Help to solve this problem

Answered

Sebastian Urbina

July 19, 2020 06:11 Edited

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I am looking for a way to solve the following problem(Balance equations)

I need to find pi that meets all the equations.(Stationary odds). 

I have doubts when defining the objective function, because I don't know very well what I want to maximize or minimize. I just want to find a solution to the equations

 

This is what I have

from gurobipy import *
mod = Model('BalanceEquations')
lamda = 5.95
mu = 6
N = 4
K = 20
p = 0.4
ro = 1-(1-p)**N

#Variables
pi={}
for x in range(N+1):
    for n in range(K+1):
        pi[x,n]=mod.addVar(vtype="C",lb=0.0000001, name="pi_%s,%s" %(x,n))

#Constraints

for x in range(1,N):
    for n in range(x,K):
        mod.addConstr((lamda+mu)*pi[x,n] == lamda*pi[x,n-1] + mu*pi[x+1,n+1])

for n in range(1,N):
    mod.addConstr(lamda*pi[0,n] == lamda*pi[0,n-1] + mu*pi[1,n+1] + mu*(1-ro)*pi[0,n+N])

for n in range(N,K+1-N):
    mod.addConstr((lamda+mu)*pi[0,n] == lamda*pi[0,n-1] + mu*pi[1,n+1] + mu*(1-ro)*pi[0,n+N])
  
for n in range(1,K+1):
    mod.addConstr((lamda+mu)*pi[N,n] == lamda*pi[N,n-1]+mu*ro*pi[0,n])   
 
mod.addConstr(lamda*pi[0,0] == mu*pi[1,1]+mu*(1-ro)*pi[0,N])

mod.addConstr(quicksum(pi[x,n] for x in range(N+1) for n in range(K+1))==1)


#Here i have doubts
#The optimal result is always 1.0 only for a value pi [i, j], which doesn't work for me
mod.update()
mod.setObjective(quicksum(pi[i,j] for i in range(N+1) for j in range(K+1)),GRB.MAXIMIZE)
mod.optimize()

The optimal result is always 1.0 only for a value pi [i, j], which doesn't work for me. What I need are the values (probabilities) that the equations meet.

 

Thanks so much

0

• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  June 23, 2025 03:09 Edited

  This post is more than three years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum. Or why not try our AI Gurobot?.
• 

  Jaromił Najman

  • Gurobi Staff

  July 27, 2020 08:26 Edited

  Dear Sebastian,

  I have executed your code for \(N=2\) and \(K=3\) and noticed that the variable \(pi_{1,3}\) appears in your objective only. Thus, Gurobi simply sets this variable to the needed value in order to fulfill the \(=1\) equality and sets every other variable to its lower bound trivially fulfilling all other equality constraints.

  I used the write function to investigate the model Gurobi is actually solving.
  
  Please double check the formulas to make sure that every variable participates in the balance equations.

  I hope this could help you in resolving the issue.

  Best regards,
  Jaromił

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