Lecture 3: Making Decisions with Multiple Objectives Under Certainty

Transcription

1 Lecture 3: Making Decisions with Multiple Objectives Under Certainty Keywords Preferential independence Additive value function Non-additive value function Bisection method Difference standard sequence technique (DSST) Direct-rating method Lecture Outline EWL 5.1 Value function and preference EWL 5.2 Methods for determining value functions EWL 6.1 Value functions for multiple attributes EWL 6.2 The additive model EWL 6.3 Requirements for the applicability of the additive model CC 5 Multi Attribute Utility Theory 1 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 1/39

2 In this lecture, we will talk about how to approach to decision problems that satisfy the following assumptions: 1. Preferences are rational (complete and transitive) 2. No uncertainty about the outcome of each alternative 3. Multiple (conflicting) objectives The procedure for solving a decision problem under certainty: 1. Determine the fundamental objectives how to measure the achievement of these objectives (attributes) the set of alternatives that might achieve these goals 2. Apply the Multi-Attribute Value Theory (MAVT) Assign value scores to each attribute level for all alternatives Determine the weight (the relative importance) of each attribute Rank all alternatives according to weighted-average total score 2 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 2/39

3 What is conflicting objectives? Example: Choosing an automobile Objectives: Minimize price, maximize life span Alternatives: Model a, b and c Example: Choosing an automobile with conflicting objectives Objectives: Minimize price, maximize life span Alternatives: Model a, b and c 3 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 3/39

4 Which automobile to choose? The answer depends on how much you are willing to pay to increase the life span. Because the alternatives with longer expected life span are more expensive, there is no clear winner among the three alternatives. The tradeoff between the objectives Price and Life span must be considered to determine the most preferred alternative. The three alternatives can be ranked only if some procedure is used to combine the two attributes into a single index of the overall desirability of an alternative. Example Ultimate objective: Best car Lower-level fundamental objectives: Minimize price and maximize life span Alternatives: Model a, Model b, Model c Attributes: Price (attribute 1) and Life span (attribute 2) Alternatives: a = (a 1,a 2 ), b = (b 1,b 2 ), c = (c 1,c 2 ) Example: a = ($17K,12 years) Attribute value functions: v 1 (.) and v 2 (.) (subjective value scores) Example: v 1 ($17K ) < v 1 ($10K ) 4 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 4/39

5 Overall value of alternative a: V (a) = f (v 1 (a 1 ),v 2 (a 2 )) Example: V (a) = f (v 1 ($17K ),v 2 (12 years)) How to aggregate attribute values? 1. Additive value function (also called weighted average method ) V (a) = w 1 v 1 (a 1 ) + w 2 v 2 (a 2 ) + w 3 v 3 (a 3 ) Simple Requires Mutual Preferential Independence 2. Non-additive value function V (a) = w 1 v 1 (a 1 ) + w 2 v 2 (a 2 ) + w 3 v 3 (a 3 ) + kw 1 w 2 v 1 (a 1 )v 2 (a 2 ) + kw 1 w 3 v 1 (a 1 )v 3 (a 3 ) + kw 2 w 3 v 2 (a 2 )v 3 (a 3 ) + k 2 w 1 w 2 w 3 v 1 (a 1 )v 2 (a 2 )v 3 (a 3 ) Permits interactions across many attributes Complex 5 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 5/39

6 When is it safe to assume an additive multi-attribute value function? If mutual preferential independence is satisfied, then preferences can be represented by an additive ordinal value function. If additive difference independence is satisfied, then preferences can be represented by an additive cardinal value function. Ordinal versus Cardinal Value Functions (See the Math Handout) If a value function is ordinal, then BMW Audi v(bmw ) > v(audi) What else? Nothing! It only ranks the alternatives, and the value differences are not meaningful. For instance, if v(audi) = 0.5,v(BMW ) = 0.7,v(Mercedes) = 0.8, then the following does not necessarily hold: Audi BMW BMW Mercedes With a cardinal value function, the value differences reflect preferences over transitions: if v(audi) = 0.5,v(BMW ) = 0.7,v(Mercedes) = 0.8, then Audi BMW BMW Mercedes 6 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 6/39

