# A Variable Block Insertion Heuristic for Solving Permutation Flow Shop Scheduling Problem with Makespan Criterion

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model Formulation

**Parameters:**

#### 2.1. The MIP Model

**Decision Variables:**

**MIP Model: Objective and Constraints:**

#### 2.2. The CP Model

**Decision Variables:**

**CP Model: Objective and Constraints:**

## 3. Meta-Heuristic Algorithms

#### 3.1. Taillard’s Speed Up Method for PFSP with Makespan Criterion

- Compute the head, ${e}_{i,k}$, which is the earliest completion time of each job on each machine. The starting time of the first job on the first machine is 0.${e}_{0,k}={e}_{i,0}=0$ $\forall i=1,\dots ,l-1;$$\forall k=1,\dots ,m$${e}_{i,k}=max\left\{{e}_{i,k-1},{e}_{i-1,k}\right\}+{p}_{{\pi}_{i},k}$ $\forall i=1,\dots ,l-1;$$\forall k=1,\dots ,m$.
- Compute the tail, ${q}_{i,k}$, which is the duration between the starting time of each job on each machine and the end of all the operations on each machine.${q}_{i,m+1}=0$ $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$${q}_{l,k}$ = 0 $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$${q}_{i,k}=max\left\{{q}_{i,k+1},{q}_{i+1,k}\right\}+{p}_{{\pi}_{i},,k}$ $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$.
- Compute the earliest relative completion time ${f}_{i,k}$ on the lth machine of job ${\pi}_{j}$ inserted at the lth position. Completion time of an inserted job on the first machine is zero.${f}_{i,0}=0$ $\forall i=1,\dots ,l$${f}_{i,k}=max\left\{{f}_{i,k-1},{e}_{i-1,k}\right\}+{p}_{{\pi}_{i},,k}$ $\forall i=1,\dots ,\mathrm{l}$; $\forall k=1,\dots ,m$.
- The value of the makespan ${C}_{max,l}$ when inserting job $j$ at the lth position is:${C}_{max,l}=ma{x}_{k}({f}_{ik}+{q}_{ik})$ $\forall i=1,\dots ,l$; $\forall k=1,\dots ,m$.

- Compute heads:${e}_{0,k}={e}_{i,0}=0$ $\forall i=1,\dots ,l-1;$$\forall k=1,\dots ,m$${e}_{i,k}=max\left\{{e}_{i,k-1},{e}_{i-1,k}\right\}+{p}_{{\pi}_{i},k}$ $\forall i=1,\dots ,l-1;$$\forall k=1,\dots ,m$${e}_{1,1}=max\left\{{e}_{1,0},{e}_{0,1}\right\}+{p}_{{\pi}_{1},1}=max\left\{{e}_{1,0},{e}_{0,1}\right\}+{p}_{1,1}=max\left\{0,0\right\}+1=1$${e}_{1,2}=max\left\{{e}_{1,1},{e}_{0,2}\right\}+{p}_{{\pi}_{1},2}=max\left\{{e}_{1,1},{e}_{0,2}\right\}+{p}_{1,2}=max\left\{1,0\right\}+8=9$${e}_{2,1}=max\left\{{e}_{2,0},{e}_{1,1}\right\}+{p}_{{\pi}_{2},1}=max\left\{{e}_{2,0},{e}_{1,1}\right\}+{p}_{2,1}=max\left\{0,1\right\}+2=3$${e}_{2,2}=max\left\{{e}_{2,1},{e}_{1,2}\right\}+{p}_{{\pi}_{2},2}=max\left\{{e}_{2,1},{e}_{1,2}\right\}+{p}_{2,2}=max\left\{3,9\right\}+9=18$.
- Compute tails:${q}_{i,m+1}=0$ $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$${q}_{l,k}$ = 0 $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$${q}_{i,k}=max\left\{{q}_{i,k+1},{q}_{i+1,k}\right\}+{p}_{{\pi}_{i},k}$ $\forall i=n,\dots ,l-1;\text{}\forall k=m,\dots ,1$${q}_{6,2}=max\left\{{q}_{6,3},{q}_{7,2}\right\}+{p}_{{\pi}_{6},2}=max\left\{{q}_{6,3},{q}_{7,2}\right\}+{p}_{6,2}=max\left\{0,0\right\}+1=1$${q}_{6,1}=max\left\{{q}_{6,2},{q}_{7,1}\right\}+{p}_{{\pi}_{6},1}=max\left\{{q}_{6,2},{q}_{7,1}\right\}+{p}_{6,1}=max\left\{1,0\right\}+7=8$${q}_{5,2}=max\left\{{q}_{5,3},{q}_{6,2}\right\}+{p}_{{\pi}_{5},2}=max\left\{{q}_{5,3},{q}_{6,2}\right\}+{p}_{4,2}=max\left\{0,1\right\}+3=4$${q}_{5,1}=max\left\{{q}_{5,2},{q}_{6,1}\right\}+{p}_{{\pi}_{5},1}=max\left\{{q}_{5,2},{q}_{6,1}\right\}+{p}_{4,1}=max\left\{4,8\right\}+5=13$${q}_{4,2}=max\left\{{q}_{4,3},{q}_{5,2}\right\}+{p}_{{\pi}_{4},2}=max\left\{{q}_{4,3},{q}_{5,2}\right\}+{p}_{5,2}=max\left\{0,4\right\}+4=8$${q}_{4,1}=max\left\{{q}_{4,2},{q}_{5,1}\right\}+{p}_{{\pi}_{4},1}=max\left\{{q}_{4,2},{q}_{5,1}\right\}+{p}_{5,1}=max\left\{8,13\right\}+5=18$${q}_{3,2}=max\left\{{q}_{3,3},{q}_{4,2}\right\}+{p}_{{\pi}_{3},2}=max\left\{{q}_{3,3},{q}_{4,2}\right\}+{p}_{3,2}=max\left\{0,8\right\}+5=13$${q}_{3,1}=max\left\{{q}_{3,2},{q}_{4,1}\right\}+{p}_{{\pi}_{3},1}=max\left\{{q}_{3,2},{q}_{4,1}\right\}+{p}_{3,1}=max\left\{13,18\right\}+7=25$.

- 5.
- Compute the earliest relative completion time ${f}_{i,k}$${f}_{i,0}=0$ $\forall i=1,\dots ,l$${f}_{i,k}=max\left\{{f}_{i,k-1},{e}_{i-1,k}\right\}+{p}_{{\pi}_{i},k}$ $\forall i=1,\dots ,\mathrm{l}$; $\forall k=1,\dots ,m$${f}_{1,1}=max\left\{{f}_{1,0},{e}_{0,1}\right\}+{p}_{{\pi}_{1},1}=max\left\{{f}_{1,0},{e}_{0,1}\right\}+{p}_{1,1}=max\left\{0,0\right\}+1=1$${f}_{1,2}=max\left\{{f}_{1,1},{e}_{0,2}\right\}+{p}_{{\pi}_{1},2}=max\left\{{f}_{1,1},{e}_{0,2}\right\}+{p}_{1,2}=max\left\{1,0\right\}+8=9$${f}_{2,1}=max\left\{{f}_{2,0},{e}_{1,1}\right\}+{p}_{{\pi}_{2},1}=max\left\{{f}_{2,0},{e}_{1,1}\right\}+{p}_{2,1}=max\left\{0,1\right\}+2=3$${f}_{2,2}=max\left\{{f}_{2,1},{e}_{1,2}\right\}+{p}_{{\pi}_{2},2}=max\left\{{f}_{2,1},{e}_{1,2}\right\}+{p}_{2,2}=max\left\{3,9\right\}+9=18$${f}_{3,1}=max\left\{{f}_{3,0},{e}_{2,1}\right\}+{p}_{{\pi}_{3},1}=max\left\{{f}_{3,0},{e}_{2,1}\right\}+{p}_{7,1}=max\left\{0,3\right\}+4=7$${f}_{3,2}=max\left\{{f}_{3,1},{e}_{2,2}\right\}+{p}_{{\pi}_{3},2}=max\left\{{f}_{3,1},{e}_{2,2}\right\}+{p}_{7,2}=max\left\{7,18\right\}+5=23$.Speed-up calculation of the complete solution is given in Figure 2.
- 6.
- The value of the makespan ${C}_{max,l}$ when inserting job ${\pi}_{i}$ at the lth position is:${C}_{max,l}=ma{x}_{k}({f}_{ik}+{q}_{ik})$ $i=l$; $\forall k=1,\dots ,m$${C}_{max,3}=ma{x}_{k}({f}_{ik}+{q}_{ik})$${C}_{max,3}=max\left\{({f}_{31}+{q}_{31}),({f}_{32}+{q}_{32})\right\}$${C}_{max,3}=max\left\{(7+25),(23+13)\right\}$${C}_{max}=max\left\{32,36\right\}=36$.

