Operations ManagementOperations ManagementLecture 7Lecture 7Linear ProgrammingLinear Programming

PowerPoint presentation to accompany PowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 7ePrinciples of Operations Management, 7eOperations Management, 9e Operations Management, 9e

RecapRecap

Implications of Quality tools and Implications of Quality tools and

Problem solving Problem solving Check SheetsCheck Sheets

Scatter Diagrams Scatter Diagrams

Cause-and-Effect DiagramsCause-and-Effect Diagrams

Pareto ChartsPareto Charts

FlowchartsFlowcharts

HistogramsHistograms

Statistical Process Control (SPC)Statistical Process Control (SPC)

OutlineOutline

Linear ProgrammingLinear Programming

Requirements of a Linear Programming Requirements of a Linear Programming ProblemProblem

Maximization ProblemsMaximization Problems

Minimization ProblemsMinimization Problems

Linear Programming MethodsLinear Programming Methods

Graphical MethodGraphical Method

Simplex Method (will do later)Simplex Method (will do later)

Outline – ContinuedOutline – Continued

Formulating Linear Programming Formulating Linear Programming ProblemsProblems Shader Electronics ExampleShader Electronics Example

Graphical Solution to a Linear Graphical Solution to a Linear Programming ProblemProgramming Problem Graphical Representation of ConstraintsGraphical Representation of Constraints

Iso-Profit Line Solution MethodIso-Profit Line Solution Method

Corner-Point Solution MethodCorner-Point Solution Method

Outline – ContinuedOutline – Continued

Linear Programming ApplicationsLinear Programming Applications Production-Mix ExampleProduction-Mix Example

Diet Problem ExampleDiet Problem Example

Labor Scheduling ExampleLabor Scheduling Example

Learning ObjectivesLearning Objectives

When you complete this module you should be able to:

1.1. Formulate linear programming models, Formulate linear programming models, including an objective function and including an objective function and constraintsconstraints

2.2. Graphically solve an LP problem with the Graphically solve an LP problem with the iso-profit line methodiso-profit line method

3.3. Graphically solve an LP problem with the Graphically solve an LP problem with the corner-point methodcorner-point method

Linear ProgrammingLinear Programming

A mathematical technique to allocate A mathematical technique to allocate limited resources to achieve an limited resources to achieve an objectiveobjective

linear programming (LP) is a linear programming (LP) is a technique for optimization of a linear technique for optimization of a linear objective function, subject to linear objective function, subject to linear equality and linear inequality equality and linear inequality constraints.constraints.

Linear ProgrammingLinear Programming

Linear programming determines the way to achieve the best outcome (such as maximum profit or minimum cost) in a given mathematical model and given some list of requirements represented as linear equations.

It is one of more powerful technique It is one of more powerful technique for managerial decisionsfor managerial decisions

Linear ProgrammingLinear Programming

• A Linear Programming model seeks to A Linear Programming model seeks to maximize or minimize a linear function, maximize or minimize a linear function, subject subject to a set of linear constraints.• The linear model consists of the The linear model consists of the followingfollowingcomponents:components:

A set of decision variables.A set of decision variables. An objective function.An objective function. A set of constraintsA set of constraints

Linear ProgrammingLinear Programming

Will find the minimum or maximum Will find the minimum or maximum value of the objectivevalue of the objective

Guarantees the optimal solution to the Guarantees the optimal solution to the model formulatedmodel formulated

LP ApplicationsLP Applications

1.1. Scheduling school buses to minimize total Scheduling school buses to minimize total distance traveled distance traveled

2.2. Allocating police patrol units to high crime Allocating police patrol units to high crime areas in order to minimize response timeareas in order to minimize response time

3.3. Scheduling tellers at banks so that needs are Scheduling tellers at banks so that needs are met during each hour of the day while met during each hour of the day while minimizing the total cost of laborminimizing the total cost of labor

LP ApplicationsLP Applications

4.4. Selecting the product mix in a factory to make Selecting the product mix in a factory to make best use of machine- and labor-hours available best use of machine- and labor-hours available while maximizing the firm’s profit while maximizing the firm’s profit

