The cutting and sewing process is a traditional flow shop scheduling problem in the real world. This twostage flexible flow shop is often commonly associated with manufacturing in the fashion and textiles industry. Many investigations have demonstrated that the ant colony optimization (ACO) algorithm is effective and efficient for solving scheduling problems. This work applies a novel effective ant colony optimization (EACO) algorithm to solve twostage flexible flow shop scheduling problems and thereby minimize earliness, tardiness, and makespan. Computational results reveal that for both small and large problems, EACO is more effective and robust than both the particle swarm optimization (PSO) algorithm and the ACO algorithm. Importantly, this work demonstrates that EACO can solve complex scheduling problems in an acceptable period of time.
Garment producers must continuously adjust their production systems, including product design, production, and distribution, to remain competitive in the market. They also use new production techniques and equipment, increase outsourcing, and increase the number of strategic alliances with logistic coordinators to shorten processing times and ensure quick responses (QR) (Uruk et al. [
In garment manufacturing, material must be cut before it is sewn. Waiting for predetermined patterns in the cutting process consumes time (Wong et al. [
Flow chart of the cutting and sewing process of general apparel manufacturing industry.
Wong et al. [
Wang and Liu [
For solving scheduling problems with time windows, this work develops an effective method that prevents unexpected delays and associated large losses during production. Based on customer responses, Solomon and Desrosiers [
The remainder of this paper is organized as follows. Section
The designated twostage flexible flow shop scheduling problem of this study is
Consider the following.
Objective function:
Equation (
Subject to:
Earliness:
Equation (
Tardiness:
Equation (
Maximum completion time and sequence constraints:
Equation (
Equation (
Equation (
Equation (
Equation (
Makespan:
Equation (
The proposed EACO algorithm modifies the state transition rule to find operations with the shortest time window and conducts it to be weighted to a definite local pheromone parameter (
The EACO algorithm utilizes similar state transition rule into original ACO (Dorigo et al. [
In (
Except when required by state transition rule, the shortest time window function also utilizes the global pheromone update rule to update pheromones on each node according to the following:
Updating the global pheromone values after each iteration reinforces the visibility of the best route, increasing the probability that a node is properly chosen to pursue the optimal solution, and the solutions converge as the amount of pheromones on the nodes on the best route increases.
Although the state transition rule and global pheromone update rule are used, the EACO computation is just the same as the ACO computation. The algorithm firstly releases a specified number of ants in one iteration, searches for the best route, and memorizes which nodes are on the best route using the global pheromone update rule. The algorithm terminates when it satisfies a stop criterion, such as a certain number of iterations or convergence.
The PSO algorithm is based on the way in which birds find food. Individual birds do not know the location of food in a particular search space. However, they move in a direction for a distance that is determined by their experiences and those of the other birds that they are following. The PSO algorithm treats each bird as a particle in which the leader is the nearest particle to food, representing the optimal solution.
Kennedy and Eberhart [
Kennedy and Eberhart [
Traditional PSO is based on iterated solutions that are found by randomly generated particles. The solution procedure consists of the following three steps: initialization, searching for and updating the optimal solution, and search termination.
In a multidimensional space, particle coordinates and velocity parameters are randomly generated. Fitness values are calculated to optimize the position for the initial coordinate of the population. Every particle then inherits the best coordinates.
The position of every particle is modified based on the following equations:
In (
After the positions of all particles are modified positions using (
The algorithm repeats Step
This study considers the twostage flexible flow shop problem. All assumptions regarding job numbers and machines used in the simulation refer to the categories for solving the overlapping shop problem proposed by Huang and Yang [
As for the issue of due window, this study adopts the formula derived by Zheng et al. [
Due window parameter settings.
Data type  DR  TF 

