Multi-Attribute Utility Models with Interactions. Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University. Introduction. Attributes Can be Substitutes to One Another
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Dr. Yan Liu
Department of Biomedical, Industrial & Human Factors Engineering
Wright State University
x, yMulti-Attribute Utility Function
When you are indifferent between A and B, EU(A) = EU(B)
p•U(x+, y+ ) + (1-p)•U (x-, y- ) = U (x, y)
p=U (x, y)
After you find the utilities for a number of (x, y) pairs, you can plot the assessed points on a graph and sketch rough indifference curves
Interaction between x and y?
e.g. X = the cost of a project ($1,000 or $2,000)
Y = time-to-completion of the project (5 days or 10 days)
If you prefer the 5-day time-to-completion to the 10-day time-to-completion no matter whether the cost is $1,000 or $2,000, then Y is preferentially independent of X
If you prefer the lower cost of the project regardless of its time-to-completion, then X is preferentially independent Y.
X and Y are mutually preferentially independent
X = the cost of a project ($1,000 or $2,000)
Y = time-to-completion of a project (5 days or 10 days)
If your CE to an option that costs $1,000 with probability 50% and $2,000 with probability 50% does not depend on the time-to-completion of the project, then X is utility independent of Y
UX (x) = utility function of X scaled so that UX (x-) =0 and UX (x+) =1
UY (y) = utility function of Y scaled so that UY (y-) =0 and UY (y+) =1
kX = U(x+, y-) NOT relative weight of UX
kY = U(x-, y+) NOT relative weight of UY
kX +kY ≠1
Lottery A: (x-, y-) with probability 0.5, (x+, y+) with probability 0.5
Lottery B: (x-, y+) with probability 0.5, (x+, y-) with probability 0.5
(additive utility function)
excellent service, excellent reliability
poor service, poor reliability
poor service, excellent reliability
excellent service, poor reliabilityAdditive Independence (Cont.)
e.g. You are considering buying a car, and reliability and quality of service are the two attributes you consider
Which assessment lottery for this car decision will you choose, A or B?
If you are indifferent between A and B, then additive independence holds for attributes reliability ad quality of service; otherwise, additive independence does not hold
If (1– kX –kY ) > 0,
so X and Y complement each other
If (1– kX –kY ) < 0,
so X and Y substitute each other
In a hospital bank it is important to have a policy for deciding how much of each type of blood should be kept on hand. For any particular year, there is a shortage rate, the percentage of units of blood demanded but not filled from stock because of shortages. Whenever there is a shortage, a special order must be placed to locate the required blood elsewhere or to locate donors. An operation may be postponed, but only rarely will a blood shortage result in a death. Naturally, keeping a lot of blood stocked means that a shortage is less likely. But there is also a rate at which it must be discarded. Although having a lot of blood on hand means a low shortage rate, it probably also would mean a high outdating rate. Of course, the eventual outcome is unknown because it is impossible to predict exactly how much blood will be demanded. Should the hospital try to keep as much blood on hand as possible so as to avoid shortages? Or should the hospital try to keep a fairly low inventory in order to minimize the amount of outdated blood discarded? How should the hospital blood bank balance these two objectives?
The final consequence at the blood bank depends on not only the inventory level chosen (high or low) but also the uncertain blood demand over the year. Therefore, this problem is a decision under risk.
Shortage rate (X) and outdating rate (Y)
Shortage rate: annual percentage of units demanded but not in stock
Outdating rate: annual percentage of units that are discarded due to aging
To choose an appropriate inventory level, we need to assess probability distributions of shortage rate and outdating rate consequences for each possible inventory level and the decision maker’s utility over these consequences.
Who is the decision maker ?
The nurse who is in charge of ordering blood is responsible for maintaining an appropriate inventory level, so the utility function will reflect his/her personal preferences
Ranges of attributes ?
The nurse judges that 0%(best case) ≤X ≤10%(worst case) and 0%(best case)≤ Y ≤10%(worst case)
Mutual Independence between X and Y ?
The nurse is asked to assess the certainty equivalent for uncertain shortage rate (X), given different fixed outdating rates (Y), say Y=0%, 2%, 5%, 8%, and 10%.
If CEX does not change for different values of Y, then X is utility independent of Y
The nurse is asked to assess the certainty equivalent for uncertain outdating rate (Y), given different fixed shortage rates (X), say X=0%, 2%, 5%, 8%, and 10%.
If CEY does not change for different values of X, then Y is utility independent of X
Suppose the nurse’s assessments suggest mutual independence between X and Y, then the utility function is of the multilinear form:
UX(x) and UY(y)?
kX and kY?
The trick is to use as much information as possible to set up equations based on indifferent outcomes and lotteries, and then to solve the equations for the weight.
There are two unknown weights in this problem, so we need to set up two equations in the two unknowns, which requires two utility assessments.
Suppose the nurse is indifferent between two consequences (X=4.75%, Y=0) and (X=0, Y=10%)
Suppose the nurse is also indifferent between the consequence (X=6%, Y=6%) and a 50-50 lottery between (X=0, Y=0) and (X=10%, Y=10%)
Solving Equations 1 and 2 simultaneously for KX and kY, we find KX=0.72 and kY=0.13
Therefore, the two-attribute utility function can be written as
Because 1- kX - kY= >0, X and Y are complements
Use the derived utility function to choose between the inventory level in the following decision tree.
EU(High) = 0.3* U(0,0) +0.7* U(0,10) = 0.3*1 + 0.7* kX = 0.3+0.7*0.72 = 0.804
EU(Low) = 0.3* U(10,0) +0.7* U(0,0) = 0.3* kY + 0.7* 1 = 0.3*0.13+0.7 = 0.739
Because EU(High)>EU(Low), the high inventory level is preferred