Suppose you want to determine the elimination$-$by$-$aspects $($EBA$)$ choice

of a consumer choosing from among P products, each having A attributes. This is a noncompensatory choice prediction model. You are given the following data structures: a P$-$byA matrix containing the consumer's rating or performance of each product on each attribute, a

vector of length A containing the consumer's importance of each attribute, a vector of length

A giving the consumer's cutoff for each attribute $($we are considering the general case where

it is possible for the consumer to have different cutoffs for different attributes$).$ Write a

function called "apply$\_$eba" that takes these three data structures as input arguments

and produces the elimination$-$by$-$aspects choice. So your function should be defined like the

following

def apply$\_$eba$($ratings$\_$matrix, importances$\_1$d$\_$array, cutoffs$\_1$d$\_$array$)$:

Your function should return an integer from the set $\{1,2,...,$P$\}$ corresponding to the product

predicted to be purchased by EBA.

In EBA, if the rating of a product on a certain attribute is greater than or equal to the cutoff

for that attribute, then that product is NOT eliminated. If the rating is strictly less than the

cutoff then the product is eliminated.

It is important to note that in the EBA algorithm, one may run into situations where there are

ties or null sets. Your code needs to handle these as follows. When picking the next$-$most

important attribute, if there is more than one attribute with the highest level of

importance, then pick one of the attributes randomly with equal probability and proceed to

eliminate products on the basis of that attribute. When eliminating products that fall below

the performance rating cutoff on a certain attribute, if none of the remaining products meet

the cutoff then pick one of those remaining products randomly with equal probability and

take the resultant product to be the final choice of that consumer.

I will give you more points if you write this function without using explicit loops like "for",

"while", "repeat" or recursion. For even more points: Write the function without using any

$"$if$"$ statement and also without using explicit loops. You can however use implicit loops like

in the "apply" family of functions in R or "map" function of python. Writing the function will

initially seem very complicated. You can write the code using explicit loops like "for",

"while", "repeat" or recursion if you don't want the extra points $($beyond the points you will

already get for doing this task$),$ but actually it is easier to write it without the explicit loops or

$"$if$"$ statements, by using array operations. My own solution program to this task has only $6$

lines of code and does not use explicit loops or the $"$if$"$ statement. I mention this to let you

know that the code is actually quite simple to write if you think carefully about EBA and how

to structure it as array operations.

Because of the tie$-$breaking, there is randomness potentially involved so that each run of the

above apply$\_$eba function may give a somewhat different output for the product chosen.

Therefore, in real$-$life applications, we usually average over multiple runs. Specifically, we

need to run the apply$\_$eba function created above a large number of times $($like $2000)$ and

then aggregate over all runs. For example, if the consumer is seen to buy product P$,$ Q$,$ R$,$ S$,$

T$,$ respectively, $620,320,400,310,350$ times out of the $2000$ runs, then we predict that the

consumer's probability of buying P$,$ Q$,$ R$,$ S$,$ T to be respectively $0\mathrm{.}31,0\mathrm{.}16,0\mathrm{.}20,0\mathrm{.}155,$

$0\mathrm{.}175$

Demonstrate that your code works correctly on the following two test cases:

Test Case $1$: Here P$=3$ and A$=4.$

The P$-$by$-$A ratings matrix is

$1,7,7,7$

$3,7,2,5$

$7,1,7,1$

The vector of length A containing the importance of each attribute is

$[9,55,12,24]$

Finally, the vector of length A giving the cutoff for each attribute is

$[2,2,2,2]$

This test case corresponds to Product$-$Price Optimization Slides $6$ and $7.$ We already know

from our class discussion that the correct answer is $2.$

Test Case $2$: Here P$=5$ and A$=7.$

The P$-$by$-$A ratings matrix is

$1,1,2,3,4,5,5$

$3,4,2,7,3,5,3$

$2,4,3,4,6,4,3$

$4,1,4,3,5,6,6$

$3,1,6,6,4,6,4$

The vector of length A containing the importance of each attribute is

$17,9,9,13,22,26,4$

Finally, the vector of length A giving the cutoff for each attribute is

$[1\mathrm{.}5,1\mathrm{.}5,3\mathrm{.}5,2\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}5,2\mathrm{.}5]$

This test case is more challenging because there is tiebreaking and randomness

involved. Hint to verify your answer. When you run your function $2000$ times, you

should find that the second product is chosen approximately $25\%$ percent of the time, so

that we predict that the consumer's probability of buying the second product to be

approximately $0\mathrm{.}25.$