Forecasting and aggregate planning

Production and Industrial Engineering (B.E./B.Tech) · Industrial engineering · free study notes for exams, placements and interviews

In short

Forecasting estimates future demand from past data and judgement; aggregate planning converts that forecast into a 6-18 month plan for total output, workforce, overtime, subcontracting, inventory and backorders at the product-family level (in "aggregate units" or standard labour hours), before it is disaggregated into a Master Production Schedule. Together they decide how much capacity a plant commits to and therefore most of its medium-term cost. They are standard in AICTE/CBCS Industrial Engineering and Operations Management syllabi and carry 4-8 marks in GATE PI/ME under "Production Planning and Inventory Control" - usually as a smoothing, regression, seasonal-index or least-cost aggregate-plan numerical. In Indian industry the same arithmetic drives monthly S&OP meetings at Maruti Suzuki, Tata Motors, Hindustan Unilever, Asian Paints, Amul and ITC, and is the core of planning roles on SAP APO/IBP, Oracle Demantra and Kinaxis.

Explained

What demand data looks like, and what is forecastable

Components of a time series: *level* (average), *trend* (long-term rise or fall), *seasonality* (fixed-period repeat - festive, monsoon, quarterly), *cycle* (multi-year, business cycle), *random/irregular noise*.
Only level, trend and seasonality are forecastable; noise is not. A method that chases noise (very small smoothing period or large α) forecasts worse, not better.
Multiplicative model: Demand = Level × Trend × Seasonal index × Error. Additive model: Demand = Level + Trend + Seasonal + Error. Multiplicative is used when seasonal swing grows with volume (true of most Indian FMCG and auto demand).
Forecast horizon: short (days-weeks, scheduling and dispatch), medium (months-year, aggregate planning, hiring, budgets), long (years, plant and capacity decisions).
Rules every exam expects: forecasts are almost always wrong, so plan for error; aggregate forecasts are more accurate than item-level ones (errors cancel); accuracy falls as the horizon lengthens; a forecast must carry a measure of error with it.

Qualitative (judgemental) methods

Delphi method: a panel of experts answers questionnaires anonymously in rounds; a coordinator feeds back the summary each round until answers converge. Removes dominance by seniority; slow and costly. Used for new technology and long-range demand.
Market survey / consumer panel: direct questioning of buyers or dealers; good for a new product with no history, expensive and biased by what people *say* they will buy.
Executive opinion (jury of executive opinion): senior managers pool estimates - fast, but optimism bias.
Sales-force composite: field staff estimate territory demand and the estimates are rolled up; good local knowledge, but staff bias forecasts down when targets follow them and up when stocks follow them.
Life-cycle analogy / historical analogy: forecast a new product on the curve of a similar past product.
Use qualitative methods when there is no data (new launch), when data is irrelevant (regime change, GST slab change, a new norm like BS-VI), or as an override on a statistical number.

Quantitative methods: time-series averaging and smoothing

Naive forecast: F(t+1) = D(t). Benchmark - any method that cannot beat it is not worth running.
Simple average: mean of all past data; suitable only for a stable, trendless series.
Simple moving average (n-period): average of the last n observations; higher n gives more smoothing and more lag. Equal weight to each of the n periods, zero to older data, and n values must be stored.
Weighted moving average: chosen weights (summing to 1) with more weight on recent periods - responds faster than the simple version.
Simple exponential smoothing (SES): F(t+1) = F(t) + α[D(t) − F(t)]. The new forecast is the old forecast corrected by α times the last error; weights on past data decay geometrically as α(1−α)^k. Needs only the last forecast and the last demand - its big practical advantage.
Choosing α: 0 < α < 1; typically 0.1-0.3 for a stable series, 0.4-0.5 when demand genuinely shifts. Equivalence used in exams: α ≈ 2/(n+1) matches an n-period moving average in average age of data.
Both SES and moving average lag behind a trend - they always under-forecast a rising series. That is why trend models are needed.
Double exponential smoothing (Holt's method): smooths level and trend separately with α and β; forecast m periods ahead = L(t) + m·T(t).
Triple exponential smoothing (Holt-Winters): adds a seasonal factor smoothed with γ - the standard production method for seasonal data.
Adaptive smoothing: α itself is adjusted (often from the tracking signal) when the series shifts.

