Modeling and AnalysisModeling and Analysis

Week 8Week 8

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Modeling and Analysis TopicsModeling and Analysis Topics Modeling for MSS (a critical component) Static and dynamic models Treating certainty, uncertainty, and risk Influence diagrams (in the posted PDF file) MSS modeling in spreadsheets Decision analysis of a few alternatives (with

decision tables and decision trees) Optimization via mathematical programming Heuristic programming Simulation Model base management

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DSS ModelingDSS Modeling A key element in most DSS Leads to reduced cost and increased

revenue DuPont Simulates Rail Transportation System

and Avoids Costly Capital Expenses

Procter & Gamble uses several DSS models collectively to support strategic decisions

Locating distribution centers, assignment of DCs to warehouses/customers, forecasting demand, scheduling production per product type, etc.

Fiat, Pillowtex (…operational efficiency)…

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P&G Used optimization models to redesign its

distribution system Several models used:

Generating model (algorithm) to estimate transportation costs

Demand forecasting model (statistics based) Distribution center location model Linear programming transportation model to

determine best shipping Financial and risk simulation model that also

considers some qualitative factors GIS for a user interface

Some built in the DSS some external and some accessed as needed

500 employees involved over the course of a year

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AMR

Used models to optimize the altitude ascent and descent profile for planes

Saved millions in fuel cost per week

Part of SABRE system that used models extensively incremental revenues eventually exceeded $1 billion annually

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Major Modeling IssuesMajor Modeling Issues Problem identification and environmental

analysis (information collection) Variable identification

Influence diagrams, cognitive maps Forecasting/predicting

More information leads to better prediction Multiple models: A DSS can include

several models, each of which represents a different part of the decision-making problem Categories of models >>>

Model management Knowledge based modeling

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Influence Diagrams Influence Diagrams Graphical representations of a model

“Model of a model” A tool for visual communication Some influence diagram packages create and

solve the mathematical model Framework for expressing DSS model

relationshipsRectangle = a decision variableCircle = uncontrollable or intermediate variableOval = result (outcome) variable: intermediate or final

Variables are connected with arrows indicates the direction of influence (relationship)

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Influence Diagrams: Influence Diagrams: RelationshipsRelationships

Amount inCDs

InterestCollected

Price

Sales

Sales

~Demand

CERTAINTY

UNCERTAINTY

RANDOM (risk) variable: Place a tilde (~) above the variable’s name

The shape of The shape of the arrow the arrow

indicates the indicates the type of type of

relationshiprelationship

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Influence Diagrams: ExampleInfluence Diagrams: Example

~Amount used inAdvertisement

Unit Price

Units Sold

Unit Cost

Fixed Cost

Income

Expenses

Profit

An influence diagram for the profit modelAn influence diagram for the profit model

Profit = Income – ExpenseProfit = Income – ExpenseIncome = UnitsSold * UnitPriceIncome = UnitsSold * UnitPriceUnitsSold = 0.5 * Advertisement ExpenseUnitsSold = 0.5 * Advertisement ExpenseExpenses = UnitsCost * UnitSold + FixedCostExpenses = UnitsCost * UnitSold + FixedCost

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Influence Diagrams: SoftwareInfluence Diagrams: Software Analytica, Lumina Decision Systems

Supports hierarchical (multi-level) diagrams DecisionPro, Vanguard Software Co.

Supports hierarchical (tree structured) diagrams DATA Decision Analysis, TreeAge Software

Includes influence diagrams, decision trees and simulation

Definitive Scenario, Definitive Software Integrates influence diagrams and Excel, also

supports Monte Carlo simulations PrecisionTree, Palisade Co.

Creates influence diagrams and decision trees directly in an Excel spreadsheet

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Analytica Influence Diagram of a Analytica Influence Diagram of a Marketing Marketing

Problem: The Marketing ModelProblem: The Marketing Model

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Analytica: The Price SubmodelAnalytica: The Price Submodel

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Analytica: The Sales SubmodelAnalytica: The Sales Submodel

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Categories of ModelsCategories of ModelsCategory Objective Techniques

Optimization of problems with few alternatives

Find the best solution from a small number of alternatives

Decision tables, decision trees

Optimization via algorithm

Find the best solution from a large number of alternatives using a step-by-step process

Linear and other mathematical programming models

Optimization via an analytic formula

Find the best solution in one step using a formula

Some inventory models

Simulation Find a good enough solution by experimenting with a dynamic model of the system

Several types of simulation

Heuristics Find a good enough solution using “common-sense” rules

Heuristic programming and expert systems

Predictive and other models

Predict future occurrences, what-if analysis, …

Forecasting, Markov chains, financial, …

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Static and Dynamic ModelsStatic and Dynamic Models

Static Analysis Single snapshot of the situation Single interval Steady state

Dynamic Analysis Dynamic models Evaluate scenarios that change over time Time dependent Represents trends and patterns over time More realistic: Extends static models

