Study on Scheduling Problems with Learning Effects and Past Sequence Delivery Times

Abstract

In this paper, we study a single-machine green scheduling problem with learning effects and past-sequence-dependent delivery times. The problem can be properly applied to tackle green manufacturing where production and delivery time are variable and highly subject to process-reengineering. Our goal is to determine the optimal sequence such that total weighted completion time and maximum tardiness are minimized. For the general case, we provide the analysis procedure of lower bound, and also propose the heuristic and branch-and-bound algorithms. Furthermore, computational experiments are conducted to demonstrate the effectiveness of our algorithms.

1. Introduction

For general scheduling problems, the processing times of the jobs are often assumed to be fixed. However, in actual production, due to the continuous repeated processing of a product by machines or workers, the processing time decreases with the increase in number of products and tends to be stable. This phenomenon is called the scheduling problem with “learning effect” (such as Biskup [1,2]; Cheng et al. [3], Cheng et al. [4]; Lu et al. [5], Lu et al. [6], Azzouz et al. [7], Geng et al. [8], Liang et al. [9], Wang et al. [10]). More recently, Liu and Jiang [11] addressed single-machine resource allocation scheduling problems with learning effects, where learning effects mean job-dependent position-based learning effects. Additionally, Muştu and Eren [12] addressed the optimization of a single-machine scheduling cases with setup times under an extension of the general learning and forgetting effects, and the objective function is to minimize the maximum completion time. They proposed an integer non-linear programming model and a dynamic programming model, which can be executed in pseudo-polynomial time. While Lin [13] addressed the parallel-machine scheduling with controllable processing times and job-dependent learning effects, the objective function is to minimize the weighted sum of total completion time, total compression cost, and total load. They showed that the problem can be solved in O($n^{m+2}$) time, where n (resp. m) is the number of jobs (resp. machines). In addition, Jiang et al. [14] considered single-machine scheduling problems with general truncated sum-of-actual processing-time-based learning effect. They showed polynomial solvable cases and approximation algorithms for these problems. Furthermore, Sun et al. [15] considered the flow shop problem of minimising the total weighted completion time in which the processing times of jobs are variable according to general position weighted learning effects. They proposed some heuristics and analysed the worst-case error bounds. While Zhao [16] addressed resource allocation flowshop scheduling with learning effect and slack due window assignment. He combined learning effect and controllable processing times, in which the flowshop has a two-machine no-wait setup, and provided a bi-criteria analysis for the scheduling and resource consumption costs. Subsequently, Zhao [17] studied scheduling problems with general truncated learning effects and past-sequence-dependent setup times on a single-machine. They demonstrated that some regular objective function minimizations can be solved in polynomial time. Yan et al. [18] illustrated the single-machine resource allocation scheduling problem with learning effects and group technology. For some special cases of the problem, they can be solved in polynomial time. For the general case, they proposed branch-and-bound algorithms.

Electronic components manufacturing is a high energy consuming industry. In practical application, an electronic component waiting to be processed may be exposed to a certain electromagnetic field and is required to neutralize the effect of electromagnetism. In this case, it needs extra time to eliminate adverse effects. As for green manufacturing environment, proces-reengineering is often introduced to reduce energy consumption and hence leads to the production as well as extra delivery time subject to learning effects. This extra time is modeled as past-sequence-dependent delivery times and is performed immediately after the component has been processed on the machine to provide a delivery to customer. In recent years, the scheduling problems with delivery times have also attracted widespread attention (see Koulamas and Kyparisis [19], Mateo et al. [20], Wang et al. [21], Pan et al. [22]). In addition, Rostami et al. [23] examined the single-machine scheduling problem of minimizing the total weighted completion and batch delivery times, and proposed a brand-and-bound algorithm as well as heuristic algorithm to specifically solve the studied problem,. Their experimental results proved that the proposed heuristic algorithm is more effective. Subsequently, Qian and Zhan [24] illustrated a single-machine scheduling problem with past-sequence-dependent delivery times and the truncated sum-of-processing-times-based learning effect. The goal is to minimize the total costs that comprise the number of early jobs, the number of tardy jobs and due date. Under the common due date, slack due date and different due date, they demonstrated that these problems are polynomial time solvable, while Qian and Han [25] studied the due date assignment with the delivery time and deteriorating jobs of scheduling problem, the goal is to minimize the total costs, they proved that these problems are polynomial time solvable and proposed the corresponding algorithms. Moreover, Toksari et al. [26] illustrated both exponential past-sequence-dependent delivery times and learning effect where the job processing time is a function based on the sum of the logarithm of processing times of jobs already processed and showed that the single-machine scheduling problems to minimize makespan, total weighted completion time, total completion time and maximum tardiness have polynomial time solutions. Wang et al. [27] addressed single-machine due-date assignment scheduling with delivery times and truncated learning effect. Ren et al. [28] studied single-machine scheduling with exponential delivery times and learning effects. Recently, Lei et al. [29] illustrated the scheduling problems where each job has a past-sequence-dependent delivery time, and they proved that the makespan and total completion time minimizations can be solved in polynomial time. Under the agreeable due date condition and agreeable weight condition, they showed that the maximum tardiness and total weighted completion time minimizations remain polynomially solvable. Hence, in this paper, we continue the work of Lei et al. [29] and the main contributions are summarized as follows:

•

For the general case of the maximum tardiness and total weighted completion time minimizations, we provide the procedure of analyzing the lower bound of the total weighted completion time and maximum tardiness;

•

Furthermore, we give the heuristic, simulated annealing, branch-and-bound algorithms and conduct numerical experiments.

The structure of this paper is arranged as follows. The scheduling model is presented in Section 2. In Section 3, we present the branch-and-bound algorithm and heuristic algorithms. Concluding remarks are provided in Section 4.

2. Problem Description

We formally state the problem as follows: There is a set of independent jobs J to be processed on a single machine. The machine can handle at most one job at a time and job preemption is not allowed. All the jobs are available for processing at time 0. The actual processing time of job j scheduled at position r is denoted by $p_{j\left[r\right]},j=1,2,...,n$, and $p_{j\left[r\right]}$ is given as follows:

$$p_{j\left[r\right]}=p_j{\left(1+\sum _{l=1}^{r-1}ln{p}_{\left[l\right]}\right)}^a(j,r=1,2,...,n),$$

(1)

where $p_j$ is the normal processing time of job j, and $a(a<0)$ is learning effect. The delivery time after the processing of job $p_{j\left[r\right]}$ is denoted by $q_{j\left[r\right]},j=1,2,...,n$. As in Lei et al. [29], the $q_{j\left[r\right]}$ is given as follows:

$$q_{j\left[r\right]}=\gamma {\left(1+\sum _{l=1}^{r-1}{p}_{\left[l\right]}\right)}^b(j,r=1,2,...,n),$$

(2)

where $0\le \gamma \le 1$ is a normalizing constant, and $b(b>0)$ is delivery times index. We study each job has a past-sequence-dependent delivery time to reduce the inventory costs for enterprises in this paper. Our goal is to determine the optimal schedule such that the maximum tardiness (i.e., $T_{max}=max\left\{T_j\right|j=1,2,\dots ,n\}$, where $T_j=max\{0,C_j-d_j\}$, $C_j$ is the completion time of job j and $d_j$ is the due date of job j) and the total weighted completion time (i.e., ${\sum }_{j=1}^n{w}_j{C}_j$, where $w_j$ is the weight of job j) are to be minimized. By the three-field notation of Graham et al. [30], the problem can be denoted as

$$1\left|LE,Q_{psd}\right|Z,$$

(3)

where the first field 1 denotes a single-machine, the second field $LE$ is learning effects (1) and $Q_{psd}$ denotes delivery times (2), the third field $Z\in \{{\sum }_{j=1}^n{w}_j{C}_j,T_{max}\}$ refers to the optimized criterion.

3. Solution Algorithms

To analyze the lower bound of ${\sum }_{j=1}^n{w}_j{C}_j$$\left(T_{max}\right)$, we give the following lemmas as the preliminaries.

 Lemma 1 

(The $SPT$ rule, Lei et al. [29]). For problem $1|LE,Q_{psd}|{\sum }_{j=1}^n{C}_j\left(C_{max}\right)$, an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of $p_j$.

 Lemma 2 

(The $WSPT$ rule, Lei et al. [29]). For problem $1|LE,Q_{psd}|{\sum }_{j=1}^n{w}_j{C}_j$, if $p_i\le p_j$ implies $w_i\ge w_j$ for all jobs $J_i$ and $J_j$, the optimal schedule can be obtained by sequencing the jobs in non-decreasing order of $p_j/w_j$.

