UShaped Assembly Line Balancing by Using Differential Evolution Algorithm
Abstract
:1. Introduction
2. Literature Review
2.1. UShape Assembly Line Balancing by Using Other Metaheuristic Methods
2.2. UShape Assembly Line Balancing by Using Differential Evolution Algorithm
2.3. Differential Evolution Algorithm for Solving Other Problems
3. Uline Assembly Line Balancing Problem Pattern and Mathematical Model
3.1. Uline Assembly Line Balancing Problem (UALBP)
 UALBP1: Given the cycle time (c), minimize the number of stations (m);
 UALBP2: Given the number of stations (m), minimize the cycle time (c).;
 UALBPE: Maximize the line efficiency (E) for c and m being variable.
3.2. Mathematical Model of the UShaped Assembly Line
3.2.1. Indices
 n denotes the index of a task, where n = 1 …. N
 k denotes the index of workstation k, where k = 1 … M
 N denotes the total number of tasks
 M denotes the total number of workstations
3.2.2. Parameter
 P_{n} denotes the processing time of task n
 CT denotes the cycle time of a workstation
 P_{nj} denotes the relationship of task n to task j
 ${F}_{nj}=\{\begin{array}{l}1\text{}\mathrm{if}\text{}\mathrm{task}\text{}n\text{}\mathrm{is}\text{}\mathrm{predecessor}\text{}\mathrm{of}\text{}\mathrm{task}\text{}j\\ 0\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{otherwise}\end{array}$
3.2.3. Decision Variables
 ${X}_{nk}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\{\begin{array}{l}1\text{}\mathrm{if}\text{}\mathrm{task}\text{}n\text{}\mathrm{is}\text{}\mathrm{assigned}\text{}\mathrm{to}\text{}\mathrm{station}\text{}k\text{}\\ 0\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{otherwise}\end{array}$
 ${Y}_{k}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\{\begin{array}{l}1\text{}\mathrm{if}\text{}\mathrm{station}\text{}k\text{}\mathrm{is}\text{}\mathrm{opened}\\ 0\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{otherwise}\end{array}$Objective function:$$Min\text{\hspace{0.17em}}Z={\displaystyle \sum _{k=1}^{M}{Y}_{k}}$$$$\sum _{k=1}^{M}{X}_{nk}}=1\text{\hspace{1em}\hspace{1em}}\forall n=1\dots N,$$$$\begin{array}{c}{\displaystyle \sum _{k=1}^{M}(K\times {X}_{jk})}(K\times {X}_{nk})\ge 0\\ \forall n=1\dots N,\text{\hspace{0.17em}\hspace{0.17em}}k=1\dots M,\text{\hspace{0.17em}}{F}_{nj}=1\end{array}$$$$\begin{array}{c}{\displaystyle \sum _{n=1}^{N}{X}_{nk}\times}{P}_{n}\le CT\times {Y}_{k}\\ \forall k=1\dots M\end{array}$$$${Y}_{k}\le {Y}_{k1}\text{\hspace{0.17em}}\forall k=2\dots M$$
4. General Differential Evolution Algorithm
4.1. Differential Evolution Algorithm
4.2. Procedure of UALB1 by Using Differential Evolution Algorithm
4.2.1. Calculation Using the General Differential Evolution Algorithm DE/rand/1 and Binomial Crossover
 V_{i,j,G} = Mutant Vector
 X_{r1,j,G}, X_{r2,j,G}, X_{r3,j,G} = Random vector from G round
 F = Scaling factor (random real number between 0 and 2)
 V_{i,j,G} = Mutant Vector
 X_{i,j,G} = Target Vector
 CR = Crossover Constant (real number in the range 0–1)
 rand j [0,1) = random real number between 0 and 1 in every position, j = 1, 2, 3, ..., G (G = number of position).
