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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
Outline
• Developing the model subsystem• Role of decision theory in DSS• Brief survey of decision analysis
and optimization methods
Unit 3 - 3 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 $3-5 Billion is a fair price• What should Hugh Liedtke, CEO of Pennzoil, do?
Unit 3 - 22 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
Result ($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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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
Thanks to Andy Loerch
Unit 3 - 27 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
LP Formulation
Thanks to Andy Loerch
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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
Thanks to Andy Loerch
Unit 3 - 34 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
In Summary...
Unit 3 - 37 -
Department of Systems Engineering and Operations Research
Copyright © 2006, Kathryn Blackmond LaskeySYST 542
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.