Priority rules used to form the probability distributions for tasks
Nowadays, assembly line balancing problems are commonly found in most manufacturing and production systems. In its basic form, an assembly line balancing problem consists of finding an assignment of tasks to a group of workstations in such a way that the precedence constraints among the tasks are maintained and the sum of the times of the task assigned to each workstation does not exceed the maximum workstation time (i.e. the cycle time). According to the objective considered, two variants of the problem are distinguished: (1) the problem aims at minimizing the number of workstations for a given cycle time and (2), given the number of workstations, the problem seeks to minimize the cycle time. Over the last years a significant amount of research work has been done towards solving assembly line balancing problems efficiently. Finding the best solution is a crucial task for maintaining the competitive advantage of industries and, in some cases, for their survival. Falkenauer (2005), pg. 360, argues that the efficiency difference between an optimal and a sub-optimal assignment can yield economies reaching millions of dollars per year. However, solving real life problems is a very difficult task for decision makers and practitioners since even the simple case is NP-hard by nature. For this reason, most assembly line balancing problems involve only a few aspects of the real systems (see, for example, (Becker & Scholl, 2006)).
In order to deal with the complexity of industrial problems, a great variety of problem definitions (i.e. generalized assembly line balancing problems) have arisen, which consider other restrictions apart from the technological ones. Most common, these include mixed models, multiple products, different line layouts, parallel workstations and multiple objectives. However, real problems require tackling many of those generalizations simultaneously (Falkenauer, 2005). Such a consideration must also be taken into account when alternatives processes are involved. Alternatives may appear when, for example, new technologies are taking place in a production system, in which different procedures are available to complete a production unit, or when the processing order affects the processing times of certain tasks; i.e., the realization of one task facilitates, or makes more difficult, the completion of other tasks (see, for example, (Scholl et al., 2008) and (Das & Nagendra, 1997)).
A novel generalized assembly line balancing problem, entitled ASALBP: the Alternative Subgraphs Assembly Line Balancing Problem, is addressed here. In this problem alternative variants for different parts of an assembly or manufacturing process are considered. Each variant is represented by a subgraph that determines the tasks required to process a particular product and the task precedence relations. Thus, alternative assembly sub-processes for a sub-assembly may involve completely different set of tasks. Consequently, in addition to cycle time or workstations requirements, subgraph constraints must also be taken into account to ensure that tasks belonging to a particular subassembly are processed considering its corresponding assembly subgraph. Furthermore, it is also considered that task processing times are not fixed, but instead are dependent on the assembly subgraph. Therefore, total processing time may vary from one processing alternative to another (even when the alternatives involve the same set of tasks). Similarly to the simple case, the ASALB problem aims at minimizing the number of required workstations for a given bound on the cycle time (i.e., ASALBP-1), or minimizing the cycle time for a given number of workstations (i.e., ASALBP-2). This work focuses on ASALBP-1. As previously discussed, to solve the ASALBP efficiently, two problems must be solved simultaneously: the decision problem, which involves selecting a single assembly subgraph for each subassembly that allows alternatives, and the balancing problem to assign the tasks to the workstations.
In practice, due to the complexity of assembly problems, a two-stage approach is usually used to solve assembly balancing problems that involve alternatives. In a first stage, an assembly variant is selected considering a given criterion, such as, for example, shortest total processing time. When the production alternative has been selected (i.e., the complete assembly process has been defined) and a single precedence graph is available, the problem is then balanced in a second stage. (Capacho & Pastor, 2008) illustrated, by means of numerical examples, how by selecting a priory an alternative, it cannot be guaranteed that an optimal solution of the global problem will be obtained, because the best solution of a problem can be discarded if it does not meet the selection criteria. For instance, it was shown that the alternative assembly variant with the longest processing time required the smallest number of workstations for a given cycle time.
The Alternative Subgraphs Assembly Line Balancing Problem was introduced by ( Capacho & Pastor, 2006 ) and was optimally solved by means of two mathematical programming models. The computational experiment carried out with the models showed that only small- and medium-scale problems could be solved optimally in significantly small computing times. Other attempts have also been done to solve the ASALBP optimally, see for example (Scholl et al., 2006). In order to solve ASALBP for practical sizes, ( Capacho et al., 2006 , 2009) proposed a group of heuristic methods based on adapting well-known priority rules (e.g. (Talbot et al., 1986), most of which are based on the precedence relations and the cycle time. Solutions of good quality were obtained for problems involving up to 305 tasks and 11 assembly subgraphs. In this work the use of weighted, rather than nominal, values for the priority rules is explored. In particular, it is considered an adaptation of a class of metaheuristic methods, namely GRASP (Greedy Randomized Adaptive Search Procedure), which has been successfully applied to hard combinatorial problems (Delorme et al., 2004).
