Aggregate Production Planning
MPS level · Tactical months · HMMS 1960
Decision class APP · LP / QP · multi-period production-inventory-workforceAggregate production planning decides how much to produce, how much to store, and how many workers to employ in each period of a medium-horizon plan — typically 6 to 18 months. Named and formalised by Holt, Modigliani, Muth & Simon (1960) at the Graduate School of Industrial Administration (now Tepper / CMU), APP is the bridge between strategic capacity and weekly scheduling, and the canonical setting for the hiring-firing vs inventory-holding trade-off that dominates seasonal manufacturing.
Where This Decision Fits
APICS planning hierarchy — the highlighted level is what this page optimises
The Problem
Twelve-month plan, six decision variables per period, one cost objective
A discrete-manufacturing company sees a 12-month demand forecast dt. Each month it chooses: regular production Pt and overtime Ot (constrained by workforce capacity), workforce Wt adjusted via hires Ht and fires Ft, and the resulting end-of-period inventory It. The objective is to meet all demand at minimum total cost, balancing regular production, overtime, holding, hiring, and firing costs.
Three pure strategies define the design space: Level (constant workforce, absorb demand swings with inventory), Chase (workforce tracks demand, zero inventory), and Mixed (base workforce plus overtime for peaks). The HMMS contribution was recognising this as an LP/QP problem and deriving closed-form linear decision rules from a quadratic cost approximation — one of the first operations-research applications of control theory.
s.t.
It = It−1 + Pt + Ot − dt // inventory balance
Wt = Wt−1 + Ht − Ft // workforce balance
Pt ≤ k · Wt // regular capacity (k units/worker)
Ot ≤ α · k · Wt // overtime cap (fraction α of regular)
Pt, Ot, It, Ht, Ft, Wt ≥ 0 // non-negativity
The LP above is the standard industrial form (Bitran & Hax 1977; Nahmias textbook). HMMS’s original quadratic form replaces the workforce-change cost with (Ht − Ft)2 and the inventory cost with (It − I*)2, yielding an unconstrained quadratic program whose optimal solution can be expressed in closed form as a linear decision rule (current decisions are a linear function of current state plus a weighted sum of forecasted demands). Modern practice uses LP directly with commercial solvers.
| Parameter | Symbol | Default | Unit |
|---|---|---|---|
| Regular production cost | cR | 50 | $ / unit |
| Overtime production cost | cO | 75 | $ / unit |
| Inventory holding cost | h | 5 | $ / unit / period |
| Hiring cost | cH | 1,500 | $ / worker |
| Firing cost | cF | 2,000 | $ / worker |
| Production per worker | k | 15 | units / month |
| Overtime ceiling (fraction) | α | 0.25 | — |
| Initial workforce | W0 | 8 | workers |
| Initial inventory | I0 | 10 | units |
Try It Yourself
Four strategies compared across the same 12-month demand forecast
APP Planner
T = 12 months · 6 decisions/periodClick “Solve & Compare” to run all four strategies on the current demand scenario.
| Strategy | Total $ | Reg. prod. | Overtime | Holding | Hire/Fire |
|---|---|---|---|---|---|
| Click Solve to run | |||||
The Strategies
Four canonical APP heuristics that span the design space
Level Production
HeuristicProduce the same amount every period, chosen so that total production equals total demand over the horizon. Workforce is hired/fired once at the start; then held constant. Absorbs demand variability entirely through inventory.
Good when: hiring and firing are expensive, storage is cheap, product is non-perishable. Bad when: demand is highly seasonal or inventory turns over fast. Classic textbook example: cement, stable consumer goods.
Chase Demand
HeuristicProduce exactly the demand each period; adjust workforce every month to match. Inventory is near zero throughout.
Good when: inventory is extremely expensive or infeasible (perishables, fashion). Bad when: hiring/firing cost or ramp time is significant, or labour market is tight. Classic example: service operations, food processing, apparel.
Mixed (Base + Overtime)
HeuristicSize the base workforce to cover the average demand and use overtime (and possibly sub-contracting) for the peaks. Workforce is held constant; overtime absorbs variability.
Good when: overtime capacity exists and is relatively cheap. Constraint: overtime is capped at α × regular capacity (commonly 20–30%). If peaks exceed the cap, the strategy must also build inventory ahead of the peak.
Smoothed (3-period moving average)
HeuristicProduce a 3-period centred moving average of demand; adjust workforce monthly to support it. Reduces both workforce volatility (vs Chase) and inventory accumulation (vs Level). This is the spirit of HMMS linear decision rules: current production is a weighted function of past and forecasted demand.
A simple smoothing rule is already quite competitive with the LP optimum on slowly-varying demand — the gap opens mainly at sharp transitions (e.g., the “demand shock” scenario).
Exact LP / QP solvers (industrial deployment)
OptimalThe formulation in Section 2 is a standard linear program with 6T variables. Open-source (HiGHS, CBC) and commercial (Gurobi, CPLEX, Xpress) solvers reach the optimum in milliseconds for T ≤ 100. HMMS’s quadratic variant is a QP solved by SLSQP or Gurobi QP.
This page demonstrates heuristic strategies to build intuition; production systems (SAP IBP, Kinaxis, o9, OMP) embed an LP/QP solver inside a rolling-horizon APP loop with re-optimisation every month on updated forecasts.
References
Foundational papers and textbooks for aggregate production planning
- (1960). Planning Production, Inventories, and Work Force. Prentice-Hall, Englewood Cliffs NJ. — HMMS: the canonical APP text, introduces linear decision rules from a quadratic cost model.
- (1955). A linear decision rule for production and employment scheduling. Management Science, 2(1), 1–30. doi:10.1287/mnsc.2.1.1 — original linear decision rule derivation.
- (1956). Production scheduling by the transportation method of linear programming. Operations Research, 4(1), 100–103. doi:10.1287/opre.4.1.100 — transportation-LP formulation.
- (1977). On the design of hierarchical production planning systems. Decision Sciences, 8(1), 28–55. doi:10.1111/j.1540-5915.1977.tb01066.x
- (1975). Hierarchical integration of production planning and scheduling. In Geisler (Ed.), Studies in Management Sciences, Vol. 1, North-Holland/American Elsevier, pp. 53–69. dspace.mit.edu/handle/1721.1/1868
- (1992). Aggregate production planning — a survey of models and methodologies. European Journal of Operational Research, 61(3), 255–272. doi:10.1016/0377-2217(92)90356-E
- (2015). Production and Operations Analysis (7th ed.). Waveland Press. — APP chapter with worked examples and LP/QP comparison.
- (1984). Production and Inventory Management. Prentice-Hall. — hierarchical production planning textbook.
- (2005). Manufacturing Planning and Control for Supply Chain Management (5th ed.). McGraw-Hill/Irwin. — APICS-aligned text, S&OP to SFC coverage.
- (2009). Planning and Scheduling in Manufacturing and Services (2nd ed.). Springer. doi:10.1007/978-1-4419-0910-7 — APP in the broader scheduling hierarchy.