5.5. Picking blends of raw materials in feed mills Picking blends of raw materials in feed mills to produce finished feed combinations at to produce finished feed combinations at minimum costsminimum costs

6.6. Determining the distribution system that will Determining the distribution system that will minimize total shipping costminimize total shipping cost

LP ApplicationsLP Applications

7.7. Developing a production schedule that will Developing a production schedule that will satisfy future demands for a firm’s product and satisfy future demands for a firm’s product and at the same time minimize total production and at the same time minimize total production and inventory costsinventory costs

Requirements of an Requirements of an LP ProblemLP Problem

1.1. There should be an objective function to There should be an objective function to maximize or minimize some quantity maximize or minimize some quantity (usually profit or cost)(usually profit or cost)

2.2. The restrictions or constraints are The restrictions or constraints are present which limit the ability to present which limit the ability to achieve objectiveachieve objective

Requirements of an Requirements of an LP ProblemLP Problem

3.3. There must be alternative courses of There must be alternative courses of action from which to chooseaction from which to choose

4.4. The objective and constraints in linear The objective and constraints in linear programming problems must be programming problems must be expressible in terms of linear equations expressible in terms of linear equations or inequalitiesor inequalities

Mathematical formulation of Linear Programming model

16

Step 1- Study the given situation- Find the key decision to be made - Identify the decision variables of the problemStep 2- Formulate the objective function to be optimizedStep 3- Formulate the constraints of the problemStep 4- Add non-negativity restrictions or constraintsThe objective function , the set of constraints and the non-

negativity restrictions together form an LP model.

Formulating LP ProblemsFormulating LP Problems

The product-mix problem at Shader ElectronicsThe product-mix problem at Shader Electronics

Two productsTwo products

1.1. Shader X-pod, a portable music playerShader X-pod, a portable music player

2.2. Shader BlueBerry, an internet-connected Shader BlueBerry, an internet-connected color telephonecolor telephone

Determine the mix of products that will Determine the mix of products that will produce the maximum profitproduce the maximum profit

Formulating LP ProblemsFormulating LP Problems

X-podsX-pods BlueBerrysBlueBerrys Available HoursAvailable HoursDepartmentDepartment (X(X11)) (X(X22)) This WeekThis Week

Hours Required Hours Required to Produce 1 Unitto Produce 1 Unit

ElectronicElectronic 44 33 240240

AssemblyAssembly 22 11 100100

Profit per unitProfit per unit $7$7 $5$5

Decision Variables:Decision Variables:XX11 = number of X-pods to be produced= number of X-pods to be produced

XX22 = number of BlueBerrys to be produced= number of BlueBerrys to be produced

Table B.1Table B.1

Formulating LP ProblemsFormulating LP Problems

Objective Function:Objective Function:

Maximize Profit = $7XMaximize Profit = $7X11 + $5X + $5X22

There are three types of constraints Upper limits where the amount used is ≤ the

amount of a resource Lower limits where the amount used is ≥ the

amount of the resource Equalities where the amount used is = the

amount of the resource

Formulating LP ProblemsFormulating LP Problems

Second Constraint:Second Constraint:

2X2X11 + 1X + 1X22 ≤ 100 (hours of assembly time) ≤ 100 (hours of assembly time)

AssemblyAssemblytime availabletime available

AssemblyAssemblytime usedtime used is ≤is ≤

First Constraint:First Constraint:

4X4X11 + 3X + 3X22 ≤ 240 (hours of electronic time) ≤ 240 (hours of electronic time)

ElectronicElectronictime availabletime available

ElectronicElectronictime usedtime used is ≤is ≤

Graphical SolutionGraphical Solution

Can be used when there are two decision Can be used when there are two decision variablesvariables

1.1. Plot the constraint equations at their limits by Plot the constraint equations at their limits by converting each equation to an equalityconverting each equation to an equality

2.2. Identify the feasible solution space Identify the feasible solution space

3.3. Create an iso-profit line based on the Create an iso-profit line based on the objective functionobjective function

4.4. Move this line outwards until the optimal Move this line outwards until the optimal point is identifiedpoint is identified

4X4X11 + 3X + 3X22 = 240-------(I) = 240-------(I)