7 Simple Preferential Independence (required for mutual independence) Example: The best meal Wine (Attribute 1): red or white Dish (Attribute 2): beef or chicken Preference Statement 1: I prefer beef over chicken (white,beef ) (white,chicken) and (red,beef ) (red,chicken) Preference Statement 2: I prefer white wine with chicken and red wine with beef (white,chicken) (red,chicken) and (red,beef ) (white,beef ) Is dish independent of wine? Yes! Is wine independent of dish? No! Definition: Simple Preferential Independence Let a = (a 1,...,a i,...,a m ) and b = (a 1,...,b i,...,a m ) be two alternatives that differ in attribute i only. Let a = (a 1,...,a i,...,a m) and b = (a 1,...,b i,...,a m) be two alternatives that also differ in attribute i only, and have the same values in attribute i as alternatives a and b respectively. We call attribute X i (simple) preferential independent of the remaining attributes if and only if a ( )b a ( )b for any a, a, b, b. Example: Attribute wine is (simple) preferential independent of dish if (red,beef ) (white,beef ) and (red,chicken) (white,chicken) 7 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 7/39

8 Mutual Preferential Independence Definition: Mutual Preferential Independence The attributes X 1,...,X m are mutually preferential independent if each possible subset of attributes is preferential independent of the complementary set. Example: Car choice with attributes color, price and maximum speed color is preferential independent of price and maximum speed. price is preferential independent of color and maximum speed. maximum speed is preferential independent of color and price. color and price are preferential independent of maximum speed. Example: (Black, $30,000, 220km/hr) (White, $20,000, 220km/hr) (Black, $30,000, 250km/hr) (White, $20,000, 250km/hr) price and maximum speed are preferential independent of color. maximum speed and color are preferential independent of price. 8 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 8/39

9 Example: Mutual Preferential Independence (White, $X, Ykm/hr) (Black, $X, Ykm/hr) for any X and Y (Zcolor, $20,000, Ykm/hr) (Zcolor, $30,000, Ykm/hr) for any Z and Y Simple preferential independence is satisfied BUT (White, $20,000, 220km/hr) (Black, $30,000, 220km/hr) (Black, $30,000, 250km/hr) (White, $20,000, 250km/hr) Mutual preferential independence is NOT REMARK: Simple preferential independence is about preferences over a single attribute whereas mutual preferential independence is about preferences over a subset of attributes. 9 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 9/39

10 Additive Difference Independence Simple preferential independence: preferences over attribute levels of a particular attribute should not depend on the level of other attributes Additive difference independence: preferences over transitions between attribute levels of a particular attribute should not depend on the level of other attributes Example: The car choice problem Example: A=(Black, $30,000, 220km/hr) B=(Black, $30,000, 250km/hr) C=(White, $20,000, 220km/hr) D=(White, $20,000, 250km/hr) The attribute maximum speed is difference independent of the attributes price and color if the additional value attached to a particular increase in maximum speed (e.g. from 200 km/hr to 220 km/hr) is independent of the vehicle being a $30,000 black car or a $20,000 white car. Definition: Additive Difference Independence Let a = (a 1,...,a i,...,a m ) and b = (a 1,...,b i,...,a m ) be two alternatives that differ in attribute i only. Let a = (a 1,...,a i,...,a m) and b = (a 1,...,b i,...,a m) be two alternatives that also differ in attribute i only, and have the same values in attribute i as alternatives a and b respectively. We call attribute X i additive difference independent of the remaining attributes if and only if (a b) (a b ) for any a, a, b, b. 10 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 10/39

11 What if preferences are not independent? Cannot use additive value functions! Try to redefine attributes in order to eliminate existing dependencies. If not, use a non-additive value function: V (a) = w 1 v 1 (a 1 ) + w 2 v 2 (a 2 ) + kw 1 w 2 v 1 (a 1 )v 2 (a 2 ) quality of life duration The last term captures the interaction between the attributes. If k is positive, then the two attributes complement each other. If k is negative, then the two attributes are substitutes. 11 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 11/39

