The two-sided assembly line balancing problem type-II (TALBP-II) is of major importance for the reconfiguration of the two-sided assembly lines which are widely utilized to assemble large-size high-volume products. The TALBP-II is NP-hard, and some assignment restrictions in real applications make this problem much more complex. This paper provides an integer programming model for solving the TALBP-II with assignment restrictions optimally and utilizes a simple and effective iterated greedy (IG) algorithm to address large-size problems. This algorithm utilizes a new local search by considering precedence relationships between tasks in order to reduce the computational time. In particular, a priority-based decoding scheme is developed to handle these assignment restrictions and reduce sequence-dependent idle times by adjusting the priority values. Experimental comparison among the proposed decoding scheme and other published ones demonstrates the efficiency of the priority-based decoding. A comprehensive computational comparison among the IG algorithm and other eight recent algorithms proves effectiveness of the proposed IG algorithm.
A two-sided assembly line consists of a set of sequential mated-stations connected by the material handling system. It is widely used to produce large-size high-volume products, such as cars, trucks, and automobiles. An example of two-sided assembly lines is shown in Figure
A typical example of two-sided assembly line.
Compared with the traditional one-sided assembly line, the two-sided assembly line balancing problem (TALBP) is more complex due to the existence of restrictions on the operation direction. Some tasks have to be operated on a predefined side of the line, while others can be allocated to any side of the line. Thus, the task preferred directions can be classified into three types, namely, L (left), R (right), and E (either). In summary, there are three restrictions that need to be satisfied, including the precedence restriction, cycle time restriction, and direction restriction.
Besides these basic restrictions, other assignment restrictions which exist in real applications need to be considered, including zoning restriction, synchronous restriction, positional restriction, distance restriction, and resource restriction [
The TALBP, so far as the objectives are concerned, can be divided into two versions: TALBP-I that minimizes the number of stations with a given cycle time, and TALBP-II that minimizes the cycle time for given mated-stations. The TALBP-I is more appropriate for the first-time installation of the assembly line, while the TALBP-II is proper for the reconfiguration of the assembly line [
To our knowledge, limited papers considered the TALBP-II [
The organization of this paper is introduced as follows. Section
The two-sided assembly line balancing has become an active field of research since the two-sided assembly line balancing problem (TALBP) was first introduced by Bartholdi [
Most attention is paid to the TALBP-I, and exact, heuristic, and meta-heuristic methods have been applied. Exact methods include station-oriented enumerative algorithm [
As for the TALBP-II, Kim et al. [
Now we focus on additional contributions dealing with assignment restrictions. Apart from the above literature of Purnomo et al. [
From the literature review, it is observed that only one paper deals with TALBP-II with assignment restrictions and no detailed encoding procedure for TALBP-II with assignment restrictions is presented. Therefore this paper focuses on the TALBP-II with assignment restrictions by utilizing a new algorithm and also provides new decoding schemes to handle assignment restrictions and reduce sequence-dependent idles by adjusting the priority values.
A single model is taken into account and the travel time between stations is ignored. Precedence diagrams are known and deterministic. The task with a positional restriction should be assigned to the predefined station. The tasks in the positive zoning restriction should be assigned to the same station. The tasks in the negative zoning restrictions are prohibited to be allocated to the same mated-station.
In the TALBP-II, minimizing the cycle time is the widely used objective [
Constraint (
The general objective of TALBP-II is minimizing the cycle time [
In the TALBP-II, the number of mated-stations is fixed and the station-oriented encoding and decoding are widely utilized [
The proposed priority-based decoding first selects the side of the current mated-station and then selects a task based on the priority values. Also, the task assignment rule is embedded into the decoding schemes to balance the workload and reduce sequence-dependent idle times. In contrast to the work of Yuan et al. [
In the proposed priority-based decoding, all the initial priority values are limited to a range of
The task assignment rule is applied to reduce the sequence-dependent idle time and reduce the workload deviation. First, we seek to reduce the sequence-dependent idle time by boosting the priorities of tasks without generating sequence-dependent idle times. Then we try to balance the workload on each station by boosting the priority values of tasks whose finishing times on the station are within a range
Calculate priority values for all the tasks.
