Dominance rule and opposition-based particle swarm optimization for two-stage assembly scheduling with time cumulated learning effect

Abstract

This paper introduces a two-stage assembly flowshop scheduling model with time cumulated learning effect, which exists in many realistic scheduling settings. By the time cumulated learning effect, we mean that the actual job processing time of a job depends on its scheduled position as well as the processing times of the jobs already processed. The first stage consists of two independently working machines where each machine produces its own component. The second stage consists of a single assembly machine. The objective is to identify a schedule that minimizes the total completion time of all jobs. With analysis on the discussed problem, some dominance rules are developed to optimize the solving procedure. Incorporating with the developed dominance rules, a dominance rule and opposition-based particle swarm optimization algorithm (DR-OPSO) and branch-and-bound are devised. Computational experiments have been conducted to compare the performances of the proposed DR-OPSO and branch-and-bound through comparing with the standard O-PSO and PSO. The results fully demonstrate the efficiency and effectiveness of the proposed DR-OPSO algorithm, providing references to the relevant decision-makers in practice.

Appendix: Dominance rule-based branch-and-bound algorithm

A branch-and-bound algorithm systematically and comprehensively searches an enumerated set of solution candidates which is consider as a full set of rooted tree. The algorithm searches out each subset (or branch) of this solution set (or tree). Each subset is examined with estimated lower and upper bounds of the optimal solution. In enumerating the subset of solution candidates, those cannot yield a solution better than the best already found by the algorithm are discarded.

The branching procedure adopts depth-first search in Fig. 6, which has the advantage of requiring the computer to store no more than (n − 1) nodes for the lower bounds throughout the procedure. Beginning with the first position, the algorithm allocates jobs in a forward manner. In the process, we select a branch and systematically search down the tree. Adopting the similar ideas from Ignall and Schrage (1965) and Lee et al. (1993), a lower bound can be obtained as follows.

$$ \begin{aligned} {\text{LB}} = & \,\hbox{max} \left\{ {t_{3k} + \mathop \sum \limits_{i = 1}^{n - k} p_{{3\left( {k + i} \right)}} \left( {1 - \frac{{T_{3}^{k} + p_{{3\left( {k + i} \right)}} }}{{T_{3*} }}} \right)^{\alpha } } \right. \\ & + \,\mathop {\hbox{min} }\limits_{{j \in AS^{c} }} \left[ {\hbox{max} \left( {t_{3k} ,t_{1k} + p_{1j} \left( {1 - \frac{{T_{1}^{k} }}{{T_{1*} }}} \right)^{\alpha } } \right),t_{2k} }\right.\\& \quad\left.{+ p_{2j} \left( {1 - \frac{{T_{2}^{k} }}{{T_{2*} }}} \right)^{\alpha } - t_{3k} } \right] , \\ & \hbox{max} \left\{ {\mathop \sum \limits_{i = 1}^{n} p_{1\left( i \right)} \left( {1 - \frac{{T_{1}^{k} + p_{{1\left( {i - 1} \right)}} }}{{T_{1*} }}} \right)^{\alpha } , }\right.\\&\quad \left.{\mathop \sum \limits_{i = 1}^{n} p_{2\left( i \right)} \left( {1 - \frac{{T_{2}^{k} + p_{{2\left( {i - 1} \right)}} }}{{T_{2*} }}} \right)^{\alpha } } \right\} \\ & \quad \left. { + \mathop {\hbox{min} }\limits_{\forall j} \left\{ {p_{3j} } \right\} \times \left( {1 - \frac{{T_{3*} - \mathop {\hbox{min} }\limits_{\forall j} \left\{ {p_{3j} } \right\}}}{{T_{3*} }}} \right)^{\alpha } } \right\} \\ \end{aligned} $$

Note that p_i(0) = 0, i = 1, 2, 3.

Fig. 6

Flowchart of dominance rule-based branch-and-bound algorithm

The branch may be eliminated by the pair of lower and upper bounds or the dominance properties. Otherwise, when reaching its final node, the sequence is likely to replace the initial solution or is removed, which depends on the quality of this current solution. The details are provided as follows:

Step 1	Initialization Perform the standard PSO algorithm to obtain a sequence as the initial solution
Step 2	Branching Use the depth-first search strategy
Step 3	Eliminating Use Properties 1 to 7 to eliminate the dominated partial sequences. For the non-dominated nodes, use Properties 8 to 10 to check whether the order of the unscheduled jobs can be directly determined by SPT
Step 4	Bounding Calculate the lower bound according to the introduced formula If the result is greater than the initial solution, eliminate that node and all nodes beyond it in the branch. Otherwise, replace it as a new solution
Step 5	Termination Repeat steps 3–4 until no more nodes left