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Operations Operations ManagementManagement
Linear ProgrammingLinear ProgrammingModule BModule B
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OutlineOutline REQUIREMENTS OF A LINEAR PROGRAMMING
PROBLEM FORMULATING LINEAR PROGRAMMING
PROBLEMS Shader Electronics example
GRAPHICAL SOLUTION TO A LINEAR PROGRAMMING PROBLEM Graphical representation of Constraints Iso-Profit Line Solution Method Corner-Point Solution Method
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Outline - ContinuedOutline - Continued
SENSITIVITY ANALYSIS Sensitivity Report Change in the Resources of the Right-Hand-Side
Values Changes in the Objective Function Coefficient
SOLVING MINIMIZATION PROBLEMS LINEAR PROGRAMMING APPLICATIONS
Production Mix Example Diet Problem Example Production Scheduling Example Labor Scheduling Example
THE SIMPLEX METHOD OF LP
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When you complete this chapter, you should be able to :
Identify or Define: Objective function Constraints Feasible region Iso-profit/iso-cost methods Corner-point solution Shadow price
Learning ObjectivesLearning Objectives
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When you complete this chapter, you should be able to :
Describe or Explain: How to formulate linear models Graphical method of linear programming How to interpret sensitivity analysis
Learning Objectives - ContinuedLearning Objectives - Continued
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Mathematical technique Not computer programming
Allocates scarce resources to achieve an objective
Pioneered by George Dantzig in World War II Developed workable solution called Simplex
Method in 1947
What is Linear Programming?What is Linear Programming?
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Scheduling school busses to minimize total distance traveled when carrying students
Allocating police patrol units to high crime areas in order to minimize response time to 911 calls
Scheduling tellers at banks to that needs are met during each hour of the day while minimizing the total cost of labor
Examples of Successful LP Examples of Successful LP ApplicationsApplications
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Examples of Successful LP Examples of Successful LP Applications - ContinuedApplications - Continued
Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs
Selecting the product mix in a factory to make best use of machine- and labor-hours available while maximizing the firm’s profit
Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company
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Requirements of a Linear Requirements of a Linear Programming ProblemProgramming Problem
1 Must seek to maximize or minimize some quantity (the objective function)
2 Presence of restrictions or constraints - limits ability to achieve objective
3 Must be alternative courses of action from which to choose
4 Objectives and constraints must be expressible as linear equations or inequalities
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Formulating Linear Programming Formulating Linear Programming ProblemsProblems
Assume: You wish to produce two products (1) Walkman
AM/FM/Cassette and (2) Watch-TV Walkman takes 4 hours of electronic work and 2 hours
assembly Watch-TV takes 3 hours electronic work and 1 hour
assembly There are 240 hours of electronic work time and 100
hours of assembly time available Profit on a Walkman is $7; profit on a Watch-TV $5
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Formulating Linear Programming Formulating Linear Programming Problems - continuedProblems - continued
Let: X1 = number of Walkmans X2 = number of Watch-TVs
Then: 4X1 + 3X2 240 electronics constraint 2 X1 + 1X2 100 assembly
constraint 7X1 + 5X2 = profit maximize profit
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Draw graph with vertical & horizontal axes (1st quadrant only)
Plot constraints as lines, then as planes Use (X1,0), (0,X2) for line
Find feasible region Find optimal solution
Corner point method Iso-profit line method
Graphical Solution MethodGraphical Solution Method
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Shader Electronic Shader Electronic CompanyCompany Problem Problem
Hours Required toProduce 1 Unit
Department X1Walkmans
X2Watch-TV’s
Available HoursThis Week
Electronic 4 3 240
Assembly 2 1 100
Profit/unit $7 $5
Constraints: 4x1 + 3x2 240 (Hours of Electronic Time)2x1 + 1x2 100 (Hours of Assembly Time)
Objective: Maximize: 7x1 + 5x2
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Shader Electronic Company Shader Electronic Company ConstraintsConstraints
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
Electronics(Constraint A)Assembly(Constraint B)
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Shader Electronic Company Shader Electronic Company Feasible RegionFeasible Region
