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SRCSP (Stochastic Resource Constrained Shortest Path)

  • Shortest Path in a graph given :
    • Weight on the vertices
    • Source node s
    • Sink node t
    • Stochastic resource consumption on a vertex (K resources)

II) Formulation Max Pr { a_k * x <= d_k | k = 1..K } with constraints: - c * x <= C - Mx = b - x in {0, 1}^n

Given that:

  • The length of x is the number of vertices (x is a binary vector, x(e) = 1 <=> e is in the path P)
  • The length of a_k is the number of vertices (a_k(e) is normally distributed and they are all independent)
  • C is the maximum allowed weight of the path
  • M is the node-arc incidence matrix (M[i,e] = 1 if e = (., i), -1 if e = (i, .), 0 otherwise)
  • b is a vector with 0 everywhere except 1 for s (source node), -1 for t (sink node)

III) K = 1 (one resource)

IV) Joint Probability Constraints

V) Individually Relaxed

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