A GRASP overview and literature reviews can be found in (Festa & Resende, 2008a, 2008b). The former research work discusses the algorithmic aspects of this type of procedures; the second one presents GRASP applications covering a wide range of optimization problems, including scheduling, routing, graph theory, partitioning, location, assignment, and manufacturing. Other examples of GRASP can be found in (Dolgui et al., 2010), which propose an algorithm for balancing transfer lines with multi-spindle machines; (Andrés et al., 2008) and (Martino & Pastor, 2007), both of which tackle problems involving setup times.
Basically, a GRASP is an iterative process in which each iteration consists of two phases: the construction phase, which generates a feasible solution, and the search phase that uses a local optimization procedure to find a local optimum in the neighbourhood of the constructed solution (Resende & Ribeiro, 2003). Feasible solutions are generated by iteratively selecting the next element to be incorporated to a partial solution. The best element is selected by ordering all candidate elements according to a greedy function. In this work, the adaptation to the ASALBP of thirteen well-known priority rules are used for selecting tasks and three priority rules are use for selecting the subgraphs (see (Capacho et al., 2009)). This resulted in a total of 39 construction methods, which are used to generate the initial feasible solutions. Furthermore, the application of two neighbourhood search strategies produces a total of 78 GRASP algorithms that are proposed, implemented and tested here. The performance of such algorithms is evaluated by considering a set of 150 medium- and large-scale problem instances; and compared with the results obtained in (Capacho et al., 2009) and with known optimal solutions (see ( Capacho & Pastor, 2006 , 2008)).
The remainder of this chapter is organized as follows: Section 2 describes the metaheuristic procedures (i.e., GRASP) that are designed and implemented here; Section 3 presents the computational experiment carried out to evaluate and compare the proposed algorithms; Section 4 presents the conclusions and proposes future research work; and, finally, Section 5 lists the References.
2. Grasp procedures for solving the ASALBP
As mentioned above, a GRASP involves a construction phase and a local search phase. In the proposed procedures (see (Capacho, 2007)), the construction phase generates a feasible solution by applying a construction method in which both the subgraphs and the assembly tasks are randomly selected following probability distributions based on weighted priority rule values. Weighted values are proportional or inversely proportional (when using a maximizing or minimizing criterion, respectively) to the values obtained considering a given priority rule. The local search phase generates and evaluates all neighbours of the current solution and maintains the best neighbour solution (i.e., the one that requires the fewest number of workstations). This process is repeated for a given length of time. The best overall solution generated is the final solution.
A solution for the Alternative Subgraphs Assembly Line Balancing Problem consists of a task sequence, a number of required workstations and a set of assembly subgraphs (i.e., one subgraph for each subassembly that allows alternatives).
2.1. The construction phase
To construct an initial solution, one subgraph is randomly selected for each available subassembly following a distribution (also referred to as an assembly route), following a probability function based on a priority rule for subgraphs. Once the subgraphs have been chosen, the set of available assembly tasks (AVT) is defined. The set of available tasks is formed with the tasks that belong to the selected subgraphs and those tasks that do not allow assembly variants. The assignable tasks are determined to form the list of candidate tasks (LCT). A task is assignable if all of its predecessors have already been assigned and the sum of its time and the time of the tasks assigned to the current workstation does not exceed the cycle time. For each task in the candidate list LCT, the priority rule value is computed to construct a probability distribution, which is then used to randomly select the next task to be assigned to the current workstation. Once a task has been assigned it is removed from AVT.
New lists LCT are generated and tasks are systematically assigned until all assembly tasks have been assigned (i.e., the sets AVT and LCT are empty) and the solution has been completed. The probability distributions used for selecting subgraphs and tasks are built using weighted values of the following priority rules.
2.1.1. Priority rules for subgraphs
The priority rules used for selecting subgraphs are the following:
Minimum NP: this rule ranks the subgraphs of the same subassembly according to ascending number of precedence relations involved in each subgraph, which is the total number of arcs entering into and within the subgraph.
