Sometimes there is a gap in an optimization model, such as:
Min cx
subject to:
AX
The simple answer would be to remodel your system with the added variable and re-optimise based on your Min Cx goal. But if you are looking for how the optimisation problem for the re-modelled system, how the optimisation would be related to the initial optimisation problem then that is another problem altogether. But is this relation what you are asking for?
You are right Dear Nyandoro,I tend to remodel and re-optimise,but I am looking for how can we allow to add new constraint ,for example feasibility of new constraint.I mean to check the feasibility we must assess whole model along with objective function?
Your question is barely clear, please, can you be more specific?
It would help if you specify what do you mean by "gap" in an optimization model. Do you mean that given your computational power you are not able to solve your model to optimality and hence you end up with a gap between the upper and lower bounds?
If that is the case you are talking about, it may be that your formulation is "hard" in the sense that commercial solvers are not able to solve them quickly by using their built-in branch-and-bound or cutting plane algorithms. Reformulating your problem may help if you are able to find a tighter formulation, tighter in the sense that the polyhedron defined by the linear constraints is closer to the convex hull defined by all feasible binary solutions.
However, "adding a binary variable Y" is too much of a general statement for us to be able to give any tips. Reasons to add a binary variable may be if your new formulation makes use of it, if you want to simplify the mathematical presentation of the model, or the model needs it to accurately represent the problem you are dealing with.
"Are we allowed to bring a new variable? How may our model change? And what conditions must be checked before adding it?" - Y
So, basically, you are allowed to do anything, provided your model represents the real world problem as accurately as possible, and the simplifications are clearly stated. How your model changes and what conditions you must check is problem dependent and there is little I could tell without further knowledge of the problem.
Dear researchers,
original problem is Mixed integer programming as below:
Max sum(i in I) a* f
subject to:{
Ai Xj
I'm afraid the question is still not clear. What do you mean by "affect on objective function", "remodel objective function", "decomposition method", "guarantee optimally problem", "update new binary variable"?
I suggest that you start by reading a decent textbook on integer programming models. The one by H.P. Williams, "Model Building in Mathematical Programming", is very good. It is now in its 5th edition.
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