from pulp import LpProblem, LpMinimize, LpVariable, lpSum, value

# Define activity data

activities $=\{$

$"$A$"$: $\{"$predecessors$"$: $[],$ "normal$\_$cost$"$: $200000,$ "normal$\_$duration": $5,$ "crash$\_$once$\_$cost$"$: $10000,$ "crash$\_$once$\_$duration": $4,$ "crash$\_$twice$\_$cost$"$: $10000,$ "crash$\_$twice$\_$duration": $3\},$

$"$B$"$: $\{"$predecessors$"$: $["$A$"],$ "normal$\_$cost$"$: $50000,$ "normal$\_$duration": $6,$ "crash$\_$once$\_$cost$"$: $15000,$ "crash$\_$once$\_$duration": $5,$ "crash$\_$twice$\_$cost$"$: $15000,$ "crash$\_$twice$\_$duration": $4\},$

$"$C$"$: $\{"$predecessors$"$: $["$A$"],$ "normal$\_$cost$"$: $250000,$ "normal$\_$duration": $4,$ "crash$\_$once$\_$cost$"$: $5000,$ "crash$\_$once$\_$duration": $3,$ "crash$\_$twice$\_$cost$"$: $5000,$ "crash$\_$twice$\_$duration": $2\},$

$"$D$"$: $\{"$predecessors$"$: $["$A$"],$ "normal$\_$cost$"$: $100000,$ "normal$\_$duration": $5,$ "crash$\_$once$\_$cost$"$: $15000,$ "crash$\_$once$\_$duration": $4,$ "crash$\_$twice$\_$cost$"$: $15000,$ "crash$\_$twice$\_$duration": $3\},$

$"$E$"$: $\{"$predecessors$"$: $["$B$"],$ "normal$\_$cost$"$: $25000,$ "normal$\_$duration": $9,$ "crash$\_$once$\_$cost$"$: $15000,$ "crash$\_$once$\_$duration": $8,$ "crash$\_$twice$\_$cost$"$: $15000,$ "crash$\_$twice$\_$duration": $7\},$

$"$F$"$: $\{"$predecessors$"$: $["$B$","$C$"],$ "normal$\_$cost$"$: $75000,$ "normal$\_$duration": $7,$ "crash$\_$once$\_$cost$"$: $7500,$ "crash$\_$once$\_$duration": $6,$ "crash$\_$twice$\_$cost$"$: $7500,$ "crash$\_$twice$\_$duration": $5\},$

$"$G$"$: $\{"$predecessors$"$: $["$C$"],$ "normal$\_$cost$"$: $125000,$ "normal$\_$duration": $6,$ "crash$\_$once$\_$cost$"$: $10000,$ "crash$\_$once$\_$duration": $5,$ "crash$\_$twice$\_$cost$"$: $10000,$ "crash$\_$twice$\_$duration": $4\},$

$"$H$"$: $\{"$predecessors$"$: $["$C$","$D$"],$ "normal$\_$cost$"$: $100000,$ "normal$\_$duration": $8,$ "crash$\_$once$\_$cost$"$: $15000,$ "crash$\_$once$\_$duration": $7,$ "crash$\_$twice$\_$cost$"$: $15000,$ "crash$\_$twice$\_$duration": $6\},$

"I": $\{"$predecessors$"$: $["$D$"],$ "normal$\_$cost$"$: $20000,$ "normal$\_$duration": $3,$ "crash$\_$once$\_$cost$"$: $10000,$ "crash$\_$once$\_$duration": $2,$ "crash$\_$twice$\_$cost$"$: $10000,$ "crash$\_$twice$\_$duration": $1\},$

$"$J$"$: $\{"$predecessors$"$: $["$E$","$F$","$G$"],$ "normal$\_$cost$"$: $75000,$ "normal$\_$duration": $7,$ "crash$\_$once$\_$cost$"$: $2500,$ "crash$\_$once$\_$duration": $6,$ "crash$\_$twice$\_$cost$"$: $2500,$ "crash$\_$twice$\_$duration": $5\},$