Minimum TT: subgraphs are ranked according to ascending total processing time ( i.e., the sum of the times of all tasks belonging to the subgraph).
Minimum NT: subgraphs are ranked according to ascending number of tasks.
Let consider, for example, a subassembly of a given manufacturing process that allows three alternative subgraphs, S1, S2 and S3, with total processing time of 30, 35 and 35 time units, respectively. Considering the priority rule TT, the cumulative probability distribution for selecting a subgraph s is as follows (r ∈ [0, 1) is a random value):
2.1.2. Priority rules for tasks
Table 1 lists the priority rules considered to build the probability distribution to select the next tasks to be assigned. It is valid to mention that priority rules 3, 4, 5, 6 and 13 of Table 1 are minimization rules while all others are maximization ones. These rules are thirteen well-known priority rules that have been adapted to the ASALBP (see (Capacho et al., 2009)) and random choice assignment. Basically, these rules are determined by considering the cycle time and task precedence relations and by measuring tasks processing times.
The following notation is considered:
nNumber of tasks
m max Upper bound on the number of workstations
t ir Duration of task i when processed through route r (i = 1,…,n ; r∈ R i )
P ir Set of immediate predecessors of task i, if processed through route r (i=1,…,n ; r∈R i )
S ir Set of all successors of task i, if it is processed through route r (i=1,…,n; r∈ R i )
Once the set of selected routes SR is known, the following values can be defined:
sub(i)Subgraph chosen for task i (); in this way it is possible to know the duration of task i (t i,sub(i) ).
Ei,LiEarliest and latest workstation to which task i can be assigned, respectively ().
SIi, SiSet of immediate and total successors of task i, respectively ().
|No.||Label||Priority rule||Computation procedure|
|1||RPW||Rank Positional Weight|
|7||TLW||Task Time divided by Latest Workstation|
|8||IS||Number of Immediate Successors|
|9||TS||Number of Total Successors|
|10||TTS||Task Time plus Total Number of Successors|
|11||STS||Average Time of Successors|
|12||TSSk||Number of Total Successors divided by the Slack|
|13||LWTS||Average Latest Workstation|
The combination of the resulting probability distributions, based on the various priority rules used for tasks and subgraphs, produced 39 constructive methods, which are presented in Table 2.
2.2. The local search phase
Two different neighbourhood search strategies based on task exchange movements are considered. At this point, it is valid to recall that a solution to the problem is represented by a sequence of tasks.
The following notation is used to describe such strategies:
m k Number of workstations required for a given sequence (solution) k
ISInitial tasks sequence generated in the construction phase
WSWorking sequence (the first WS is IS)
SSStored sequence (the first SS is IS)
Slk j Slack of workstation j (i.e., cycle time minus workstation time)
|Weighted rules for subgraphs|
|No.||Heuristic Name||No.||Heuristic Name||No.||Heuristic Name|
The proposed local optimization procedures generate the neighbourhood of the working sequence WS by using a transformation or exchange movement (which are described in what follows). Each exchange generates a neighbour sequence NS, which is evaluated and compared with the stored sequence SS. If NS improves the stored sequence SS (i.e., it requires fewer workstations) the neighbour sequence becomes the stored sequence SS.
When a neighbour sequence requires the same number of workstations as the store sequence (situation that may frequently occur in line balancing problems), a secondary objective function (3) is used as tie-breaker. This function creates solutions by attempting to load the first workstations at maximum capacity and the last ones at minimum capacity. To achieve this objective, the weight parameter α of f has been set to 10. It was verified that equivalent results can be obtained by using α=10 e , being e an integer greater than 1.
The local search ends when, for each task in WS, all feasible exchanges have been made; i.e., all neighbours have been generated and evaluated. For the next iteration, the stored sequence SS is assigned to the working sequence WS. The entire procedure is repeated until a predetermined computing time has been completed. The final solution is the best of all solutions that have been generated.
An adaptation of two classical transformations (see (Armentano & Basi, 2006)) has been considered to generate the neighbourhood of a given solution, as follows.
a. The exchange of the positions in WS of a pair of tasks
In this case, the exchange movement tries to exchange the position, in the working sequence WS, of any two tasks i and k, provided it is feasible; i.e., the precedence relations amongst the tasks are maintained. Furthermore, both task i and task k should have been assigned to different workstations. When task i and task k belong to the same subgraph s, new neighbour sequences are searched by interchanging s with each one of the remaining subgraphs available for such tasks.
b. The movement of task i to another position of the working sequence WS
In this type of movement a task is yielded to a different workstation. A task i can be moved to the position of task k when the tasks precedence relations are maintained and when task k and task i have been assigned to different workstations. In this case, all tasks between the positions of task i and k, including task k itself, are moved in the sequence one position backwards. Furthermore, for each movement, neighbour sequences are generated by interchanging the alternative subgraphs available for the moved task.
