#
Complexity Constraint in the Distributor’s Pallet Loading Problem

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. The Role of the Complexity Constraint in the Pallet Loading Problem

#### 2.2. A Typology for C&P Problems by Wäscher

- Dimensionality (one, two, three, and as a problem variation, more than three dimensions);
- Type of assignment (output value maximization, where a set of large items is insufficient to accommodate a set of small items ready to be packed, forcing the usage of all large objects while removing the need for any selection, and input value minimization, where a set of large objects can fully accommodate all smaller ones, which changes the goal towards minimizing the value of large items to be used);
- The shape of small items (regular items, such as rectangular or circular shaped items, and irregular items, which do not have a well-known geometrical shape);
- The assortment of large objects (a single element, where this scenario can have two variations: (1) when the extension of the large object may be fixed in all dimensions; (2) when at least one dimension is variable in its extension, and several large objects);
- The assortment of small items (identical small items of similar shape and size, weakly heterogeneous assortment, where the small items can be grouped into a low number of classes when compared to the total number of items, and strongly heterogeneous assortment, where nearly all items are treated as an individual entity).

#### Applying Wäscher’s Typology to the Pallet Loading Problem

## 3. Materials and Methods

#### 3.1. Metrics to Analyse Complexity

#### 3.2. Sample Retrieval

- Each worker, upon completing the loading process of a pallet, will print and a label with (a) a barcode and (b) other necessary numerical information to be stuck to the packed pallet prior to shipping. Two of such numbers will help to differentiate each packed pallet and each order.
- At the end of the day, the software system will create an Excel file with all the data collected by the workers.
- During the box loading process, each box is scanned so that the Excel file contains information such as the time the box was picked, the quantity, and its dimensions.
- After compiling data from the two numbers affixed to each pallet from multiple packed pallets, each picker would rate the complexity of the pallet he/she packed from 1 to 10.

## 4. Results

_{0}, the y-intercept, and ε, the error term that captures errors in measurement of y and the effect on y of any variables missing from the equation that would contribute to explaining variations in y [35].

_{1}represents the Box Quantities component. In the end, this regression allowed us to conclude that the component Box Quantities, which contains the variables Number of Column Piles, Number of Box Types, Number of Boxes, Time Spent Packing and Percentage of Fragile Boxes, is the component that explains the variance of the Evaluation variable better, while the Box Dimensions component is not very relevant towards explaining the dependant variable.

_{1}represents the variable “Number of Column Piles” and the β is 3.637. The variable that affects the complexity of the pallet loading problem, according to this model, is the Number of Column Piles. This multiple linear regression matches what was seen in the principal components analysis: the Box Quantities component was the most significant. This test showed that, statistically, only one variable is responsible for explaining the Evaluation variable. However, the latter variable is subjective, which means that other parameters can affect this variable, depending on the perceptions of the different workers.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Sample\Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Evaluation |

