Placing stock across a food-distribution network

The cost of playing it safe

A food distributor's inventory decisions ripple into centre capacity, transport cost, stock-outs and overstock all at once. A conservative policy — keep plenty of every SKU close to where it sells — protects service but fills storage, multiplies safety stock, and moves goods between centres that did not need to hold them.

After interviews with the business on ordering logic, inventory policy, capacity and transport conditions, the objective was set as improving capacity utilization and reducing transport cost while keeping supply stable. That made it an optimization problem rather than a forecasting one: given what each centre is likely to sell, where should stock live?

Three decisions, one model

For every SKU the model decides together:

• Storage — which centres hold it, up to a limit per SKU; centres differ by site and storage condition, and some can only receive.
• Sourcing — which storage centre serves each demand point, one source per point, avoiding lanes that cannot carry the item.
• Size — how many whole pallets each storage centre keeps: lead-time demand plus a safety buffer of three standard deviations, never below the supplier's minimum order, and within the centre's capacity.

The objective is transport cost over four weeks plus a weighted holding cost per pallet. The interesting part is the safety buffer. When a hub serves several demand points, their uncertainties pool: the buffer grows with the square root of the summed variances, not with their sum. Consolidating a SKU into fewer hubs therefore saves pallets even when it adds transport — and the model has to weigh exactly that trade, SKU by SKU, under shared capacity.

y_icj ∈ {0,1}   centre c supplies demand point j with item i      (single source)
z_ic  ∈ {0,1}   item i stocked at centre c          p_ic ∈ ℤ₊   pallets

min   Σ cost_cj · w_i · μ_ij · H · y_icj   +   w_s · Σ p_ic
s.t.  Σ_c y_icj = 1,       y_icj ≤ z_ic,       Σ_c z_ic ≤ K
      S_ic² = LT_i · Σ_j σ_ij² · y_icj                         pooled safety stock
      R_ic ≥ LT_i · Σ_j μ_ij · y_icj + 3·S_ic,      R_ic ≥ MOQ_i · b_i · z_ic
      b_i·p_ic ≥ R_ic,     b_i·p_ic ≤ R_ic + b_i − 1           p = ⌈R / b⌉
      Σ_i p_ic ≤ cap_c

Inside the model

A square root inside an integer program

Written directly, the pooled buffer is a quadratic equality — the squared buffer equals the lead-time-weighted sum of variances of the points assigned to the hub — which makes the model a non-convex mixed-integer quadratically constrained program. It was solved with SCIP, which branches on the non-convexity; the same constraint can also be written as a second-order cone, which keeps it convex and is the direction for scaling up. Linking constraints tie assignment to storage, force the rounded-up pallet count, and enforce capacity and self-supply.

Methods compared

The study did not assume the MIP was the answer. Mathematical programming, search-based optimization and reinforcement learning were reviewed against each other, and a simulation structure was built to evaluate how a change in policy moves capacity, cost and service level — so the business could test alternatives rather than accept a single number.

How the demo solves it

The live model keeps the same ingredients but solves a small network a different way: for each SKU it enumerates every feasible set of storage centres, routes each demand point to its cheapest open hub, and sizes the pooled pallets; a small integer program then picks one plan per SKU subject to centre capacity. For a network this size that gives the same optimum in milliseconds, and it makes the trade-off visible.

What it showed

As a feasibility study, the result was a case rather than a deployment: the optimization confirmed room to secure centre capacity and cut transport cost, and it proposed moving from conservative stocking to demand-based inventory operation. Just as important, it left a decision environment — data marts joining inventory, demand and capacity, plus the model and simulation — in which inventory policy alternatives can be tested before they are tried.

Limitations

• This was a feasibility study; the policy was proposed, not rolled out, and no operational savings are claimed here.
• Single sourcing per demand point and a fixed service factor (three standard deviations) are modelling choices, not business constraints.
• The demand statistics are recency-weighted history; there is no forecast model for promotions or new items.
• The live model uses plan enumeration and a linear integer program instead of the original non-convex solver; it is exact only because the network is small.

About the demo and confidentiality

The centres, SKUs, demand histories, lanes and costs in the embedded model are generated from a seed. No network layout, item master, cost table or result from the real study appears here.