12 What if preferences are not independent? Example 1: Medical treatment with attributes quality of life and duration A = (good,long) B = (good,short) V (A) = w 1 v 1 (good) + w 2 v 2 (long) > w 1 v 1 (good) + w 2 v 2 (short) = V (B) C = (bad,short) D = (bad,long) V (C) = w 1 v 1 (bad) + w 2 v 2 (short) > w 1 v 1 (bad) + w 2 v 2 (long) = V (D) CONTRADICTION! MUTUAL PREFERENTIAL INDEPENDENCE IS VIOLATED Example 2: Medical treatment with attributes quality of life and duration A = (good,10years) B = (good,12years) C = (bad,10years) D = (bad,12years) V (B) V (A) = w 2 [v 2 (12years) v 2 (10years)] > V (D) V (C) = w 2 [v 2 (12years) v 2 (10years)] CONTRADICTION! ADDITIVE DIFFERENCE INDEPENDENCE IS VIOLATED REMARK: In general, we observe preferences satisfying mutual preferential independence more often than additive difference independence. 12 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 12/39

13 Additive value function n V (a) = w r v r (a r ) = w 1 v 1 (a 1 ) w n v n (a n ) r=1 w r : Compare different attributes in terms of importance n r=1 w r = 1 and w r > 0 v r : Compare the value of different attribute levels v r (x r ) = 0 and v r (x + r ) = 1 Is this a sensible approach? Concepts of preference independence: Ordinal: Mutual Preferential Independence Cardinal: Additive Difference Independence Where do v r come from? Value function elicitation methods:? Bisection method DSST Direct-Rating method Where do the w r come from?? 13 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 13/39

14 Common forms of attribute value functions Attribute value functions convert attribute levels into levels of desirability, worth, utility There is no right or wrong value function for an attribute The shape of a value function depends on the decision maker s personal preferences. While attribute levels are objective, the level of desirability is inherently subjective. Value functions do not need to be perfect but they should capture the preferences of the DM well enough to understand and analyze the current situation. 14 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 14/39

15 General procedure for deriving value functions 1. Choose x min and x max (also called x + and x ) 2. Determine some points on the value function curve 3. Use these data points to generate the complete curve Linear interpolation Best fitting curve Note: We will use linear interpolation throughout the course! 4. Normalize the function to the interval [0, 1] 5. Check for consistency 15 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 15/39

16 Methods for determining attribute value functions 1. Bisection method (mid-value splitting technique): find the attribute level whose value is halfway between the most and the least preferred attribute level 2. Difference standard sequence technique (DSST): find the attribute levels x 0,x 1,...,x n, such that the increments in the strength of preference from x i to x i+1 are equal for all i = 0,...,n 1 3. Direct rating method: directly assign values to the attribute levels The assumptions : Preferences are rational Value functions are cardinal (therefore value differences reflect preferences over transitions) Attribute values are continuous (What if they are discrete?) Monotonically increasing (or decreasing) value functions (What if they are non-monotonic?) 16 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 16/39

17 1) Bisection Method: Value function for the attribute Salary Normalization condition yields 2 data points on the original value function 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

18 1) Bisection Method: Value function for the attribute Salary We are looking for 3 more points that satisfy the original function 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

19 1) Bisection Method: Value function for the attribute Salary If the original value function is linear, then we know those 3 data points 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

20 1) Bisection Method: Value function for the attribute Salary But the original value function is not necessarily linear 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

21 1) Bisection Method: Value function for the attribute Salary Check if the value function is linear: (30K 55K )? (55K 80K ) 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

22 1) Bisection Method: Value function for the attribute Salary If (30K 55K ) (55K 80K ), then adjust the transitions until the DM is indifferent between the two: (30K 50K )? (50K 80K ) 17 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 17/39

23 1) Bisection Method: Value function for the attribute Salary 1. Determine the most-preferred outcome (x + = 80,000) and the least-preferred (x = 30,000) outcome. 2. Normalize the value function by assuming v(30,000) = 0 and v(80,000) = 1. (Why do we normalize the value function? What kind of a transformation is this?) 3. Ask the DM for the outcome x 0.5 such that: (30,000 x 0.5 ) (x ,000), e.g. x 0.5 = 50,000 Let M = (Best level + Worst level)/2 = $55,000 = midpoint of total range Ask the DM which change produces a greater value improvement: - Change 1: Improve from $55,000 to $80,000 - Change 2: Improve from $30,000 to $55,000 If, for example, the answer is that Change 2 has a greater impact, this implies v(55,000) - v(30,000) > v(80,000) - v(55,000) (How is this related to cardinality?) Repeat the question with a salary less than $55,000 until the DM is indifferent between Change 1 and 2 (Let s assume that the DM is indifferent if we replace $55,000 with $50,000). 4. Assign the evaluation 0.5 to this outcome v(50,000) v(30,000) = v(80,000) v(50,000) v(50,000) = / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 18/39

