PopulationBased MetaHeuristic Algorithms for Integrated Batch Manufacturing and Delivery Scheduling Problem
Abstract
:1. Introduction
2. Literature Review
3. MixedInteger Linear Programming Model
Indices  
$i,j$  jobs  
$f$  families  
$k,l$  production batches  
$m,n$  delivery batches  
$u$  buckets  
$t$  trucks  
$c$  customers  
Parameters  
$J$  set of jobs  
$F$  set of families  
${B}^{M}$  set of production batches  
$U$  set of buckets  
${B}^{D}$  set of delivery batches  
$T$  set of trucks  
$C$  set of customers  
${p}_{f}$  production time of family $f\in F$  
${F}_{j}^{J}$  family of job $j\in J$  
${F}_{k}^{B}$  family of production batches $k\in {B}^{M}$  
${h}_{c}$  delivery time for customer $c\in C$  
${R}_{jc}^{C}$  1 if job $j\in J$ is required by customer $c\in C;$ 0 otherwise  
${v}_{j}$  volume of job $j\in J$  
${d}_{j}$  due time of job $j\in J$  
$DR$  deterioration rate  
${V}^{M}$  machine capacity  
$Q$  RMA processing time  
${V}^{T}$  truck capacity  
$M$  a large number  
Continuous variables  
${x}_{k}$  production starting time of production batch $k$  
${I}_{k}$  time interval between starting time of production batch $k$ and completion time of the most recent RMA before production batch $k$  
${c}_{u}$  completion time of bucket $u$  
${r}_{m}$  shipping starting time of delivery batch $m$  
${\mathsf{\tau}}_{j}$  tardiness of job $j$  
Binary variables  
${y}_{ik}^{{B}^{M}}$  1 if production batch $k$ assigns job $i;$ 0 otherwise  
${y}_{ku}^{U}$  1 if bucket $u$ assigns production batch $k;$ 0 otherwise  
${z}_{klu}^{U}$  1 if production batch $k$ immediately precedes production batch $l$ at bucket $u;$ 0 otherwise  
${y}_{im}^{{B}^{D}}$  1 if delivery batch $m$ assigns job $i;$ 0 otherwise  
${y}_{mt}^{T}$  1 if truck $t$ assigns delivery batch $m;$ 0 otherwise  
${y}_{mc}^{C}$  1 if customer $c$ assigns delivery batch $m;$ 0 otherwise  
${z}_{mnt}^{T}$  1 if delivery batch $m$ immediately precedes delivery batch $n$ in truck $t;$ 0 otherwise 
$$\sum}_{\begin{array}{c}k\in {B}^{M}\\ {F}_{k}^{B}={F}_{i}^{J}\end{array}}{y}_{ik}^{{B}^{M}}=1$$

$$\forall i\in J$$
 (2) 
$$\sum}_{i\in J}{v}_{i}\xb7{y}_{ik}^{{B}^{M}}\le {V}^{M$$

$$\forall k\in {B}^{M}$$
 (3) 
$$\sum}_{i\in J}{y}_{ik}^{{B}^{M}}\le M\xb7{\displaystyle \sum}_{u\in U}{y}_{ku}^{U$$

$$\forall k\in {B}^{M}$$
 (4) 
$$\sum}_{u\in U}{y}_{ku}^{U}\le 1$$

$$\forall k\in {B}^{M}$$
 (5) 
$$\sum}_{l\in {B}^{M}}{z}_{lku}^{U}={y}_{ku}^{U$$

$$\forall k\in {B}^{M};u\in U$$
 (6) 
$$\sum}_{\begin{array}{c}l\in {B}^{M}\\ l\ne k\end{array}}{z}_{klu}^{U}\le {y}_{ku}^{U$$

$$\forall k\in {B}^{M};u\in U$$
 (7) 
$$\sum}_{k\in {B}^{M}}{z}_{kku}^{U}\le 1$$

$$\forall u\in U$$
 (8) 
$${I}_{k}\xb7\left(1+DR\right)+{p}_{{F}_{k}^{B}}\le {I}_{l}+M\xb7\left(1{\displaystyle \sum}_{u\in U}{z}_{klu}^{U}\right)$$

$$\forall k,l\in {B}^{M};k\ne l$$
 (9) 
$${I}_{k}\xb7\left(1+DR\right)+{p}_{{F}_{k}^{B}}\le {c}_{u}+M\xb7\left(1{y}_{ku}^{U}\right)$$