When a movement exchange type a is used in the local search phase, the local optimization procedure is regarded as LOP-1; otherwise, it is regarded as LOP-2.
Examples of exchange movements
Let consider an example (see (Capacho et al., 2009)) of an ASALB problem involving 11 tasks and 7 alternative subgraphs, which represent the assembly variants that are allowed for three parts of a manufacturing system: alternative S1 and S2, for subassembly one; S3 and S4, for the second part; and S5, S6 and S7, for the third subassembly. Then, if the NR (i.e., minimum number of tasks) is the rule applied, subgraphs S1, S3 and S5 are selected for subassembly 1, 2 and 3, respectively. By choosing such subgraphs, the precedence relations presented in Table 3 are determined.
|Immediate predecessors||-||-||-||1, 2||3||2, 5||4||6||7||8||9|
Then, by applying rule RPW for selecting the next task to be assigned, the solution presented in Table 4 is obtained. Table 4 includes the number of required workstations m, the tasks assigned to each workstation, and the corresponding workstation time wt.
|Tasks||2, 1||3||4, 5||6, 7||8, 10||9, 11|
As can be observed in Table 4, tasks are assigned to the workstations in a particular order, which defines the tasks sequence that is used as the initial sequence ISq, as follows:
ISq = 2, 1, 3, 4, 5, 6, 7, 8, 10, 9, 11.
If a transformation type a is applied, the neighbour solution of Figure 1 is obtained by interchanging the positions of task 2 and task 3. This movement is possible since, as can be seen in Table 3, neither task 2 nor task 1 are predecessors of task 3, and neither task 1 nor task 3 are successors of task 2 (i.e., the precedence constraints are maintained). Moreover, as shown in Table 4, the tasks are assigned to different workstations: task 2 is assigned to workstation I and task 3 is assigned to workstation II.
If transformation b is considered, the neighbour sequence is generated by moving task 2 to the position of task 3; and by moving task 1 and task 3 one position backwards in the sequence. The resulting neighbour sequence is as follows.
3. Computational experiment
To evaluate and compare the performance of the proposed GRASP procedures described in the previous section, a computational experiment was carried out considering medium- and large-scale ASALBP. The data sets (see Table 5) used in the computational experiment are base on the following 10 benchmark problems: Gunther, Kilbrid, Hann, Warnecke, Tongue, Wee-Mag, Lutz3, Arcus2, Bartholdi and Scholl; with 35, 45, 53, 58, 70, 75, 89, 111, 149 and 297 tasks, respectively. Each benchmark problem is subdivided into two, three and four subassemblies; involving five, eight and eleven subgraphs, respectively. For each problem instance five cycle time values were considered. Benchmarks are available online at the web page for assembly line balancing research: www.assembly-line-balancing.de.
A total of 150 problem instances, involving from 37 to 305 tasks, were solved considering a computation time of 0.1 second on a Pentium IV, 3GHZ CPU with 512 Mb of RAM with each of the 78 proposed algorithms. All heuristic methods were implemented using C++.
4. Analysis of the results
To present the results obtained in the computational experiment, the following notation is used: NI, number of the tested instances; CT, computation time; NBS, number of best solutions obtained; PBS, percentage of best solutions; Δ max , Δ av, Δ min , maximal, average and minimal deviation from the best solution, respectively. For each problem instance, the relative deviation from the best solution BS, of each heuristic solution HS, is computed as follows:
|Problem||Cycle time values||Number of subgraphs|
|Number of tasks|
The overall performance of all methods is evaluated by considering the number of best solutions provided by the methods. The best solution for a problem instance, the basis for the comparative analysis, is the best of all solutions found by the proposed heuristic methods. The results are also compared with previous results obtained by methods using nominal, rather than weighted, values of the priority rules (e.g., Capacho et al., 2009).