 U_{i,j,G} = Trial Vector
 X_{i,j,G+1} = Target Vector in the next generation, i = 1,2,...n
4.2.2. Procedure of UALB1 by Using the Improved Differential Evolution Algorithm
Algorithm 1. Pseudocode of the DE for (UALBP1) 
Setup initial DE parameter 
Do while from first iteration to final iteration 
Do while from first DE to final DE 
Setup initial parameters: cycle time, remaining time, station number 
Do while from first task to final task 
Find start/following task with task time is less than or equal to 
Remaining time, and proper precedence to data list 
Input scaling factor, crossover rate and NP to data list 
Select task randomly to list 
Update remaining time/station number 
Produce the four Mutation Equations
$${V}_{i,j,G}={X}_{best,j,G}+F({X}_{r1,j,G}{X}_{r2,j,G})$$
$${V}_{i,j,G}={X}_{i,j,G}+F\left({X}_{best,j,G}{X}_{i,j,G}\right)+F\left({X}_{r1,j,G}{X}_{r2,j,G}\right)$$
$${V}_{i,j,G}={X}_{best,j,G}+F\left({X}_{r1,j,G}{X}_{r2,j,G}\right)+F\left({X}_{r3,j,G}{X}_{r4,j,G}\right)$$
$${V}_{i,j,G}={X}_{r1,G}+F\left({X}_{r2,G}{X}_{r2,G}\right)+F\left({X}_{r4,G}{X}_{r5,G}\right)$$

Developed by applying the two Crossover or Recombination Equations
$${U}_{i,j,G}=\{\begin{array}{l}{V}_{i,j,G}\text{}\mathrm{when}\text{}rand{b}_{i}\le j\\ {X}_{i,j,G}\text{}if\text{}rand{b}_{i}j\end{array}$$
$${U}_{i,j,G}=\{\begin{array}{l}{V}_{i,j,G}\text{}\mathrm{when}\text{}\mathrm{j}\le rand{b}_{i,1}\text{}and\text{}j\ge rand{b}_{i,2}\\ {X}_{i,j,G}\text{}\mathrm{when}\text{}rand{b}_{i,1}\text{}jrand{b}_{i,2}\end{array}$$

Produce new target vector (selection\process)
$${X}_{i,j,G+1}=\{\begin{array}{l}{U}_{i,j,G}iff({U}_{i,j,G})\le f({X}_{i,j,G})\\ {X}_{i,j,G}\text{\hspace{0.05em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}}otherwise\end{array}$$

End do 
End do 
Select best solution from all DE in the iteration 
End do 
Show/select best solution from all DE in all iteration 
5. Analysis of the Results from the Experiment on DE for Solving UALBP
 DE/rand/1 to Binomial Crossover (Basic)
 DE/rand/1 to Exponential Crossover 1 Position (improved)
 DE/rand/1 to Exponential Crossover 2 Position (improved)
 DE/ randtobest/1 to Binomial Crossover (improved)
 DE/Best/2 to Exponential Crossover 1 Position (improved)
 DE/Best/2 to Exponential Crossover 2 Position (improved)
6. The Results from the Comparison of DE Algorithm and Other Metaheuristic Methods
7. Conclusions and Suggestions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Vector 1  Station  1  2  3  4  5 
Work  1, 8  6, 4  2  3, 5  7  
Time  11, 3  12, 5  17  9, 8  10  
Target Vector  0.30, 0.57  0.44, 0.61  0.72  0.53, 0.68  0.92  
Vector 2  Station  1  2  3  4  5 
Work  7, 5  1, 8  6  3  4, 2  
Time  10, 8  11, 3  12  12  5, 11  
Target Vector  0.57, 0.32  0.74, 0.92  0.21  0.44  0.69, 0.82  
Vector 3  Station  1  2  3  4  5 
Work  1, 8  7, 5  6, 4  3  2  
Time  11, 3  10, 8  12, 5  9  17  
Target Vector  0.51, 0.96  0.88, 0.67  0.84, 0.62  0.41  0.92  
Vector 4  Station  1  2  3  4  5 
Work  1, 8  7, 5  2  3, 4  6  
Time  11, 3  10, 8  17  9, 5  12  
Target Vector  0.13, 0.26  0.58, 0.81  0.64  0.42, 0.21  0.69  
