1. Multi-Attribute Decision Making MADM
Many decisions involve consideration of multiple attributes
Another term: multiple criteria
Examples:
Purchasing a car, boat, house, production equipment
Deciding among several designs of system or system improvement

2. Multi-Attribute Utility functions
Very often these decision involve conflicting objectives (criteria)
Conflicts and trade-offs for other situations:
Car, house, etc.
Rarely does one alternative provide the optimum of all desired criteria
Use multi-attribute utility functions to model these decisions

3. Multi-Attribute Utility functions
Determine the objectives of the decision
Determine the attributes (criteria)
Search for alternative
Collect data, raw/natural values of each alternative for each criteria
Convert raw data to Utilities (0 to 1)
Assessing the relative value of each criteria, weights
Evaluate the Total Utility of each alternative
Perform Sensitivity
Choose best

4. Common Decision situations
Single future with multiple criteria: 2 dimensions Wj
Aj Ci W1 C1 W2 C2 Wn Cn A1
v11
v12
v1n A2
v21
v22
v2n Am
vm1
vm2
vmn

5. Additive utility function
Criteria: C1, C2, …, Cn , or Ci, i = 1 to n
Alternatives: A1, A2, …, Am , or Aj, j = 1 to m
You have a utility function for each: U1(x1), U2(x2), …, Um(xm), on a scale of 0 to 1.
For best x, U(x) = 1, For worst x, U(x) = 0
Utility of vij of xj = uij
TU(xj) = w1 U(x11)+ w2 U(x21)+ … + wn U(xnm)
= wi U(xij)
wi is the weight of the ith attribute, wi = 1

6. Objectives and Attributes
Goals & Objectives  Attributes, Criteria
Goals are usually non-measurable, qualitative language
Objectives are measure
measurement scales of the attributes
$$, MPG, Ratings, Speed, etc.

7. Objectives and Attributes
Essential aspects of objectives
The set of objectives should represent the overall goals.
The objectives in the set should not be redundant
Objectives converted to measurable attributes
Attribute scales must be operational – provide an easy way to measure and obtain evaluate on outcomes

8. Example Automobile example:
Three alternatives
five attributes (criteria):
Price($), fuel efficiency(mpg), Safety(Rating), Comfort/Ride(Rating), Color
Min price(Less is better), Max Others(more is better)
Relatively easy to evaluate
More complex with more attributes and different measurement scales

9. Additive utility function
Consider:
How do you compare preference of attributes with different metrics? (apples and oranges)
How do you compare the attributes in terms of importance to the decision?
Safety is twice as important as price??
Utility function to model preferences

10. Additive utility function
Criteria: C1, C2, …, Cn , or Ci, i = 1 to n
Alternatives: A1, A2, …, Am , or Aj, j = 1 to m
You have a utility function for each: U1(x1), U2(x2), …, Um(xm), on a scale of 0 to 1.
For best x, U(x) = 1, For worst x, U(x) = 0
Utility of vij of xj = uij
U(xj) = w1 U(x11)+ w2 U(x21)+ … + wn U(xnm)
= wi U(xij)
wi is the weight of the ith attribute, wi = 1

11. Raw or natural scores (measurements)
Example
Scores – determine scores on the same attribute scale (utility) for valid comparison
Assessing weights – determine the importance of each weight relative to the others.
Example: Choosing an Automobile
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Prexel
$22,000 32
8.5
6.7
Red
Criston
$25,000 38
8.2
7.9
Black
Thrush
$27,000 35
9.6
9.2
Blue
Raw or natural scores (measurements)

12. Utility function Example: More is Better, Choosing an Automobile
determine the best and worst raw scores on each attribute
Fuel Effic.: Best: Criston (38), Worst: Prexel (32)
Utility of Best, Worst: U(38) = 1, U(32) = 0
Utility of Thrush (35mpg)? Linear scaling
ui(x) = (x – Worst Value) / (Best Value – Worst Value)
ui(35) = (35 – 32) / (38 – 32) = 3 / 6= 0.50

13. Utility function Example: Less is Better, Choosing an Automobile
determine the best and worst raw scores on each attribute
Price: Best: Prexel ($22), Worst: Thrush ($27)
Utility of price: U(Best[min]) = U(22) = 1, ….and U(Worst[max]) = U(27) = 0
Utility of Criston($25)? Linear scaling
ui(x) = (Worst Value - X) / (Worst Value – Best Value)
ui(25) = ( ) / (27 – 22) = 2 / 5= 0.4

14. Assessing qualitative attributes
Some attributes may not have natural scales of measurement or may be qualitative
This is a subjective evaluation.
Example: color of automobile
Three possible colors: Red, Blue, or Black
Determine most preferred and least preferred
Then scale to utility of 0 to 1
U(most preferred) = 1, U(least preferred) = 0
Determine where the intermediate colors are on this 0 to 1 scale.
U(Red) = 1, U(Black) = 0
U(Blue) = ??
Subjective: U(Blue) = 0.7

15. Decision Table with Utilities
After all raw scores are converted to utilities, this is the decision table.
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Prexel 1
0.21
Criston
0.4
0.48
Thrush
0.5
0.7

16. Assessing weights
Determine the relative importance of each criteria (attribute)
There are several methods.
Subjective ranking and evaluating
Pricing out
Swing Weighting
Determine the weights, wi, and

17. Assessing weights Subjective ranking and evaluating
Rank the criteria in importance
Give higher weights to more important and lower weights to less important.
Remember
Example: price(c1), fuel efficiency(c2), safety(c3), comfort/ride(c4), and color(c5)
Ranking: [C2 and C3], C4, [C1 and C5].
Weights: C1 = 0.1, C2 = 0.3, C3 = 0.3, C4 = 0.2, C5 = 0.1

18. Applying the Weights – Total Utility
TU(xj) = w1 U(x11)+ w2 U(x21)+ … + wn U(xnm)
= wi U(xij)
For A1, the Prexel, the Total Utility Score =
(0.1)(1) + (0.3)(0) + (0.3)(0.21) + (0.2)(0) + (0.1)(1) =
10%
30%
20%
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Prexel 1
0.21
Criston
0.4
0.48
Thrush
0.5
0.7

19. Evaluating all Alternatives
10%
30%
20%
100%
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Total Score
Prexel 1
0.21
0.263
Criston
0.4
0.48
0.436
Thrush
0.5
0.7
0.720

20. Sensitivity Analysis Play with the weights, adjust up and down
Find where alternatives are equal in total utility
How do the adjusted weights feel compared to the original set of weights?

21. Sensitivity Example Equal weights More weight on Price 20% 100% Price
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Total Score
Prexel 1
0.21
0.442
Criston
0.4
0.48
0.376
Thrush
0.5
0.7
0.640
40%
20%
10%
100%
Price
Fuel Effic. (MPG)
Safety Rating
Comfort/Ride
Color
Total Score
Prexel 1
0.21
0.542
Criston
0.4
0.48
0.408
Thrush
0.5
0.7
0.47