#### 3.2. IG Algorithms

_{RS}is proposed by [4]. In this algorithm, the initial solution is constructed by the NEH heuristic in [37]. In the destruction step, $dS$ jobs are randomly removed from the solution $\pi $ without repetition and stored in ${\pi}_{D}$. The remaining jobs are also stored in ${\pi}_{P}$ that represents the partial solution. In the construction step, each job in ${\pi}_{D}$ is inserted into the partial solution ${\pi}_{P}$, in the order in which they were removed, until a complete solution of $n$ jobs is constructed. Having carried out the destruction and construction procedure, a local search is employed to further enhance solution quality. After a local search, if the solution is better than or equal to the incumbent solution, it is accepted. Otherwise, it is accepted with a simple simulated annealing-type acceptance criterion, which is suggested by [38]:

_{RS}is given in Algorithm 1, where $r$ is a uniform random number between 0 and 1.

Algorithm 1: Traditional IG_{RS} algorithm |

$\pi =NEH$ |

${\pi}^{best}=\pi $ |

$while\text{}(NotTermination)\text{}do$ |

${\pi}_{D}=Destruction(\pi )$ |

${\pi}^{1}=\mathrm{Construction}({\pi}_{D},{\pi}_{P})$ |

${\pi}^{1}=LocalSearch({\pi}^{1})$ $//Algorithm4$ |

$if\text{}(f({\pi}^{1})\le f(\pi ))then$ |

$\pi ={\pi}^{1}$ |

$if\text{}(f({\pi}^{1})f({\pi}^{best}))then$ |

${\pi}^{best}={\pi}^{1}$ |

$endif$ |

$else$$if\text{}(rexp\left\{-(f({\pi}^{1})-f(\pi ))/T\right\})\text{}then$ |

$\pi ={\pi}^{1}$ |

$endif$ |

$endwhile$ |

$return{\pi}^{best}\text{}and\text{}f({\pi}^{best})$ |

_{RS}algorithm for the PFSP under makespan minimization employs an initial solution generated by the NEH heuristic. In addition, the NEH heuristic was extended to the FRB5 heuristic with a local search on the partial solutions [39]. Both heuristics are simple and very effective for minimizing the makespan, and its pseudo-code is given in Algorithm 2. In the first phase, the sum of the processing times on all machines are calculated for each job. Then, jobs are sorted in decreasing order to obtain $\delta $. In the second phase, the first job in $\delta $ is selected to establish a partial solution ${\pi}_{1}$. The remaining jobs in $\delta $ are inserted in the partial solution one by one. After each iteration, optionally, a local search is applied to the partial solution. Local search is implemented as long as the partial solution is improved. After having inserted all jobs, a complete solution is obtained. Note that the NEH heuristic is denoted as FRB5 heuristic with an optional local search to partial solutions.

Algorithm 2: NEH and FRB5 constructive heuristics |

$\delta =DecreasingOrder({\displaystyle \sum}_{k=1}^{m}{p}_{ik})$ |

${\pi}_{1}={\delta}_{1}$ |

$fori=2tondo$ |

${\pi}_{i}=InsertJobInBestPosition({\pi}_{i},{\delta}_{i})$ |

${\pi}_{i}=ApplyLocalSearch({\pi}_{\mathrm{i}},\text{}f({\pi}_{\mathrm{i}}))$ $//Algorithm3forFRB5heuristic$ |

$endfor$ |

$return\text{}\pi withnjobsandf(\pi )$ |

_{RS}algorithm employs insertion neighborhood structure as a local search after destruction and construction procedure. Insertion neighborhood is very effective with the speed-up method explained in Section 3.1 for makespan minimization. Insertion neighborhood can be deterministic or stochastic depending on the decision of choosing a job from solution to be removed. The deterministic variant is given in Algorithm 3. This procedure removes ${\pi}_{i}$ from the solution $\pi $ and inserts it into all possible positions of the incumbent solution $\pi $. When the best-improving insertion position is found, job ${\pi}_{i}$ is inserted into that position. These steps are repeated for all jobs. If an improvement is observed, the local search is re-run until no better solution is obtained.

Algorithm 3: First improvement insertion neighborhood(π) |

$for\text{}i=1\text{}to\text{}n\text{}do$ |

${\pi}^{*}=InsertJobInBestPosition(\pi ,\text{}{\pi}_{i})$ |

$if\text{}(f({\pi}^{*})f(\pi ))\text{}then\text{}do\text{}$ |

$\pi ={\pi}^{*}$ |

$end\text{}if$ |

$end\text{}for$ |

$return\pi \text{}and\text{}f(\pi )$ |

Algorithm 4: First improvement insertion neighborhood(π) |

$for\text{}i=1\text{}to\text{}n\text{}do$ |

${\pi}_{P}=Removejob{\pi}_{k}fromsolution\pi randomlyandwithoutrepetition$ |

${\pi}^{*}=InsertJobInBestPosition({\pi}_{P},\text{}{\pi}_{k})$ |

$if\text{}(f({\pi}^{*})f(\pi ))\text{}then\text{}do$ |

$\pi ={\pi}^{*}$ |

$end\text{}if$ |

$end\text{}for$ |

$return\pi \text{}and\text{}f(\pi )$ |

_{ALL}algorithm has been presented in the literature [5] with excellent results for the PFSP with makespan minimization. The difference between IG

_{ALL}and IG

_{RS}is that IG

_{ALL}applies an additional local search to partial solutions after destruction, which substantially enhances solution quality. In the IG

_{RS}algorithm, local search is applied to the complete solution after the construction phase to improve the current candidate solution whereas, in IG

_{ALL}algorithm, local search is applied to a partial solution after destruction phase. This idea is applied in heuristic algorithms by Reference [39]. They study on vehicle routing problem and apply local search on the routes in the construction phase. Applying local search to the partial solution is more advantageous in terms of computational time and providing different search directions. Due to having a partial solution, a local search is applied to the smaller size of the complete solution so that the search procedure can be conducted quickly. Another difference between IG

_{RS}and IG

_{ALL}is due to the fact that the initial solution is constructed by FRB5 heuristic. The pseudo code of IG

_{ALL}algorithm is presented in Algorithm 5.

Algorithm 5: IG_{ALL} algorithm |

$\pi =\mathrm{FRB}5$ |

${\pi}^{best}=\pi $ |

$While\text{}(NotTermination)do$ |

${\pi}_{D}=Destruction(\pi )$ |

${\pi}_{P}=LocalSearchToPartialSolution({\pi}_{P})$ $//Algorithm4$ |

${\pi}^{1}=Construction({\pi}_{P},{\pi}_{D})$ |

${\pi}^{1}=LocalSearchToCompleteSolution({\pi}^{1})$ $//Algorithm4$ |

$if\text{}f({\pi}^{1})\le f(\pi )then\text{}do$ |

$\pi ={\pi}^{1}$ |

$if\text{}f({\pi}^{1})f({\pi}^{best})then\text{}do$ |

${\pi}^{best}={\pi}^{1}$ |

$endif$ |

$elseif\text{}(rexp\left\{-(f({\pi}^{1})-f(\pi ))/T\right\})$ |

$\pi ={\pi}^{1}$ |

$endif$ |

$endif$ |

$endwhile$ |

$return\text{}{\pi}^{best}\text{}and\text{}f({\pi}^{best})$ |

$endprocedure$ |

#### 3.3. Variable Block Insertion Algorithm

_{ALL}algorithm, it applies the local search in Algorithm 4 to the partial solution. Then, it makes a number, $n-b+1,$ of block insertion moves sequentially in the partial solution. It chooses the best one amongst those solutions from block insertion moves. Well-known RIS local search in the literature is applied to the complete solution found after block insertion moves. If the new solution obtained after the local search is better than or equal to the current solution, it replaces the current solution. As long as it improves, it retains the same block size ($i.e.,\text{}b=b$). Otherwise, the block size is incremented by one ($i.e.,\text{}b=b+1$) and a simulated annealing-based acceptance criterion, similar to IG

_{RS}and IG

_{ALL}algorithms, is employed to accept the new solution to escape from local minima. This process is repeated until the block size reaches its maximum limit $(i.e.,\text{}b\le {b}_{max})$. The outline of the VBIH algorithm is given in Algorithm 6. Note that ${\pi}^{R}$ is the reference sequence; $tP$ is temperature parameter for the acceptance criterion and r is a uniform random number between 0 and 1.

Algorithm 6: VBIH algorithm |

$\pi =FRB5$ |

${\pi}^{best}=\pi $ |

${\pi}^{R}={\pi}^{best}$ |

$while(NotTermination)$ |

$b={b}_{min}=2$ |

$do$ |

${\pi}_{b}=Removeblock{\pi}_{b}from\pi $ |

${\pi}_{P}=LocalSearchToPartialSolution({\pi}_{P})$ $//Algorithm4$ |

${\pi}^{1}=InsertBlockInBestPosition({\pi}_{P},{\pi}_{b})$ |

${\pi}^{1}=RISLocalSearchToCompleteSolution({\pi}^{1})$ $//Algorithm5$ |

$if(f({\pi}^{1})f(\pi ))thendo$ |

$\pi ={\pi}^{1}$ |

$b=b$ |

$if(f({\pi}^{1})f({\pi}_{best}))thendo$ |

${\pi}^{best}={\pi}^{1}$ |

${\pi}^{R}={\pi}^{best}$ |

$endif$ |

$else$ |

$b=b+1$ |

$if(rexp\left\{-(f({\pi}^{1})-f(\pi ))/T\right\})$ |

$\pi ={\pi}^{1}$ |

$endif$ |

$endif$ |

$while((b\le {b}_{max})$ |

$endwhile$ |

$return{\pi}^{best}andf({\pi}^{best})$ |

_{RS}and IG

_{ALL}algorithms. Note that Taillard’s speed-ups are employed wherever possible in our code.