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
FeasibleRegion
Electronics(Constraint A)Assembly(Constraint B)
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Shader Electronic CompanyShader Electronic CompanyIso-Profit LinesIso-Profit Lines
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
7*X1 + 5*X
2 = 210
7*X1 + 5*X2 = 420
Electronics(Constraint A)Assembly(Constraint B)
Iso-profit line
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Shader Electronic Company Shader Electronic Company Corner Point SolutionsCorner Point Solutions
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
Iso-profit line
Electronics(Constraint A)Assembly(Constraint B)
Possible Corner Point Solution
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Shader Electronic Company Shader Electronic Company Optimal SolutionOptimal Solution
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
Optimal solution
Iso-profit line
Electronics(Constraint A)Assembly(Constraint B)
Possible Corner Point Solution
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Shader Electronic Company Shader Electronic Company Optimal SolutionOptimal Solution
0
20
40
60
80
100
120
0 10 20 30 40 50 70 80
Number of Walkmans (X1)
Num
ber o
f Wat
ch-T
Vs (X
2)
Optimal solution
Iso-profit line
Electronics(Constraint A)Assembly(Constraint B)
Possible Corner Point Solution
X1 = 30
X2 = 40
60
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Decision variables X1 = tons of BW chemical produced X2 = tons of color chemical produced
Objective Minimize Z = 2500X1 + 3000X2
Constraints X1 30 (BW); X2 20 (Color) X1 + X2 60 (Total tonnage) X1 0; X2 0 (Non-negativity)
Formulation of SolutionFormulation of Solution
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Simplex Steps for MaximizationSimplex Steps for Maximization1. Choose the variable with the greatest positive Cj- Zj to enter
the solution2. Determine the row to be replaced by selecting that one with
the smallest (non-negative) quantity-to-pivot column ratio3. Calculate the new values for the pivot row4. Calculate the new values for the other row(s)5. Calculate the Cj and Cj-Zj values for this tableau.
If there are any Cj-Zj numbers greater than zero, return to step 1.
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Simplex Steps for MinimizationSimplex Steps for Minimization1 Choose the variable with the greatest negative Cj- Zj to
enter the solution2 Determine the row to be replaced by selecting that one
with the smallest (non-negative) quantity-to-pivot column ratio
3 Calculate the new values for the pivot row4 Calculate the new values for the other row(s)5 Calculate the Cj and Cj-Zj values for this tableau. If there
are any Cj-Zj numbers less than zero, return to step 1.
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Sensitivity AnalysisSensitivity Analysis
Projects how much a solution might change if there were changes in variables or input data.
Shadow price (dual) - value of one additional unit of a resource
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You’re an analyst for a division of Kodak, which makes BW & color chemicals. At least 30 tons of BW and at least 20 tons of color must be made each month. The total chemicals made must be at least 60 tons. How many tons of each chemical should be made to minimize costs?
BW: $2,500 BW: $2,500 manufacturing cost manufacturing cost per monthper month
Color: $ 3,000 manufacturing cost per month
© 1995 Corel Corp.
Minimization ExampleMinimization Example
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Graphical SolutionGraphical Solution
X1
Feasible Region
0
20
40
60
80
0
Tons
, Col
or C
hem
ical (
XTo
ns, C
olor
Che
mica
l (X 22))
20 40 60 80Tons, BW Chemical (X1)
BW
Color
Total
Find values for X1 + X2 60.
X1 30, X2 20.
X1
X2
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Optimal Solution: Optimal Solution: Corner Point MethodCorner Point Method
Feasible Region
0
20
40
60
80
0
Tons
, Col
or C
hem
ical
Tons
, Col
or C
hem
ical
20 40 60 80Tons, BW Chemical
BW
Color
Total
A
B
Find corner points
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Assembly Constraint RHS Assembly Constraint RHS Increased by 10Increased by 10
X10
20
40
60
80
100
0 20 40 60
Original assembly constraint
Assembly constraint increased by 10
Sol’n
Sol’n
X2
Original Solution
Electronics Constraint
New Solution
Feasible Region
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Assembly Constraint RHS Assembly Constraint RHS Decreased by 10Decreased by 10
X10
20
40
60
80
100
0 20 40 60
Original assembly constraint
Sol’n
Sol’n
X2
Assembly constraint
decreased by 10
Original Solution
New Solution
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0
10
20
30
40
50
60
0 10 20 30 40 50 60
A Minimization ProblemA Minimization Problem
Feasible region
X1 = 30 X2 = 20
x1 + x2 = 60
a
b