24 1) Bisection Method: Value function for the attribute Salary 5. Determine the outcomes x 0.25 and x 0.75 in the same way. v($30,000)= v(x 0.25 ) v($50,000)=0.5 v(x 0.75 ) v($80,000)=1 (30,000 40,000) (40,000 50,000) and (50,000 65,000) (65,000 80,000) }{{}}{{}}{{}}{{} =0 =0.5 =0.5 =1 v(50,000)-v(40,000) = v(40,000)-v(30,000) and v(65,000)-v(50,000) = v(80,000)-v(65,000) v(40,000) = 0.25 v(65,000) = Use linear interpolation to complete the value function. 7. Check consistency Use a different method Ask additional questions: (40,000 a) (a 65,000) Is a close to x 0.5? 19 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 19/39

25 Exercise: Bisection Method An entrepreneur has made the following statements concerning his value function for profits: Given that the profits will lie within the range between $4 million and $8 million, the value of the difference between $4 million and $5 million is the same as the value of the transition from $5 million to $8 million. Furthermore, the transition from $4 million to $4.25 million is valued the same as the transition from $4.25 to $5 million. Similarly, the transition from $5 million to $6.25 million is valued the same as the transition from $6.25 million to $8 million. a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? 20 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 20/39

26 Exercise: Bisection Method (Cont.) An entrepreneur has made the following statements concerning his value function for profits: Given that the profits will lie within the range between $4 million and $8 million, the value of the difference between $4 million and $5 million is the same as the value of the transition from $5 million to $8 million. Furthermore, the transition from $4 million to $4.25 million is valued the same as the transition from $4.25 to $5 million. Similarly, the transition from $5 million to $6.25 million is valued the same as the transition from $6.25 million to $8 million. Bisection method 0 v($4m) v($4.25m) v($5m) v($6.25m) 1 v($8m) ($4M $5M) ($5M $8M) v($5m) v($4m) = v($8m) v($5m) v($5m) = 0.5 ($4M $4.25M) ($4.25M $5M) v($4.25m) v($4m) = v($5m) v($4.25m) v($4.25m) = / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 21/39

27 Exercise: Bisection Method (Cont.) a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? v($4m)=0 v($5m)=0.5 v($8m)=1 22 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 22/39

28 Exercise: Bisection Method (Cont.) a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? v($4.25m)=0.25? v($6.25m)= / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 22/39

29 Exercise: Bisection Method (Cont.) a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? v($4.25m)=0.25? v($6.25m)=0.75 ($4.25M?) (? $6.25M) 22 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 22/39

30 Exercise: Bisection Method (Cont.) a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? v($4.25m)=0.25? v($6.25m)=0.75 ($4.25M?) (? $6.25M)? roughly = $5M 22 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 22/39

31 2) DSST Method: Value function for the attribute Salary 1. Determine the most-preferred outcome (x + = 80,000) and the least-preferred (x = 30,000) outcome. 2. Define (as the researcher) a unit that is approximately 1/5 of the length of the interval [30,000,80,000] (10,000). Define x 1 = 30, ,000 = 40,000. $30,000 $40,000 $50,000 $60,000 $70,000 $80,000 = $10, Ask the DM for the outcome x 2 that satisfies the following indifference statement: (30,000 40,000) (40,000 x 2 ) x x v($30,000)=0 v($40,000) v(x 2 ) (30,000 40,000) (40,000 50,000) 23 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 23/39

32 2) DSST Method: Value function for the attribute Salary 4. Ask for the outcome x 3 with: (40,000 50,000) (50,000 x 3 ) x x x v($30,000)=0 v($40,000) v($50,000) v(x 3 ) (40,000 50,000) (50,000 65,000) 5. Proceed with the same procedure until x + is reached or exceeded. x x x x v($30,000)=0 v($40,000) v($50,000) v($65,000) v(x 4 ) (50,000 65,000) (65,000 80,000) 6. v($80,000) = 1 4x = 1 x = v($30,000)=0 v($40,000) v($50,000) v($65,000) v($80,000)=1 24 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 24/39