Algorithm 7: Referenced insertion neighborhood(π) |

$Count=1$ |

$pos=1$ |

${\pi}^{R}={\pi}^{best}$ |

$while(Count\le n)do$ |

$k=1$ |

$while({\pi}_{k}!={\pi}_{Pos}^{R})k=k+1;endwhile$ $//Findjob{\pi}_{k}atpositionposin{\pi}^{R}$ |

$pos=pos+1$ |

$if(pos=n+1)then$ |

$pos=1$ |

$endif$ |

${\pi}_{P}=remove{\pi}_{k}from\pi $ |

${\pi}^{*}=InsertJobInBestPosition({\pi}_{P},{\pi}_{k})$ |

$if(f({\pi}^{*})f(\pi ))thendo$ |

$\pi ={\pi}^{*}$ |

$Count=1$ |

$end$ |

$Count=Count+1$ |

$endif$ |

$endwhile$ |

$return\pi andf(\pi )$ |

## 4. Design of Experiment for Parameter Tuning

## 5. Computational Results

_{RS}and IG

_{ALL}algorithms. In addition, the results of these algorithms are obtained without the Taillard’s speed up method, and they are denoted as IG

_{RS}*, IG

_{ALL}* and VBIH*. Regarding parameters of them with, destruction size $ds,$ and temperature adjustment factor, $tP$ are taken as $ds=4$ and $tP=0.4$ for IG

_{RS}and IG

_{RS}* as suggested in [4]. They are taken as $ds=2$ and $tP=0.7$ for IG

_{ALL}and IG

_{ALL}* as indicated in [5]. As explained in the previous section DOE is conducted for the VBIH algorithm and its parameters are determined as follows: $bMax=2$, $\tau P=0.5$, and $pL=1$, which are also used for the VBIH* algorithm.

#### 5.1. Small VRF Instances

#### 5.1.1. MIP Versus CP

#### 5.1.2. Comparison of Heuristic Algorithms with Exact Solutions

_{ALL}, IG

_{RS}, and VBIH algorithms are run for five replications with three different time limits $15$, $30$, and $45\times n\times m$. As expected, the performance of these algorithms is much better than those by CP exact model, and they improve the upper bounds provided in [9], which means that the proposed algorithm and other IG algorithms can find good (optimal in some cases) solutions in a very short time. As the solution time increases, the solution quality of VBIH algorithm increases and according to the RPD, it gives the best solutions amongst all other algorithms. It should be noted that the VBIH algorithm further improves 64 out 240 upper bounds for small VRF instances within a very short time.

#### 5.2. Large VRF Instances

_{ALL}and VBIH algorithms employ the FRB5 heuristic for constructing initial solution whereas IG

_{RS}uses the traditional NEH heuristic. For the large VRF instances, Table 5 summarizes the ARPD values of heuristic methods such as NEH, NEH without speed-up, denoted as NEH*, and extended NEH heuristic with a local search on partial solutions denoted as FRB5.

#### 5.3. Computational Results of Metaheuristics

_{RS}and IG

_{ALL}, from the literature. All algorithms are run five replications to solve the large VRF instances. In Table 6, we present average, minimum and maximum ARPD values for the CPU time limit ${T}_{max}=15\times n\times m$ milliseconds.

_{RS}and IG

_{ALL}algorithms because their confidence intervals do not coincide.

_{ALL}algorithm was also remarkable. Briefly, both VBIH and IG

_{ALL}outperformed IG

_{RS}in almost each problem set.

_{ALL}algorithms against IG

_{RS}algorithm because their confidence intervals do not coincide with IG

_{RS}. In other words, VBIH and IG

_{ALL}algorithms were statistically equivalent but significantly superior to IG

_{RS}.

_{RS}and IG

_{ALL}algorithms with respect to average, minimum and maximum RPD values on the overall average. On overall average, it was able to further improve the upper bounds by −0.25% on the average value, −0.36% on the minimum value, and its worst-case performance was −0.13%. These statistics indicate that VBIH generated much better results than both the IG

_{RS}and IG

_{ALL}algorithms.

_{RS}and IG

_{ALL}algorithms because their confidence intervals do not coincide. In other words, VBIH algorithm was statistically superior to both IG

_{RS}and IG

_{ALL}algorithm.

_{ALL}and VBIH algorithms. The VBIH algorithm further improves 230 out of 240 instances. In addition, 173 out of 240 instances are improved by the IG

_{RS}algorithm, while the IG

_{ALL}algorithm further improves 222 out of 240 instances. The IG

_{ALL}algorithm improves six instances that are not improved by VBIH algorithm. Ultimately, 236 out of 240 instances are further improved by all algorithms within $45\times n\times m$ time limits with the remaining four solutions being equal.

_{ALL}algorithm is presented in [5], where they analyzed the performances of IG

_{RS}and IG

_{ALL}on both Taillard’s [42] and large VRF instances. They observed that the results obtained by using Taillard’s benchmark set, both algorithms do not present very significant differences with respect to the RPDs obtained. In fact, they have shown that both algorithms did not show any statistically significant differences. However, statistically significant differences between IG

_{RS}and IG

_{ALL}have been shown when large VRF instances are employed. In order to validate this observation, we have run three algorithms on Taillard’s benchmark set with a stopping criterion ${T}_{max}=45\times n\times m$ milliseconds. Furthermore, we run three algorithms without the Taillard’s speed up method and they are denoted as IG

_{RS}*, IG

_{ALL}* and VBIH*. The computational results are given in Table 9. As seen in Table 9, VBIH produced much better RPDs than IG

_{RS}and IG

_{ALL}algorithms when the Taillard’s speed up method is employed since its overall RPD was 0.17 from the best-known solutions. However, IG

_{RS}and IG

_{ALL}algorithms do not show so many differences in terms of RPDs. Interval plots of the algorithms in Figure 10 show that differences in RFDs are not statistically significant because their confidence intervals do coincide. This suggests a fact that researches on PFSP and its variants should employ VRF benchmark suite to see differences in algorithms newly presented. Figure 10 also shows that the Taillard’s speed up method is significantly effective for all three algorithms. During these runs, we were also able to find 3 new best-known solutions for the Taillard’s benchmark suite (ta054 = 3719, ta55 = 3610, ta56 = 3680) and their permutations are also provided in the Supplementary Materials.

## 6. Conclusions

_{ALL}algorithms for the first time in this paper with remaining solutions being equal, which are also given in Appendix B (Table A5). Furthermore, three instances of Taillard’s benchmark suite are also further improved for the first time in this paper since 1993.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Jobs | Machines | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

1 | 456 | 654 | 852 | 145 | 632 | 425 | 214 | 654 |

2 | 789 | 123 | 369 | 678 | 581 | 396 | 123 | 789 |

3 | 654 | 123 | 632 | 965 | 475 | 325 | 456 | 654 |

4 | 321 | 456 | 581 | 421 | 32 | 147 | 789 | 123 |

5 | 456 | 789 | 472 | 365 | 536 | 852 | 654 | 123 |

6 | 789 | 654 | 586 | 824 | 325 | 12 | 321 | 456 |

7 | 654 | 321 | 320 | 758 | 863 | 452 | 456 | 789 |

8 | 789 | 147 | 120 | 639 | 21 | 863 | 789 | 654 |

${\mathit{e}}_{\mathit{j},\mathit{k}}$ | Machines | ||||||||
---|---|---|---|---|---|---|---|---|---|

Job | Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

7 | 1 | 654 | 975 | 1295 | 2053 | 2916 | 3368 | 3824 | 4613 |

3 | 2 | 1308 | 1431 | 2063 | 3028 | 3503 | 3828 | 4284 | 5267 |

8 | 3 | 2097 | 2244 | 2364 | 3667 | 3688 | 4691 | 5480 | 6134 |

5 | 4 | 2553 | 3342 | 3814 | 4179 | 4715 | 5567 | 6221 | 6344 |

1 | 5 | 3009 | 3996 | 4848 | 4993 | 5625 | 6050 | 6435 | 7089 |

6 | 6 | 3798 | 4650 | 5434 | 6258 | 6583 | 6595 | 6916 | 7545 |

4 | 7 | 4119 | 5106 | 6015 | 6679 | 6711 | 6858 | 7705 | 7828 |

${\mathit{f}}_{\mathit{j},\mathit{k}}$ | Machines | ||||||||
---|---|---|---|---|---|---|---|---|---|

Job | Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

7 | 1 | 654 | 975 | 1295 | 2053 | 2916 | 3368 | 3824 | 4613 |

3 | 2 | 1308 | 1431 | 2063 | 3028 | 3503 | 3828 | 4284 | 5267 |

8 | 3 | 2097 | 2244 | 2364 | 3667 | 3688 | 4691 | 5480 | 6134 |

5 | 4 | 2553 | 3342 | 3814 | 4179 | 4715 | 5567 | 6221 | 6344 |

2 | 5 | 3342 | 3465 | 4183 | 4861 | 5442 | 5936 | 6344 | 7133 |

${\mathit{q}}_{\mathit{j},\mathit{k}}$ | Machines | ||||||||
---|---|---|---|---|---|---|---|---|---|

Job | Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1 | 6 | 4942 | 4486 | 3832 | 2649 | 2504 | 1872 | 1447 | 1233 |

6 | 7 | 4423 | 3634 | 2980 | 2394 | 1570 | 1245 | 1233 | 579 |

4 | 8 | 2870 | 2549 | 2093 | 1512 | 1091 | 1059 | 912 | 123 |

## Appendix B

**Table A5.**New best solutions of our algorithms for Large VRF Instances (The bolds shows the new best known solutions).

