Nonserial dynamic programming with applications in smart home appliances scheduling - Part II: Nonserial dynamic programming

Abstract

In this paper a dynamic programming (DP) solution approach to a nonconvex resource allocation problem (RAP) is presented. The problem in this paper generalizes the smart home appliances scheduling problem introduced in the companion paper (i.e., Part I). The computation difficulty with solving the RAP depends on the decision variable coupling, which can be described by an interaction graph. This paper proposes a DP algorithm to solve the RAP in the special setting where the interaction graph is a tree. This extends the applicability result of DP to RAP beyond the standard serial case where the interaction graph is a line. The extension of the result is achieved by establishing that even in the tree case DP computation effort is polynomial in the number of decision variables. For RAP in its general form, this paper proposes a modification to the nonserial DP procedure originally introduced by Bertele and Brioschi. Contrary to that of the previous method, the computation effort of the proposed method can be characterized by a well-known minimum feedback vertex set problem. Case studies on known examples indicate that both nonserial DP methods are similarly efficient.

Minimum equivalent precedence relation systems

“…Therefore, w is in fact a walk from u to v, which is denoted as w uv and stored in W in step 6. Furthermore, (13) and (15) together suggest that after the first pass of the first while-loop the degree counters (in particular those of u) are updated to…”

“…In addition, x i 's are precedence-constrained as in (1). It was shown in [15] that the computation effort for solving the resource allocation problem using dynamic programming increases exponentially with the cardinality of the minimum feedback vertex set of the undirected version of graph (V, E). By solving the precedence relation system simplification problems in this paper, and replacing (V, E) with an equivalent (V, E ′ ) with |E ′ | < |E|, it is possible to reduce the computation effort for solving the resource allocation problem in [15].…”

“…Hence, algorithms that require more than O(|V | 3 ) time for problems involving precedence constraints in (1) can potentially benefit from the simplification results in this paper. For example, [15] considered a nonconvex resource allocation problem where the decision variables x i 's are the start times of tasks to be scheduled. In addition, x i 's are precedence-constrained as in (1).…”

Nonserial dynamic programming with applications in smart home appliances scheduling - Part I: Precedence graph simplification

“…One motivation for the simplification is that by describing (1) with a more compact system of time precedence constraints, the computation effort for the nonserial DP procedure presented in the companion paper [9] can be reduced. As an alternative motivation, the time precedence constraint system simplification problem is a generalization of a precedence relation simplification problem considered in the literature (e.g., parallel computing [10]- [12] and manufacturing system [13], [14]).…”

“…The main objective of this and the companion paper [9] is to describe a modified DP approach to solve (1) with nonserial time precedence constraints. The approach should provide the exact optimal solution to (1), and the computation should be as efficient as possible.…”