Master Production Scheduling
LP-Based Capacitated Lot Scheduling (CLSP)
A factory’s master production schedule determines how many units of each product family to produce in each week over a rolling 12-week horizon, balancing inventory holding costs against capacity constraints. Poor MPS leads to 20–30% excess inventory or costly emergency overtime.
Where This Decision Fits
Manufacturing planning chain — the highlighted step is what this page optimizes
The Problem
Minimize total setup and holding cost over a multi-period horizon
A manufacturing plant produces 3 product families — Automotive Parts, Appliance Components, and Electronics Housing — over an 8-period planning horizon. Each product family has known demand per period, a fixed setup cost incurred whenever production occurs, and a per-unit holding cost for end-of-period inventory. The plant has a shared capacity limit of production units per period.
The planner must decide how much of each family to produce in each period to satisfy all demand on time while minimizing the sum of setup costs and inventory holding costs, subject to the shared production capacity constraint.
subject to
Ii,t-1 + xit - Iit = dit // inventory balance ∀ i, t
Σi xit ≤ Ct // capacity limit per period
xit ≤ M · yit // setup indicator linking
xit, Iit ≥ 0; yit ∈ {0,1} // domains
Where xit is production quantity for family i in period t, Iit is ending inventory, yit is a binary setup indicator, si is the setup cost, hi is the holding cost per unit per period, and Ct is the production capacity in period t.
See Linear Programming theoryTry It Yourself
Schedule production across periods to minimize total cost
MPS Optimizer
3 Families · 8 Periods| Family | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | Setup $ | Hold $/u |
|---|
Ready. Click “Solve & Compare All Algorithms” to run.
| Algorithm | Setup Cost | Holding Cost | Cap. Util % | Total Cost |
|---|---|---|---|---|
| Click Solve & Compare All Algorithms | ||||
The Algorithms
Three approaches to lot sizing in master production scheduling
Lot-for-Lot (L4L)
O(n · T) | Zero inventory policyThe simplest lot-sizing rule: produce exactly what is needed in each period. This eliminates all holding costs but incurs a setup cost every period for every product family. It serves as a useful upper bound on setup costs and a lower bound on holding costs. L4L is optimal when setup costs are zero or negligible relative to holding costs.
Silver-Meal Heuristic
O(n · T²) | Near-optimal for stationary demandA forward-looking heuristic that groups consecutive periods into a single production lot. Starting from the first uncovered period, it extends the lot one period at a time, computing the average cost per period (setup amortised plus cumulative holding). When the average cost increases, it closes the current lot and starts a new one. Silver-Meal tends to produce good solutions with moderate lot sizes.
LP Relaxation
O(n · T) | Lower bound + roundingRelax the binary setup variables yit to continuous [0, 1] and solve the resulting linear program. The LP relaxation provides a lower bound on the optimal MILP cost. For many practical instances, the LP solution is near-integer and can be rounded to a feasible schedule with small optimality gap. This approach scales well and gives a quality benchmark for heuristics.
Real-World Complexity
Why practical MPS goes beyond the basic CLSP model
Beyond Basic Lot Sizing
- Sequence-dependent setups — Changeover time and cost depend on which product was produced previously, turning the problem into a scheduling+lot-sizing hybrid
- Multi-level BOM — Components feed into subassemblies that feed into finished goods; lot-sizing decisions cascade through the bill of materials
- Overtime and subcontracting — When regular capacity is insufficient, overtime at premium rates or external subcontractors can be modelled as additional capacity tiers
- Demand uncertainty — Forecast errors grow with the horizon; safety stocks, rolling horizons, and stochastic models hedge against demand volatility
- Perishability and shelf life — Some products have limited shelf life, adding constraints on maximum inventory age
- Minimum lot sizes — Tooling or process constraints may require a minimum batch quantity once a setup is performed
- Backlogging — Some models allow demand to be met late at a penalty cost, adding flexibility but complicating the objective
Key References
Foundational works in production planning and lot sizing
- (1958). “Dynamic Version of the Economic Lot Size Model.” Management Science, 5(1), 89–96. doi:10.1287/mnsc.5.1.89
- (1973). “A Heuristic for Selecting Lot Size Quantities for the Case of a Deterministic Time-Varying Demand Rate and Discrete Opportunities for Replenishment.” Production and Inventory Management, 14(2), 64–74.
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