CHAPTER 19 Linear Programming Mc GrawHillIrwin Operations Management

CHAPTER 19 Linear Programming Mc. Graw-Hill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright © 2005 by The Mc. Graw-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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours Assembly 2 X 1 + 1 X 2 <= 22 hours Inspection 3 X 1 + 3 X 2 <= 39 cubic feet Storage X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours Assembly 2 X 1 + 1 X 2 <= 22 hours Inspection 3 X 1 + 3 X 2 <= 39 cubic feet Storage X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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 =60 X 1 + 50 X 2 · Subject to 4 X 1 + 10 X 2 <= 100 hours 2 X 1 + 1 X 2 <= 22 hours 3 X 1 + 3 X 2 <= 39 cubic feet X 1, X 2 >= 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