Distributionally Robust Decision Model
Discrete Polyhedral Ambiguity Set
To solve the best worst-case expected value, we must form the discrete polyhedral ambiguity set as discussed in Uncertainty Sets for the Probabilities of a robust chance node $i∈C.$
We denote the probability of state $s_i∈S_i$ with information path $s_{I(i)}∈𝐒_{I(i)}$ as $ℙ(X_i=s_i∣X_{I(i)}=s_{I(i)})$ and difference as $𝔻(X_i=s_i∣X_{I(i)}=s_{I(i)}).$ Then, we can denote the probability vector of given information path as
\[𝐩(s_{I(i)})=(ℙ(X_i=s_i∣X_{I(i)}=s_{I(i)})∣s_i∈S_i)\]
Furthermore, we can use cross-assignment on the probability vector to form the optimal Local Polyhedral Ambiguity Set
\[𝐃_{𝐩(s_{I(i)})}.\]
We should ignore all inactive chance states when forming the ambiguity set. The elements of the ambiguity set are difference vectors, denoted as $𝐝∈𝐃_{𝐩(s_{I(i)})}$ where
\[𝐝=(𝔻(X_i=s_i∣X_{I(i)}=s_{I(i)})∣s_i∈S_i).\]
We will refer to them as uncertainty.
Path Probability with Uncertainty
Before we can formulate the expected value, we must formulate the path probability with uncertainty by reformulating the Path Probability such that we add the difference $𝐝_{𝐬_i}$ to the probabilities of robust chance node $i.$ Then, the upper bound of path probability with uncertainty becomes
\[p(𝐬,i,𝐝) = (ℙ(X_i=𝐬_i∣X_{I(i)}=𝐬_{I(i)})+𝐝_{𝐬_i}) ⋅ ∏_{j∈C∖\{i\}} ℙ(X_j=𝐬_j∣X_{I(j)}=𝐬_{I(j)}).\]
Furthermore, we have the path probability with uncertainty as
\[ℙ(X=𝐬∣Z,i,𝐝)=p(𝐬,i,𝐝)⋅q(𝐬∣Z),\]
Partial Expected Value
We can formulate the partial expected value for all information path $s_{I(i)}∈𝐒_{I(i)},$ in terms of the path probability with uncertainty and path utility
\[𝔼^{′}(X∣Z,i,𝐝,s_{I(i)})= ∑_{𝐬∈𝐒,\, 𝐬_{I(i)}=s_{I(i)}} ℙ(X=𝐬∣Z,i,𝐝)⋅\mathcal{U}(𝐬).\]
The expected value is the sum of partial expected values over all information paths.
In relation to the notation used when defining the expected value, we have the elements of discrete probabilities $𝐪$ as
\[ℙ(X_i=𝐬_i∣X_{I(i)}=𝐬_{I(i)})+𝐝_{𝐬_i},\]
and elements of utility vector $𝐮$ as
\[∏_{j∈C∖\{i\}} ℙ(X_j=𝐬_j∣X_{I(j)}=𝐬_{I(j)})⋅q(𝐬∣Z)⋅\mathcal{U}(𝐬).\]
Path Probability Variables with Uncertainty
To formulate the linear optimization model for the partial expected value, we need to define the path probability variables with uncertainty as equivalent to the path probability with uncertainty similar to the definition of Path Probability Variables.
\[0≤π(𝐬,i,𝐝)≤p(𝐬,i,𝐝),\quad ∀𝐬∈𝐒\]
\[π(𝐬,i,𝐝)≤z(𝐬_j∣𝐬_{I(j)}),\quad ∀j∈D,𝐬∈𝐒\]
\[π(𝐬,i,𝐝)≥p(𝐬,i,𝐝)+∑_{j∈D}z(𝐬_j∣𝐬_{I(j)})-|D|,\quad ∀𝐬∈𝐒\]
The symbol $z(𝐬_j∣𝐬_{I(j)})$ denotes the decision variables.
Maximin Expected Value
As defined in the Best Worst-Case Expected Value, we can define the maximin expected value and minimax regret formulations of the robust decision model.
We maximize the minimum expected value over all possible combinations of difference vectors
\[\underset{Z∈ℤ}{\text{maximize}} \min_{𝐃∈𝐃^{×}} ∑_{s_{I(i)}∈𝐒_{I(i)}} 𝔼^{′}(X∣Z,i,𝐃_{s_{I(i)}},s_{I(i)})\]
where the product ambiguity set is
\[𝐃^{×}=∏_{s_{I(i)}∈𝐒_{I(i)}}𝐃_{𝐩(s_{I(i)})}.\]
The linearized maximin is
\[\underset{Z∈ℤ}{\text{maximize}} ∑_{s_{I(i)}∈𝐒_{I(i)}} x_{s_{I(i)}}\]
\[x_{s_{I(i)}} ≤ 𝔼^{′}(X∣Z,i,𝐝,s_{I(i)}),\quad ∀𝐝∈𝐃_{𝐩(s_{I(i)})},\, s_{I(i)}∈𝐒_{I(i)}\]
By substituting the path probability variables with uncertainty to the definition of partial expected value and, we obtain the linearized maximin in linear optimization model form as
\[\underset{Z∈ℤ}{\text{maximize}} ∑_{s_{I(i)}∈𝐒_{I(i)}} x_{s_{I(i)}}\]
\[x_{s_{I(i)}} ≤ ∑_{𝐬∈𝐒,\, 𝐬_{I(i)}=s_{I(i)}} π(𝐬,i,𝐝)⋅\mathcal{U}(𝐬),\quad ∀𝐝∈𝐃_{𝐩(s_{I(i)})},\, s_{I(i)}∈𝐒_{I(i)}.\]