Unit 10 – Linear Inequalities & Linear Programmin: Basics of Statistics and Mathematics

Vocabulary flashcards covering key terms from Unit 10 on Linear Inequalities, Linear Programming, and the Simplex Method.

Inequality

A mathematical statement relating two numbers or expressions with

Strict Inequality

An inequality that uses < or >, indicating the two sides are never equal.

Slack Inequality

An inequality that uses ≤ or ≥, allowing the two sides to be equal as well as unequal.

Linear Inequality in One Variable

An inequality of the form ax + b < 0, > 0, ≤ 0, or ≥ 0 where a ≠ 0 and x is the only variable.

Linear Inequality in Two Variables

An inequality of the form ax + by < c, > c, ≤ c, or ≥ c with a and b not both zero.

Solution of an Inequality

Any value of the variable(s) that makes the inequality a true statement.

Solution Set

The complete set of all values that satisfy a given inequality or system of inequalities.

Feasible Solution

A set of decision-variable values that satisfies every constraint of a linear programming problem.

Optimal Solution

A feasible solution that yields the best (maximum or minimum) value of the objective function.

Decision Variable

A variable whose value is to be determined in order to optimize the objective function in an LP model.

Objective Function

The linear function of decision variables that is to be maximized or minimized in linear programming.

Constraint

A linear equality or inequality that limits the values the decision variables may assume.

Non-Negativity Restriction

Requirement that all decision variables must be zero or positive, reflecting real-world quantities.

Linear Programming (LP)

A mathematical technique for optimizing a linear objective function subject to linear constraints.

Standard Form of an LPP

A form in which the objective function is maximization, all constraints are equalities, and variables are non-negative.

Simplex Method

An iterative algorithm that moves from one basic feasible solution to another to find the optimal LP solution.

Slack Variable

A non-negative variable added to a ≤ constraint to convert it into an equality in standard form.

Surplus Variable

A non-negative variable subtracted from a ≥ constraint to convert it into an equality.

Reduced Cost

For a variable at zero level, the amount its objective coefficient must improve before it can enter the optimal basis.

Dual Price (Shadow Price)

The improvement in the objective value per one-unit relaxation of a binding constraint.

Feasible Region

The set of all points that simultaneously satisfy every constraint in a linear programming model.

Basic Solution

A solution obtained by setting n − m variables to zero (where m is number of constraints) so that the remaining variables solve the equalities.

Basic Feasible Solution

A basic solution that also satisfies all non-negativity restrictions, making it feasible.

Pivot Column

In the simplex tableau, the column with the most positive (for maximization) objective coefficient, indicating entering variable.

Pivot Row

The row with the smallest non-negative ratio of rhs to the pivot column entry, indicating leaving variable.

Pivot Element

The entry at the intersection of the pivot row and column, used to perform row operations in simplex.

Graphical Method

LP solution technique that plots constraints, finds the feasible region, and evaluates the objective at its corner points.

Dual Problem

The LP problem derived from the primal, interchanging roles of constraints and variables while reversing optimization direction.