# A Genetic Hyper-Heuristic for an Order Scheduling Problem with Two Scenario-Dependent Parameters in a Parallel-Machine Environment

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

_{v}, a ready time ${r}_{i}^{s}$, and a due date ${d}_{i}^{s}$ in scenario s, where s = 1, 2. Our objective here was to formulate a robust policy that minimized the maximum of the weighted number of tardy orders in two scenario-dependent environments. In other words, the aim was to identify a job sequence ${\sigma}^{*}$ such that ${\sigma}^{*}=\mathrm{arg}\{mi{n}_{\sigma \in \mathsf{\Omega}}\{ma{x}_{s=1,2}\sum _{i=1}^{n}{w}_{i}N{T}_{i}^{s}\left(\sigma \right)\}\}$, where $\mathsf{\Omega}$ is the set of all possible permutation schedules, and $N{T}_{i}^{s}\left(\sigma \right)=1$ if customer order $i$ is tardy in scenario $s$ in $\sigma $ and is 0 otherwise. When m = 1, the problem with the one-scenario environment is NP-hard, as demonstrated by Karp [43]; the same COSP problem with one scenario was addressed by Lin et al. [28]. Thus, the problem considered in the present study was NP-hard as well.

## 3. Branch-and-Bound Method

_{v}, v = 1, 2, …, m and s = 1, 2. According the definition, the completion times of order i and order j in $\sigma $ and ${\sigma}^{\prime}$ are given as:

**Property**

**1.**

- Case (i)
- ${r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}$, ${r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}$ and ${r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}{d}_{j}^{s}$.
- Case (ii)
- ${r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}{d}_{j}^{s}$, and${r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}$.
- Case (iii)
- ${d}_{j}^{s}{r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}$and${r}_{j}^{s}++{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}$.

**Proof:**

_{j}in sequence $\sigma $ and O

_{i}in sequence ${\sigma}^{\prime}$ are, respectively:

**Property**

**2.**

- Case (i):
- ${r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}{d}_{j}^{s}$, and${r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}{r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}$.
- Case (ii):
- ${r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}{d}_{j}^{s}{r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}$, and${w}_{i}>{w}_{j}$.
- Case (iii):
- ${r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}{d}_{j}^{s}$, and${r}_{i}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}{d}_{i}^{s}$.
- Case (iv):
- ${d}_{j}^{s}{r}_{j}^{s}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{iv}^{s}\right\}+{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{t}_{jv}^{s}\right\}$.

**Property**

**3.**

- Case (i):
- ${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}+{t}_{jv}^{s}\right\}{d}_{j}^{s}$, and${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}+{t}_{iv}^{s}\right\}{d}_{i}^{s}{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}\right\}.$.
- Case (ii):
- ${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}\right\}{d}_{j}^{s}$, ${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}\right\}{d}_{i}^{s}$and${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}+{t}_{iv}^{s}\right\}{d}_{i}^{s}$.
- Case (iii):
- ${w}_{i}>{w}_{j}$, and${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}+{t}_{jv}^{s}\right\}{d}_{j}^{s}{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}\right\}$.
- Case (iv):
- ${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}\right\}{d}_{j}^{s},$and${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}\right\}{d}_{i}^{s}$.
- Case (v):
- ${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{i}^{s}\right\}+{t}_{iv}^{s}+{t}_{jv}^{s}\right\}{d}_{j}^{s},$and${\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{\mathrm{max}}_{v\in {\mathsf{\Omega}}_{M}}\left\{{z}_{v}^{s},{r}_{j}^{s}\right\}+{t}_{jv}^{s}+{t}_{iv}^{s}\right\}{d}_{i}^{s}$.