33 Exercise: DSST Method An entrepreneur has made the following statements concerning his value function for profits ranging between -$10 million and $10 million: the value of the difference between -$10 million and -$7 million is the same as the values of the transitions from -$7 million to -$5 million, from -$5 million to -$3 million, from -$3 million to $0 million, from $0 million to $1 million, from $1 million to $5 million, and from $5 million to $10 million. a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? 25 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 25/39

34 Exercise: DSST Method (Cont.) An entrepreneur has made the following statements concerning his value function for profits ranging between -$10 million and $10 million: the value of the difference between -$10 million and -$7 million is the same as the values of the transitions from -$7 million to -$5 million, from -$5 million to -$3 million, from -$3 million to $0 million, from $0 million to $1 million, from $1 million to $5 million, and from $5 million to $10 million. The difference standard sequence technique (DSST) has been used. 0 v(-10) x x x x x x x v(-7) v(-5) v(-3) v(0) v(1) v(5) 1 v(10) x = 1 7 Each transition has a value of 1/7. 26 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 26/39

35 Exercise: DSST Method (Cont.) a) Draw the value function for the interval [0,1]. Which method is used? b) How would you check for consistency? Change the size of the first transition: e.g. instead of =3 (-10 to -7), set =4 What is the attribute level x that satisfies the following? ( 10 6) ( 6 x). REMARK: The number of transitions (and the number of data points) depends on the DM s preferences, therefore is not known until the interview is completed. 27 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 27/39

36 DSST Method: Attribute range Local versus global attribute range It is often preferable to use a (global) range that is wider than the minimum and maximum values of the alternatives (local range). If the last question in the interview results in an x value that is greater than x + (maximum attribute level): Expand the attribute range or Ask for the value of the transition from the last x elicited to x +, relative to the previous transitions 28 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 28/39

37 Exercise: Attribute Range Now, assume that all the transitions mentioned in the previous question hold, except the last one such that the transition from $5 million to $10 million creates a smaller value than the other transitions. Can you still generate the value function for the profits from -$10 million to $10 million if the entrepreneur makes one of the following statements? (i) The transition from $1 million to $5 million generates the same value as the transition from $5 million to $12 million. 0 x x x x x x x v(-10) v(-7) v(-5) v(-3) v(0) v(1) v(5) 1 v(12) (ii) The transition from $5 million to $10 million generates half as much value as the transition from $1 million to $5 million. 0 1 x x x x x x x/2 v(-10) v(-7) v(-5) v(-3) v(0) v(1) v(5) v(10) 29 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 29/39

38 Exercise: Attribute Range (Cont.) Now, assume that all the transitions mentioned in the question hold, except the last one such that the transition from $5 million to $10 million creates a smaller value than the other transitions. Can you still generate the value function for the profits from -$10 million to $10 million if the entrepreneur makes one of the following statements? (i) The transition from $1 million to $5 million generates the same value as the transition from $5 million to $12 million. 0 v(-10) x x x x x x x v(-7) v(-5) v(-3) v(0) v(1) v(5) 1 v(12) v(12) = 1,v(5) = = v(10) = = 0.96 (ii) The transition from $5 million to $10 million generates half as much value as the transition from $1 million to $5 million. 0 v(-10) 1 x x x x x x x/2 v(-7) v(-5) v(-3) v(0) v(1) v(5) v(10) v( 10) = 0,v( 7) = 1 6.5,...,v(5) = = 0.923,v(10) = 1 30 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 30/39

39 3) Direct Rating Method: Value function for the attribute Salary 1. Determine the most-preferred outcome (x + = 80,000) and the least-preferred (x = 30,000) outcome. 2. Order the outcomes of all alternatives (a:30,000, b:50,000, c:65,000, d:40,000, e:80,000) from the most preferred to the least preferred. (e c b d a) 3. Assign 100 and 0 points to the best and worst outcomes. 4. Assign points to the intermediate outcomes, s.t. the point differences truly reflect the strength of preference. e:100, c:75, b:50, d:25, a:0 5. Normalize: Divide points by 100 to receive evaluations. e:1, c:0.75, b:0.5, d:0.25, a:0 6. Check consistency Use a different method 31 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 31/39

40 What if the assumptions do not hold? Attribute values are continuous (What if they are discrete?) Bisection method and DSST cannot be used Use the direct method Example: Car choice problem Can you use DSST in order to elicit the attribute value function for Color? Monotonically increasing (or decreasing) value functions (What if they are non-monotonic?) Split the objective into monotonic lower-level objectives Or split the interval into subintervals on which the value function is monotonically increasing or decreasing. Then, apply the value function elicitation methods separately on both intervals. 32 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 32/39