Set
If
If
After Step 4, there are three conditions:
In Step 2, only direction and precedence restrictions need to be satisfied while
Yuan et al. [
After adjusting the priorities, when the former mated-station is the current mated-station,
Yuan et al. [
For task
The procedure of the decoding scheme is introduced as follows.
Set the initial cycle time
Open a new mated-station.
Choose a side with greater capacity of the current mated-station. If the capacities of both sides are equal, select the left side by default.
Update the priority values with task assignment rule.
If task
If
If
If task
If no task whose priority value is larger than one exists, select another side and execute Steps 2–7. If no tasks can be assigned to either side, go to Step 2; else, select the task with the largest priority and allocate the tasks to the selected station, and then go to Step 10.
If some tasks are still not assigned, go to Step 3; if all the tasks are allocated, go to Step 11.
Choose the largest finishing time of the stations as the current cycle time and update the initial cycle time.
In this procedure, Step 4 considers the cycle time, direction, and precedence restrictions. Steps 5, 6, 7, and 8 deal with negative zoning restrictions, positional restrictions, positive zoning restrictions, and synchronous restrictions, respectively. After executing Steps 4–8, the task can be assigned only if
In the decoding scheme, the synchronism restriction and negative zoning restriction are satisfied while positive zoning restriction and positional restrictions may be violated. Therefore, the two factors are considered in the cost function as follows:
The iterated greedy (IG) algorithm is a local search method and it is can be regarded as a simple approach with less sophisticated parameters than other hybrid algorithms [
IG starts with constructing a high performing initial solution. Then random destruction is proposed to remove some tasks and reconstruction is utilized to reinserts these tasks back. After reconstruction, a different solution can be obtained and then the acceptance criterion is applied to determine whether the incumbent can be substituted. Optionally, after the acceptance criterion phase, a local search procedure aims at finding better solutions locally. The random destruction, reconstruction, acceptance criterion, and local search together make up a loop. The procedure of IG for TALBP-II with assignment restrictions is explained as follows.
The NEH heuristic [
Generate an initial task permutation
Remove the 2nd task
Remove the remaining tasks
The mNEH differs from the original NEH in two aspects:
The insert operator is selected as the neighbor structure after checking the swap operator and insert operator. Based on the work of Ruiz and Stützle [
improve improve remove a task improve
The local search of Algorithm
Remove task Continue;
In order to escape local optima, destruction and reconstruction are applied to obtain a new solution. Then, the acceptance criterion is utilized to determine whether the new solution replaces the incumbent. In the destruction phase of Ruiz and Stützle [
Note that the number of removed tasks
The computational tests are carried out to prove the effectiveness of the priority-based decoding schemes for TALBP-II with assignment restrictions and the high performance of the improved IG algorithm. All the benchmark problems with different cycle times are solved, which range from small-size problems, P9, P12, P16, and P24, to large-size problems, P65, P148, and P205. P9, P12, and P24 are taken from Kim et al. [
List of assignment restrictions.
Problem | Number of mated-station | Positional restriction | Positive zoning restriction | Negative zoning restriction | Synchronism restriction |
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P9 | 2 |
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P12 | 2 |
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P16 | 2 |
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P24 | 2 |
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P65 | 4 |
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P148 | 5 |
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P205 | 6 |
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This section tests the performance of the proposed priority-based decoding scheme by comparing it with the reported ones. There are two components to be considered:
For the first component, two genetic algorithms (GA) with different decoding schemes are compared and all the algorithms have the same parameters. They are GA1 with the station-oriented decoding scheme [
In order to check whether there is statically significant difference between the two decoding schemes, we use the parametric
Mean plot of average ranks with 95% confidence intervals.