 Lemma 3 

(The $EDD$ rule, Lei et al. [29]). For problem $1|LE,Q_{psd}|T_{max}$, if $p_i\le p_j$ implies $d_i\le d_j$ for all jobs $J_i$ and $J_i$, the optimal schedule can be obtained by sequencing the jobs in non-decreasing order of $d_j$.

To solve the problem $1|LE,Q_{psd}|Z$, a branch-and-bound (B&B) algorithm and heuristic algorithms are presented as follows.

3.1. Lower Bounds

3.1.1. For the Total Weighted Completion Time (i.e., ${\sum }_{j=1}^n{w}_j{C}_j$)

We assume that $\pi =[\overbrace{{\pi }_{PS}},\overbrace{{\pi }_{US}}]$ be a sequence, where $\overbrace{{\pi }_{PS}}$ ($\overbrace{{\pi }_{US}}$) is the scheduled (unscheduled) part, and there are r jobs already scheduled. So, for the $r+1$th ($r+j$th) job, we have

$$\begin{array}{ccc}\hfill C_{[r+1]}\left(\overbrace{{\pi }_{US}}\right)& =& \sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a+p_{[r+1]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}\right)}^a+\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}\right)}^b\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill & & C_{[r+j]}\left(\overbrace{{\pi }_{US}}\right)\hfill \\ & =& \sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a+\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a\hfill \\ & & +\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b,\hfill \end{array}$$

(4)

where $1\le j\le n-r.$ Therefore,

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^n{w}_j{C}_j\left(\pi \right)\hfill \\ & =& \sum _{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+\sum _{j=r+1}^n{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{US}}\right)\hfill \\ & =& \sum _{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+\sum _{j=r+1}^n{w}_{\left[j\right]}\left[\sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a\right]\hfill \\ & & +\sum _{j=1}^{n-r}{w}_{[r+j]}\left[\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a\right]\hfill \\ & & +\gamma \sum _{j=1}^{n-r}{w}_{[r+j]}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b.\hfill \end{array}$$

(5)

We assume $\widehat{w_{min}}=min\left\{w_{\left[j\right]}\right|j=r+1,r+2,\dots ,n\}$, hence

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^n{w}_j{C}_j\left(\pi \right)\hfill \\ & \ge & \sum _{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+\sum _{j=r+1}^n{w}_{\left[j\right]}\left[\sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a\right]\hfill \\ & & +\widehat{w_{min}}\sum _{j=1}^{n-r}\left[\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a\right]\hfill \\ & & +\gamma \widehat{w_{min}}\sum _{j=1}^{n-r}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b.\hfill \end{array}$$

Obviously, ${\sum }_{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+{\sum }_{j=r+1}^n{w}_{\left[j\right]}\left[{\sum }_{h=1}^r{p}_{\left[h\right]}{\left(1+{\sum }_{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a\right]$ is a fixed constant, and

$$\begin{array}{ccc}\hfill & & \widehat{w_{min}}\sum _{j=1}^{n-r}\left[\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a\right]+\gamma \widehat{w_{min}}\sum _{j=1}^{n-r}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b\hfill \end{array}$$

can be minimized by Lemma 1. Hence, the first lower bound of $1|LE,Q_{psd}|{\sum }_{j=1}^n{w}_j{C}_j$ is:

$$\begin{array}{ccc}\hfill \widehat{L{B}_i\left(TW\right)}& =& \sum _{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+\sum _{j=r+1}^n{w}_{\left[j\right]}\left[\sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a\right]\hfill \\ & & +\widehat{w_{min}}\sum _{j=1}^{n-r}\left[\sum _{h=1}^j{p}_{<r+h>}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{<r+l>}\right)}^a\right]\hfill \\ & & +\gamma \widehat{w_{min}}\sum _{j=1}^{n-r}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{<r+h>}\right)}^b,\hfill \end{array}$$

(6)

where $p_{<r+1>}\le p_{<r+2>}\le \dots \le p_{<n>}$.

Similarly, if $p_i\le p_j$ implies $w_i\ge w_j$ for all the jobs $J_i$ and $J_j$,

$$\begin{array}{ccc}\hfill & & \sum _{j=1}^{n-r}{w}_{[r+j]}\left[\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a\right]\hfill \\ & & +\gamma \sum _{j=1}^{n-r}{w}_{[r+j]}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b\hfill \end{array}$$

is minimized by Lemma 2, we have

$$\begin{array}{ccc}\hfill \widehat{L{B}_{ii}\left(TW\right)}& =& \sum _{j=1}^r{w}_{\left[j\right]}{C}_{\left[j\right]}\left(\overbrace{{\pi }_{PS}}\right)+\sum _{j=r+1}^n{w}_{\left[j\right]}\left[\sum _{h=1}^r{p}_{\left[h\right]}{\left(1+\sum _{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a\right]\hfill \\ & & +\sum _{j=1}^{n-r}{w}_{(r+j)}\left[\sum _{h=1}^j{p}_{<r+h>}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{<r+l>}\right)}^a\right]\hfill \\ & & +\gamma \sum _{j=1}^{n-r}{w}_{(r+j)}{\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{<r+h>}\right)}^b,\hfill \end{array}$$

(7)

where $w_{(r+1)}\ge w_{(r+2)}\ge \dots \ge w_{\left(n\right)}$ and $p_{<r+1>}\le p_{<r+2>}\le \dots \le p_{<n>}$ (note that $w_{\left(j\right)}$ and $p_{<j>}$ do not necessarily correspond to the same job)).

To make the lower bound tighter by both $\widehat{L{B}_i\left(TW\right)}$ and $\widehat{L{B}_{ii}\left(TW\right)}$, we have

$$\widehat{LB\left(TW\right)}=max\{\widehat{L{B}_i\left(TW\right)},\widehat{L{B}_{ii}\left(TW\right)}\}.$$

(8)

3.1.2. For the Maximum Tardiness (i.e., $T_{max}=max\left\{T_j\right|j=1,2,\dots ,n\}$)

Let $D={\sum }_{h=1}^r{p}_{\left[h\right]}{\left(1+{\sum }_{l=1}^{h-1}ln{p}_{\left[l\right]}\right)}^a$ and $\widehat{d_{max}}=max\left\{d_{\left[j\right]}\right|j=r+1,r+2,\dots ,n\}$, from Equation (4), there is

$$\begin{array}{ccc}\hfill & & T_{[r+j]}\left(\pi \right)\hfill \\ & =& max\left\{D+\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a+\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b-d_{[r+j]},0\right\}\hfill \\ & \ge & max\left\{D+\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a+\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b-\widehat{d_{max}},0\right\},\hfill \end{array}$$

(9)

where $1\le j\le n-r.$

It is noticed that

$$\begin{array}{ccc}\hfill & & D+\sum _{h=1}^j{p}_{[r+h]}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{[r+l]}\right)}^a+\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{[r+h]}\right)}^b\hfill \end{array}$$

can be minimized by Lemma 1 of unscheduled jobs. Therefore, we obtain the first lower bound:

$$\begin{array}{ccc}\hfill \widehat{L{B}_i\left(T_{max}\right)}& =& max\{T_{\left[1\right]},T_{\left[2\right]},\dots ,T_{\left[r\right]},D+\sum _{h=1}^{n-r}{p}_{<r+h>}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{<r+l>}\right)}^a\hfill \\ & & +\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{n-r-1}{p}_{<r+h>}\right)}^b-\widehat{d_{max}}\},\hfill \end{array}$$

(10)

where $p_{<r+1>}\le p_{<r+2>}\le \dots \le p_{<n>}$.

Similarly, if $p_j$ and $d_j(j=r+1,r+2,\dots ,n)$ are agreeable (i.e., Lemma 3), from Equation (9), the second lower bound of the problem $1|LE,Q_{psd}|T_{max}$ is:

$$\begin{array}{ccc}\hfill \widehat{L{B}_{ii}\left(T_{max}\right)}& =& max\{T_{\left[1\right]},T_{\left[2\right]},\dots ,T_{\left[r\right]},D+\sum _{h=1}^j{p}_{<r+h>}{\left(1+\sum _{l=1}^{r}ln{p}_{\left[l\right]}+\sum _{l=1}^{h-1}ln{p}_{<r+l>}\right)}^a\hfill \\ & & +\gamma {\left(1+\sum _{l=1}^r{p}_{\left[l\right]}+\sum _{h=1}^{j-1}{p}_{<r+h>}\right)}^b-\widehat{d_{<<r+j>>}}|j=1,2,\dots ,n-r\},\hfill \end{array}$$

(11)

where $d_{<<r+1>>}\le d_{<<r+2>>}\le \dots \le d_{<<n>>}$ and $p_{<r+1>}\le p_{<r+2>}\le \dots \le p_{<n>}$ (note that $d_{<<j>>}$ and $p_{<j>}$ do not necessarily correspond to the same job).