$$\forall k\in {B}^{M};u\in U$$
 (10) 
$$\begin{array}{c}{I}_{k}+{\displaystyle \sum}_{\begin{array}{c}v\in U\\ v<u\end{array}}{c}_{v}+Q\xb7{\displaystyle \sum}_{\begin{array}{c}w\in U\\ w<u\end{array}}{\displaystyle \sum}_{l\in {B}^{M}}{z}_{llw}^{U}\\ \hfill \le {x}_{k}+M\xb7\left(1{y}_{ku}^{U}\right)\end{array}$$

$$\forall k\in {B}^{M};u\in U$$
 (11) 
$$\sum}_{m\in {B}^{D}}{y}_{im}^{{B}^{D}}=1$$

$$\forall i\in J$$
 (12) 
$$\sum}_{i\in J}{v}_{i}\xb7{y}_{im}^{{B}^{D}}\le {V}^{T$$

$$\forall m\in {B}^{D}$$
 (13) 
$${y}_{im}^{{B}^{D}}+{y}_{jm}^{{B}^{D}}\le 1+{\displaystyle \sum}_{c\in C}{R}_{ic}^{C}\xb7{R}_{jc}^{C}$$

$$\forall m\in {B}^{D};i,j\in J;i<j$$
 (14) 
$${r}_{m}\ge {x}_{k}+{I}_{k}\xb7DR+{p}_{{F}_{k}^{B}}M\xb7\left(2{y}_{im}^{{B}^{D}}{y}_{ik}^{{B}^{M}}\right)$$

$$\forall i\in J;m\in {B}^{D};k\in {B}^{M}$$
 (15) 
$$\sum}_{c\in C}{y}_{mc}^{C}\le 1$$

$$\forall m\in {B}^{D}$$
 (16) 
$${y}_{mc}^{C}\ge {R}_{ic}^{C}\xb7{y}_{im}^{{B}^{D}}$$

$$\forall i\in J;m\in {B}^{D};c\in C$$
 (17) 
$${r}_{m}+{\displaystyle \sum}_{c\in C}{h}_{c}\xb7{y}_{mc}^{C}\le {r}_{n}+M\xb7\left(1{\displaystyle \sum}_{t\in T}{z}_{mnt}^{T}\right)$$

$$\forall m,n\in {B}^{D};m\ne n$$
 (18) 
$$\sum}_{t\in T}{y}_{mt}^{T}=1$$

$$\forall m\in {B}^{D}$$
 (19) 
$$\sum}_{n\in {B}^{D}}{z}_{nmt}^{T}={y}_{mt}^{T$$

$$\forall m\in {B}^{D};t\in T$$
 (20) 
$$\sum}_{\begin{array}{c}n\in {B}^{D}\\ n\ne m\end{array}}{z}_{mnt}^{T}\le {y}_{mt}^{T$$

$$\forall m\in {B}^{D};t\in T$$
 (21) 
$$\sum}_{m\in {B}^{D}}{z}_{mmt}^{T}\le 1$$

$$\forall t\in T$$
 (22) 
$${r}_{m}+{\displaystyle \sum}_{c\in C}{h}_{c}\xb7{y}_{mc}^{C}{d}_{i}\le {\tau}_{i}+M\xb7\left(1{y}_{im}^{{B}^{D}}\right)$$

$$\forall i\in J;m\in {B}^{D}$$
 (23) 
$${x}_{k},{I}_{k},{c}_{u},{r}_{m},{\mathsf{\tau}}_{i}\ge 0$$

$$\forall k\in {B}^{M};u\in U;\phantom{\rule{0ex}{0ex}}\text{}m\in {B}^{D};t\in T;c\in C$$
 (24) 
$${y}_{ik}^{{B}^{M}},{y}_{ku}^{U},{y}_{im}^{{B}^{D}},{y}_{mt}^{T},{y}_{mc}^{C}=0\mathrm{or}1$$

$$\forall i\in J;k\in {B}^{M};u\in U;\phantom{\rule{0ex}{0ex}}m\in {B}^{D};t\in T;c\in C$$
 (25) 
$${z}_{klu}^{U},{z}_{mnt}^{T}=0\mathrm{or}1$$