## 4. Three Modified Moore’s Heuristics

- Moore_pi_M heuristic:
- 01:
- Let ${t}_{i}^{*}={\mathrm{max}}_{1\le v\le m}\left\{{t}_{iv}^{1}+{r}_{i}^{1},{t}_{iv}^{2}+{r}_{i}^{2}\right\}$ and ${d}_{i}^{*}=\mathrm{max}\left\{{d}_{i}^{1},{d}_{i}^{2}\right\}$ be the processing time and due date, respectively, for order O
_{i}, i = 1, 2, …, n, and ${O}_{N}=$ {O_{1}, O_{2}, …, O_{n}}. - 02:
- Form a schedule σ by following the earliest due dates (EDD) rule based on ${O}_{N}$.
- 03:
- Let the current schedule σ* = σ and the corresponding set of orders S* = ${\mathrm{O}}_{N}$
- 04:
- Compute the completion times of the orders in σ until a tardy order is found and delete it from
- 05:
- σ* (and from S*) to form a new current schedule σ* and new S*. Find an order O*
- 06:
- with the largest processing time in S*.
- 07:
- Delete order O* from σ* and S*. Repeat Steps 04–07 until no tardy order remains.
- 08:
- Output the final schedule ${\sigma}_{M}=\left({\sigma}^{*},{\sigma}^{\prime}\right)$, where ${\sigma}^{*}$ denotes the sequence of orders
- 09:
- completed on time and scheduled using the EDD rule, and ${\sigma}^{\prime}$ denotes an arbitrary
- 10:
- sequence of orders that are tardy under ${\sigma}_{M}$.
- 11:
- Execute ${\sigma}_{M}$ by using the pairwise interchange improvement method and output the final solution.

- Moore_pi_m heuristic:
- 01:
- Let ${t}_{i}^{*}={\mathrm{min}}_{1\le v\le m}\left\{{t}_{iv}^{1}+{r}_{i}^{1},{t}_{iv}^{2}+{r}_{i}^{2}\right\}$ and ${d}_{i}^{*}=\mathrm{min}\left\{{d}_{i}^{1},{d}_{i}^{2}\right\}$ be the processing time and due date, respectively, for order O
_{i}, i = 1, 2, …, n, and ${\mathrm{O}}_{N}=$ {O_{1}, O_{2}, …, O_{n}}. - 02–11:
- are the same as those in the Moore_pi_M heuristic.

- Moore_pi_mean heuristic:
- 01:
- Let ${t}_{i}^{*}=\mathrm{max}\{{r}_{i}^{1}+\sum _{v=1}^{m}\frac{{t}_{iv}^{1}}{m},{r}_{i}^{2}+\sum _{v=1}^{m}\frac{{t}_{iv}^{2}}{m}\}$ and ${d}_{i}^{*}=({d}_{i}^{1}+{d}_{i}^{2})/2$ be the processing time and due date, respectively, for order O
_{i}, i = 1, 2, …, n and ${\mathrm{O}}_{N}=$ {O_{1}, O_{2}, …, O_{n}}. - 02–11:
- are the same as those in the Moore_pi_M heuristic.

## 5. A Genetic and a Genetic Hyper-Heuristic

**Steps of genetic algorithm**:

- 00:
**Input**Pop, P, IT_GA.- 01:
- Generate a series of Pop initial parents (schedules) and find their fitness values.
- 02:
**Do**i = 1, IT_GA- 03:
- Choose two parents from Pop populations by using the roulette wheel method and employ a linear order crossover to reproduce a set of Pop offspring.
- 04:
- For each offspring, generate a random number u (0 < u < 1) if u < P; then, create a new
- 05:
- offspring by inserting a displacement mutation.
- 06:
- Record the best one (schedule) and replace these Pop parents with their offspring.
- 07:
**End do**/* for the number of iterations (IT_GA) is fulfilled */- 08:
**Output**the final best schedule and its fitness value.

_{1}, LH

_{2}, …, and LH

_{7}. The details of the proposed seven low-level heuristics are as follows:

- LH
_{1}: - Two-order swap heuristic: randomly select two orders (e.g., ${O}_{2}$ and ${O}_{4}$) in a schedule $\sigma $ and swap the selected two orders, resulting in a new schedule $\sigma \prime $. For example, $\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{4},{O}_{3},{O}_{2},{O}_{5}\right)$.
- LH
_{2}: - One step to the right heuristic: randomly select one order (e.g., ${O}_{2}$) in a schedule $\sigma $, extract order ${O}_{2}$ from its position, move it one position toward the right, and reinsert it to obtain a new schedule ${\sigma}^{\prime}$. For example, $\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{3},{O}_{2},{O}_{4},{O}_{5}\right)$.
- LH
_{3}: - Two steps to the right heuristic: randomly select one order (e.g., ${O}_{3}$) in a schedule $\sigma $, extract order ${O}_{3}$ from its position, move it two positions toward the right, and reinsert it, resulting in a new schedule ${\sigma}^{\prime}$. For example, $\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{2},{O}_{4},{O}_{5},{O}_{3}\right)$.
- LH
_{4}: - One step to the left heuristic: randomly select one order (e.g., ${O}_{4}$) in a schedule $\sigma $, extract job ${O}_{4}$ from its position, move it one position toward the left, and reinsert it, resulting in the new schedule ${\sigma}^{\prime}$. For example,$\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{2},{O}_{4},{O}_{3},{O}_{5}\right)$.
- LH
_{5}: - Two steps to the left heuristic: randomly select one order (e.g., ${O}_{5}$) in a schedule $\sigma $, extract order ${O}_{5}$ from its position, move it two positions toward the left, and reinsert it to obtain a new schedule ${\sigma}^{\prime}$. For example,$\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{2},{O}_{5},{O}_{3},{O}_{4}\right)$.
- LH
_{6}: - Pulling-out and onward-moved reinsertion heuristic: Randomly select two orders (e.g., ${O}_{2}$ and ${O}_{5}$) in a schedule $\sigma $, extract order ${O}_{2}$ (the leftward of the two selected orders) from its position, and onward reinsert it just after ${O}_{5}$ to obtain a new schedule ${\sigma}^{\prime}$. For example, $\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{3},{O}_{4},{O}_{5},{O}_{2}\right)$.
- LH
_{7}: - Pulling-out and backward-moved reinsertion heuristic: randomly select two orders (e.g., ${O}_{2}$ and ${O}_{5}$) in a schedule $\sigma $, extract order ${O}_{5}$ (the rightward of the two selected orders) from its position, and backward reinsert it just before ${O}_{2}$ to obtain a new schedule ${\sigma}^{\prime}$. For example,$\sigma =\left({O}_{1},{O}_{2},{O}_{3},{O}_{4},{O}_{5}\right)$, ${\sigma}^{\prime}=\left({O}_{1},{O}_{5},{O}_{2},{O}_{3},{O}_{4}\right)$.

_{6}and LH

_{7}differ from the other five heuristics, especially when n = 100 or 200.

_{l}; l = 1, 2, …, 7. Assume that ${\pi}_{l}$ is the recorded total frequency of obtaining a superior solution when cyclically executing LH

_{l}. To ensure that all of the seven low-level heuristics in the pool were in the GAHH, we set ${\pi}_{l}=\mathrm{max}\left\{1,{\pi}_{l}\right\}$. The procedures of the GAHH were as follows:

**Steps of genetic algorthim hyper-heuristic**:

- 00:
**Input**Pop, P, ITRN, L_no.- 01:
- Generate a series of Pop initial parents and find their fitness values.
- 02:
**Do**c = 1, ITRN- 03:
- set${f}_{l}=1/7$, l = 1, 2, …, 7.
- 04:
**Do**i = 1, pop /* for each parent ${\sigma}_{i}$- 05:
**Do**k = 1, L_no- 06:
- Select an $L{H}_{l}$ by using the roulette wheel method based on the value
- 07:
- of ${f}_{l}$ to improve ${\sigma}_{i}$ for generating another new schedule ${\sigma}_{t}$.
- 08:
- Replace ${\pi}_{l}={\pi}_{l}+1$ with LH
_{j}if RC(${\mathsf{\sigma}}_{t}$) < RC(${\sigma}_{i}$). - 09:
- Retain each superior current parent ${\sigma}_{i}$
- 10:
**End do**/* for the low-level heuristics */- 11:
**End do**/* i = 1, 2, …, Pop */- 12:
- Update the probabilities $\{{f}_{l}$, l = 1, 2, …, 7} of ${\mathrm{LH}}_{1}$, ${\mathrm{LH}}_{2}$, …, and ${\mathrm{LH}}_{7}$ according
- 13:
- their past records as {${f}_{l}={\pi}_{l}/\sum _{j=1}^{7}{\pi}_{j},l=1,\dots $,7}
- 14:
- Select two parents from Pop populations by using the roulette wheel method and
- 15:
- employ a linear order crossover to reproduce a set of Pop offspring.
- 16:
- For each offspring, generate a random number u (0 < u < 1) if u < P; then, create a
- 17:
- new offspring by inserting a displacement mutation.
- 18:
- Retain the best offspring, and replace the parents of this Pop with their offspring.
- 19:
**End do**/*when the iterative number of high-level cycles (ITRN) is */- 20:
**Output**the final best sequence and its fitness value.