I  0.6  0.2 
II  1.0  0.2 
III  0.6  0.5 
IV  1.0  0.5 
In the weighting of parameters for measuring solutions whether they meet stop criterion, this work incorporates decision processes that are used by enterprises to increase the weight of the cost of inventory to reduce stock pressure or increase the cost of tardiness for important orders and simultaneously to minimize makespan. This study thus adopts all possible conditions not just using unique weight cost but divides weighted costs into three categories
The settings of the parameters in the algorithm significantly influence the execution results. This work optimizes the parameter settings following a literature review and a pretest adjustment:
To test the efficiency of the EACO algorithm, the EACO algorithm is used to solve a distinct problem to verify its effectiveness and compare it with the effectiveness of other methods. The formulae for measuring effectiveness in small problems are as follows.
Improvement ratio of effectiveness:
To demonstrate how the EACO algorithm outperforms PSO and ACO algorithms for small problems, 30 test datasets were calculated 30 times to compare computational times, mean solutions, and percentage difference among IP and the EACO, PSO, and ACO algorithms. With respect to the solving of the large problem, preliminary test results demonstrated that IP computational time increased exponentially with the number of jobs and stages (Huang et al. [
Effectiveness test results for small problems (
Data type 

IP  ACO  PSO  EACO  

Average  CPU time (second)  Average  CPU time (second)  Average  CPU time (second)  Average  CPU time (second)  
3 

5.01 

0.81 

0.82 

0.81  
4 

56.14 

1.03 

1.02 

1.05  
I  5 

262.32 

1.86 

1.84 

1.85 
6 

402.16 

0.85 

0.83 

0.84  
7 

555.08 

1.95 

1.92 

1.93  
8 

832.31 

1.99 

1.97 

1.98  
 
3 

5.12 

0.84 

0.83 

0.83  
4 

55.43 

1.12 

1.03 

1.05  
II  5 

264.66 

1.86 

1.84 

1.83 
6 

305.92 

0.85 

0.83 

0.83  
7 

552.41 

1.94 

1.92 

1.92  
8 

753.85 

2.02 

2.24 

2.28  
 
3 

5.24 

0.84 

0.82 

0.83  
4 

53.75 

1.13 

1.1 

1.11  
III  5 

241.22 

1.86 

1.84 

1.85 
6 

345.19 

0.87 

0.83 

0.83  
7 

564.32 

1.97 

1.92 

1.92  
8 

613.93 

2.08 

2.12 

2.06  
 
3 

3.97 

0.84 

0.83 

0.83  
4 

51.32 

1.12 

1.03 

1.05  
IV  5 

246.73 

1.86 

1.83 

1.84 
6 

365.08 

0.88 

0.84 

0.85  
7 

526.04 

1.98 

1.92 

1.92  
8 

728.46 

2.01 

1.98 

2.02 
This study also tests the weight cost conditions of (0.50, 0.25, 0.25) and (0.25, 0.25, 0.50). Table
Effectiveness test results for small problems contrast to PSO and ACO.
Data type 






EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus  
3  −25.66%  −11.31%  34.26%  33.64%  −21.21%  −10.33%  
4  −13.20%  −5.31%  18.51%  21.27%  −9.64%  12.88%  
I  5  17.80%  18.55%  46.28%  37.25%  44.39%  37.15% 
6  21.69%  19.91%  25.41%  38.25%  38.19%  29.65%  
7  16.48%  40.80%  56.13%  48.26%  27.74%  28.62%  
8  16.23%  13.40%  31.26%  27.05%  30.25%  43.27%  
 
3  30.71%  18.07%  42.52%  30.13%  18.22%  37.67%  
4  27.18%  13.95%  13.29%  10.47%  27.02%  32.01%  
II  5  43.73%  5.65%  33.15%  53.24%  29.41%  45.84% 
6  23.63%  14.28%  43.19%  38.53%  18.46%  52.61%  
7  2.58%  10.19%  32.03%  41.21%  52.25%  65.48%  
8  7.10%  14.11%  18.24%  30.23%  29.24%  39.47%  
 