Trend, regression and seasonality

Trend projection by least squares: fit D = a + bt to the data; coding time so Σt = 0 (t = ..., −2, −1, 0, 1, 2 ...) makes the arithmetic trivial: b = ΣtD/Σt², a = ΣD/n.
Simple linear regression on a causal variable: demand regressed on an independent variable (tractor sales on monsoon rainfall, cement on construction index, two-wheeler sales on rural income). Correlation coefficient r measures strength; r² is the fraction of variation explained.
Multiple regression / econometric models: several causal variables; used for national-level forecasts.
Seasonal index (ratio-to-moving-average method), step by step:
1. Compute the centred moving average (CMA) over one full season (4 for quarters, 12 for months).
2. Ratio = actual demand / CMA for each period.
3. Average the ratios for each season position; this gives the raw seasonal index.
4. Normalise so the indices sum to the number of periods (4 or 12), i.e. average 1.0.
5. Deseasonalise: deseasonalised demand = actual / index. Fit the trend to the deseasonalised series.
6. Forecast = trend value × seasonal index.
Decomposition is the name of the whole procedure (trend + seasonal + cyclical + irregular separated, forecast, then recombined).
Focus forecasting: run several simple rules on recent history and use whichever would have performed best last period.

Measuring and controlling forecast error

Error (deviation) e(t) = D(t) − F(t). Positive means under-forecast.
Bias (mean error / CFE, cumulative forecast error): average of signed errors; should be near zero. A persistent bias means the model itself is wrong (usually a missed trend).
MAD (mean absolute deviation), MSE (mean squared error - penalises large errors), RMSE (√MSE), MAPE (mean absolute percentage error - unit-free, used to compare items of different sizes; fails when demand is near zero).
Tracking signal = CFE / MAD. A control device: if it goes outside about ±4 MAD (some texts ±3 to ±6), the forecast is out of control and the model or parameters must be revised.
Relation used in inventory work: σ ≈ 1.25 MAD when errors are normally distributed - this links forecast error to safety stock.
Fitting versus forecasting: a model can be tuned to fit history perfectly and still forecast badly; error should ideally be measured on data held back.

Aggregate planning: purpose, inputs and pure strategies

Definition: medium-range (typically 6-18 months, monthly or quarterly buckets) planning of production rate, workforce, inventory, overtime, subcontracting and backlog for *product families* expressed in a common aggregate unit - equivalent units, tonnes, litres, or standard labour hours.
Position in the hierarchy: business plan → aggregate production plan (from Sales & Operations Planning) → master production schedule (MPS) → MRP → detailed scheduling and shop-floor control. Aggregate planning fixes capacity; MPS and MRP fix items and components.
Inputs: demand forecast by period, current workforce and inventory, normal and overtime capacity per worker, cost rates (regular time, overtime, hire, layoff, holding, backorder, subcontract), and policy limits (no-layoff policy, maximum overtime).
Why aggregate: family-level forecasts are more accurate; the plan is about *capacity*, which is shared across items; and a monthly aggregate plan is tractable where item-level planning for thousands of SKUs is not.
Demand options (influencing demand): pricing and discounts in the lean season, promotion, back-ordering/advance booking, counter-seasonal (complementary) products - a fan maker adding room heaters.
Capacity options (supplying demand): hire and layoff, overtime and idle time, inventory build-up, subcontracting (vendors/job work), part-time and contract labour, multi-skilling.
Pure strategies:
- Level (constant production) strategy: output and workforce held constant; demand swings absorbed by inventory and backorders. Stable employment and smooth operation, high holding cost; suited to process plants and companies with strong unions.
- Chase (matching) strategy: production matched to demand each period by changing workforce or hours; little inventory, high hire/layoff and training costs, quality and morale risk. Suited to services and short shelf-life goods.
- Mixed strategy: a combination, usually cheapest in practice - the level base load plus overtime and subcontracting for peaks.
Overtime and contract labour are regulated. Working hours, overtime wages (at twice the ordinary rate) and spread-over are governed by the Occupational Safety, Health and Working Conditions Code, 2020 and the Code on Wages, 2019, which came into force on 21 November 2025 and replaced the Factories Act, 1948 and the Minimum Wages Act, 1948; limits are set in the Codes and the rules notified under them, and the figures differ by State, so check the applicable State rules rather than assuming a number. An aggregate plan that relies on heavy overtime or layoffs must be legally and contractually feasible, not merely cheap.

