Post on 08-Mar-2019
11/5/2008
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Modeling Modeling Modeling Modeling
& Decision Analysis& Decision Analysis& Decision Analysis& Decision Analysis
DR. MOHAMMAD ABDUL MUKHYI, SE., MM.
Rabu, 05 Nopember 2008 1METODE KUANTITATIF
Metode KuantitatifSpektrum situasi keputusan:
- Terstruktur/terprogram
- Tak terstruktur/tak terprogram
- Sebagian terstruktur
Pembagian masalah :
- Certainty : semua alternatif tindakan diketahui dan hanya terdapatsatu konsekuansi untuk masing-masing tindakan.
- Risk : apabila terdapat lebih dari satu konsekuensi atau alternatifdan pengambil keputusan mengetahui probabilitaskonsekuensinya.
- Ucertainty :apabila jumlah kemungkinan konsekuensi tidak diketahuioleh pengambil keputusan.
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• We face numerous decisions in life & business.
• We can use computers to analyze the potential outcomes of decision alternatives.
• Spreadsheets are the tool of choice for today’s managers.
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What is Management Science?
• A field of study that uses computers, statistics, and mathematics to solve business problems.
• Also known as:
– Operations research
– Decision science
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Introduction
We all face decision about how to use limited resources such as:
– Oil in the earth
– Land for dumps
– Time
– Money
– Workers
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Home Runs
in Management Science
• Motorola
– Procurement of goods and services account for 50% of its costs
– Developed an Internet-based auction system for negotiations with suppliers
– The system optimized multi-product, multi-vendor contract awards
– Benefits:
$600 million in savings
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Home Runs
in Management Science
• Waste Management
– Leading waste collection company in North America
– 26,000 vehicles service 20 million residential & 2 million commercial customers
– Developed vehicle routing optimization system
– Benefits:
Eliminated 1,000 routes
Annual savings of $44 million
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Mathematical Programming
MP is a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual of a business.
• Optimization
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Applications of Optimization
• Determining Product Mix
• Manufacturing
• Routing and Logistics
• Financial Planning
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What is a “Computer Model”?
• A set of mathematical relationships and logical assumptions implemented in a computer as an abstract representation of a real-world object of phenomenon.
• Spreadsheets provide the most convenient way for business people to build computer models.
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The Modeling Approach
to Decision Making
• Everyone uses models to make decisions.
• Types of models:
– Mental (arranging furniture)
– Visual (blueprints, road maps)
– Physical/Scale (aerodynamics, buildings)
– Mathematical (what we’ll be studying)
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Characteristics of Models
• Models are usually simplified versions of the things they represent
• A valid model accurately represents the relevant characteristics of the object or decision being studied
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Benefits of Modeling
• Economy - It is often less costly to analyze decision problems using models.
• Timeliness - Models often deliver needed information more quickly than their real-world counterparts.
• Feasibility - Models can be used to do things that would be impossible.
• Models give us insight & understandingthat improves decision making.
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Example of a Mathematical Model
Profit = Revenue - Expenses
or
Profit = f(Revenue, Expenses)
or
Y = f(X1, X2)
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A Generic Mathematical Model
Y = f(X1, X2, …, Xn)
Y = dependent variable
(aka bottom-line performance measure)
Xi = independent variables (inputs having an impact on Y)
f(.) = function defining the relationship between the Xi & Y
Where:
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Mathematical Models & Spreadsheets
• Most spreadsheet models are very similar to our generic mathematical model:
Y = f(X1, X2, …, Xn)
� Most spreadsheets have input cells (representing Xi) to which mathematical functions ( f(.)) are applied to compute a
bottom-line performance measure (or Y).
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Categories of Mathematical Models
Prescriptive known, known or under LP, Networks, IP,
well-defined decision maker’s CPM, EOQ, NLP,
control GP, MOLP
Predictive unknown, known or under Regression Analysis,
ill-defined decision maker’s Time Series Analysis,
control Discriminant Analysis
Descriptive known, unknown or Simulation, PERT,well-defined uncertain Queueing,
Inventory Models
Model Independent OR/MS
Category Form of f(.) Variables Techniques
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The Problem Solving Process
Identify Problem
Formulate & Implement
ModelAnalyze Model
Test Results
Implement Solution
unsatisfactoryresults
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The Psychology of Decision Making
• Models can be used for structurable aspects of decision problems.
• Other aspects cannot be structured easily, requiring intuition and judgment.
• Caution: Human judgment and intuition is not always rational!
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Anchoring Effects
• Arise when trivial factors influence initial thinking about a problem.
• Decision-makers usually under-adjust from their initial “anchor”.
• Example:
– What is 1x2x3x4x5x6x7x8 ?
– What is 8x7x6x5x4x3x2x1 ?
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Framing Effects
• Refers to how decision-makers view a problem from a win-loss perspective.
