Multi-plant allocation

A company with several plants making similar products has to decide each month who makes what. The cheapest plant per unit isn’t automatically the answer, because freight, duties, capacity and changeovers all count.

Take shampoo made in two plants:

	Cost to make	Freight to North India	Freight to South India	Monthly capacity
Plant A (Baddi)	₹40	₹4	₹14	60,000
Plant B (Chennai)	₹43	₹15	₹3	50,000

North needs 50,000 units and South 40,000. Plant A is cheaper to run, but for the South it lands at ₹54 a unit against Plant B’s ₹46. The sensible split is that A serves the North (₹44 landed) and B serves the South (₹46 landed). Both fit within capacity, and the total comes to ₹40.4 lakh. Sending everything from A would cost more and would also exceed its capacity.

With dozens of products, plants and markets, this becomes a linear program: choose quantities that minimize total landed cost (production + freight + duties) while meeting demand and staying within each plant’s capacity. Real models add setup costs and minimum runs, which turn it into a mixed-integer problem, plus things like keeping a second source qualified.

Writing it as a linear program

• Decision: $x_{ij}$, how many units plant $i$ sends to market $j$.
• Objective: minimize total landed cost, $\sum (c_i + t_{ij})\,x_{ij}$, where $c_i$ is the plant’s variable production cost and $t_{ij}$ the freight and duties to market $j$.
• Constraints: each market’s demand is met, $\sum_i x_{ij} \ge D_j$; each plant stays within capacity, $\sum_j x_{ij} \le K_i$; and quantities are never negative.

Spreadsheet solvers handle small cases; planning systems and optimization solvers handle thousands of products, plants and markets.

Landed costs per unit: Plant A to North ₹44, A to South ₹54, Plant B to North ₹58, B to South ₹46. Capacities: A 60,000, B 50,000.

Now North needs 70,000 and South 40,000: 110,000 in total, exactly the combined capacity.

• A can’t cover the whole North, so the cheapest plan is: A sends 60,000 North, B sends 10,000 North and 40,000 South.
• Total cost: 60,000 × ₹44 + 10,000 × ₹58 + 40,000 × ₹46 = ₹50.6 lakh.
• Any other split costs more. Sending 10,000 from A to the South and 20,000 from B to the North, for example, costs ₹52.8 lakh.

Shadow prices: what more capacity is worth

The solver also reports a shadow price for each constraint: how much the total cost would fall if that constraint were relaxed by one unit. Here, one more unit of capacity at Plant A would replace a unit B sends North at ₹58 with one from A at ₹44, so A’s capacity is worth ₹14 a unit. If overtime at A costs less than ₹14 a unit extra, it pays. That turns a cost model into a capacity-investment argument.

When the model needs integers

Real allocation decisions include yes-or-no choices: whether to run a line this month (with a fixed cost), whether a product’s run meets its minimum batch, whether a market must be served by a single plant. These turn the linear program into a mixed-integer program. Solvers handle them well at planning scale, but solution times grow, so models are kept as simple as the decision allows.

Which costs belong in the model

Use the costs that actually change with the decision. For a monthly allocation, that is variable production cost (materials, energy, variable labour), freight, duties and taxes. A plant’s “standard cost” often includes fixed overheads such as depreciation, management salaries and rent, which don’t change with this month’s volume. Allocating on full standard cost can send volume away from a plant that is actually cheaper at the margin. Fixed costs matter in longer-term decisions, such as closing a line or a plant, which belong to network design.

Adding time

A multi-period model plans several months together: build stock at a plant with spare capacity in quiet months to cover a peak, weighed against holding cost. This links plant allocation with the level-versus-chase decision in S&OP, and the same solver can answer both.

Constraints that aren’t about cost

• Regulatory approvals: in pharmaceuticals and food, a plant may be approved only for certain markets or products.
• Service: some customers need short delivery times only a nearby plant can meet.
• Risk: keeping a second plant qualified and producing at a minimum level protects supply if the main plant fails.
• Taxes and duties: cross-border flows can change the landed cost sharply, and rules change.
• Carbon: some companies add emissions limits or an internal carbon price to the objective.

Making the answer usable

Planners need to trust the model. Show the landed cost per lane, the binding constraints and their shadow prices, and how the plan changes under different demand scenarios. A model that only outputs numbers, without the reasons, tends to be overridden.