$$\forall k,l\in {B}^{M};u\in U;\phantom{\rule{0ex}{0ex}}m,n\in {B}^{D};t\in T$$
 (26) 
4. MetaHeuristic Algorithms
4.1. Solution Representation and Decoding Method
4.2. Particle Swarm Optimization (PSO)
Algorithm 1: The PSO procedure  
1  Input iteration (Iter), population size (${S}_{p}$), and acceleration weight (${c}_{1}$) and (${c}_{2}$). 
2  Randomly generate initial positions and velocities through $U(0,1$). 
3  While ($g\le Iter$) 
4  $g\leftarrow g+1$ 
5  For (i = 1 to ${S}_{p}$) 
6  If (${X}_{i}<{P}_{i})$ 
7  ${P}_{i}\leftarrow {X}_{i}$ 
8  End if 
9  If (${X}_{i}<{P}_{g})$ 
10  ${P}_{g}\leftarrow {X}_{i}$ 
11  End if 
12  End for 
13  For (i = 1 to ${S}_{p}$) 
14  ${v}_{i}\leftarrow {v}_{i}+{c}_{1}\xb7U\left(0,1\right)\xb7\left({P}_{i}{X}_{i}\right)+{c}_{2}\xb7U\left(0,1\right)\xb7\left({P}_{g}{X}_{i}\right)$ 
15  ${X}_{i}\leftarrow {X}_{i}+{v}_{i}$ 
16  End for 
17  End while 
4.3. Imperialist Competitive Algorithm (ICA)
Algorithm 2: The ICA procedure  
1 

2  Generate initial countries and determine the imperialist countries and colonies. 
3  While ($g\le Iter$) 
4  $g\leftarrow g+1$ 
5  For (i = 1 to ${S}_{p}$) 
6  Move to the colony toward its imperialist. 
7  If ($U\left(0,1\right)<{p}_{r})$ 
8  Conduct revolution. 
9  End if 
10  If $\mathrm{f}\left({X}^{i}\right)\le \mathrm{f}\left({X}_{n}^{I}\right)$ 
11  ${X}_{n}^{I}\leftarrow {X}^{i}$ 
12  End if 
13  End for 
14  Calculate the total cost of empires. 
15  Conduct imperialistic competition. 
16  End while 
4.4. Genetic Algorithm (GA)
Algorithm 3: The GA procedure  
1  Input generation size (${S}_{g}$), population size (${S}_{p}$), crossover rate (${p}_{c}$), and mutation rate (${p}_{m})$. 
2 

3  While ($g\le {S}_{g}$) 
4  For (i = 1 to ${S}_{p}$) 
5  If ($U\left(0,1\right)\text{}{p}_{c}$) 
6  Perform the crossover operator for two different randomly selected chromosomes. 
7  End if 
8  End for 
9  For (i = 1 to ${S}_{p}$) 
10  For (n = 1 to $N$) 
11  If ($U\left(0,1\right)\text{}{p}_{m}$) 
12  Perform the mutation operation. 
13  End if 
14  End for 
15  End for 
16  Perform the roulette wheel selection. 
17  $g\leftarrow g+1$ 
18  End while 
5. Computational Results
5.1. Calibration of the Algorithm Parameters
5.2. Setting of the Problem Parameters
5.3. Experimental Results in the Small Problem Instances
5.4. Experimental Results in the Large Problem Instances
6. Sensitivity Analysis
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Compatibility in Production Batches  Vehicles Number  Deterioration  RMA  Shipping Method  

Incompatible Product  Incompatible Family  Incompatible Customer  DirectShipping  VRP  
Liu [6]  1  ✓  
Jia et al. [2]  Limited  ✓  
Selvarajah and Steiner [7]  ✓  ✓  1  ✓  
Gao et al. [8]  1  ✓  
Cheng et al. [9]  1  ✓  
Cheng et al. [10]  1  ✓  
Jia et al. [11]  Limited  ✓  
Li et al. [12]  ✓  1  ✓  
Li et al. [13]  Limited  ✓  
Zhang et al. [14]  1  ✓  
Nogueira et al. [15]  Limited  ✓  
Feng and Xu [16]  Unlimited  ✓  
Jia et al. [17]  Limited  ✓  
He et al. [18]  ✓  Limited  ✓  
Li et al. [19]  ✓  Limited  ✓  
Karaoğlan and Kesen [20]  1  ✓  
Low et al. [21]  Unlimited  ✓  
Low et al. [22]  Limited  ✓  
Li et al. [23]  Limited  ✓  
Kong et al. [24]  Unlimited  ✓  ✓  
Liu et al. [25]  1  ✓  ✓  
Yin et al. [26]  Unlimited  ✓  ✓  
This study  ✓  Limited  ✓  ✓  ✓ 
Parameters  Levels  

