linear programming mcgraw-hill/irwin copyright © 2012 by the mcgraw-hill companies, inc. all rights...

Linear Programming

Chapter 19

McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 19: Learning Objectives

You should be able to:1. Describe the type of problem that would lend itself to

solution using linear programming2. Formulate a linear programming model from a

description of a problem3. Solve simple linear programming problems using the

graphical method4. Interpret computer solutions of linear programming

problems5. Do sensitivity analysis on the solution of a linear

programming problem

Instructor Slides 19-2

Linear Programming (LP)LP

A powerful quantitative tool used by operations and other manages to obtain optimal solutions to problems that involve restrictions or limitationsApplications include:

Establishing locations for emergency equipment and personnel to minimize response time

Developing optimal production schedulesDeveloping financial plansDetermining optimal diet plans


LP ModelsLP Models

Mathematical representations of constrained optimization problems

LP Model Components:Objective function

A mathematical statement of profit (or cost, etc.) for a given solution

Decision variablesAmounts of either inputs or outputs

ConstraintsLimitations that restrict the available alternatives

ParametersNumerical constants


6S-5

Used to obtain optimal solutions to problems that involve restrictions or limitations, such as:MaterialsBudgetsLaborMachine time

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

LP AssumptionsIn order for LP models to be used

effectively, certain assumptions must be satisfied: Linearity

The impact of decision variables is linear in constraints and in the objective function

DivisibilityNoninteger values of decision variables are

acceptable Certainty

Values of parameters are known and constant Nonnegativity

Negative values of decision variables are unacceptable


Model Formulation1. 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


Example– LP Formulation

0,,

units 10 1Product

pounds 100567 Material

hours 250842 Labor

Subject to

(profit) 485 Maximize

produce to3product ofQuantity



VariablesDecision

321

1

321

321

321

3

2

1

xxx

x

xxx

xxx

xxx

x

x

x

(Objective function)

(Constraints)

(Nonnegativity constraints)


Graphical LPGraphical LP

A method for finding optimal solutions to two-variable problems

Procedure1. Set up the objective function and the constraints in

mathematical format2. Plot the constraints3. Indentify the feasible solution space

The set of all feasible combinations of decision variables as defined by the constraints

4. Plot the objective function5. Determine the optimal solution


6S-10

Linear Programming Example

Assembly Time/Unit

Inspection Time/Unit

Storage Space/Unit Profit/Unit

Model A 4 2 3 60$

Model B 10 1 3 50$

Available 100 hours 22 hours 39 cubic feet

6S-11


Find the quantity of each model to produce in order to maximize the profit

6S-12

LP Example – Decision Variables

Decision Variables

A: # of model A product to be built

B: # of model B product to be built

Example– Graphical LP: Step 1

0,

feet cubic 3933 Storage

hours 2212 Inspection

hours 100104 Assembly

Subject to

5060 Maximize

produce to2 typeofquantity

produce to1 typeofquantity VariablesDecision

21

21

21

21

21

2

1

xx

xx

xx

xx

xx

x

x


6S-14

LP Example – Objective FunctionProfit:

Profit = profit from model A + profit from model B

(profit/model A) x (# of model A) + (profit/model B) x (# of model B)

Z = 60A + 50B (Profit)

Obejective Function:

MaximizeZ = 60A + 50B

6S-15

LP Example – Objective Function

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11

A

B60A+50B=600

60A+50B=1200

Example– Graphical LP: Step 2Plotting constraints:

Begin by placing the nonnegativity constraints on a graph


Example– Graphical LP: Step 2Plotting constraints:

1. Replace the inequality sign with an equal sign.2. Determine where the line intersects each axis3. Mark these intersection on the axes, and connect

them with a straight line4. Indicate by shading, whether the inequality is

greater than or less than5. Repeat steps 1 – 4 for each constraint


Example– Graphical LP: Step 24A+10B<100

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

A

B Feasible

Infeasible

6S-23

Storage Space ConstraintStorage space

3A + 3B < 39 cubic feet

Storage Space/Unit

3

3

39 cubic feet

6S-24

Storage Space Constraint3A + 3B < 39

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A

BFeasible

Infeasible

6S-25

Objective - profitMaximize Z=60A + 50B

Subject toAssembly 4A + 10B <= 100 hoursInspection 2A + 1B <= 22 hoursStorage 3A + 3B <= 39 cubic feetA, B >= 0

