Operations management by stevenson 9th student slides supplement 6

List the constraints – Right hand side value – Relationship symbol ≤, ≥, or = – Left Hand Side • The variables subject to the constraint, and their coefficients that indicate how much o

Supplement 6

Linear Programming

Supplement 6: Learning Objectives

• You should be able to:

– Describe the type of problem that would be appropriately solved

using linear programming

– Formulate a linear programming model

– Solve simple linear programming problems using the graphical

method

– Interpret computer solutions of linear programming problems

– Do sensitivity analysis on the solution of a linear programming

problem

Linear Programming (LP)

• LP

– A powerful quantitative tool used by operations and other

manages to obtain optimal solutions to problems that

involve restrictions or limitations

• Applications include:

– Establishing locations for emergency equipment and

personnel to minimize response time

– Developing optimal production schedules

– Developing financial plans

– Determining optimal diet plans

Model Formulation

1 List and define the decision variables (D.V.)

– These typically represent quantities

2 State the objective function (O.F.)

– It includes every D.V in the model and its contribution to profit (or cost)

3 List the constraints

– Right hand side value – Relationship symbol (≤, ≥, or =) – Left Hand Side

• The variables subject to the constraint, and their coefficients

that indicate how much of the RHS quantity one unit of the D.V represents

4 Non-negativity constraints

Computer Solutions

using its Solver routine

– Enter the problem into a worksheet

– You must designate the cells where you want the

optimal values for the decision variables

Computer Solutions

• Click on Tools on the top of the worksheet, and in the

drop-down menu, click on Solver

• Begin by setting the Target Cell

– This is where you want the optimal objective function value to be recorded

– Highlight Max (if the objective is to maximize)

– The changing cells are the cells where the optimal values of the

decision variables will appear

Computer Solutions

• Add the constraint, by clicking add

– For each constraint, enter the cell that contains the left-hand side

for the constraint

– Select the appropriate relationship sign (≤, ≥, or =)

– Enter the RHS value or click on the cell containing the value

• Repeat the process for each system constraint

Computer Solutions

• For the nonnegativity constraints, enter the range of

cells designated for the optimal values of the decision

variables

– Click OK, rather than add

– You will be returned to the Solver menu

• Click on Options

– In the Options menu, Click on Assume Linear Model

– Click OK; you will be returned to the solver menu

• Click Solve

Solver Results

– You will have one of two results

• A Solution

– In the Solver Results menu Reports box

» Highlight both Answer and Sensitivity

» Click OK

• An Error message

– Make corrections and click solve

Solver Results

• Solver will incorporate the optimal values of the decision variables

and the objective function into your original layout on your

worksheets

Sensitivity Analysis

– Assessing the impact of potential changes to the

numerical values of an LP model – Three types of changes

• Objective function coefficients

• Right-hand values of constraints

• Constraint coefficients

O.F Coefficient Changes

• A change in the value of an O.F coefficient can cause a change in the optimal solution of a problem

• Not every change will result in a changed solution

• Range of Optimality

– The range of O.F coefficient values for which the optimal values of the decision variables will not change

Basic and Non-Basic Variables

– Decision variables whose optimal values are non-zero

– Decision variables whose optimal values are zero

– Reduced cost

• Unless the non-basic variable’s coefficient increases by more than its reduced cost, it will continue to be non-basic

RHS Value Changes

– Amount by which the value of the objective function

would change with a one-unit change in the RHS value of a constraint

– Range of feasibility

• Range of values for the RHS of a constraint over which

the shadow price remains the same

Binding vs Non-binding Constraints

• Non-binding constraints

– have shadow price values that are equal to zero

– have slack (≤ constraint) or surplus (≥ constraint)

– Changing the RHS value of a non-binding constraint (over its range of

feasibility) will have no effect on the optimal solution

– have shadow price values that are non-zero

– have no slack (≤ constraint) or surplus (≥ constraint)

– Changing the RHS value of a binding constraint will lead to a change in

the optimal decision values and to a change in the value of the objective function