## 6. Tuning Genetic Algorithm Hyper-Heuristic Parameters

_{i}is the objective solution searched using the GAHH, and ${B}_{i}^{*}$ is the optimal objective value obtained using the B&B method for each instance i. With reference to the designs of Leung et al. [10,11,12,13,14,16], Lee [20], Lin et al. [28], and Yang and Yu [33], the weights ${w}_{i}$ were generated from the uniform distribution U (1, 100). The component processing time ${t}_{iv}^{\left(1\right)}$ and ready time ${r}_{i}^{\left(1\right)}$ of an order were generated from the uniform distributions U (1, 100) and U (1, $100\times \mathrm{n}\times \mathsf{\lambda}$); the due dates of an order were generated from the uniform distribution U (TPTbar(1) (1 − τ − ρ/2), TPTbar(1) (1 − τ + ρ/2)) in Scenario 1. In Scenario 2, the component processing time ${t}_{iv}^{\left(2\right)}$ and ready time ${r}_{i}^{\left(2\right)}$ of an order were generated from the uniform distributions U (1, 200) and U (1, $200\times \mathrm{n}\times \mathsf{\lambda}$), and the due dates of an order were generated from the uniform distribution U (TPTbar(2) (1 − τ − ρ/2), TPTbar(2) (1 – τ + ρ/2)), where $TPTbar\left(s\right)=\sum _{v=1}^{m}\sum _{i=1}^{n}{t}_{iv}^{\left(s\right)}/m$; τ and ρ describe the tardiness factor and range of due dates, respectively; and $0<\mathsf{\lambda}<1$ is a controllable parameter. For simplification, we set n = 10, m = 3, τ = 0.5, ρ = 0.5, and $\mathsf{\lambda}=0.3$ and generated 100 problem instances for testing.

## 7. Simulation Study

_{i}of each order was randomly generated from another U (1, 100) independently. In addition, we generated the due dates of the orders from another U (TPTbar(s) (1 − τ − ρ/2), TPTbar(s) (1 – τ + ρ/2)), where $TPTbar\left(s\right)=\sum _{v=1}^{m}\sum _{i=1}^{n}{t}_{iv}^{\left(s\right)}/m$, τ denotes the tardiness factor, and ρ denotes range of due dates. The combinations of (τ, ρ) included (0.25, 0.25), (0.25, 0.5), (0.5, 0.75), (0.5, 0.5), (0.5, 0.25), and (0.25, 0.75). With reference to the design of Reeves [50], we generated the ready times from U (1, 100nλ) for Scenario 1 and U (1, 200nλ) for Scenario 2, respectively, where λ is the control variable. The value of λ was set to 0.1, 0.3, and 0.5. Herein, the two parts of the simulation study were designed to address small- and large-sized orders.

#### 7.1. Results Obtained for Small-Sized Orders

^{8}.

_{3}was called most frequently and was typically followed by HL

_{1}. However, HL

_{4}, HL

_{5}, HL

_{6}, and HL

_{7}were rarely called.

#### 7.2. Results for Large-Sized Orders

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Node | CPU_Time | ||||
---|---|---|---|---|---|

