Aggregate Planning

378. Aggregate Planning

A mid-term (3–18 month) production / staffing plan that matches aggregated output to aggregated demand, deciding levels of:

Formulated as a linear program.

For periods $𝑡 = 1, \dots, 𝑇$ :

Variables:

Costs per unit / hire / firing:

Objective:

min \sum_{𝑡} (𝑐_{𝑃} 𝑃_{𝑡} + 𝑐_{𝑂} 𝑂_{𝑡} + 𝑐_{𝐻} 𝐻_{𝑡} + 𝑐_{𝐹} 𝐹_{𝑡} + 𝑐_{𝐼} 𝐼_{𝑡} + 𝑐_{𝐵} 𝐵_{𝑡})

Constraints (workforce balance, production capacity, inventory balance):

𝑊_{𝑡} = 𝑊_{𝑡 - 1} + 𝐻_{𝑡} - 𝐹_{𝑡} (workforce balance)

𝑃_{𝑡} \leq 𝑘 𝑊_{𝑡} ($k$ = units per worker per period)

𝑂_{𝑡} \leq 𝑜_{max} 𝑊_{𝑡} (overtime cap)

𝐼_{𝑡} - 𝐵_{𝑡} = 𝐼_{𝑡 - 1} - 𝐵_{𝑡 - 1} + 𝑃_{𝑡} + 𝑂_{𝑡} - 𝐷_{𝑡} (net inventory balance)

𝐼_{𝑡}, 𝐵_{𝑡}, 𝑃_{𝑡}, 𝑂_{𝑡}, 𝐻_{𝑡}, 𝐹_{𝑡}, 𝑊_{𝑡} \geq 0

Strategy	Description	Best when
Chase	Match production to demand each period; vary workforce	Low inventory cost, high hire/fire flexibility, low workforce wages
Level	Constant production; let inventory + backorder absorb demand fluctuations	Stable workforce, low inventory cost, low backorder cost
Mixed	Combination of both, plus overtime / contract	Real-world; what the LP discovers automatically

Pure chase or pure level rarely optimal in practice — LP finds the right mixed strategy.

12-month plan for a manufacturer. Demand peaks in November. Cost parameters set; LP gives:

vs pure chase (high hiring/firing): $1.5$ M vs pure level (high inventory): $1.4$ M

Aggregate planning sets the envelope — workforce, total monthly output. The detail of which specific products gets handled by:

MPS (Master Production Schedule) — disaggregates aggregate plan into product-week
MRP (Material Requirements Planning) — explodes MPS into parts / materials

See MPS and MRP.

Linear costs assumed — real overtime / hiring may have step functions
Aggregate units — assumes products are interchangeable, ignores changeover times
Single facility — multi-plant versions exist but more complex
Deterministic demand — stochastic versions = stochastic programming (see here)