rm(list = ls())
library(distr)
T=360
B=1
c=0.05
L=900
MM=400
Cp=1
Ci=2
Cf=3
nmax=15


tau1=5
Target1=function(alpha){

Pn_func1=function(n){
  integrand1=function(x){
   (1- pgamma(L-x,shape = alpha*tau1,rate = B))*
      dgamma(x,shape = alpha*(n-1)*tau1,rate = B)
  }
  p1=integrate(integrand1,lower = 0,upper = L)$value
  p1
}
Pn_func1=Vectorize(Pn_func1,"n")

  n=1:nmax
  Pn1=Pn_func1(n)
  En1=sum(n*Pn1)
######################Enp######
Np_func1=function(n){
  i=1:(n-1)
  Np1=sum(
    pgamma(L,shape = alpha*i*tau1,rate = B)-
      pgamma(MM,shape = alpha*i*tau1,rate = B)
  )
  Np1
}
Np_func1=Vectorize(Np_func1,"n")

  n=1:nmax
  Np1=Np_func1(n)
  #Pn=Pn_func1(n, alpha, B, T, L)
  Enp1=sum(Np1*Pn1)
 
#####################lL##############

   L1=1000*(Cf+Ci*En1+Cp*Enp1)/(tau1*En1)
 
 return(L1)

}
Target1=Vectorize(Target1,"alpha")

optimize(Target1,c(1,100))
plot(Target1,12,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))


tau2=10
Target2=function(alpha){

Pn_func2=function(n){
  integrand2=function(x){
   (1- pgamma(L-x,shape = alpha*tau2,rate = B))*
      dgamma(x,shape = alpha*(n-1)*tau2,rate = B)
  }
  p2=integrate(integrand2,lower = 0,upper = L)$value
  p2
}
Pn_func2=Vectorize(Pn_func2,"n")

  n=1:nmax
  Pn2=Pn_func2(n)
  En2=sum(n*Pn2)
######################Enp######
Np_func2=function(n){
  i=1:(n-1)
  Np2=sum(
    pgamma(L,shape = alpha*i*tau2,rate = B)-
      pgamma(MM,shape = alpha*i*tau2,rate = B)
  )
  Np2
}
Np_func2=Vectorize(Np_func2,"n")

  n=1:nmax
  Np2=Np_func2(n)
  #Pn=Pn_func2(n, alpha, B, T, L)
  Enp2=sum(Np2*Pn2)
 
#####################lL##############

   L=1000*(Cf+Ci*En2+Cp*Enp2)/(tau2*En2)
 
 return(L)

}
Target2=Vectorize(Target2,"alpha")

optimize(Target2,c(1,100))
plot(Target2,6,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))


tau3=15
Target3=function(alpha){

Pn_func3=function(n){
  integrand3=function(x){
   (1- pgamma(L-x,shape = alpha*tau3,rate = B))*
      dgamma(x,shape = alpha*(n-1)*tau3,rate = B)
  }
  p3=integrate(integrand3,lower = 0,upper = L)$value
  p3
}
Pn_func3=Vectorize(Pn_func3,"n")

  n=1:nmax
  Pn3=Pn_func3(n)
  En3=sum(n*Pn3)
######################Enp######
Np_func3=function(n){
  i=1:(n-1)
  Np=sum(
    pgamma(L,shape = alpha*i*tau3,rate = B)-
      pgamma(MM,shape = alpha*i*tau3,rate = B)
  )
  Np
}
Np_func3=Vectorize(Np_func3,"n")

  n=1:nmax
  Np3=Np_func3(n)
  #Pn=Pn_func3(n, alpha, B, T, L)
  Enp3=sum(Np3*Pn3)
 
#####################lL##############

   L=1000*(Cf+Ci*En3+Cp*Enp3)/(tau3*En3)
 
 return(L)

}
Target3=Vectorize(Target3,"alpha")

optimize(Target3,c(1,100))
plot(Target3,3.9,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))

tau=20
Target4=function(alpha){

Pn_func4=function(n){
  integrand4=function(x){
   (1- pgamma(L-x,shape = alpha*tau,rate = B))*
      dgamma(x,shape = alpha*(n-1)*tau,rate = B)
  }
  p=integrate(integrand4,lower = 0,upper = L)$value
  p
}
Pn_func4=Vectorize(Pn_func4,"n")

  n=1:nmax
  Pn=Pn_func4(n)
  En=sum(n*Pn)
######################Enp######
Np_func4=function(n){
  i=1:(n-1)
  Np=sum(
    pgamma(L,shape = alpha*i*tau,rate = B)-
      pgamma(MM,shape = alpha*i*tau,rate = B)
  )
  Np
}
Np_func4=Vectorize(Np_func4,"n")

  n=1:nmax
  Np=Np_func4(n)
  #Pn=Pn_func4(n, alpha, B, T, L)
  Enp=sum(Np*Pn)
 
#####################lL##############

   L=1000*(Cf+Ci*En+Cp*Enp)/(tau*En)
 
 return(L)

}
Target4=Vectorize(Target4,"alpha")

optimize(Target4,c(1,100))
plot(Target4,2.9,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))


par(mfrow=c(2,2))
plot(Target1,12,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))
legend("topleft",c(expression("tau=5")))
plot(Target2,6,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))
legend("topleft",c(expression("tau=10")))
plot(Target3,3.9,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))
legend("topleft",c(expression("tau=15")))
plot(Target4,2.9,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(eta(tau,alpha)))
legend("topleft",c(expression("tau=20")))




<pre>An Error occurred while handling another error:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/ErrorHandler.php(136): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/ErrorHandler.php(135): yii\web\ErrorHandler-&gt;renderException()
#3 [internal function]: yii\base\ErrorHandler-&gt;handleException()
#4 {main}
Previous exception:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(656): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/models/Files.php(696): yii\base\Application-&gt;end()
#3 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/controllers/FilesController.php(484): faravaghi\filemanager\models\Files-&gt;getFile()
#4 [internal function]: faravaghi\filemanager\controllers\FilesController-&gt;actionGetFile()
#5 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/InlineAction.php(57): call_user_func_array()
#6 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Controller.php(180): yii\base\InlineAction-&gt;runWithParams()
#7 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Module.php(528): yii\base\Controller-&gt;runAction()
#8 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Application.php(103): yii\base\Module-&gt;runAction()
#9 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(386): yii\web\Application-&gt;handleRequest()
#10 /var/www/html/prof-homepages/frontend/web/index.php(18): yii\base\Application-&gt;run()
#11 {main}</pre>