Model Formation Decision Variables - Decision variables are ...

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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
.