modeling & decision analysis - gunadarma...

11/5/2008

1

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.

Rabu, 05 Nopember 2008

METODE KUANTITATIF 2

11/5/2008

2

• 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.



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



11/5/2008

3

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:

$600 million in savings



11/5/2008

4

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



11/5/2008

5

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.


10METODE KUANTITATIF

11/5/2008

6

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)



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



11/5/2008

7

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.



Example of a Mathematical Model

Profit = Revenue - Expenses

or

Profit = f(Revenue, Expenses)

or

Y = f(X1, X2)



11/5/2008

8

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:



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).



11/5/2008

9

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

ModelAnalyze Model

Test Results

Implement Solution

unsatisfactoryresults



11/5/2008

10

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 ?



11/5/2008

11

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



11/5/2008

12

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.



11/5/2008

13

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



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



11/5/2008

14

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).



Ada 3 (tiga) cara pemecahan PL

1. Cara dengan menggunakan grafik

2. Cara dengan substitusi (cara matematik/Aljabar)

3. Cara simplex



11/5/2008

15

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



Linear Programming (LP) Problems

MAX (or MIN): c1X1 + c2X2 + … + cnXn

Subject to: a11X1 + a12X2 + … + a1nXn <= b1

:

ak1X1 + ak2X2 + … + aknXn >=bk

:

am1X1 + am2X2 + … + amnXn = bm



11/5/2008

16

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



11/5/2008

17

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


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



11/5/2008

18

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


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.



11/5/2008

19

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

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



11/5/2008

20

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



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



11/5/2008

21

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



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



11/5/2008

22

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


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 = $60,900obj. value = $0

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



11/5/2008

23

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



11/5/2008

24

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 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



11/5/2008

25

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



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



modeling & decision analysis - gunadarma...

Documents