Pertemuan 1 - Intro to LP (Contoh Formulasi)
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Transcript of Pertemuan 1 - Intro to LP (Contoh Formulasi)
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Course Objectives
Engineers and managers are constantly attempting to
optimize, particularly in the design, analysis, and operation
of complex systems. The course seeks to:
to present a range of applications of linear programming and
network optimization problem in many scientific domains and
industrial setting;
provide an in-depth understanding of the underlying theory of
linear programming and network flows;
to present a range of algorithms available to solve such
problems;
to give exposure to the diversity of applications of these
problems in engineering and management;
to help each student develop his or her intuition about algorithm
design, development and analysis.
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Course Topics
Linear Programming Formulating linear programs Applications of linear programming Linear algebra Simplex algorithm Duality theory Sensitivity analysis Integer programming: Applications and algorithms
Network Optimization
Shortest path problem Minimum spanning tree problem Maximum flow problem Minimum cost flow problem
Text Books: M.S. Bazaraa, J. J. Jarvis, and H.D. Sherali, “Linear Programming
and Network Flows : Second Edition," ISBN: 0-471-63681-9 Winston, Wayne L., “Operations Research : Applications and
Algoritms”, Fourth Edition, Thomson, 2004
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Grading and Regrading
Grading Tugas: 20%
Quiz: 30%
Midterm Exam: 25%
Final Exam: 25%
Regrading
If I have made a mistake in grading something, I will be happy
to correct it.
In order to receive a re-grade, you must contact/email me
within 48 hours of my handing back the test.
If a test is submitted for regrading, I have the right to regrade
the entire test. So it is possible to lose additional points.
Therefore, it is strongly recommended that you do not ask for
regrading unless you have substantial reason to believe that I
made a mistake when originally grading the test.
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OR Definition
Scientific approach to solve decision
making problem for finding the best system
design and operation, with limited resources
(can also be defined as Management
Science)
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Characteristics of OR
Decision making is the main focus
Economy si the effective criteria.
Depends on the formal mathematical model
Depends on computer
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The steps of OR study
1. Model formulation:
Objective function (Max atau Min)
Decision variables (controllable)
Parameters (uncontrollable)
Constraints
2. Build mathematical model
3. Do analytical
4. Validity test of the model and solution
5. Conduct sensitivitis analysis
6. Implementation
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TI091306/OR1/sew/2012/#1
The steps of OR study
SYSTEM
Management
Problems Mathematic
Model Solu-
tion ?
C
O
N
T
R
O
L
IMPLEMENTATION
Observation
Data Collection
Y
N
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INTRODUCTION TO LINEAR PROGRAMMING
CONTENTS
Introduction to Linear Programming
Applications of Linear Programming
Reference: Chapter 1 in BJS book.
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A Typical Linear Programming Problem
Linear Programming Formulation: Minimize c1x1 + c2x2 + c3x3 + …. + cnxn
subject to a11x1 + a12x2 + a13x3 + …. + a1nxn b1
a21x1 + a22x2 + a23x3 + …. + a2nxn b2 : : am1x1 + am2x2 + am3x3 + …. + amnxn bm x1, x2, x3 , …., xn 0 or,
Minimize j=1, n cjxj
subject to
j=1, n aijxj +xn+i = bi for all i = 1, …, m
xj 0 for all j =1, …, n+m Note:- xn+i is the slack variable corresponding to ith equation.
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Matrix Notation
Minimize cx
subject to
Ax = b
x 0
where
a11 a12 ….. a1n 1
a21 a22 ….. a2n 1
A = ::
::
::
::
am1 am2 amn 1
b1
b2
b = ::
bm
x1
x2
::
x = xn
Xn+1
::
Xn+m
c1
c2
::
c = cn
0
::
0
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Features of a Linear Programming (LP) Problem
Decision Variables
We minimize (or maximize) a linear function of decision
variables, called objective function.
The decision variables must satisfy a set of constraints.
Decision variables have sign restrictions.
