Most supply chain planning models run on linear programming (LP) and mixed integer programming (MIP), because both solve quickly at scale and can prove how close a plan is to the best one. Many real supply chain costs are nonlinear, though: freight rates drop at volume breaks, purchase prices fall with quantity, and safety stock climbs steeply as service targets approach 100%. Linear vs nonlinear optimization is a trade-off between solve speed and cost accuracy. Good planning models keep the speed of a linear model, represent those curves accurately where they change the decision, and solve network flows and planning decisions in the same model.
Linear vs Nonlinear Optimization: What Each Means in a Supply Chain Model
In a linear model, every cost and constraint is a straight-line function of the decisions. Shipping 200 pallets on a lane costs twice as much as shipping 100, and a warehouse has a fixed capacity ceiling. A mixed integer model adds whole-number and yes-or-no decisions: open a distribution center or not, send three trucks or four, run a production batch of at least the minimum size.
A nonlinear model allows costs and constraints to curve. The usual sources in supply chain planning are:
- Transportation: cost per unit falls as a shipment moves from parcel to less-than-truckload to a full truck, and the steps are not proportional.
- Purchasing: all-units and incremental quantity discounts lower the unit price once an order crosses a threshold.
- Inventory: under common assumptions, safety stock grows with the square root of lead time and in proportion to demand variability. Each extra point of service level costs more stock than the one before it.
- Production: changeovers add a fixed cost per batch, so the cost per unit depends on batch size.
- Warehousing: handling cost per unit tends to rise when a site runs close to its capacity, because overtime and overflow space cost more.
Why Do Planning Models Stay Linear?
Linear programs solve quickly even at very large scale, and the solver returns a provable optimum. Mixed integer programs are harder, but solvers still report an optimality gap: a bound on how far the current plan can be from the best possible one. Planners can act on a plan when they know that number.
General nonlinear problems lose that guarantee. Volume discounts and economies of scale are nonconvex, because unit cost falls as volume rises, and nonconvex problems can have many local optima. A nonlinear solver can stop at a plan that only looks best from where it started, and it cannot always say how much better the true answer is.
That is why planning tools have often ignored the curves. They used an average freight rate per unit, a single discount tier, or a safety stock figure calculated in a spreadsheet after the plan was built. Each shortcut moves the cost of the nonlinearity somewhere else. An average freight rate hides the saving from consolidating orders into full trucks. A fixed safety stock number ignores how buffers change when flows change.
How Nonlinear Costs Get Into a Linear Model
The practical answer is to keep a linear or mixed integer core and represent the nonlinear parts carefully. Four techniques do most of the work:
- Piecewise linear approximation splits a curve into straight segments. In a cost-minimizing model, convex curves, such as rising handling cost near capacity, fit without extra variables. Nonconvex curves, such as discounts, need binary variables that pick the active segment.
- Integer variables handle step costs. Truck counts, fixed facility costs, minimum order quantities and batch sizes become integer or binary decisions, so the model sees the real cost of one more truck.
- Precomputed trade-off curves cover inventory. The safety stock needed for each service level is calculated per item and location from forecast error and lead time variability, then passed to the optimizer as cost segments or constraints.
- Iteration links the pieces: the model solves flows, recalculates inventory targets for the new flows, and solves again until the plan stops changing.
Each technique adds accuracy at a price. More segments and more binary variables mean a larger model and longer solve times. Deciding which curves matter enough to model in detail, and which can stay linear, is the main judgment in building a planning model.
Why Network Flow Belongs Underneath Planning Decisions
Most planning decisions can be modeled as flows through a network. Store replenishment is flow from a distribution center to a store. A production plan is flow from a line to a warehouse. A purchase order is flow from a supplier into the network. The network model already describes the lanes, capacities and costs each of those flows uses.
Network flow models have usually been used for strategic design, run once or twice a year to decide where facilities should sit. Planning then happens in separate tools that treat sourcing and lanes as fixed inputs. The gaps show up in daily operations: a replenishment plan that meets store demand but sends half-empty trucks, or a production plan that fits the lines but supplies a region from the wrong plant.
When the flow model sits under the planning decisions, every plan is checked against the same network. Take inbound order frequency to a distribution center. Fewer, larger orders cut freight cost per unit, which is a nonlinear effect, but they raise cycle stock and use more storage. The right frequency depends on both sides at once, so it has to be found in one model.
Where Solving It Together Changes the Answer
Three planning decisions come out differently when flows and nonlinear costs are solved together:
- Store replenishment and truck loading. Rounding store quantities to fill trucks can cost less than shipping exact quantities in partial loads, as long as the extra stock fits the store and stays within its shelf life.
- Safety stock across echelons. Holding buffer stock at a distribution center pools variability across the stores it serves, so total stock falls. How much it falls depends on which stores the center supplies, which is a flow decision.
- Production and distribution. Larger batches cut changeover cost but can add transport cost if the plant that makes the batch is far from the demand. Batch size and sourcing have to be set together.
A sequential process sets the first decision and hands it on as fixed, so the second step can only work around it.
How Solvoyo Approaches Linear and Nonlinear Planning
Solvoyo, a supply chain decision automation platform, was built by operations research scientists, and its proprietary mathematical optimization engine weighs Total Cost to Serve against service levels in one model. Supply planning runs on a network model that includes plant and distribution center capacities, multiple transport modes, tariffs and shelf life. Its capacitated DRP weighs space, flow constraints and cost alongside demand. Inventory planning sets safety stock and multi-echelon targets from the service level strategy, using actual forecast error and lead time variability. Strategic network design uses the same Total Cost to Serve logic to compare network scenarios before capital is committed.
A model that prices freight breaks, discounts and buffers correctly produces plans planners have little reason to change, and across Solvoyo deployments over 95% of plans execute untouched. For more on the network side, read our posts on network design for Total Cost to Serve and CO2 and strategic network design and scenario planning, or request a demo to see the engine on your own network.
Frequently Asked Questions
In linear optimization, costs and constraints change in direct proportion to the decisions, so twice the volume costs twice as much. Nonlinear optimization allows curved relationships, such as freight rates that fall at volume breaks, quantity discounts, and safety stock that rises steeply as service targets increase. Linear models solve faster and prove optimality; nonlinear models describe real costs more closely.
Yes, with approximation. Volume discounts and step costs are modeled with piecewise linear segments and binary variables that select the active price tier. Safety stock is usually calculated per item and location from forecast error and lead time variability, then passed to the model as cost segments or constraints. More segments improve accuracy but increase solve time.
Every replenishment, production and purchasing decision moves goods through the same lanes and capacities. When planning runs separately from the network model, each step treats earlier decisions as fixed, which leads to half-empty trucks, buffers in the wrong place, or the wrong plant supplying a region. Solving them together lets the model trade freight, inventory and capacity costs against each other.