To make the lower bound tighter, combining $\widehat{L{B}_i\left(T_{max}\right)}$ and $\widehat{L{B}_{ii}\left(T_{max}\right)}$, the lower bound of $1|LE,Q_{psd}|T_{max}$ is

$$\widehat{LB\left(T_{max}\right)}=max\left\{\widehat{L{B}_i\left(T_{max}\right)},\widehat{L{B}_{ii}\left(T_{max}\right)}\right\}.$$

(12)

3.2. Algorithms

The NEH algorithm is a high-performing heuristic for solving the flow shop problem, hence, the NEH heuristic (Nawaz et al. [31]) can be adopted for the the general problems $1|LE,Q_{psd}|{\sum }_{j=1}^n{w}_j{C}_j$ and $1|LE,Q_{psd}|T_{max}$, where NEH with an initial solution sorted by the WSPT/EDD rule (WSPT for Min ${\sum }_{j=1}^n{w}_j{C}_j$, and EDD for Min $T_{max}$). Therefore, the pseudo code of the NEH (WSPT) algorithm is summarized as follows (i.e., Algorithm 1). Simulated annealing (SA) is a global optimization algorithm whose basic idea is to avoid falling into a local optimum solution by accepting bad solutions with a certain probability. Therefore, the SA is also a solution to solve $1|LE,Q_{psd}|{\sum }_{j=1}^n{w}_j{C}_j$ and $1|LE,Q_{psd}|T_{max}$. The pseudo code of the SA is summarized as follows (i.e., Algorithm 2). Branch-and-bound (B&B) algorithm is a type of algorithm for solving optimization problems. It improves the efficiency of the algorithm by constantly breaking the problem down into smaller subproblems and setting bounds for each subproblem to prune search paths that are unlikely to be optimal solutions. The pseudo code of the B&B algorithm is summarized as follows (i.e., Algorithm 3).

Algorithm 1: Heuristic

Algorithm 2: Simulated Annealing

Algorithm 3: B&B

3.3. Computational Experiments

In order to assess the efficiency of the proposed heuristic and B&B algorithms, we write code in C++ and run on a PC with 4 GB of RAM on a Windows 10 OS. Processing time $p_j$ is generated from a uniform discrete distribution with an interval [5, 20]; the weight $w_j$ is uniformly distributed over [1, 10]; the due dates $d_j$ is generated from a uniform discrete distribution with an interval [1, $\frac{1}{2}{C}_{max}$] (where $C_{max}$ can be obtained by the SPT rule); The index of delivery times $b=1.1,1.3,1.5$ and the learning index $a=-0.35,-0.3,-0.25$. For each parameters combination (n, $a,b$), 20 randomly generated instances were evaluated.

3.3.1. Small-Sized Instances

For small-scale examples ($n=$ 8, 9, 10, 11, 12), it is defined that “CPU time (s)” as the running time of the WSPT, EDD, NEH, SA, B&B algorithm, time unit is second, and the error of the solution produced by heuristics was calculated as $\frac{\sum w_j{C}_j\left(G\right)}{\sum w_j{C}_j\left(G^{*}\right)}\left(\frac{T_{max}\left(G\right)}{T_{max}\left(G^{*}\right)}\right)$ for ${\sum }_{j=1}^n{w}_j{C}_j$($T_{max}$), respectively, where G is a schedule obtained by the WSPT(EDD), NEH, SA and the optimal schedule $G^{*}$ is obtained by the B&B algorithm. The corresponding results were summarized in

Table 1. CPU time results of the algorithms for ${\sum }_{j=1}^n{w}_j{C}_j$.

			WSPT		NEH [SPT]		NEH [WSPT]		SA		B&B
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	0.0040	0.008	0.0076	0.010	0.0071	0.011	0.0532	0.059	0.2319	0.405
8	−0.35	1.3	0.0041	0.006	0.0071	0.012	0.0062	0.010	0.0547	0.064	0.2317	0.333
		1.5	0.0049	0.008	0.0060	0.010	0.0064	0.009	0.0558	0.070	0.1806	0.314
		1.1	0.0042	0.007	0.0061	0.010	0.0056	0.008	0.0528	0.056	0.3060	0.432
8	−0.3	1.3	0.0050	0.008	0.0066	0.014	0.0071	0.018	0.0527	0.058	0.2641	0.357
		1.5	0.0046	0.008	0.0067	0.010	0.0072	0.009	0.0543	0.058	0.2041	0.367
		1.1	0.0045	0.009	0.0079	0.015	0.0080	0.013	0.0517	0.057	0.3456	0.492
8	−0.25	1.3	0.0055	0.007	0.0060	0.009	0.0056	0.009	0.0506	0.052	0.3302	0.443
		1.5	0.0045	0.009	0.0066	0.012	0.0067	0.010	0.0526	0.056	0.2832	0.420
		1.1	0.005	0.011	0.0076	0.015	0.0072	0.014	0.0658	0.069	3.6724	4.652
9	−0.35	1.3	0.0040	0.007	0.0073	0.012	0.0067	0.010	0.0682	0.084	3.0988	4.049
		1.5	0.0049	0.010	0.0074	0.009	0.0067	0.009	0.0669	0.077	2.7130	3.910
		1.1	0.0039	0.007	0.0055	0.008	0.0072	0.010	0.0647	0.066	3.5200	4.563
9	−0.3	1.3	0.0050	0.009	0.0062	0.010	0.0075	0.009	0.0650	0.067	3.6839	4.565
		1.5	0.0063	0.008	0.0051	0.008	0.0052	0.010	0.0651	0.071	2.4250	4.557
		1.1	0.0044	0.009	0.0067	0.010	0.0071	0.011	0.0679	0.084	3.8984	4.829
9	−0.25	1.3	0.0046	0.009	0.0082	0.023	0.0064	0.009	0.0678	0.090	3.3172	4.225
		1.5	0.0041	0.007	0.0059	0.010	0.0069	0.010	0.0668	0.071	2.0806	4.654
		1.1	0.0039	0.009	0.0071	0.010	0.0080	0.019	0.0824	0.099	48.7101	58.454
10	−0.35	1.3	0.0062	0.012	0.0085	0.022	0.0070	0.017	0.0818	0.094	40.8171	54.900
		1.5	0.0048	0.010	0.0068	0.012	0.0066	0.012	0.0819	0.090	27.7426	45.406
		1.1	0.0022	0.006	0.0061	0.018	0.0047	0.009	0.0958	0.139	52.8939	107.098
10	−0.3	1.3	0.0032	0.010	0.0105	0.040	0.0058	0.011	0.1184	0.202	18.8109	48.119
		1.5	0.0018	0.005	0.0035	0.006	0.0033	0.005	0.0860	0.090	57.9169	69.869
		1.1	0.0032	0.008	0.0052	0.014	0.0091	0.045	0.1178	0.190	11.5889	49.168
10	−0.25	1.3	0.0025	0.004	0.0035	0.008	0.0036	0.012	0.0950	0.113	42.5347	62.936
		1.5	0.0031	0.008	0.0045	0.010	0.0056	0.013	0.0885	0.093	65.2307	151.610
		1.1	0.0024	0.005	0.0032	0.005	0.0042	0.005	0.1115	0.133	727.0171	846.621
11	−0.35	1.3	0.0022	0.004	0.0055	0.013	0.0045	0.013	0.1093	0.113	509.7685	927.570
		1.5	0.0014	0.002	0.0030	0.005	0.0032	0.005	0.1112	0.120	564.4302	799.208
		1.1	0.0015	0.003	0.0034	0.006	0.0034	0.007	0.1181	0.143	768.5305	886.115
11	−0.3	1.3	0.0020	0.004	0.0071	0.033	0.0034	0.005	0.1148	0.123	525.4747	908.383
		1.5	0.0026	0.010	0.0048	0.007	0.0033	0.005	0.1162	0.136	533.2328	748.700
		1.1	0.0050	0.033	0.0049	0.019	0.0102	0.075	0.1164	0.137	784.9697	928.329
11	−0.25	1.3	0.0026	0.004	0.0055	0.017	0.0054	0.020	0.1128	0.117	704.5131	883.534
		1.5	0.0023	0.004	0.0034	0.006	0.0035	0.005	0.1153	0.166	502.6579	801.074
		1.1	0.0018	0.005	0.0050	0.016	0.0050	0.014	0.1435	0.146	3600.0000	3600.000
12	−0.35	1.3	0.0019	0.008	0.0040	0.005	0.0038	0.006	0.1450	0.158	3005.2937	3600.000
		1.5	0.0026	0.010	0.0037	0.004	0.0043	0.006	0.1451	0.151	3600.0000	3600.000
		1.1	0.0014	0.003	0.0045	0.012	0.0045	0.009	0.1310	0.134	3600.0000	3600.000
12	−0.3	1.3	0.0021	0.004	0.0039	0.007	0.0040	0.005	0.1376	0.187	3600.0000	3600.000
		1.5	0.0015	0.003	0.0042	0.006	0.0054	0.016	0.1289	0.145	3600.0000	3600.000
		1.1	0.0018	0.003	0.0051	0.017	0.0048	0.019	0.1303	0.135	3600.0000	3600.000
12	−0.25	1.3	0.0044	0.028	0.0036	0.006	0.0037	0.005	0.1315	0.143	3600.0000	3600.000
		1.5	0.0016	0.004	0.0038	0.006	0.0037	0.006	0.1309	0.151	3600.0000	3600.000

,

Table 2. Error results of the algorithms for ${\sum }_{j=1}^n{w}_j{C}_j$.