Instance | Cmax | Best | Instance | Cmax | Best | Instance | Cmax | Best |
---|---|---|---|---|---|---|---|---|

100_20_1 | 6198 | 6173 | 300_60_1 | 20522 | 20483 | 600_40_1 | 33839 | 33683 |

100_20_2 | 6306 | 6267 | 300_60_2 | 20399 | 20249 | 600_40_2 | 33467 | 33405 |

100_20_3 | 6238 | 6221 | 300_60_3 | 20434 | 20328 | 600_40_3 | 33866 | 33713 |

100_20_4 | 6245 | 6227 | 300_60_4 | 20395 | 20293 | 600_40_4 | 33693 | 33584 |

100_20_5 | 6296 | 6264 | 300_60_5 | 20341 | 20200 | 600_40_5 | 33553 | 33401 |

100_20_6 | 6321 | 6285 | 300_60_6 | 20388 | 20280 | 600_40_6 | 33809 | 33626 |

100_20_7 | 6434 | 6401 | 300_60_7 | 20457 | 20358 | 600_40_7 | 33686 | 33545 |

100_20_8 | 6104 | 6074 | 300_60_8 | 20410 | 20319 | 600_40_8 | 33482 | 33298 |

100_20_9 | 6354 | 6328 | 300_60_9 | 20549 | 20405 | 600_40_9 | 33697 | 33567 |

100_20_10 | 6145 | 6125 | 300_60_10 | 20472 | 20385 | 600_40_10 | 33642 | 33473 |

100_40_1 | 7881 | 7846 | 400_20_1 | 21120 | 21042 | 600_60_1 | 36198 | 35976 |

100_40_2 | 8007 | 7976 | 400_20_2 | 21457 | 21411 | 600_60_2 | 36184 | 35923 |

100_40_3 | 7935 | 7894 | 400_20_3 | 21441 | 21428 | 600_60_3 | 36201 | 35917 |

100_40_4 | 7932 | 7913 | 400_20_4 | 21247 | 21237 | 600_60_4 | 36136 | 36000 |

100_40_5 | 8011 | 7997 | 400_20_5 | 21553 | 21528 | 600_60_5 | 36153 | 36004 |

100_40_6 | 8023 | 7993 | 400_20_6 | 21214 | 21188 | 600_60_6 | 36116 | 35943 |

100_40_7 | 8006 | 7980 | 400_20_7 | 21625 | 21599 | 600_60_7 | 36179 | 35965 |

100_40_8 | 7979 | 7957 | 400_20_8 | 21277 | 21264 | 600_60_8 | 36185 | 35894 |

100_40_9 | 7931 | 7888 | 400_20_9 | 21346 | 21293 | 600_60_9 | 36195 | 35987 |

100_40_10 | 7952 | 7917 | 400_20_10 | 21538 | 21526 | 600_60_10 | 36163 | 35943 |

100_60_1 | 9395 | 9353 | 400_40_1 | 23578 | 23393 | 700_20_1 | 36394 | 36388 |

100_60_2 | 9596 | 9567 | 400_40_2 | 23456 | 23380 | 700_20_2 | 36337 | 36316 |

100_60_3 | 9349 | 9349 | 400_40_3 | 23575 | 23467 | 700_20_3 | 36568 | 36519 |

100_60_4 | 9426 | 9403 | 400_40_4 | 23409 | 23269 | 700_20_4 | 36452 | 36380 |

100_60_5 | 9465 | 9431 | 400_40_5 | 23339 | 23213 | 700_20_5 | 36584 | 36556 |

100_60_6 | 9667 | 9630 | 400_40_6 | 23444 | 23298 | 700_20_6 | 36671 | 36645 |

100_60_7 | 9391 | 9346 | 400_40_7 | 23556 | 23415 | 700_20_7 | 36624 | 36597 |

100_60_8 | 9534 | 9523 | 400_40_8 | 23411 | 23290 | 700_20_8 | 36522 | 36492 |

100_60_9 | 9527 | 9488 | 400_40_9 | 23637 | 23424 | 700_20_9 | 36329 | 36315 |

100_60_10 | 9598 | 9572 | 400_40_10 | 23720 | 23606 | 700_20_10 | 36417 | 36386 |

200_20_1 | 11305 | 11272 | 400_60_1 | 25607 | 25395 | 700_40_1 | 38964 | 38767 |

200_20_2 | 11265 | 11240 | 400_60_2 | 25656 | 25549 | 700_40_2 | 38775 | 38560 |

200_20_3 | 11327 | 11294 | 400_60_3 | 25821 | 25707 | 700_40_3 | 38621 | 38460 |

200_20_4 | 11208 | 11188 | 400_60_4 | 25837 | 25638 | 700_40_4 | 38785 | 38597 |

200_20_5 | 11208 | 11143 | 400_60_5 | 25877 | 25669 | 700_40_5 | 38671 | 38490 |

200_20_6 | 11367 | 11310 | 400_60_6 | 25536 | 25407 | 700_40_6 | 38710 | 38440 |

200_20_7 | 11380 | 11365 | 400_60_7 | 25600 | 25415 | 700_40_7 | 38585 | 38355 |

200_20_8 | 11141 | 11128 | 400_60_8 | 25800 | 25603 | 700_40_8 | 39059 | 38817 |

200_20_9 | 11123 | 11091 | 400_60_9 | 25882 | 25673 | 700_40_9 | 38814 | 38569 |

200_20_10 | 11310 | 11294 | 400_60_10 | 25767 | 25658 | 700_40_10 | 38850 | 38712 |

200_40_1 | 13132 | 13124 | 500_20_1 | 26411 | 26374 | 700_60_1 | 41436 | 41192 |

200_40_2 | 13102 | 13049 | 500_20_2 | 26681 | 26641 | 700_60_2 | 41375 | 41002 |

200_40_3 | 13264 | 13222 | 500_20_3 | 26409 | 26359 | 700_60_3 | 41317 | 41173 |

200_40_4 | 13232 | 13163 | 500_20_4 | 26124 | 26080 | 700_60_4 | 41401 | 41120 |

200_40_5 | 13043 | 12974 | 500_20_5 | 26781 | 26759 | 700_60_5 | 41262 | 41167 |

200_40_6 | 13124 | 13061 | 500_20_6 | 26443 | 26411 | 700_60_6 | 41340 | 41159 |

200_40_7 | 13299 | 13220 | 500_20_7 | 26433 | 26409 | 700_60_7 | 40876 | 40734 |

200_40_8 | 13238 | 13132 | 500_20_8 | 26318 | 26305 | 700_60_8 | 41474 | 41305 |

200_40_9 | 13166 | 13033 | 500_20_9 | 26442 | 26430 | 700_60_9 | 41291 | 41111 |

200_40_10 | 13228 | 13146 | 500_20_10 | 26072 | 26034 | 700_60_10 | 41377 | 41186 |

200_60_1 | 14990 | 14906 | 500_40_1 | 28548 | 28402 | 800_20_1 | 41558 | 41479 |

200_60_2 | 14954 | 14909 | 500_40_2 | 28793 | 28613 | 800_20_2 | 41407 | 41345 |

200_60_3 | 15200 | 15134 | 500_40_3 | 28607 | 28526 | 800_20_3 | 41425 | 41399 |

200_60_4 | 15044 | 14968 | 500_40_4 | 28828 | 28615 | 800_20_4 | 41426 | 41426 |

200_60_5 | 15130 | 15042 | 500_40_5 | 28683 | 28579 | 800_20_5 | 41710 | 41705 |

200_60_6 | 15035 | 14996 | 500_40_6 | 28524 | 28432 | 800_20_6 | 42010 | 41961 |

200_60_7 | 15040 | 15006 | 500_40_7 | 28760 | 28553 | 800_20_7 | 41425 | 41395 |

200_60_8 | 14968 | 14894 | 500_40_8 | 28698 | 28488 | 800_20_8 | 41492 | 41435 |

200_60_9 | 15022 | 14925 | 500_40_9 | 28870 | 28640 | 800_20_9 | 41796 | 41783 |

200_60_10 | 15000 | 14908 | 500_40_10 | 28758 | 28644 | 800_20_10 | 41574 | 41568 |

