embed_youtube("NT-mVVprnbs")Assembly Lines Notes Ch. 15
Manual Assembly lines
Line balancing is the science of distributing taskss across a set of workstations to minimize their number, and maximize utilization while still meeting the TAKT time.
Line balancing is all about maximizing output to the TAKT time.
\[ TAKT=\frac{AT}{\overline{D}} \tag{1}\]
Where \(AT\) is the available working time, and \(\overline{D}\) is the average demand.
Example of balancing process.
embed_youtube("PJg1DyZElvc")Example of instructions in an assembly line
embed_youtube("LvD2-_Tl7j4")Line balancing: How to.
Quick and dirty video:
embed_youtube("QU-Jwf1-Tyc")When designing and managing a manual line-assembly, industrial engineers strive to satisfy objectives such as maximizing line throughput, minimizing the number of stations, maintaining a balance of work across stations, satisfying delivery rates, accommodating product mix changes, and more. Next, we’ll estimate how many stations are required and how we assign tasks to those stations using an heuristic.
Heuristic1 for minimization of number of stations
We use heuristics because line balancing problems are NP-Hard2. In this case, we present a modified Heuristic based on Kilbridge and Wester (Hopp and Spearman 2011).
The objective is to obtain the minimum number of stations such that labor is used efficiently and the workload per station is fair or “balanced.”
Let \(n\) be the number of tasks to be performed and identified by number: \(i=1,…,n\) .
Each task has a processing time \(t_i\)
The goal is to assign tasks to \(k\) workstations such that \(k<n\).
\(TAKT\) is the upper limit of time for each station, otherwise we will not satisfy average demand. Because all tasks have variability, we would like to set up stations with a cycle time \(c<TAKT\) . Thus \(c\) is the maximum time allowed for task assignment into the workstations. In other words, we’d like to have a maximum utilization \(u=\frac{c}{TAKT}\) for all work stations such that we always have \(u<1\).
Some measures of performance when balancing manual assembly lines are the total idle time:
\[ Tot_{idle}=kc-\sum_{i=1}^{n} t_i \]A related measurement is the “balance delay:”
\[ b= \frac{kc-\sum_{i=1}^{n} t_i}{kc}*100 \]
Heuristics have to satisfy precedence constraints where some tasks must be done before others. In the figure below, the nodes are the tasks and the edges represent the precedence constraints. The numbers inside the nodes identify the tasks by number.
For the figure above, the table below describe the tasks and their processing time as well as the number of successors per task.
| Task | Processing time \(t_i\) | Number of successors |
|---|---|---|
| 1 | 5 | 7 |
| 2 | 3 | 6 |
| 3 | 6 | 4 |
| 4 | 8 | 5 |
| 5 | 10 | 3 |
| 6 | 7 | 3 |
| 7 | 1 | 2 |
| 8 | 5 | 0 |
| 9 | 3 | 0 |
Heuristic pre-processing
- Select a value for \(c\) the cycle time per workstation using the following rules:
\[ \max{(t_i)}\le c \le \sum t_i \]
In the example \(10 \le c \le 48\)
- Factor \(\sum t_i\) between 10 and 48:
2x2x3=12
2x2x2x2=16
2x2x2x3=24
2x2x2x2x3=48
These are the cycle times that might give us a balanced line without delays3
- The number of workstations with potentially no idle time are:
\(\frac{48}{48}=1\) workstation with \(c=48\)
\(\frac{48}{24}=2\) workstations with \(c=24\)
\(\frac{48}{16}=3\) workstations with \(c=16\)
\(\frac{48}{16}=4\) workstations with \(c=12\)
If our \(TAKT=16\) then we can only design workstations with \(c=16\) or \(c=12\) . Which cycle time will provide more throughput?
Heuristic steps
Let \(N\) be the current station number; \(T\) the set of tasks assigned to the current station; \(S\) the set of available tasks to be assigned4; and \(A\) the time available to be assigned at the current station.
- Set \(N=1\)
- Set \(A=c\), and \(T=\emptyset\)
- Get candidates for assignment \(S\) such that:
All predecessors of the candidates are assigned, or the candidate has no predecessors.
Their processing time does not exceed the available time \(A\).
- Chose a task \(j\) from set \(S\) using the following rules, and place the selected task in \(T\):
The task that has the largest number of successors.
Break ties choosing the task with the largest processing time.
- Update \(A=A-t_j\). Remove task \(j\) from set \(S\).
- Repeat steps 3, 4, and 5 until \(S\) is empty or \(A=0\).
- If there are remaining tasks, increment \(N=N+1\), and go to step 2. Otherwise stop.
Note: If the sum of the remaining tasks is less than \(A=c\) then group them.
Simulation for complex systems
embed_youtube("NnEX4AoAPn0")Another example video
embed_youtube("H5M2hopvWFw")References
Hopp, Wallace J., and Mark L. Spearman. 2011. Factory physics. 3rd ed. Long Grove, Ill: Waveland Press.
Footnotes
Heuristics are experience-based techniques that help in problem solving. A heuristic method is created and used to rapidly come to a solution that is hoped to be close to the best possible answer, or optimal solution.↩︎
There are a number of problems that are classified as NP-hard, meaning that they are difficult to solve using traditional methods of computation (optimization algorithms). Some examples of these problems include the travelling salesman problem, the knapsack problem, and the line balancing problem. These problems are difficult to solve because they involve a large number of potential solutions that must be checked in order to find the best one. This can be a time-consuming process, especially when the problem is large and complex.↩︎
Not guaranteed because the precedence constraints.↩︎
Tasks whose precedence constraints have been satisfied and whose processing times fit within the remaining time.↩︎