Modeling Decision Process

Chapter 5

The What's & Whys of Modeling What is a model?

A replica of a real system or object. An abstraction of reality

Model formats: Physical Graphical Verbal Mathematical

The What's & Whys of Modeling Why do we use models:

Understanding through simplification. Demonstrating and evaluating cause and

effect relationships. Experimenting with decision alternatives

on the real system is infeasible, too expensive, too dangerous, or just plain impossible.

Need for time compression for analysis of a system or prediction of future values.

The Whats & Whys of Modeling 3 conditions under which models

operate: Certainty: outcome of each alternative is

known Uncertainty: possible outcomes of each

alternative can be identified. Cannot estimate the probability of occurrence of the possible outcomes

Risk: possible outcomes of each alternative can be identified with probabilities attached

Basic Model Types

A survey of Models

Basic Model Types Descriptive/Predictive/Prescriptive Static/Dynamic

Static – no explicit acknowledgement of time

Dynamic – explicit inclusion of time as an element (time dependent)

Deterministic/Stochastic (based on the use of random numbers and probability statistics to investigate problems.)

Decision Model Classification Deterministic – optimization, linear

programming, financial planning, production planning, convex programming.

Probabilistic – queuing theory, linear regression, logic analysis, path analysis, time series.

Simulation – production modeling, transportation and logistics analysis, econometrics.

Modeling Steps Define & analyze the problem Select and/or construct the model

Variables: controllable Parameters: not controllable Objectives: singular or multiple Constraints: limits on possible solution

The model establishes relationships among variables, parameters, objectives, and constraints

Modeling Steps Validate the model: does the model

accurately represent the real system? Compare model output with historical or

real world data Have model evaluated by experts Have model evaluated by decision-makers Compare model output with expectations

based on experience & expertise

Modeling Steps Acquire input data

Input data must be accurate & timely. Use data to design modeling experiments

Solve the model / develop the solution Test the model solution

Is it realistic ? Is it valid? Sensitivity analysis of modeling

results Implementation of modeling results

Modeling & Decision-Making Strategies

Optimization Economic Optimization Utility Optimization

Satisficing “Good enough” solution Application of Heuristics

Elimination-by-Aspects Stepwise application of decision criteria

Modeling & Decision-Making Strategies

Incrementalism Decision are based on past decision

outcomes Mixed Scanning

Elimination of alternatives through increasing amounts of information gathering

Influence Diagrams and Decision Trees

Influence Diagram A simple graphical representation of a

model Decision Tree

Complement influence diagram Modeling of choices and uncertainties

Components of Influence Diagrams and Decision Trees

DecisionsDecisions UncertaintiesUncertainties Final Outcomes

DecisionDecision

Alternative AAlternative A

Alternative B

Alternative CAlternative C

Alternative DAlternative D

Outcome AOutcome A

Outcome BOutcome B

Outcome COutcome C

Outcome DOutcome D

Uncertainty Model with Outcomes

Sales VolumeSales Volume

Low 0.30Low 0.30

Medium 0.50Medium 0.50

High 0.20High 0.20

Simple Decision Tree

Enter Contest

Do Not Enter Contest

Win ContestWin Contest

Lose Contest

Win large return on wager

Lose wager

Lose/Gain nothing

Basic Risky Decision

Decision

Uncertainty

ObjectiveObjective

Buy Stock

Do Not Buy Stock

Price goes upPrice goes up

Price goes down

Gain

Loss

Lose/Gain nothing

Decision Tree for Odds Forecasting Method

Bet on Vikes

Bet Against VikesBet Against Vikes

Vikes Win

Vikes Win

Vikes Lose

Vikes Lose

$X

-$X

-$Y

$Y

Decision Tree for Comparison Forecasting Method

Uncertainty Game

Reference GameReference Game

Win

(P)

Lose

(1 – P)

European Vacation

-$100

European Vacation

-$100

A Variety of Models Decision Tables Game Theory Mathematical & Linear

Programming Simulation Forecasting Analytic Hierarchy Process

Decision Tables Decision Alternatives

Controllable State of Nature

Not controllable Uncertainty or Risk

Payoffs Product of Decision Alternative and

states of Nature

Decision Tables Decision Goal: what new store to

open

State of Nature Alternative recision recovery economic boom

Stereo Eqpmt 10,000 30,000 60,000Book Store 30,000 45,000 20,000Food Store 55,000 30,000 10,000

Decision Tables / Uncertainty

State of Nature

Alternative recision recovery economic boom

Stereo Eqpmt 10,000 30,000 60,000

Book Store 30,000 45,000 20,000

Food Store 55,000 30,000 10,000

• Optimistic Criterion: Stereo Equipment

•Highest payoff in table

• Pessimistic criterion: Book Store

•Take best of the worst payoffs of each alternatives

• Equal likelihood Criterion: Stereo Eqpmt.