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Mathematical Models

4 basic components Result variables

Reflect the level of effectiveness of a system Decision variables

Alternative courses of action Uncontrollable variables

constraints Intermediate result variables

Intermediate outcomes

Mathematical relationships link the components together

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Examples of the Components of Models

Area Decision variables

Result variables Uncontrollable variables

Financial investment

Investment alternatives and amounts

Total profit, risk, ROI, EPS, liquidity

Inflation rate, prime rate, competition

Marketing Ad budget, where to advertise

Market share, customer satisfaction

Customer’s income, competitor’s actions

Manufacturing What and how much to produce, inventory levels

Total cost, quality level

Machine capacity, material prices

Accounting Audit schedule Error rate Tax rates, legal requirements

Transportation Shipments schedule, use of smart cards

Total transportation cost

Delivery distance, regulations

Services Staffing levels Customer satisfaction

Demand for services

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Certainty, Uncertainty and RiskCertainty, Uncertainty and Risk

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Decision Making:Decision Making:Treating Certainty, Uncertainty and Treating Certainty, Uncertainty and RiskRisk Certainty Models

Assume complete knowledge All potential outcomes are known May yield optimal solution

Uncertainty Several outcomes for each decision Probability of each outcome is unknown Knowledge would lead to less uncertainty

Risk analysis (probabilistic decision making) Probability of each of several outcomes

occurring Level of uncertainty => Risk (expected value)

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DSS Modeling with SpreadsheetsDSS Modeling with Spreadsheets Spreadsheet: most popular end-user modeling

tool Flexible and easy to use Powerful functions

Add-in functions and solvers Programmability (via macros) What-if analysis Goal seeking Simple database management Seamless integration of model and data Incorporates both static and dynamic models Examples: Microsoft Excel, Lotus 1-2-3

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Excel spreadsheet - static model example: Excel spreadsheet - static model example: Simple loan calculation of monthly Simple loan calculation of monthly paymentspayments

1)1(

)1(

)1(

n

n

n

i

iiPA

iPF

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Excel spreadsheet - Excel spreadsheet - Dynamic model Dynamic model example: example: Simple loan Simple loan calculation of calculation of monthly payments monthly payments and effects of and effects of prepaymentprepayment

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Decision Analysis: A Few Decision Analysis: A Few AlternativesAlternativesSingle Goal Situations Decision tables

Multiple criteria decision analysis

Features include decision variables (alternatives), uncontrollable variables, result variables

Decision trees Graphical representation of

relationships Multiple criteria approach Demonstrates complex

relationships Cumbersome, if many

alternatives exists

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Decision TablesDecision Tables Investment example

One goal: maximize the yield after one year

Yield depends on the status of the economy (the state of nature) Solid growth Stagnation Inflation

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Investment Example: Investment Example: Possible SituationsPossible Situations

1. If solid growth in the economy, bonds yield 12%; stocks 15%; time deposits 6.5%

2. If stagnation, bonds yield 6%; stocks 3%; time deposits 6.5%

3. If inflation, bonds yield 3%; stocks lose 2%; time deposits yield 6.5%

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Payoff Decision variables (alternatives) Uncontrollable variables (states of economy) Result variables (projected yield)

Tabular representation:

Investment Example: Investment Example: Decision TableDecision Table

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Investment Example: Investment Example: Treating UncertaintyTreating Uncertainty Optimistic approach Pessimistic approach Treating Risk:

Use known probabilities Risk analysis: compute expected values

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Decision Analysis: A Few Decision Analysis: A Few AlternativesAlternatives

Other methods of treating risk Simulation, Certainty factors, Fuzzy

logic Multiple goals

Yield, safety, and liquidity

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DSS Mathematical ModelsDSS Mathematical Models

Decision Decision VariablesVariables

MathematicalMathematicalRelationshipsRelationships

UncontrollableUncontrollableVariablesVariables

Result Result VariablesVariables

Non-Quantitative Models (Qualitative) Captures symbolic relationships between decision variables,

uncontrollable variables and result variables Quantitative Models: Mathematically links decision

variables, uncontrollable variables, and result variables

Decision variables describe alternative choices. Uncontrollable variables are outside decision-maker’s control Result variables are dependent on chosen combination of decision

variables and uncontrollable variables

Independent Variables

Independent Variables

Dependent VariablesDependent Variables

IntermediateIntermediateVariablesVariables

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Optimization Optimization via Mathematical Programmingvia Mathematical Programming Mathematical Programming

A family of tools designed to help solve managerial problems in which the decision maker must allocate scarce resources among competing activities to optimize a measurable goal

Optimal solution: The best possible solution to a modeled problem Linear programming (LP): A mathematical

model for the optimal solution of resource allocation problems. All the relationships are linear

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LP Problem CharacteristicsLP Problem Characteristics

