Solution technique

5.6.4 Solution technique

The path-based train slot generation formulation in (8)-(11) is a binary integer program with a block-angular structure. If constraints (9) were omitted, this problem formulation can be separated into a subproblem for each drop-off/pick-up terminal associated with a certain time. This structure can be exploited by a column generation technique that can quickly generate a near optimal solution. Column generation has been successfully applied to solve many large-scale optimization problems in, for example, vehicle routing (Desrochers et

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al., 1992), air crew scheduling (Lavoie et al., 1988; Vance, 1997), lot sizing and scheduling (Cattrysse et al., 1993).

Column generation is a price-directive method that decomposes the multicommodity flow problem into single commodity network flows. Tolls (or prices) are placed on the bundle constraints that complicate finding a solution. This approach exploits the fact that constraints (9) are independent for each k ∈ K and only constraints (10) are dependent among each k ∈ K .

The key idea in column generation is to never explicitly list all of the columns (i.e. decision variables) of the problem formulation, but rather to generate them only “as needed.”

A column for the given formulation represents a train slot. Only a subset of the columns (train slots) are considered at each iteration. This smaller program is referred to as the restricted master problem. The restricted master problem is solved to optimality by the simplex method. Whether the solution is optimal for the original program or if additional columns must be added to improve the solution is assessed at the end of each iteration. A subproblem for each commodity is used to generate a new column for the restricted master problem and solution of the subproblem is used to prove optimality of the current solution. The potential column (train slot) with the most negative cost in each subproblem will be added to the restricted master program.

The column generation procedure is given as follows (see Ahuja et al., 1993 and Hu T. C., 1963 for additional detail). Text in bold will be discussed in detail in the following subsections.

Step 1. Initialization. Choose a set of train slots as an initial basic feasible solution.

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Step 2. Solve the restricted master problem. Solve the problem by using the simplex method to determine the value of the dual variables. Step 3. Check if a new column can be generated. Use the dual variable values of the solved restricted master problem to update the cost coefficient of the subproblem. Get new columns with largest negative reduced costs based on subproblem solutions and add the new columns to the restricted master problem. Return to Step 2 if a new column is found. Otherwise, terminate the procedure. The optimal solution is obtained.

Master problem

The goal of the master problem is to obtain the value of the dual variables so that the reduced cost for each train slot can be calculated for the subproblems. Since the train slot generation formulation shown in section 5.6.2.2 has a block-angular structure, the formulation associated with a smaller set of variables in which the integrality constraints (11) are relaxed can be treated as the restricted master problem.

Subproblem

The goal of the subproblem is to find the column (train slot) with the minimum reduced cost to be added to the master problem. If the minimum reduced cost is nonnegative, then we can terminate the column generation procedure and the problem is solved to optimality. Let k σ denote the dual variable corresponding to each route k in constraints (9)

and ϖ lt denote the dual variable corresponding to each terminal l at time t in constraints (10). The reduced cost, k λ , of the column corresponding to the master problem is given by (12).

Each route k has its own subproblem. The value of reduce cost of each column, k λ can be

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treated as the benefit (i.e. reducing the train operational and penalty costs) obtained by using

a train slot i on the route k.

δ lt ( y i ) ϖ lt − σ , ∀ i ∈ I ∀ k ∈ ∑∑ K (12)

Hence, equation (12) is considered for each train slot i operated on the route k to check if the reduced cost of any column is negative. The column (train slot) with the most negative value of the reduced cost will be added into the restricted master problem. Thus, the train operational and penalty costs incurred from delivery delay can further be reduced by selecting this train slot to transport the shipments.