What is optimization?
Optimization, in the mathematical sense, is the method of finding the best values for a set of decisions given a goal and a set of limits. Every optimization problem has three parts. Decision variables are the quantities to be chosen, such as how many units to ship from each warehouse to each store. The objective function is the measure to maximize or minimize, such as total transport cost. Constraints are the conditions any solution must satisfy, such as warehouse capacity and store demand. A solver searches the space of feasible solutions for the one with the best objective value.
When the objective and constraints are linear, the problem is a linear program, which solvers handle at enormous scale. When some variables must be whole numbers (a company cannot open half a distribution center or dispatch a third of a truck), it becomes an integer or mixed-integer program, which is harder but routinely solved at large scale.
Why optimization matters
Optimization replaces decisions made by judgment and spreadsheet with decisions made consistently against stated goals and rules. It is the engine behind production scheduling, vehicle routing, inventory planning, staff rostering, network design, and pricing. Where machine learning predicts what will happen, optimization decides what to do about it, which is why the two are paired in prescriptive analytics: a forecast model supplies the demand estimate and the optimizer chooses the replenishment plan.
Standard optimization assumes its inputs are known. Real demand, lead times, and prices are uncertain, and a plan that is optimal for the forecast can be fragile when the forecast is wrong. Stochastic optimization addresses this by modeling the uncertainty directly and finding plans that perform well across many scenarios rather than best in one.
Optimization in practice
A beverage distributor redesigns its delivery network with a mixed-integer model. Decision variables cover which of twelve candidate depots to operate, which customers each depot serves, and the fleet assigned to each. Constraints include depot capacity, driver hours, delivery windows, and a service-level requirement for key accounts. The objective is total annual cost. The solver evaluates combinations no planning team could enumerate, and the team stress-tests the result against high and low demand scenarios before committing.
What to watch for
- State the objective precisely, since a solver optimizes exactly what it is given and will exploit anything left out.
- Check the constraints with the people who run the operation, because an unrealistic constraint produces a plan nobody can execute.
- Test the solution against uncertainty in the inputs, or use stochastic methods when the inputs are volatile.
- Prefer a good solution delivered in time over a provably optimal one delivered too late.