$"$K$"$: $\{"$predecessors$"$: $["$F$","$G$","$H$"],$ "normal$\_$cost$"$: $50000,$ "normal$\_$duration": $4,$ "crash$\_$once$\_$cost$"$: $10000,$ "crash$\_$once$\_$duration": $3,$ "crash$\_$twice$\_$cost$"$: $10000,$ "crash$\_$twice$\_$duration": $2\},$

$"$L$"$: $\{"$predecessors$"$: $["$H$",$ "I"$],$ "normal$\_$cost$"$: $20000,$ "normal$\_$duration": $1,$ "crash$\_$once$\_$cost$"$: $0,$ "crash$\_$once$\_$duration": $0,$ "crash$\_$twice$\_$cost$"$: $0,$ "crash$\_$twice$\_$duration": $0\},$

$"$M$"$: $\{"$predecessors$"$: $["$K$","$L$"],$ "normal$\_$cost$"$: $25000,$ "normal$\_$duration": $2,$ "crash$\_$once$\_$cost$"$: $25000,$ "crash$\_$once$\_$duration": $1,$ "crash$\_$twice$\_$cost$"$: $0,$ "crash$\_$twice$\_$duration": $0\},$

$\}$

# Define predecessors

predecessors $=\{$

$"$A$"$: $[],$

$"$B$"$: $["$A$"],$

$"$C$"$: $["$A$"],$

$"$D$"$: $["$A$"],$

$"$E$"$: $["$B$"],$

$"$F$"$: $["$B$","$C$"],$

$"$G$"$: $["$C$"],$

$"$H$"$: $["$C$","$D$"],$

"I": $["$D$"],$

$"$J$"$: $["$E$","$F$","$G$"],$

$"$K$"$: $["$F$","$G$","$H$"],$

$"$L$"$: $["$H$",$ "I"$],$

$"$M$"$: $["$K$","$L$"],$

$\}$

# Function to find the critical path

def find$\_$critical$\_$path$($activities$,$ predecessors$)$:

critical$\_$path $=[]$

visited $=$ set$()$

def dfs$($activity$)$:

visited.add$($activity$)$

if not predecessors$[$activity$]$:

return $[($activity$,$ activities$[$activity$]["$normal$\_$duration"$])]$

max$\_$duration $=0$

max$\_$path $=[]$

for predecessor in predecessors$[$activity$]$:

if predecessor not in visited:

path $=$ dfs$($predecessor$)$

duration $=$ sum$([$p$[1]$ for p in path$])+$ activities$[$activity$]["$normal$\_$duration"$]$

if duration $>$ max$\_$duration:

max$\_$duration $=$ duration

max$\_$path $=$ path $+[($activity$,$ activities$[$activity$]["$normal$\_$duration"$])]$

return max$\_$path

for activity in activities:

path $=$ dfs$($activity$)$

if sum$([$p$[1]$ for p in path$])>$ sum$([$p$[1]$ for p in critical$\_$path$])$:

critical$\_$path $=$ path

return $[$p$[0]$ for p in critical$\_$path$]$

# Define the problem

prob $=$ LpProblem$("$ProjectCrashing$",$ LpMinimize$)$

# Define decision variables

crash$\_$once $=$ LpVariable.dicts$("$CrashOnce$",$ activities.keys$(),$ lowBound$=0,$ upBound$=1,$ cat$=$'Binary'$)$

crash$\_$twice $=$ LpVariable.dicts$("$CrashTwice$",$ activities.keys$(),$ lowBound$=0,$ upBound$=1,$ cat$=$'Binary'$)$

# Define objective function

prob $+=$ lpSum$($

crash$\_$once$[$activity$]*$ activities$[$activity$]["$crash$\_$once$\_$cost$"]+$

crash$\_$twice$[$activity$]*$ activities$[$activity$]["$crash$\_$twice$\_$cost$"]$

for activity in activities

$)$

# Define constraints

for activity in activities:

for predecessor in predecessors$[$activity$]$:

prob $+=($

$($activities$[$activity$]["$normal$\_$duration"$]-$ crash$\_$once$[$activity$]*$ activities$[$activity$]["$crash$\_$once$\_$duration"$]-$ crash$\_$twice$[$activity$]*$ activities$[$activity$]["$crash$\_$twice$\_$duration"$])<=$

$($activities$[$predecessor$]["$normal$\_$duration"$]-$ crash$\_$once$[$predecessor$]*$ activities<

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