# Robust Scheduling of Waste Wood Processing Plants with Uncertain Delivery Sources and Quality

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## Abstract

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## 1. Introduction

## 2. Problem Definition

#### 2.1. General Process and Infrastructure

#### 2.2. Uncertainty and Objective

#### 2.3. Formal Problem Definition

#### 2.3.1. Job Related Data

- J is the finite set of jobs/deliveries.
- ${J}^{\mathtt{B}},{J}^{\mathtt{H}}$ are the sets of jobs with building/industrial and household deliveries, respectively. ${J}^{\mathtt{B}}\cap {J}^{\mathtt{H}}=\varnothing $ and ${J}^{\mathtt{B}}\cup {J}^{\mathtt{H}}=J$
- ${J}^{\mathtt{S}},{J}^{\mathtt{D}}$ are the sets of jobs with solid and derived wood deliveries, respectively. ${J}^{\mathtt{S}}\cap {J}^{\mathtt{D}}=\varnothing $ and ${J}^{\mathtt{S}}\cup {J}^{\mathtt{D}}=J$
- ${d}_{j}^{a}$$\in {\mathbb{Z}}^{0,+}$ is the day of arrival of the delivery for job $j\in J$. $\left(\right)$
- ${d}_{j}^{s}$$\in {\mathbb{Z}}^{0,+}$ is the day of shipping of the product for job $j\in J$. $\left(\right)$
- ${p}_{j}$$\in {\mathbb{R}}^{0,+}$ is the priority of job $j\in J$. $\left[-\right]$
- ${m}_{j}$$\in {\mathbb{R}}^{0,+}$ is the total mass of the delivery for job $j\in J$. $\left[t\right]$

#### 2.3.2. Infrastructure Related Data

- ${c}^{\mathtt{IS}},{c}^{\mathtt{CR}},{c}^{\mathtt{MS}}$$\in {\mathbb{R}}^{0,+}$ are the throughput capacities for inspection and separation, coating removal, and manual metal separation at the facility, respectively. $\left(\right)$
- M is the finite set of machines, the union of the following pairwise disjoint sets:
- ${M}^{\mathtt{MS}}$ is the finite set of machines for automated metal separation.
- ${M}^{\mathtt{PS}}$ is the finite set of machines for pre-shredding.
- ${M}^{\mathtt{SH}}={M}^{\mathtt{RS}}$ is the finite set of machines for shredding and re-shredding.
- ${M}^{\mathtt{SC}}$ is the finite set of machines for screening.

- ${c}_{m,j}$$\in {\mathbb{R}}^{0,+}$ is the throughput capacity of a machine $m\in M$ for job $j\in J$. $\left(\right)$
- ${e}_{m}$$\in {\mathbb{R}}^{0,+}$ is the electrical power consumption of a machine $m\in M$. $\left(\right)$
- ${w}_{m}$$\in {\mathbb{R}}^{0,+}$ is the total amount of electrical energy needed to shutdown and then start up machine $m\in M$. $\left(\right)$
- s$\in {\mathbb{R}}^{0,+}$ is the length of the shifts. $\left[h\right]$

#### 2.3.3. Uncertain Data

- ${p}_{X,Y}^{\mathtt{CR}}$, ${p}_{X,Y}^{\mathtt{RS}}$$\in [0,1]$ are the percentages requiring coating removal and re-shredding for jobs with origins $X\in \{\mathtt{B},\mathtt{H}\}$ and material type $Y\in \{\mathtt{S},\mathtt{D}\}$ in the robust case. $\left[-\right]$
- ${\overline{p}}_{X,Y}^{\mathtt{CR}}$, ${\overline{p}}_{X,Y}^{\mathtt{RS}}$$\in [0,1]$ are the percentages requiring coating removal and re-shredding for jobs with origins $X\in \{\mathtt{B},\mathtt{H}\}$ and material type $Y\in \{\mathtt{S},\mathtt{D}\}$ in the worst-case scenario. $\left[-\right]$

