syst 542 decision support systems engineeringseor.vse.gmu.edu/~klaskey/syst542/dss_unit3.pdf ·...

Unit 3 - 1 -

Department of Systems Engineering and Operations Research

Copyright © 2006, Kathryn Blackmond LaskeySYST 542

SYST 542Decision Support Systems

EngineeringInstructor: Kathryn Blackmond Laskey

Spring Semester, 2006

Unit 3: DSS Elements:The Model Subsystem (1)

Decision Analysis & Optimization

Unit 3 - 2 -



Outline

• Developing the model subsystem• Role of decision theory in DSS• Brief survey of decision analysis

and optimization methods

Unit 3 - 3 -



Models and DSS• A model is a representation of a system which

can be used to answer questions about thesystem

• A DSS uses computer models in conjunctionwith human judgment

– Performs computations that assist user with decision problem– Design is based on a model of how human user does / ought to solve

decision problem

• Model subsystem can be:– completely automated– partially automated– manual with automated support for information entry,

retrieval and display

Unit 3 - 4 -



DSS and Exploratory Models• DSS modeling is by definition exploratory

– Human remains in the loop

• Consolidative model may be possible for partsof problem

– Avoid the temptation to pour too many resources into thepart you know how to model!

• Good DSS helps DM make use of partialinformation

– to generate hypotheses about system behavior– to demonstrate occurrence of types of behavior under not-

too-implausible assumptions– to explore possible risks / failure modes– to determine regions of parameter space in which certain

qualitative behaviors occur

Unit 3 - 5 -



Issues for Exploratory Modeling• Representing the ensemble of models

– internal system representation– decision maker’s mental model– language for communicating with decision maker

• Tools for allowing DM to explore alternativemodeling assumptions

– what-if analysis– sensitivity analysis– exploring different parts of parameter space– exploring different combinations of modeling assumptions

• Techniques for helping DM assess consequences ofalternative assumptions

– summaries of high-dimensional data– graphical displays

Unit 3 - 6 -



Steps in Developingthe Model Subsystem

1. Map functions in decision process ontomodels

2. Determine input / output requirements formodels

3. Develop interface specifications for modelswith each other and with dialog and datasubsystemsthis step may result in additional modeling activity

4. Obtain / develop software realizations of themodels and interfaces

Unit 3 - 7 -



Decision Process (review)Identify Problem

Identify Objectives (values)

Identify Alternatives

Decompose and Model Problem – Structure – Uncertainty – Preference

Choose Best Alternative

Sensitivity Analysis

MoreAnalysis Needed

Make Recommendation

Yes

No

{Use ofDSS

Thanks to Andy Loerch

Unit 3 - 8 -



Some Typical Problems to Model• Evaluate benefits of proposed policy against costs• Forecast value of variable at some time in the

future• Evaluate whether likely return justifies investment• Decide where to locate a facility• Decide how many people to hire & where to assign

them• Plan activities and resources for a project• Develop repair, replacement & maintenance policy• Develop inventory control policy

Unit 3 - 9 -



A Brief Tour of Modeling Options• A wide variety of modeling approaches is

available• DSS developer must be familiar with broad

array of methods• It is important to know the class of problems

for which each method is appropriate• It is important to know the limitations of each

method• It is important to know the limitations of your

knowledge and when to call in an expert

Unit 3 - 10 -



Decision Theory• Formal theory to support GOOD-D process• Goals (What do I want?)

– Begin with value-focused thinking– Quantify values with utility function

• Options (What can I do?)• Outcomes (What might happen?)

– Quantify uncertainty with probability distribution• Decide:

– Develop a mathematical model of expected utility for each option– Model recommends the option for which expected utility is greatest– In a good decision analysis, model building process increases

understanding of decision problem– The model gives insight but the decision maker makes the final choice

• Do it!– Discussion and evaluation of options should consider issues of

implementation

Unit 3 - 11 -



Role of Decision Theory in DSS• Avoid “elicit model out of decision maker’s

head, push the button and solve for the correctanswer” mentality

• Decision theoretic models are appropriatewhen:– We can quantify values and uncertainties to a reasonable

approximation– It is useful to suggest potentially optimal solutions and/or

to weed out clearly suboptimal solutions

• Useful outputs (in addition to recommendedsolution)– Explanation of results– Sensitivity analysis– Visualization of feasible region

