Modeling & Decision Analysis

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Modeling Modeling Modeling Modeling

& Decision Analysis

DR. MOHAMMAD ABDUL MUKHYI, SE., MM.

Rabu, 05 Nopember 2008 METODE KUANTITATIF 1

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

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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.

Rabu, 05 Nopember 2008

METODE KUANTITATIF 3

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

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:

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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

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

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.

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, …, X_n)

Y = dependent variable

(aka bottom-line performance measure)

X_i = independent variables (inputs having an impact on Y) f(^.) = function defining the relationship between the X_i & Y Where:

Mathematical Models & Spreadsheets

• Most spreadsheet models are very similar to our generic mathematical model:

Y = f(X₁, X₂, …, X_n)

Most spreadsheets have input cells

(representing X_i) 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

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!

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

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

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 PL Cari : x₁, x₂, x₃, ……… , x_n.

Fungsi Tujuan : Z = c₁x₁+ c₂x₂+ c₃x₃+ …… + c_nx_n optimum (max/min) (srs)

Fungsi Kendala : a₁₁x₁+ a₁₂x₂ + a₁₃x₃+ ……… + a_1nx_n >< h₁. (F. Pembatas) a₂₁x₁+ a₂₂x₂ + a₂₃x₃+ ……… + a_2nx_n >< h₂. (dp) a₃₁x₁+ a₃₂x₂ + a₃₃x₃+ ……… + a_3nx_n >< h₃.

↓ …. ↓ …. ↓ ...………… ↓ .. ↓ a_m1x₁+ a_m2x₂+ a_m3x₃+ ….……… + a_mnx_n >< h_m.

x_j> 0 j = 1 , 2, 3 ………n nonnegativity consraint

srs : sedemikian rupa sehingga dp : dengan pembatas

ada n macam barang yg akan diproduksi masing2 sbesar x₁,x₂, … , x_n x_j: banyaknya barang yang diproduksi ke j , j = 1, 2, 3, …. , n c_j: harga per satuan barang ke j , j = 1, 2, 3, ………. , n

ada m macam bahan mentah, masing2 tersedia h₁, h₂, h₃, …., h_m h_i: banyaknya bahan mentah ke i , i = 1, 2, 3, ………. , m a_ij: banyaknya bahan mentah ke i yg digunakan utk memproduksi 1

satuan barang ke j

x_j> 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

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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 sum- ber daya al-am yang terbatas atau fasilitas yang tersedia, 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 nilai tujuan dari permasalahan.

4. Ulanglah langkah 3 hingga tercapai tujuan optimal (hanya satu yang bernilai tertinggi atau terendah).

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, …, X_n)

Subject to: f₁(X₁, X₂, …, X_n)<=b₁ :

f_k(X₁, X₂, …, X_n)>=b_k :

f_m(X₁, X₂, …, X_n)=b_m

Note: If all the functions in an optimization are linear, the problem is a Linear Programming (LP) problem

Linear Programming (LP) Problems

MAX (or MIN): c₁X₁+ c₂X₂+ … + c_nX_n

Subject to: a₁₁X₁+ a₁₂X₂+ … + a_1nX_n<= b₁ :

a_k1X₁ + a_k2X₂+ … + a_knX_n >=b_k :

a_m1X₁ + a_m2X₂ + … + a_mnX_n= b_m

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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

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>= 0

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 X₁is 174 and the total profit is $350*174 + $300*0 = $60,900

• This solution is feasible, but is it optimal?

• No!

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

X₂

250

200

150

100

50

(0, 261)

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

X₂

Plotting A Level Curve of the Objective Function

250

200

150

100

50

0

0 100 150

(0, 116.67)

(100, 0)

objective function 350X₁+ 300X₂= 35000

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A Second Level Curve of the Objective Function

X₂

X₁

250

200

150

100

50

0

0 50 100 150 200 250

(0, 175)

(150, 0) objective function

350X₁+ 300X₂= 35000

Using A Level Curve to Locate the Optimal Solution

X₂

250

200

150

100

50

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 X₂in (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

Enumerating The Corner Points

X₂

250

200

150

100

50

0

0 100 150

(0, 180)

(174, 0) (122, 78) (80, 120)

(0, 0)

obj. value = $54,000

obj. value = $64,000

obj. value = $66,100

obj. value = $60,900 obj. 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

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

Example of a Redundant Constraint

X₂

250

200

150

100

50

0

0 100 150

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 Solution

X₂

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

Example of Infeasibility

X₂

250

200

150

100

50

X₁+ X₂= 200

feasible region for second constraint

feasible region for first