Basis factorization method for multistage linear programming problems with an application to optimal energy plant operation

An algorithm based on the basis factorization method for solving multistage linear programming problems is presented. In applying linear programming for optimal dynamic-plant operation, the constraint appears in the form of a staircase structure. When pivotings occur within each stage, the inversion of the basis matrix is easily obtained using inversions of small square diagonal blocks. In general cases, pivotings across stages occur. However, this basis matrix can be easily transformed into the former square block triangular form using the correction factor. The correction factor means multiple pivotings among the bases and contains structural columns corresponding to different columns among these bases. When these structural columns are maintained in computer storage, the relative cost factor for general bases is easily obtained using a set of elementary column operations. A program code using the above idea is developed and applied to in-plant energy operating systems. A computational example shows that the computing time of the proposed algorithm is less than 10 percent of the general purpose LP code MPS-II.