3  10.59%  6.69%  35.41%  40.62%  50.14%  2.02%  
4  25.76%  29.34%  27.56%  32.26%  21.52%  23.97%  
III  5  37.75%  49.63%  31.26%  42.25%  43.25%  58.25% 
6  11.83%  −0.62%  28.15%  22.56%  −15.59%  8.03%  
7  5.34%  9.09%  −21.13%  61.51%  34.28%  44.96%  
8  15.85%  34.46%  29.24%  42.87%  29.16%  26.47%  
 
3  35.35%  43.03%  39.15%  65.02%  59.38%  75.43%  
4  39.91%  33.85%  −10.67%  35.38%  24.91%  35.17%  
IV  5  26.40%  46.95%  53.55%  56.19%  33.28%  32.32% 
6  21.36%  16.57%  30.57%  32.57%  61.45%  65.16%  
7  7.16%  22.04%  25.73%  28.82%  26.41%  47.29%  
8  36.66%  13.27%  27.23%  33.53%  33.72%  34.84%  
 
Avg. improvement  18.43%  19.02%  28.76%  37.63%  28.59%  36.18%  
 
Total improvement (EACO versus PSO)  25.26%  
 
Total improvement (EACO versus ACO)  30.94% 
The EACO algorithm is better than the PSO and ACO algorithms in finding the mean solutions to 30 problems. Under distinct conditions of due window and cost weighting, the total improvements in contrast to PSO and ACO over mean solutions are calculated to be 25.26% and 30.94%, respectively. Accordingly, the improvements in contrast to PSO and ACO over mean solutions that are achieved by the EACO algorithm in solving small problems exceed those obtained using the PSO and ACO algorithms.
To examine further whether the EACO algorithm is more robust than the PCO and ACO algorithms, this work determines the improvement in robustness for small and large problems.
Improvement ratio of robustness:
A randomly generated problem is tested 30 times using the EACO, PSO, and ACO algorithms under three cost conditions. The IP is presented once to compare the deviation of solutions with that of the PSO and ACO algorithms. This work calculates the worst solution, best solution, mean solution, standard deviations, and CPU time (in seconds) for each scenario. Table
Robustness test results for small problems (
Data type 