Techniques for solving an aggregate plan

Graphical / trial-and-error (tabular) method: cost out two or three candidate plans and pick the cheapest. Not optimal, but transparent, easily modified, and the method actually used in industry - and the one GATE tests.
Cumulative (cumulative demand versus cumulative production) chart: the vertical gap is inventory (production above demand) or shortage; a straight cumulative production line is the level plan.
Transportation model (Bowman's method): a linear-programming special case; sources are regular time, overtime and subcontract capacity in each period, destinations are period demands, cell cost includes production plus holding cost for carrying forward (and backorder cost if backorders are allowed). Solved by the transportation algorithm; optimal within its linear-cost assumptions.
Linear programming (general LP): minimises total cost with decision variables for production, inventory, workforce, hiring, firing and overtime, subject to inventory-balance and capacity constraints. Optimal, but assumes linear costs and continuous (fractional) workers.
Linear Decision Rule (Holt-Modigliani-Muth-Simon): quadratic cost functions give two linear rules for production rate and workforce size; optimal under its assumptions but the quadratic cost coefficients are hard to estimate.
Management Coefficients Model (Bowman): fits the firm's *own* past decisions by regression on the belief that managers are good but inconsistent; the fitted rule is applied consistently.
Search Decision Rule / simulation: computer search over a complex, realistic cost function; no optimality guarantee.
Heuristics in software: SAP IBP/APO, Oracle, Kinaxis and Excel Solver all implement the LP/heuristic versions used in practice.
Disaggregation: the family plan is split into end items (MPS) using planning bills, past mix ratios or run-out/cut-off rules, then rough-cut capacity planning (RCCP) checks the MPS against key work centres.

Key points

Forecast horizon - short for scheduling, medium for aggregate planning, long for capacity and plant decisions; method must match horizon.
Aggregation rule - a family or plant-level forecast is more accurate than an SKU forecast because errors cancel.
Moving average - equal weights on the last n periods; larger n means smoother but more lagged.
Exponential smoothing - F_new = F_old + α(error); weights decay geometrically; needs only two numbers stored.
α ≈ 2/(n+1) - the smoothing constant equivalent to an n-period moving average.
Trend lag - SES and moving averages systematically under-forecast a growing series; use Holt's double smoothing.
Holt-Winters - three constants α (level), β (trend), γ (seasonality); the standard seasonal method.
Seasonal index - demand of a period ÷ deseasonalised average; indices average 1.0 and sum to 4 or 12.
MAD, MSE, MAPE - absolute, squared and percentage error; MAPE is unit-free, MSE punishes big misses.
Tracking signal = CFE/MAD - outside roughly ±4, the forecasting model is out of control.
σ ≈ 1.25 MAD - converts forecast error into the standard deviation used for safety stock.
Aggregate plan - 6-18 month plan of output, workforce, overtime, inventory and subcontracting for product families in common units.
Level strategy - constant output; inventory and backorders absorb variation; stable employment, high holding cost.
Chase strategy - output follows demand; minimum inventory, high hiring/firing and training cost.
Mixed strategy - level base plus overtime and subcontracting; usually the lowest total cost.
Transportation model - optimal aggregate plan when production, overtime, subcontract and holding costs are linear.
Hierarchy - S&OP aggregate plan → MPS → MRP → scheduling; RCCP validates the MPS.
Demand management - off-season discounts, advance booking and counter-seasonal products reduce the capacity swing.

Formulas and examples

Moving average: F(t+1) = [D(t) + D(t−1) + ... + D(t−n+1)] / n (D = demand in units, n = periods)
Weighted MA: F(t+1) = Σ wᵏ D(t−k+1), with Σ wᵏ = 1
Simple exponential smoothing: F(t+1) = αD(t) + (1−α)F(t) = F(t) + α·e(t), 0 < α < 1
Holt's double smoothing: L(t) = αD(t) + (1−α)[L(t−1)+T(t−1)]; T(t) = β[L(t)−L(t−1)] + (1−β)T(t−1); F(t+m) = L(t) + m·T(t)
Holt-Winters (multiplicative): F(t+m) = [L(t) + m·T(t)] × S(t+m−s), s = season length
Least-squares trend: D = a + bt; b = [nΣtD − ΣtΣD]/[nΣt² − (Σt)²]; a = D̄ − b·t̄. With coded t so Σt = 0: b = ΣtD/Σt², a = ΣD/n
Seasonal index Iₖ = average(actual ÷ centred moving average) for season k, normalised so ΣI = s; Forecast = trend × Iₖ
Errors: e = D − F; MAD = Σ|e|/n; MSE = Σe²/n; MAPE = (100/n)·Σ|e/D| %; CFE = Σe; Tracking signal = CFE/MAD; σ ≈ 1.25 MAD
Aggregate plan costs (₹): Total = Σ[regular-time + overtime + subcontract + hiring + layoff + holding + backorder]
Workers needed in a period = (units required × standard hours per unit) / (working hours per worker in the period)
Inventory balance: I(t) = I(t−1) + P(t) − D(t); cumulative production ≥ cumulative demand for no shortage