Linear Programming Formulation

Nonnegativity Condition

Example– Graphical LP: Step 3Feasible Solution Space

The set of points that satisfy all constraints simultaneously


6S-28

Feasible Solution Space

0

5

10

15

20

25

1 3 5 7 9 11 13 15 17 19 21 23 25

A

B

Assembly

Inspection

Storage

Feasible Region

Example– Graphical LP: Step 4Plotting the objective function line

This follows the same logic as plotting a constraint line There is no equal sign, so we simply set the objective

function to some quantity (profit or cost) The profit line can now be interpreted as an isoprofit

lineEvery point on this line represents a combination of

the decision variables that result in the same profit (in this case, to the profit you selected)


6S-31

LP Example – Objective Function

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11

A

B60A+50B=600

60A+50B=1200

Example– Graphical LP: Step 4As we increase the value for the objective

function: The isoprofit line moves further away from the origin The isoprofit lines are parallel


Example– Graphical LP: Step 5 Where is the optimal solution?

The optimal solution occurs at the furthest point (for a maximization problem) from the origin the isoprofit can be moved and still be touching the feasible solution space

This optimum point will occur at the intersection of two constraints:

Solve for the values of x1 and x2 where this occurs


Redundant ConstraintsRedundant constraints

A constraint that does not form a unique boundary of the feasible solution space

Test:A constraint is redundant if its removal does not

alter the feasible solution space


Solutions and Corner PointsThe solution to any problem will occur at one

of the feasible solution space corner pointsEnumeration approach

Determine the coordinates for each of the corner points of the feasible solution spaceCorner points occur at the intersections of

constraints Substitute the coordinates of each corner point into the

objective function The corner point with the maximum (or minimum,

depending on the objective) value is optimal


6S-36

The intersection of inspection and storage

Solve two equations with two unknowns 2A + 1B = 22

3A + 3B = 39

A = 9B = 4Z = $740

Optimal Solution

Slack and Surplus Binding Constraint

If a constraint forms the optimal corner point of the feasible solution space, it is binding

It effectively limits the value of the objective function If the constraint could be relaxed, the objective function could

be improved Surplus

When the value of decision variables are substituted into a ≥ constraint the amount by which the resulting value exceeds the right-hand side value

Slack When the values of decision variables are substituted into a ≤

constraint, the amount by which the resulting value is less than the right-hand side


6S-38


Labor (hrs) Stone (tons) Lumber (board feet) Profit/Unit

Model A 4,000 2 2,000 3,000$

Model B 10,000 3 2000 6,000$

Available 400,000 150 200,000

The Simplex MethodSimplex method

A general purpose linear programming algorithm that can be used to solve problems having more than two decision variables


Computer SolutionsMS Excel can be used to solve LP

problems using its Solver routineEnter the problem into a worksheetWhere there is a zero in Figure 19.15, a

formula was enteredSolver automatically places a value of zero after

you input the formulaYou must designate the cells where you want

the optimal values for the decision variables


Computer Solutions


Computer SolutionsIn Excel 2010, click on Tools on the top of the

worksheet, and in that menu, click on SolverBegin 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 SolutionsAdd a 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 SolutionsFor 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


Computer Solutions


The Solver Results menu will appearYou will have one of two results

A SolutionIn the Solver Results menu Reports box

Highlight both Answer and Sensitivity Click OK

An Error messageMake corrections and click solve

Solver Results


Solver Results Solver will incorporate the optimal values of the decision

variables and the objective function into your original layout on your worksheets


Answer Report


Sensitivity Report


Sensitivity AnalysisSensitivity Analysis

Assessing the impact of potential changes to the numerical values of an LP model

Three types of changesObjective function coefficientsRight-hand values of constraintsConstraint coefficients

We will consider these


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 OptimalityThe range of O.F. coefficient values for which the

optimal values of the decision variables will not change

O.F. Coefficient Changes


Basic and Non-Basic VariablesBasic variables

Decision variables whose optimal values are non-zero

Non-basic variablesDecision variables whose optimal values are

zeroReduced cost

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


RHS Value ChangesShadow price

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 feasibilityRange of values for the RHS of a constraint over

which the shadow price remains the same


Binding vs. Non-binding ConstraintsNon-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 solutionBinding constraint

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


linear programming mcgraw-hill/irwin copyright © 2012 by the mcgraw-hill companies, inc. all rights...

Documents

profit slide

linear programming model

optimal solutions

impact of decision variables

unacceptable instructor

variables subject

graphical lp

constraints right hand