# Using Optimization Models for Scheduling in Enterprise Resource Planning Systems

## Abstract

**:**

## 1. Introduction

## 2. Real-World Problem

## 3. A Linear Optimization Model

#### 3.1. Literature Review

#### 3.2. Model

- M:
- number of stations; $1\le \mathrm{j}\le \mathrm{M}$.
- N:
- number of jobs; $1\le \mathrm{i}\le \mathrm{N}$.
- NE:
- number of artificial jobs; $1\le \mathrm{i}\le \mathrm{N}$.
- ${\mathrm{a}}_{\mathrm{i}}$:
- release date of job i for all $1\le \mathrm{i}\le \mathrm{N}$.
- ${\mathrm{f}}_{\mathrm{i}}$:
- due date of job i for all $1\le \mathrm{i}\le \mathrm{N}$.
- ${\mathrm{o}}_{\mathrm{i},\mathrm{j}}$:
- operation j of job i $\left(1\le \mathrm{i}\le \mathrm{N}\right)$ which is worked on station j for all $1\le \mathrm{i}\le \mathrm{N}$ and $1\le \mathrm{j}\le \mathrm{M}$.
- ${\mathrm{t}}_{\mathrm{i},\mathrm{j}}$:
- duration of operation j of job i $\left({\mathrm{o}}_{\mathrm{i},\mathrm{j}}\right)$ which is worked on station j for all $1\le \mathrm{i}\le \mathrm{N}$ and $1\le \mathrm{j}\le \mathrm{M}$

- ${\mathrm{x}}_{\mathrm{i},\mathrm{p}}$:
- position of job i in the permutation: job i is at position p for all $1\le \mathrm{i},\mathrm{p}\le \mathrm{N}$.

- $\mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$:
- upper bound of the realised finish time of the job at position p in a permutation on station j for all $1\le \mathrm{p},\mathrm{j}\le \mathrm{N}$.
- $\mathrm{F}{\mathrm{T}}_{\mathrm{p},0}$:
- release date of the job at position p for all $1\le \mathrm{p}\le \mathrm{N}$.
- $\mathrm{S}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$:
- earliest starting time for job p in a permutation on station j for all $1\le \mathrm{p}\le \mathrm{N}$; i.e., $\mathrm{S}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}=\mathrm{F}{\mathrm{T}}_{\mathrm{p}-1,\mathrm{j}}$.
- $\mathrm{S}\mathrm{R}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$:
- realized starting and finish time of the job at position p in a permutation on station j for all $1\le \mathrm{p},\mathrm{j}\le \mathrm{N}$.
- $\mathrm{F}\mathrm{R}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$:
- realized starting and finish time of the job at position p in a permutation on station j for all $1\le \mathrm{p},\mathrm{j}\le \mathrm{N}$.
- $\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p},\mathrm{k}}^{\mathrm{O}\mathrm{F}}$:
- job at position p in the permutation has type k and is an outflow filter for all $1\le \mathrm{p}\le \mathrm{N}$ and for all $1\le \mathrm{k}\le 10$.
- ${\mathrm{T}}_{\mathrm{p}}$:
- upper bound for the tardiness of the job at position p for all $1\le \mathrm{p}\le \mathrm{N}$.

- (1)
- ${\mathrm{x}}_{\mathrm{i},\mathrm{p}}=\{\begin{array}{cc}1,& \mathrm{j}\mathrm{o}\mathrm{b}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{i}\text{\hspace{0.05em}\hspace{0.05em}}\mathrm{i}\mathrm{s}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{a}\mathrm{t}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{i}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$ for all $1\le \mathrm{i},\mathrm{p}\le \mathrm{N}$.

- (2)
- $\sum _{\mathrm{p}=1}^{\mathrm{N}}{\mathrm{x}}_{\mathrm{i},\mathrm{p}}}=1$ for all $1\le \mathrm{i}\le \mathrm{N}$.
- (3)
- $\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{x}}_{\mathrm{i},\mathrm{p}}}=1$ for all $1\le \mathrm{p}\le \mathrm{N}$.