Example 1 - moving average, weighted MA, SES and error measures. Monthly demand of a pump (units): Jan 420, Feb 450, Mar 435, Apr 480, May 465. Forecast June.

3-month MA (Mar, Apr, May) = (435 + 480 + 465)/3 = 1,380/3 = 460 units.
Weighted MA with weights 0.5 (May), 0.3 (Apr), 0.2 (Mar) = 0.5(465) + 0.3(480) + 0.2(435) = 232.5 + 144 + 87 = 463.5 units.
SES with α = 0.2 and F(Jan) = 420:

F(Feb) = 420 + 0.2(420 − 420) = 420.0 F(Mar) = 420 + 0.2(450 − 420) = 426.0 F(Apr) = 426 + 0.2(435 − 426) = 427.8 F(May) = 427.8 + 0.2(480 − 427.8) = 438.24 F(Jun) = 438.24 + 0.2(465 − 438.24) = 443.6 units

Errors of the SES forecasts for Feb-May: 450 − 420 = 30; 435 − 426 = 9; 480 − 427.8 = 52.2; 465 − 438.24 = 26.76.

CFE = 117.96; MAD = 117.96/4 = 29.49; MSE = (900 + 81 + 2,724.84 + 716.10)/4 = 4,421.94/4 = 1,105.5; RMSE = 33.25. MAPE = 100/4 × (30/450 + 9/435 + 52.2/480 + 26.76/465) = 25 × (0.06667 + 0.02069 + 0.10875 + 0.05755) = 25 × 0.25366 = 6.34%. Tracking signal = 117.96/29.49 = +4.0 - all errors positive, the model is lagging a rising trend and is on the control limit. Raise α or switch to Holt's method.

Example 2 - least-squares trend and seasonal indices. Quarterly demand of a ceiling fan family (thousand units): Q1 60, Q2 100, Q3 70, Q4 50; annual total 280, average 70.

Raw seasonal indices from one year of data: 60/70 = 0.857; 100/70 = 1.429; 70/70 = 1.000; 50/70 = 0.714. Sum = 4.000 ✓ (already normalised).
Yearly totals for 5 years (thousand units): 240, 258, 272, 280, 300. Code t = −2, −1, 0, 1, 2 (Σt = 0, Σt² = 10).

ΣD = 1,350 → a = 1,350/5 = 270. ΣtD = (−2)(240) + (−1)(258) + 0 + (1)(280) + (2)(300) = −480 − 258 + 0 + 280 + 600 = 142 → b = 142/10 = 14.2 thousand units per year.

Next year is t = 3: annual forecast = 270 + 14.2(3) = 312.6 thousand units → quarterly average 312.6/4 = 78.15.
Quarterly forecasts: Q1 = 78.15 × 0.857 = 66.98; Q2 = 78.15 × 1.429 = 111.68; Q3 = 78.15 × 1.000 = 78.15; Q4 = 78.15 × 0.714 = 55.80 thousand units. Total = 312.61 ✓.

Example 3 - level versus chase plan (full cost comparison). Four-month forecast (units): M1 1,200; M2 1,600; M3 2,000; M4 1,200 - total 6,000. One worker makes 100 units/month. Opening workforce 12, opening inventory 0, closing inventory may be 0. Costs: regular time ₹18,000 per worker-month; hiring ₹6,000 per worker; layoff ₹10,000 per worker; holding ₹60 per unit per month on closing stock. No shortages allowed.

*Level plan:* 6,000/4 = 1,500 units/month → 15 workers throughout; hire 3 at the start.

