Line Balancing

392. Line Balancing

The task of allocating work to stations on a production line so each station’s work time is below the takt time, minimizing idle time and station count.

392.1. Setup

A product is assembled via $𝑛$ tasks with processing times $𝑡_{1}, \dots, 𝑡_{𝑛}$
Tasks have precedence constraints: task $𝑖$ must finish before task $𝑗$ starts
Tasks are grouped into stations; each station’s total time $\leq takt$

Goals (often trade-offs):

Minimize number of stations (capital cost)
Minimize cycle time (= maximum station time)
Smooth station loads (workforce equity)

392.2. Theoretical bounds

Minimum stations:

𝑁^{*} \geq ⌈ \frac{\sum_{𝑖} 𝑡_{𝑖}}{takt} ⌉

(Total work ÷ takt time, rounded up.)

Minimum cycle time: ${max}_{𝑖} 𝑡_{𝑖}$ (longest indivisible task sets a floor).

392.3. Balance efficiency

Balance efficiency = \frac{\sum_{𝑖} 𝑡_{𝑖}}{𝑁 \cdot cycle time}

100% means every station fully loaded — no idle time. Lower efficiency means wasted capacity.

392.4. Heuristics

The problem (Assembly Line Balancing Problem, ALBP) is NP-hard. Common heuristics:

Longest Task Time (LTT): pick assignable tasks (precedences satisfied, fits in remaining station time) in decreasing order of $𝑡_{𝑖}$ .

Ranked Positional Weight (RPW, Helgeson-Birnie 1961):

Compute each task’s positional weight = $𝑡_{𝑖}$ + sum of $𝑡_{𝑗}$ for all successors $𝑗$
Assign tasks in decreasing positional weight, respecting precedences

COMSOAL (Arcus 1966): Random assignment with backtracking. Multi-start.

392.5. Worked example

10 tasks with times $(5, 3, 4, 6, 4, 5, 4, 3, 5, 4)$ , total $= 43$ minutes. Takt time $= 8$ minutes.

Minimum stations: $⌈ \frac{43}{8} ⌉ = 6$ .

LTT heuristic with simple precedence chain: place tasks into stations one at a time, picking the longest available task that fits.

Result might give 6 or 7 stations, balance efficiency 89%–100%.

392.6. Mixed-model line balancing

Real lines produce multiple products with different task sets. Mixed-model balancing:

Define product mix (proportion of each)
Compute expected task times weighted by mix
Balance using weighted times
Sequence products through stations to avoid bottleneck overload

392.7. Single-product vs U-shaped lines

Layout	Pros	Cons
Straight line	simple, fast	workers can’t help each other
U-shaped	workers reach multiple stations, can help bottleneck	more complex, requires multi-skilled workers
Cell	flexible, small batches	coordinating multiple cells

U-shaped (cellular) layout is the lean ideal — natural for CONWIP / kanban control.

392.8. Variability and balance

Even a perfectly balanced deterministic line behaves much worse than the formula suggests when variability is added (per VUT). Real balancing requires both:

Mean balance: station load averages match
Variance balance: station variability matches

Otherwise the high-variance station becomes a “virtual bottleneck” that drives queueing throughout.

392.9. See also

Takt Time — the constraint
Best/Worst/PWC
Factory Physics
CONWIP