			WSPT		NEH [SPT]		NEH [WSPT]		SA
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	1.153793	1.295931	1.008399	1.024746	1.011077	1.032914	1.001287	1.006465
8	−0.35	1.3	1.134968	1.245271	1.001498	1.007663	1.006699	1.035161	1.001024	1.002747
		1.5	1.112083	1.234499	1.005805	1.015160	1.006708	1.021868	1.000849	1.006536
		1.1	1.109401	1.158144	1.005588	1.014658	1.009061	1.031351	1.001473	1.004281
8	−0.3	1.3	1.123022	1.302447	1.006828	1.024521	1.015471	1.054399	1.000709	1.003919
		1.5	1.122659	1.381033	1.004426	1.016365	1.008292	1.035747	1.000011	1.000780
		1.1	1.116102	1.196116	1.007807	1.023802	1.003098	1.009004	1.000946	1.004389
8	−0.25	1.3	1.133121	1.367961	1.003683	1.010209	1.001278	1.007803	1.000349	1.003281
		1.5	1.136821	1.249175	1.004952	1.011526	1.017395	1.087800	1.000000	1.000000
		1.1	1.156571	1.299052	1.004717	1.010790	1.010768	1.056839	1.002830	1.008667
9	−0.35	1.3	1.145262	1.268797	1.005978	1.013920	1.012631	1.056035	1.001156	1.005027
		1.5	1.152582	1.277769	1.001882	1.009849	1.006569	1.017948	1.000351	1.001436
		1.1	1.130162	1.246478	1.010669	1.021749	1.009885	1.041701	1.002883	1.010331
9	−0.3	1.3	1.094137	1.127839	1.003580	1.013867	1.005748	1.022403	1.001004	1.004694
		1.5	1.128695	1.242345	1.003321	1.008475	1.012372	1.048688	1.000246	1.001292
		1.1	1.137278	1.223085	1.003682	1.011862	1.008655	1.044267	1.002133	1.005024
9	−0.25	1.3	1.158883	1.366356	1.005186	1.012680	1.002271	1.007475	1.001628	1.003969
		1.5	1.139216	1.323293	1.004950	1.008945	1.010153	1.044249	1.000092	1.000236
		1.1	1.114850	1.203954	1.001548	1.005676	1.010642	1.034419	1.004832	1.008004
10	−0.35	1.3	1.124982	1.297028	1.003348	1.007410	1.002078	1.007135	1.000882	1.003733
		1.5	1.161921	1.285757	1.003060	1.006878	1.007517	1.028547	1.000217	1.001583
		1.1	1.134824	1.237106	1.002389	1.008488	1.004457	1.037028	1.000519	1.010865
10	−0.3	1.3	1.160045	1.231845	1.000272	1.000578	1.005628	1.007707	1.002843	1.007881
		1.5	1.153393	1.376685	1.002740	1.011537	1.006211	1.015333	1.004082	1.012140
		1.1	1.165836	1.240345	1.006832	1.009313	1.000449	1.001523	1.000670	1.002829
10	−0.25	1.3	1.167985	1.279389	1.005328	1.022303	1.005879	1.018318	1.003713	1.008853
		1.5	1.174685	1.260239	1.005599	1.011322	1.010267	1.046237	1.004259	1.012722
		1.1	1.181602	1.227735	1.004034	1.010475	1.012151	1.042239	1.010139	1.019491
11	−0.35	1.3	1.124929	1.243339	1.003483	1.020451	1.004506	1.035540	1.001605	1.010377
		1.5	1.168116	1.364536	1.002212	1.006295	1.004298	1.014586	1.000165	1.000907
		1.1	1.137376	1.245895	1.001831	1.007394	1.002247	1.008825	1.006110	1.016095
11	−0.3	1.3	1.171947	1.304076	1.009608	1.010689	1.000667	1.018131	1.009211	1.011384
		1.5	1.181579	1.257707	1.001542	1.004800	1.007411	1.023322	1.000081	1.000682
		1.1	1.120671	1.250543	1.002479	1.015090	1.003541	1.010141	1.005726	1.010955
11	−0.25	1.3	1.102724	1.205647	1.000798	1.004841	1.007567	1.017888	1.002190	1.005589
		1.5	1.147618	1.266045	1.001281	1.010437	1.004608	1.019019	1.003464	1.014524
		1.1	1.143930	1.326818	1.007937	1.016891	1.001999	1.019799	1.002545	1.006871
12	−0.35	1.3	1.203649	1.390974	1.002139	1.013408	1.004875	1.028680	1.001423	1.009115
		1.5	1.141869	1.235865	1.000954	1.002979	1.000084	1.005370	1.008734	1.013572
		1.1	1.118866	1.190562	1.001012	1.015384	1.000891	1.007737	1.002098	1.009420
12	−0.3	1.3	1.121215	1.231603	1.000644	1.004012	1.000866	1.004081	1.000830	1.009272
		1.5	1.142815	1.268542	1.001076	1.005068	1.004947	1.028670	1.001428	1.012770
		1.1	1.123243	1.189385	1.001120	1.005239	1.001363	1.020022	1.000538	1.008453
12	−0.25	1.3	1.135348	1.200315	1.002112	1.008383	1.001599	1.009211	1.000944	1.007106
		1.5	1.137938	1.216416	1.000779	1.006002	1.005620	1.037257	1.000817	1.002448

,

Table 3. CPU time results of the algorithms for $T_{max}$.

