operations management lecture 7 linear programming powerpoint presentation to accompany...
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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
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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 –
–
80 80 –
–
60 60 –
–
40 40 –
–
20 20 –
–
–| | | | | | | | | | |
00 2020 4040 6060 8080 100100
Num
ber
of B
lueB
erry
sN
umbe
r of
Blu
eBer
rys
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
100 –
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80 80 –
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60 60 –
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40 40 –
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20 20 –
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00 2020 4040 6060 8080 100100
Num
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of W
atch
TV
sN
umbe
r of
Wat
ch T
Vs
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
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
100 –
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80 80 –
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60 60 –
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00 2020 4040 6060 8080 100100
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Number of X-podsNumber of X-pods
XX11
XX22
Figure B.4Figure B.4
(0, 42)
(30, 0)(30, 0)
$210 = $7X$210 = $7X11 + $5X + $5X22
Graphical SolutionGraphical Solution
100 –
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80 80 –
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60 60 –
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40 40 –
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20 20 –
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00 2020 4040 6060 8080 100100
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eryy
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Number of X-podsNumber of X-pods
XX11
XX22
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
100 –
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80 80 –
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60 60 –
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40 40 –
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20 20 –
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00 2020 4040 6060 8080 100100
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Number of X-podsNumber of X-pods
XX11
XX22
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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Number of X-podsNumber of X-pods
XX11
XX22
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