Objectives, Decision Variables, Constraints, and Feasibility

A packaging plant runs two products on shared equipment. Every pallet of product 1 consumes 2 machine-hours and 1 labour-hour; every pallet of product 2 consumes 1 machine-hour and 2 labour-hours. Today the plant has 8 machine-hours and 8 labour-hours available, product 1 earns $40 per pallet, and product 2 earns $30 per pallet. The plant manager does not want a prediction of tomorrow's demand. She wants an answer to a decision question: how many pallets of each product should we build with the hours available?

Answering that question requires three separate pieces of work. First we must translate the situation into mathematics: name the quantities we control, write an expression that scores each alternative, and state the rules alternatives must obey. This is formulation. Second we search for a good or provably best alternative, by hand on a small problem or with a solver on a larger one. This is optimization. Third we independently check the returned decision against the model and operational requirements. This is solution validation. A solver status describes its numerical search; it cannot establish that the model represents the plant correctly.

This chapter introduces the formulation vocabulary used throughout constrained optimization: decision variables, domains, objectives, hard constraints, soft penalties, and feasible sets. Factory planning, minimum-cost flow, portfolio optimization, rostering, and routing all require these choices. Their variable domains and constraint functions differ; they are not interchangeable coefficient matrices.

Consider how this differs from unconstrained least-squares fitting in Linear and Generalized Linear Models and Gradient Descent and Numerical Optimization. In that fitting problem the decisions are coefficients $\theta$, the objective measures residual error, and the domain permits any real coefficient vector. Operational optimization instead chooses actions such as production quantities, with explicit resource limits and a value assigned to each action. Fitting can also include constraints, so the distinction is the meaning of the decisions and objective rather than whether constraints exist. A good prediction alone does not specify a feasible production plan.