AD ALTA 

 

JOURNAL OF INTERDISCIPLINARY RESEARCH

 

 

 

ARFM40 10 7 ∞ 

5119.0775 1038 5119.0019 0.06640 5119.0118 0.01112 5119.0775 0.01240 5118.4428 

0.00000 

ARFM41 15 2 0 910.402 853 910.3212 0.01092 910.2115 0.02143 910.402 0.04236 910.402 

0.04236 

ARFM42 15 2 1 898.723 761 898.666 0.03348 898.594 0.28039 898.723 0.29478 897.2341 

0.12863 

ARFM43 15 2 2 898.723 761 896.195 0.28842 896.9124 0.08005 898.2150 0.22540 898.723 

0.28208 

ARFM44 15 2 4 898.723 761 896.012 0.00000 896.8150 0.13122 896.7112 0.11964 897.1943 

0.17357 

ARFM45 15 2 ∞ 

898.723 

761 

895.1963 0.04157 898.723 0.40162 896.1542 0.11464 895.1280 

0.00000 

ARFM46 15 3 0 1801.24 1125 1801.24 

0.00000 

1801.24 

0.09150 

1800.8141 

0.06784 

1801.24 

0.09150 

ARFM47 15 3 1 1786.9466 1022 1783.9370 0.09150 1783.6050 0.07247 1785.3387 0.16975 1784.6243 0.12966 
ARFM48 15 3 2 1784.4 

973 

1780.19 0.09110 1779.6112 0.02339 1779.1950 0.00000 1780.4370 

0.06981 

ARFM49 15 3 4 1781.27 1119 1779.2219 0.05592 1779.3114 0.00000 1779.2219 0.00000 1781.27 

0.11511 

ARFM50 15 3 ∞ 

1781.27 

1119 1779.1871 0.00000 1778.1741 0.00000 1781.27 0.17411 1778.237 

0.00000 

ARFM51 15 5 0 1945.5209 1287 1944.922 0.05697 1944.1774 0.01217 1944.2050 0.01359 1945.388 

0.07444 

ARFM52 15 5 1 1842.97 1062 1840.568 0.05047 1840.7544 0.26503 1839.371 0.18968 1839.5367 0.19870 
ARFM53 15 5 2 1756.62 1052 1744.9271 0.25488 1740.9066 0.44497 1739.7095 0.37590 1739.637 

0.37172 

ARFM54 15 5 4 1737.433 1063 1735.906 0.67694 1735.7441 0.18636 1733.7536 0.07147 1735.196 

0.15472 

ARFM55 15 5 ∞ 

1734.45 

960 

1732.186 0.19570 1731.295 0.03074 1732.687 0.11116 1731.225 

0.02669 

ARFM56 15 7 0 7113.25 1567 7108.713 0.08222 7108.435 0.04221 7106.5365 0.01549 7105.4357 0.00000 
ARFM57 15 7 1 6463.77 1380 6442.763 0.04612 6441.75 0.12973 6435.2766 0.02911 6441.4266 0.12470 
ARFM58 15 7 2 6451.37 1508 6421.401 0.14547 6419.7475 0.24574 

6417.0431 

3313131

0.20351 

6419.23 

0.23766 

ARFM59 15 7 4 6443.723 1480 6368.6273 0.27156 6365.5124 0.20524 6368.476 0.25190 6367.538 

0.23713 

ARFM60 15 7 ∞ 6372.13 1288 6344.21 

0.25428 

6341.143 

0.01798 6343.14 0.04948 6344.151 

0.06543 

Mean 

  

 

 

 

 0.05152 

 0.05111 

 0.05250 

 0.05557 

 

The mathematical model is coded and solved by the modeling 
language Lingo 9.0. Meta-heuristic algorithms are coded in 
Matlab software, version 2013. A personal computer with the 
configuration of Core i5 2.5 GHz and 4 GB Ram is applied to 
solve the test problems. 

In the next part the best answers obtained from lingo software 
and two GA and SA algorithms in the 8 to 10 tables are 
examined, and the percentage of using stand by and turn off/turn 
on in the answers according to the factor of the buffer numbers 
in theses tables are presented. As shown in the tables increasing 
the amount of buffer factor the number of setting up decreases. 
So the number of stand by increases and the number of turn 
off/turn on decreases. 

Table 9: Percentage of using stand by and turn off/turn on in 
small size problems according to the factor of the buffer numbers 
 

Buffer Stand 

by 

turn 
off/turn on 

0 24% 

76% 

1 32% 

68% 

2 44% 

56% 

4 59% 

41% 

∞ 86% 

14% 

 
 
Table 10: Percentage of using stand by and turn off/turn on in 
medium size problems according to the factor of the buffer 
numbers 
 

Buffer Stand 

by 

turn 
off/turn on 

0 27% 

73% 

1 35% 

65% 

2 42% 

58% 

4 63% 

37% 

∞ 91% 

9% 

 

 
 
Table 11: Percentage of using stand by and turn off/turn on in 
large size problems according to the factor of the buffer numbers 
 

Buffer Stand 

by 

turn 
off/turn on 

0 21% 

79% 

2 52% 

48% 

4 67% 

33% 

∞ 97% 

3% 

  
5. Conclusion 
 
In this paper we investigated the permutation flow shop 
scheduling problem with limited buffers and the objectives of the 
minimization of total energy consumption and makespan. We 
formulated a mathematical model for the described problem. 
Since the proposed problem is NP-hard, so two well-known 
meta-heuristics namely; genetic algorithm and simulated 
annealing, have been used to produce approximate solutions in a 
reasonable time. We generated three different sizes of the 
problem, small, medium and large size problems. Lingo was able 
to give us the exact solution for all small size problems in time 
limit of 300 minutes, but for medium and large scale problems, 
Lingo is inefficient, so GA and SA have been used to reach near 
optimal solutions. The computational experiments show that 
with the used parameter settings of the algorithms for all 
problem sizes GA outperforms SA. At the end the best answers 
obtained from lingo software and two GA and SA algorithms in 
the 8 to 10 tables are examined, and the percentage of using 
stand by and turn off/turn on in the answers according to the 
factor of the buffer numbers in theses tables are presented. For 
future work it's suggested to use some other metaheuristic 
methods to solve the problem and compare the solutions with the 
existing ones, or maybe suggest a new heuristic for the problem. 
In our future research, the proposed algorithm might be extended 
to other machine environments such as job shop. Another 
extension is considering multi-objective optimization method

 

such as the Non-dominated Sorting Genetic Algorithm-II 
(NSGA-II) for the problem. 
 

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