Vector 5  Station  1  2  3  4  
Work  8, 6, 4  7, 5  3, 1  2  
Time  3, 12, 5  10, 8  9, 11  17  
Target Vector  0.84, 0.41, 0.59  0.91, 0.76  0.32, 0.98  0.48 
Vector 1  Position  1  2  3  4  5  6  7  8 
Work  1  8  6  4  2  3  5  7  
Time  11  3  12  5  17  9  8  10  
Target Vector  0.3  0.57  0.44  0.61  0.72  0.53  0.68  0.92  
Vector 2  Position  1  2  3  4  5  6  7  8 
Work  7  5  1  8  6  3  4  2  
Time  10  8  11  3  12  12  5  11  
Target Vector  0.57  0.32  0.74  0.92  0.21  0.44  0.69  0.82  
Vector 3  Position  1  2  3  4  5  6  7  8 
Work  1  8  7  5  6  4  3  2  
Time  11  3  10  8  12  5  9  17  
Target Vector  0.51  0.96  0.88  0.67  0.84  0.62  0.41  0.92  
Mutant Vector 1  0.35  0.06  0.33  0.81  0.22  0.39  0.90  0.84 
Vector  Position  1  2  3  4  5  6  7  8 
1  Work  1  8  6  4  2  3  5  7 
Time  11  3  12  5  17  9  8  10  
Target Vector  0.3  0.57  0.44  0.61  0.72  0.53  0.68  0.92  
Vector  Position  1  2  3  4  5  6  7  8 
1  Mutant Vector  0.35  0.06  0.33  0.81  0.22  0.39  0.90  0.84 
rand(j)  0.40  0.40  0.06  0.96  0.47  0.40  0.94  0.33  
Trial Vector  0.35  0.06  0.33  0.61  0.22  0.39  0.68  0.84 
Vector 1  Station  1  2  3  4  5 
Work  8, 6  1, 3  2  4, 5  7  
Time  3, 12  11, 9  17  5, 8  10  
Trial Vector  0.06, 0.33  0.35, 0.39  0.22  0.61, 0.68  0.84 
Problem  c  m*  DE1^{*}  DE2^{**}  DE3^{***}  DE4^{****}  DE5^{*****}  DE6^{******}  

m  Cal Time (s)  m  Cal Time (s)  m  Cal Time (s)  m  Cal Time (s)  m  Cal Time (s)  m  Cal Time (s)  
Sawyer 30  25  14  14  1.87  14  0.67  14  0.11  14  1.87  14  0.07  14  0.08 
27  13  13  1.11  13  0.44  13  0.13  13  0.98  13  0.12  13  0.09  
30  11  11  1.23  11  0.83  11  0.42  11  0.77  11  0.08  11  0.10  
33  10  10  1.11  10  0.67  10  0.67  10  0.44  10  0.18  10  0.11  
36  10  10  1.34  10  0.55  10  0.55  10  0.34  10  0.05  10  0.15  
41  8  8  1.96  8  0.59  8  0.59  8  0.50  8  0.04  8  0.19  
54  6  6  1.61  6  0.78  6  0.38  6  0.61  6  0.09  6  0.38  
75  5  5  1.94  5  0.53  5  0.07  5  0.57  5  0.06  5  0.35  
Arcus 183  3786  21  21  0.91  21  0.31  21  0.54  21  0.31  21  0.30  21  0.14 
3985  20  20  0.89  20  0.89  20  0.17  20  0.89  20  0.06  20  0.11  
3786  21  21  0.91  21  0.31  21  0.54  21  0.31  21  0.30  21  0.14  
4454  18  18  0.85  18  0.55  18  0.31  18  0.55  18  0.10  18  0.98  
4732  17  17  0.46  17  0.98  17  0.89  17  0.46  17  0.28  17  0.16  
5048  16  16  0.31  16  0.77  16  0.64  16  0.31  16  0.08  16  0.25  
5408  15  15  0.89  15  0.44  15  0.06  15  0.89  15  0.12  15  0.06  
5824  14  14  0.64  14  0.34  14  1.98  14  0.64  14  0.04  14  1.98  
5853  13  13  0.06  13  0.50  13  0.24  13  0.06  13  0.06  13  0.24  
6309  13  13  1.98  13  0.61  13  0.55  13  1.98  13  0.09  13  0.55  
6842  12  12  0.64  12  0.57  12  0.54  12  0.24  12  0.19  12  0.54  
6883  12  12  0.79  12  0.99  12  0.47  12  0.59  12  0.38  12  0.16  
7571  11  11  0.74  11  0.44  11  0.98  11  0.54  11  0.35  11  0.25  
8412  10  10  1.97  10  0.34  10  1.98  10  0.47  10  0.22  10  0.06  
8898  9  9  1.89  9  0.50  9  0.24  9  0.51  9  0.45  9  1.66  