1  2  3  4  5  
${G}_{s}$  200  400  600  800  1000 
${P}_{s}$  20  40  60  80  100 
${p}_{c}$  0.1  0.3  0.5  0.7  0.9 
${p}_{m}$  0.001  0.002  0.003  0.004  0.004 
Run  ${\mathit{G}}_{\mathit{s}}$  ${\mathit{P}}_{\mathit{s}}$  ${\mathit{p}}_{\mathit{c}}$  ${\mathit{p}}_{\mathit{m}}$ 

1  ${G}_{s}\left(1\right)$  ${P}_{s}\left(1\right)$  ${p}_{c}\left(1\right)$  ${p}_{m}\left(1\right)$ 
2  ${G}_{s}\left(1\right)$  ${P}_{s}\left(2\right)$  ${p}_{c}\left(2\right)$  ${p}_{m}\left(2\right)$ 
3  ${G}_{s}\left(1\right)$  ${P}_{s}\left(3\right)$  ${p}_{c}\left(3\right)$  ${p}_{m}\left(3\right)$ 
4  ${G}_{s}\left(1\right)$  ${P}_{s}\left(4\right)$  ${p}_{c}\left(4\right)$  ${p}_{m}\left(4\right)$ 
5  ${G}_{s}\left(1\right)$  ${P}_{s}\left(5\right)$  ${p}_{c}\left(5\right)$  ${p}_{m}\left(5\right)$ 
6  ${G}_{s}\left(2\right)$  ${P}_{s}\left(1\right)$  ${p}_{c}\left(2\right)$  ${p}_{m}\left(3\right)$ 
7  ${G}_{s}\left(2\right)$  ${P}_{s}\left(2\right)$  ${p}_{c}\left(3\right)$  ${p}_{m}\left(4\right)$ 
8  ${G}_{s}\left(2\right)$  ${P}_{s}\left(3\right)$  ${p}_{c}\left(4\right)$  ${p}_{m}\left(5\right)$ 
9  ${G}_{s}\left(2\right)$  ${P}_{s}\left(4\right)$  ${p}_{c}\left(5\right)$  ${p}_{m}\left(1\right)$ 
10  ${G}_{s}\left(2\right)$  ${P}_{s}\left(5\right)$  ${p}_{c}\left(1\right)$  ${p}_{m}\left(2\right)$ 
11  ${G}_{s}\left(3\right)$  ${P}_{s}\left(1\right)$  ${p}_{c}\left(3\right)$  ${p}_{m}\left(5\right)$ 
12  ${G}_{s}\left(3\right)$  ${P}_{s}\left(2\right)$  ${p}_{c}\left(4\right)$  ${p}_{m}\left(1\right)$ 
13  ${G}_{s}\left(3\right)$  ${P}_{s}\left(3\right)$  ${p}_{c}\left(5\right)$  ${p}_{m}\left(2\right)$ 
14  ${G}_{s}\left(3\right)$  ${P}_{s}\left(4\right)$  ${p}_{c}\left(1\right)$  ${p}_{m}\left(3\right)$ 
15  ${G}_{s}\left(3\right)$  ${P}_{s}\left(5\right)$  ${p}_{c}\left(2\right)$  ${p}_{m}\left(4\right)$ 
16  ${G}_{s}\left(4\right)$  ${P}_{s}\left(1\right)$  ${p}_{c}\left(4\right)$  ${p}_{m}\left(2\right)$ 
17  ${G}_{s}\left(4\right)$  ${P}_{s}\left(2\right)$  ${p}_{c}\left(5\right)$  ${p}_{m}\left(3\right)$ 
18  ${G}_{s}\left(4\right)$  ${P}_{s}\left(3\right)$  ${p}_{c}\left(1\right)$  ${p}_{m}\left(4\right)$ 
19  ${G}_{s}\left(4\right)$  ${P}_{s}\left(4\right)$  ${p}_{c}\left(2\right)$  ${p}_{m}\left(5\right)$ 
20  ${G}_{s}\left(4\right)$  ${P}_{s}\left(5\right)$  ${p}_{c}\left(3\right)$  ${p}_{m}\left(1\right)$ 
21  ${G}_{s}\left(5\right)$  ${P}_{s}\left(1\right)$  ${p}_{c}\left(5\right)$  ${p}_{m}\left(4\right)$ 
22  ${G}_{s}\left(5\right)$  ${P}_{s}\left(2\right)$  ${p}_{c}\left(1\right)$  ${p}_{m}\left(5\right)$ 
23  ${G}_{s}\left(5\right)$  ${P}_{s}\left(3\right)$  ${p}_{c}\left(2\right)$  ${p}_{m}\left(1\right)$ 
24  ${G}_{s}\left(5\right)$  ${P}_{s}\left(4\right)$  ${p}_{c}\left(3\right)$  ${p}_{m}\left(2\right)$ 
25  ${G}_{s}\left(5\right)$  ${P}_{s}\left(5\right)$  ${p}_{c}\left(4\right)$  ${p}_{m}\left(3\right)$ 
Parameters  SS  df  V  ${\mathit{F}}_{0}$  pValue 

