1Homework 3
ISE 562
A multi-attribute decision problem is shown below for a decision to select an apartment. The
following six attributes were defined as important to choosing the apartment.
Rental cost per month per tenant, R, dollars
Deposit required, D, dollars
Distance to university, M, miles
Noise level, 6pm-6am, N, subjective scale (0-10)
Length of lease required, L, months
Media (internet/cable) service, I, indicator scale (0= none; 1= internet or cable; 2= both)
The data table for the alternatives and attributes is:
______________________Attributes_________________________________________
Apt Rent, R Deposit, D Distance, M Noise, N Lease, L Media
A $450 $1100 5 5 12 1
B 380 850 4 3 12 0
C 550 1000 0.8 8 9 1
D 640 900 0.5 9 9 2____
After some reflection you and your roommates come up with the following attribute utility functions.
YD ? ? ? D ? '%!? $)! ? V ? '%!V "260 R for
YDH? ? ? DH ? ""!!? )&! ? H ? ""!!"#&! for
YDQ? ? ? DQ ? &? !T& ? Q ? &#* for
YDR? ? DR ? *? $ ? R ? *"$' # for
YDP? ? D"# ? P? * ? P ? "#"* # for
YDM? ? M ! ? M ? #"% # for
You determine that your own tradeoff scaling constants (you are decision maker 1) are:
k =0.40 k =0.40 k =0.10 k =0.20 k =0.60 k =0.60R D Mà à à à àR P M
1. Using the data in the table, find the master scaling constant for the attributes to 6 decimal'
places. Plot the master scaling constant function over the appropriate range.
2. Compute the multiattribute utility and rankings for the four alternative apartments using the
data for decision maker 1. What is your recommended choice? Why is it in first place (don't
say “because it has the largest utility.”)
3. Compute the additive model equivalent of question 2 by normalizing the attribute tradeoff
scaling constants to 1.0. Does it make a difference?
24. Now suppose your friends who will share the apartment with you want to input their own
attribute scaling constants to the problem. The value data for decision makers 2, 3, and 4 are
shown in the following table. Using the utility functions above, compute the rankings for
decision makers 2, 3, and 4.
Roommate k k k k k k
2 .6 .5 .2 .2 .2 .2
3 .3 .4 .1 .05 .5 .5
4 .2 .2 .05 .05 .3 .7
R D M N L I
5. Compute the concordance of the rankings for the 4 roommates. Do they agree?
6. Compute the group ranking for the four sets of rankings from question 4 using the Nash
Bargaining, Borda, and Additive Utility rules. You will have three group rankings.
7. Compute the concordance of the rankings for the 3 group decision rules. Is there concordance
(agreement)?
Now suppose the roommate who will provide the deposit says, “the only thing that really matters
here is the rent and how far we have to go to campus.” After a long discussion four of the attributes
are discarded. There is also uncertainty in what the actual rent will be due to a number of factors.
The problem has now been reduced to a two attribute problem:
Attributes
Apt Rent, R Distance, M
A $440-460 5
B 380-420 4
C 540-560 0.8
D 630-640 0.5____
The ranges can be represented as uniform probability distributions. For example, the pdf for
apartment A would be represented as
f(R )= for .E " "%'!?%%! #!? %%! ? V ? %'!
To obtain the CDF (cumulative distribution function) we would integrate from the lower end of the
distribution to an arbitrary point like this:-
for .JDV? ? 0D ?. ? . ? DV ? %%!?? %%! ? V ? %'!? ? ? ?%%! %%!V V " "#! #!7 7 7
Since a stream of random numbers can be scaled between zero and one, if we set the CDF equal to a
random number, r , we can solve for the simulated value of :3 V
or JDV? ? < ? DV ? %%!? V ? #!< ? %%!T3 3"#!
Now, if we feed a stream of random numbers into the above equation, we will get a stream of < V3values between and corresponding to the uniform distributions.%%! %'!
38. Find the simulation functions for rent cost for apartments 2-4.
9. Find the value of the master scaling constant, K, using the attribute tradeoff scaling constants for
rent and distance above. There will be one K-value for each roommate.
Using the multiplicative multiattribute decision model:
i.e., k 1.0, then: ( ) 1+K k u ( ) 1 Eq 1? ? ??n=1 n=1N n n n n1K Ná ê ? ò ? ? ó ?t” ( (
10. For roommate, perform the following in an Excel worksheet ):each
a) compute 200 simulated rents for each apartment
b) compute the corresponding 200 utility function values
c) compute the multiattribute expected utility (Eq 1) by combining the simulated utilities for rent
and the actual utilities for time to produce 200 multiattribute utility values for each contractor.
d) Now compute the mean and standard deviations of the multiattribute utilities for each of the
roomates and graph on a chart where the x-axis is the multiattribute utility and the y-axis is
the standard deviation (MAU 1 .? 5
(e) Which apartment should be selected and why?
11. Compute the concordance of the 4 roommates rankings using the mean values obtained in
10d. Do they agree? Why or why not?