- (4)
- $\mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{j}-1}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i},\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i},\mathrm{p}}}\le \mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$ for all $1\le \mathrm{p}\le \mathrm{N}$ and for all $1\le \mathrm{j}\le \mathrm{M}$ (remember: $\mathrm{F}{\mathrm{T}}_{\mathrm{p},0}$ is the release date of the job at position p).

- (5)
- $\mathrm{F}{\mathrm{T}}_{\mathrm{p}-1,\mathrm{j}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i},\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i},\mathrm{p}}}\le \mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$ for all $1\le \mathrm{p}\le \mathrm{N}$ and for all $1\le \mathrm{j}\le \mathrm{M}$ with $\mathrm{F}{\mathrm{T}}_{0,\mathrm{j}}$ being the availability of station j for all $1\le \mathrm{j}\le \mathrm{M}$.

- (6)
- $\mathrm{F}{\mathrm{T}}_{\mathrm{p}+\left(\mathrm{j}-1\right),\mathrm{M}-\left(\mathrm{j}-1\right)}=\mathrm{F}{\mathrm{T}}_{\mathrm{p}+\mathrm{j},\mathrm{M}-\mathrm{j}}$ for all $1\le \mathrm{p}\le \mathrm{N}-\mathrm{M}+1$ and for all $1\le \mathrm{j}\le \mathrm{M}-1$.

- (7)
- $\mathrm{F}{\mathrm{T}}_{1+\left(\mathrm{j}-1\right),\left(\mathrm{M}-\mathrm{p}\right)-\left(\mathrm{j}-1\right)}=\mathrm{F}{\mathrm{T}}_{1+\mathrm{j},\left(\mathrm{M}-\mathrm{p}\right)-\left(\mathrm{j}-1\right)-1}$ for all $1\le \mathrm{p}<\left(\mathrm{M}-1\right)$ and for all $1\le \mathrm{j}<\mathrm{M}-\mathrm{p}$.

- (8)
- $\mathrm{F}{\mathrm{T}}_{\mathrm{p}+\left(\mathrm{j}-1\right),\mathrm{M}-\left(\mathrm{j}-1\right)}=\mathrm{F}{\mathrm{T}}_{\mathrm{p}+\mathrm{j},\mathrm{M}-\mathrm{j}}$ for all $\mathrm{N}-\mathrm{M}+1<\mathrm{p}<\mathrm{N}$ and for all $1\le \mathrm{j}\le \mathrm{N}-\mathrm{p}$.

- (9)
- $\mathrm{S}\mathrm{R}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i},\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i},\mathrm{p}}}=\mathrm{F}\mathrm{R}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}\le \mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$ for all $1\le \mathrm{p}\le \mathrm{N}$.
- (10)
- $\mathrm{S}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}=\mathrm{S}\mathrm{R}{\mathrm{T}}_{\mathrm{p},\mathrm{j}}$ for all $1\le \mathrm{j}\le \mathrm{M}$.

- (11)
- $\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p},\mathrm{k}}^{\mathrm{O}\mathrm{F}}=\{\begin{array}{cc}1,& \mathrm{j}\mathrm{o}\mathrm{b}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{a}\mathrm{t}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{i}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{h}\mathrm{a}\mathrm{s}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{k}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{a}\mathrm{n}\mathrm{d}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{i}\mathrm{s}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{a}\mathrm{n}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$ for all $1\le \mathrm{p}\le \mathrm{N}$ and for all $1\le \mathrm{k}\le 10$.The above condition is not met, if $\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p},\mathrm{k}}^{\mathrm{O}\mathrm{F}}+\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p}+1,\mathrm{k}}^{\mathrm{O}\mathrm{F}}>1$ for one k, $1\le \mathrm{k}\le 10$, and one p, $1\le \mathrm{p}\le \mathrm{N}$.