Month	Production	Demand	Closing stock
M1	1,500	1,200	300
M2	1,500	1,600	200
M3	1,500	2,000	−300 → shortage
M4	1,500	1,200	0

M3 ends 300 short, so the level plan at exactly 1,500/month is infeasible without backorders. Build ahead instead: run 16 workers (1,600/month) in M1-M3 and 12 workers' worth of output in M4.

Production 1,600, 1,600, 1,600, 1,200 = 6,000. Closing stock: 400, 400, 0, 0 → total stock-months 800. Hiring = 4 workers × ₹6,000 = ₹24,000. Layoff at M4 = 4 × ₹10,000 = ₹40,000. Regular time = (16 + 16 + 16 + 12) × ₹18,000 = 60 × 18,000 = ₹10,80,000. Holding = 800 × ₹60 = ₹48,000. Total = 10,80,000 + 24,000 + 40,000 + 48,000 = ₹11,92,000.

*Chase plan:* workers = demand/100 → 12, 16, 20, 12. Regular time = (12 + 16 + 20 + 12) × ₹18,000 = 60 × 18,000 = ₹10,80,000. Hiring = 4 (M2) + 4 (M3) = 8 × ₹6,000 = ₹48,000. Layoff = 8 (M4) × ₹10,000 = ₹80,000. Holding = 0. Total = 10,80,000 + 48,000 + 80,000 = ₹12,08,000.

Plan	Regular time	Hire	Layoff	Holding	Total
Level (build ahead)	₹10,80,000	₹24,000	₹40,000	₹48,000	₹11,92,000
Chase	₹10,80,000	₹48,000	₹80,000	₹0	₹12,08,000

Regular-time cost is identical because total output is the same; the level plan wins by ₹16,000 because ₹48,000 of holding is cheaper than the extra ₹64,000 of hire-and-fire churn. Raise holding cost to ₹150/unit/month and holding becomes ₹1,20,000 - the chase plan then wins. That trade-off is the whole of aggregate planning.

Example 4 - overtime and subcontracting (mixed plan). Demand for a month: 2,400 units. Regular capacity 1,800 units at ₹400/unit; overtime up to 400 units at ₹560/unit; subcontract unlimited at ₹620/unit; holding ₹30/unit/month; 300 units are in opening stock and may be used.

Use opening stock 300 → net requirement 2,100.
Regular 1,800 × ₹400 = ₹7,20,000. Overtime 300 × ₹560 = ₹1,68,000. Subcontract 0.
Total = ₹8,88,000 for 2,100 units → average ₹422.86/unit. Cheapest source order is always regular → overtime → subcontract here; if subcontract were below ₹560 the order would reverse.

Example 5 - transportation-model cost cells. Period-1 regular production costs ₹400/unit; carrying to period 2 costs ₹30/unit-month. So the cell cost (source: period-1 regular time → destination: period-2 demand) is 400 + 30 = ₹430; to period 3 it is 400 + 60 = ₹460. Period-1 overtime (₹560) serving period-3 demand costs 560 + 60 = ₹620 - equal to fresh subcontracting, so the model is indifferent between them. Unused capacity is assigned to a dummy destination at zero cost; backorder cells (a later period supplying an earlier demand) carry the backorder penalty per period.

Method	Data needed	Handles trend	Handles seasonality	Typical use
Naive	Last value	No	No	Benchmark, very stable items
Moving average (n)	Last n values	Poorly (lags)	No	Stable, low-value items
SES (α)	Last forecast	Poorly (lags)	No	Thousands of SKUs, stock control
Holt (α, β)	Level + trend	Yes	No	Growing product lines
Holt-Winters (α, β, γ)	Level, trend, season	Yes	Yes	Seasonal FMCG, fans, coolers
Regression	Causal variable series	Yes	Via dummies	Cement, tractors, commodity demand
Decomposition	2-3 seasons of history	Yes	Yes	Quarterly/monthly planning
Delphi / survey	Expert or buyer input	n/a	n/a	New product, long range