Units | Kg | % | m | mins | m | |||||

1 | 55 | 2.38 | 20.0 | 0.170 | 5 | 0 | 4 | −1.087 | 0 | 4 |

2 | 121 | 3.09 | 54.5 | 0.310 | 11 | 0 | 9 | 1.340 | 0 | 4 |

3 | 99 | 2.05 | 77.8 | 0.210 | 9 | 0 | 6 | −0.652 | 0 | 4 |

4 | 174 | 2.36 | 2.3 | 0.170 | 4 | 0 | 15 | −0.617 | 0 | 2 |

5 | 5 | 2.61 | 0 | 0.153 | 1 | 0 | 1 | −1.294 | 0 | 1 |

6 | 87 | 3.61 | 36.8 | 0.361 | 25 | >9 | 42 | 1.310 | 11 | 9 |

7 | 199 | 3.79 | 38.7 | 0.290 | 18 | 0 | 8 | 1.340 | 11 | 4 |

8 | 450 | 2.48 | 13.3 | 0.150 | 8 | 0 | 31 | 1.340 | 0 | 3 |

9 | 16 | 23.40 | 0 | 0.570 | 2 | 0 | 16 | −0.756 | 16 | 2 |

10 | 50 | 5.86 | 0 | 0.530 | 1 | 0 | 2 | 1.340 | 0 | 4 |

11 | 176 | 2.57 | 24.4 | 0.220 | 54 | >9 | 48 | 1.220 | 11 | 9 |

12 | 32 | 12.25 | 0 | 0.290 | 9 | 9 | 9 | −0.566 | 22 | 6 |

13 | 97 | 9.14 | 1.0 | 0.310 | 10 | >9 | 22 | 1.220 | 32 | 7 |

14 | 78 | 2.41 | 3.8 | 0.390 | 13 | >9 | 24 | 0.323 | 0 | 5 |

15 | 153 | 2.61 | 0 | 0.150 | 1 | 0 | 8 | −0.774 | 0 | 8 |

16 | 375 | 2.91 | 0 | 0.410 | 3 | 0 | 13 | 1.340 | 0 | 3 |

17 | 31 | 23.40 | 0 | 0.560 | 2 | 0 | 6 | 0.106 | 31 | 7 |

18 | 86 | 7.96 | 0 | 0.270 | 3 | 0 | 8 | 0.140 | 6 | 5 |

19 | 33 | 5.31 | 0 | 0.330 | 2 | 0 | 4 | −1.126 | 18 | 2 |

20 | 25 | 10.60 | 100 | 0.280 | 1 | 0 | 3 | −0.888 | 25 | 2 |

21 | 52 | 2.43 | 61.5 | 0.270 | 14 | >9 | 19 | −0.839 | 0 | 6 |

22 | 143 | 1.90 | 21 | 0.160 | 10 | >9 | 13 | −0.885 | 0 | 7 |

23 | 108 | 2.28 | 27.8 | 0.170 | 24 | >9 | 21 | −0.930 | 0 | 7 |

24 | 91 | 1.83 | 49.5 | 0.220 | 15 | >9 | 15 | −0.924 | 0 | 8 |

25 | 106 | 1.60 | 12.3 | 0.190 | 13 | >9 | 17 | −0.865 | 0 | 6 |

26 | 24 | 5.86 | 0 | 0.530 | 1 | 0 | 3 | −0.096 | 0 | 5 |

27 | 32 | 17.72 | 0 | 0.340 | 1 | 0 | 7 | −0.258 | 32 | 6 |

28 | 24 | 14.07 | 0 | 0.390 | 1 | 0 | 9 | −0.726 | 24 | 3 |

29 | 90 | 1.63 | 52.2 | 0.200 | 16 | >9 | 16 | −0.770 | 0 | 5 |

30 | 162 | 2.23 | 100 | 0.270 | 1 | 0 | 12 | 0.051 | 0 | 6 |

31 | 130 | 1.89 | 53.8 | 0.160 | 22 | >9 | 25 | −0.832 | 0 | 3 |

32 | 99 | 10 | 0 | 0.600 | 1 | 0 | 5 | 1.220 | 99 | 3 |

33 | 95 | 3.88 | 10.5 | 0.240 | 16 | > 9 | 27 | 1.220 | 17 | 6 |

34 | 115 | 2.31 | 87.8 | 0.320 | 7 | 7 | 20 | 1.340 | 16 | 8 |

35 | 34 | 4.37 | 17.6 | 0.390 | 4 | 4 | 10 | −0.672 | 14 | 3 |

36 | 1550 | 2.62 | 41.9 | 0.200 | 14 | 0 | 90 | 1.340 | 120 | 4 |

37 | 49 | 4.95 | 14.3 | 0.260 | 6 | 6 | 5 | −0.713 | 12 | 8 |

