Problem $5(10$ pts$).$ In this problem, we compare efficiency of various pivot rules on randomly

generated linear optimization problems. For your submission, include your code for parts $($a$)-($c$),$

the histograms from $($c$),$ and your written response for $(d).$

$($a$)$ Write modified simplex algorithms pivots$\_$LC and pivots$\_$B that only output the number

of pivots used when following the largest coefficient rule or Bland's rule. One way of doing

this is to start with $t=0,$ add $1$ to $t$ each time a pivot is performed, and return $t$ at the

end of the algorithm.

$($b$)$ The largest increase rule selects an entering variable that increases the objective function

the most during the pivot. One $($inefficient$)$ way of implementing this rule that does not

require modifying our earlier code is shown below, and you are free to use this.n $=$ len$($T$[-1,$:$])-1$ #total num of variablesr $=$ pivot$\_$row$($T$,$index$)$ #corresponding pivot rowfor j in range$($index $+1,$n$)$: r $=$ pivot$\_$row$($T$,$j$)$ #find corresponding pivot row return j #case of infinite increase increase $=-$T$[-1,$j$]*$T$[$r$,-1]/$T$[$r$,$j$]$ #when j is pivot column max$\_$increase $=$ increase #overwrite previous bestreturn index #outputs index for maximum increaseRepeat part $($a$)$ by writing an algorithm pivots$\_$LI that counts the number of pivots needed

when using this rule.

$($c$)$ The code below generates $1000$ problems where $Ain{R}^{5050}$ and $b,cin{R}^{50}$ consist of integers

between $0$ and $200,$ solves them using the simplex algorithm with largest coefficient rule,

records the number of pivots needed in data$\_$LC$,$ and plots the resulting data in a histogram.data$\_$LC $=$ numpy.zeros$(1000)$

for i in range$(1000)$:T$[0$:$50,0$:$50]=$ numpy.random.randint $(0,201,(50,50))$T$[50,0$:$50]=-1*$numpy$.$random.randint $(0,201,50)$plt$.$hist$($data$\_$LC$,$bins$=$range$(0,51))$Adapt the above code to generate histograms for the number of iterations needed when

using Bland's rule or the largest increase rule. Then produce the three histograms, one for

each rule.

$($d$)$ Which of the three pivot rules appears to be most efficient? Why do you think this rule

usually performs better than the others?