10816  7  7  1.98  7  0.61  7  0.55  7  0.98  7  0.71  7  0.78  
Total Optimal Solutions Found from 24 Problem Instances  24  1.16  24  0.61  24  0.53  24  0.67  24  0.17  24  0.40 
Problems  Size  Cycle Time  IP* Solution  MMAS**  DE1***  

m*  m  %  cal. Time (s)  E  m  %  cal. Time (s)  E  
Mitchell  21  14  8  8  0  1.50  93.75  8  0  0.05  93.75 
15  8  8  0  1.63  87.75  8  0  0.03  87.75  
21  5  5  0  5.43  100.00  5  0  0.01  100.00  
Heskiaoff  28  114  9  10  11.11  30.00  79.33  9  11.11  0.04  89.82 
128  8  9  12.50  30.00  81.28  8  12.50  0.08  88.89  
138  8  8  0  1.71  92.75  8  0  0.09  92.75  
205  5  5  0  1.68  99.99  5  0  0.07  99.99  
216  5  5  0  2.36  94.81  5  0  0.13  94.81  
256  4  4  0  5.52  100.00  4  0  0.16  100.00  
324  4  4  0  2.35  79.01  4  0  0.09  79.01  
342  3  3  0  2.45  99.81  3  0  0.12  99.81  
Sawyer  30  25  14  14  0  1.22  92.57  14  0  0.07  92.57 
27  13  13  0  0.96  92.31  13  0  0.12  92.31  
30  11  11  0  7.48  98.18  11  0  0.08  98.18  
33  10  10  0  20.15  98.18  10  0  0.18  98.18  
36  10  10  0  2.08  90.00  10  0  0.05  90.00  
41  8  8  0  14.67  98.78  8  0  0.04  98.78  
54  6  6  0  5.56  100.00  6  0  0.09  100.00  
75  5  5  0  2.97  86.84  5  0  0.06  86.84  
Kilbridge and Wester  45  57  10  10  0  1.33  96.84  10  0  0.11  96.84 
79  7  8  14.28  30.00  90.67  7  0  0.12  99.82  
92  6  7  16.67  30.00  89.28  6  0  0.14  100.00  
110  6  6  0  1.48  83.64  6  0  0.10  83.64  
138  4  4  0  4.08  100.00  4  0  0.14  100.00  
184  3  3  0  6.73  100.00  3  0  0.15  100.00  
Total Optimal Solution (or Lower Bound) Found from 25 Problem Instances  21  2.18/inst.  9.19/inst.  95.51/inst.  25  0.94/inst.  0.10/inst.  97.84/inst. 
Problems  Size  Cycle Time  IP^{*} Solution  MMAS^{**}  DE1^{***}  

m*  m  %  cal. Time (s)  E  m  %  cal. Time(s)  E  
Weemag  75  28  63  63  0  1.19  93.75  63  0  0.12  93.75 
29  63  63  0  1.19  87.75  63  0  0.14  87.75  
30  62  62  0  1.23  100.00  62  0  0.07  100.00  
31  62  62  0  1.14  89.82  62  0  0.05  89.82  
32  61  61  0  1.16  88.89  61  0  0.10  88.89  
33  61  61  0  1.22  92.75  61  0  0.28  92.75  
34  61  61  0  1.22  99.99  61  0  0.08  99.99  
35  60  60  0  1.22  94.81  60  0  0.12  94.81  
36  60  60  0  1.25  100.00  60  0  0.04  100.00  
37  60  60  0  1.23  79.01  60  0  0.06  79.01  
38  60  60  0  1.19  99.81  60  0  0.22  99.81  
39  60  60  0  1.14  92.57  60  0  0.08  92.57  
40  60  60  0  1.25  92.31  60  0  0.19  92.31  
41  59  59  0  1.19  98.18  59  0  0.10  98.18  
42  55  55  0  1.16  98.18  55  0  0.11  98.18  
43  50  50  0  1.30  90.00  50  0  0.15  90.00  
45  38  38  0  4.31  98.78  38  0  0.19  98.78  
46  34  34  0  3.59  100.00  34  0  0.38  100.00  
52  31  31  0  2.59  86.84  31  0  0.35  86.84  
54  31  31  0  1.17  96.84  31  0  0.22  96.84  
Weemag  75  56  30  30  0  1.55  99.82  30  0  0.45  99.82 
Arcus 1  83  3786  21  21  0  1.03  100.00  21  0  0.30  100.00 
3985  20  20  0  1.00  83.64  20  0  0.06  83.64  
4206  19  19  0  1.08  89.82  19  0  0.07  89.82  
4454  18  18  0  1.03  88.89  18  0  0.10  88.89  