IP  ACO  PSO  EACO  

Optimal  CPU  Best  Avg.  CPU  Best  Avg.  CPU  Best  Avg.  CPU  
time 


time 


time 


time  
3  79.20  5  80.10  85.07  0.8  80.30  85.02  0.8  80.20  85.36  0.81  







4  148.50  52  157.20  162.3  1.02  158.50  161.96  1  158.90  159.6  1  







5  274.50  246  282.00  292.37  2.01  289.30  292.44  2  291.50  285.25  2  
I 







6  379.00  413  389.20  397.23  0.85  381.20  399.33  0.83  389.20  405.23  0.83  







7  498.10  582  515.00  529.33  1.97  504.20  524.76  1.93  514.20  534.72  1.93  







8  587.90  849  626.40  641.03  2.01  614.70  649.74  2  604.30  657.78  2  







 
3  138.90  6  143.90  149.07  0.84  141.80  148.02  0.83  141.20  148.15  0.83  







4  249.60  59  254.30  261.93  1.21  252.60  269.36  1.21  259.60  271.24  1.21  







5  389.20  248  392.10  409.73  1.98  390.10  401.8  1.94  397.50  405.04  1.94  
II 







6  528.90  321  569.70  581.27  0.95  558.90  589.07  0.93  558.90  581.98  0.93  







7  736.00  572  846.40  862.73  1.9  831.50  858.39  1.9  846.40  861.6  1.9  







8  832.60  751  915.80  931.17  2.03  903.10  923.07  2.01  901.20  936.52  2.01  







 
3  80.80  5  89.10  98.08  0.86  102.80  108.27  0.85  101.80  102.96  0.85  







4  192.70  52  203.60  210.93  1.1  200.60  208.04  1.03  197.20  206.77  1.05  







5  319.90  256  381.10  396.9  2  399.30  411.67  2  370.10  387.72  2  
III 







6  410.80  361  461.30  501.23  0.85  452.80  511.63  0.83  450.80  495.91  0.83  







7  523.20  589  561.40  591.17  1.99  568.40  578.09  1.92  573.90  576.45  1.92  







8  638.60  625  683.50  709.9  2.01  674.50  721.59  2  679.10  712.77  2  







 
3  150.20  4  161.20  168.55  0.84  160.00  170.3  0.83  160.20  169.27  0.83  







4  268.30  50  289.20  303.67  1.12  291.30  313.09  1.11  298.30  302.9  1.11  







5  415.70  259  487.30  528.57  1.86  478.40  533.3  1.84  472.90  524.31  1.84  
IV 







6  579.00  392  617.30  669.87  0.82  609.00  678.24  0.81  595.00  680.11  0.81  







7  724.70  554  786.50  795.16  1.95  778.50  788.74  1.94  765.00  780.89  1.94  







8  828.20  714  872.00  897.83  2.21  881.10  890.08  1.99  887.50  892.52  1.99  






This study also tests the weight cost conditions of (0.50, 0.25, 0.25) and (0.25, 0.25, 0.50). Table
Improvement ratio of robustness using EACO in small problems, in contrast to PSO and ACO.
Data type 






EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus  
3  15.97%  22.56%  18.27%  25.16%  41.01%  43.16%  
4  40.66%  29.68%  51.41%  25.68%  29.16%  38.48%  
I  5  39.91%  37.02%  32.01%  34.42%  32.07%  45.64% 
6  −24.15%  5.24%  40.11%  53.42%  34.42%  15.83%  
7  −49.57%  −12.38%  6.82%  23.57%  −24.21%  2.86%  
8  15.39%  20.50%  32.18%  31.06%  40.25%  49.47%  
 
3  57.07%  31.76%  32.65%  30.56%  22.37%  34.54%  
4  −41.02%  −19.58%  34.95%  45.87%  21.47%  36.01%  
II  5  32.08%  −9.12%  41.53%  42.48%  32.49%  71.26% 
6  19.46%  22.51%  −30.23%  31.75%  52.51%  46.51%  
7  19.01%  11.40%  25.07%  −18.23%  25.04%  31.86%  
8  7.34%  22.94%  27.65%  15.26%  48.47%  37.61%  
 
3  51.61%  25.20%  21.21%  21.02%  −31.26%  45.67%  
4  28.57%  14.91%  37.25%  16.21%  61.42%  35.07%  
III  5  31.37%  27.29%  32.47%  22.06%  51.37%  47.33% 
6  46.81%  18.54%  −22.09%  45.19%  9.53%  50.76%  
7  32.05%  10.21%  31.20%  31.63%  −25.76%  8.62%  
8  10.57%  22.86%  54.19%  43.68%  54.79%  38.65%  
 
3  41.76%  47.44%  40.71%  23.64%  22.92%  21.48%  
4  8.65%  28.81%  −23.82%  0.00%  37.68%  43.64%  
IV  5  42.69%  21.69%  52.08%  29.69%  60.42%  48.92% 
6  38.44%  29.26%  44.32%  48.13%  50.62%  52.39%  
7  28.87%  21.24%  18.11%  23.02%  43.16%  31.01%  
8  43.50%  22.07%  51.36%  37.19%  62.25%  58.42%  
 
Avg. improvement  22.38%  18.83%  27.06%  28.44%  31.34%  38.97%  
 
Total improvement (EACO versus PSO)  26.93%  
 
Total improvement (EACO versus ACO)  28.75% 
The test results demonstrate that improvement ratios of robustness are excellent and outstanding in each test set within 2.3 s. The total improvements from using the EACO algorithm in the robustness ratios in contrast to the PSO and ACO algorithms are 26.93% and 28.75%, respectively. These ratios demonstrate that the EACO algorithm is more robust than the PSO and ACO algorithms in solving small twostage flexible flow shop scheduling problems.
In real scheduling problems, the complexity of amount of jobs is multiplied by various orders. Therefore, following the small problem testing, large problems are tested. Pretesting demonstrates that the CPU time that is required to solve IPs increases exponentially with the total number of jobs. The CPU time for IP is significantly large, so the optimal solution is unidentified preventing the optimal solution from being found. Thus, the IP solution is omitted in tests that involve large problems. This work also compares the solving capacity of the EACO algorithm with those of the PSO and ACO algorithms when they are applied to large problems. Table
Robustness test results for large problems (
ACO  PSO  EACO  

