5 Optimization Basics for Supply Chains

What an optimization model represents

Supply-chain optimization selects decisions to meet a defined goal while respecting operational limits. Decisions can include how much to produce, where to hold inventory, which supplier to use, or how to route shipments. The model’s recommendation depends on its objective, constraints, and input data.

Variables, objectives, and constraints

A model usually has three main parts:

• Decision variables represent choices the model can make, such as the units shipped from each warehouse to each store.

• Objective function measures what the model seeks to minimize or maximize, such as total cost, delivery time, or profit.

• Constraints express requirements and limits, such as customer demand, supplier capacity, warehouse space, or a budget.

A Feasible solution satisfies every constraint. An Optimal solution is a feasible solution with the best objective value among all feasible choices.

A Linear program (LP) has a linear objective and linear constraints, with continuous decision variables. A standard form minimizes or maximizes a linear expression subject to linear equalities or inequalities and variable bounds.

A shipment-planning model

Consider two suppliers serving two distribution centers. Supplier 1 can provide up to 40 units and Supplier 2 up to 30 units. Center A requires 20 units, and Center B requires 50 units. The shipping costs per unit are as follows:

• Supplier 1 to Center A: 2 dollars; Supplier 1 to Center B: 4 dollars.

• Supplier 2 to Center A: 3 dollars; Supplier 2 to Center B: 1 dollar.

Let $x_{ij}$ be the quantity shipped from supplier $i$ to center $j$. The total transportation cost is minimized:

The supply limits and demand requirements are:

One minimum-cost plan ships 20 units from Supplier 1 to Center A, 20 units from Supplier 1 to Center B, and 30 units from Supplier 2 to Center B. This plan meets both demands and stays within both suppliers’ capacities. Its cost is $2(20)+4(20)+1(30)=150$ dollars.

This transportation LP uses shipment quantities as decision variables, shipping expense as the objective, and supply and demand requirements as constraints.

Model assumptions and decision types

A useful model depends on a sound formulation and suitable input data. Give each decision variable a clear unit, make sure each cost or capacity uses a matching unit, and represent each business requirement as an objective term or constraint. For example, use an equality when demand must be met exactly; use a greater-than-or-equal constraint when shipments may exceed a minimum requirement.

Basic LP assumes proportional, additive relationships. Shipping twice as much on a route doubles its modeled cost, and total cost is the sum of route costs. It also allows fractional quantities unless integrality is specified. This makes it suitable for divisible flows, but it may not suit choices such as opening a warehouse, selecting a carrier, or dispatching whole truckloads. Those choices may require integer or binary variables, producing an Integer or mixed-integer optimization model.

Validate the recommendation

Optimization supports operational judgment rather than replacing it. Forecasts, costs, lead times, and capacity data may be uncertain or out of date. Teams should validate a recommended plan, test how it changes under different assumptions, and account for service and risk as well as the numerical objective.

Supply-chain optimization translates operational choices into decision variables, a measurable objective, and constraints that represent real requirements and limits. Linear programming is a useful starting point for continuous allocation problems such as shipment planning, but the model’s data, units, and business rules should be checked before its recommendation is implemented.