linear programming - dronacharyagn.dronacharya.info/.../unit-1/linearprogramming.pdf · a...
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Operations Management - 5th Edition Operations Management - 5th Edition
Linear ProgrammingLinear Programming
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Lecture OutlineLecture Outline
Model Formulation
Graphical Solution Method
Linear Programming Model
Solution
Solving Linear Programming Problems
with Excel
Sensitivity Analysis
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A model consisting of linear relationships
representing a firm’s objective and resource
constraints
Linear Programming (LP)
LP is a mathematical modeling technique used to
determine a level of operational activity in order to
achieve an objective, subject to restrictions called
constraints
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Types of LPTypes of LP
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Types of LP (cont.)Types of LP (cont.)
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Types of LP (cont.)Types of LP (cont.)
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LP Model FormulationLP Model Formulation
Decision variables mathematical symbols representing levels of activity of an
operation
Objective function a linear relationship reflecting the objective of an operation
most frequent objective of business firms is to maximize profit
most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost
Constraint a linear relationship representing a restriction on decision
making
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LP Model Formulation (cont.)LP Model Formulation (cont.)
Max/min z = cMax/min z = c11xx11 + c+ c22xx22 + ... + c+ ... + cnnxxnn
subject to:subject to:
aa1111xx11 + a+ a1212xx22 + ... + a+ ... + a1n1nxxnn (≤, =, ≥) b(≤, =, ≥) b11
aa2121xx11 + a+ a2222xx22 + ... + a+ ... + a2n2nxxnn (≤, =, ≥) b(≤, =, ≥) b22
::
aam1m1x1 + ax1 + am2m2xx22 + ... + a+ ... + amnmnxxnn (≤, =, ≥) b(≤, =, ≥) bmm
xxjj = decision variables= decision variables
bbii = constraint levels= constraint levels
ccj j = objective function coefficients= objective function coefficients
aaijij = constraint coefficients= constraint coefficients
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LP Model: ExampleLP Model: Example
LaborLabor ClayClay RevenueRevenue
PRODUCTPRODUCT (hr/unit)(hr/unit) (lb/unit)(lb/unit) ($/unit)($/unit)
BowlBowl 11 44 4040
MugMug 22 33 5050
There are 40 hours of labor and 120 pounds of clay There are 40 hours of labor and 120 pounds of clay
available each dayavailable each day
Decision variablesDecision variables
xx11 = number of bowls to produce= number of bowls to produce
xx22 = number of mugs to produce= number of mugs to produce
RESOURCE REQUIREMENTSRESOURCE REQUIREMENTS
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LP Formulation: ExampleLP Formulation: Example
Maximize Maximize ZZ = $40 = $40 xx11 + 50 + 50 xx22
Subject toSubject to
xx11 ++ 22xx22 40 hr40 hr (labor constraint)(labor constraint)
44xx11 ++ 33xx22 120 lb120 lb (clay constraint)(clay constraint)
xx1 1 , , xx22 00
Solution is Solution is xx11 = 24 bowls = 24 bowls xx2 2 = 8 mugs= 8 mugs
Revenue = $1,360Revenue = $1,360
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Graphical Solution MethodGraphical Solution Method
1.1. Plot model constraint on a set of coordinates Plot model constraint on a set of coordinates
in a planein a plane
2.2. Identify the feasible solution space on the Identify the feasible solution space on the
graph where all constraints are satisfied graph where all constraints are satisfied
simultaneouslysimultaneously
3.3. Plot objective function to find the point on Plot objective function to find the point on
boundary of this space that maximizes (or boundary of this space that maximizes (or
minimizes) value of objective functionminimizes) value of objective function
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Graphical Solution: ExampleGraphical Solution: Example
4 4 xx11 + 3 + 3 xx2 2 120 lb120 lb
xx11 + 2 + 2 xx2 2 40 hr40 hr
Area common toArea common to both constraintsboth constraints
50 50 –
40 40 –
30 30 –
20 20 –
10 10 –
0 0 – |