38 | 77 | 5.92 | 15.6 | 0.390 | 18 | >9 | 19 | 1.220 | 38 | 6 |

Sample\Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Evaluation |

1 | 8 | 2 | 4 | 1 | 3 | 1 | 1 | 1 | 1 | 4 |

2 | 10 | 3 | 10 | 2 | 6 | 1 | 2 | 10 | 1 | 4 |

3 | 10 | 2 | 10 | 2 | 5 | 1 | 2 | 2 | 1 | 4 |

4 | 10 | 2 | 1 | 1 | 2 | 1 | 3 | 2 | 1 | 2 |

5 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

6 | 10 | 3 | 8 | 3 | 10 | 10 | 9 | 10 | 6 | 9 |

7 | 10 | 3 | 8 | 2 | 8 | 1 | 1 | 10 | 6 | 4 |

8 | 10 | 2 | 3 | 1 | 4 | 1 | 7 | 10 | 1 | 3 |

9 | 4 | 10 | 1 | 4 | 1 | 1 | 4 | 2 | 8 | 2 |

10 | 7 | 4 | 1 | 3 | 1 | 1 | 1 | 10 | 1 | 4 |

11 | 10 | 2 | 5 | 2 | 10 | 10 | 10 | 10 | 6 | 9 |

12 | 6 | 9 | 1 | 2 | 5 | 9 | 2 | 2 | 10 | 6 |

13 | 10 | 7 | 1 | 2 | 5 | 10 | 5 | 10 | 10 | 7 |

14 | 10 | 2 | 1 | 3 | 7 | 10 | 5 | 9 | 1 | 5 |

15 | 10 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 8 |

16 | 10 | 2 | 1 | 3 | 2 | 1 | 3 | 10 | 1 | 3 |

17 | 6 | 10 | 1 | 4 | 1 | 1 | 2 | 5 | 10 | 7 |

18 | 10 | 6 | 1 | 2 | 2 | 1 | 2 | 5 | 3 | 5 |

19 | 6 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 9 | 2 |

20 | 5 | 8 | 10 | 2 | 1 | 1 | 1 | 1 | 10 | 2 |

21 | 8 | 2 | 10 | 2 | 7 | 10 | 4 | 2 | 1 | 6 |

22 | 10 | 2 | 5 | 1 | 5 | 10 | 3 | 1 | 1 | 7 |

23 | 10 | 2 | 6 | 1 | 10 | 10 | 5 | 1 | 1 | 7 |

24 | 10 | 2 | 9 | 2 | 7 | 10 | 3 | 1 | 1 | 8 |

25 | 10 | 2 | 3 | 2 | 7 | 10 | 4 | 2 | 1 | 6 |

26 | 5 | 4 | 1 | 3 | 1 | 1 | 1 | 2 | 1 | 5 |

27 | 6 | 10 | 1 | 2 | 1 | 1 | 2 | 2 | 10 | 6 |

28 | 5 | 9 | 1 | 3 | 1 | 1 | 2 | 2 | 10 | 3 |

29 | 10 | 2 | 10 | 2 | 8 | 10 | 4 | 2 | 1 | 5 |

30 | 10 | 2 | 10 | 2 | 1 | 1 | 3 | 4 | 1 | 6 |

31 | 10 | 2 | 10 | 1 | 10 | 10 | 5 | 2 | 1 | 3 |

32 | 10 | 7 | 1 | 4 | 1 | 1 | 1 | 10 | 10 | 3 |

33 | 10 | 3 | 3 | 2 | 8 | 10 | 6 | 10 | 9 | 6 |

34 | 10 | 2 | 10 | 2 | 4 | 7 | 4 | 10 | 8 | 8 |

35 | 6 | 3 | 4 | 3 | 2 | 4 | 2 | 2 | 7 | 3 |

36 | 10 | 2 | 9 | 2 | 7 | 1 | 10 | 10 | 10 | 4 |

37 | 7 | 4 | 3 | 2 | 3 | 6 | 1 | 2 | 6 | 8 |

38 | 9 | 4 | 4 | 3 | 8 | 10 | 4 | 10 | 10 | 6 |

Variable Code | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |

1 | 1.000 | |||||||||

2 | −0.525 | 1.000 | ||||||||

3 | 0.379 | −0.428 | 1.000 | |||||||

4 | 0.571 | −0.458 | 0.516 | 1.000 | ||||||

5 | −0.235 | 0.526 | −0.261 | −0.240 | 1.000 | |||||