If XIf X22 = 0, X = 0, X11 = 60 = 60

Point (60, 0)Point (60, 0)

If XIf X11 = 0, X = 0, X22 = 80 = 80

Point (0, 80) Point (0, 80)

2X1 + 1X2 = 100-----(II)2X1 + 1X2 = 100-----(II)

If XIf X22 = 0, X = 0, X11 = 50 = 50

Point (50, 0)Point (50, 0)

If XIf X11 = 0, X = 0, X22 = 100 = 100

Point (0, 100) Point (0, 100)

Graphical SolutionGraphical Solution

100 –

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80 80 –

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60 60 –

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00 2020 4040 6060 8080 100100

Num

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Number of X-podsNumber of X-pods

XX11

XX22

Assembly (constraint B)Assembly (constraint B)

Electronics (constraint A)Electronics (constraint A)Feasible region

Figure B.3Figure B.3

I (60, 0)I (60, 0)

I(0, 80)I(0, 80)

II(0, 100)II(0, 100)

II(50, 0)II(50, 0)

Graphical SolutionGraphical Solution

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Number of X-podsNumber of X-pods

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Assembly (constraint B)Assembly (constraint B)

Electronics (constraint A)Electronics (constraint A)Feasible region

Figure B.3Figure B.3

Iso-Profit Line Solution Method

Choose a possible value for the objective function

$210 = 7X1 + 5X2

Solve for the axis intercepts of the function and plot the line

X2 = 42 X1 = 30

Graphical SolutionGraphical Solution

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Figure B.4Figure B.4

(0, 42)

(30, 0)(30, 0)

$210 = $7X$210 = $7X11 + $5X + $5X22

Graphical SolutionGraphical Solution

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Figure B.5Figure B.5

$210 = $7X$210 = $7X11 + $5X + $5X22

$350 = $7X$350 = $7X11 + $5X + $5X22

$420 = $7X$420 = $7X11 + $5X + $5X22

$280 = $7X$280 = $7X11 + $5X + $5X22

Graphical SolutionGraphical Solution

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Figure B.6Figure B.6

$410 = $7X$410 = $7X11 + $5X + $5X22

Maximum profit lineMaximum profit line

Optimal solution pointOptimal solution point(X(X11 = 30, X = 30, X22 = 40) = 40)

Corner-Point MethodCorner-Point Method

Figure B.7Figure B.7 1

2

3

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4

Corner-Point MethodCorner-Point Method The optimal value will always be at a corner

point

Find the objective function value at each corner point and choose the one with the highest profit

Point 1 1 : (X1 = 0, X2 = 0) Profit $7(0) + $5(0) = $0

Point 2 : (X1 = 0, X2 = 80) Profit $7(0) + $5(80) = $400

Point 4 : (X1 = 50, X2 = 0) Profit $7(50) + $5(0) = $350

Corner-Point MethodCorner-Point Method The optimal value will always be at a corner

point

Find the objective function value at each corner point and choose the one with the highest profit

Point 1 : (X1 = 0, X2 = 0) Profit $7(0) + $5(0) = $0

Point 2 : (X1 = 0, X2 = 80) Profit $7(0) + $5(80) = $400

Point 4 : (X1 = 50, X2 = 0) Profit $7(50) + $5(0) = $350

Solve for the intersection of two constraints

2X1 + 1X2 ≤ 100 (assembly time)

4X1 + 3X2 ≤ 240 (electronics time)

4X1 + 3X2 = 240

- 4X1 - 2X2 = -200

+ 1X2 = 40

4X1 + 3(40) = 240

4X1 + 120 = 240

X1 = 30

Corner-Point MethodCorner-Point Method The optimal value will always be at a corner

point

Find the objective function value at each corner point and choose the one with the highest profit

Point 1 : (X1 = 0, X2 = 0) Profit $7(0) + $5(0) = $0

Point 2 : (X1 = 0, X2 = 80) Profit $7(0) + $5(80) = $400

Point 4 : (X1 = 50, X2 = 0) Profit $7(50) + $5(0) = $350

Point 3 : (X1 = 30, X2 = 40) Profit $7(30) + $5(40) = $410