Table 6 shows the results obtained by applying all proposed procedures to solve all data sets. As observed in Table 6, better results were obtained by methods using local procedure LOP-2, which in most cases outperformed methods applying LOP-1; i.e., 24 methods performed better, in two cases both performed the same, and for 3 procedures it behave worse that LOP-1. On the other hand, all methods using LOP-2 provided best solutions in more than 54% of the cases.
The best performance was recorded for the method that employed W-NT_TSSk in the construction phase and applied LOP-2; which generated best solutions in 85,3% of the cases (this represents a 3.3% of improvement comparing with the same method when applying LOP-1). Furthermore, method W-NT_TSSk yielded a comparatively small value of Δ max (16.7%), and the smallest value of Δ av (1%).
Regarding LOP-1, the method that performed the best was W-NT_TSSk, which generated best solutions in 82,7% of the cases. Method W-NT_TSSk performed the same, generating best solutions in 82.7% of the cases regardless of the local optimization procedure applied.
Other methods that also performed well are those that employed W-TT_LWTS, W-TT_LWTS, W-NT_TS, W-TT_TS and applied LOP-2, all of which generated the best solutions in 78.7% of cases. Table 6 also shows that for most methods Δ max is significantly high.
As it can be observed in Table 6, priority rule EW has a very poor performance, regardless of the rule used for subgraphs and the local optimization procedure applied; generating best solutions, at best, in 56 % of the cases.
|NBS||PBS||Δ max||Δ av||NBS||PBS||Δ max||Δ av|
|Δ min = 0 in all cases|
On the other hand, Table 6 also shows that methods employing different local search procedures behave very similarly when the same heuristic method is used to build the initial solution. This means that when a constructive method performs well when applying LOP-1, it also performs well (or better) when applying LOP-2. This behaviour can also be observed in Figure 3, which shows the percentage of improvement (in one workstation) of the proposed procedures comparing with other multi-pass methods.
The results obtained with the proposed GRASP methods were also compared with the results obtained by multi-pass methods, in which the assembly subgraphs are randomly selected and tasks are assigned according to nominal values of the priority rules of Table 1. Table 7 shows the percentages of improvement in the solution provided by the proposed GRASP methods comparing with multi-pass ones (e.g., Capacho et al., 2009), considering a CT=0.1 seconds. Data in Table 7 includes Imp max , Imp av , Imp min , maximal, average, and minimum percentage of improvement, respectively. It also shows the percentage of cases in which the best solution found requires 1, 2 or 3 workstations less (%1ws, %2ws, %3ws), respectively, than the best result provided by the corresponding multi-pass methods. As can be seen in Table 7, the best results were obtained when LOP-2 was applied. This strategy provided an additional 12.9% of best solutions, getting in some cases up to 35.6%. Table 7 also reveals that both types of procedures were able to generate improved solutions in which up to three fewer workstations were required; i.e., in 0.1% and 0.3% of the cases with LOP-1 and LOP-2, respectively.
5. Conclusions and further research work
In this chapter, the metaheuristic approach GRASP was used to solve the Alternative Subgraphs Assembly Line Balancing Problem (ASALBP). Thirty-nine construction methods, based on weighted priority rule values, and two local search strategies (LOP-1 and LOP-2) were proposed, implemented and evaluated.
|Local method||Impmax||Impav||Impmin||% 1 ws||% 2 ws||% 3 ws|
The results obtained showed that methods that used LOP-2 performed better than those that used LOP-1, achieving best solutions in up to 85.3% of all cases, considering all the proposed construction methods. Nevertheless, some methods applying LOP-1 (i.e., W-NT_TSSk and W-TT_TSSk) also performed well, providing best solutions in up to 82.7%.
The results also showed that a significant improvement can be obtained in comparison to previous results obtained using multi-pass methods based on single priority rule values and using random choice for subgraphs. Improved solutions were obtained in which the number of workstations required was reduced by one, two or even three, which represent the best results obtained with any method developed up to now to solve the ASALBP. Thus, all proposed methods that used LOP-2 could be applied to solve an Alternative Subgraphs Assembly Line Balancing Problem to select the best overall solution.
Further research involves exploring other methods to solve the ASALBP. The growing interest on using Evolutionary Algorithms to solve optimization problems in industry makes the use of such procedures an attractive approach, which, in addition, has been successfully applied to complex assembly line balancing problems. On the other hand, in order to increase the practicality of the problem, its definition can be extended by including new features such as, for example, stochastic processing times, setups, and different line layouts.
Supported by the University of Los Andes, Mérida – Venezuela.