19 Linear Programming CHAPTER Operations Management, Eighth Edition, by William J. Stevenson Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

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 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 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 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 Assembly 2X1 + 1X2 <= 22 hours Inspection 3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

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 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 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 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 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 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 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 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 Assembly 2X1 + 1X2 <= 22 hours Inspection 3X1 + 3X2 <= 39 cubic feet Storage X1, X2 >= 0

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

Constraints Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space Binding constraint: a constraint that forms the optimal corner point of the feasible solution space

Example

Example