300_20_1 | 16149 | 16089 | 500_60_1 | 30861 | 30682 | 800_40_1 | 43671 | 43466 |

300_20_2 | 16512 | 16483 | 500_60_2 | 30828 | 30664 | 800_40_2 | 43746 | 43575 |

300_20_3 | 16173 | 16129 | 500_60_3 | 31125 | 30852 | 800_40_3 | 43749 | 43596 |

300_20_4 | 16181 | 16168 | 500_60_4 | 30928 | 30793 | 800_40_4 | 43892 | 43743 |

300_20_5 | 16342 | 16307 | 500_60_5 | 30935 | 30763 | 800_40_5 | 43905 | 43794 |

300_20_6 | 16137 | 16095 | 500_60_6 | 31027 | 30788 | 800_40_6 | 43811 | 43638 |

300_20_7 | 16266 | 16244 | 500_60_7 | 30928 | 30826 | 800_40_7 | 43766 | 43484 |

300_20_8 | 16416 | 16369 | 500_60_8 | 30988 | 30837 | 800_40_8 | 43839 | 43666 |

300_20_9 | 16376 | 16324 | 500_60_9 | 30978 | 30805 | 800_40_9 | 43879 | 43643 |

300_20_10 | 16899 | 16798 | 500_60_10 | 31050 | 30866 | 800_40_10 | 43861 | 43630 |

300_40_1 | 18298 | 18199 | 600_20_1 | 31433 | 31372 | 800_60_1 | 46470 | 46279 |

300_40_2 | 18454 | 18373 | 600_20_2 | 31418 | 31397 | 800_60_2 | 46493 | 46232 |

300_40_3 | 18457 | 18348 | 600_20_3 | 31429 | 31429 | 800_60_3 | 46389 | 46258 |

300_40_4 | 18351 | 18227 | 600_20_4 | 31547 | 31487 | 800_60_4 | 46457 | 46261 |

300_40_5 | 18484 | 18343 | 600_20_5 | 31448 | 31407 | 800_60_5 | 46401 | 46164 |

300_40_6 | 18449 | 18340 | 600_20_6 | 31717 | 31696 | 800_60_6 | 46421 | 46288 |

300_40_7 | 18419 | 18396 | 600_20_7 | 31527 | 31527 | 800_60_7 | 46319 | 46061 |

300_40_8 | 18392 | 18290 | 600_20_8 | 31564 | 31523 | 800_60_8 | 46474 | 46257 |

300_40_9 | 18394 | 18261 | 600_20_9 | 31577 | 31532 | 800_60_9 | 46538 | 46279 |

300_40_10 | 18401 | 18286 | 600_20_10 | 31130 | 31107 | 800_60_10 | 46244 | 46211 |

## References

- Fernandez-Viagas, V.; Ruiz, R.; Framinan, J.M. A new vision of approximate methods for the permutation flowshop to minimise makespan: State-of-the-art and computational evaluation. Eur. J. Oper. Res.
**2017**, 257, 707–721. [Google Scholar] [CrossRef] - Pinedo, M.L. Scheduling: Theory, Algorithms, and Systems; Springer: New York, NY, USA, 2008. [Google Scholar]
- Garey, M.R.; Johnson, D.S.; Sethi, R. The Complexity of Flowshop and Jobshop Scheduling. Math. Oper. Res.
**1976**, 1, 117–129. [Google Scholar] [CrossRef] - Ruiz, R.; Stützle, T. A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. Eur. J. Oper. Res.
**2007**, 177, 2033–2049. [Google Scholar] [CrossRef] - Dubois-Lacoste, J.; Pagnozzi, F.; Stützle, T. An iterated greedy algorithm with optimization of partial solutions for the makespan permutation flowshop problem. Comput. Oper. Res.
**2017**, 81, 160–166. [Google Scholar] [CrossRef] - Ruiz, R.; Stützle, T. An Iterated Greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives. Eur. J. Oper. Res.
**2008**, 187, 1143–1159. [Google Scholar] [CrossRef] - Fernandez-Viagas, V.; Framinan, J. On insertion tie-breaking rules in heuristics for the permutation flowshop scheduling problem. Comput. Oper. Res.
**2014**, 45, 60–67. [Google Scholar] [CrossRef] - Pan, Q.-K.; Tasgetiren, M.F.; Liang, Y.-C. A discrete differential evolution algorithm for the permutation flowshop scheduling problem. Comput. Ind. Eng.
**2008**, 55, 795–816. [Google Scholar] [CrossRef] - Vallada, E.; Ruiz, R.; Framinan, J.M. New hard benchmark for flowshop scheduling problems minimising makespan. Eur. J. Oper. Res.
**2015**, 240, 666–677. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Pan, Q.-K.; Kizilay, D.; Velez-Gallego, M.C. A variable block insertion heuristic for permutation flowshops with makespan criterion. In Proceedings of the 2017 IEEE Congress on Evolutionary Computation (CEC), San Sebastian, Spain, 5–8 June 2017. [Google Scholar]
- Shao, W.; Pi, D.; Shao, Z. Optimization of makespan for the distributed no-wait flow shop scheduling problem with iterated greedy algorithms. Knowl. Based Syst.
**2017**, 137, 163–181. [Google Scholar] [CrossRef] - Ding, J.-Y.; Song, S.; Gupta, J.; Zhang, R.; Chiong, R.; Wu, C. An improved iterated greedy algorithm with a Tabu-based reconstruction strategy for the no-wait flowshop scheduling problem. Appl. Soft Comput.
**2015**, 30, 604–613. [Google Scholar] [CrossRef] - Li, X.; Yang, Z.; Ruiz, R.; Chen, T.; Sui, S. An iterated greedy heuristic for no-wait flow shops with sequence dependent setup times, learning and forgetting effects. Inf. Sci.
**2018**, 453, 408–425. [Google Scholar] [CrossRef] - Ribas, I.; Companys, R.; Tort-Martorell, X. An iterated greedy algorithm for the flowshop scheduling problem with blocking. Omega
**2011**, 39, 293–301. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Kizilay, D.; Pan, Q.-K.; Suganthan, P.N. Iterated greedy algorithms for the blocking flowshop scheduling problem with makespan criterion. Comput. Oper. Res.
**2017**, 77, 111–126. [Google Scholar] [CrossRef] - Fernandez-Viagas, V.; Leisten, R.; Framinan, J. A computational evaluation of constructive and improvement heuristics for the blocking flow shop to minimise total flowtime. Expert Syst. Appl.
**2016**, 61, 290–301. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Pan, Q.-K.; Kizilay, D.; Suer, G. A populated local search with differential evolution for blocking flowshop scheduling problem. In Proceedings of the 2015 IEEE Congress on Evolutionary Computation (CEC), Sendai, Japan, 25–28 May 2015. [Google Scholar]
- Ying, K.-C.; Lin, S.-W.; Cheng, C.-Y.; He, C.-D. Iterated reference greedy algorithm for solving distributed no-idle permutation flowshop scheduling problems. Comput. Ind. Eng.
**2017**, 110, 413–423. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Pan, Q.-K.; Suganthan, P.N.; Buyukdagli, O. A variable iterated greedy algorithm with differential evolution for the no-idle permutation flowshop scheduling problem. Comput. Oper. Res.
**2013**, 40, 1729–1743. [Google Scholar] [CrossRef] - Pan, Q.-K.; Ruiz, R. An effective iterated greedy algorithm for the mixed no-idle permutation flowshop scheduling problem. Omega
**2014**, 44, 41–50. [Google Scholar] [CrossRef] - Ding, J.-Y.; Song, S.; Wu, C. Carbon-efficient scheduling of flow shops by multi-objective optimization. Eur. J. Oper. Res.
**2016**, 248, 758–771. [Google Scholar] [CrossRef] - Öztop, H.; Tasgetiren, M.F.; Eliiyi, D.T.; Pan, Q.-K. Green Permutation Flowshop Scheduling: A Trade- off- Between Energy Consumption and Total Flow Time. In Intelligent Computing Methodologies; Springer: Cham, Switzerland, 2018; pp. 753–759. [Google Scholar]
- Minella, G.; Ruiz, R.; Ciavotta, M. Restarted Iterated Pareto Greedy algorithm for multi-objective flowshop scheduling problems. Comput. Oper. Res.
**2011**, 38, 1521–1533. [Google Scholar] [CrossRef] - Ciavotta, M.; Minella, G.; Ruiz, R. Multi-objective sequence dependent setup times permutation flowshop: A new algorithm and a comprehensive study. Eur. J. Oper. Res.
**2013**, 227, 301–312. [Google Scholar] [CrossRef] - Pan, Q.-K.; Wang, L. Effective heuristics for the blocking flowshop scheduling problem with makespan minimization. Omega
**2012**, 40, 218–229. [Google Scholar] [CrossRef] - Karabulut, K. A hybrid iterated greedy algorithm for total tardiness minimization in permutation flowshops. Comput. Ind. Eng.
**2016**, 98, 300–307. [Google Scholar] [CrossRef] - Fernandez-Viagas, V.; Valente, J.M.S.; Framinan, J. Iterated-greedy-based algorithms with beam search initialization for the permutation flowshop to minimise total tardiness. Expert Syst. Appl.
**2018**, 94, 58–69. [Google Scholar] [CrossRef] - Pan, Q.-K.; Ruiz, R. Local search methods for the flowshop scheduling problem with flowtime minimization. Eur. J. Oper. Res.
**2012**, 222, 31–43. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Pan, Q.; Ozturkoglu, Y.; Chen, A.H.L. A memetic algorithm with a variable block insertion heuristic for single machine total weighted tardiness problem with sequence dependent setup times. In Proceedings of the 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, Canada, 24–29 July 2016; pp. 2911–2918. [Google Scholar]
- Subramanian, A.; Battarra, M.; Potts, C.N. An Iterated Local Search heuristic for the single machine total weighted tardiness scheduling problem with sequence-dependent setup times. Int. J. Prod. Res.
**2014**, 52, 2729–2742. [Google Scholar] [CrossRef] - Xu, H.; Lü, Z.; Cheng, T.C.E. Iterated Local Search for single-machine scheduling with sequence-dependent setup times to minimize total weighted tardiness. J. Sched.
**2014**, 17, 271–287. [Google Scholar] [CrossRef] - Fernández, M.Á.G.; Palacios, J.; Vela, C.; Hernández-Arauzo, A. Scatter search for minimizing weighted tardiness in a single machine scheduling with setups. J. Heuristics
**2017**, 23, 81–110. [Google Scholar] - Tasgetiren, M.F.; Pan, Q.-K.; Kizilay, D.; Gao, K. A Variable Block Insertion Heuristic for the Blocking Flowshop Scheduling Problem with Total Flowtime Criterion. Algorithms
**2016**, 9, 71. [Google Scholar] [CrossRef] - Manne, A.S. On the Job-Shop Scheduling Problem. Oper. Res.
**1960**, 8, 219–223. [Google Scholar] [CrossRef] - Taillard, E. Some efficient heuristic methods for the flow shop sequencing problem. Eur. J. Oper. Res.
**1990**, 47, 65–74. [Google Scholar] [CrossRef] - Johnson, S.M. Optimal Two and Three Stage Production Schedules with Set-Up Time Included. Nav. Res. Logist. Q.
**1954**, 1, 61–68. [Google Scholar] [CrossRef] - Nawaz, M.; Enscore, E.E.; Ham, I. A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega
**1983**, 11, 91–95. [Google Scholar] [CrossRef] - Osman, I.; Potts, C.N. Simulated Annealing for Permutation Flow-Shop Scheduling. Omega
**1989**, 17, 551–557. [Google Scholar] [CrossRef] - Rad, S.F.; Ruiz, R.; Boroojerdian, N. New high performing heuristics for minimizing makespan in permutation flowshops. Omega
**2009**, 37, 331–345. [Google Scholar] [CrossRef] - Tasgetiren, M.F.; Pan, Q.-K.; Suganthan, P.N.; Chua, T.J. A differential evolution algorithm for the no-idle flowshop scheduling problem with total tardiness criterion. Int. J. Prod. Res.
**2011**, 49, 5033–5050. [Google Scholar] [CrossRef] - Montgomery, D.C. Design and Analysis of Experiments, 2nd ed.; Wiley: New York, NY, USA, 1984. [Google Scholar]
- Taillard, E. Benchmarks for basic scheduling problems. Eur. J. Oper. Res.
**1993**, 64, 278–285. [Google Scholar] [CrossRef]

