387. Best Worst PWC

For any production line with bottleneck rate ${𝑟}_{𝑏}$, raw processing time ${𝑇}_0$, and WIP level $𝑤$, throughput is bounded by three curves (Hopp & Spearman). Real lines fall between these curves.

387.1. Three curves

where ${𝑊}_0={𝑟}_{𝑏}{𝑇}_0$ is the critical WIP.

387.2. Interpretation

Best case: deterministic line, no variability. Throughput grows linearly until hitting bottleneck rate. Cycle time stays at ${𝑇}_0$ until WIP exceeds capacity then grows linearly.

Worst case: pathological — all work at one station, batched. Throughput stays at $\frac{1}{{𝑇}_0}$ no matter how much WIP. Useless line.

PWC: a “balanced random” baseline. Each station equally loaded with exponential variability. Real production lines should land near PWC if well-managed.

387.3. Critical WIP

The WIP at which best-case throughput reaches ${𝑟}_{𝑏}$ and best-case cycle time equals ${𝑇}_0$. Above ${𝑊}_0$:

• Best case: cycle time grows linearly with $𝑤$, throughput stays at ${𝑟}_{𝑏}$
• PWC: cycle time grows linearly too, throughput approaches ${𝑟}_{𝑏}$ asymptotically

387.4. Take-aways

• If your real line’s throughput is well below the PWC curve, you have more variability than even PWC assumes — improve setups, breakdowns, batching
• If real cycle time is well above PWC, same — variability is the culprit
• Critical WIP ${𝑊}_0$ is the sweet spot for CONWIP (constant WIP) control

387.5. Plot intuition

Throughput
 r_b ┤  best ───────────────────────────────────────────────
     │ /  ╱ PWC asymptote
     │/  ╱
     │  ╱
     │ ╱
     │╱  worst  ──────────────────────────────────────────
 1/T_0
     └─────────────────────────────────────────── WIP w
                W_0

387.6. See also

• Critical WIP
• Factory Physics overview
• VUT — the underlying variability driver
• CONWIP