			EDD		NEH [SPT]		NEH [EDD]		SA		BNB
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	0.0065	0.010	0.0107	0.013	0.0111	0.016	0.1159	0.151	0.5271	0.607
8	−0.35	1.3	0.0070	0.011	0.0095	0.018	0.0091	0.012	0.1119	0.134	0.5125	0.597
		1.5	0.0080	0.016	0.0111	0.015	0.0105	0.016	0.1086	0.115	0.5024	0.592
		1.1	0.0068	0.011	0.0091	0.013	0.0085	0.014	0.1086	0.151	0.5090	0.608
8	−0.3	1.3	0.0076	0.012	0.0104	0.011	0.0091	0.012	0.1037	0.110	0.5068	0.576
		1.5	0.0073	0.011	0.0100	0.014	0.0097	0.011	0.1063	0.117	0.4876	0.570
		1.1	0.0080	0.012	0.0099	0.014	0.0123	0.030	0.1061	0.115	0.5373	0.708
8	−0.25	1.3	0.0068	0.010	0.0087	0.013	0.0105	0.014	0.1058	0.122	0.4939	0.576
		1.5	0.0073	0.010	0.0106	0.015	0.0112	0.016	0.1061	0.118	0.5199	0.574
		1.1	0.0078	0.013	0.0098	0.014	0.0115	0.014	0.1417	0.153	5.2138	6.452
9	−0.35	1.3	0.0081	0.014	0.0111	0.021	0.0101	0.021	0.1317	0.136	5.4292	6.197
		1.5	0.0102	0.016	0.0122	0.014	0.0150	0.020	0.2946	1.608	5.7520	11.942
		1.1	0.0065	0.011	0.0093	0.012	0.0111	0.015	0.1371	0.164	4.5770	5.930
9	−0.3	1.3	0.0079	0.013	0.0107	0.015	0.0101	0.015	0.1340	0.163	4.8075	5.644
		1.5	0.0072	0.010	0.0091	0.017	0.0109	0.021	0.1355	0.154	4.8945	5.893
		1.1	0.0075	0.011	0.0108	0.014	0.0102	0.017	0.1335	0.148	4.8332	5.732
9	−0.25	1.3	0.0078	0.014	0.0091	0.011	0.0091	0.013	0.1327	0.137	4.9119	6.089
		1.5	0.0069	0.010	0.0104	0.015	0.0095	0.012	0.1361	0.160	4.9378	6.307
		1.1	0.0068	0.010	0.0113	0.017	0.0103	0.014	0.1833	0.239	51.4277	67.427
10	−0.35	1.3	0.0084	0.013	0.0151	0.060	0.0101	0.013	0.1707	0.188	52.3317	69.006
		1.5	0.0086	0.018	0.0104	0.014	0.0106	0.016	0.1718	0.184	48.0880	66.805
		1.1	0.0063	0.015	0.0092	0.012	0.0102	0.012	0.1782	0.275	57.9763	67.831
10	−0.3	1.3	0.0078	0.010	0.0098	0.013	0.0099	0.014	0.1685	0.198	59.1357	72.411
		1.5	0.0069	0.010	0.0100	0.018	0.0097	0.013	0.1650	0.174	52.4632	65.451
		1.1	0.0051	0.008	0.0059	0.011	0.0063	0.012	0.1867	0.246	63.5020	89.227
10	−0.25	1.3	0.0033	0.005	0.0040	0.006	0.0041	0.008	0.1900	0.287	56.5084	69.985
		1.5	0.0035	0.005	0.0047	0.008	0.0046	0.007	0.1827	0.208	56.4558	78.742
		1.1	0.0073	0.028	0.0091	0.032	0.0072	0.011	0.2651	0.363	712.5808	1031.763
11	−0.35	1.3	0.0086	0.023	0.0151	0.062	0.0126	0.071	0.2811	0.404	757.4174	1038.250
		1.5	0.0046	0.009	0.0054	0.009	0.0074	0.019	0.2737	0.337	854.5852	1014.339
		1.1	0.0051	0.011	0.0085	0.024	0.0073	0.022	0.3045	0.520	725.1830	927.480
11	−0.3	1.3	0.0056	0.017	0.0064	0.011	0.0063	0.018	0.2809	0.399	812.3170	1040.331
		1.5	0.0063	0.009	0.0072	0.012	0.0062	0.015	0.2509	0.270	848.7941	1072.108
		1.1	0.0049	0.011	0.0093	0.017	0.0064	0.011	0.2654	0.324	866.4369	1064.095
11	−0.25	1.3	0.0044	0.007	0.0095	0.026	0.0077	0.014	0.2729	0.329	931.0045	1149.633
		1.5	0.0074	0.026	0.0065	0.009	0.0063	0.010	0.2530	0.283	718.2544	1034.691
		1.1	0.0026	0.005	0.0061	0.020	0.0055	0.018	0.2685	0.339	3600.0000	3600.000
12	−0.35	1.3	0.0028	0.004	0.0035	0.005	0.0039	0.005	0.2632	0.270	3600.0000	3600.000
		1.5	0.0027	0.006	0.0040	0.005	0.0038	0.005	0.2653	0.279	3600.0000	3600.000
		1.1	0.0037	0.015	0.0049	0.014	0.0043	0.013	0.2763	0.321	3600.0000	3600.000
12	−0.3	1.3	0.0026	0.007	0.0039	0.005	0.0046	0.006	0.2691	0.276	3600.0000	3600.000
		1.5	0.0024	0.004	0.0035	0.005	0.0039	0.008	0.2680	0.280	3600.0000	3600.000
		1.1	0.0102	0.077	0.0278	0.243	0.0232	0.194	0.2714	0.276	3600.0000	3600.000
12	−0.25	1.3	0.0032	0.007	0.0055	0.010	0.0050	0.010	0.2758	0.292	3600.0000	3600.000
		1.5	0.0026	0.004	0.0041	0.006	0.0043	0.006	0.2735	0.305	3600.0000	3600.000

and

Table 4. Error results of the algorithms for $T_{max}$.

			EDD		NEH [SPT]		NEH [EDD]		SA
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	1.391985	1.696948	1.000845	1.004362	1.000009	1.000088	1.000551	1.005426
8	−0.35	1.3	1.502860	1.971499	1.001549	1.008998	1.005438	1.031705	1.001073	1.006287
		1.5	1.524976	2.101402	1.000228	1.002283	1.000228	1.002283	1.001686	1.008312
		1.1	1.378595	1.995094	1.000899	1.008995	1.000899	1.008995	1.002128	1.008982
8	−0.3	1.3	1.380949	1.737122	1.001307	1.013072	1.000215	1.002152	1.001172	1.005177
		1.5	1.402049	1.982816	1.001727	1.017275	1.003809	1.038085	1.002507	1.007036
		1.1	1.389848	2.048553	1.002504	1.014560	1.000625	1.006252	1.001696	1.006978
8	−0.25	1.3	1.447619	1.958566	1.002759	1.027594	1.002759	1.027594	1.000408	1.004078
		1.5	1.579351	2.161202	1.000750	1.007502	1.000750	1.007502	1.001027	1.006463
		1.1	1.442345	1.961234	1.003413	1.022266	1.005356	1.026331	1.005508	1.013646
9	−0.35	1.3	1.516955	1.998692	1.001710	1.008447	1.005736	1.032017	1.007354	1.014214
		1.5	1.722542	2.258243	1.001126	1.006600	1.001805	1.006790	1.006587	1.027363
		1.1	1.551717	2.120692	1.001856	1.010495	1.001856	1.010495	1.007568	1.016909
9	−0.3	1.3	1.469008	1.708295	1.000396	1.003960	1.000396	1.003960	1.006551	1.016286
		1.5	1.709125	2.456867	1.001634	1.011738	1.001174	1.011738	1.007813	1.019808
		1.1	1.411960	1.802741	1.000342	1.003417	1.001503	1.015028	1.004869	1.013625
9	−0.25	1.3	1.416398	1.746858	1.001777	1.011064	1.001784	1.011064	1.006863	1.013219
		1.5	1.623309	2.017430	1.002379	1.011785	1.002195	1.011785	1.005921	1.013746
		1.1	1.455887	1.745446	1.000234	1.002340	1.000234	1.002340	1.010044	1.037316
10	−0.35	1.3	1.565331	1.997918	1.000464	1.004640	1.000464	1.004640	1.015864	1.034781
		1.5	1.584080	2.141175	1.000465	1.004646	1.000465	1.004646	1.005883	1.017630
		1.1	1.238595	1.425475	1.006448	1.031679	1.001288	1.007398	1.009499	1.021841
10	−0.3	1.3	1.288117	1.514335	1.001713	1.014802	1.001382	1.008260	1.008825	1.016427
		1.5	1.486307	1.779684	1.003633	1.016038	1.001586	1.008210	1.018826	1.036123
		1.1	1.396746	1.919641	1.004264	1.031699	1.001000	1.007284	1.014586	1.024169
10	−0.25	1.3	1.526785	1.951016	1.001354	1.007907	1.000878	1.005637	1.011859	1.026652
		1.5	1.619648	2.072637	1.000492	1.004921	1.002769	1.013436	1.006504	1.015283
		1.1	1.340860	1.588910	1.001787	1.007769	1.000594	1.003627	1.022935	1.054960
11	−0.35	1.3	1.570589	1.888829	1.000334	1.003338	1.000502	1.003338	1.019197	1.031687
		1.5	1.620125	2.283564	1.003848	1.019394	1.004453	1.019394	1.021817	1.039559
		1.1	1.424601	1.687650	1.001473	1.008906	1.001271	1.008906	1.019632	1.037068
11	−0.3	1.3	1.535443	1.987049	1.001130	1.011304	1.001722	1.009317	1.020158	1.037680
		1.5	1.527245	2.180450	1.000000	1.000000	1.000000	1.000000	1.010958	1.018582
		1.1	1.341775	1.644813	1.000000	1.000000	1.001438	1.008571	1.022547	1.041254
11	−0.25	1.3	1.401902	1.993975	1.001235	1.003687	1.001106	1.003687	1.014975	1.032908
		1.5	1.540078	2.475227	1.001551	1.015513	1.001551	1.015513	1.015537	1.046111
		1.1	1.478510	1.904917	1.000188	1.016961	1.007880	1.024763	1.023930	1.048241
12	−0.35	1.3	1.460320	1.752007	1.002501	1.038033	1.003738	1.073533	1.020798	1.038565
		1.5	1.413032	1.731996	1.001328	1.004962	1.004484	1.017246	1.023011	1.042456
		1.1	1.329080	1.592992	1.001255	1.004114	1.005515	1.008019	1.025919	1.045782
12	−0.3	1.3	1.435380	1.729377	1.000660	1.010773	1.000497	1.015057	1.022584	1.036376
		1.5	1.588068	1.918008	1.000840	1.012516	1.001242	1.012516	1.031792	1.062825
		1.1	1.413564	1.823790	1.009342	1.010195	1.009475	1.010195	1.020642	1.035687
12	−0.25	1.3	1.425990	1.825923	1.004648	1.006965	1.008004	1.010111	1.029889	1.067646
		1.5	1.481751	1.861310	1.007327	1.035780	1.001397	1.002828	1.019218	1.044245

.

