# Optimization and Objective Function

An Introduction to

Linear Programming

21

KEY CONCEPTS

ILLUSTRATED ANSWERED CONCEPT PROBLEMS PROBLEMS

Formulation 7,8 15,16

Minimization 2,5 9,10,15

Standard Form 1 14

Slack/Surplus Variables 1 16

Equal-to Constraints 3,5 14

Redundant Constraints 5,7 12,13

Extreme Points 2,5 11,16

Alternative Optimal Solutions 7 10,11,15

Infeasibility 4 14

Unbounded 4 11

Spreadsheet Example2

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1. A mathematical programming problem is one that seeks to maximize an objective function subject to constraints. If both the objective function and the constraints are linear, the problem is referred to as a linear programming problem.

2. Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0).

3. Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant.

4. The maximization or minimization of some quantity is the objective in all linear programming problems.

5. A feasible solution satisfies all the problem's constraints.

6. A linear program which is overconstrained so that no point satisfies all the constraints is said to be infeasible. Changes to the objective function coefficients do not affect the feasibility of the problem.

7. An optimal solution is a feasible solution that results in the largest possible objective function value, z, when maximizing or smallest possible z when minimizing.

8. A graphical solution method can be used to solve a linear program with two variables.

9. If a linear program possesses an optimal solution, then an extreme point will be optimal.

10. If a constraint can be removed without affecting the shape of the feasible region, the constraint is said to be redundant. If changes are anticipated to the linear programming model, constraints which were redundant in the original formulation may not be redundant in the revised formulation.

11. In the graphical method, if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternative optimal solutions, with all points on this line segment being optimal.

12. A feasible region may be unbounded and yet there may be optimal solutions. This is common in minimization problems and is possible in maximization problems.

13. The feasible region for a two-variable linear programming problem can be: a) nonexistent, b) a single point, c) a line, d) a polygon, or e) an unbounded area.

14. Any linear program either (a) is infeasible, (b) has a unique optimal solution or alternate optimal solutions, or (c) has an objective function that can be increased without bound.

15. A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in standard form. Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to"...

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