1.Limited quantity of economic resources2.Resources are used in the production of

products or services3.Two or more ways (solutions, programs)

to use the resources4.Each activity (product or service) yields

a return in terms of the goal5.Allocation is usually restricted by

constraints

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LineLine

Linear Programming StepsLinear Programming Steps 1. Identify the …

Decision variables Objective function Objective function coefficients Constraints

Capacities / Demands

2. Represent the model LINDO: Write mathematical formulation EXCEL: Input data into specific cells in Excel

3. Run the model and observe the results

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LP ExampleLP Example

The Product-Mix Linear Programming Model MBI Corporation Decision: How many computers to build next month? Two types of mainframe computers: CC7 and CC8 Constraints: Labor limits, Materials limit, Marketing

lower limits

CC7 CC8 Rel LimitLabor (days) 300 500 <= 200,000 /moMaterials ($) 10,000 15,000 <= 8,000,000 /moUnits 1 >= 100Units 1 >= 200Profit ($) 8,000 12,000 Max

Objective: Maximize Total Profit / Month

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LP SolutionLP Solution

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LP SolutionLP Solution

Decision Variables:X1: unit of CC-7

X2: unit of CC-8 Objective Function:

Maximize Z (profit)Z=8000X1+12000X2

Subject To300X1 + 500X2 200K

10000X1 + 15000X2 8000K

X1 100

X2 200

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Sensitivity, What-if, and Sensitivity, What-if, and Goal Seeking AnalysisGoal Seeking Analysis Sensitivity

Assesses impact of change in inputs on outputs Eliminates or reduces variables Can be automatic or trial and error

What-if Assesses solutions based on changes in

variables or assumptions (scenario analysis) Goal seeking

Backwards approach, starts with goal Determines values of inputs needed to achieve

goal Example is break-even point determination

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Heuristic ProgrammingHeuristic Programming

Cuts the search space Gets satisfactory solutions

more quickly and less expensively

Finds good enough feasible solutions to very complex problems

Heuristics can be Quantitative Qualitative (in ES)

Traveling Salesman Problem >>>

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Heuristic Programming - SEARCHHeuristic Programming - SEARCH

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Traveling Salesman ProblemTraveling Salesman Problem What is it?

A traveling salesman must visit customers in several cities, visiting each city only once, across the country. Goal: Find the shortest possible route

Total number of unique routes (TNUR):TNUR = (1/2) (Number of Cities – 1)!Number of Cities TNUR

5 12 6 60 9 20,160

20 1.22 1018

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When to Use HeuristicsWhen to Use Heuristics

When to Use Heuristics Inexact or limited input data Complex reality Reliable, exact algorithm not available Computation time excessive For making quick decisions

Limitations of Heuristics Cannot guarantee an optimal solution

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SimulationSimulation

Technique for conducting experiments with a computer on a comprehensive model of the behavior of a system

Frequently used in DSS tools

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Imitates reality and capture its richness Technique for conducting experiments Descriptive, not normative tool Often to “solve” very complex problems

Simulation is normally used only when a problem is too complex to be treated using numerical optimization techniques

Major Characteristics of Major Characteristics of SimulationSimulation

!!

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Advantages of SimulationAdvantages of Simulation The theory is fairly straightforward Great deal of time compression Experiment with different alternatives The model reflects manager’s perspective Can handle wide variety of problem types Can include the real complexities of

problems Produces important performance measures Often it is the only DSS modeling tool for

non-structured problems

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Limitations of SimulationLimitations of Simulation

Cannot guarantee an optimal solution Slow and costly construction process Cannot transfer solutions and inferences

to solve other problems (problem specific)

So easy to explain/sell to managers, may lead overlooking analytical solutions

Software may require special skills

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Simulation MethodologySimulation Methodology Model real system and conduct repetitive experiments. Steps:

1. Define problem 5. Conduct experiments2. Construct simulation model 6. Evaluate results3. Test and validate model 7. Implement solution4. Design experiments

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Simulation TypesSimulation Types Stochastic vs. Deterministic Simulation

In stochastic simulations: We use distributions (Discrete or Continuous probability distributions)

Time-dependent vs. Time-independent Simulation

Time independent stochastic simulation via Monte Carlo technique (X = A + B)

Discrete event vs. Continuous simulation

Simulation Implementation Visual simulation

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Visual interactive modeling (VIM)Also called Visual interactive problem solving Visual interactive modeling Visual interactive simulation

Uses computer graphics to present the impact of different management decisions

Often integrated with GIS Users perform sensitivity analysis Static or a dynamic (animation) systems

Visual Interactive Modeling (VIM) Visual Interactive Modeling (VIM) / Visual Interactive Simulation / Visual Interactive Simulation (VIS)(VIS)

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Model Base ManagementModel Base Management MBMS: capabilities similar to that of

DBMS But, there are no comprehensive model

base management packages Each organization uses models

somewhat differently There are many model classes

Within each class there are different solution approaches

Relations MBMS Object-oriented MBMS