n | m | Mean | Max | Mean | Max |

9 | 2 | 102,150 | 35,6594 | 0.56 | 1.25 |

3 | 108,842 | 35,9043 | 0.68 | 1.48 | |

4 | 116,378 | 386,330 | 0.82 | 1.79 | |

11 | 2 | 606,5047 | 25,997,813 | 57.42 | 193.89 |

3 | 6,701,240 | 27,681,786 | 74.84 | 235.82 | |

4 | 6,952,756 | 27,613,651 | 90.24 | 272.42 | |

Mean | 3,341,069 | 1,373,2536 | 37.43 | 117.78 | |

n | $\mathsf{\lambda}$ | Mean | Max | Mean | Max |

9 | 0.1 | 63,883 | 15,8272 | 0.52 | 1.11 |

0.3 | 84,005 | 374,265 | 0.61 | 1.56 | |

0.5 | 179,482 | 569,430 | 0.93 | 1.86 | |

11 | 0.1 | 316,9517 | 9,901,037 | 47.60 | 131.62 |

0.3 | 4,378,853 | 21,526,663 | 61.14 | 236.57 | |

0.5 | 12,170,673 | 49,865,551 | 113.76 | 333.93 | |

Mean | 3,341,069 | 13,732,536 | 37.43 | 117.78 | |

n | $\tau $ | Mean | Max | Mean | Max |

9 | 0.25 | 73,284 | 287,313 | 0.55 | 1.40 |

0.50 | 144,963 | 447,331 | 0.82 | 1.62 | |

11 | 0.25 | 3,734,194 | 17,472,356 | 53.12 | 200.44 |

0.50 | 9,411,835 | 36,723,144 | 95.21 | 267.64 | |

Mean | 3,341,069 | 13,732,536 | 37.43 | 117.78 | |

n | $\rho $ | Mean | Max | Mean | Max |

9 | 0.25 | 120,476 | 374,104 | 0.71 | 1.49 |

0.50 | 105,302 | 379,570 | 0.67 | 1.51 | |

0.75 | 101,593 | 348,294 | 0.68 | 1.53 | |

11 | 0.25 | 7,668,642 | 29,579,795 | 80.48 | 255.49 |

0.50 | 6,353,625 | 26,905,933 | 71.67 | 225.25 | |

0.75 | 5,696,776 | 24,807,523 | 70.35 | 221.39 | |

Mean | 3,341,069 | 13,732,536 | 37.43 | 117.78 |

Moores_pi_M | Moores_pi_m | Moores_pi_Mean | GA | GAHH | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | m | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

9 | 2 | 63.19 | 684.27 | 63.39 | 851.01 | 65.54 | 650.55 | 18.47 | 278.18 | 0.77 | 26.61 |

3 | 47.03 | 335.87 | 45.22 | 355.35 | 47.88 | 284.18 | 15.21 | 130.37 | 0.35 | 14.42 | |

4 | 45.80 | 319.15 | 43.15 | 237.90 | 45.31 | 307.29 | 13.79 | 92.95 | 0.32 | 14.15 | |

11 | 2 | 81.46 | 478.10 | 78.19 | 448.14 | 83.70 | 448.73 | 26.99 | 226.09 | 2.45 | 69.87 |

3 | 68.64 | 467.13 | 64.53 | 460.21 | 70.29 | 473.65 | 22.55 | 121.51 | 1.89 | 33.21 | |

4 | 66.44 | 432.34 | 62.17 | 326.86 | 66.07 | 340.21 | 21.62 | 125.73 | 1.55 | 27.67 | |

Mean | 62.09 | 452.81 | 59.44 | 446.58 | 63.13 | 417.44 | 19.77 | 162.47 | 1.22 | 30.99 | |

$n$ | $\lambda $ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

9 | 0.1 | 107.49 | 1009.37 | 103.09 | 1121.00 | 108.48 | 927.90 | 27.76 | 348.97 | 1.10 | 34.63 |

0.3 | 38.61 | 220.82 | 37.83 | 248.44 | 39.25 | 234.59 | 14.14 | 106.10 | 0.26 | 16.50 | |

0.5 | 9.92 | 75.27 | 10.84 | 74.82 | 11.01 | 79.53 | 5.57 | 46.44 | 0.07 | 4.04 | |

11 | 0.1 | 148.71 | 990.73 | 134.73 | 822.99 | 148.81 | 891.54 | 41.34 | 303.50 | 4.46 | 98.83 |

0.3 | 53.71 | 301.37 | 54.70 | 323.03 | 55.24 | 282.37 | 21.32 | 116.75 | 1.12 | 18.92 | |

0.5 | 14.12 | 85.47 | 15.46 | 89.19 | 16.01 | 88.67 | 8.50 | 53.08 | 0.30 | 13.01 | |

Mean | 62.09 | 447.17 | 59.44 | 446.58 | 63.13 | 417.43 | 19.77 | 162.47 | 1.22 | 30.99 | |

n | τ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

9 | 0.25 | 84.43 | 796.07 | 82.39 | 871.91 | 85.42 | 727.47 | 24.21 | 271.87 | 0.75 | 26.91 |