For ${\sum }_{j=1}^n{w}_j{C}_j$, from Table 1, we can see that the CPU time of WSPT changes very little as n increases, but B&B increases rapidly, i.e., grows exponentially. Table 2 shows that the error of the SA algorithm is somewhat smaller than that of the WSPT and NEH algorithms, indicating that SA appears to be more accurate than the WSPT and NEH (WSPT). In conclusion, in terms of both time and error dimensions, algorithm B&B is more accurate although takes longer to execute.

For $T_{max}$, again the CPU time of B&B grows exponentially as n increases. and it can be seen from Table 4 that the algorithm NEH appears to be more accurate than SA.

3.3.2. Large-Sized Instances

For large-scale examples ($n=100,125,150,175,200$), it is defined that “CPU time” (s) as the running time of the WSPT (EDD), NEH, SA, B&B algorithm, time unit is second, and the performance of the heuristics WSPT, EDD, NEH [SPT], NEH [WSPT], NEH [EDD] and SA was verified by the ratios $\frac{{\sum }_{j=1}^n{w}_j{C}_j\left(G\right)}{{\sum }_{j=1}^n{w}_j{C}_j\left(Best\right)}$ and $\frac{T_{max}\left(G\right)}{T_{max}\left(Best\right)}$, where G was a sequence obtained by the WSPT, EDD, NEH [SPT], NEH [WSPT], NEH [EDD] and SA, and $Best$ denotes the best solution found by any of the heuristic procedures in each of the problem instances. The corresponding results were summarized in

Table 5. CPU time of the algorithms for ${\sum }_{j=1}^n{w}_j{C}_j$.

			WSPT		NEH [SPT]		NEH [WSPT]		SA
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	0.0043	0.015	32.1174	36.554	32.8210	39.555	42.0538	47.751
100	−0.35	1.3	0.0019	0.005	30.5495	30.942	30.5970	30.919	40.4003	40.781
		1.5	0.0022	0.005	30.5546	31.161	30.7483	31.702	41.4554	46.573
		1.1	0.0037	0.008	32.8921	38.296	32.4227	37.241	43.3568	49.491
100	−0.3	1.3	0.0016	0.003	31.1891	31.456	31.1909	31.595	40.4172	40.966
		1.5	0.0021	0.004	31.2537	32.466	31.7434	34.458	41.1115	45.163
		1.1	0.0027	0.006	33.2766	37.293	33.3414	36.736	38.9160	47.744
100	−0.25	1.3	0.0015	0.002	32.3208	34.450	32.1044	33.168	39.6546	42.437
		1.5	0.0020	0.003	32.1615	33.796	32.1032	33.145	39.6327	41.574
		1.1	0.0075	0.040	100.8299	102.498	104.1485	126.708	89.5918	100.188
125	−0.35	1.3	0.0029	0.005	101.1583	102.822	101.5433	106.839	88.4149	91.589
		1.5	0.0050	0.010	101.5997	102.911	102.0005	109.248	88.6029	90.131
		1.1	0.0103	0.068	101.9015	108.984	101.6730	106.675	85.5008	87.885
125	−0.3	1.3	0.0107	0.041	102.3991	108.330	101.7231	103.316	85.3295	86.877
		1.5	0.0056	0.021	95.2835	102.307	93.7359	102.330	78.6950	86.964
		1.1	0.0020	0.006	89.8276	98.492	90.7208	102.276	74.7486	77.613
125	−0.25	1.3	0.0026	0.011	88.7815	89.728	89.2368	89.982	74.1769	75.801
		1.5	0.0027	0.007	88.7236	89.636	89.1936	92.303	74.1995	78.704
		1.1	0.0035	0.007	224.8844	234.479	225.7813	234.541	127.5309	128.795
150	−0.35	1.3	0.0056	0.027	223.4674	226.843	223.4250	225.230	129.0307	132.514
		1.5	0.0031	0.010	224.1562	226.971	223.1970	225.067	128.1263	134.168
		1.1	0.0019	0.005	218.1563	226.579	218.2396	220.007	129.9573	131.464
150	−0.3	1.3	0.0038	0.012	217.9165	223.287	217.1279	220.100	130.2998	133.138
		1.5	0.0030	0.008	217.0470	222.205	216.1124	219.064	129.0232	131.122
		1.1	0.0032	0.009	249.1793	268.382	247.3459	257.998	146.3248	152.955
150	−0.25	1.3	0.0132	0.099	232.9484	254.353	233.4128	248.585	137.1951	147.897
		1.5	0.0055	0.011	248.2794	254.415	247.3834	250.640	145.5044	147.773
		1.1	0.0042	0.022	441.3598	481.858	438.2147	454.027	182.7255	184.840
175	−0.35	1.3	0.0022	0.003	436.1500	438.778	436.5397	439.222	182.5295	183.987
		1.5	0.0030	0.006	437.0817	443.313	435.2025	439.747	180.6198	185.014
		1.1	0.0023	0.004	434.8344	469.997	432.3358	448.198	182.9713	185.022
175	−0.3	1.3	0.0024	0.004	430.3138	432.819	430.5209	432.549	182.8286	184.508
		1.5	0.0025	0.005	431.5106	437.022	430.5689	433.309	182.9001	184.248
		1.1	0.0022	0.004	432.4441	454.244	431.5986	447.575	183.0795	184.984
175	−0.25	1.3	0.0027	0.005	429.7236	432.266	429.8054	431.737	182.6727	184.558
		1.5	0.0036	0.007	430.6976	436.018	429.7777	432.268	182.9039	184.164
		1.1	0.0024	0.003	817.0101	880.226	812.6110	819.578	272.7165	274.651
200	−0.35	1.3	0.0025	0.004	811.2221	813.743	810.9756	814.021	272.9047	274.930
		1.5	0.0029	0.006	812.0180	818.339	813.2050	816.986	271.5289	274.671
		1.1	0.0025	0.004	815.2231	871.821	811.5153	818.395	266.3699	268.005
200	−0.3	1.3	0.0025	0.004	810.4137	813.732	809.9646	812.664	266.1901	268.020
		1.5	0.0032	0.006	810.8555	816.802	812.0056	813.409	267.2094	271.522
		1.1	0.0022	0.004	813.8868	856.458	811.3914	817.977	268.2697	269.884
200	−0.25	1.3	0.0023	0.003	810.3899	812.691	809.8105	812.808	268.8522	271.181
		1.5	0.0029	0.005	811.0429	818.118	811.8657	814.180	268.9761	271.981

,

Table 6. Error results of the algorithms for ${\sum }_{j=1}^n{w}_j{C}_j$.