•Highest average payoff per alternative

Decision Tables / Risk State of Nature

Alternative recision recovery economic boom

Stereo Eqpmt 10,000 30,000 60,000

Book Store 30,000 45,000 20,000

Food Store 55,000 30,000 10,000

• Expected Value = Sum(Payoff * respective Prob.)

• Expected Value Criterion: Book Store

•E.V. Stereo Equipment = $30,000

•E.V. Book Store = $35,500

•E.V. Discount Foods = $33,500

Game Theory Two (or more) players. Players act in self-interest only. Players have full information on

each other’s strategies or payoffs. Zero-Sum Game: one player’s

profit is the other player’s loss Non-Zero-Sum Game: both players

may win or lose simultaneously.

Mathematical Programming Modeling using mathematical

equations Usually requires solving for variables

and for simultaneous equations Linear Programming

Standard, programmable solution techniques

Non-Linear Programming Usually requires mathematical expertise

Linear Programming Furniture Makers Production Mix

Problem: Which production combination yields the

highest profit?

Tables Chairs Hours Avail.Carpentry 4 hrs 3 hrs 240 hours Painting 2 hrs 1 hr 100 hoursProfit/Unit $7 $5

Linear Programming Objective Function Max 7 T + 5 C

Constraints: Carpentry: 4 T + 3 C <= 240 Painting: 2 T + 1 C <=100 Non-negativity T,C >=0 Optimal Solution: Tables = 30 Chairs = 40 Revenue = 410

Simulation “The use of a model to represent the

critical characteristics of a system and to observe the system’s operations over time.”

Most common dynamic process modeling type.

Given heavy use of computers, simulation now very much resembles programming!

Monte Carlo Simulation Simulation of randomness into a

system, using Random Number Generator Cumulative Probability Distribution

Monte Carlo Simulation The Bakery Problem: how many

chocolate donuts to bake each day? Gather sales data for 100 days Sales Frequency Probability

30 20 days 20 %

31 35 days 35 %32 25 days 25 %33 15 days 15 %34 5 days 5 %

Monte Carlo Simulation Put the probabilities on the

roulette wheel…

79

80 94

95

99

0

79

20

19

79

Sales = 30Sales = 31Sales = 32Sales = 33Sales = 34

Monte Carlo Simulation …and start simulating

Generate a random number: 00-99 Find this number on the roulette-

wheel. Find the matching sales-levels Random Number Sales Level

35 31 donuts 82 33 donuts 01 30 donuts

Forecasting The prediction of future values,

based on past experience. Prediction based on personal

expertise. Prediction based on a mathematical

model.

Mathematical Forecasting A variety of techniques

Linear & Nonlinear regression Time Series / Box –Jenkins Technique Etc

These techniques differ in predictive quality, applicability, and ease of use

Forecasting - Regression The fitting of a line to a cloud of

observation-points, based on minimizing the distance between the line and the set of points

Dependentvariable

Independent variable

Forecasting - Regression Standard linear regression function:

Y = a + bX Y = dependent(forecast) variable X = independent variable a = intercept b = slope

Forecasting - Regression Multiple regression function:

Y = a + b1 X1 + b2 X2 + b3 X3 Y = dependent(forecast) variable X = independent variable a = intercept b = slope

Analytic Hierarchy Process

Method to solve Multiple-criteria decision-making

Specifies: Decision goal Decision Criteria Decision Alternatives

Real world decision problems multiple, diverse criteria qualitative as well as quantitative

information

Analytic Hierarchy Process

Comparing apples and oranges?Spend on defense or agriculture?Open the refrigerator - apple or orange?