Where it is used

Automotive and auto components: Maruti Suzuki, Tata Motors, Bajaj and their Tier-1 suppliers run monthly S&OP, aggregate the forecast into shifts and manpower, and release it to vendors as a rolling schedule; the pre-festive build-up is a classic level-strategy inventory build.
Seasonal consumer durables: fans, air coolers and air conditioners (Crompton, Havells, Voltas, Blue Star) produce through the winter against a summer peak; room heaters are the counter-seasonal product.
FMCG and dairy: Hindustan Unilever, ITC, Dabur and Amul forecast at the family level for plant and depot capacity, with monsoon and festive seasonality and heavy use of contract packers (subcontracting).
Process industries: cement, steel, fertiliser and paints run near-level plans because shutting down and restarting is costly; demand swings are absorbed by stock and by dispatch planning.
Job roles: demand planner, production planner, PPC engineer, supply-chain analyst, S&OP manager - most working in SAP APO/IBP, Oracle or Kinaxis, with Excel for the cost comparison.
Services and utilities: manpower planning in BPO/IT services, hospital staffing, Indian Railways coach and seat planning, electricity load forecasting by SLDCs - all aggregate planning with manpower or capacity as the variable.

Interview questions

Q: Why is an aggregate forecast more accurate than an item-level forecast? A: Random errors in individual items are partly independent and cancel when added, so the relative error (coefficient of variation) of the total is smaller.
Q: What does the smoothing constant α do? A: It sets how much of the latest error is corrected into the forecast - a high α responds quickly to real change but also chases noise; a low α smooths heavily but lags.
Q: Why do simple exponential smoothing and moving averages fail on trending demand? A: Both are averages of past values, so on a rising series the forecast sits below the latest demand and shows a persistent positive bias; Holt's method adds a trend term to remove it.
Q: MAD, MSE or MAPE - which would you report and why? A: MAPE to compare items of different volumes because it is unit-free, MSE when large misses are disproportionately costly, MAD because it is simple and converts to σ ≈ 1.25 MAD for safety stock.
Q: What is a tracking signal used for? A: It is the ratio of cumulative error to MAD and is monitored like a control chart; crossing about ±4 signals bias and triggers a review of the model or its constants.
Q: Difference between a level and a chase strategy? A: Level keeps output and workforce constant and absorbs demand variation in inventory and backorders; chase varies output to match demand and absorbs variation in workforce and hours.
Q: Which strategy suits an Indian cement plant, and which a food delivery operation? A: Cement is level - the kiln cannot be throttled and stock is cheap to hold; delivery is chase - the output cannot be stored at all, so capacity (riders) must follow demand.
Q: How does the transportation model solve an aggregate plan? A: Period capacities (regular, overtime, subcontract) are sources, period demands are destinations, and each cell cost is production cost plus holding cost for the months carried, so the transportation algorithm returns the least-cost assignment.
Q: What is the difference between the aggregate plan and the MPS? A: The aggregate plan sets total output and capacity for product families in monthly or quarterly buckets; the MPS disaggregates it into specific end items in weekly buckets and drives MRP.
Q: Give one demand-side option in aggregate planning. A: Off-season discounting or advance booking, which shifts demand from the peak to the trough and reduces the capacity needed - the standard tactic for fans and air conditioners.
Q: When is the Delphi method preferred over regression? A: When there is no relevant history or the relationship has broken - a new product, a technology change or a policy shift - and expert judgement is the only input available.
Q: Your plan needs 20% overtime for three months. What do you check before approving it? A: Legal and contractual feasibility - overtime hours, spread-over and double-rate overtime wages under the labour Codes in force since 21 November 2025 and the applicable State rules - plus fatigue, quality and whether subcontracting is cheaper.

Common mistakes

Treating the smoothing formula as applying α to the old forecast - α multiplies the *latest demand* (or, equivalently, the latest error).
Using the initial forecast's own period as an error term, or computing MAD over periods with no forecast - the denominator must be the number of forecast-error pairs.
Averaging seasonal ratios without normalising them to sum to 4 (quarters) or 12 (months).
Fitting the trend to raw seasonal data instead of deseasonalised data, then multiplying by the index again.
Reporting a near-zero mean error as "accurate" when large positive and negative errors are cancelling - always report MAD or MSE alongside bias.
Costing an aggregate plan without the inventory-balance check, ending a period with negative stock and no backorder cost charged.
Comparing plans on only one cost head (usually payroll) when total output, and hence regular-time cost, is identical and the decision lies entirely in hire/layoff, overtime and holding.
Forgetting fractional workers must be rounded up (and the resulting extra output carried as inventory) in a chase plan.