0.50 | 19.58 | 100.94 | 18.78 | 90.93 | 20.40 | 100.54 | 7.44 | 62.47 | 0.20 | 9.87 | |

11 | 0.25 | 116.61 | 813.22 | 110.02 | 722.44 | 117.08 | 735.28 | 35.40 | 260.65 | 3.16 | 70.70 |

0.50 | 27.75 | 105.17 | 26.57 | 101.04 | 29.63 | 106.44 | 12.04 | 54.90 | 0.76 | 16.47 | |

Mean | 62.09 | 453.85 | 59.44 | 446.58 | 63.13 | 417.43 | 19.77 | 162.47 | 1.22 | 30.99 | |

$n$ | $\rho $ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

9 | 0.25 | 38.27 | 233.89 | 37.68 | 247.73 | 38.07 | 212.74 | 11.73 | 75.19 | 0.47 | 19.79 |

0.50 | 53.90 | 296.54 | 51.15 | 289.09 | 54.89 | 294.14 | 16.97 | 122.25 | 0.51 | 18.54 | |

0.75 | 63.85 | 816.19 | 62.93 | 907.44 | 65.77 | 735.13 | 18.77 | 304.06 | 0.45 | 16.85 | |

11 | 0.25 | 51.33 | 272.43 | 52.18 | 292.91 | 52.07 | 280.87 | 17.25 | 97.82 | 1.01 | 21.54 |

0.50 | 75.80 | 463.10 | 72.77 | 441.39 | 77.43 | 416.31 | 25.34 | 169.34 | 1.81 | 31.39 | |

0.75 | 89.40 | 642.04 | 79.93 | 500.92 | 90.56 | 565.40 | 28.57 | 206.17 | 3.07 | 77.82 | |

Mean | 62.09 | 454.03 | 59.44 | 446.58 | 63.13 | 417.43 | 19.77 | 162.47 | 1.22 | 30.99 |

Small Job Size n | Large Job Size n | |||
---|---|---|---|---|

Pairwise Comparison between Algorithm | |Pairwise Rank-Sum Difference| | Difference > 64.4 * | |Pairwise Rank-Sum Difference| | Difference > 64.4 * |

Moores_pi_M vs. Moores_pi_m | |401.0–414.0| | NO | |359.0–496.0| | YES |

Moores_pi_M vs. Moores_pi_mean | |401.0–472.0| | YES | |359.0–402.0| | NO |

Moores_pi_M vs. GA | |401.0–223.0| | YES | |359.0–255.0| | YES |

Moores_pi_M vs. GAHH | |401.0–110.0| | YES | |359.0–108.0| | YES |

Moores_pi_m vs. Moores_pi_mean | |414.0–472.0| | NO | |496.0–402.0| | YES |

Moores_pi_m vs. GA | |414.0–223.0| | YES | |496.0–255.0| | YES |

Moores_pi_m vs. GAHH | |414.0–110.0| | YES | |496.0–108.0| | YES |

Moores_pi_mean vs. GA | |472.0–223.0| | YES | |402.0–255.0| | YES |

Moores_pi_mean vs. GAHH | |472.0–110.0| | YES | |402.0–108.0| | YES |

GA vs. GAHH | |223.0–110.0| | YES | |255.0–108.0| | YES |

Moores_pi_M | Moores_pi_m | Moores_pi_Mean | GA | GAHH | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | m | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 5 | 117.98 | 190.76 | 134.45 | 225.48 | 115.52 | 194.47 | 79.90 | 143.19 | 0.00 | 0.00 |

10 | 111.11 | 175.04 | 127.34 | 213.33 | 108.83 | 174.50 | 74.67 | 126.33 | 0.00 | 0.00 | |

15 | 107.61 | 196.72 | 123.20 | 207.02 | 105.71 | 201.19 | 72.78 | 130.91 | 0.00 | 0.00 | |

200 | 5 | 105.19 | 154.86 | 127.30 | 180.75 | 105.31 | 156.38 | 81.17 | 123.84 | 0.00 | 0.00 |

10 | 100.35 | 146.16 | 120.94 | 172.32 | 99.54 | 151.17 | 77.28 | 114.04 | 0.00 | 0.00 | |