Data type 

Best  Avg.  CPU time  Best  Avg.  CPU time  Best  Avg.  CPU 






time  
50  2142  2245.73  9.06  2102.7  2181.07  9.05  2014.3  2093.8  10.03  







100  3862  4052.17  12.73  3787.5  3945.02  12.7  3460.3  3704.16  12.7  
I 







150  6342.6  6962.7  23.26  6361.4  6680.35  23.2  5952.3  6311.41  23.2  







200  10252.2  10725.04  34.05  9749.65  10232.95  34.02  9865.2  10677.51  34.04  







 
50  2412.3  2537.45  9.07  2375.5  2592.08  9.01  2357  2392.8  9.03  







100  4225.2  4411.61  12.53  4057.9  4437.43  12.5  3972.2  4407.94  12.51  
II 







150  7119  7588.2  23.16  7084  7451.48  22.1  7387.9  7801.52  23.4  







200  9526  10216.47  34.03  9431.2  9863.23  34.15  9321.7  9931.53  34.41  







 
50  2502  2606.72  11.1  2488.1  2587.94  11.9  2391.1  2517.68  11.02  







100  4371.5  4913.59  15.63  4368  4897.42  14.6  4826.2  4822.51  15.6  
III 







150  6362  6862.47  23.06  6272.4  6980.27  23.5  6116.2  6742.05  23.8  







200  11687  11897.02  45.08  10725.1  11956.01  46.02  11218.5  11723.49  45.03  







 
50  2757.5  2872.46  9.06  2601.8  2712.39  9.87  2605.2  2807.25  9.53  







100  4752  4982.57  12.15  4718.3  4854.81  12.33  4519.1  4851.33  12.04  
IV 







150  6852.1  7611.53  23.22  6927  7528.47  23.17  6803.2  7482.16  23.03  







200  10714  11375.54  34.05  10521.5  11421.79  34.27  10817.6  11302.58  34.68  






This study also tests the weight cost conditions of (0.50, 0.25, 0.25) and (0.25, 0.25, 0.50). Table
Improvement ratio of robustness using EACO in large problems, in contrast to PSO and ACO.
Data type 






EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus 
EACO versus  
50  29.76%  23.86%  22.07%  24.06%  31.27%  60.87%  
I  100  27.24%  49.15%  29.02%  42.48%  31.38%  67.51% 
150  44.66%  69.97%  −33.19%  61.75%  54.63%  42.09%  
200  29.59%  35.58%  32.19%  37.13%  49.23%  32.83%  
 
50  −85.35%  −10.96%  27.56%  39.49%  30.72%  −62.49%  
II  100  21.68%  41.22%  30.12%  −49.03%  24.75%  52.66% 
150  −16.49%  49.09%  35.25%  72.19%  32.24%  43.84%  
200  44.54%  85.76%  41.11%  82.72%  48.05%  61.25%  
 
50  −13.91%  −105.57%  27.26%  52.42%  38.25%  45.34%  
III  100  27.46%  71.34%  32.23%  74.15%  −52.87%  51.94% 
150  38.93%  25.33%  32.46%  60.02%  37.59%  55.42%  
200  40.51%  72.65%  49.28%  77.19%  41.32%  40.89%  
 
50  51.43%  37.42%  −21.54%  34.26%  42.17%  58.04%  
IV  100  33.58%  57.58%  32.27%  53.66%  36.26%  78.54% 
150  68.18%  51.83%  38.64%  67.81%  37.31%  −56.45%  
200  34.41%  69.19%  31.33%  −76.82%  46.13%  48.73%  
 
