16.2 High-Yield Calculations Drill

High-Yield Math on a Concept Exam

CPIM is not an engineering calculus test, but ECM 9.0 still expects clean arithmetic on inventory, forecasting error, and time-phased planning. This drill reviews the formulas that appear most often, with multiple worked examples. On exam day, write the formula, substitute carefully, and watch units (days vs. weeks, annual vs. period demand).

EOQ — Economic Order Quantity

Formula: EOQ = √(2DS / H)
where D = annual demand, S = ordering (or setup) cost per order, H = annual holding cost per unit.

Example A. D = 8,000 units/year, S = $50, H = $2/unit/year.
EOQ = √((2 × 8,000 × 50) / 2) = √400,000 ≈ 632 units.

Example B. Same demand, but H rises to $8 because warehouse space is tight.
EOQ = √((2 × 8,000 × 50) / 8) = √100,000 ≈ 316 units.
Insight: higher holding cost → smaller orders, more frequent replenishment.

Example C — setup cost change. A work center reduces setup from $200 to $50; D = 12,000, H = $4.
Old EOQ = √((2 × 12,000 × 200) / 4) = √1,200,000 ≈ 1,095.
New EOQ = √((2 × 12,000 × 50) / 4) = √300,000 ≈ 548.
Lean setup reduction enables smaller lots—connect this to lot sizing and capacity discussions elsewhere on the exam.

Exam traps: mixing monthly demand with annual H; using purchase price instead of holding cost; forgetting that EOQ assumes relatively stable demand and known costs.

ROP — Reorder Point (Continuous Review)

Basic formula (with safety stock): ROP = dL + SS
where d = demand per period, L = lead time in the same period units, SS = safety stock.

Example D — no variability given. Demand averages 40 units/day, lead time 5 days, safety stock 60.
ROP = 40 × 5 + 60 = 260. When on-hand + on-order falls to 260, place an order.

Example E — service-level style inputs. Average weekly demand 200, lead time 3 weeks, safety stock 150.
ROP = 200 × 3 + 150 = 750.

Example F — unit consistency. Annual demand 10,400, lead time 10 days, SS = 80, assume 260 operating days/year.
Daily demand = 10,400 / 260 = 40.
ROP = 40 × 10 + 80 = 480.

If the question gives demand during lead time already, do not multiply again—add safety stock only once.

Inventory Turns and Days of Supply

Turns: Turns = Annual COGS (or annual usage at cost) ÷ Average inventory at cost

Days of supply: Days = 365 ÷ Turns (or use 52 if the exam frames weeks).

Example G. COGS $2,400,000; average inventory $400,000.
Turns = 2,400,000 / 400,000 = 6. Days = 365 / 6 ≈ 61 days.

Example H. Usage 18,000 units/year; average inventory 1,500 units (unit-based turn).
Turns = 18,000 / 1,500 = 12. Days = 365 / 12 ≈ 30 days.

Exam cue: higher turns usually mean leaner inventory but may raise stockout risk if lead times or forecast error are high—tie answers to service policy, not “turns good / inventory bad” slogans.

MAD and MAPE — Forecast Error

MAD (Mean Absolute Deviation): MAD = Σ|Actual − Forecast| ÷ n

MAPE (Mean Absolute Percentage Error): MAPE = (100% / n) × Σ|(Actual − Forecast) / Actual|
(Only defined when actuals ≠ 0.)

Example I — MAD. Actuals: 100, 120, 90. Forecasts: 110, 110, 100.
Errors: |−10|, |10|, |−10| → 10, 10, 10.
MAD = 30 / 3 = 10.

Example J — MAPE. Same data:
|−10|/100 = 0.10, 10/120 ≈ 0.083, 10/90 ≈ 0.111.
Average ≈ 0.098 → MAPE ≈ 9.8%.

Example K — tracking signal setup. If sum of forecast errors (with signs) = +40 over 8 periods and MAD = 10, tracking signal = 40/10 = 4. Large absolute tracking signals suggest bias (consistent over- or under-forecasting). CPIM may ask whether to investigate the forecasting process when the tracking signal exceeds control limits.

Projected Available Balance (PAB) and ATP Sketch

Time-phased MPS logic uses projected available balance:

PAB(t) = PAB(t−1) + MPS(t) − Forecast(t)
(or orders, depending on the rule stated).

Available-to-promise (ATP) (discrete, simple form between MPS receipts):
For a period with an MPS receipt, ATP ≈ MPS quantity minus customer orders already booked until the next MPS receipt (methods vary—use the method in the stem).

Example L — PAB walk. Beginning inventory 50.

If the system consumes the greater of forecast vs. orders in some weeks, PAB would use 50 in week 3 instead of 40, yielding PAB 30—always follow the stated consumption rule.

Example M — ATP sketch. Week 2 MPS = 100. Customer orders before next MPS: week 2 orders 35 + week 3 orders 50 = 85.
Rough discrete ATP in week 2 ≈ 100 − 85 = 15 (plus any leftover ATP from prior periods if the method carries it forward). Sales should not promise more than ATP without changing the MPS.

Gross-to-Net MRP Logic

For a component in a period:

1. Gross requirements (from parent plans / dependent demand)
2. Subtract scheduled receipts
3. Subtract on-hand (applied in the first period / as available)
4. Result = net requirements (if positive)
5. Apply lot size → planned order receipt
6. Offset by lead time → planned order release

Example N — single-level netting.

Week 1 on hand 35. Week 2: 35 − 50 + 40 scheduled receipt = 25 on hand. Week 3: 25 − 50 = −25 → net requirement 25 before lot sizing. If lot size = 100, planned order receipt in week 3 = 100; with lead time 1 week, planned order release in week 2 = 100. Ending on hand week 3 becomes 75, then week 4: 75 − 60 = 15.

Example O — parent drives child. Parent MPS receipts of 20 in week 5; usage is 2 components per parent; component lead time 2 weeks; on hand 10; no scheduled receipts. Gross in week 5 = 40. Net = 30. If lot-for-lot, planned receipt week 5 = 30 → release week 3 = 30.

Mixed Drill Set (Work Before Looking)

1. D = 5,000, S = $40, H = $5 → EOQ = ?
2. Demand 25/day, L = 8 days, SS = 40 → ROP = ?
3. COGS $900k, avg inventory $150k → turns and days = ?
4. Actual 200, 180; forecast 210, 170 → MAD = ?
5. On hand 100; gross 70, 80 in weeks 1–2; LT = 1; L4L → release week for week-2 need?

Answers: (1) √((2 × 5,000 × 40) / 5) = √80,000 ≈ 283. (2) 25 × 8 + 40 = 240. (3) turns = 6; days ≈ 61. (4) MAD = (10 + 10) / 2 = 10. (5) Week 1 on hand after gross = 30; week 2 net = 50; release in week 1 for L4L receipt week 2.

Calculation Exam Habits

• Circle the time unit before multiplying for ROP.
• For EOQ, confirm annual demand and annual holding cost.
• For PAB/ATP, rewrite the table; do not compute in your head.
• For MRP, separate gross, net, receipt, and release—those four words are often the distractors.
• If two methods are possible (forecast vs. orders in PAB), the stem’s rule wins.