Tri-directional Scheduling Scheme: Theory and Computation

Abstract

In this paper we introduce a new scheduling scheme based on so called tri-directional scheduling strategy to solve the well known resource constrained project scheduling problem. In order to demonstrate the effectiveness of tri-directional scheduling scheme, it is incorporated into a priority rule based parallel scheduling scheme. Theoretical and numerical investigations show that the tri-directional scheduling scheme outperforms forward, backward and even bidirectional schemes depending on the problem structure and the priority rule used. Based on empirical evidence, it seems that as the number of activities are increased, the tri-directional scheduling scheme performs better irrespective of the priority rule used. This suggests that tri-directional scheme should also be applied within the category of heuristic methods.

Appendix

Proof of Lemma 1

Without loss of generality, suppose that only one resource type is required by the activities of P. Let A _N be the set of activities of S whose finish time is T. Let u _j1 is constant per unit of time resource requirements (resource 1) of activity j and P _j denotes the set of predecessors of activity j. Suppose that, there are w (w ≥ 1) gaps in the feasible schedule S. Gap is defined as a space in S where there is availability of resource such that part of some eligible activities can be executed. The amount of available resource is called depth of the gap.

For the k-th gap; Gap _k , k = 1,...,w define:

$$ Acti\emph{v}e\left( {{\rm {\bf A}}_N ,k} \right)=\left\{ {i\in {\rm {\bf J}}=\left\{ {1,...,N} \right\}\,\,\left| {\,\,SG_k <F_i \le \max \left\{ {S_i \left| {\,i\in {\rm {\bf A}}_N } \right.} \right\}\,\,} \right.} \right\} $$

in which S _i and F _i are the start time and the finish time of activity i, i = 1,...,N respectively in schedule S, and Gap _k occurs in the time interval $[ {SG_k ,SG_k +d_k^{\prime} } ]$ where $SG_k +d_k^{\prime} <T$. If there is a $Gap_{i_0 }$ subject to ∀ i ∈ Active (A _N , i _o ) such that i ∉ P _j ; ∀ j ∈ A _N , and $\sum\limits_{j\in {\rm {\bf A}}_N } {u_{j1} } \;\le g_{i_0 }$ in which $g_{i_0 }$ is the depth of $Gap_{i_0 }$, then by transferring a part of all activities that belong to A _N onto $Gap_{i_0 }$ we have T ^′ < T.□

Proof of Lemma 2

By Lemma 1, when PS _f and PS _b are interlinked, the activities of PS _b that are left shifted towards PS _f may fill some or all of the possible gaps that exist in PS _f and PS _b . This may reduce the gaps in the complete schedule and will result in a schedule with smaller makespan, i.e. T ₂ ≤ T ₁.□

Proof of Lemma 3

In tri-directional scheme, we have three sub-schedules, namely PS _f , PS _m and PS _b . When PS _m is interlinked to PS _f , according to Lemma 1, the makespan of the resulting schedule i.e. PS _fm can not be longer than the makespan of the schedule obtained when activities of PS _f and PS _m are linked together as two independent blocks of activities. The same argument can be applied to PS _fm and PS _b , resulting in a schedule, PS _fmb . Thus the makespan PS _fmb is not worse than T ₂, i.e. T ₃ ≤ T ₂.□