6 | 0.371 | −0.270 | 0.251 | 0.771 | −0.140 | 1.000 | ||||

7 | 0.458 | −0.299 | 0.262 | 0.663 | −0.080 | 0.480 | 1.000 | |||

8 | 0.456 | −0.108 | 0.015 | 0.280 | 0.314 | 0.055 | 0.460 | 1.000 | ||

9 | −0.282 | 0.692 | −0.192 | −0.133 | 0.453 | −0.044 | 0.072 | 0.230 | 1.000 | |

0 | 0.416 | −0.118 | 0.166 | −0.046 | 0.431 | 0.591 | 0.356 | 0.155 | 0.029 | 1.000 |

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Parameter | Scaling, Rating (Range of Values) and Unit of Measurement |
---|---|

1-Number of boxes: | 1 (up to 5 boxes), 2 (6 to 10 boxes), 3 (11 to 15 boxes), 4 (16 to 20 boxes), 5 (21 to 30 boxes), 6 (31 to 40 boxes), 7 (41 to 50 boxes), 8 (51 to 60 boxes), 9 (61 to 80 boxes), 10 (over 80 boxes). |

2-Average box weight: | 1 (up to 1.5kg), 2 (1,6 to 3kg), 3 (3,1 to 4.5kg), 4 (4.6 to 6kg), 5 (6.1 to 7.5kg), 6 (7.6 to 9kg), 7 (9.1 to 10.5kg), 8 (10.6 to 12kg), 9 (12.1 to 15kg), 10 (over 15 boxes). |

3-Percentage of fragile boxes: | 1 (up to 5%), 2 (6 to 10%), 3 (11 to 15%), 4 (16 to 20%), 5 (21 to 25 %), 6 (26 to 30%), 7 (31 to 35%), 8 (36 to 40%), 9 (41 to 50%), 10 (over 50%). |

4-Average box maximum width: | 1 (up to 18 cm), 2 (19 to 36 cm), 3 (37 to 54 cm), 4 (55 to 72 cm), 5 (73 to 90 cm), 6 (91 to 108 cm), 7 (109 to 126 cm), 8 (127 to 144 cm), 9 (145 to 162 cm), 10 (over 162 cm). |

5-Number of box types: | 1 (1 or 2 types), 2 (3 or 4 types), 3 (5 or 6 types), 4 (7 or 8 types), 5 (9 or 10 types), 6 (11 or 12 types), 7 (13 to 15 types), 8 (16 to 18 types), 9 (19 or 20 types), 10 (over 20 types). |

6-Number of box types to pack in columns: | 1 (does not apply), 2 (does not apply), 3 (up to 4 types), 4 (4 types), 5 (5 types), 6 (6 types), 7 (7 types), 8 (8 types), 9 (9 types), 10 (over 10 types). |

7-Time spent loading a pallet: | 1 (up to 5 min), 2 (6 to 10mins), 3 (11 to 15mins), 4 (16 to 20mins), 5 (21 to 25mins), 6 (26 to 30mins), 7 (31 to 35mins), 8 (36 to 40mins), 9 (41 to 45mins), 10 (over 45mins). |

8-Height difference between worker and pile: | 1 (top of the pile below waist level), 2 (top of the pile between waist level and worker’s height), 3 (pile up to 5cm taller than worker), 4 (pile 6 to 10cm taller), 5 (pile 11 to 15cm taller), 6 (pile 16 to 20cm taller), 7 (pile 21 to 25cm taller), 8 (pile 26 to 30cm taller), 9 (pile 31 to 35cm taller), 10 (pile over 35cm taller). |