In fact, the original station-oriented decoding scheme cannot reduce the sequence-dependent idle time effectively since it only uses a heuristic method. The proposed priority-based decoding, on the contrary, takes the sequence-dependent idle times and the balance of the stations into account, which can further reduce the idle times on the stations.
For the method to deal with assignment restrictions, we also utilized two genetic algorithms:
In this part, we calibrate the iterated greedy algorithm and determine
Since different problems are involved, we select the best combination for each problem. Taking the largest-size problem, P205 [
After carrying out all the runs, the analysis of variance (ANOVA) technique [
ANOVA results for RDI values.
Source of variation | Df | Sum of squares | Mean square |
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3 | 1.375 | 0.458 | 8.151 |
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3 | 0.102 | 0.034 | 0.604 | 0.615 |
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9 | 0.491 | 0.0546 | 0.971 | 0.472 |
Residual | 64 | 3.599 | 0.0562 | ||
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Total | 79 | 5.568 | 0.0705 |
Mean plot for the factors
Notice that it is more convenient to divide all the problems into several groups and decide the parameters for each group. This paper selects the parameters for each problem due to different performance of an algorithm on different problems, and the homogeneity of variance and normality of the residuals are violated if we calibrate the parameters for several problems together.
This section aims at showing the effectiveness of the IG on TALBP-II and TALBP-II with additional restrictions. The IG is compared with several recent algorithms: a tabu search algorithm (TS) [
Since the IG algorithms have not been applied to the TALBP-II without additional restrictions, we first compare the performances of these algorithms on the TALBP-II without additional restrictions. All the cases are solved for 20 times, and the average results on 14 small-size cases are reported in Table
Average result comparison for small-size problems.
Problem | nm | OPT | TS | SA | LAHC | 2-ANTBAL | PSONG | GA | TLBO | BA | IG1 | IG2 |
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P9 | 2 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
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3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
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P12 | 2 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
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3 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
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4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
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5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
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P16 | 2 | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22 |
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3 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 |
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4 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |
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5 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 |
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P24 | 2 | 35 | 35 | 35 | 35 | 35 | 35 | 35 | 35 | 35 | 35 |
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3 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
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4 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 |
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5 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 |
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As for the large-size problem, the average RDI value of each case is reported in Table
Average relative deviation index (RDI) for the large-size cases. Best results in bold.
Problem | nm | TS | SA | LAHC | 2-ANTBAL | PSONG | GA | TLBO | BA | IG1 | IG2 |
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65 | 4 | 100.00 | 75.00 | 51.09 | 38.04 | 4.35 | 18.48 | 15.22 | 21.74 | 13.04 |
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5 | 100.00 | 73.79 | 43.45 | 15.86 |
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10.34 | 11.03 | 13.10 | 16.55 | 1.38 | |
6 | 100.00 | 93.87 | 50.92 | 15.34 | 5.52 | 23.93 | 0.00 | 33.74 | 13.50 |
