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 Wj Aj Ci W1 C1 W2 C2 Wn Cn A1 v11 v12 v1n A2 v21 v22 v2n Am vm1 vm2 vmn

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

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: 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

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)

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

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 - 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 Rating Comfort/Ride Color Prexel 1 0.21 Criston 0.4 0.48 Thrush 0.5 0.7

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

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

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) = 0.263. 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

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

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   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