Unit 3 - 12 -



Decision Analysis• Collection of analytic and heuristic procedures for

developing decision theoretic model• Goals of decision analysis

– Organize or structure complex problems for analysis– Deal with tradeoffs between multiple objectives– Identify and quantify sources of uncertainty– Incorporate subjective judgments

• Decision analysis methods help to:– decompose problem into subproblems which are easier to solve– detect and resolve inconsistencies in solutions to the

subproblems– aggregate solutions to subproblems into a consistent action

recommendation for the original problem

Unit 3 - 13 -



Decision Analysis Methods• Value Models: Multiattribute Utility• Uncertainty Models: Decision Trees

– A structured representation for options and outcomes– A computational architecture for solving for expected utility– Best with “asymmetric” problems (different actions lead to qualitatively

different worlds)• Uncertainty Models: Influence Diagrams

– A structured representation for options, outcomes and values– A computational architecture for solving for expected utility– Best with “symmetric” problems (different actions lead to worlds with

qualitatively similar structure)• Decision analysis software:

– http://faculty.fuqua.duke.edu/daweb/dasw.htm (there are some brokenlinks)

Unit 3 - 14 -



Example: Patient TreatmentA patient is suspected of having a disease. Treated patientsrecover quickly from the illness, but the treatment hasunpleasant side effects. Untreated patients suffer a longand difficult illness but eventually recover.

Disease

Treatment

Outcome Utility

Influence Diagram

UT

UD

UN

Treat

Don’ttreat

Disease

No disease

Decision Tree

Utility

Speed ofRecovery

Side Effects

MultiattributeHierarchy

Goals:• Recovery• Freedom from side effects

Options:• Treat of don’t treat

Outcomes:• Sick/Well• Side Effects / No Side Effects

Unit 3 - 15 -



Value Model

• Objectives related to alternatives by Attributes• Attributes are measures of achievement of objectives

– Quantitative– Reflect consequences

• Usually decision maker has multiple objectives– Objectives are often in conflict– Value model incorporates tradeoffs among objectives

• Types of value model– Ordinal - ranking only– Measurable value function - strength of preference– Utility function - includes risk attitude

• Medical example:– Need to assess relative degree of misery of side effects vs illness– Need utility model to trade off chance of illness against cost of

side effects

Unit 3 - 16 -



Constructing a Value Model• Decompose objectives

– Independent components of value (avoid double-counting)– Begin with fundamental objective and decompose into important means

objectives• Find ways to measure objectives

– Natural attribute (e.g., cost in dollars, weight in pounds)– Constructed attribute (e.g., consumer price index for inflation)– Proxy attribute (e.g., sulfur dioxide emissions for erosion of monuments

from acid rain)• Combine objectives

– Turn attribute scores into value function» Better options have higher value» Equal differences in value function are equally valued by DM

– Functional form depends on relationship between attributes» Most common combination method is linear additive with cutoffs» Justification depends on independence assumptions

– Weights trade off objectives against each other» Subjective» Need to consider range of weights

• Adjust for risk attitude if necessary

Unit 3 - 17 -



Linear Additive Value Function• Value function is weighted sum of single-

attribute value functions– v(x1, …, xn) = w1v1(x1) + … + wnvn(xn)

• Requires attributes to be preferentiallyindependent:– Preference order between levels of any pair Xi and Xj of

attributes does not depend on levels of other attributes

• Much simpler to specify and use than morecomplex functional forms

• Try to specify attributes to be preferentiallyindependent

Unit 3 - 18 -



Example Multiattribute Hierarchy:Buying a Beach House

Total

Utility

Financial

Enjoyment

Initial

Investment

NPV

Time Spent

Luxury

Ocean access

Walking time

View

• Decompose value into attributes– nonoverlapping– cover all important aspects of value– bottom level attributes are measurable

• Assess function for combiningattributes at each level (usually linearweighted average)

• Compute utilities of all options– score on bottom-level attributes– compute overall score

Unit 3 - 19 -



Assessing Weights: Swing Weight Method

• First weight– Imagine all attributes are at worst level (may be imaginary)– Which would you choose to increase to best level?– Assign this attribute weight of 1

• Rest of weights– All attributes are at worst level again– Pick another attribute to move to best level– What % of value of moving first to its best level?

