CHAPTER Linear Programming McGraw-Hill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright © 2005 by The McGraw-Hill Companies, Inc.
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Transcript CHAPTER Linear Programming McGraw-Hill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright © 2005 by The McGraw-Hill Companies, Inc.
CHAPTER
19
Linear
Programming
McGraw-Hill/Irwin
Operations Management, Eighth Edition, by William J. Stevenson
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Linear Programming
Used to obtain optimal solutions to problems
that involve restrictions or limitations, such
as:
Materials
Budgets
Labor
Machine time
Linear Programming
Linear programming (LP) techniques
consist of a sequence of steps that will lead
to an optimal solution to problems, in cases
where an optimum exists
Linear Programming Model
Objective: the goal of an LP model is maximization
or minimization
Decision variables: amounts of either inputs or
outputs
Feasible solution space: the set of all feasible
combinations of decision variables as defined by the
constraints
Constraints: limitations that restrict the available
alternatives
Parameters: numerical values
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Linear Programming Model
Objective: the goal of an LP model is maximization or
minimization
Decision variables: amounts of either inputs or
outputs
Feasible solution space: the set of all feasible
combinations of decision variables as defined by the
constraints
Constraints: limitations that restrict the available
alternatives
Parameters: numerical values
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Linear Programming Model
Objective: the goal of an LP model is maximization or
minimization
Decision variables: amounts of either inputs or
outputs
Feasible solution space: the set of all feasible
combinations of decision variables as defined by the
constraints
Constraints: limitations that restrict the available
alternatives
Parameters: numerical values
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
Assembly
2X1 + 1X2 <= 22 hours
Inspection
3X1 + 3X2 <= 39 cubic feet Storage
X1, X2 >= 0
Linear Programming Model
Objective: the goal of an LP model is maximization or
minimization
Decision variables: amounts of either inputs or
outputs
Feasible solution space: the set of all feasible
combinations of decision variables as defined by the
constraints
Constraints: limitations that restrict the available
alternatives
Parameters: numerical values
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Linear Programming Assumptions
Linearity: the impact of decision variables is
linear in constraints and objective function
Divisibility: non-integer values of decision
variables are acceptable
Certainty: values of parameters are known and
constant
Non-negativity: negative values of decision
variables are unacceptable
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the
Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each
constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
Assembly
2X1 + 1X2 <= 22 hours
Inspection
3X1 + 3X2 <= 39 cubic feet Storage
X1, X2 >= 0
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in each
Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Formulating Linear Programming
Define the Objective
Define the Decision Variable
Write the mathematical Function for the Objective
Write one or two word description for each constraint
RHS for each constraint, including unit measure
Write = , <=, or >= for each constraint
Write in all of the decision variables on the LHS of
Constraint
Write Coefficient for each decision variable in
each Constraint
Linear Programming Example
Objective - profit
Maximize Z =60X1 + 50X2
Subject to
4X1 + 10X2 <= 100 hours
2X1 + 1X2 <= 22 hours
3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
Assembly
Inspection
Storage
Slack and Surplus
Surplus: when the optimal values of decision
variables are substituted into a greater than or
equal to constraint and the resulting value
exceeds the right side value
Slack: when the optimal values of decision
variables are substituted into a less than or equal
to constraint and the resulting value is less than
the right side value
Example
Example