Line balancing is usually treated as an art. A supervisor watches the line, spots the operation where work-in-progress piles up, moves an operator, and repeats. That works, slowly, and it stops working the moment the style changes. The same problem has a clean mathematical statement, and on a line of thirty to sixty operations it solves in under a second.

The decision you are actually making

You have a set of operations, each with a standard time. You have a set of workstations. Each operation must be assigned to exactly one station, precedence must be respected, and no station may exceed the cycle time implied by your target output. The objective is either to minimise the number of stations for a fixed cycle time (Type-1) or to minimise cycle time for a fixed number of stations (Type-2). Type-2 is the one factories usually want, because the manpower is already hired.

The formulation

Let i index operations, k index stations, t(i) be the standard time of operation i in minutes, and P be the set of precedence pairs. Define a binary variable x(i,k) that is 1 when operation i is assigned to station k, and a continuous variable C for the cycle time.

minimise   C

subject to
  (1)  sum over k of x(i,k) = 1                     for every operation i
  (2)  sum over i of t(i) * x(i,k) <= C             for every station k
  (3)  sum over k of k * x(a,k) <= sum over k of k * x(b,k)
                                                    for every precedence pair (a,b) in P
  (4)  x(i,k) in {0,1},  C >= 0

Constraint (1) is completeness: every operation gets a home. Constraint (2) is the workload ceiling that defines the bottleneck. Constraint (3) is the precedence rule written in the standard "station index ordering" form — if operation a must precede b, then a cannot sit in a later station than b. That single line replaces the entire informal argument about what can move where.

The constraints nobody writes down but everybody needs

A model that only contains the four constraints above will hand you an answer that the floor rejects. Three additions make it usable.

  • Machine compatibility. An overlock operation cannot be assigned to a station equipped with a flatlock. Add x(i,k) = 0 for every incompatible pair. This is the constraint that removes most of the "the model is wrong" objections.
  • Skill matrix. If station k is staffed by an operator whose grade cannot perform operation i, force x(i,k) = 0. In practice you build this as a matrix once per line and reuse it.
  • Zoning. Some operations must stay together — for example, a set of matching topstitch operations on the same panel. Add x(i,k) = x(j,k) for every pair that must share a station.

A worked check

Take a basic polo shirt line: 42 operations, total standard time of 18.6 minutes, and 36 operators available. The theoretical minimum cycle time is 18.6 / 36 = 0.517 minutes, which implies a ceiling of about 116 pieces per hour. A solved model on the same data typically lands at a cycle time of 0.55 to 0.58 minutes once precedence and machine constraints bite, giving 103 to 109 pieces per hour. The gap between 116 and 105 is the real, structural loss of the style — not something a supervisor can walk off.

Reading the balance loss

Balance loss is (n × C − total time) / (n × C), where n is the number of stations. At 36 stations and C = 0.56, that is (20.16 − 18.6) / 20.16 = 7.7 per cent. Anything under 10 per cent on a real garment is a good result. If your model reports 2 per cent, check your precedence set — you have probably left constraints out.

Where this stops working

Two situations break the classical model. The first is high operator variability: if your standard times carry a coefficient of variation above roughly 0.25, a deterministic balance will be optimistic and you need buffers between stations, which pushes you toward a stochastic or simulation-based approach. The second is mixed-model lines, where several styles run through the same stations. That is a different formulation — you balance on a weighted average of the models with an additional smoothing term — and it deserves its own treatment.

Practical next step

Build the model on your worst-performing style, not your best. Feed it the standard times you already trust from your SMV study, add machine compatibility, and compare the solved cycle time with the actual. If the model says 105 pieces per hour and the line runs 78, the difference is not the balance — it is absenteeism, feeding, or quality rework, and you have just proved it.