Avg. improvement  23.51%  38.97%  25.38%  40.84%  33.03%  38.81%  
 
Total improvement (EACO versus PSO)  27.31%  
 
Total improvement (EACO versus ACO)  39.54% 
For large problems, test results for all test problems indicate that the EACO algorithm is more robust than the PSO or ACO algorithm. The total improvements in robustness are 27.31% and 39.54%, respectively. Thus, the EACO algorithm is more robust than the PSO or ACO algorithm for large twostage flexible flow shop scheduling problems and it exhibits excellent stability.
Table
Total test results for small and large problems.
Avg. improvement ratio of effectiveness  Avg. improvement ratio of robustness 




Small problems  EACO versus PSO  25.26%  26.93%  *  * 
EACO versus ACO  30.94%  28.75%  *  *  
 
Large problems  EACO versus PSO  —  27.31%  *  * 
EACO versus ACO  —  39.54%  *  * 
Owing to the limitations of IP testing, this work cannot compare the effectiveness of IP with that of other algorithms. The EACO algorithm provides mean improvements in robustness of 27.31% and 39.54% over the PSO and ACO algorithms in solving large problems, respectively, verifying that the EACO algorithm has an excellent capacity to solve such scheduling problems.
A good fashion and textiles operational control systems management can efficiently reduce the manufacturing duration, support the justintime shipping of garments, and strengthen competitiveness of enterprises. This study is to propose a novel algorithm to solve the cutting and sewing processes in a traditional flow shop scheduling problem that comprises justintime and costs to minimize earliness, tardiness, and makespan. Since the problem is NP hard, exactsolution methods cannot satisfy the agile requirements for solving the complex problems.
Numerous investigations demonstrate that the ACO and PSO are effective for solving scheduling problems. Hence, this study aims at the characteristic of cutting and sewing processes constructing an EACO algorithm to significantly solve the FFSP in the fashion and textiles industry. These heuristics cannot ensure getting the optimal solution; however, the proposed method must test both effectiveness and robust performance to solve the attempted problems. The results of data testing show that the proposed EACO algorithm is superior to the basic ACO and PSO algorithms. By using the EACO algorithm, this study obtains excellent results and provides the production managers and scheduling operators with an alternative way to deal with the attempted problems.
In the manufacturing of garments, the garments must be cut before they are sewn, and the cutting process consuming time requires that time be taken in waiting for patterns. Producers depend only on manual scheduling when planning production activities, likely causing excessive makespan and overhead costs. This work applies the EACO algorithm to solve this scheduling problem efficiently.
Following careful testing, this work solves the twostage flexible flow shop scheduling problem with due window constraints. The objective function minimizes total weighted earliness, tardiness, and makespan. Previous studies have demonstrated that PSO and ACO algorithms can solve scheduling problems, but that they have limited effectiveness and robustness. This work demonstrates that the twostage flexible flow shop scheduling problem with due window constraints is an NPhard problem.
In this work, a novel EACO algorithm, based on the ACO algorithm, is utilized to solve the twostage flexible flow shop scheduling problem. The EACO algorithm is 25.26% and 30.94% more effective for solving small problems than the PSO and ACO algorithms, respectively. EACO is much more effective, on average, than the other two algorithms in solving large problems. Furthermore, average improvements in robustness in solving small problems are 26.93% and 28.75%, respectively, confirming that the EACO algorithm significantly outperforms the PSO and ACO algorithms in both effectiveness and efficiency. Enterprises can exploit the advantages of the proposed algorithm to generate profits and reduce overhead manufacturing costs.
This work offers IP solutions for the best solutions within a satisfactory time and the EACO algorithms provide solutions for quick response to market within a short time. Computational results prove that the EACO algorithm has a higher solving capacity than the common ACO and PSO algorithms. The proposed method reduces waiting and tardiness costs, helping enterprises simultaneously shorten makespan, increase profits, and lower overhead costs.
The authors thank the Editor and anonymous reviewers for their valuable comments and suggestions for the improvement of this paper. They would like to thank the National Science Council, Taiwan, for supporting this research under Contract no. NSC 1002221E030005.