The characteristics discussed in section 1 can be induced by following specific steps.
3.1. Variables selection
Three sets of variables need be defined. Each such set consists of one vector of variables per scenario-tree node (i.e., one vector for each combination of time period and scenario realization). The first set represents the state of the system describing the application under consideration. Members of this set are binary. The second describes the recourse actions that can be taken once uncertainty unfolds. Members of this set are continuous variables. The third set describes the necessary actions to transition from one state variable set (of the first variables set) to another. Members of this set are binary.
3.2. Scenario-tree formulation
The scenario-tree node formulation of a problem can be constructed in two steps. First, mathematically formulate each nodal subproblem by a SMIP for which the first-stage decision involves a single period and the second-stage decisions involves a single period. The first-stage decisions represent the state of the system before the
unfolding of uncertainty. The second-stage decisions represent the recourse actions that could be implemented after the unfolding of uncertainty.
The second step involves the stage coupling constraints. These constraints establish the interdependencies between scenario-tree nodes by describing the actions necessary to transition for the state of the system in a parent node to that of its descendant. In other words, these constraints establish the relationship between the first and third of variables.
Following these steps results into constraints indexed over scenario-tree nodes.
3.3. Relatively-complete recourse
The relatively complete recourse property refers to the existence of a feasible second stage solution for any feasible first stage solution. This property enables using a solution of one nodal subproblem to jump start another nodal subproblem (see section 3 of Chapter VII for more details regarding the role of relatively-complete recourse in solving the nodal subproblems).
Inducing the relatively-complete recourse property into a second-stage of a SMIP can be done by introducing an extra set of variables. These variables act as surplus or slack variables for constraints that could otherwise be violated. Pe- nalizing these variables in the objective function guide the second-stage to choose a solution that sets these variables to zero over one that doesn’t. These variables cary practical meaning depending on the application. For example, in production and distribution applications, a set of variables representing the shortage in fulfilling customers’ demand artificially guarantees the feasibility of a constraint mandating that all demand be satisfied.
3.4. Nodal decomposition
By relaxing the stage coupling constraints, the problem decomposes into a number of independent nodal subproblems. Each nodal subproblem is a SMIP enjoying the property of relatively complete recourse.
The master problem re-establishes the interdependencies among these nodal subproblems. The state variables are now decided in the nodal subproblems. Con- sequently, each state variable is replaced by all its feasible values, each weighted by a binary indicator variable.
3.5. Master problem reformulation
The master problem is reformulated into a Leontief substitution flow system (LSFS). This is best achieved by using a hypergraph. In this hypergraph, a vertex represents a possible solution for a nodal subproblem. A hyperarc represents possible transition between nodal subproblem solutions. Each such transition combine several of the third variable type (as many as the number of descendants for a node).
The weights associated with a hyperarc are carefully chosen to induce equiva- lence between a transition cost in the hypergraph and a transition cost in the master problem. This is achieved by calculating the weighted sum of the transition costs to each of the descendant nodes that a hyperac leads to and the state variable cost for these nodes, where each cost is weighted by the arc probability of its corresponding node.
3.6. Solution space approximation
Approximating the solution space is achieved by applying an iterative algorithm on each nodal subproblem. If these iterative solutions are feasible (i.e., resulting from a
primal algorithm like the L-shaped method) they form vertices for the hypergraph. If they are not feasible (i.e., resulting from a dual algorithm like the Lagrangian relaxation) a feasible solution need to be constructed using these infeasible solution. This process will differ from one application to another. However, usually dual algorithms involves finding such a feasible solution to use its value as a bound. This implies that no additional effort is expended to find these feasible solutions.
The generated iterative solutions serve multiple functions:
1. They populate the columns associated with their node in the restricted master problem.
2. They also serve as columns for all other nodes as a result of the relatively complete recourse property of the subproblems.
3. They jump start the L-shape procedure for all following subproblems, as de- scribed in section 4 of Chapter VI.
As a result, as the solution procedure progresses, the required computation effort declines.
3.7. Global solution selection
Once the problem is expressed as an LSFP and its vertices are generated using nodal subproblems, it becomes amenable to solution using the customized shortest hyper- path algorithm summarized in Fig. 7 of Chapter VII. This algorithm constructs a global solution by selecting the best combination of generated nodal solutions.
CHAPTER IX
CONCLUSION
This chapter lists the conclusions, contributions, and future research.