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Linear Programming: Model Formation, Graphical Solution, and Sensitivity Analysis, Study notes of Introduction to Business Management

Virginia Polytechnic Institute and State University (Virginia Tech)Introduction to Business Management

This study guide covers the fundamentals of linear programming, including model formation, graphical solutions, and sensitivity analysis. Topics include objective functions, decision variables, constraints, feasible and infeasible solutions, slack and surplus variables, and sensitivity analysis for objective function coefficients and constraint quantity values. The guide also introduces the Simplex method and shadow prices.

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Fall 2007 BIT 2406
Exam 2
Study Guide
Chapter 2
Linear Programming: Model Formation and Graphical Solution
oObjectives of a business frequently are to maximize profit or minimize
cost
oLinear Programming - model that consists of linear relationships
representing a firm's decision (s), given an objective and resource
constraints
•Model Formation
oModel components include decision variables, an objective function,
and model constraints
oDecision variables - mathematical solutions that represent levels of
activity
oObjective function - linear relationship symbols that represent levels of
activity
•Always consists of either maximizing or minimizing some value
oConstraint - linear relationship that represents a restriction on decision
making
•Restrictions can be in the form of limited resources or
restrictive guidelines
oParameters - numerical values that are included in the objective
functions and constraints
oModel constraints
•Nonnegativity restraints - restrict the decision variables to zero
or positive variables
X1 ≥ 0, X2 ≥ 0
•Feasable solution - does not violate any of the constraints
•Infeasable problem - violates at least one of the constraints
•Graphical Solutions of Linear Programming Models
oGraphical solutions are limited to linear programming problems with
only two decision variables
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Fall 2007 BIT 2406

Exam 2

Study Guide Chapter 2 Linear Programming: Model Formation and Graphical Solution o (^) Objectives of a business frequently are to maximize profit or minimize cost o (^) Linear Programming - model that consists of linear relationships representing a firm's decision (s), given an objective and resource constraints

  • (^) Model Formation o (^) Model components include decision variables, an objective function, and model constraints o (^) Decision variables - mathematical solutions that represent levels of activity o (^) Objective function - linear relationship symbols that represent levels of activity - Always consists of either maximizing or minimizing some value o (^) Constraint - linear relationship that represents a restriction on decision making - Restrictions can be in the form of limited resources or restrictive guidelines o (^) Parameters - numerical values that are included in the objective functions and constraints o (^) Model constraints - Nonnegativity restraints - restrict the decision variables to zero or positive variables  X1 ≥ 0, X2 ≥ 0 - Feasable solution - does not violate any of the constraints - (^) Infeasable problem - violates at least one of the constraints
  • Graphical Solutions of Linear Programming Models o (^) Graphical solutions are limited to linear programming problems with only two decision variables

o (^) The graphical method provides a picture of how a solution is obtained for a linear programming problem o (^) Graphical solution of a Maximization Model

  • Constraint lines are plotted as equations  X1 + X2 = 40
  • The feasable solution area is an area on the graph that is bounded by the constrain equations
  • (^) Optimal solution point - best feasable solution  point in the feasable solution area that will result in the greatest total profit  Graph from pg. 40
  • (^) The solution values  The optimal solution point is the last point the objective function touches as it leaves the feasable solution area  Graph from pg. 41  Extreme points - corner points on the boundary of the feasable solution area  Constraint equations are solved simultaneously at the optimal extreme point to determine the variable solution values  Slope - computed as the "rise" over the "run"  Sensitivity analysis - used to analyze changes in the model parameters  Multiple optimal solution - can occur when the objective function is parallel to a constraint line

o (^) An unbounded problem

  • In an unbounded problem the objective function can increase indefinitely without reaching a maximum value
  • (^) The solution space is not completely closed in
  • Graph from pg. 55
  • (^) Characteristics of Linear Programming Problems o (^) The component of a linear programming model are an objective function, decision variables, and constraints o (^) Properties of linear programming models
  • Proportionality means the slope of a constraint or objective function line is constant
  • (^) The terms in the objective function or constraints are additive
  • Values of decision variables are continuous or divisible
  • (^) All model parameters are assumed to be known with certainty Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis o (^) Simplex method - procedure involving a set of mathematical steps to solve linear programming problems o (^) Marginal value - dollar amount one would be willing to pay for one additional resource unit
  • Sensitivity analysis o (^) Sensitivity analysis - analysis of the effect of parameter changes on the optimal solution

o (^) Changes in objective function coefficients

  • Sensitivity range for an objective coefficient is the range of values over which the current optimal solution point will remain optimal
  • (^) Increasing the x1 coefficient makes the objective line steeper, so much that the optimal solution point changes from B to C
  • (^) Increasing the x2 coefficient make the objective line flatter, to the extent that point A would become optimal o (^) Changes in constraint quantity values
  • (^) Sensitivity range for a right-hand-side value is the range of values over which the quantity values can change without changing the solution variable mix, including slack variables o (^) Other forms of sensitivity analysis
  • Changing constraint value parameter values
  • Adding new constraints
  • Adding new variables o (^) Shadow prices
  • Shadow price (dual value) - marginal value of one additional unit of resource
  • (^) Sensitivity range for a constraint quantity is also the range over which the shadow price is valid Chapter 4 Linear Programming: Model Examples ** SEE BOOK FOR EXAMPLES OF DIFFERENT LINEAR PROGRAMMING MODELS** Chapter 5 Integer Programming

o (^) A rounded-down integer solution can result in a less than optimal (suboptimal) solution Chapter 6 Transportation, Transshipment, and Assignment Problems

  • The Transportation Model o (^) Transportation problem - items are allocated from sources to destinations at minimum cost o (^) The linear programming model for a transportation problem has constraints for supply at each source and demand at each destination o (^) Balanced transportation model - supply equals demand, all constraints are equalities o (^) Unbalanced transportation model - supply is greater than demand or demand is greater than supply
  • (^) The Transshipment Model o (^) Transshipment model - includes intermediate points between sources and destinations - (^) Example of a transshipment point is a distribution center or a warehouse
  • (^) The Assignment Model o (^) Assignment model - for a special form of transportation problem in which all supply and demand values equal 1