• The way a problem is framed often influences choices in irrational ways…
• Suppose you’ve been given $1000 and must choose between:
– A. Receive $500 more immediately
– B. Flip a coin and receive $1000 more if heads occurs or $0 more if tails occurs
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Framing Effects (Example)
• Now suppose you’ve been given $2000 and must choose between:
– A. Give back $500 immediately
– B. Flip a coin and give back $0 if heads occurs or give back $1000 if tails occurs
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A Decision Tree for Both Examples
Initial state
$1,500
Heads (50%)
Tails (50%)
$2,000
$1,000
Alternative A
Alternative B
(Flip coin)
Payoffs
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Good Decisions vs. Good Outcomes
• Good decisions do not always lead to good outcomes...
� A structured, modeling approach to decision making helps us make good decisions, but can’t guarantee good outcomes.
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2. Perumusan PL
Ada tiga unsur dasar dari PL, ialah:
1. Fungsi Tujuan
2. Fungsi Pembatas (set ketidak samaan/pembatas strukturis)
3. Pembatasan selalu positip.
Bentuk umum persoalan PLCari : x1 , x2 , x3 , …………… , xn.Fungsi Tujuan : Z = c1x1 + c2x2 + c3x3 + …… + cnxn optimum (max/min) (srs)Fungsi Kendala : a11x1 + a12x2 + a13x3 + …………… + a1nxn >< h1.(F. Pembatas) a21x1 + a22x2 + a23x3 + …………… + a2nxn >< h2.(dp) a31x1 + a32x2 + a33x3 + …………… + a3nxn >< h3.
↓ …. ↓ …. ↓ ...………… ↓ .. ↓
am1x1 + am2x2 + am3x3 + ….……… + amnxn >< hm.
xj > 0 j = 1 , 2, 3 ………n nonnegativity consraint
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srs : sedemikian rupa sehinggadp : dengan pembatasada n macam barang yg akan diproduksi masing2 sbesar x1,x2, … , xn
xj : banyaknya barang yang diproduksi ke j , j = 1, 2, 3, …. , n cj : harga per satuan barang ke j , j = 1, 2, 3, ……………. , n
ada m macam bahan mentah, masing2 tersedia h1 , h2 , h3 , …., hm
hi : banyaknya bahan mentah ke i , i = 1, 2, 3, ……………. , maij : banyaknya bahan mentah ke i yg digunakan utk memproduksi 1
satuan barang ke j
xj > 0 , j = 1, 2,…,n ; cj tdk boleh neg, paling kecil 0 (nonnegativity consraint)
Maksimumdp < h1 artinya, pemakaian input tidak boleh melebihi h1
Minimumdp > h1 artinya, pemakaian input paling tidak dipenuhi h1
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3. Langkah-langkah dan teknik pemecahan
Dasar dari pemecahan PL adalah suatu tindakan yang berulang (Inter-active search) dengan sekelompok carauntuk mencapai suatu hasil optimal. Tidakan dilakukandengan cara sistimatis.
Selanjutnya langkah2 dari tindakan berulang adl sebagai:
1. Tentukan kemungkinan2 kombinasi yang baik dari sum-ber daya al-am yang terbatas atau fasilitas yang terse-dia, yang disebut se-bagai initial feasible solution.
2. Selesaikan persamaan pembatasan strukturil untuk me-ndapatkan titik2 ekstreem (dis sebagai ‘basic feasible solution’).
3. Tentukanlah nilai dari titik2 ekstreem yang akan merupa-kan nilai2 pilihan, yang telah disesuaikan dengan nilaitujuan dari permasalahan.
4. Ulanglah langkah 3 hingga tercapai tujuan optimal (ha-nya satu yang bernilai tertinggi atau terendah).
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Ada 3 (tiga) cara pemecahan PL
1. Cara dengan menggunakan grafik
2. Cara dengan substitusi (cara matematik/Aljabar)
3. Cara simplex
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General Form of an Optimization Problem
MAX (or MIN): f0(X1, X2, …, Xn)
Subject to: f1(X1, X2, …, Xn)<=b1
:
fk(X1, X2, …, Xn)>=bk
:
fm(X1, X2, …, Xn)=bm
Note: If all the functions in an optimization are linear, the problem is a Linear Programming (LP) problem
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Linear Programming (LP) Problems
MAX (or MIN): c1X1 + c2X2 + … + cnXn
Subject to: a11X1 + a12X2 + … + a1nXn <= b1
:
ak1X1 + ak2X2 + … + aknXn >=bk
:
am1X1 + am2X2 + … + amnXn = bm
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An Example LP Problem
Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes.
There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available.