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7 | 84.35 | 100.00 | 74.78 | 8.70 | 9.57 | 23.48 | 13.04 | 31.30 | 20.87 |
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8 | 82.28 | 100.00 | 72.15 | 17.72 | 3.80 | 23.42 | 5.70 | 22.15 | 15.82 |
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148 | 4 | 57.58 | 100.00 | 18.18 |
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10.61 | 0.00 | 0.00 | 4.55 |
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5 | 100.00 | 96.43 | 50.00 |
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26.79 | 0.00 | 8.93 | 21.43 | 10.71 | |
6 | 89.89 | 100.00 | 13.30 | 1.06 |
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6.38 | 0.00 | 0.00 | 3.19 | 2.13 | |
7 | 72.16 | 100.00 | 47.42 |
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16.49 | 0.00 | 0.00 | 12.37 | 11.34 | |
8 | 100.00 | 98.65 | 59.46 |
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6.76 | 41.89 | 20.27 | 21.62 | 28.38 | 18.92 | |
9 | 96.04 | 100.00 | 60.40 | 2.97 |
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21.78 | 2.97 | 0.00 | 10.89 | 8.91 | |
10 | 100.00 | 82.91 | 64.56 |
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33.54 | 0.00 | 0.00 | 21.52 | 18.99 | |
11 | 79.77 | 100.00 | 63.58 | 1.16 |
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15.03 | 2.89 | 3.47 | 14.45 | 8.67 | |
12 | 81.44 | 100.00 | 70.10 | 1.03 |
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25.77 | 6.70 | 0.00 | 18.56 | 17.01 | |
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205 | 4 | 100.00 | 84.57 | 56.35 | 17.24 | 33.48 | 14.34 | 17.51 | 48.91 | 54.08 |
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5 | 95.66 | 100.00 | 63.30 | 13.98 | 23.61 | 26.80 | 5.63 | 27.41 | 43.76 |
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6 | 100.00 | 80.81 | 50.36 | 15.06 | 20.24 | 21.70 | 28.74 | 27.85 | 40.81 |
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7 | 100.00 | 73.04 | 68.97 | 16.30 | 16.22 | 14.42 | 26.18 | 45.69 | 33.62 |
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8 | 84.32 | 100.00 | 73.64 | 3.36 | 11.11 | 13.18 | 20.84 | 6.55 | 33.07 |
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9 | 83.55 | 100.00 | 56.52 | 18.36 | 15.90 | 14.39 | 17.65 | 40.22 | 41.97 |
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10 | 95.99 | 100.00 | 72.43 |
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9.42 | 30.63 | 17.89 | 23.91 | 42.93 | 4.54 | |
11 | 96.91 | 100.00 | 65.40 | 6.65 | 7.21 | 14.89 | 8.08 | 11.24 | 23.20 |
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12 | 100.00 | 78.69 | 71.81 | 16.43 | 13.84 | 18.63 | 20.92 | 32.77 | 40.04 |
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13 | 100.00 | 93.14 | 88.76 | 0.15 | 0.00 | 3.21 | 0.00 | 0.00 | 15.04 |
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14 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
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Avg | 88.00 | 89.24 | 56.28 | 8.38 | 7.24 | 18.80 | 9.65 | 16.82 | 23.35 |
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Best results of large-size problems. Best results in bold.
Problem | nm | LB | n-GA | TS | SA | LAHC | 2-ANTBAL | PSONG | GA | TLBO | BA | IG1 | IG2 |
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65 | 4 | 638 | 641 | 641 |
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639 | 639 |
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5 | 510 | 515 | 514 | 512 | 513 | 512 |
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512 | 513 | 512 | 512 | |
6 | 425 | 432 | 431 | 427 | 427 | 427 | 427 |
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427 | 430 |
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7 | 365 | 372 | 369 |
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368 | 368 | 368 | 370 | 368 |
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8 | 319 | 327 | 326 | 322 | 323 | 322 |
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148 | 4 | 641 |
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5 | 513 | 514 |
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6 | 427 |
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7 | 366 | 368 |
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8 | 321 | 323 | 323 |
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322 | 322 |
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9 | 285 | 287 | 287 |
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10 | 257 | 259 | 259 |
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11 | 233 | 237 | 236 | 235 | 236 |
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235 | 235 |