4732  17  17  0  1.02  92.75  17  0  0.28  92.75  
5048  16  16  0  1.05  99.99  16  0  0.08  99.99  
5408  15  15  0  1.08  94.81  15  0  0.12  94.81  
5824  14  14  0  1.10  100.00  14  0  0.04  100.00  
5853  13  14  7.69  1.05  79.01  13  0  0.06  89.63  
6309  13  13  0  1.07  99.81  13  0  0.09  99.81  
6842  12  12  0  1.03  92.57  12  0  0.19  92.57  
6883  12  12  0  1.03  92.31  12  0  0.38  92.31  
7571  11  11  0  1.04  98.18  11  0  0.35  98.18  
8412  10  10  0  1.05  98.18  10  0  0.22  98.18  
8898  9  9  0  1.03  90.00  9  0  0.45  90.00  
10816  7  8  14.29  1.04  98.78  8  14.29  0.71  98.78  
Scholl  297  1394  50  51  2.00  12.08  100.00  51  2.00  12.28  100.00 
1452  48  49  2.08  16.10  86.84  49  2.08  11.08  86.84  
1483  47  48  2.13  11.30  96.84  48  2.13  11.30  96.84  
1515  46  47  2.17  11.55  99.82  47  2.17  11.55  99.82  
1548  45  46  2.22  11.92  89.82  46  2.22  11.92  89.82  
1584  44  45  2.27  11.63  88.89  45  2.27  11.63  88.89  
1620  43  44  2.33  11.91  92.75  44  2.33  11.91  92.75  
1659  42  43  2.38  11.97  99.99  43  2.38  11.97  99.99  
1699  41  42  2.44  12.28  94.81  42  2.44  12.28  94.81  
1742  40  41  2.50  11.08  100.00  41  2.50  11.08  100.00  
1787  39  40  2.56  11.96  79.01  40  2.56  11.96  79.01  
1834  38  39  2.63  11.45  99.81  39  2.63  11.55  99.81  
1883  37  38  2.70  12.22  92.57  38  2.70  12.30  92.57  
1935  36  37  2.77  12.30  92.31  37  2.77  12.10  92.31  
>1991  >35  >36  >2.86  >12.10  >98.18  >36  >2.86  >11.55  >98.18  
2049  34  35  2.94  11.55  98.18  35  2.94  12.03  98.18  
2111  33  34  3.03  12.30  90.00  34  3.03  12.85  90.00  
Scholl  297  2177  32  33  3.13  12.10  98.78  33  3.13  11.55  98.78 
2247  31  32  3.23  11.55  100.00  32  3.23  11.92  100.00  
2322  30  31  3.33  12.03  86.84  31  3.33  11.63  86.84  
2402  29  30  3.45  12.85  96.84  30  3.45  11.91  96.84  
2488  28  29  3.57  12.84  99.82  29  3.57  11.97  99.82  
2580  27  28  3.70  12.81  92.57  28  3.7  12.28  92.57  
2680  26  27  3.85  11.77  92.31  27  3.85  11.08  92.31  
2787  25  26  4.00  12.63  98.18  26  4.00  11.99  98.18  
Total Optimal Solution (or Lower Bound) Found from 62 Problem Instances  35  1.49/inst.  5.70/inst.  94.61/inst.  36  0.94/inst.  4.88/inst.  95.51/inst. 
© 2018 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/).
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Sresracoo, P.; Kriengkorakot, N.; Kriengkorakot, P.; Chantarasamai, K. UShaped Assembly Line Balancing by Using Differential Evolution Algorithm. Math. Comput. Appl. 2018, 23, 79. https://doi.org/10.3390/mca23040079
Sresracoo P, Kriengkorakot N, Kriengkorakot P, Chantarasamai K. UShaped Assembly Line Balancing by Using Differential Evolution Algorithm. Mathematical and Computational Applications. 2018; 23(4):79. https://doi.org/10.3390/mca23040079
Chicago/Turabian StyleSresracoo, Poontana, Nuchsara Kriengkorakot, Preecha Kriengkorakot, and Krit Chantarasamai. 2018. "UShaped Assembly Line Balancing by Using Differential Evolution Algorithm" Mathematical and Computational Applications 23, no. 4: 79. https://doi.org/10.3390/mca23040079