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

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

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

Common Decision situations Single future with multiple criteria: 2 dimensions W j A j C i W1C1W1C1 W2C2W2C2 WnCnWnCn A1A1 v 11 v 12 v 1n A2A2 v 21 v 22 v 2n AmAm v m1 v m2 v mn

Additive utility function Criteria: C 1, C 2, …, C n, or C i, i = 1 to n Alternatives: A 1, A 2, …, A m, or A j, j = 1 to m You have a utility function for each: U 1 (x 1 ), U 2 (x 2 ), …, U m (x m ), on a scale of 0 to 1. For best x, U(x) = 1, For worst x, U(x) = 0 Utility of v ij of x j = u ij TU(x j ) = w 1 U(x 11 )+ w 2 U(x 21 )+ … + w n U(x nm ) = w i U(x ij ) w i is the weight of the i th attribute, w i = 1

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.

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

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

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

Additive utility function Criteria: C 1, C 2, …, C n, or C i, i = 1 to n Alternatives: A 1, A 2, …, A m, or A j, j = 1 to m You have a utility function for each: U 1 (x 1 ), U 2 (x 2 ), …, U m (x m ), on a scale of 0 to 1. For best x, U(x) = 1, For worst x, U(x) = 0 Utility of v ij of x j = u ij U(x j ) = w 1 U(x 11 )+ w 2 U(x 21 )+ … + w n U(x nm ) = w i U(x ij ) w i is the weight of the i th attribute, w i = 1

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 Raw or natural scores (measurements) PriceFuel Effic. (MPG)Safety RatingComfort/RideColor Prexel$22, Red Criston$25, Black Thrush$27, Blue

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 u i (x) = (x – Worst Value) / (Best Value – Worst Value) u i (35) = (35 – 32) / (38 – 32) = 3 / 6= 0.50

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 u i (x) = (Worst Value - X) / (Worst Value – Best Value) u i (25) = ( ) / (27 – 22) = 2 / 5= 0.4

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

Decision Table with Utilities After all raw scores are converted to utilities, this is the decision table. Price Fuel Effic. (MPG) Safety RatingComfort/RideColor Prexel Criston Thrush

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, w i, and

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(c 1 ), fuel efficiency(c 2 ), safety(c 3 ), comfort/ride(c 4 ), and color(c 5 ) Ranking: [C 2 and C 3 ], C 4, [C 1 and C 5 ]. Weights: C 1 = 0.1, C 2 = 0.3, C 3 = 0.3, C 4 = 0.2, C 5 = 0.1

Applying the Weights – Total Utility TU(x j ) = w 1 U(x 11 )+ w 2 U(x 21 )+ … + w n U(x nm ) = w i U(x ij ) For A 1, the Prexel, the Total Utility Score = (0.1)(1) + (0.3)(0) + (0.3)(0.21) + (0.2)(0) + (0.1)(1) = %30% 20%10% Price Fuel Effic. (MPG) Safety RatingComfort/RideColor Prexel Criston Thrush

Evaluating all Alternatives 10%30% 20%10%100% Price Fuel Effic. (MPG) Safety RatingComfort/RideColor Total Score Prexel Criston Thrush

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?

Sensitivity Example Equal weights More weight on Price 20% 100% Price Fuel Effic. (MPG) Safety Rating Comfort/RideColor Total Score Prexel Criston Thrush %20% 10% 100% Price Fuel Effic. (MPG) Safety Rating Comfort/RideColor Total Score Prexel Criston Thrush