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12 | 214 | 218 | 218 |
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216 |
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205 | 4 | 2919 | 2946 | 2943 | 2945 | 2941 | 2933 | 2946 | 2933 | 2937 | 2948 | 2947 |
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5 | 2335 | 2364 | 2369 | 2360 | 2347 | 2350 | 2362 | 2351 | 2349 | 2353 | 2359 |
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6 | 1946 | 1984 | 1995 | 1966 | 1969 | 1961 | 1969 | 1966 | 1972 | 1975 | 1968 |
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7 | 1668 | 1709 | 1692 | 1691 | 1690 | 1689 | 1690 | 1687 | 1698 | 1709 | 1692 |
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8 | 1460 | 1507 | 1485 | 1479 | 1486 |
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1482 | 1481 | 1490 | 1481 | 1486 | 1474 | |
9 | 1297 | 1337 | 1338 | 1325 | 1323 | 1325 | 1319 | 1314 | 1325 | 1340 | 1328 |
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10 | 1168 | 1189 | 1214 | 1193 | 1196 | 1182 | 1188 | 1197 | 1195 | 1201 | 1198 |
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11 | 1062 | 1095 | 1093 | 1082 | 1101 | 1081 | 1082 | 1082 | 1085 | 1089 | 1082 |
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12 | 973 | 1039 | 1002 | 995 | 1000 | 995 | 994 | 994 | 999 | 1004 | 1000 |
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13 | 944 |
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953 |
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14 | 944 |
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Based on the above computational results, we can draw the following conclusions. First, some swarm-based algorithms cannot obtain satisfying results on the best solutions since they lack the ability of strong local search. Second, the local search methods, TS, SA, and LAHC, perform worse than population-based methods on the average results. Third, the destruction and reconstruction phase are an effective method to escape from local optima, which is confirmed by the superiority of the IG2 over TS, SA, and LAHC about the average results. Fourth, the new local search is proved efficient by the superiority of the IG2 over the IG1.
In order to confirm that the IG2 is statistically better than the compared ones for the average results, we also carry out statistical analysis. Since the normality of the average RDI values is violated, we carry out a nonparameter Friedman rank-based test [
Means plot of the ranks of the average solutions with 95% confidence level.
As you see, a total of 10 algorithms are compared. If ordering these algorithms from the best ranking to the worst ranking, an order of algorithms can be achieved: IG2, PSONG, 2-ANTBAL, TLBO, BA, GA, IG1, LAHC, TS, and SA. As stated above, the local search algorithms, SA, TS, and LAHC, are statistically worse than the swarm-based algorithms. Still, as a local search method, the IG2 is statistically better than all the compared algorithms on the average results. According to these results, it is suggested that the proposed IG2 statistically outperforms all the compared algorithms.
As for TALBP-II with additional restrictions, we also utilize all the algorithms to solve the TALBP-II with additional restrictions. The computational results are also analyzed with statistical techniques and these algorithms show similar results on the TALBP-II without additional restrictions. In fact, only the decoding schemes of the algorithms are slightly changed, and other parts of the algorithms remain unaltered. For simplification, we only show the best results by IG2 in Table
Result comparisons forassignment restrictions.
Problem | nm | LB | Without restrictions | With restrictions | |||||
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IG2 | GAMS | IG2 | |||||||
Min | Mean | Min | CPU (s) | Min | Mean | CPU (s) | |||
P9 | 2 | 4.25 | 5 | 5 | 5 | <1 | 5 | 5 | <1 |
3 | 2.83 | 3 | 3 | 4 | <1 | 4 | 4 | <1 | |
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P12 | 2 | 6.25 | 7 | 7 | 7 | <1 | 7 | 7 | <1 |
3 | 4.17 | 5 | 5 | 5 | <1 | 5 | 5 | <1 | |
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P16 | 2 | 20.5 | 22 | 22 | 24 | <1 | 24 | 24 | <1 |
3 | 13.67 | 16 | 16 | 16 | <1 | 16 | 16 | <1 | |
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P24 | 2 | 35 | 35 | 35 | 36 | >7200 | 36 | 36 | <1 |
3 | 23.33 | 24 | 24 | 25 | >7200 | 25 | 25 | <1 | |
4 | 17.5 | 18 | 18 | 18 | 152 s | 18 | 18 | <1 | |
