Introduction to Management Science, 10
th
Edition by Bernard W. Taylor III ISOM 3123 (Spring 2013)
Model Formation
Decision Variables

Decision variables
are mathematical symbols that represent levels of activity.
Objective Function

The
objective function
is a linear relationship that reflects the objective of an operation.
Constraint

A
constraint
is a linear relationship that represents a restriction on decision making.
Parameters

Parameters
are numerical values that are included in the objective functions and constraints.
A Maximization Model Example
Step 1: Define the decision variables
Step 2:
Define the objective function
Step 3: Define the constraints
Nonnegativity Constraints

Nonnegativity constraints
restrict the decision variables to zero or positive values.
Feasible Solution

A
feasible solution
does not violate any of the constraints.
I
nfeasible Problem

An
infeasible problem
(solution) violates at least one of the constraints.
Graphical Solution to a Maximization Model
Graphical solutions are limited to linear programming problems with only two decision variables.
The graphical method p
rovides a picture of how a solution is obtained for a linear programming problem.
Constraint lines are plotted as (linear) equations.
The feasible solution area is an area on the graph that is bounded by the constraint equations.
Optimal Solution

The
optimal solution
is the best feasible solution.
The optimal solution point is the last point the objective function touches as it leaves the feasible solution
area.
Extreme Points

Extreme points
are corner points on the boundary of the feasible solution a
rea.
Constraint equations are solved simultaneously at the optimal extreme point to determine the variable
solution values.
Slope

The
slope
is computed as the “rise” (vertical) over the “run” (horizontal).
Sensitivity Analysis

Sensitivity analysis
is use
d to analyze changes in model parameters.
Multiple Optimal Solutions

Multiple optimal solutions
can occur when the objective function is parallel to a
constraint line.
A slack variable is added to a ≤
constraint to convert it to an equation (=).
Slack Var
iable

A
slack variable
represents unused resources.
A slack variable contributes nothing to the objective function value.
Introduction to Management Science, 10
th
Edition by Bernard W. Taylor III ISOM 3123 (Spring 2013)
A Minimization Model Example
Step 1: Define the decision variables
Step 2: Define the objective function
Step 3: Define the
constraints
The three types of linear programming constraints are ≤, =, and ≥.
The optimal solution of a minimization problem is at the extreme point closest to the origin.
A surplus variable is subtracted from a ≥ constraint to convert it to an equation.
Surplus Variable

A
surplus variable
represents an excess above a constraint level.
Irregular Types of Linear Programming Problems
For some linear programming models, the general rules do not always apply.
Alternate Optimal Solutions

Alternate optimal sol
utions
are at the endpoints of the constraint line segment
that the objective function parallels.
Multiple optimal solutions provide greater flexibility to the decision maker.
Infeasible Problem

An
infeasible problem
has no feasible solution area; every possible solution point violates
one or more constraints.
Unbounded Problem

In an unbounded problem the objective function can increase indefinitely without
reaching a maximum value. The solution space is not complet
ely closed in.
Characteristics of Linear Programming Problems
The components of a linear programming model are an objective function, decision variables, and
constraints.
Proportionality

Proportionality
means the slope of a constraint or objective functio
n line is constant.
Additivity

The terms in the objective function or constraints are
additive
.
Divisibility

The values of decision variables are continuous or
divisible
.
Certainty

All model parameters are assumed to be known with
certainty
.
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