Assembly line balancing

Why it matters

On an assembly line every unit passes every station, so the line can only run as fast as its most heavily loaded station. Line balancing decides how many stations are needed and which tasks go where, which fixes the line's output, labour cost and idle time. It is one of the most frequently examined numerical topics in production engineering.

Key ideas

The problem. A product's total work is divided into small tasks (work elements), each with a standard time and precedence constraints (some tasks must be finished before others can start), shown in a precedence diagram. Tasks are grouped into stations so that:

• the work at each station does not exceed the cycle time,
• precedence is respected,
• the number of stations (or the idle time) is minimised.

Zoning constraints (tasks that must or must not be together, e.g. painting and electrical work) may also apply.

Cycle time and takt time. Takt time is the pace demanded by the customer: available time divided by required output. The design cycle time C is normally set equal to takt time (or a little less to allow for losses). The actual cycle time of a balanced line is the largest station time; it must not exceed C.

Key measures

• Work content T_wc = Σ t_i.
• Theoretical minimum number of stations N_min = ⌈T_wc / C⌉. A real balance may need more because of precedence and indivisible tasks.
• Line (balance) efficiency: the fraction of paid station time that is productive.
• Balance delay: the idle fraction, 1 − efficiency.
• Smoothness index: how evenly work is spread; 0 means perfect balance.

Heuristic methods (exact solutions are hard for large problems)

• Largest candidate rule: list tasks in descending order of time; repeatedly assign the largest task whose predecessors are done and that fits the remaining station time.
• Kilbridge and Wester (column) method: arrange tasks in columns of the precedence diagram (column 1 has no predecessors, etc.) and assign column by column, choosing larger tasks first within a column.
• Ranked positional weight (RPW, Helgeson and Birnie): a task's positional weight is its own time plus the times of all tasks that follow it in the precedence network. Rank by weight and assign in that order, subject to precedence and the cycle time. Tasks with long chains behind them go early.

If a task is longer than the cycle time: duplicate (parallel) the station, split the task if possible, use faster tooling or a more skilled operator, or accept a longer cycle time (overtime, extra shift).

Other issues: mixed-model lines (several models on one line, with model sequencing), variable task times (stochastic lines need some slack), and U-shaped lines, which allow a worker to take tasks from both ends and often give better balance.

Formulas

Takt time C = Available time per period / Required output per period

Work content T_wc = Σ t_i

N_min = ⌈T_wc / C⌉ (round up)

Line efficiency η = T_wc / (n × C_a) × 100 %

• n = actual number of stations; C_a = actual cycle time = largest station time (use the design C if the question says so; state which).

Balance delay d = 1 − η (also written (n·C_a − T_wc)/(n·C_a))

Smoothness index SI = √Σ (S_max − S_k)²

• S_k = time of station k; S_max = largest station time.

Output per period = Available time / C_a

Worked examples

Example 1 (standard). A line must make 400 units in a 480-min shift. Work content is 4.0 min per unit. Find the takt time, the minimum number of stations, and the efficiency if the line needs 5 stations at the takt time.

1. C = 480 / 400 = 1.2 min.
2. N_min = ⌈4.0 / 1.2⌉ = ⌈3.33⌉ = 4 stations.
3. With 5 stations: η = 4.0 / (5 × 1.2) = 4.0 / 6.0 = 66.7 % (balance delay 33.3 %).

Example 2 (GATE level). Tasks (time in min; immediate predecessors): A 0.5 (–); B 0.3 (A); C 0.8 (A); D 0.4 (B); E 0.6 (B, C); F 0.2 (C); G 0.7 (D, E); H 0.5 (F, G). C = 1.2 min. Balance by RPW and find the efficiency and smoothness index.

1. Positional weights (own time + all followers): A 4.0; C 0.8 + E, F, G, H = 2.8; B 0.3 + D, E, G, H = 2.5; E 0.6 + G, H = 1.8; D 0.4 + G, H = 1.6; G 1.2; F 0.7; H 0.5. Rank: A, C, B, E, D, G, F, H.
2. Station 1: A (0.5, 0.7 left); C does not fit; B (0.3, 0.4 left); E not yet available (needs C); D (0.4, 0 left). S1 = {A, B, D}, 1.2 min.
3. Station 2: C (0.8, 0.4 left); E does not fit; F (0.2, 0.2 left). S2 = {C, F}, 1.0 min.
4. Station 3: E (0.6, 0.6 left); G (0.7) does not fit; H needs G. S3 = {E}, 0.6 min.
5. Station 4: G (0.7), H (0.5). S4 = {G, H}, 1.2 min.
6. n = 4 = N_min; η = 4.0 / (4 × 1.2) = 83.3 %; balance delay 16.7 %.
7. SI = √(0² + 0.2² + 0.6² + 0²) = √0.40 = 0.63 min.

Common mistakes

• Rounding N_min down, or treating it as the actual number of stations.
• Confusing cycle time (interval between units) with throughput time (time for one unit to pass the whole line).
• Assigning a task before all its predecessors are assigned.
• Positional weight with only immediate followers; it must include all tasks that follow, directly or indirectly.
• Using the design cycle time in efficiency when the question asks for the actual (largest station) time, or vice versa.

For GATE PI

This topic is a regular source of NAT questions: takt time, minimum stations, line efficiency and balance delay, output per shift from station times, and full RPW or largest-candidate balances on 6–10 tasks. Practise computing positional weights and building stations neatly, checking precedence at every step.

Quick check

1. 300 units are needed in 450 min. What is the takt time?
2. Work content 9 min, C = 2 min. Minimum stations?
3. Station times 4, 5, 3 and 4 min. Output in 480 min and efficiency?
4. What is the positional weight of the last task in a line?

Answers: 1. 1.5 min; 2. ⌈4.5⌉ = 5; 3. 480/5 = 96 units, η = 16/(4 × 5) = 80 %; 4. Its own task time.