## Problem Reframed Determine the net daily throughput of a warehouse picking operation, accounting for re-pick time consumed by error correction. ## Given & Assumed | Variable | Value | Unit | Source | |---|---|---|---| | Inventory capacity | 12,000 | SKUs | Given | | Pick rate | 340 | picks/hr | Given | | Shifts per day | 2 | shifts | Given | | Hours per shift | 8 | hr/shift | Given | | Error rate | 5% | % of picks | Given | | Re-pick requirement | 1 additional pick per error | picks | Assumed — each error requires exactly one re-pick | | Pickers | Not specified | — | ⚠️ Gap (see below) | **⚠️ Data Gap (R5):** The number of pickers is not provided. This solution treats **340 picks/hr as the total warehouse pick rate** (aggregate), not a per-picker rate. If it is a per-picker rate, the total picker count is needed. Both interpretations are addressed under G4 below. --- ## Method Selection - **Branch A — Gross-then-net deduction:** Calculate gross picks from available hours, then subtract re-pick overhead implicitly baked into those same hours. Simple but risks double-counting time. - **Branch B — Capacity-consumption model:** Total available pick-hours are a fixed resource. Errors consume capacity re-picking, leaving less capacity for *net new* picks. More precise. **Selected: Branch B** — because re-picks consume the same picker capacity as first picks; errors reduce net output, they don't add hours. --- ## Formula $$\text{Gross Picks} = R \times H_{shift} \times N_{shifts}$$ $$\text{Net Throughput} = \frac{\text{Gross Picks}}{1 + e}$$ Where: - $R$ = pick rate (picks/hr) - $H_{shift}$ = hours per shift - $N_{shifts}$ = number of shifts per day - $e$ = error rate (decimal) — each error consumes one re-pick from the same capacity pool - Dividing by $(1+e)$ allocates the fixed capacity between first-picks and re-picks **Dimensional check:** picks/hr × hr/shift × shifts = picks ✓ --- ## Calculation **Step 1 — Total available pick capacity per day:** $$340 \text{ picks/hr} \times 8 \text{ hr/shift} \times 2 \text{ shifts} = 5{,}440 \text{ picks/day}$$ **Step 2 — Apportion capacity between first picks and re-picks:** Every 1 net unit picked also generates 0.05 re-picks, so each net pick costs 1.05 pick-operations. $$\text{Net Throughput} = \frac{5{,}440}{1.05} = 5{,}180.95...$$ **Step 3 — Apply precision (nearest 10):** $$\boxed{5{,}180 \text{ units/day}}$$ **Step 4 — Implied re-pick volume (sanity check):** $$5{,}180 \times 0.05 = 259 \text{ re-picks}$$ $$5{,}180 + 259 = 5{,}439 \approx 5{,}440 \text{ ✓}$$ --- ## LoT Chain Check | Link | Check | |---|---| | Given rate × hours → gross capacity | ✓ | | Error rate consumes capacity → reduces net picks | ✓ | | Division by (1+e) correctly partitions capacity | ✓ | | Final → nearest 10 rounding applied | ✓ | --- ## Verification **Method: Inverse (work backward)** If net throughput = 5,180, then re-picks = 5,180 × 0.05 = 259. Total operations = 5,180 + 259 = **5,439** — within 1 unit of capacity (5,440). Rounding artifact only. ✓ **Agreement: ✓ within precision target (nearest 10)** --- ## Answer in Context Under current operating conditions — two 8-hour shifts at 340 picks/hour — the warehouse delivers **5,180 net picked units per day**. Approximately 260 of the 5,440 daily pick operations are consumed correcting the 5% error rate, representing **roughly 46 minutes of lost productive capacity per shift**. The 12,000-unit inventory figure establishes that a full inventory cycle would require approximately **2.3 days** at this throughput rate (12,000 ÷ 5,180), useful for replenishment planning. --- ## Confidence & Caveats - **Confidence: HIGH** — arithmetic is closed-form with all required inputs present - **Assumptions needing validation:** - 340 picks/hr is the *aggregate* warehouse rate, not per-picker - Each error requires exactly one re-pick (not zero or multiple) - Both shifts operate at the same rate with no downtime/breaks reducing net hours - **Conditions where answer changes:** - If error rate rises to 10%: net throughput drops to ~4,950 (−230 units) - If pick rate is per-picker: total picker count must be supplied to solve - Breaks, shift changeover time, or equipment downtime would reduce the 16 available hours --- **G4 — Alternate interpretation:** If 340 picks/hr is a *per-picker* rate, the formula becomes identical in structure but $R$ becomes $340 \times N_{pickers}$. The problem is unsolvable without $N_{pickers}$; that value must be supplied.