			$\frac{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{WSPT}\right)}{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{Best}\right)}$		$\frac{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{NEH}\left[\mathit{SPT}\right]\right)}{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{Best}\right)}$		$\frac{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{NEH}\left[\mathit{WSPT}\right]\right)}{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{Best}\right)}$		$\frac{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{SA}\right)}{{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{j}}{\mathit{C}}_{\mathit{j}}\left(\mathit{Best}\right)}$
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	1.036954	1.054012	1.021573	1.045366	1.000000	1.000000	1.003519	1.006554
100	−0.35	1.3	1.044962	1.060569	1.025172	1.039109	1.000148	1.000941	1.000559	1.001195
		1.5	1.050418	1.064382	1.025248	1.047217	1.000274	1.001805	1.000153	1.000294
		1.1	1.035566	1.044404	1.013494	1.036464	1.000000	1.000000	1.001537	1.005140
100	−0.3	1.3	1.036240	1.048568	1.016733	1.047571	1.000274	1.002152	1.000255	1.000593
		1.5	1.055272	1.069122	1.036045	1.067625	1.000156	1.001559	1.001219	1.005149
		1.1	1.034751	1.045295	1.018090	1.037085	1.000566	1.002543	1.000672	1.001724
100	−0.25	1.3	1.046670	1.059142	1.031779	1.053538	1.000348	1.002361	1.000987	1.004724
		1.5	1.046329	1.060851	1.019240	1.031961	1.000911	1.004362	1.000023	1.000117
		1.1	1.041622	1.053573	1.017301	1.032824	1.000350	1.002534	1.000672	1.003258
125	−0.35	1.3	1.044984	1.066139	1.031357	1.047600	1.000291	1.002869	1.001902	1.006682
		1.5	1.051531	1.076385	1.027787	1.046987	1.000707	1.006116	1.000409	1.002287
		1.1	1.037141	1.046269	1.022844	1.056697	1.000118	1.000793	1.000356	1.000674
125	−0.3	1.3	1.046439	1.058477	1.028552	1.045449	1.000826	1.006412	1.000533	1.002495
		1.5	1.047775	1.058128	1.022050	1.044407	1.000281	1.002506	1.000423	1.003361
		1.1	1.036338	1.045623	1.018068	1.038139	1.000000	1.000000	1.006351	1.012230
125	−0.25	1.3	1.039846	1.046813	1.022218	1.044372	1.000216	1.000818	1.000746	1.002067
		1.5	1.047987	1.072990	1.029656	1.052340	1.000115	1.000528	1.000528	1.002905
		1.1	1.038792	1.049708	1.024257	1.045010	1.000164	1.000917	1.001737	1.004585
150	−0.35	1.3	1.043335	1.049759	1.018966	1.037425	1.000092	1.000556	1.000625	1.003164
		1.5	1.046706	1.058638	1.033135	1.044877	1.000136	1.000677	1.000466	1.002139
		1.1	1.035897	1.042391	1.015651	1.031913	1.000316	1.002022	1.001093	1.002656
150	−0.3	1.3	1.043970	1.049197	1.021245	1.031130	1.000697	1.002385	1.000647	1.005178
		1.5	1.051268	1.060151	1.028940	1.045466	1.000181	1.000784	1.001252	1.004006
		1.1	1.039513	1.053879	1.021009	1.040601	1.000970	1.005116	1.000420	1.002948
150	−0.25	1.3	1.044452	1.055217	1.017391	1.038033	1.000536	1.002863	1.000157	1.000436
		1.5	1.053730	1.062573	1.030865	1.051478	1.001292	1.004868	1.000678	1.005139
		1.1	1.036283	1.042407	1.019139	1.042097	1.000006	1.000058	1.001327	1.003982
175	−0.35	1.3	1.045164	1.052823	1.024624	1.040228	1.000252	1.002495	1.000969	1.003571
		1.5	1.053126	1.066924	1.029755	1.043504	1.000090	1.000899	1.000762	1.002772
		1.1	1.039030	1.051165	1.017138	1.027578	1.000758	1.002126	1.000291	1.002757
175	−0.3	1.3	1.040464	1.055880	1.021767	1.037030	1.000247	1.001310	1.000191	1.000556
		1.5	1.050189	1.057683	1.021367	1.046832	1.000124	1.000840	1.001091	1.003998
		1.1	1.034653	1.053831	1.014328	1.031598	1.000135	1.001346	1.005274	1.036593
175	−0.25	1.3	1.040749	1.050106	1.017471	1.042529	1.000368	1.003061	1.001858	1.011035
		1.5	1.047830	1.055207	1.029811	1.051316	1.000000	1.000000	1.001697	1.006083
		1.1	1.035588	1.045371	1.015440	1.024891	1.000258	1.001673	1.003623	1.034281
200	−0.35	1.3	1.042460	1.048864	1.023037	1.035818	1.000676	1.003039	1.000100	1.004084
		1.5	1.050197	1.063436	1.018771	1.024690	1.000066	1.000481	1.000638	1.003463
		1.1	1.036709	1.047059	1.025257	1.063084	1.000578	1.002239	1.000364	1.001912
200	−0.3	1.3	1.043742	1.052294	1.025257	1.063084	1.000578	1.002239	1.000364	1.001912
		1.5	1.046824	1.058465	1.024041	1.041735	1.000105	1.000623	1.000454	1.001806
		1.1	1.037993	1.047319	1.019775	1.035887	1.000389	1.001368	1.001711	1.015238
200	−0.25	1.3	1.044992	1.060204	1.023553	1.034875	1.000288	1.001973	1.001092	1.006503
		1.5	1.046474	1.063612	1.024352	1.050963	1.000224	1.001590	1.000563	1.004302

,

Table 7. CPU time of the algorithms for $T_{max}$.

			EDD		NEH [SPT]		NEH [EDD]		SA
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	0.0049	0.014	35.7995	44.037	35.5141	40.394	104.9574	115.665
100	−0.35	1.3	0.0055	0.017	34.1109	38.102	33.8245	36.840	100.1980	106.372
		1.5	0.0045	0.015	31.6923	40.995	31.9844	36.162	94.0071	102.430
		1.1	0.0038	0.009	36.5595	45.508	36.0100	38.865	80.8120	91.208
100	−0.3	1.3	0.0044	0.008	35.1985	37.784	34.6108	36.195	77.5120	82.227
		1.5	0.0030	0.006	32.9251	34.742	32.6552	33.407	74.2111	84.192
		1.1	0.0038	0.010	41.4234	44.642	42.1242	49.681	95.7675	101.616
100	−0.25	1.3	0.0032	0.007	40.1205	49.229	39.5503	44.546	90.0902	97.322
		1.5	0.0030	0.006	37.0282	39.212	36.9542	37.794	85.1035	88.136
		1.1	0.0026	0.006	93.6647	102.042	93.5877	105.691	147.3931	150.990
125	−0.35	1.3	0.0032	0.008	92.1210	93.211	92.3207	94.025	148.1259	150.224
		1.5	0.0038	0.008	104.4802	175.005	100.7225	157.957	155.1233	192.530
		1.1	0.0055	0.027	95.3332	105.785	94.2544	100.591	148.5498	151.490
125	−0.3	1.3	0.0029	0.008	93.5627	95.330	93.1668	95.606	149.6621	154.573
		1.5	0.0051	0.025	98.2485	122.848	98.0707	132.257	168.3613	250.543
		1.1	0.0061	0.034	105.3366	119.881	104.6303	114.299	139.4032	153.027
125	−0.25	1.3	0.0022	0.004	91.5431	91.733	91.6551	92.096	123.6464	125.502
		1.5	0.0022	0.004	91.6460	91.966	91.7085	92.295	123.2681	126.132
		1.1	0.0036	0.007	227.4756	232.860	226.6170	230.864	256.2105	259.224
150	−0.35	1.3	0.0034	0.006	226.9224	232.162	227.1699	231.748	258.5131	262.945
		1.5	0.0032	0.007	208.1170	226.679	207.0839	229.431	234.2087	256.957
		1.1	0.0155	0.110	260.2437	302.185	255.7124	258.976	285.8951	292.131
150	−0.3	1.3	0.0041	0.010	245.1265	264.954	244.8100	264.567	275.2374	290.767
		1.5	0.0038	0.008	229.3247	260.306	229.1621	256.510	256.7482	290.538
		1.1	0.0049	0.007	260.6876	296.354	257.3179	261.548	286.6712	292.955
150	−0.25	1.3	0.0037	0.007	245.3216	264.160	245.4663	265.695	275.6782	292.135
		1.5	0.0035	0.007	231.7387	261.185	231.8750	269.221	257.4253	301.127
		1.1	0.0054	0.013	469.1597	509.933	464.2641	468.257	381.6888	387.467
175	−0.35	1.3	0.0029	0.006	464.7862	469.037	463.8316	467.706	384.6989	396.320
		1.5	0.0031	0.007	463.6323	466.778	462.8959	466.322	381.9178	390.827
		1.1	0.0161	0.130	467.6403	499.727	464.1768	466.146	380.0000	384.436
175	−0.3	1.3	0.0041	0.008	460.5128	465.980	460.2540	464.600	377.3023	385.459
		1.5	0.0029	0.005	461.7312	467.387	461.7700	465.680	380.1052	390.331
		1.1	0.0155	0.127	511.7128	526.430	508.3289	511.821	410.6986	426.447
175	−0.25	1.3	0.0041	0.011	508.6204	513.282	507.4111	515.918	409.4637	417.110
		1.5	0.0042	0.008	494.1450	508.634	490.0148	509.834	394.7240	414.508
		1.1	0.0040	0.009	1017.7486	1050.312	1011.2776	1016.040	540.0450	548.385
200	−0.35	1.3	0.0031	0.005	1009.8994	1011.659	1011.4599	1015.956	539.8576	553.313
		1.5	0.0027	0.004	971.8302	1031.903	969.4584	1013.336	510.1022	560.211
		1.1	0.0024	0.005	850.0115	879.971	845.7156	852.730	534.0446	557.581
200	−0.3	1.3	0.0029	0.004	846.1232	858.366	843.3768	854.918	534.2898	553.874
		1.5	0.0078	0.048	852.9483	892.596	846.7610	856.320	535.3189	556.104
		1.1	0.0026	0.004	842.6111	860.951	839.8413	847.021	535.3703	558.302
200	−0.25	1.3	0.0051	0.013	838.4530	852.184	837.4318	849.783	532.8262	545.375
		1.5	0.0036	0.005	837.8098	846.619	840.6965	872.608	532.3132	541.777

and

Table 8. Error results of the algorithms for $T_{max}$.