Analytic Hierarchy Process

Goal

Criterion

Criterion Criterion

Alt. 1 Alt. 1 Alt. 1Alt. 2 Alt. 2 Alt. 2

Alt. 3 Alt. 3Alt. 3

Analytic Hierarchy Process Each criterion is rated against each

other criterion for its importance in achieving the goal

For each criterion separately, each alternative is rated against each other alternative for its capacity for satisfying the criterion

For large decisions, this will involve a large number of pair-wise comparisons

Analytic Hierarchy Process AHP computer programs determine

the consistency of the pair-wise comparisons. Sometimes, the comparison-phase

will need to be repeated If consistent, the AHP program will

provide a rank-order of the alternatives

AHP

Information is decomposed into a hierarchy of alternatives and criteria

Information is then synthesized to determine relative ranking of alternatives

Both qualitative and quantitative information can be compared using informed judgements to derive weights and priorities

Example: Car Selection

Objective Selecting a car

Criteria Style, Reliability, Fuel-economy

Cost? Alternatives

Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

Hierarchical tree

S tyle R e lia b ility F u e l E con o m y

S e lec tinga N e w C ar

- Civic- Saturn- Escort- Miata

- Civic- Saturn- Escort- Miata

- Civic- Saturn- Escort- Miata

Ranking of criteria

Weights? AHP

pair-wise relative importance [1:Equal, 3:Moderate, 5:Strong, 7:Very strong,

9:Extreme]

Style Reliability Fuel Economy

Style

Reliability

Fuel Economy

1/1 1/2 3/1

2/1 1/1 4/1

1/3 1/4 1/1

Ranking of priorities

Eigenvector [Ax = x]Iterate

1. Take successive squared powers of matrix2. Normalize the row sums

Until difference between successive row sums is

less than a pre-specified value

1 0.5 32 1 40.333 0.25 1.0

3.0 1.75 8.05.3332 3.0 14.01.1666 0.6667 3.0

squared

Row sums 12.75 22.3332 4.8333

39.9165

NormalizedRow sums 0.3194 0.5595 0.1211

1.0

• New iteration gives normalized row sum 0.3196 0.5584 0.1220

• Difference is: - 0.3194 0.5595 0.1211

0.3196 0.5584 0.1220

= - 0.0002 0.0011 - 0.0009

Preference Style .3196 Reliability .5584 Fuel Economy .1220

S tyle.3 196

R e lia b ility.5 584

F u e l E con o m y.1 220

S e lec tinga N e w C ar

1 .0

Ranking alternatives

Style

Civic

Saturn

Escort

1/1 1/4 4/1 1/6

4/1 1/1 4/1 1/4

1/4 1/4 1/1 1/5

Miata 6/1 4/1 5/1 1/1

Civic Saturn Escort Miata

Miata

Reliability

Civic

Saturn

Escort

1/1 2/1 5/1 1/1

1/2 1/1 3/1 2/1

1/5 1/3 1/1 1/4

Miata 1/1 1/2 4/1 1/1

Civic Saturn Escort Miata

.1160

.2470

.0600

.5770

Eigenvector

.3790

.2900

.0740

.2570

Fuel Economy(quantitative information)

Civic

Saturn

Escort

MiataMiata

34

27

24

28 113

Miles/gallon Normalized

.3010

.2390

.2120

.2480 1.0

S tyle.3 196

R e lia b ility.5 584

F u e l E con o m y.1 220

S e lec tinga N e w C ar

1 .0

- Civic .1160- Saturn .2470- Escort .0600- Miata .5770

- Civic .3790 - Saturn .2900- Escort .0740- Miata .2570

- Civic .3010- Saturn .2390- Escort .2120- Miata .2480

Ranking of alternatives

Style Reliability Fuel Economy

Civic

EscortMiataMiata

Saturn

.1160 .3790 .3010

.2470 .2900 .2390

.0600 .0740 .2120

.5770 .2570 .2480

* .3196

.5584

.1220

= .3060

.2720

.0940

.3280

Handling Costs

Dangers of including Cost as another criterion political, emotional responses?

Separate Benefits and Costs hierarchical trees

Costs vs. Benefits evaluation Alternative with best benefits/costs ratio

Cost vs. Benefits

MIATA $18K .333.9840

CIVIC $12K .222 1.3771

SATURN $15K .2778.9791

ESCORT $9K .1667.5639

CostNormalized Cost

Cost/Benefits Ratio

Complex decisions•Many levels of criteria and sub-criteria

Application areas strategic planning resource allocation source selection, program selection business policy etc., etc., etc..

AHP software (ExpertChoice) computations sensitivity analysis graphs, tables

Group AHP

Model Management Model Base Management System Basic Features:

Tracking a large variety of models, model-types, model-versions, purposes, etc.

Provide access to model & model descriptions.

Provide for new models and model updates to be placed in Model Base.

Model Management Model Base Management System Advanced Features Suggest appropriate models to

decision maker. Relate models to required data

from DSS database Allow decision maker to customize

models or build new models