Aggregate production planning

Why it matters

Demand for most products rises and falls through the year, but a plant's workforce and equipment cannot be changed overnight. Aggregate production planning (APP) decides, month by month for the next 3–18 months, how much to produce, how many people to employ, and how much inventory, overtime or subcontracting to use so that forecast demand is met at the lowest total cost. Every master schedule is built inside the limits it sets.

Key ideas

What is "aggregate". Products are grouped into families and measured in one common unit (units of a representative product, tonnes, labour-hours or rupees). Time is in months or quarters. Detail by individual item comes later in the MPS.

Inputs. Demand forecast by period, opening inventory and workforce, productivity (units per worker per period), regular and overtime capacity, subcontracting availability, and costs: regular time, overtime, hiring, firing (layoff), inventory holding, backorder or lost-sale (shortage) and subcontracting.

Decision options (supply side).

• Vary the workforce size by hiring and firing.
• Vary utilisation with overtime or idle time.
• Build or draw down inventory (anticipation stock).
• Subcontract part of the work.
• Use part-time or temporary workers. Demand-side options: pricing and promotion to shift demand into slack periods, backorders (accepted delays), and counter-seasonal product mixes.

Pure strategies.

• Level strategy: constant production rate and workforce; inventory absorbs the difference between demand and production (built in low months, used in high months). Low hiring/firing cost, stable labour, but high holding cost and possible shortages if the rate is too low.
• Chase strategy: production equals demand each period by hiring and firing (or overtime/idle time). Very little inventory, but high workforce-change costs, lower morale and training problems. Common in services where output cannot be stored.
• Mixed (hybrid) strategy: combines two or more options, for example a steady core workforce plus overtime and subcontracting in peaks. The least-cost plan is usually mixed.

Inventory balance. In every period: closing inventory = opening inventory + production − demand. A negative value is a backorder (if allowed) or a lost sale.

Level production rate. With no backorders and a required closing inventory, the constant rate is (total demand + closing inventory − opening inventory)/number of periods, but it must also be high enough that cumulative production never falls below cumulative demand in any period. If an early period has high demand, the rate (or opening stock) must be raised.

Solution methods.

• Trial-and-error (charting/spreadsheet): cost out a few candidate plans (level, chase, mixed) and pick the cheapest. Fast but not guaranteed optimal.
• Transportation method (Bowman): treat each capacity source (regular, overtime, subcontract in each period) as a supply and each period's demand as a destination; the "transport" cost from period i to period j is the production cost plus holding cost for j − i periods. Valid when hiring/firing costs are not involved. Solved by the least-cost or optimal transportation algorithm.
• Linear programming: handles workforce changes, backorders and capacity limits together and gives the true optimum for linear costs.
• Others: linear decision rule (quadratic costs), management coefficients, search heuristics.

Disaggregation. The aggregate plan is split by product mix into the MPS for individual end items, which then drives MRP.

Formulas

I(t) = I(t−1) + P(t) − D(t)

• I = closing inventory (units; negative = backorder), P = production (units), D = demand (units), t = period.

P_level = [ΣD(t) + I_end − I_0] / N

• P_level = constant production per period (units/period), I_0 = opening inventory, I_end = required closing inventory, N = number of periods.

W(t) = P(t) / k

• W = workers needed, k = output per worker per period (units/worker·period).

Total cost = Σ [c_r·R(t) + c_o·O(t) + c_s·S(t) + c_h·I⁺(t) + c_b·I⁻(t) + c_H·H(t) + c_F·F(t)]

• R, O, S = regular, overtime, subcontracted units; I⁺ = inventory held, I⁻ = backorders (units); H, F = workers hired and fired; c_r, c_o, c_s = cost per unit (₹/unit), c_h = holding cost (₹/unit·period), c_b = backorder cost (₹/unit·period), c_H, c_F = cost per worker hired or fired (₹/worker). Holding cost is charged on closing (or average) inventory – state which.

cost of making in period i for period j = c_i + c_h·(j − i)

• Cell cost in the transportation method (no backorders), with j ≥ i.

