LP Graphical Method
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Transcript of LP Graphical Method
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Linear Programming Linear Programming Models: Graphical MethodsModels: Graphical Methods
What is Linear Programming?What is Linear Programming?
• Linear Programming (LP) is a technique that helps in resource allocation decisions.• Programming refers to modelling and solving a
problem mathematically
• Linear programming can solve two variable problems (graphical method); or more than two variable problems (simplex method)
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Requirements of a Linear Requirements of a Linear Programming ProblemProgramming Problem
• All problems seek to maximize or minimize some quantity (the objective function).
• The presence of restrictions or constraints, limits the degree to which we can pursue our objective.
• There must be alternative courses of action to choose from.
• The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Examples of Successful LP Examples of Successful LP ApplicationsApplications
1. Development of a production schedule that will satisfy
future demands for a firm’s production and at the same
time minimize total production and inventory costs
2. Selection of the product mix in a factory to make best
use of machine-hours and labor-hours available while
maximizing the firm’s products
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Examples of Successful LP Examples of Successful LP ApplicationsApplications
3. Determination of grades of petroleum products to yield the
maximum profit
4. Selection of different blends of raw materials to feed mills
to produce finished feed combinations at minimum cost
5. Determination of a distribution system that will minimize
total shipping cost from several warehouses to various
market locations
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Basic Assumptions of Linear Basic Assumptions of Linear ProgrammingProgramming
• Certainty – coefficients in the objective function and constraints are known with certainty and do not change during the period being studied.
• Proportionality-in the objective function and constraints. If one product uses 5 hours of a machine resource, then making 10 of that product uses 50 hours of machine time.
• Additivity- the total of all activitiesequals the sum of each individual activity
• Divisibility – solutions need not be integers but may also
be fractional values
• Nonnegativity – all answers and variable values are
either zero or positive.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Equations and InequalitiesEquations and Inequalities
• Solve for X1 and X2 and graph solution
1. X1 ≥ 80
2. X2 ≤ 100
3. 3X1+ 2X2 ≤ 240
4. 2X1+ 1X2 ≥ 140
5. 1X1 + 2X2 = 16
6. 2X1 - X2 ≥ 2
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Company Data - Flair Furniture Company Data - Table Table 7.17.1
Hours Required to Produce One Unit
Department TTables
CChairs
AvailableHours This
Week
CarpentryPainting &Varnishing
42
31
240100
Profit Amount $7 $5
Constraints: 4T + 3C 240 (Carpentry)
2T + 1C 100 (Paint & Varnishing)
Objective: Max: 7T + 5C
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Corner Point Solution MethodCorner Point Solution Method
• Find the objective function and constraints• Define the decision variables• Use the decision variables to write mathematical
expressions for the objective function and constraints• Solve for unknown decision variables in each constraint
and graph the solution• Find the value of the variables at each corner point to
form the feasible region or feasible line segment• Test corner points by substituting value of decision
variables and corresponding profit or cost.• Do simultaneous equation for corner points with
intersecting constraint lines• Select the corner point with the highest (profit) or lowest
(cost) objective coefficient values as optimal solution
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Company Flair Furniture Company ConstraintsConstraints
Number of Tables
120
100
80
60
40
20
0
Num
ber
of C
hair
s
20 40 60 80 100
Painting/Varnishing
Carpentry
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Company Feasible Flair Furniture Company Feasible RegionRegion
120
100
80
60
40
20
0
Num
ber
of C
hair
s
20 40 60 80 100Number of Tables
Painting/Varnishing
CarpentryFeasibleRegion
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Company Flair Furniture Company Optimal SolutionOptimal Solution
Num
ber
of C
hair
s
120
100
80
60
40
20
0
20 40 60 80 100Number of Tables
Painting/Varnishing
Carpentry
Solution(T = 30, C = 40)
Corner PointsCorner Points
1
2
3
4
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Holiday Meal Turkey Ranch p. Holiday Meal Turkey Ranch p. 290 (11290 (11thth ed) or p. 296 (10 ed) or p. 296 (10thth ed) ed)
(C)
(B)
toSubject
:Minimize
½ X
XX
A)(XX:
XX
INGREDIENTS BRAND 1 FEED BRAND 2 FEED MINIMIMUM REQUIREMENT PER TURKEY (OZ)
A 5 10 90
B 4 3 48
C 0.5 0 1.5
COST PER POUND 2 cents 3 cents
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Holiday Meal Turkey Problem Holiday Meal Turkey Problem
Corner PointsCorner Points
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
LP-GraphicalLP-Graphical
• SEATWORK
Solve 7-18 and 7-19 p319 10th and 11th ed p313
• HOMEWORK
Solve 7-27 p. 320 10th ed.; p314 for 11th edition
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special Cases in LPSpecial Cases in LP
• Infeasibility
• Unbounded Solutions
• Redundancy
• Degeneracy
• More Than One Optimal Solution
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
A Problem with No Feasible A Problem with No Feasible SolutionSolution
X2
X1
8
6
4
2
02 4 6 8
Region Satisfying3rd Constraint
Region Satisfying First 2 Constraints
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
A Solution Region That is A Solution Region That is Unbounded to the RightUnbounded to the Right
X2
X1
15
10
5
05 10 15
Feasible Region
X1 > 5 X2 < 10
X1 + 2X2 > 10
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
A Problem with a Redundant A Problem with a Redundant ConstraintConstraint
X2
X1
30
25
20
15
10
5
05 10 15 20 25 30
Feasible Region
2X1 + X2 < 30
X1 < 25
X1 + X2 < 20
RedundantConstraint
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
An Example of Alternate Optimal An Example of Alternate Optimal SolutionsSolutions
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8
Optimal Solution Consists of All Combinations of X1 and X2 Along the AB Segment
Isoprofit Line for $12Overlays Line Segment
Isoprofit Line for $8A
BAB
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Sensitivity AnalysisSensitivity Analysis
• Changes in the Objective Function
Coefficient
• Changes in Resources (RHS)
• Changes in Technological Coefficients
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Changes in the Technological Coefficients Changes in the Technological Coefficients for High Note Sound Co.for High Note Sound Co.
Ste
reo
Rec
eive
rs
X1
60
40
20
0
CD Players
20 40
X2
(a) Original Problem
3X1 + 1X2 < 60
Optimal Solution
a2X1 + 4X2 < 80
b
c
X2
(b) Change in CircledCoefficient
Still Optimal
a2X1 + 4X2 < 80
d
e
2X1 + 1X2 < 60
20 40 X130
CD Players
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
7-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Changes in the Technological Coefficients Changes in the Technological Coefficients for High Note Sound Co.for High Note Sound Co.
X1
Ste
reo
Rec
eive
rs
60
40
20
0
CD Players
20 40
X2
(a) Original Problem
3X1 + 1X2 < 60
Optimal Solution
a2X1 + 4X2 < 80
b
c
20 40
X2
X1
(c) Change in CircledCoefficient
3X1 + 1X2 < 60
Optimal Solution
f2X1 + 5X2 < 80
g
c
CD Players