- (12)
- $\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p},\mathrm{k}}^{\mathrm{O}\mathrm{F}}+\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p}+1,\mathrm{k}}^{\mathrm{O}\mathrm{F}}\le 1$ for all $1\le \mathrm{p}<\mathrm{N}-1$ and for all $1\le \mathrm{k}\le 10$.
- (13)
- ${\mathrm{x}}_{\mathrm{i},\mathrm{p}}\in \left\{0,1\right\}$ for all $1\le \mathrm{i}\le \mathrm{N}$ and for all $1\le \mathrm{p}\le \mathrm{N}$.
- (14)
- $\mathrm{T}\mathrm{y}{\mathrm{p}}_{\mathrm{p},\mathrm{k}}^{\mathrm{O}\mathrm{F}}\in \left\{0,1\right\}$ for all $1\le \mathrm{p}\le \mathrm{N}$ and for all $1\le \mathrm{k}\le 10$.

- (15)
- ${\mathrm{T}}_{\mathrm{p}}\ge 0$ for all $1\le \mathrm{p}\le \mathrm{N}$.
- (16)
- ${\mathrm{T}}_{\mathrm{p}}\ge \mathrm{F}{\mathrm{T}}_{\mathrm{p},\mathrm{M}}-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{f}}_{\mathrm{i}}\cdot {\mathrm{x}}_{\mathrm{i},\mathrm{p}}}$ for all $1\le \mathrm{p}\le \mathrm{N}$.

- (17)
- Minimize $\left({\displaystyle \sum _{\mathrm{p}=1}^{\mathrm{N}}{\mathrm{T}}_{\mathrm{p}}}\right)$

- (18)
- ${\mathrm{x}}_{\mathrm{p},\mathrm{p}}=1$ for all $1\le \mathrm{p}\le \left(\mathrm{M}-1\right)$

- (19)
- ${\mathrm{x}}_{\mathrm{N}+\left(\mathrm{M}-1\right)+\mathrm{N}\mathrm{E}+\mathrm{p},\mathrm{N}+\left(\mathrm{M}-1\right)+\mathrm{N}\mathrm{E}+\mathrm{p}}=1$ for all $1\le \mathrm{p}\le \left(\mathrm{M}-1\right)$

#### 3.3. Model Extension for a Rolling Planning Environment

_{1}, …, A

_{M − 1}) are partially part of the first (M − 1) cycles of NS. A

_{1}, …, A

_{M − 1}determine a minimum length of the first (M − 1) cycles of NS. For cycle j, $1\le \mathrm{j}\le \left(\mathrm{M}-1\right)$, of these cycles, this minimum length is the maximum of the durations of the jobs in A

_{1}, …, A

_{M − 1}which are executed in cycle j and it is denoted by $\mathrm{T}\mathrm{Z}{\mathrm{B}}_{\mathrm{j}}$. In addition, these first (M − 1) cycles of NS determine the (realized) due dates of A

_{1}, …, A

_{M − 1}which is denoted by $\mathrm{S}\mathrm{E}{\mathrm{T}}_{\mathrm{i}}$ for job A

_{i}, $1\le \mathrm{i}\le \left(\mathrm{M}-1\right)$.

_{1}, …, A

_{M − 1}: last (M − 1)-jobs in P.

_{i}for all $1\le \mathrm{i}\le \left(\mathrm{M}-1\right)$.

_{1}, …, A

_{M − 1}are not scheduled, but substituted by the above mentioned artificial jobs. So, the positions M until (M+ M − 1− 1) have to be regarded and the finishing time is, related to station 1, i.e., $\left(\mathrm{F}{\mathrm{T}}_{\left(\mathrm{M}-1\right)+\mathrm{j},1}\right)$. This corresponds to the first (M − 1) cycles of NS—without the (M − 1) artificial jobs. Then the linear inequalities

- (20)
- $\mathrm{T}\mathrm{Z}{\mathrm{B}}_{\mathrm{j}}\le \mathrm{F}{\mathrm{T}}_{\left(\mathrm{M}-1\right)+\mathrm{j},1}-\mathrm{F}{\mathrm{T}}_{\left(\mathrm{M}-1\right)+\left(\mathrm{j}-1\right),1}$ for all $1\le \mathrm{j}\le \left(\mathrm{M}-1\right)$

_{1}, …, A

_{M − 1}, so that their adapted delay have to be integrated in the objective function. With $\mathrm{S}\mathrm{E}{\mathrm{T}}_{\mathrm{i}}$ being the due date of the job which is finished in cycle i (of NS) – i.e., A

_{i}—the linear inequalities (21, 22) ensure upper bounds for the variable tardiness $\left({\mathrm{T}}_{\mathrm{i}}\right)$ or $\left({\mathrm{T}}_{{\mathrm{A}}_{\mathrm{i}}}\right)$, respectively.