			$\frac{{\mathit{T}}_{max}\left(\mathit{EDD}\right)}{{\mathit{T}}_{max}\left(\mathit{Best}\right)}$		$\frac{{\mathit{T}}_{max}\left(\mathit{NEH}\left[\mathit{SPT}\right]\right)}{{\mathit{T}}_{max}\left(\mathit{Best}\right)}$		$\frac{{\mathit{T}}_{max}\left(\mathit{NEH}\left[\mathit{EDD}\right]\right)}{{\mathit{T}}_{max}\left(\mathit{Best}\right)}$		$\frac{{\mathit{T}}_{max}\left(\mathit{SA}\right)}{{\mathit{T}}_{max}\left(\mathit{Best}\right)}$
n	a	b	Mean	Max	Mean	Max	Mean	Max	Mean	Max
		1.1	1.473754	1.653052	1.020897	1.061052	1.000121	1.001214	1.292191	1.364199
100	−0.35	1.3	1.511293	1.679862	1.022395	1.057531	1.000346	1.003462	1.100052	1.123072
		1.5	1.563430	1.800869	1.006935	1.020075	1.000023	1.000234	1.021721	1.030132
		1.1	1.492235	1.643907	1.016645	1.053443	1.001942	1.016024	1.191750	1.234499
100	−0.3	1.3	1.556259	1.795146	1.009639	1.030086	1.001161	1.011528	1.213167	1.407604
		1.5	1.574323	1.670788	1.008739	1.063792	1.001855	1.010011	1.093213	1.144223
		1.1	1.472540	1.532818	1.018223	1.073833	1.001380	1.013764	1.190574	1.207461
100	−0.25	1.3	1.530074	1.691434	1.003921	1.016537	1.001270	1.003316	1.245192	1.351231
		1.5	1.598949	1.774946	1.020953	1.038657	1.000000	1.000000	1.058741	1.087734
		1.1	1.490413	1.552454	1.018157	1.073742	1.000227	1.001414	1.248805	1.312706
125	−0.35	1.3	1.522274	1.559265	1.019339	1.065854	1.000249	1.002363	1.084719	1.361410
		1.5	1.590207	1.752900	1.015250	1.035104	1.000069	1.000691	1.079213	1.209878
		1.1	1.524563	1.621970	1.018973	1.043267	1.000000	1.000000	1.240576	1.289821
125	−0.3	1.3	1.554150	1.650697	1.018699	1.049223	1.000000	1.000000	1.079476	1.305713
		1.5	1.625665	1.743180	1.019719	1.050929	1.000074	1.000564	1.073347	1.176114
		1.1	1.455516	1.545623	1.020136	1.068915	1.000000	1.000000	1.265651	1.315263
125	−0.25	1.3	1.544356	1.717690	1.025510	1.057638	1.000335	1.003349	1.061383	1.074242
		1.5	1.586829	1.712102	1.012191	1.058153	1.000078	1.000775	1.105849	1.301847
		1.1	1.491936	1.566608	1.039236	1.073057	1.000000	1.000000	1.251022	1.371957
150	−0.35	1.3	1.511540	1.598136	1.015010	1.049847	1.000000	1.000000	1.104730	1.361085
		1.5	1.583574	1.668974	1.024485	1.065390	1.002250	1.017197	1.023661	1.051295
		1.1	1.507705	1.564044	1.028743	1.067513	1.000122	1.001219	1.269116	1.375915
150	−0.3	1.3	1.530170	1.664550	1.016270	1.041276	1.000000	1.000000	1.120929	1.319993
		1.5	1.552592	1.651253	1.018249	1.036614	1.000104	1.000528	1.032040	1.107554
		1.1	1.500467	1.550770	1.018323	1.049755	1.000736	1.004015	1.267174	1.387317
150	−0.25	1.3	1.558558	1.623855	1.020049	1.045220	1.000000	1.000000	1.124601	1.349715
		1.5	1.549457	1.647546	1.010298	1.045583	1.000103	1.000867	1.064911	1.353000
		1.1	1.546401	1.632883	1.029337	1.056355	1.000000	1.000000	1.236135	1.386901
175	−0.35	1.3	1.542372	1.594353	1.019601	1.065573	1.000000	1.000000	1.068689	1.182530
		1.5	1.563982	1.634967	1.022558	1.042928	1.001225	1.010330	1.035085	1.128978
		1.1	1.518948	1.644169	1.027663	1.061846	1.000000	1.000000	1.231353	1.389865
175	−0.3	1.3	1.524679	1.583958	1.033369	1.057156	1.000000	1.000000	1.069979	1.205596
		1.5	1.560340	1.651844	1.023858	1.081949	1.000004	1.000039	1.044736	1.188887
		1.1	1.478379	1.563133	1.029333	1.055492	1.000004	1.000038	1.216890	1.307547
175	−0.25	1.3	1.531086	1.635649	1.024749	1.060072	1.000000	1.000000	1.109001	1.351927
		1.5	1.561837	1.630478	1.022139	1.051489	1.000127	1.001275	1.016039	1.056480
		1.1	1.517268	1.581725	1.032981	1.058222	1.000000	1.000000	1.201538	1.385585
200	−0.35	1.3	1.532300	1.625112	1.033554	1.076344	1.000000	1.000000	1.094805	1.378270
		1.5	1.553501	1.607619	1.020570	1.058106	1.000000	1.000000	1.075708	1.428918
		1.1	1.484898	1.596637	1.026437	1.061230	1.000000	1.000000	1.224258	1.382158
200	−0.3	1.3	1.545884	1.646671	1.028041	1.064949	1.000949	1.009495	1.067740	1.201090
		1.5	1.606757	1.721663	1.024916	1.059894	1.000406	1.002605	1.051122	1.307438
		1.1	1.507418	1.608975	1.022933	1.066916	1.000538	1.005236	1.221224	1.373607
200	−0.25	1.3	1.531956	1.650285	1.023385	1.057931	1.000000	1.000000	1.070749	1.203446
		1.5	1.539974	1.732727	1.014916	1.066775	1.000115	1.001140	1.024880	1.077879

.

For ${\sum }_{j=1}^n{w}_j{C}_j$, it can be seen from Table 5 that the CPU times of NEH (SPT) and NEH (WSPT) are longer than that of SA. When $n=100$, the CPU times of NEH (SPT), NEH (WSPT) and SA grow faster, and when $n=200$, their CPU times have reached more than 800 s, while SA is only about 270 s, indicating that the algorithm SA is slightly more accurate than NEH in terms of CPU time when calculating large-scale. From Table 6, we can see that the error of WSPT, NEH (SPT) is relatively large and unsuitable for computing this problem. the error of NEH (WSPT) and SA is relatively smaller, but there are still differences between them, and obviously the error of NEH (WSPT) is a little smaller, and at this time, the NEH (WSPT) algorithm is better.

For $T_{max}$, as n increases, the CPU time of the NEH algorithm is somewhat longer than that of SA, but the difference is not very large relative to ${\sum }_{j=1}^n{w}_j{C}_j$. And it is obvious from Table 8 that the error of NEH (EDD) is significantly smaller than the other algorithms. In short, from the two dimensions of time and error, NEH (EDD) is superior compared to NEH (SPT) and SA.

4. Conclusions

This article studied scheduling problems with general learning effects and where each job has a past-sequence-dependent delivery time on a single machine. For the total weighted completion time and maximum tardiness minimizations, we proposed the heuristic, simulated annealing, branch-and-bound algorithms and conducted numerical experiments. Future research might focus on analyzing the above problems with m-machine flow shops and m unrelated parallel machines (Sun et al. [32]), extending the models to scheduling with variable processing times (Wang et al. [33]), or considering the lateness/earliness objective functions (Liang et al. [34], Wu et al. [35], Wu et al. [36], Sun et al. [37], Lv et al. [38], Wang et al. [39]).