15 | 97.85 | 144.41 | 118.07 | 166.99 | 97.77 | 139.84 | 74.94 | 109.50 | 0.00 | 0.00 | |

Mean | 106.68 | 167.99 | 125.22 | 194.32 | 105.45 | 169.59 | 76.79 | 124.64 | 0.00 | 0.00 | |

n | λ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.1 | 184.26 | 323.65 | 235.51 | 406.75 | 174.74 | 320.61 | 114.44 | 212.68 | 0.00 | 0.00 |

0.3 | 109.43 | 168.40 | 103.81 | 162.00 | 109.91 | 175.22 | 71.13 | 118.47 | 0.00 | 0.00 | |

0.5 | 43.02 | 70.46 | 45.67 | 77.09 | 45.42 | 74.33 | 41.78 | 69.28 | 0.00 | 0.00 | |

200 | 0.1 | 161.27 | 245.04 | 210.16 | 302.04 | 158.99 | 243.33 | 124.74 | 189.48 | 0.00 | 0.00 |

0.3 | 101.68 | 139.58 | 110.93 | 150.51 | 101.66 | 141.29 | 68.45 | 98.37 | 0.00 | 0.00 | |

0.5 | 40.45 | 60.81 | 45.22 | 67.52 | 41.97 | 62.76 | 40.21 | 59.54 | 0.00 | 0.00 | |

Mean | 106.68 | 167.99 | 125.22 | 194.32 | 105.45 | 169.59 | 76.79 | 124.64 | 0.00 | 0.00 | |

n | τ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.25 | 158.29 | 274.79 | 190.88 | 332.02 | 155.80 | 284.64 | 106.99 | 195.69 | 0.00 | 0.00 |

0.50 | 66.18 | 100.22 | 65.78 | 98.53 | 64.24 | 95.47 | 44.58 | 71.27 | 0.00 | 0.00 | |

200 | 0.25 | 139.15 | 209.37 | 170.33 | 245.76 | 139.91 | 212.30 | 110.48 | 167.20 | 0.00 | 0.00 |

0.50 | 63.12 | 87.58 | 73.87 | 100.95 | 61.83 | 85.96 | 45.12 | 64.39 | 0.00 | 0.00 | |

Mean | 106.68 | 167.99 | 125.22 | 194.32 | 105.45 | 169.59 | 76.79 | 124.64 | 0.00 | 0.00 | |

n | ρ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.25 | 93.79 | 156.60 | 108.21 | 173.59 | 94.62 | 160.03 | 70.15 | 119.17 | 0.00 | 0.00 |

0.50 | 117.33 | 211.28 | 134.50 | 232.41 | 115.43 | 217.54 | 79.01 | 145.33 | 0.00 | 0.00 | |

0.75 | 125.60 | 194.63 | 142.28 | 239.83 | 120.02 | 192.59 | 78.19 | 135.93 | 0.00 | 0.00 | |

200 | 0.25 | 86.77 | 128.64 | 106.01 | 146.45 | 89.63 | 129.30 | 73.60 | 107.26 | 0.00 | 0.00 |

0.50 | 105.66 | 155.60 | 126.86 | 181.39 | 105.13 | 156.28 | 81.10 | 122.02 | 0.00 | 0.00 | |

0.75 | 110.97 | 161.18 | 133.44 | 192.22 | 107.86 | 161.80 | 78.70 | 118.10 | 0.00 | 0.00 | |

Mean | 106.68 | 167.99 | 125.22 | 194.32 | 105.45 | 169.59 | 76.79 | 124.64 | 0.00 | 0.00 |

Moore_pi_M | Moore_pi_m | Moore_pi_Mean | GA | GAHH | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | m | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 5 | 0.04 | 0.05 | 0.04 | 0.05 | 0.04 | 0.05 | 0.01 | 0.02 | 0.51 | 0.58 |