${G}_{s}$  4.0983  4  1.0246  3.6219  0.1202 
${P}_{s}$  5.1190  4  1.2797  4.5239  0.0865 
${p}_{c}$  2.3853  4  0.5963  2.1080  0.2439 
${p}_{m}$(error)  1.1315  4  0.2829     
Total  12.7342  16       
Group  PH  $\left\mathit{J}\right$  ${\mathit{p}}_{\mathit{f}}$  ${\mathit{h}}_{\mathit{c}}$  ${\mathit{v}}_{\mathit{j}}$  ${\mathit{V}}^{\mathit{M}}$  ${\mathit{V}}^{\mathit{T}}$  $\mathit{D}\mathit{R}$  $\mathit{Q}$  ${\mathit{d}}_{\mathit{j}}$  

$\mathit{\tau}=0.6$  $\mathit{\tau}=0.3$  
Small problem instances  480  5  [65,100]  [130,200]  [5,10]  20  20  0.3  165  [48,336]  NA 
6  [55,90]  [110,180]  [5,10]  20  20  0.3  145  
Large problem instances  2400  200  [30,45]  [60,90]  [5,10]  50  20  0.3  75  [240,1680]  [420,2940] 
250  [25,35]  [50,70]  [5,10]  50  20  0.3  60  
300  [20,30]  [40,60]  [5,10]  50  20  0.3  50 
$\left\mathit{J}\right$  $\left\mathit{T}\right$  $\left\mathit{C}\right$  $\left\mathit{F}\right$  CPLEX  PSO  ICA  GA  

$\mathit{O}\mathit{p}\mathit{t}.$  Time  $\mathit{O}\mathit{b}{\mathit{j}}_{\mathit{s}\mathit{o}\mathit{l}}$  Time  $\mathit{O}\mathit{b}{\mathit{j}}_{\mathit{s}\mathit{o}\mathit{l}}$  Time  $\mathit{O}\mathit{b}{\mathit{j}}_{\mathit{s}\mathit{o}\mathit{l}}$  Time  
5  1  1  1  584.00  1.05  584.00  0.26  584.00  0.11  584.00  0.27 
2  844.00  2.03  844.00  0.26  844.83  0.11  846.66  0.28  
2  1  484.00  0.77  484.00  0.25  484.00  0.11  484.00  0.27  
2  980.00  0.34  980.00  0.25  980.00  0.11  980.00  0.28  
2  1  1  675.80  4.57  675.80  0.24  675.80  0.11  675.80  0.26  
2  949.40  25.03  949.40  0.24  949.40  0.11  949.40  0.26  
2  1  365.00  27.30  365.00  0.26  365.00  0.11  365.00  0.28  
2  437.00  11.60  437.00  0.25  437.00  0.11  437.00  0.28  
6  1  1  1  633.00  4.13  633.00  0.31  639.72  0.14  633.00  0.33 
2  770.00  11.83  770.00  0.31  771.47  0.14  770.00  0.33  
2  1  629.00  1.62  629.00  0.32  629.43  0.14  629.00  0.34  
2  821.00  2.67  821.00  0.30  821.00  0.13  821.00  0.32  
2  1  1  NA  7200.00++  1031.33  0.32  1031.33  0.14  1031.33  0.33  
2  671.03  104.65  671.03  0.31  671.03  0.14  671.03  0.34  
2  1  NA  7200.00++  886.98  0.35  887.83  0.14  886.98  0.34  
2  NA  7200.00++  608.70  0.35  618.07  0.15  599.32  0.36  
Average  710.64  0.29  711.87  0.13  710.22  0.30 
$\left\mathit{J}\right$  $\left\mathit{T}\right$  $\left\mathit{C}\right$  $\mathit{\tau}=0.6$  $\mathit{\tau}=0.3$  