41 Exercise: Generating non-monotonic value functions You are thinking about buying a house in a nice village. This village, as most of the villages in this area, consists of only one street that goes alongside a canal for about 10 km. However, in the middle of the village there s a new highway connecting the village with the bigger cities around. The distance of a house to this highway is an important objective that determines your choice of a house. Assume that your valuation of distance to highway is non-monotonic such that you find a distance of 500 m optimal and better than the extremes 20 m and 5 km. Why is it impossible to represent your preferences using a monotone value function? How could non-monotonicity of the value function be resolved? Canal 5km 5km Highway 33 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 33/39

42 Exercise: Generating non-monotonic value functions (Cont.) Canal 5km 5km Highway Splitting up the objective distance to highway could resolve non-monotonicity. There are two underlying fundamental objectives, noise and time to reach high-way. A higher discomfort due to noise would always lead to lower valuations. 34 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 34/39

43 Exercise: The Additive Model Flight time Paid to home Salary vacation days New York 9 hours $5,000 0 days Paris 3 hours $2, days Dubai 5 hours $6, days For the value function of the attribute Paid vacation days, you collected the following indifference statements regarding the transitions between different attribute levels: (0 days 12 days) (12 days 30 days) (0 days 5 days) (5 days 12 days) (12 days 20 days) (20 days 30 days) a) Plot the attribute value function by using linear interpolation. Which method did you use? Find v 3 (22 days). b) While you were running several consistency checks, you obtained the following preference statements. Can you still use the additive model? Why or why not? (i) (,$6,000,30 days) (,$6,000,22 days) and (,$2,000,30 days) (,$2,000,22 days) (ii) (5 hours,$6,000, ) (3 hours,$6,000, ) (5 hours,$2,000, ) (3 hours,$2,000, ) 35 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 35/39

44 Exercise: The Additive Model (Cont.) Flight time Paid to home Salary vacation days New York 9 hours $5,000 0 days Paris 3 hours $2, days Dubai 5 hours $6, days a) For the value function of the attribute Paid vacation days, you collected the following indifference statements regarding the transitions between different attribute levels: (0 days 12 days) (12 days 30 days) x 0.5 = 12 (0 days 5 days) (5 days 12 days) x 0.25 = 5 (12 days 20 days) (20 days 30 days) x 0.75 = 20 Bisection method has been used. v 3 (22 days) = (22 20) = / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 36/39

45 Exercise: The Additive Model (Cont.) b) While you were running several consistency checks, you obtained the following preference statements. Can you still use the additive model? Why or why not? Explain. (i) (,$6,000,30 days) (,$6,000,22 days) and (,$2,000,30 days) (,$2,000,22 days) w 3 v 3 (30) > w 3 v 3 (22), w 3 v 3 (30) = w 3 v 3 (22) contradiction! Violates simple (and mutual) preferential independence (ii) (5 hours,$6,000, ) (3 hours,$6,000, ) (5 hours,$2,000, ) (3 hours,$2,000, ) w 1 v 1 (3) w 1 v 1 (5) > w 1 v 1 (3) w 1 v 1 (5) contradiction! Violates additive difference independence 37 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 37/39

46 Answer the following questions: 1. What does conflicting objective mean? 2. How does a non-additive value function differ from an additive value function? 3. What are the requirements for representing preferences with an ordinal or a cardinal attribute value function? 4. What are the methods for determining value functions? Compare and contrast each method. 5. What is the problem with eliciting the value function for a discrete attribute? 6. How can you elicit non-monotonic attribute value functions? 38 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 38/39

47 Reading EWL Chapter 5 and Chapter 6 and CC Chapter 5 Supplementary Reading (Optional) (ILIAS) Ulvila, Snider, 1980: Negotiation of International Oil Tanker Standards: An Application of Multiattribute Value Theory. Operations Research, Vol. 28, pp Treadwell, 1998: Tests of Preferential Independence in the QALY Model. Medical Decision Making, Vol. 18, pp Next Lecture Lecture 4: Determination of the weights Reading: EWL Chapter 6 and CC Chapter 5 39 / 39 Irem Demirci CC 501 Decison Anlaysis Lecture 3: Making Decisions with Multiple Objectives Under Certainty 39/39