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An Example of a LP
Giapetto’s woodcarving manufactures two types of wooden toys: soldiers and trains
Constraints:
100 finishing hour per week available 80 carpentry hours per week available produce no more than 40 soldiers per week
Objective: maximize profit
Soldier Train
Selling Price $27 $21
Raw Material
required
$10 $9
Variable Cost $14 $10
Finishing Labor
required
2 hrs 1 hr
Carpenting labor
required
1 hr 1 hr
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An Example of a LP (cont.)
Linear Programming formulation:
Maximize z = 3x1 + 2x2 (Obj. Func.)
subject to
2x1 + x2 100 (Finishing constraint)
x1 + x2 80 (Carpentry constraint)
x1 40 (Bound on soldiers)
x1 0 (Sign restriction)
x2 0 (Sign restriction)
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Assumptions of Linear Programming
Proportionality Assumption
Contribution of a variable is proportional to its value.
Additivity Assumptions
Contributions of variables are independent.
Divisibility Assumption
Decision variables can take fractional values.
Certainty Assumption
Each parameter is known with certainty.
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Linear Programming Modeling and Examples
Stages of an application:
Problem formulation
Mathematical model
Deriving a solution
Model testing and analysis
Implementation
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Capital Budgeting Problem
Five different investment opportunities are available for
investment.
Fraction of investments can be bought and cannot more
than one
Money available for investment:
Time 0: $40 million
Time 1: $20 million
Maximize the NPV of all investments.
Inv.1 Inv. 2 Inv. 3 Inv. 4 Inv. 5
Time 0 cash Outflow
$11 $5 $5 $5 $29
Time 1 cash Outflow
$3 $6 $5 $1 $3
NPV $17 $16 $16 $14 $39
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Solution: Capital Budgeting Problem
Decision Variables:
xi: fraction of investment i purchased
Formulation:
Maximize z = 13x1 + 16x2 + 16x3 + 14x4 + 39x5
subject to
11x1 + 53x2 + 5x3 + 5x4 + 29x5 40
3x1 + 6x2 + 5x3 + x4 + 34x5 20
x1 1
x2 1
x3 1
x4 1
x5 1
x1, x2, x3, x4, x5 0
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Transportation Problem
The Brazilian coffee company processes coffee beans into
coffee at m plants. The production capacity at plant i is ai.
The coffee is shipped every week to n warehouses in major
cities for retail, distribution, and exporting. The demand at
warehouse j is bj.
The unit shipping cost from plant i to warehouse j is cij.
It is desired to find the production-shipping pattern xij from
plant i to warehouse j, i = 1, .. , m, j = 1, …, n, that minimizes
the overall shipping cost.
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Solution: Transportation Problem
Decision Variables: xij: amount shipped from plant i to warehouse j Formulation: Minimize z =
subject to
= ai, i = 1, … , m bj, j = 1, … , n xij 0, i = 1, … , m, j = 1, … , n
m n
ij iji=1j=1
c x
n
ijj=1
x
m
iji=1
x
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Static Workforce Scheduling
Number of full time employees on different days of the week
are given below.
Each employee must work five consecutive days and then
receive two days off.
The schedule must meet the requirements by minimizing
the total number of full time employees.
Day 1 = Monday 17
Day 2 = Tues. 13
Day 3 = Wedn. 15
Day 4 = Thurs. 19
Day 5 = Friday 14
Day 6 = Satur. 16
Day 7 = Sunday 11
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Solution: Static Workforce Scheduling
LP Formulation:
Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7
subject to
x1 + x4 + x5 + x6 + x7 17
x1+ x2 + x5 + x6 + x7 13
x1+ x2 + x3 + x6 + x7 15
x1+ x2 + x3 + x4 + x7 19
x1+ x2 + x3 + x4 + x5 14
x2 + x3 + x4 + x5 + x6 16
x3 + x4 + x5 + x6 + x7 11
x1, x2, x3, x4, x5, x6, x7 0
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Multi-Period Financial Models
Determine investment strategy for the next three years
Money available for investment at time 0 = $100,000
Investments available : A, B, C, D & E
No more than $75,000 in one invest
Uninvested cash earns 8% interest
Cash flow of these investments:
0 1 2 3A -1 + 0.5 + 1 0
B 0 -1 + 0.5 + 1
C -1 + 1.2 0 0
D -1 0 0 + 1.9
E 0 0 -1 + 1.5
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Multi-Period Workforce Scheduling
Requirement of skilled repair time (in hours) is given below.
