Empirical study on comparison of two
different modeling approaches for the
robust shortest path problem with
interval data.
Zuhiar Iftikhar
Robustness criteria [Karasan, Pinar & Yaman-01]
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The robustness criteria we use are the minimax (robust path) criterion and
minimax regret (robust deviation) criterion.
The robust path length problem is defined as finding among all paths the
one that minimizes the maximum path length from origin to destination
over all realizable input data are costs.
– This criterion will select a path for which the maximum path length taken
across all possible realizations is as low as possible.
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The robust path deviation problem is defined as finding among all paths
the one that, over all realizable input data, minimizes the maximum
deviation of the path length from the optimal path length of the
corresponding realization.
– In other words, we wish to find a path such that the maximum difference
between the length of this path and the length of the shortest path over all
realizations of input data is smallest, i.e., a solution that exhibits the smallest
worst case deviation from optimality over all potential realizations.
Definitions
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Definition: The robust deviation for a path p is defined as the difference between
the length of path p and the length of the shortest path in the graph for a specific
realization of arc lengths, i.e., deviation p = length p − length p*
where dp denotes the robust deviation and p∗ denotes the shortest path in the
graph.
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Definition: A path p is said to be a robust deviation path if it has the least
maximum robust deviation among all paths, i.e., robust deviation path
pr ∈ arg min p∈P max s∈S(length p − length p*(s))
where p∗(s) denotes the shortest path in scenario s.
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In order to find a robust deviation path in a graph, we need to only consider the
scenario which makes the robust deviation maximum. Such a scenario is given in
the proposition described in next slides
Relative Robust Shortest Path problem (RelRSP)
Originally proposed in [Karasan, Pinar & Yaman,04], targets minimization of the
maximum deviation from the optimal shortest path over all realizations of
interval arc costs. We will refer to this model as relative robust shortest path
problem. This case is principally different from the absolute robust shortest
problem [Kouvelis & Yu,97] which is to select a path for which the maximum
path length taken across all possible scenarios is minimal – a variant which can
be easily solved in case of interval data. This problem is significantly harder than
the conventional shortest path problem that deals with fixed and static positive
values of weights associated with every arc.
Flexible Robust Shortest Path problem (FlexRSP)
First mentioned in [Bertsimas&Sim,02]. When only the arc costs are subject to
uncertainty in a network with m arcs, the procedure of solving the robust
shortest path problem involves solving up to m + 1 instances of the shortest path
problem with modified costs. As a consequence, we get a robust counterpart
model which is polynomially solvable. The presence of some integer parameter
Γ in the robust counterpart model, which is used to restrict uncertainty, can be
used to control the degree of robustness of the given solution.
Basic setup
Let G = (V,A) be a digraph, where V = {v1, ..., vn}, |V | = n, is the set of nodes and A =
{a1, ..., am}, |A| = m, m ≤ n(n − 1), is the set of arcs.
The source node v1 and the destination node vn are given. An ordered chain of
alternating nodes and arcs such that every two neighboring nodes vk and vk+1 are
connected with arc (vk, vk+1) є A is said to be a path p from v1 to vn.
Moreover, if all the nodes in a chain are different, the path is called simple. Let P be
the set of all paths from v1 to vn. With each arc (vi, vj) є A we associate nominal costs
c0ij
Simple network
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NOMINAL SHORTEST PATH PROBLEM (SP)
Deviation values:
Now we assume that the costs may deviate from their original values with
corresponding deviation values dij ≥ 0. i.e. with each arc (vi, vj) є A we
associate a cost interval [lij , uij],
0 < lij < uij < c, lij = c0ij ,
uij = c0ij + dij ,
i.e. for each arc (vi, vj) є A its cost csij is not fixed and belongs to [lij , uij ].
A realization of all arc costs is called a scenario s. The set S is the set of all
possible scenarios. Let s be a realization of arc costs, i.e. csij є [lij , uij].
We denote by cps the total cost of path p in scenario s, which is defined as:
where Ap is the set of arcs that the path p consists of.
Robust deviations, worst case scenario, minmax regret shortest path
Relative robust shortest path problem (RelRSP)
Absoulute robust shortest path problem (AbsRSP)
Integer programming formulation of the FlexRSP
Flexible Robust Shortest Path problem (FlexRSP) [Bertsimas&Sim,02]
When only the arc costs are subject to uncertainty in a network with m arcs, the
procedure of solving the robust shortest path problem involves solving up to m
+ 1 instances of the shortest path problem with modified costs. As a
consequence, we get a robust counterpart model which is polynomially
solvable.
Flexible robust shortest path problem (FlexRSP)
Test graphs
Graph network examples
Simple network
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2 subgraphs
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Karasan network examples
Karasan graph. Width = 2
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Karasan graph. Width = 3
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Some observations
Conclusions