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P65 | 4 | 637.4 | 638 | 638.80 | — | — | 640 | 643.75 | 63.375 |
6 | 424.9 | 427 | 427.55 | — | — | 428 | 430.60 | 63.375 | |
8 | 318.7 | 321 | 323.60 | — | — | 324 | 324.95 | 63.375 | |
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P148 | 5 | 512.4 | 513 | 513.00 | — | — | 515 | 515.85 | 328.56 |
7 | 366 | 367 | 367.35 | — | — | 368 | 369.9 | 328.56 | |
9 | 284.7 | 286 | 286.40 | — | — | 287 | 287.7 | 328.56 | |
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P205 | 6 | 1945.4 | 1962 | 1969.20 | — | — | 1968 | 1980.65 | 630.375 |
8 | 1459.1 | 1479 | 1487.10 | — | — | 1487 | 1492.65 | 630.375 | |
10 | 1167.3 | 1187 | 1199.70 | — | — | 1187 | 1200.5 | 630.375 |
For small-size problems, the results by IG2 are compared with optimal results by GAMS and the runs are interrupted when the optimal solution is discovered. For large-size problems, the results of TALBP-II with additional restrictions are compared with those of TALBP-II without additional restrictions. As can be seen in Table
In this paper, a simple and effective iterated greedy (IG) algorithm is developed for the two-sided assembly line balancing problem type-II (TALBP-II) with assignment restrictions. The NEH heuristic is modified for TALBP-II as the initialization procedure. A new local search with referred permutation is developed, and acceleration by eliminating the insert operator that conflicted with precedence restrictions is developed to speed up the search process while preserving the ability of finding a local optimum.
A new priority-based decoding scheme is also proposed to reduce sequence-dependence idle time, balance the workload, and deal with assignment restrictions. To be specific, the task assignment rules are embedded into the decoding scheme to reduce sequence-dependent idle time and balance the workload on each station. A new method to deal with the positional restriction increase the possibility of finding a feasible solution by preventing assigning the tasks with positional restriction to the former mated-station of the predetermined one. And a new method to deal with positive zoning restriction is applied to further reduce the sequence-dependent idle time by allocating the tasks with positive zoning restriction separately.
Computational studies are carried out to demonstrate the superiority of the developed priority-based decoding scheme and the effectiveness of the proposed IG algorithm. The priority-based decoding scheme is compared with published ones, and computational results show that the priority-based decoding scheme can reduce the sequence-dependent idle times, balance the workload, and deal with assignment restrictions effectively. The computational results of IG on the TALBP-II are compared with those of eight recent algorithms, and the IG obtains superior results on both the average results and the best results, which prove its strong local search and the remarkable ability of escaping from local optima. As for the TALBP-II with additional restrictions, IG can find all the optimal solutions for small-size problems and also obtain promising results for large-size problems.
Future research can apply the IG algorithm to solve other complex assembly line balancing problems, such as the mixed-model assembly line. The priority-based encoding and decoding, especially the methods to reduce idle times and deal with assignment restrictions, may be modified to address other two-sided assembly line problem.
Tasks
Mated-stations
A side of the line;
Station of mated-station
Number of tasks
Number of mated-stations
Cycle time
Set of tasks,
Set of mated-stations;
Processing time of task
Set of tasks with left direction,
Set of tasks with right direction,
Set of tasks either direction,
Set of tasks that have no immediate predecessors
Set of immediate predecessors of the task
Set of all predecessors of the task
Set of successors of the task
Set of immediate successors of the task
Large positive number
Set of tasks whose operation directions are opposite to that of task
Set of integers which indicate the preferred directions of the task
Set of pairs of tasks and predetermined station for positional restriction
Set of pairs of tasks for positive zoning restriction
Set of pairs of tasks for negative zoning restriction
Set of pair of tasks for synchronism restriction.
1, if task
Finishing time of task
1, if task
The authors declare that they have no competing interests.
This research work is funded by the National Natural Science Foundation of China (Grant nos. 51275366 and 51305311).