388. Critical WIP

The amount of work-in-process inventory at which a deterministic line just barely achieves its bottleneck throughput.

388.1. Why it’s “critical”

• Below ${𝑊}_0$: line is starved — bottleneck idle some of the time, throughput limited by WIP rather than capacity
• At ${𝑊}_0$: bottleneck just fully utilized, cycle time = ${𝑇}_0$
• Above ${𝑊}_0$: bottleneck still at max throughput, but cycle time grows (excess WIP just sits in queues)

For deterministic lines, ${𝑊}_0$ is the minimum WIP that achieves maximum throughput. Beyond ${𝑊}_0$, you’re tying up capital without throughput gain.

388.2. Real lines need more WIP

Variability makes the curves wider — to achieve close-to-${𝑟}_{𝑏}$ throughput on a real (variable) line, you need WIP somewhat above ${𝑊}_0$. How much above depends on variability — see best/worst/PWC curves.

PWC throughput at $𝑤$ WIP:

For $𝑤=2{𝑊}_0$: TH = $\left(2\frac{{𝑊}_0}{2{𝑊}_0+{𝑊}_0-1}\right){𝑟}_{𝑏}\approx \left(\frac{2}{3}\right){𝑟}_{𝑏}$. So real lines need $𝑤\gg {𝑊}_0$ to approach ${𝑟}_{𝑏}$.

388.3. Usage

• CONWIP control: set WIP cap at ${𝑤}^{\ast }>{𝑊}_0$, chosen to hit target throughput
• Lean assessment: cycle time / ${𝑇}_0$ ratio tells you how variability-burdened your line is
• Capacity planning: increasing ${𝑟}_{𝑏}$ (de-bottlenecking) lowers ${𝑊}_0$ → less WIP needed

388.4. Worked example

Line: bottleneck produces 100 units/hour (${𝑟}_{𝑏}=100$), raw processing time ${𝑇}_0=3$ hours.

Need at least 300 units of WIP in the line to keep the bottleneck fed.

For a real (variable) line, set CONWIP cap at ${𝑤}^{\ast }=400-500$ units to hit close to ${𝑟}_{𝑏}$.

388.5. See also

• Best/Worst/PWC
• CONWIP
• Factory Physics
• Little’s Law