Modeling & Decision Analysis

11/5/2008

Modeling & Decision Analysis DR. MOHAMMAD ABDUL MUKHYI, SE., MM.

Rabu, 05 Nopember 2008

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Metode Kuantitatif Spektrum situasi keputusan: - Terstruktur/terprogram - Tak terstruktur/tak terprogram - Sebagian terstruktur Pembagian masalah : - Certainty : semua alternatif tindakan diketahui dan hanya terdapat satu konsekuansi untuk masing-masing tindakan. - Risk : apabila terdapat lebih dari satu konsekuensi atau alternatif dan pengambil keputusan mengetahui probabilitas konsekuensinya. - Ucertainty :apabila jumlah kemungkinan konsekuensi tidak diketahui oleh pengambil keputusan. Rabu, 05 Nopember 2008

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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, multivendor contract awards – Benefits: $600 million in savings Rabu, 05 Nopember 2008

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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 & understanding that improves decision making. Rabu, 05 Nopember 2008

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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) Where:

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 Rabu, 05 Nopember 2008

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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). Rabu, 05 Nopember 2008

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Categories of Mathematical Models Model Category Prescriptive

Form of

Independent Variables

f(.)

OR/MS Techniques

known, well-defined

known or under decision maker’s control

LP, Networks, IP, CPM, EOQ, NLP, GP, MOLP

Predictive

unknown, ill-defined

known or under decision maker’s control

Regression Analysis, Time Series Analysis, Discriminant Analysis

Descriptive

known, well-defined

unknown or uncertain

Simulation, PERT, Queueing, Inventory Models


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The Problem Solving Process

Identify Problem

Formulate & Implement Model

Analyze Model

Test Results

Implement Solution

unsatisfactory results


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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 Rabu, 05 Nopember 2008

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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

Payoffs $1,500

Alternative A Initial state Heads (50%) Alternative B (Flip coin)


Tails (50%)

$2,000 $1,000

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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. Rabu, 05 Nopember 2008

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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 PL Cari : 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 Rabu, 05 Nopember 2008

j = 1 , 2, 3 ………n nonnegativity consraint METODE KUANTITATIF

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srs : sedemikian rupa sehingga dp : dengan pembatas ada 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, ……………. , m aij : 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) Maksimum dp < h1 artinya, pemakaian input tidak boleh melebihi h1 Minimum dp > 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 cara untuk mencapai suatu hasil optimal. Tidakan dilakukan dengan cara sistimatis. Selanjutnya langkah2 dari tindakan berulang adl sebagai: 1. Tentukan kemungkinan2 kombinasi yang baik dari sumber daya al-am yang terbatas atau fasilitas yang tersedia, yang disebut se-bagai initial feasible solution. 2. Selesaikan persamaan pembatasan strukturil untuk mendapatkan titik2 ekstreem (dis sebagai ‘basic feasible solution’). 3. Tentukanlah nilai dari titik2 ekstreem yang akan merupakan nilai2 pilihan, yang telah disesuaikan dengan nilai tujuan dari permasalahan. 4. Ulanglah langkah 3 hingga tercapai tujuan optimal (hanya satu yang bernilai tertinggi atau terendah). Rabu, 05 Nopember METODE KUANTITATIF 27 2008

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 Rabu, 05 Nopember 2008

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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. Pumps Labor Tubing Unit Profit

Aqua-Spa 1 9 hours 12 feet $350

Hydro-Lux 1 6 hours 16 feet $300

There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. Rabu, 05 Nopember 2008

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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. } pumps 1X1 + 1X2 <= 200 9X1 + 6X2 <= 1566 } labor 12X1 + 16X2 <= 2880 } tubing 5. Identify any upper or lower bounds on the decision variables. X1 >= 0 X2 >= 0 Rabu, 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!


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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. Rabu, 05 Nopember 2008

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Plotting the First Constraint

X2 250

(0, 200) 200

boundary line of pump constraint X1 + X2 = 200

150

100

50 (200, 0) 0 0

50


100

150

200

250

X1

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Plotting the Second Constraint

X2

(0, 261)

250 boundary line of labor constraint 9X1 + 6X2 = 1566

200

150

100

50 (174, 0) 0 0


50

100

150

200

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X1

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Plotting the Third Constraint

X2 250

(0, 180) 200

150 boundary line of tubing constraint 12X1 + 16X2 = 2880

100 Feasible Region

50 (240, 0) 0 0


50

100

150

200

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X1

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Plotting A Level Curve of the Objective Function

X2 250

200 (0, 116.67)

objective function

150

350X1 + 300X2 = 35000 100 (100, 0)

50

0 0


50

100

150

200

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X1

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A Second Level Curve of the Objective Function

X2 250

(0, 175)

objective function 350X1 + 300X2 = 35000

200


150

100 (150, 0)

50

0 0


50

100

150

200

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X1

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Using A Level Curve to Locate the Optimal Solution

X2 250


200

150 optimal solution 100 objective function 350X1 + 300X2 = 52500

50

0 0


50

100

150

200

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X1

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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 = $66,100 Rabu, 05 Nopember 2008

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Enumerating The Corner Points

X2 250

obj. value = $54,000 (0, 180)

200

Note: This technique will not work if the solution is unbounded.

obj. value = $64,000

150

(80, 120) obj. value = $66,100 (122, 78)

100

50

obj. value = $60,900 (174, 0)

obj. value = $0 (0, 0)

0 0

50


100

150

200

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X1

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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 250

objective function level curve 450X1 + 300X2 = 78300

200

150

100

alternate optimal solutions

50

0 0

100

50


150

200

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X1

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Example of a Redundant Constraint X2 250 boundary line of tubing constraint 200 boundary line of pump constraint 150 boundary line of labor constraint

100 Feasible Region

50

0 0


50

100

150

200

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X1

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Example of an Unbounded Solution X2 1000

objective function X1 + X2 = 600

800

-X1 + 2X2 = 400

objective function X1 + X2 = 800

600

400

200 X1 + X2 = 400

0 0


200

400

600

800 METODE KUANTITATIF

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X1

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Example of Infeasibility

X2 250

200

X1 + X2 = 200

feasible region for second constraint

150

100 feasible region for first constraint

50

X1 + X2 = 150

0 0


50

100

150

200

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X1

50

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Modeling & Decision Analysis

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