1010
|
6060
|
5050
|
2020
|
3030
|
4040 xx11
xx22
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Computing Optimal ValuesComputing Optimal Values
xx11 ++ 22xx22 == 4040
44xx11 ++ 33xx22 == 120120
44xx11 ++ 88xx22 == 160160
--44xx11 -- 33xx22 == --120120
55xx22 == 4040
xx22 == 88
xx11 ++ 2(8)2(8) == 4040
xx11 == 2424
4 4 xx11 + 3 + 3 xx2 2 120 lb120 lb
xx11 + 2 + 2 xx2 2 40 hr40 hr
40 40 –
30 30 –
20 20 –
10 10 –
0 0 – |
1010
|
2020
|
3030
|
4040
xx11
xx22
ZZ = $50(24) + $50(8) = $1,360= $50(24) + $50(8) = $1,360
24
8
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Extreme Corner PointsExtreme Corner Points
xx11 = 224 bowls= 224 bowls
xx2 2 ==8 mugs8 mugs
ZZ = $1,360= $1,360 xx11 = 30 bowls= 30 bowls
xx2 2 ==0 mugs0 mugs
ZZ = $1,200= $1,200
xx11 = 0 bowls= 0 bowls
xx2 2 ==20 mugs20 mugs
ZZ = $1,000= $1,000
AA
BB
CC |
2020
|
3030
|
4040
|
1010 xx11
xx22
40 40 –
30 30 –
20 20 –
10 10 –
0 0 –
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44xx11 + 3+ 3xx2 2 120 lb120 lb
xx11 + 2+ 2xx2 2 40 hr40 hr
40 40 –
30 30 –
20 20 –
10 10 –
0 0 –
BB
|
1010
|
2020
|
3030
|
4040 xx11
xx22
CC
AA
ZZ = 70= 70xx11 + 20+ 20xx22 Optimal point:Optimal point:
xx11 = 30 bowls= 30 bowls
xx2 2 ==0 mugs0 mugs
ZZ = $2,100= $2,100
Objective FunctionObjective Function
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Minimization ProblemMinimization Problem
CHEMICAL CONTRIBUTIONCHEMICAL CONTRIBUTION
BrandBrand Nitrogen (lb/bag)Nitrogen (lb/bag) Phosphate (lb/bag)Phosphate (lb/bag)
GroGro--plusplus 22 44
CropCrop--fastfast 44 33
Minimize Minimize ZZ = $6x= $6x11 + $3x+ $3x22
subject tosubject to
22xx11 ++ 44xx22 16 lb of nitrogen16 lb of nitrogen
44xx11 ++ 33xx22 24 lb of phosphate24 lb of phosphate
xx11, , xx22 00
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14 14 –
12 12 –
10 10 –
8 8 –
6 6 –
4 4 –
2 2 –
0 0 – |
22
|
44
|
66
|
88
|
1010
|
1212
|
1414 xx11
xx22
A
B
C
Graphical SolutionGraphical Solution
x1 = 0 bags of Gro-plus
x2 = 8 bags of Crop-fast
Z = $24
Z = 6x1 + 3x2
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Simplex MethodSimplex Method
A mathematical procedure for solving linear programming A mathematical procedure for solving linear programming problems according to a set of stepsproblems according to a set of steps
Slack variables added to ≤ constraints to represent unused Slack variables added to ≤ constraints to represent unused resourcesresources xx11 + 2x+ 2x22 + s+ s11 == 40 hours of labor40 hours of labor
4x4x11 + 3x+ 3x22 + s+ s22 == 120 lb of clay120 lb of clay
Surplus variables subtracted from ≥ constraints to represent variables subtracted from ≥ constraints to represent excess above resource requirement. For exampleexcess above resource requirement. For example 22xx11 + 4+ 4xx22 ≥ ≥ 16 is transformed into16 is transformed into
22xx11 + 4+ 4xx22 -- ss11 = = 1616
Slack/surplus variables have a 0 coefficient in the objective Slack/surplus variables have a 0 coefficient in the objective functionfunction Z = $40xZ = $40x11 + $50x+ $50x22 + 0s+ 0s11 + 0s+ 0s22
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Solution
Points with
Slack
Variables
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Solution
Points with
Surplus
Variables
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Solving LP Problems with ExcelSolving LP Problems with Excel
Click on “Tools” to invoke “Solver.”
Objective function
Decision variables – bowls (x1)=B10; mugs (x2)=B11
=C6*B10+D6*B11
=C7*B10+D7*B11
=E6-F6
=E7-F7
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Solving LP Problems with Excel Solving LP Problems with Excel (cont.)(cont.)
After all parameters and constraints have been input, click on “Solve.”
Objective function
Decision variables
C6*B10+D6*B11≤40
C7*B10+D7*B11≤120
Click on “Add” to insert constraints
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Solving LP Problems with Excel Solving LP Problems with Excel (cont.)(cont.)
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Sensitivity AnalysisSensitivity Analysis
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Sensitivity Range for Labor
Hours
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Sensitivity Range for Bowls