Instance | Optimal Solution with ${\mathit{C}}_{\mathit{m}\mathit{a}\mathit{x}}=36$ | |||||
---|---|---|---|---|---|---|

Jobs | M1 | M2 | Jobs | Position | M1 | M2 |

1 | 1 | 8 | 1 | 1 | 1 | 8 |

2 | 2 | 9 | 2 | 2 | 2 | 9 |

3 | 7 | 5 | 7 | 3 | 4 | 5 |

4 | 5 | 3 | 3 | 4 | 7 | 5 |

5 | 5 | 4 | 5 | 5 | 5 | 4 |

6 | 7 | 1 | 4 | 6 | 5 | 3 |

7 | 4 | 5 | 6 | 7 | 7 | 1 |

Source | DF | Seq SS | Adj SS | Adj MS | F | p-Value |
---|---|---|---|---|---|---|

$bMax$ | 6 | 0.0086 | 0.0086 | 0.0014 | 33.370 | 0.000 |

$tP$ | 4 | 0.0090 | 0.0090 | 0.0022 | 52.080 | 0.000 |

$pL$ | 1 | 5.5441 | 5.5441 | 5.5441 | 129,096.720 | 0.000 |

$bMax\times tP$ | 24 | 0.0010 | 0.0010 | 0.0000 | 0.990 | 0.505 |

$bMax\times pL$ | 6 | 0.0025 | 0.0025 | 0.0004 | 9.830 | 0.000 |

$tP\times pL$ | 4 | 0.0090 | 0.0090 | 0.0022 | 52.100 | 0.000 |

Error | 24 | 0.0010 | 0.0010 | 0.0000 | ||

Total | 69 | 5.5752 |

**Table 3.**MIP and CP results for VRF small benchmarks with 3600 s time limit (The number in bold shows the total optimal solutions).

n × m | CP | MIP | ||||||
---|---|---|---|---|---|---|---|---|

nOpt | ARPD | CPU | GAP | nOpt | RPD | CPU | GAP | |

10 × 5 | 10 | 0 | 14.03 | 0 | 10 | 0 | 2.68 | 0 |

10 × 10 | 10 | 0 | 102.13 | 0 | 10 | 0 | 4.35 | 0 |

10 × 15 | 10 | 0 | 256.45 | 0 | 10 | 0 | 5.68 | 0 |

10 × 20 | 10 | 0 | 452.79 | 0 | 10 | 0 | 9.59 | 0 |

20 × 5 | 10 | 0 | 2.49 | 0 | 0 | 0.58 | 3600.18 | 0.37 |

20 × 10 | 6 | 0.11 | 2250.09 | 0.03 | 0 | 2.24 | 3600.51 | 0.32 |

20 × 15 | 0 | 0.53 | 3600.05 | 0.13 | 0 | 2.54 | 3600.06 | 0.29 |

20 × 20 | 0 | 0.48 | 3600.07 | 0.17 | 40 | 2.61 | 3600.06 | 0.25 |

30 × 5 | 10 | 0 | 5.82 | 0 | Na | Na | Na | Na |

30 × 10 | 2 | 0.47 | 3191.89 | 0.05 | Na | Na | Na | Na |

30 × 15 | 0 | 1.29 | 3600.14 | 0.11 | Na | Na | Na | Na |

30 × 20 | 0 | 1.63 | 3600.13 | 0.15 | Na | Na | Na | Na |

40 × 5 | 10 | 0 | 15.03 | 0 | Na | Na | Na | Na |

40 × 10 | 3 | 0.22 | 3113.36 | 0.03 | Na | Na | Na | Na |

40 × 15 | 0 | 2.16 | 3600.10 | 0.10 | Na | Na | Na | Na |

40 × 20 | 0 | 2.11 | 3600.16 | 0.13 | Na | Na | Na | Na |

50 × 5 | 10 | 0 | 11.64 | 0 | Na | Na | Na | Na |

50 × 10 | 3 | 0.19 | 2939.96 | 0.02 | Na | Na | Na | Na |

50 × 15 | 0 | 2.28 | 3600.22 | 0.08 | Na | Na | Na | Na |

50 × 20 | 0 | 2.73 | 3600.22 | 0.12 | Na | Na | Na | Na |

60 × 15 | 10 | 0 | 6.44 | 0 | Na | Na | Na | Na |

60 × 10 | 4 | 0.19 | 3158.95 | 0.01 | Na | Na | Na | Na |

60 × 15 | 0 | 1.98 | 3600.19 | 0.07 | Na | Na | Na | Na |

60 × 20 | 0 | 2.82 | 3600.29 | 0.10 | Na | Na | Na | Na |

Overall Avg. | 108 | 0.80 | 2146.78 | 0.05 | 40 | 2.61 | 3600.06 | 0.25 |

Instance | CP | 15 × n × m | 30 × n × m | 45 × n × m | ||||||
---|---|---|---|---|---|---|---|---|---|---|

IG_{RS} | IG_{ALL} | VBIH | IG_{RS} | IG_{ALL} | VBIH | IG_{RS} | IG_{ALL} | VBIH | ||

10 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

10 × 10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

10 × 15 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00 | 0.00 | 0.02 | 0.00 | 0.00 | 0.02 |

10 × 20 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

20 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

20 × 10 | 0.11 | 0.04 | 0.00 | 0.04 | 0.03 | 0.00 | 0.04 | 0.02 | 0.00 | 0.04 |

20 × 15 | 0.53 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

20 × 20 | 0.48 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

30 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

30 × 10 | 0.47 | 0.06 | 0.04 | 0.05 | 0.01 | 0.03 | 0.01 | 0.01 | 0.03 | −0.01 |

30 × 15 | 1.29 | 0.03 | 0.02 | 0.03 | 0.02 | −0.02 | 0.02 | 0.02 | −0.02 | 0.02 |

30 × 20 | 1.63 | 0.02 | 0.00 | 0.03 | 0.02 | 0.00 | 0.02 | 0.02 | 0.00 | 0.02 |

40 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

40 × 10 | 0.22 | 0.06 | 0.02 | 0.03 | 0.02 | 0.01 | −0.01 | 0.00 | 0.00 | −0.01 |

40 × 15 | 2.16 | 0.09 | 0.05 | 0.04 | 0.04 | 0.02 | −0.02 | −0.01 | −0.05 | −0.05 |

40 × 20 | 2.11 | 0.10 | −0.08 | −0.04 | 0.04 | −0.08 | −0.05 | −0.01 | −0.08 | −0.07 |

50 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

50 × 10 | 0.19 | 0.16 | 0.14 | 0.04 | 0.11 | 0.11 | 0.00 | 0.08 | 0.08 | −0.03 |

50 × 15 | 2.28 | 0.24 | 0.18 | 0.10 | 0.15 | 0.14 | 0.05 | 0.10 | 0.09 | 0.02 |

50 × 20 | 2.73 | 0.17 | 0.02 | 0.00 | 0.07 | −0.08 | −0.10 | 0.04 | −0.11 | −0.10 |

60 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

60 × 10 | 0.19 | 0.07 | 0.11 | −0.01 | −0.04 | 0.08 | −0.03 | −0.06 | 0.05 | −0.05 |

60 × 15 | 1.98 | 0.21 | 0.09 | 0.10 | 0.12 | 0.06 | 0.01 | 0.08 | 0.06 | −0.04 |

60 × 20 | 2.81 | 0.20 | 0.01 | 0.00 | 0.03 | −0.07 | −0.12 | −0.03 | −0.08 | −0.17 |

Avg. | 0.80 | 0.06 | 0.02 | 0.02 | 0.03 | 0.01 | −0.01 | 0.01 | 0.00 | −0.02 |

**Table 5.**Comparison of ARPD and computation time (CPU) for constructive heuristic methods (The number in bold shows better results).