• Scale all weights to sum to 1

Attribute 1

Worst

Best

Attribute 2

Worst

Best70% w2 = 0.7 w1w1 + w2 = 1

w1 = 0.59w2 = 0.41

Beware: Some commonlyused weight assessmentmethods ignore absolute

scale of attributes and canlead to preference reversals.

Unit 3 - 20 -



Analytic Hierarchy Process• Popular method for building a preference

model• Problem decomposition into multiattribute

hierarchy is same as for multiattribute utility• Method of assigning weights is different

– Based on paired comparisons– Pairs of options are compared on scale from 0 to 9– Ratings are used to develop weights for the value function

• Comments– Method is popular because paired comparisons are natural

and intuitive to many decision makers– Theoretical justification of the MAU “swing weight”

assessment is lacking– Can have preference reversals when options are added or

removed from the option set (i.e., whether we prefer A to Bmay depend on whether or not C is under consideration)

Unit 3 - 21 -



Decision Analysis Example: Texaco vs Pennzoil (1984)

• Pennzoil and Getty agreed to merge• Texaco made Getty a better offer - Getty reneges• Pennzoil sues, wins case in 1985, get $11.1 Billion• Texas appeals court reduces judgment by $2 Billion

– With court costs and interest $10.3 Billion– Texaco threatened to bankrupt and go to Supreme Court

• 1987, before Pennzoil starts issuing liens Texaco offers to settlefor $2 Billion

• Pennzoil thinks $3-5 Billion is a fair price• What should Hugh Liedtke, CEO of Pennzoil, do?

Unit 3 - 22 -



Result ($B)Accept $2 Billion

Texaco Accepts $5 Billion

Final CourtDecision

Final CourtDecision

TexacoRefusesCounteroffer

Counteroffer$5 Billion

TexacoCounteroffers$3 Billion

Accept $3 Billion

2

5

10.3

5

0

3

10.3

5

0

Decision Tree for Pennzoil’s Problem(simplified model)

How could this model be made more complex?

0.2

0.5

0.3

0.2

0.5

0.3

0.17

0.33

0.50

Unit 3 - 23 -



Influence Diagram

Net

Value

Economic

Value

Cancer

Cost

Usage

Decision

Exposure

Test

Activity

Test

Human

Exposure

Carcinogenic

Activity

• Alternative representation ofdecision problem

– Ovals are “chance nodes”– Boxes are “decision nodes”– Rounded boxes are “value nodes”– Arcs show influences

• Formally equivalent to decisiontree

– Probability and utility values areencapsulated inside the nodes

– Some software packages switch backand forth between views

• Dotted lines are information arcs• Whether to collect information can be represented as a decision problem• Note: influence diagram represents multiattribute utility function explicitly

Unit 3 - 24 -



Some Simple Qualitative Rules

• Dominance– If Option X is at least as good as Option Y on all attributes

of value, Option X is at least as good as Option Y– If Option X is at least as good as Option Y for each

possible outcome, then Option X is at least as good asOption Y

• Useless Information: If information gatheringis costly and the result would not change yourdecision, then do not gather the information

Unit 3 - 25 -



Mathematical Programming• Constrained optimization problems:

– Maximize or minimize objective function– Subject to constraints defining feasible region of solution space

• Solution methods:– Linear programming (LP)

» Objective function and constraints are linear– Nonlinear programming (NLP)

» Objective function and/or some constraints are nonlinear– Integer programming (IP)

» Feasible space consists of integer variables– Mixed integer programming (MIP)

» Feasible space consists of some integer and some real variables– Goal programming (GP)

» Try to find at least one solution in feasible region– Dynamic programming (DP)

» Find optimal policy in sequential decision making problem

• Traditional mathematical programming ignores uncertainty

Unit 3 - 26 -



LP Example• A company makes 3 types of furniture:

Type Profit Labor Materials Minimum /item Required Required Qty

(hours) (sq ft)Chair $50 10.5 5 5Bench $100 15 15 7Table $75 17 10 5

° Objective: Find the highest profit combination of items to manufacture° Constraints:

- Labor hours available = 400- Lumber available = 300- Must make at least minimum quantity of each item


Unit 3 - 27 -



LP Formulation


Maximize 50 c + 100 b + 75 t profit

s.t. 10.5 c + 100 b + 17 t ≤ 400 labor

5 c + 15 b + 10 t ≤ 300 lumber

c ≥ 5 chairs

b ≥ 7 benches

t ≥ 5 tables

Unit 3 - 28 -



Solving Linear Programs• Simplex method - developed by Dantzig in 1940’s

– Standard method– Exponential in number of variables– Guaranteed to give optimal solution– Searches extreme points in feasible region

• Karmarkar’s algorithm - 1980’s– Polynomial time– Very fast on large problems– Limited ability to do sensitivity analysis

• Specialty algorithms exploit special casestructures– Transportation method– Network simplex

Unit 3 - 29 -



Goal Programming• Define goals (aspiration levels) as constraints:

– f(x) ≥ b; f(x) ≤ b; f(x) = b

• In standard LP these would be constraintsdefining feasible region

• In GP we try to minimize deviation from goal– Minimize weighted sum of goal deviations– Minimize some other function of goal deviations– Minimize worst deviation– Lexicographically minimize ordered set of goal deviations

Unit 3 - 30 -



Solving Integer Programs• Most IPs and MIPs are binary

– General integers expressed as sums of binaries withrounding

• Standard method: Branch and bound– Solve LP with integer constraints relaxed– Choose a variable to branch on

» Make 2 problems - set chosen variable to 1 or 0» Solve both relaxed problems

– Repeat till best integer solution is found– Worst case: 2n LPs to solve

» Can explode rapidly

Unit 3 - 31 -



Solving Nonlinear Programs• Standard methods

– Steepest descent– Conjugate gradient

• Convexity is important– Using standard NLP solvers on non-convex problems can

give local (not global) optimum!!– Stay tuned (next week) for more on non-convex problems!

g(x)

x

Non-convex Function

x*

Local minGlobal min

Unit 3 - 32 -



Solving Mathematical Programs• Special purpose optimization packages

– e.g., OSL, CPLEX– Linear, nonlinear, integer programs

• Spreadsheet add-ins– e.g., Excel’s solver– Easily available, don’t need to learn new package or

interface to external software– Usually limited (e.g., LP only; size limits)

• Many problems cannot be solved exactly– Heuristic methods are used– Interface between AI and OR/MS

Unit 3 - 33 -



Solving LP Using Excel Solver(1) Logically organize data (label, etc.)

• Coefficients for objective function• Coefficients for constraints• RHS of the constraints

(2) Reserve cells for the decision variables– Called Changing Cells

(3) Create formula in a cell for the objective function– Called Target Cell

(4) Create a formula for the LHS of each constraint(5) Open Solver Dialog box (Tools menu)(6) Enter the appropriate info and run Solver


Unit 3 - 34 -



Sensitivity Analysis• One-variable sensitivity analysis

– How sensitive is solution to change in parameter (weight inobjective function or constraint value)?

– Simplex method can produce one-variable sensitivityanalysis as a by-product

• Parametric analysis– Specify range of values for parameter or parameters

(weight on objective function; value of constraint;probability)

– Evaluate change in solution as parameters vary throughrange

Unit 3 - 35 -



Visualizing Sensitivity Analysis Results

• Tornado Diagram– Visualizes result of varying a

set of parameter throughspecified ranges on anoutput of interest

• Strategy Region Graph– Visualizes changes in

optimal strategy as 2parameters are variedthrough a range

D

S

P

L

R

Sensitivities to Parameters

Para

met

er 1

Parameter 2

Unit 3 - 36 -



In Summary...

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References• Anderson, D., Williams, T., and Sweeney, T., An Introduction to Management

Science: Quantitative Approaches to Decision Making, Southwestern, 1999.• Clemen, R. Making Hard Decisions: An Introduction to Decision Analysis,

Duxbury, 1997.• Winston, W. Operations Research Applications and Algorithms, Duxbury, 1997.

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