9-Number of heavy boxes to pack: | 1 (1 or 2 boxes), 2 (3 or 4 boxes), 3 (5 or 6 boxes), 4 (7 or 8 boxes), 5 (9 or 10 boxes), 6 (11 or 12 boxes), 7 (13 or 14 boxes), 8 (15 or 16 boxes), 9 (17 or 18 boxes), 10 (over 18 boxes). |

Variable | Mean | Correspondence | Std. Deviation | N |
---|---|---|---|---|

0 | 5.03 | 2.16 | 38 | |

1 | 8.39 | 51 to 60 | 2.33 | 38 |

2 | 3.92 | 3 to 4.5 kg | 2.74 | 38 |

3 | 4.47 | 16 to 20% | 3.73 | 38 |

4 | 2.16 | 19 to 36 cm | 0.86 | 38 |

5 | 4.39 | 7 or 8 | 3.21 | 38 |

6 | 4.66 | 4 | 4.28 | 38 |

7 | 3.37 | 11 to 15 min | 2.44 | 38 |

8 | 4.95 | 6 to 10 cm | 3.96 | 38 |

9 | 4.66 | 7 or 8 | 3.98 | 38 |

Component | Initial Eigenvalues | ||
---|---|---|---|

Total | % of Variance | Cumulative % | |

1 | 3.641 | 40.453 | 40.453 |

2 | 2.003 | 22.250 | 62.704 |

3 | 1.070 | 11.892 | 74.596 |

4 | 0.731 | 8.127 | 82.723 |

5 | 0.567 | 6.296 | 89.019 |

6 | 0.431 | 4.786 | 93.805 |

7 | 0.276 | 3.066 | 96.872 |

8 | 0.171 | 1.902 | 98.774 |

9 | 0.110 | 1.226 | 100.000 |

Variable | Component | ||
---|---|---|---|

1 | 2 | 3 | |

6 | 0.891 | −0.023 | −0.031 |

5 | 0.890 | −0.213 | 0.256 |

7 | 0.666 | 0.008 | 0.496 |

3 | 0.497 | −0.397 | 0.053 |

9 | 0.088 | 0.880 | 0.048 |

2 | −0.247 | 0.850 | −0.255 |

4 | −0.214 | 0.713 | 0.290 |

8 | 0.072 | 0.184 | 0.932 |

1 | 0.387 | −0.443 | 0.628 |

Dimension | Cronbach’s Alpha | Variance Accounted For | |
---|---|---|---|

Total (Eigenvalue) | % of Variance | ||

1 | 0.836 | 3.725 | 41.387 |

2 | 0.718 | 2.521 | 28.007 |

Total | 0.945 | 6.245 | 69.394 |

Variable | Component | |
---|---|---|

1 | 2 | |

5 | 0.930 | −0.091 |

7 | 0.867 | 0.144 |

6 | 0.791 | −0.042 |

1 | 0.773 | −0.286 |

3 | 0.624 | −0.330 |

4 | −0.132 | 0.827 |

9 | −0.133 | 0.826 |

2 | −0.535 | 0.750 |

8 | 0.416 | 0.608 |

Rotated Cronbach’s Alpha | 0.836 | 0.718 |

% of variance | 41.387 | 28.007 |

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**MDPI and ACS Style**

Barros, H.; Pereira, T.; Ramos, A.G.; Ferreira, F.A. Complexity Constraint in the Distributor’s Pallet*Mathematics* **2021**, *9*, 1742.
https://doi.org/10.3390/math9151742

**AMA Style**

Barros H, Pereira T, Ramos AG, Ferreira FA. Complexity Constraint in the Distributor’s Pallet*Mathematics*. 2021; 9(15):1742.
https://doi.org/10.3390/math9151742

**Chicago/Turabian Style**

Barros, Hugo, Teresa Pereira, António G. Ramos, and Fernanda A. Ferreira. 2021. "Complexity Constraint in the Distributor’s Pallet*Mathematics* 9, no. 15: 1742.
https://doi.org/10.3390/math9151742