Aqua-Spa Hydro-Lux
Pumps 1 1
Labor 9 hours 6 hours
Tubing 12 feet 16 feet
Unit Profit $350 $300
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5 Steps In Formulating LP Models:
1. Understand the problem.
2. Identify the decision variables.
X1=number of Aqua-Spas to produce
X2=number of Hydro-Luxes to produce
3. State the objective function as a linear combination of the decision variables.
MAX: 350X1 + 300X2
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5 Steps In Formulating LP Models(continued)
4. State the constraints as linear combinations of the decision variables.
1X1 + 1X2 <= 200 } pumps
9X1 + 6X2 <= 1566 } labor
12X1 + 16X2 <= 2880 } tubing
5. Identify any upper or lower bounds on the decision variables.
X1 >= 0
X2 >= 0Rabu, 05 Nopember 2008
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LP Model for
Blue Ridge Hot Tubs
MAX: 350X1 + 300X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1 >= 0
X2 >= 0
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Solving LP Problems: An Intuitive Approach
• Idea: Each Aqua-Spa (X1) generates the highest unit profit ($350), so let’s make as many of them as possible!
• How many would that be?
– Let X2 = 0
• 1st constraint: 1X1 <= 200
• 2nd constraint: 9X1 <=1566 or X1 <=174
• 3rd constraint: 12X1 <= 2880 or X1 <= 240
• If X2=0, the maximum value of X1 is 174 and the total profit is $350*174 + $300*0 = $60,900
• This solution is feasible, but is it optimal?
• No!Rabu, 05 Nopember 2008
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Solving LP Problems:
A Graphical Approach
• The constraints of an LP problem defines its feasible region.
• The best point in the feasible region is the optimal solution to the problem.
• For LP problems with 2 variables, it is easy to plot the feasible region and find the optimal solution.
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X2
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 200)
(200, 0)
boundary line of pump constraint
X1 + X2 = 200
Plotting the First Constraint
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X2
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 261)
(174, 0)
boundary line of labor constraint
9X1 + 6X2 = 1566
Plotting the Second Constraint
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X2
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 180)
(240, 0)
boundary line of tubing constraint
12X1 + 16X2 = 2880
Feasible Region
Plotting the Third Constraint
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X2Plotting A Level Curve of the
Objective Function
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 116.67)
(100, 0)
objective function
350X1 + 300X2 = 35000
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A Second Level Curve of the Objective FunctionX2
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 175)
(150, 0)
objective function
350X1 + 300X2 = 35000
objective function 350X1 + 300X2 = 52500
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Using A Level Curve to Locate the Optimal SolutionX2
X1
250
200
150
100
50
0
0 50 100 150 200 250
objective function
350X1 + 300X2 = 35000
objective function
350X1 + 300X2 = 52500
optimal solution
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Calculating the Optimal Solution
• The optimal solution occurs where the “pumps” and “labor” constraints intersect.
• This occurs where:
X1 + X2 = 200 (1)
and 9X1 + 6X2 = 1566 (2)
• From (1) we have, X2 = 200 -X1 (3)
• Substituting (3) for X2 in (2) we have,
9X1 + 6 (200 -X1) = 1566
which reduces to X1 = 122
• So the optimal solution is,
X1=122, X2=200-X1=78
Total Profit = $350*122 + $300*78 = $66,100Rabu, 05 Nopember 2008
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Enumerating The Corner PointsX2
X1
250
200
150
100
50
0
0 50 100 150 200 250
(0, 180)
(174, 0)
(122, 78)
(80, 120)
(0, 0)
obj. value = $54,000
obj. value = $64,000
obj. value = $66,100
obj. value = $60,900obj. value = $0
Note: This technique will not work if the solution is unbounded.
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Summary of Graphical Solution
to LP Problems
1. Plot the boundary line of each constraint
2. Identify the feasible region
3. Locate the optimal solution by either:
a. Plotting level curves
b. Enumerating the extreme points
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Special Conditions in LP Models
• A number of anomalies can occur in LP problems:
– Alternate Optimal Solutions
– Redundant Constraints
– Unbounded Solutions
– Infeasibility
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Example of Alternate Optimal Solutions
X2
X1
250
200
150
100
50
0
0 50 100 150 200 250
450X1 + 300X2 = 78300
objective function level curve
alternate optimal solutions
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Example of a Redundant ConstraintX2
X1
250
200
150
100
50
0
0 50 100 150 200 250
boundary line of tubing constraint
Feasible Region
boundary line of pump constraint
boundary line of labor constraint
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Example of an Unbounded SolutionX2
X1
1000
800
600
400
200
0
0 200 400 600 800 1000
X1 + X2 = 400
X1 + X2 = 600
objective function
X1 + X2 = 800
objective function
-X1 + 2X2 = 400
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Example of InfeasibilityX2
X1
250
200
150
100
50
0
0 50 100 150 200 250
X1 + X2 = 200
X1 + X2 = 150
feasible region for second constraint
feasible region for first constraint
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