Instance | NEH | NEH * | FRB5 | |||
---|---|---|---|---|---|---|

ARPD | CPU(s) | ARPD | CPU(s) | ARPD | CPU(s) | |

100 × 20 | 5.82 | 0.00 | 5.82 | 0.01 | 2.45 | 0.10 |

100 × 40 | 5.30 | 0.00 | 5.30 | 0.03 | 2.57 | 0.21 |

100 × 60 | 4.89 | 0.00 | 4.89 | 0.05 | 2.19 | 0.32 |

200 × 20 | 4.15 | 0.00 | 4.15 | 0.10 | 1.42 | 0.89 |

200 × 40 | 4.81 | 0.01 | 4.81 | 0.23 | 1.67 | 1.91 |

200 × 60 | 4.48 | 0.01 | 4.48 | 0.39 | 1.56 | 2.73 |

300 × 20 | 3.17 | 0.01 | 3.17 | 0.33 | 0.80 | 2.75 |

300 × 40 | 4.05 | 0.02 | 4.05 | 0.79 | 1.07 | 6.45 |

300 × 60 | 3.94 | 0.03 | 3.94 | 1.31 | 1.23 | 9.85 |

400 × 20 | 2.44 | 0.01 | 2.44 | 0.80 | 0.50 | 6.27 |

400 × 40 | 3.80 | 0.03 | 3.80 | 1.91 | 0.82 | 15.83 |

400 × 60 | 3.42 | 0.04 | 3.42 | 3.14 | 0.75 | 24.39 |

500 × 20 | 2.06 | 0.02 | 2.06 | 1.53 | 0.43 | 12.10 |

500 × 40 | 3.17 | 0.04 | 3.17 | 3.75 | 0.63 | 31.73 |

500 × 60 | 3.27 | 0.06 | 3.27 | 6.05 | 0.57 | 47.97 |

600 × 20 | 1.70 | 0.03 | 1.70 | 2.60 | 0.24 | 20.76 |

600 × 40 | 2.96 | 0.06 | 2.96 | 6.34 | 0.53 | 54.97 |

600 × 60 | 2.97 | 0.09 | 2.97 | 10.31 | 0.37 | 82.27 |

700 × 20 | 1.42 | 0.04 | 1.42 | 4.13 | 0.25 | 31.50 |

700 × 40 | 2.80 | 0.08 | 2.80 | 10.06 | 0.26 | 84.38 |

700 × 60 | 2.66 | 0.13 | 2.66 | 17.22 | 0.32 | 249.99 |

800 × 20 | 1.35 | 0.04 | 1.35 | 6.06 | 0.21 | 42.31 |

800 × 40 | 2.45 | 0.10 | 2.45 | 15.48 | 0.24 | 125.13 |

800 × 60 | 2.74 | 0.16 | 2.74 | 26.17 | 0.31 | 195.41 |

Avg | 3.33 | 0.04 | 3.33 | 4.95 | 0.89 | 43.76 |

**Table 6.**Computational results of algorithms with ${T}_{max}=15\times n\times m$ milliseconds (The bolds show better results).

Instance | IG_{RS} | IG_{ALL} | VBIH | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | Min | Max | Avg. | Min | Max | Avg. | Min | Max | |

100 × 20 | 0.45 | 0.13 | 0.74 | 0.12 | −0.07 | 0.33 | 0.00 | −0.21 | 0.23 |

100 × 40 | 0.56 | 0.26 | 0.90 | 0.28 | 0.04 | 0.49 | 0.13 | −0.09 | 0.37 |

100 × 60 | 0.50 | 0.22 | 0.78 | 0.23 | 0.02 | 0.42 | 0.27 | 0.05 | 0.54 |

200 × 20 | 0.42 | 0.24 | 0.61 | 0.19 | 0.04 | 0.35 | 0.03 | −0.14 | 0.17 |

200 × 40 | 0.47 | 0.25 | 0.68 | 0.14 | −0.01 | 0.31 | 0.01 | −0.21 | 0.24 |

200 × 60 | 0.46 | 0.24 | 0.65 | 0.17 | −0.01 | 0.37 | 0.05 | −0.15 | 0.22 |

300 × 20 | 0.22 | 0.06 | 0.35 | 0.10 | −0.03 | 0.21 | −0.03 | −0.17 | 0.11 |

300 × 40 | 0.35 | 0.15 | 0.56 | 0.04 | −0.16 | 0.25 | −0.18 | −0.35 | −0.02 |

300 × 60 | 0.36 | 0.16 | 0.56 | 0.12 | −0.06 | 0.27 | −0.03 | −0.20 | 0.15 |

400 × 20 | 0.20 | 0.11 | 0.33 | 0.09 | 0.01 | 0.18 | 0.03 | −0.03 | 0.10 |

400 × 40 | 0.31 | 0.12 | 0.50 | 0.01 | −0.11 | 0.14 | −0.17 | −0.32 | −0.03 |

400 × 60 | 0.27 | 0.08 | 0.46 | −0.02 | −0.17 | 0.12 | −0.16 | −0.27 | −0.05 |

500 × 20 | 0.15 | 0.06 | 0.26 | 0.12 | 0.07 | 0.18 | 0.03 | −0.05 | 0.12 |

500 × 40 | 0.29 | 0.12 | 0.45 | 0.00 | −0.10 | 0.11 | −0.19 | −0.30 | −0.07 |

500 × 60 | 0.33 | 0.15 | 0.51 | −0.06 | −0.20 | 0.08 | −0.19 | −0.31 | −0.06 |

600 × 20 | 0.11 | 0.03 | 0.18 | 0.02 | −0.03 | 0.07 | 0.01 | −0.05 | 0.06 |

600 × 40 | 0.38 | 0.23 | 0.54 | 0.03 | −0.07 | 0.13 | −0.05 | −0.17 | 0.06 |

600 × 60 | 0.30 | 0.12 | 0.50 | −0.05 | −0.18 | 0.05 | −0.13 | −0.23 | −0.04 |

700 × 20 | 0.11 | 0.05 | 0.18 | 0.04 | −0.01 | 0.08 | 0.03 | −0.03 | 0.08 |

700 × 40 | 0.24 | 0.13 | 0.37 | −0.11 | −0.20 | 0.00 | −0.21 | −0.28 | −0.12 |

700 × 60 | 0.26 | 0.09 | 0.46 | −0.05 | −0.15 | 0.04 | −0.13 | −0.24 | −0.03 |

800 × 20 | 0.07 | 0.02 | 0.14 | 0.06 | 0.02 | 0.12 | 0.01 | −0.04 | 0.05 |

800 × 40 | 0.22 | 0.09 | 0.36 | −0.06 | −0.14 | 0.02 | −0.25 | −0.33 | −0.17 |

800 × 60 | 0.40 | 0.25 | 0.57 | 0.02 | −0.04 | 0.08 | −0.19 | −0.29 | −0.10 |

Avg | 0.31 | 0.14 | 0.48 | 0.06 | −0.06 | 0.18 | −0.05 | −0.18 | 0.08 |

**Table 7.**Computational results of algorithms with ${T}_{max}=30\times n\times m$ milliseconds (The bolds show better results).

n × m | IG_{RS} | IG_{ALL} | VBIH | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | Min | Max | Avg. | Min | Max | Avg. | Min | Max | |