- (21)
- ${\mathrm{T}}_{\mathrm{i}}\ge 0$ for all $1\le \mathrm{i}\le \left(\mathrm{M}-1\right)$.
- (22)
- ${\mathrm{T}}_{\mathrm{i}}\ge \mathrm{F}{\mathrm{T}}_{\left(\mathrm{M}-1\right)+\mathrm{i},1}-\mathrm{S}\mathrm{E}{\mathrm{T}}_{\mathrm{i}}$ for all $1\le \mathrm{i}\le \left(\mathrm{M}-1\right)$.Then, the sum of these tardiness variables has to be added in the objective function:
- (23)
- Minimize $\left({\displaystyle \sum _{\mathrm{p}=1}^{\mathrm{N}}{\mathrm{T}}_{\mathrm{p}}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{M}-1}{\mathrm{T}}_{\mathrm{i}}}\right)$.

_{1}, …, A

_{M − 1}which are executed in cycle j. For this, there exists a so called simulation algorithm which simulates the execution of a permutation of jobs on the flow shop. This simulation algorithm can be used to ensure restrictions outside the optimization model; then the simulation algorithm ensures a feasible schedule and calculates the correct completion times for example. This is helpful for restrictions in industrial practice which are difficult to formulate by a linear inequality (or linear inequalities) or for a linear inequality (or linear inequalities) which increases the runtime significantly.

## 4. Optimal and Heuristic Scheduling

## 5. Results and Discussion

## 6. Conclusions

## Conflicts of Interest

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**Figure 2.**Results by the priority rules and optimal scheduling for the real world application (

**a**) for ${\mathrm{T}}_{\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}}$ and (

**b**) for ${\mathrm{T}}_{\mathrm{R}\mathrm{M}\mathrm{S}}$.

Filter Type | Station 1 | Station 2 | Station 3 | Station 4 | Sum of Processing Times | ||
---|---|---|---|---|---|---|---|

if | of | if | of | ||||

1 | 65 | 5 | 15 | 20 | 10 | 95 | 100 |

2 | 300 | 50 | 55 | 60 | 15 | 420 | 425 |

3 | 255 | 125 | 60 | 67 | 75 | 515 | 522 |

4 | 295 | 45 | 245 | 259 | 15 | 600 | 614 |

5 | 135 | 195 | 295 | 313 | 105 | 730 | 748 |

6 | 75 | 305 | 20 | 25 | 305 | 705 | 710 |

7 | 255 | 285 | 205 | 215 | 75 | 820 | 830 |

8 | 255 | 275 | 195 | 209 | 230 | 955 | 969 |

9 | 50 | 290 | 250 | 263 | 290 | 880 | 893 |

10 | 270 | 300 | 235 | 247 | 300 | 1105 | 1117 |

**Table 2.**Relative differences to r = 1 with “+” means a higher value (worse) and “−“ means a lower value (better) for the six demand scenarios.

OS | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Historical |
---|---|---|---|---|---|---|

r = 0 | 20.8% | 23.9% | 30.4 | 39.1% | 29.9% | 32.8% |

r = 2 | 0% | −0.01% | −0.01% | 0.1% | 0 | 0% |

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Herrmann, F.
Using Optimization Models for Scheduling in Enterprise Resource Planning Systems. *Systems* **2016**, *4*, 15.
https://doi.org/10.3390/systems4010015

**AMA Style**

Herrmann F.
Using Optimization Models for Scheduling in Enterprise Resource Planning Systems. *Systems*. 2016; 4(1):15.
https://doi.org/10.3390/systems4010015

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Herrmann, Frank.
2016. "Using Optimization Models for Scheduling in Enterprise Resource Planning Systems" *Systems* 4, no. 1: 15.
https://doi.org/10.3390/systems4010015