At the beginning of the period, 50 skilled technicians are
available.
Each technician is paid $2,000 and works up to 160 hrs per
month.
Each month 5% of the technicians leave.
A new technician needs one month of training, is paid
$1,000 per month, and requires 50 hours of supervision of a
trained technician.
Meet the service requirement at minimum cost.
Month 1 Month 2 Month 3 Month 4 Month 5
6,000 7,000 8,000 9,500 11,000
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Solution: Multiperiod Financial Model
Decision Variables:
A, B, C, D, E : Dollars invested in the investments A, B, C, D, and E
St: Dollars invested in money market fund at time t (t = 0, 1, 2)
Formulation:
Maximize z = B + 1.9D + 1.5E + 1.08S2
subject to
A + C + D + S0 = 100,000
0.5A + 1.2C + 1.08S0 = B + S1
A + 0.5B + 1.08S1 = E + S2
A 75,000
B 75,000
C 75,000
D 75,000
E 75,000
A, B, C, D, E, S0, S1, S2 0
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Solution: Multiperiod Workforce Scheduling
Decision Variables:
xt: number of technicians trained in period t
yt: number of experienced technicians in period t
Formulation:
Minimize z = 1000(x1 + x2 + x3 + x4 + x5) + 2000(y1 + y2 + y3 + y4 + y5)
subject to
160y1 - 50 x1 6000 y1 = 50
160y2 - 50 x2 7000 0.95y1 + x1 = y2
160y3 - 50 x3 8000 0.95y2 + x2 = y3
160y4 - 50 x4 9500 0.95y3 + x3 = y4
160y5 - 50 x5 11000 0.95y4 + x4 = y5
xt, yt 0, t = 1, 2, 3, … , 5
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Cutting Stock Problem (contd.)
Formulation:
Minimize i=1,n xi
subject to
i=1,n aij xi bi i = 1, …, m
xi 0 j = 1, …, n
xi integer j = 1, …, n
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Feasible Region
Feasible Region: Set
of all points satisfying
all the constraints
and all the sign
restrictions
Example:
Max. z = 3x1 + 2x2
subject to
2x1 + x2 100
x1 + x2 80
x1 40
x1 0
x2 0
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Example 1
Maximize z = 50x1 + 100x2
subject to
7x1 + 2x2 28
2x1 + 12x2 24
x1, x2 0
Feasible region in this example is unbounded.
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Example 2
Maximize z = 3x1+ 2x2
subject to
1/40x1 + 1/60x2 1
1/50x1 + 1/50x2 1
x1 30
x2 20
x1, x2 0
This linear program does not have any feasible solutions.
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Example 3
Max. z = 3x1 + 2x2
subject to 1/40 x1 + 1/60x2 1 1/50 x1 + 1/50x2 1 x1, x2 0
This linear program has multiple or alternative optimal
solutions.
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Cutting Stock Problem
A manufacturer of metal sheets produces rolls of standard
fixed width w and of standard length l.
A large order is placed by a customer who needs sheets of
width w and varying lengths. He needs bi sheets of length
li, i = 1, …, m.
The manufacturer would like to cut standard rolls in such a
way as to satisfy the order and to minimize the waste.
Since scrap pieces are useless to the manufacturer, the
objective is to minimize the number of rolls needed to
satisfy the order.
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Cutting Stock Problem (contd.)
Given a standard sheet of length l, there are many ways of
cutting it. Each such way is called a cutting pattern.
The jth cutting pattern is characterized by the column
vector aj, where the ith component, namely, aij, is a
nonnegative integer denoting the number of sheets of
length li in the jth pattern.
Note that the vector aj represents a cutting pattern if and
only if i=1,n aijli l and each aij is a nonnegative number.