Business Mathematics and Statistics · Ch 10 — Operations Research (Linear Programming Problem, Network Analysis)
Introduction to Operations Research and Linear Programming
Introduction to Operations Research and Linear Programming
Every earlier chapter of this course dealt with a single formula applied to a fixed set of numbers. Real management decisions are rarely that simple — a factory manager deciding how much of two products to make is constrained by limited machine hours, limited labour, and limited raw material all at once, and must find the single best combination among countless possible ones. Operations Research (OR) is the branch of applied mathematics that builds precise, quantitative models for exactly this class of decision problem, and this chapter covers its two best-known techniques: Linear Programming (allocating scarce resources optimally) and Network Analysis (planning and scheduling a project made of many interdependent activities).
A linear programming problem (LPP) seeks the maximum or minimum value of a linear function, subject to a collection of linear constraints in the same variables. Formulating an LPP from a word problem always follows the same four steps:
The Four Pieces of an LPP
- Decision variables — the unknown quantities actually being decided upon (conventionally and for two products).
- Objective function — the linear expression to be maximised (e.g. profit) or minimised (e.g. cost).
- Constraints — one linear inequality per limited resource or requirement: an at most statement becomes , an at least statement becomes .
- Non-negativity restrictions — , since a negative quantity of a physical product has no meaning.
This structured way of turning a business word problem into a precise mathematical statement — decision variables, objective function, constraints, non-negativity — is the universal first step of linear programming taught identically across Indian commerce and operations-research curricula.
The branch of applied mathematics that builds quantitative models to support optimal decision-making under limited resources, covering techniques including Linear Programming and Network Analysis.
A problem of maximising or minimising a linear objective function , subject to linear structural constraints and non-negativity restrictions, in two decision variables within this chapter's scope.
The linear expression an LPP asks to be maximised (e.g. profit) or minimised (e.g. cost).
The decision variables () are the quantities a business controls; the non-negativity restrictions reflect that a negative quantity of a physical good or resource has no meaning.