Linear Programming

Mathematical Modeling · Axiom Academy

LESSON Linear Programming - Mathematical Modeling Optimization with Linear Objectives and Constraints Linear Programming (LP) is one of the most powerful and widely-used optimization techniques in mathematical modeling. It helps us find the best outcome when: Resources are limited: Budget, time, materials, labor Decisions must be made: How much to produce, where to allocate resources Relationships are linear: Costs, profits, and resource usage scale proportionally Applications span every industry: supply chain optimization, production planning, portfolio optimization, transportation logistics, workforce scheduling, and more. The Standard Form of Linear Programming A linear programming problem consists of three components: decision variables, an objective function, and constraints. A linear programming problem in standard form is: We can express LP problems compactly using matrices and vectors: is the vector of decision variables is the objective coefficient vector Example 1: Production Planning Problem: A furniture company makes chairs and tables. Each chair requires 2 hours of carpentry and 1 hour of finishing. Each table requires 3 hours of carpentry and 2 hours of finishing. The company has 120 hours of carpentry time and 80 hours of finishing time available per week. Chairs sell for 40 profit and tables for 60 profit. How many of each should be produced to maximize profit? Let x_1 = number of chairs, x_2 = number of tables Geometric Interpretation: The Feasible Region

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.