10 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.91 | 1.01 | |

15 | 0.10 | 0.13 | 0.10 | 0.13 | 0.10 | 0.13 | 0.01 | 0.04 | 1.33 | 1.52 | |

200 | 5 | 0.28 | 0.35 | 0.28 | 0.33 | 0.29 | 0.35 | 0.01 | 0.03 | 1.78 | 1.98 |

10 | 0.50 | 0.67 | 0.49 | 0.57 | 0.49 | 0.56 | 0.02 | 0.05 | 3.11 | 3.47 | |

15 | 0.90 | 1.13 | 0.83 | 1.05 | 0.83 | 1.07 | 0.03 | 0.08 | 5.21 | 6.46 | |

mean | 0.32 | 0.40 | 0.30 | 0.37 | 0.30 | 0.38 | 0.02 | 0.04 | 2.14 | 2.50 | |

$n$ | $\lambda $ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.1 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.90 | 1.01 |

0.3 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.89 | 1.01 | |

0.5 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.02 | 0.91 | 1.02 | |

200 | 0.1 | 0.51 | 0.62 | 0.49 | 0.59 | 0.51 | 0.61 | 0.03 | 0.06 | 3.22 | 3.65 |

0.3 | 0.54 | 0.69 | 0.52 | 0.64 | 0.53 | 0.66 | 0.03 | 0.07 | 3.42 | 4.26 | |

0.5 | 0.61 | 0.78 | 0.56 | 0.68 | 0.55 | 0.67 | 0.01 | 0.03 | 3.28 | 3.80 | |

mean | 0.31 | 0.39 | 0.30 | 0.36 | 0.30 | 0.37 | 0.02 | 0.04 | 2.10 | 2.46 | |

n | τ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.25 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.93 | 1.05 |

0.50 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.91 | 1.02 | |

200 | 0.25 | 0.55 | 0.69 | 0.53 | 0.64 | 0.53 | 0.64 | 0.03 | 0.06 | 3.39 | 4.00 |

0.50 | 0.58 | 0.75 | 0.54 | 0.66 | 0.54 | 0.68 | 0.02 | 0.05 | 3.35 | 3.94 | |

mean | 0.32 | 0.41 | 0.30 | 0.37 | 0.30 | 0.38 | 0.02 | 0.04 | 2.15 | 2.50 | |

$n$ | $\rho $ | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max |

100 | 0.25 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.92 | 1.05 |

0.50 | 0.07 | 0.10 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.91 | 1.02 | |

0.75 | 0.07 | 0.09 | 0.07 | 0.09 | 0.07 | 0.09 | 0.01 | 0.03 | 0.93 | 1.05 | |

200 | 0.25 | 0.56 | 0.70 | 0.52 | 0.63 | 0.53 | 0.65 | 0.02 | 0.05 | 3.34 | 3.97 |

0.50 | 0.55 | 0.70 | 0.53 | 0.63 | 0.53 | 0.66 | 0.02 | 0.06 | 3.38 | 3.99 | |

0.75 | 0.58 | 0.75 | 0.54 | 0.68 | 0.55 | 0.67 | 0.03 | 0.06 | 3.38 | 3.96 | |

mean | 0.32 | 0.41 | 0.30 | 0.37 | 0.30 | 0.38 | 0.02 | 0.04 | 2.14 | 2.51 |

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## Share and Cite

**MDPI and ACS Style**

Li, L.-Y.; Xu, J.-Y.; Cheng, S.-R.; Zhang, X.; Lin, W.-C.; Lin, J.-C.; Wu, Z.-L.; Wu, C.-C. A Genetic Hyper-Heuristic for an Order Scheduling Problem with Two Scenario-Dependent Parameters in a Parallel-Machine Environment. *Mathematics* **2022**, *10*, 4146.
https://doi.org/10.3390/math10214146

**AMA Style**

Li L-Y, Xu J-Y, Cheng S-R, Zhang X, Lin W-C, Lin J-C, Wu Z-L, Wu C-C. A Genetic Hyper-Heuristic for an Order Scheduling Problem with Two Scenario-Dependent Parameters in a Parallel-Machine Environment. *Mathematics*. 2022; 10(21):4146.
https://doi.org/10.3390/math10214146

**Chicago/Turabian Style**

Li, Lung-Yu, Jian-You Xu, Shuenn-Ren Cheng, Xingong Zhang, Win-Chin Lin, Jia-Cheng Lin, Zong-Lin Wu, and Chin-Chia Wu. 2022. "A Genetic Hyper-Heuristic for an Order Scheduling Problem with Two Scenario-Dependent Parameters in a Parallel-Machine Environment" *Mathematics* 10, no. 21: 4146.
https://doi.org/10.3390/math10214146