.97 | 11.86 | 11.97 | 11.35 |

a = 0.4, b = 0.6 | 11.98 | 11.87 | 11.98 | 11.36 | |

a = 0.4, b = 0.8 | 12.00 | 11.89 | 12.00 | 11.37 | |

0.99 | a = 0.4, b = 0.4 | 12.76 | 12.66 | 12.76 | 12.14 |

a = 0.4, b = 0.6 | 12.77 | 12.67 | 12.77 | 12.15 | |

a = 0.4, b = 0.8 | 12.78 | 12.67 | 12.78 | 12.16 |

^{1}Units of total inventory costs: Million.

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**MDPI and ACS Style**

Lei, M.; Li, S.; Yu, S. Demand Forecasting Approaches Based on Associated Relationships for Multiple Products. *Entropy* **2019**, *21*, 974.
https://doi.org/10.3390/e21100974

**AMA Style**

Lei M, Li S, Yu S. Demand Forecasting Approaches Based on Associated Relationships for Multiple Products. *Entropy*. 2019; 21(10):974.
https://doi.org/10.3390/e21100974

**Chicago/Turabian Style**

Lei, Ming, Shalang Li, and Shasha Yu. 2019. "Demand Forecasting Approaches Based on Associated Relationships for Multiple Products" *Entropy* 21, no. 10: 974.
https://doi.org/10.3390/e21100974