PSO  ICA  GA  PSO  ICA  GA  
RDI  Time  RDI  Time  RDI  Time  RDI  Time  RDI  Time  RDI  Time  
200  10  10  0.90  61.01  0.43  35.95  0.09  61.41  0.90  61.44  0.60  36.22  0.08  61.52 
15  0.91  61.32  0.43  36.30  0.06  61.30  0.91  61.78  0.56  36.19  0.07  61.63  
20  0.89  61.58  0.44  36.37  0.05  61.88  0.93  61.87  0.60  36.15  0.06  61.87  
15  10  0.89  61.66  0.46  36.51  0.12  62.42  0.92  62.08  0.58  36.75  0.07  62.57  
15  0.90  62.15  0.41  37.11  0.07  62.59  0.93  62.44  0.61  36.95  0.06  62.66  
20  0.87  62.23  0.46  37.01  0.08  62.36  0.91  62.60  0.65  36.97  0.08  63.47  
20  10  0.90  62.35  0.45  36.84  0.07  62.79  0.92  62.48  0.58  36.94  0.08  62.55  
15  0.93  62.59  0.44  37.01  0.11  63.25  0.93  63.07  0.65  37.06  0.08  62.69  
20  0.91  62.91  0.50  37.28  0.10  63.29  0.93  63.05  0.64  37.24  0.11  62.78  
250  10  10  0.92  92.04  0.48  54.58  0.09  92.22  0.92  92.22  0.56  54.31  0.09  91.90 
15  0.91  92.29  0.46  54.80  0.05  92.04  0.91  92.84  0.59  54.65  0.08  92.39  
20  0.93  92.56  0.44  54.71  0.06  92.71  0.94  93.15  0.54  54.82  0.09  92.60  
15  10  0.92  92.71  0.38  55.46  0.10  93.34  0.95  93.12  0.59  55.23  0.09  93.55  
15  0.89  92.86  0.43  54.99  0.08  93.40  0.92  93.69  0.61  55.57  0.11  93.55  
20  0.93  94.02  0.42  55.56  0.10  93.64  0.94  94.12  0.62  55.64  0.10  93.10  
20  10  0.88  92.97  0.39  55.52  0.11  93.88  0.91  93.72  0.55  55.59  0.04  93.50  
15  0.93  93.95  0.43  56.12  0.11  94.18  0.93  94.67  0.57  56.05  0.09  94.78  
20  0.88  94.15  0.42  56.08  0.05  94.42  0.93  94.51  0.58  56.00  0.07  94.28  
300  10  10  0.92  128.67  0.42  76.30  0.07  129.12  0.94  129.57  0.54  76.79  0.07  129.18 
15  0.90  129.55  0.44  76.98  0.06  129.14  0.89  130.33  0.47  76.68  0.05  130.27  
20  0.94  130.14  0.39  77.45  0.10  130.51  0.93  130.49  0.53  76.80  0.07  130.07  
15  10  0.91  129.47  0.39  77.02  0.08  129.98  0.94  130.03  0.58  77.43  0.07  130.23  
15  0.95  130.62  0.37  78.09  0.10  132.10  0.93  131.17  0.59  77.20  0.08  130.83  
20  0.92  130.78  0.38  78.12  0.06  131.46  0.93  131.81  0.55  77.85  0.08  131.70  
20  10  0.92  130.49  0.39  77.74  0.08  131.14  0.95  131.34  0.54  78.01  0.07  132.78  
15  0.91  131.13  0.35  78.68  0.09  132.44  0.93  131.89  0.57  77.93  0.11  133.99  
20  0.92  131.58  0.36  78.08  0.10  132.49  0.94  132.35  0.58  78.51  0.08  133.41  
Average  0.91  95.10  0.42  56.42  0.08  95.54  0.93  95.62  0.58  56.50  0.08  95.70 
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Kim, Y.J.; Kim, B.S. PopulationBased MetaHeuristic Algorithms for Integrated Batch Manufacturing and Delivery Scheduling Problem. Mathematics 2022, 10, 4127. https://doi.org/10.3390/math10214127
Kim YJ, Kim BS. PopulationBased MetaHeuristic Algorithms for Integrated Batch Manufacturing and Delivery Scheduling Problem. Mathematics. 2022; 10(21):4127. https://doi.org/10.3390/math10214127
Chicago/Turabian StyleKim, YongJae, and ByungSoo Kim. 2022. "PopulationBased MetaHeuristic Algorithms for Integrated Batch Manufacturing and Delivery Scheduling Problem" Mathematics 10, no. 21: 4127. https://doi.org/10.3390/math10214127