Worked examples

Example 1 (standard) – level vs chase. Demand for six months (units): 800, 1000, 1200, 1400, 1400, 1400 (total 7200). Opening inventory 0, required closing inventory 0. Each worker makes 20 units/month; opening workforce 60 workers. Hiring costs ₹3000 per worker, firing ₹5000 per worker, holding ₹20 per unit per month on closing inventory. Regular wages are the same under both plans and can be ignored. Level plan

1. P_level = 7200/6 = 1200 units/month = 60 workers, so no hiring or firing.
2. Closing inventory by month: 400, 600, 600, 400, 200, 0 (never negative, so no shortages).
3. Holding cost = (400 + 600 + 600 + 400 + 200 + 0) × 20 = 2200 × 20 = ₹44,000. Chase plan
4. Workers needed = D/20: 40, 50, 60, 70, 70, 70.
5. Changes from 60: fire 20 in month 1; hire 10 in each of months 2, 3 and 4.
6. Cost = 20 × 5000 + 30 × 3000 = 1,00,000 + 90,000 = ₹1,90,000.
7. The level plan is cheaper: ₹44,000 against ₹1,90,000.

Example 2 (GATE level) – transportation method with overtime and subcontracting. Demand for three months: 900, 1200, 1500 units. Each month: regular capacity 1000 units at ₹100/unit, overtime capacity 200 units at ₹130/unit, subcontracting unlimited at ₹150/unit. Holding cost ₹15 per unit per month; no backorders; no opening or closing stock.

1. Cell cost = production cost + 15 × (months held). For example regular month 1 used in month 3 costs 100 + 30 = ₹130.
2. All 3000 regular units are cheapest, so use them: month 1 needs only 900, leaving 100 units of month-1 regular time.
3. Month 2 shortfall = 1200 − 1000 = 200 units. Cheapest sources: month-1 regular carried 1 month (₹115) for 100 units, then month-2 overtime (₹130) for 100 units.
4. Month 3 shortfall = 1500 − 1000 = 500 units. Cheapest: month-3 overtime ₹130 (200 units), month-2 overtime carried 1 month ₹145 (the remaining 100 units), then subcontracting ₹150 (200 units), which is cheaper than month-1 overtime carried 2 months (₹160).
5. Cost = 3000 × 100 + 100 × 15 + 200 × 130 + 100 × 15 + 200 × 130 + 200 × 150 = 3,00,000 + 1500 + 26,000 + 1500 + 26,000 + 30,000.
6. Minimum total cost = ₹3,85,000. (An alternative plan – month-2 overtime fully used in month 2 and the spare month-1 regular units carried to month 3 – costs the same; the optimum is not unique. A full enumeration confirms no cheaper plan exists.)

Common mistakes

• Using total demand ÷ periods as the level rate without checking cumulative production against cumulative demand; early peaks cause shortages.
• Charging holding cost on opening inventory in one period and closing in another; pick one basis and state it.
• Forgetting the opening workforce when counting hires and fires in the chase plan.
• In the transportation method, allowing production in a later period to satisfy earlier demand when backorders are not permitted.
• Ignoring the end condition: a chase plan that ends with more workers than it started with hides a future cost.
• Treating APP as item-level scheduling; it works on families and aggregate units.

For GATE PI

Questions give a demand table with costs and ask for the cost of a level, chase or mixed plan, the level production rate, the inventory at the end of a given period, or the minimum-cost allocation by the transportation method. Conceptual questions test the features of pure strategies and where APP sits in the planning hierarchy. Practise cumulative-demand tables and transportation cell costs until they are quick and error-free.

Quick check

1. Opening inventory 100 units, production 500, demand 450. What is the closing inventory?
2. Total demand 4800 units over 4 months, opening and closing inventory zero. What is the level rate?
3. Which pure strategy has the highest inventory holding cost?
4. Regular cost ₹80/unit, holding ₹6/unit·month. What is the cell cost of month-1 regular production used in month 4?

Answers: 1. 150 units. 2. 1200 units per month (if cumulative demand never exceeds cumulative production). 3. Level strategy. 4. 80 + 3 × 6 = ₹98 per unit.