Mathematical optimization in supply chain planning finds the plan that scores best on one objective, such as total cost, while meeting every constraint the model states. “Optimal” has a precise meaning here: no other feasible plan scores better, and the solver can prove it or report how far its best plan could be from the best possible one. That proof separates an optimized plan from a plan that is merely better than last week’s.
The Three Parts of Mathematical Optimization in Supply Chain
Every optimization model, from a five-year network study to tomorrow’s replenishment run, is built from the same three parts.
- Decision variables: the quantities the model is free to choose. How many cases plant A ships to DC 3 next week, whether a DC is open, how many units line 2 produces on Tuesday.
- Constraints: the limits every plan has to respect. Plant capacity, DC throughput, truck size, shelf life, minimum order quantities, service level targets and the basic rule that stock in equals stock out plus what stays on hand.
- Objective: the single number the model tries to make as small or as large as possible. Usually total cost at a required service level, sometimes margin, sometimes a weighted mix of goals.
Take a simple case: two plants supplying three DCs. The variables are the weekly flows on each of the six lanes and the production volume at each plant. The constraints say each plant cannot exceed its capacity and each DC must receive what its demand requires. The objective adds up production and freight cost. A linear programming (LP) solver will return the cheapest set of flows that satisfies every DC, and it will return it in a fraction of a second.
Real models add yes-or-no choices (open this DC or not, run this changeover or not, send a full truck or nothing). Those make it a mixed integer program (MIP), which is harder to solve. Our post on linear vs nonlinear optimization covers how quantity discounts, freight breaks and other non-straight-line costs get into these models.
What Optimal Means in Mathematical Optimization
A plan is feasible if it breaks no constraint. It is optimal if it is feasible and no other feasible plan has a better objective value. The word gets used loosely in sales decks and meetings for “good,” “feasible” or “better than what we did before,” and those plans may well be useful, but they are not optimal in the mathematical sense.
Two qualifiers matter in practice.
- Optimal is always relative to the model. If the model leaves out a real constraint, such as a dock that can only unload four trucks an hour, the optimal plan can be one nobody can execute. If the objective prices a stockout too cheaply, the optimal plan will run lean. Model quality decides whether optimal on paper is good in the warehouse.
- For LP models, a solver finds the global optimum and can prove it. For MIP models it searches a tree of possibilities and proves optimality by showing that no unexplored branch can beat the best plan found so far. On large problems that proof can take longer than the planning window allows, which is where the optimality gap comes in.
What Is the Optimality Gap?
While a MIP solver works, it tracks two numbers. The incumbent is the best feasible plan it has found so far. The bound is a value no feasible plan can beat, usually obtained by relaxing the yes-or-no choices into fractions and solving the easier problem. The true optimum sits somewhere between them.
The optimality gap (often called the MIP gap) is the distance between the two, divided by the incumbent. Suppose the best plan found costs 1,000,000 and the bound says no plan can cost less than 990,000. The gap is 1%. In plain terms: the solver cannot promise this is the cheapest plan, but it can promise that the cheapest plan saves at most 10,000 more.
| Term | What it means | What a planner can read from it |
|---|---|---|
| Feasible plan | Breaks no constraint | It can be executed, if the model is right |
| Incumbent | Best feasible plan found so far | The plan you would run if the solver stopped now |
| Bound | A value no feasible plan can beat | The best result any plan could reach |
| Optimality gap | Distance between incumbent and bound, relative to the incumbent | The most that is still left on the table |
| Optimal | Gap closed to zero, or below the tolerance set for the run | No better plan exists for this model |
Solvers stop when the gap falls below a tolerance. A common default is 0.01%. For a large daily run a team can loosen it, to 0.5% or 1% for example, so the plan arrives on time. A heuristic can find a good plan quickly too, but it produces no bound, so nobody knows how much it leaves behind.
Why Proven-Optimal Matters to Planners
The gap gives a planner a number to check and defend, which a plan from a spreadsheet or a heuristic cannot offer.
- It sizes the opportunity. A plan with a 0.2% gap is close enough that a planner should not spend the afternoon looking for a better one by hand. A plan with a 12% gap says the problem needs more time, a better formulation or a different decomposition.
- It makes scenarios comparable. When two network or capacity scenarios are each solved to a tight gap, the cost difference between them comes from the business, not from the solver stopping early on one of them.
- It explains the limits. An LP model, or a MIP re-solved with its yes-or-no choices fixed, can report which constraints are binding and what each one costs. If the plan is expensive because one DC is at its throughput limit, the model can show how much an extra shift there would be worth.
- It prices overrides. When a planner changes a recommended plan, the model can re-solve with the change fixed and show what it costs. Some overrides are worth it, and the number makes that conversation short.
Optimization Beyond Network Design
For a long time the classic home of supply chain optimization was strategic network design: a yearly or one-off study to decide where plants and DCs should sit and which customers each one serves. That work still pays off, and our post on strategic network design and scenario planning covers it.
The same mathematics applies to tactical and operational decisions, which repeat every week. Our post on strategic, tactical and operational supply chain planning covers how the model moves down those levels.
How Solvoyo Approaches Optimization
Solvoyo, a supply chain decision automation platform, combines a proprietary mathematical optimization engine with commercial solvers, goal programming and heuristics, and it solves demand, inventory, production, replenishment, fulfillment, transportation and S&OP together so every constraint holds at the same moment.
The models run automatically on a daily or weekly schedule, or on demand, and re-run when a forecast or parameter changes. Because each plan meets every constraint and its gap is known, a planner can see how little a manual search could still find, and over 95% of Solvoyo’s plans execute untouched. See it in supply planning and strategic network design, or request a demo.
Frequently Asked Questions
A plan is optimal when it meets every constraint in the model and no other plan that meets those constraints has a better objective value, such as lower total cost at the required service level. The term is precise: a plan that is feasible or better than last month is not optimal unless the solver can show that nothing better exists.
The optimality gap, often called the MIP gap, is the distance between the best plan a solver has found and a bound that no feasible plan can beat, divided by the value of the best plan. A 1% gap on a plan costing 1,000,000 means the best possible plan saves at most 10,000 more. Solvers stop when the gap falls below a set tolerance.
Only if the model is right. Optimal is always relative to the model, so a missing constraint or a badly priced objective can produce an optimal plan that cannot be executed or that runs too lean. Good optimization depends on capturing real capacities, costs and service targets, and on clean data every run.