## 3. Proposed Approach

#### 3.1. General Structure and Derived Sets

- $\mathtt{IS}$ Inspection and separation
- $\mathtt{CR}$ Coating removal
- $\mathtt{MS}$ Metal separation
- $\mathtt{PS}$ Pre-shredding
- $\mathtt{SH}$ Shredding
- $\mathtt{RS}$ Re-shredding
- $\mathtt{SC}$ Screening

#### 3.2. Basic Scheduling Variables

#### 3.3. Constraints

#### 3.3.1. Logical and Balance Constraints

#### 3.3.2. Processing Time Constraints

#### 3.3.3. Production Precedence Constraints

#### 3.3.4. Scheduling Precedence Constraints

#### 3.4. Objective Functions

#### 3.4.1. Priority Weighted Lateness Minimization

#### 3.4.2. Electrical Footprint Minimization

#### 3.5. Overview of the Mathematical Model

## 4. Numerical Results

#### 4.1. Single Objective Optimization

#### 4.2. Bi-Objective Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The possible flow of waste wood through the different transformation processes (green: manual steps, gray: machine steps).

**Figure 2.**Probability density function for the ratio of raw material needing coating removal based on statistical data.

BASE MODEL | ||
---|---|---|

Input data | sets | ${J}^{\mathtt{B}}$, ${J}^{\mathtt{H}}$, ${J}^{\mathtt{S}}$, ${J}^{\mathtt{D}}$, ${M}^{\mathtt{MS}}$, ${M}^{\mathtt{PS}}$, ${M}^{\mathtt{SH}}$, ${M}^{\mathtt{SC}}$ |

parameters | ${d}_{j}^{a}$, ${d}_{j}^{s}$, ${m}_{j}$, ${c}^{\mathtt{IS}}$, ${c}^{\mathtt{CR}}$, ${c}^{\mathtt{MS}}$, ${c}_{m,j}$, s, ${p}_{X,Y}^{\mathtt{CR}}$, ${p}_{X,Y}^{\mathtt{RS}}$ | |

Derived sets | J, M, ${M}^{\mathtt{RS}}$, ${S}^{\mathtt{C}}$, ${S}^{\mathtt{M}}$, S, $\overline{S}$, P, T, A, ${C}^{\mathtt{M}}$, ${C}^{\mathtt{C}}$ | |

Decision variables | continuous | ${t}_{j,s}^{s}$, ${t}_{j,s}^{c}$, ${q}_{j,s,m}$ |

binary | ${d}_{j}$, ${a}_{j,s,m}$, ${b}_{{j}_{1},{s}_{1},{j}_{2},{s}_{2}}^{\mathtt{M}}$, ${b}_{{j}_{1},s,{j}_{2}}^{\mathtt{C}}$ | |

PRIORITY WEIGHTED LATENESS MINIMIZATION | ||

Additional input data | ${p}_{j}$, ${\overline{p}}_{X,Y}^{\mathtt{CR}}$, ${\overline{p}}_{X,Y}^{\mathtt{RS}}$ | |

Additional variables | continuous | ${\overline{t}}_{j,s}^{s}$, ${\overline{t}}_{j,s}^{c}$, ${\overline{q}}_{j,s,m}$ |

discrete | ${\overline{l}}_{j}$ | |

Objective function | ${\sum}_{j\in J}{p}_{j}\xb7{\overline{l}}_{j}$ | |

ELECTRICAL FOOTPRINT MINIMIZATION | ||

Additional input data | ${H}_{j}$, H, ${e}_{m}$, ${w}_{m}$ | |

Additional variables | binary | ${u}_{m,d}$, ${y}_{j,s,m,d}$ |

Objective function | ${\sum}_{(j,s,m)\in A}\frac{{q}_{j,s,m}\xb7{e}_{m}}{{c}_{m,j}}+{\sum}_{m\in M,d\in H}{u}_{m,d}\xb7{w}_{m}$ |

Deliveries | Size | Lateness Sol. Time (s) | Energy Sol. Time (s) |
---|---|---|---|

5 | S | 0.10 | 0.15 |

5 | L | 0.15 | 0.55 |

10 | S | 1.62 | 2.21 |

10 | L | 1.06 | 7.23 |

15 | S | 2.73 | 23.20 |

15 | L | 9.70 | 154.24 |

20 | S | 2.86 | 339.56 |

20 | L | - | - |

Deliveries | Size | Lateness Sol. Time (s) | Energy Sol. Time (s) |
---|---|---|---|