100 × 20 | 0.25 | −0.02 | 0.54 | 0.03 | −0.11 | 0.16 | −0.05 | −0.25 | 0.16 |

100 × 40 | 0.38 | 0.08 | 0.68 | 0.05 | −0.14 | 0.23 | 0.07 | −0.15 | 0.33 |

100 × 60 | 0.36 | 0.13 | 0.63 | 0.05 | −0.17 | 0.23 | 0.21 | −0.02 | 0.51 |

200 × 20 | 0.28 | 0.12 | 0.45 | 0.07 | −0.05 | 0.22 | 0.00 | −0.16 | 0.14 |

200 × 40 | 0.30 | 0.06 | 0.51 | −0.08 | −0.25 | 0.08 | −0.04 | −0.25 | 0.16 |

200 × 60 | 0.26 | 0.05 | 0.51 | −0.04 | −0.19 | 0.13 | 0.02 | −0.17 | 0.19 |

300 × 20 | 0.12 | −0.01 | 0.23 | 0.01 | −0.10 | 0.14 | −0.06 | −0.21 | 0.08 |

300 × 40 | 0.17 | −0.03 | 0.41 | −0.22 | −0.37 | −0.04 | −0.23 | −0.39 | −0.07 |

300 × 60 | 0.18 | −0.03 | 0.42 | −0.08 | −0.25 | 0.12 | −0.09 | −0.24 | 0.07 |

400 × 20 | 0.12 | 0.04 | 0.19 | 0.03 | −0.04 | 0.09 | 0.01 | −0.06 | 0.09 |

400 × 40 | 0.16 | −0.03 | 0.37 | −0.20 | −0.38 | −0.07 | −0.22 | −0.36 | −0.08 |

400 × 60 | 0.08 | −0.11 | 0.24 | −0.22 | −0.37 | −0.07 | −0.20 | −0.31 | −0.11 |

500 × 20 | 0.11 | 0.02 | 0.20 | 0.07 | 0.01 | 0.13 | 0.02 | −0.06 | 0.10 |

500 × 40 | 0.13 | −0.05 | 0.32 | −0.16 | −0.26 | −0.06 | −0.24 | −0.36 | −0.12 |

500 × 60 | 0.15 | −0.03 | 0.32 | −0.22 | −0.35 | −0.09 | −0.23 | −0.35 | −0.10 |

600 × 20 | 0.07 | −0.02 | 0.15 | −0.01 | −0.06 | 0.04 | −0.02 | −0.07 | 0.03 |

600 × 40 | 0.20 | 0.04 | 0.36 | −0.11 | −0.19 | −0.02 | −0.19 | −0.29 | −0.07 |

600 × 60 | 0.13 | −0.03 | 0.32 | −0.23 | −0.37 | −0.11 | −0.26 | −0.37 | −0.15 |

700 × 20 | 0.08 | 0.01 | 0.16 | 0.02 | −0.03 | 0.06 | −0.01 | −0.07 | 0.03 |

700 × 40 | 0.09 | −0.01 | 0.19 | −0.27 | −0.38 | −0.15 | −0.34 | −0.42 | −0.27 |

700 × 60 | 0.07 | −0.11 | 0.23 | −0.21 | −0.28 | −0.13 | −0.28 | −0.39 | −0.19 |

800 × 20 | 0.04 | −0.01 | 0.09 | 0.02 | −0.01 | 0.05 | 0.00 | −0.04 | 0.04 |

800 × 40 | 0.07 | −0.07 | 0.21 | −0.20 | −0.30 | −0.11 | −0.28 | −0.35 | −0.21 |

800 × 60 | 0.22 | 0.10 | 0.40 | −0.13 | −0.22 | −0.04 | −0.23 | −0.32 | −0.13 |

Avg | 0.17 | 0.00 | 0.34 | −0.08 | −0.20 | 0.03 | −0.11 | −0.24 | 0.02 |

**Table 8.**Computational results of algorithms with ${T}_{max}=45\times n\times m$ milliseconds (The bolds show better results).

n × m | IG_{RS} | IG_{ALL} | VBIH | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | Min | Max | Avg. | Min | Max | Avg. | Min | Max | |

100 × 20 | 0.13 | −0.14 | 0.39 | −0.04 | −0.21 | 0.1 | −0.25 | −0.44 | −0.03 |

100 × 40 | 0.29 | 0.02 | 0.59 | −0.05 | −0.25 | 0.13 | −0.18 | −0.35 | −0.01 |

100 × 60 | 0.26 | 0.03 | 0.48 | −0.03 | −0.28 | 0.17 | −0.02 | −0.17 | 0.19 |

200 × 20 | 0.21 | 0.05 | 0.37 | 0 | −0.14 | 0.12 | −0.12 | −0.27 | 0.03 |

200 × 40 | 0.21 | 0.01 | 0.4 | −0.2 | −0.36 | −0.03 | −0.3 | −0.53 | −0.07 |

200 × 60 | 0.14 | −0.07 | 0.37 | −0.14 | −0.3 | 0.02 | −0.27 | −0.43 | −0.1 |

300 × 20 | 0.07 | −0.06 | 0.17 | −0.04 | −0.18 | 0.1 | −0.15 | −0.26 | −0.05 |

300 × 40 | 0.06 | −0.13 | 0.27 | −0.33 | −0.47 | −0.17 | −0.45 | −0.56 | −0.28 |

300 × 60 | 0.08 | −0.14 | 0.34 | −0.24 | −0.4 | −0.04 | −0.32 | −0.47 | −0.17 |

400 × 20 | 0.09 | 0 | 0.17 | −0.03 | −0.12 | 0.02 | −0.05 | −0.12 | 0.01 |

400 × 40 | 0.09 | −0.09 | 0.3 | −0.44 | −0.57 | −0.3 | −0.41 | −0.52 | −0.28 |

400 × 60 | −0.03 | −0.23 | 0.16 | −0.48 | −0.64 | −0.31 | −0.41 | −0.52 | −0.32 |

500 × 20 | 0.07 | −0.02 | 0.18 | 0.02 | −0.06 | 0.08 | −0.04 | −0.11 | 0.06 |

500 × 40 | 0.04 | −0.16 | 0.21 | −0.41 | −0.53 | −0.29 | −0.42 | −0.5 | −0.29 |

500 × 60 | 0.02 | −0.14 | 0.17 | −0.44 | −0.56 | −0.3 | −0.41 | −0.54 | −0.29 |

600 × 20 | 0.04 | −0.04 | 0.13 | −0.04 | −0.08 | 0.01 | −0.05 | −0.08 | −0.01 |

600 × 40 | 0.11 | −0.05 | 0.29 | −0.32 | −0.41 | −0.21 | −0.27 | −0.39 | −0.15 |

600 × 60 | 0.03 | −0.12 | 0.22 | −0.45 | −0.6 | −0.33 | −0.35 | −0.44 | −0.23 |

700 × 20 | 0.06 | −0.02 | 0.14 | 0 | −0.05 | 0.05 | −0.03 | −0.08 | 0.02 |

700 × 40 | 0.01 | −0.11 | 0.13 | −0.36 | −0.48 | −0.24 | −0.42 | −0.5 | −0.35 |

700 × 60 | −0.01 | −0.2 | 0.16 | −0.3 | −0.4 | −0.22 | −0.37 | −0.48 | −0.25 |

800 × 20 | 0.02 | −0.04 | 0.07 | 0.01 | −0.03 | 0.04 | −0.01 | −0.06 | 0.03 |

800 × 40 | −0.01 | −0.15 | 0.12 | −0.27 | −0.36 | −0.17 | −0.36 | −0.43 | −0.29 |

800 × 60 | 0.13 | 0 | 0.31 | −0.21 | −0.3 | −0.14 | −0.32 | −0.4 | −0.22 |

Average | 0.09 | −0.07 | 0.26 | −0.20 | −0.32 | −0.08 | −0.25 | −0.36 | −0.13 |

**Table 9.**Computational results of Taillard’s instances with ${T}_{max}=45\times n\times m$ milliseconds (The bolds show better results).

IG_{RS} | IG_{RS} * | IG_{ALL} | IG_{ALL} * | VBIH | VBIH * | |
---|---|---|---|---|---|---|

Avg | Avg | Avg | Avg | Avg | Avg | |

20 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

20 × 10 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 |

20 × 20 | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.01 |

50 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

50 × 10 | 0.34 | 0.43 | 0.40 | 0.43 | 0.26 | 0.31 |

50 × 20 | 0.57 | 0.79 | 0.53 | 0.71 | 0.33 | 0.53 |

100 × 5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

100 × 10 | 0.10 | 0.19 | 0.04 | 0.11 | 0.02 | 0.09 |

100 × 20 | 0.82 | 1.33 | 0.89 | 1.23 | 0.54 | 0.94 |

200 × 10 | 0.05 | 0.14 | 0.03 | 0.05 | 0.03 | 0.05 |

200 × 20 | 1.04 | 1.46 | 0.82 | 1.29 | 0.55 | 1.02 |

500 × 20 | 0.47 | 0.92 | 0.35 | 0.75 | 0.26 | 0.64 |

Overall Avg. | 0.28 | 0.44 | 0.26 | 0.38 | 0.17 | 0.30 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kizilay, D.; Tasgetiren, M.F.; Pan, Q.-K.; Gao, L.
A Variable Block Insertion Heuristic for Solving Permutation Flow Shop Scheduling Problem with Makespan Criterion. *Algorithms* **2019**, *12*, 100.
https://doi.org/10.3390/a12050100

**AMA Style**

Kizilay D, Tasgetiren MF, Pan Q-K, Gao L.
A Variable Block Insertion Heuristic for Solving Permutation Flow Shop Scheduling Problem with Makespan Criterion. *Algorithms*. 2019; 12(5):100.
https://doi.org/10.3390/a12050100

**Chicago/Turabian Style**

Kizilay, Damla, Mehmet Fatih Tasgetiren, Quan-Ke Pan, and Liang Gao.
2019. "A Variable Block Insertion Heuristic for Solving Permutation Flow Shop Scheduling Problem with Makespan Criterion" *Algorithms* 12, no. 5: 100.
https://doi.org/10.3390/a12050100