CHAPTER19

Linear Programming

McGraw-Hill/IrwinOperations Management, Eighth Edition, by William J. StevensonCopyright © 2005 by The McGraw-Hill Companies, Inc. All rights

reserved.

· 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

· 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 Model

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

· 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 Model

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

· 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 Model

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

· 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 Model

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

· 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

Linear Programming Assumptions

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

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

· Objective - profitMaximize Z =60X1 + 50X2

· Subject to4X1 + 10X2 <= 100 hours

Assembly 2X1 + 1X2 <= 22 hours Inspection

3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

Linear Programming Example

· 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

Slack and Surplus

Example

Example