5 | S | 0.09 | 0.13 |

5 | L | 0.13 | 0.49 |

10 | S | 0.30 | 0.46 |

10 | L | 0.54 | 8.62 |

15 | S | 1.46 | 5.35 |

15 | L | 38.36 | 743.70 |

20 | S | 7.45 | 86.17 |

20 | L | 979.11 * | 1243.16 * |

Deliveries | Size | Lateness Sol. Time (s) | Energy Sol. Time (s) |
---|---|---|---|

20 | S | 3.59 | 72.37 |

20 | L | 21.59 * | 494.32 * |

25 | S | 10.02 | 113.25 |

25 | L | 431.46 * | 1655.49 * |

30 | S | 17.23 * | 549.61 * |

30 | L | - | - |

Deliveries | Size | Lateness Sol. Time (s) | Energy Sol. Time (s) |
---|---|---|---|

20 | S | 1.68 | 4.80 |

20 | L | 9.10 | 329.92 |

25 | S | 5.28 | 36.56 |

25 | L | 144.57 * | 2445.34 * |

30 | S | 17.45 | 776.60 |

30 | L | - | - |

Deliveries | Size | Limit Reached | Division 5 | Division 10 | ||
---|---|---|---|---|---|---|

Front | Time (s) | Front | Time (s) | |||

5 | S | 0 | 1.20 | 0.72 | 1.20 | 0.89 |

5 | L | 0 | 1.00 | 1.13 | 1.00 | 1.07 |

10 | S | 0 | 1.60 | 7.43 | 1.70 | 11.32 |

10 | L | 0 | 2.10 | 172.48 | 2.10 | 303.07 |

15 | S | 0 | 3.10 | 46.61 | 3.20 | 80.23 |

15 | L | 3 | 3.71 | 1321.81 | 3.86 | 2079.09 |

20 | S | 0 | 3.00 | 109.42 | 3.20 | 181.13 |

20 | L | 4 | 2.50 | 1331.42 | 2.67 | 1981.28 |

25 | S | 0 | 3.10 | 593.51 | 3.50 | 1043.97 |

25 | L | 10 | - | - | - | - |

Deliveries | Size | Limit Reached | Division 5 | Division 10 | ||
---|---|---|---|---|---|---|

Front | Time (s) | Front | Time (s) | |||

5 | S | 0 | 1.10 | 0.62 | 1.10 | 0.70 |

5 | L | 0 | 1.00 | 0.92 | 1.00 | 0.90 |

10 | S | 0 | 1.80 | 12.44 | 1.80 | 18.75 |

10 | L | 0 | 1.40 | 185.86 | 1.40 | 316.93 |

15 | S | 0 | 2.60 | 50.88 | 2.80 | 85.67 |

15 | L | 4 | 3.67 | 2314.70 | 4.33 | 3952.80 |

20 | S | 0 | 2.50 | 75.03 | 2.60 | 118.81 |

20 | L | 4 | 3.83 | 1809.51 | 4.17 | 3213.87 |

25 | S | 0 | 3.40 | 631.90 | 3.70 | 896.69 |

25 | L | 10 | - | - | - | - |

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## Share and Cite

**MDPI and ACS Style**

Dávid, B.; Ősz, O.; Hegyháti, M.
Robust Scheduling of Waste Wood Processing Plants with Uncertain Delivery Sources and Quality. *Sustainability* **2021**, *13*, 5007.
https://doi.org/10.3390/su13095007

**AMA Style**

Dávid B, Ősz O, Hegyháti M.
Robust Scheduling of Waste Wood Processing Plants with Uncertain Delivery Sources and Quality. *Sustainability*. 2021; 13(9):5007.
https://doi.org/10.3390/su13095007

**Chicago/Turabian Style**

Dávid, Balázs, Olivér Ősz, and Máté Hegyháti.
2021. "Robust Scheduling of Waste Wood Processing Plants with Uncertain Delivery Sources and Quality" *Sustainability* 13, no. 9: 5007.
https://doi.org/10.3390/su13095007