modeling flow management problem
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thanks to help me encode variable dependency for flow management problem:
i' have constraints on capacity on each arcs
at each step some flow should be migrated ; but i need to express non violation at transition during a \delta(t)
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Hi Bahri,
Your question reads as if the audience is familiar with your problem, and your model formulation (but they are not).
I think we will need to bridge this gap in order for anybody to be able to help. Feel free to post your formulation and a brief description of the problem, variables, and constraints, and illustrate where the issue is with a small example.
- Riley
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let be K=[k1,k2,k3..km] a set of commodities of demand D=[d1..dm] to be migrated from thier old path Pold=[Po1,Po2..Pom] to thier final Path Pf=[Pf1,Pf2..Pfm]
y[k,t] binary denote that flow k is migrated at step t
A set of arcs
C=[ Cij i j in A ] set of capacities on arc ij
the objective is to find minimal flow management plan such that transition do not create overload on arcs
contraints :
on capacity on each arc at each step t on each arcs ij : sum( not migrated flow) + sum( of migrated flow) < C(i;j)
each flow is migrated once. for each k sum of (y[k,t] over t) = 1
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I think some details are still missing, but I'd guess you could introduce continuous variables: z_{k,i,j,t} to represent the flow of commodity k along arc (i,j) at timestep t.
Then you could link it to y with
\[z_{k,i,j,t} \leq \min\{F_k, C_{i,j}\} y_{k,t} \quad \quad \forall k,i,j,t: (i,j) \in A_k\]
where F_k is the total flow required by F, and A_k is the arcs that can be used by commodity k.
Then to observe capacity constraints at each t:
\[\sum_{k : (i,j) \in A_k} z_{k,i,j,t} \leq C_{i,j} \quad \quad \forall (i,j) \in A, t\]- Riley
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