Level scheduling under limited resequencing flexibility

Abstract

A mixed-model assembly line requires the solution of a short-term sequencing problem, which decides on the succession of different models launched down the line. A famous solution approach stemming from the Toyota Production System is the so-called Level Scheduling (LS), which aims to distribute the part consumption induced by a model sequence evenly over the planning horizon. LS attracted a multitude of different researchers, who, however, invariably treat initial sequence planning where all degrees of freedom in assigning models to production cycles exist. In the real-world, conflicting objectives and restrictions of preceding production stages, i.e., body and paint shop, simultaneously need to be considered and perturbations of an initial sequence will regularly occur, so that the sequencing problem often becomes a resequencing problem. Here, a given model sequence is to be reshuffled with the help of resequencing buffers (denoted as pull-off tables). This paper shows how to adapt famous solution approaches for alternative LS problems, namely the Product-Rate-Variation (PRV) and the Output-Rate-Variation (ORV) problem, if the (re-)assignment of models to cycles is restricted by the given number of pull-off tables. Furthermore, the effect of increasing re-sequencing flexibility is investigated, so that the practitioner receives decision support for buffer dimensioning, and the ability of the PRV in reasonably approximating the more detailed ORV in a resequencing environment is tested.

Appendix proof of feasibility condition

Proposition

A sequence π =  < 1,…,T > can be resequenced to a new sequence represented by mapping $\sigma:\{1,\ldots,T\}\rightarrow \{1,\ldots,T\}$ by use of K pull-off tables if and only if$\sigma(i) {\geq} i - K \quad \forall i=1,\ldots,T$.

Proof

Necessity of the condition follows directly from the fact that if σ(i) < i − K for any model i then i − σ(i) > K, which means that more than K models would need to be stored simultaneously in order for model i to arrive at its position in σ.

In order to prove sufficiency, we will use a recursive argument to show that a feasible mapping for any sequence π can be readily constructed on the basis of a shorter sequence for which a feasible mapping is known. Specifically, we show that σ is a feasible mapping of π if $1 {\geq}\sigma^{-1}(1)-K$, with σ⁻¹(1) denoting the model assigned to the first position of the new sequence, and it further holds that $\sigma'\{1,\ldots,\sigma^{-1}(1)-1,\sigma^{-1}(1)+1,\ldots,T\}\rightarrow\{1,\ldots,T-1\}$ constitutes a feasible mapping of the reduced sequence $\pi'=<1,\ldots,\sigma^{-1}(1)-1,\sigma^{-1}(1)+1,\ldots,T>$ with $\sigma'(i)=\sigma(i)-1 {\quad}{\forall}i=1,{\ldots,T}{\land}i\,{\neq}\,\sigma^{-1}(1)$.

Suppose that σ′ is a feasible mapping for π′ then it follows from $K+1 {\geq}\sigma^{-1}(1)$ that the first σ⁻¹(1) − 1 models of sequence π′ can be stored in K pull-off tables. Since models from pull-off tables can be accessed in arbitrary order, there in fact has to exist a feasible resequencing strategy which stores the first σ⁻¹(1) − 1 models of π′ in pull-off tables and then inserts them in accordance with their sequence position in σ′.

With regard to π, assigning model σ⁻¹(1) to the first position in the new sequence cannot cause infeasibility as long as $1 {\geq}\sigma^{-1}(1)-K$ holds. A feasible mapping of π thus exists, if there is a resequencing strategy which assigns the remaining models to the T − 1 remaining sequencing positions under the condition that the first σ⁻¹(1) − 1 models of π have to be pulled off-line. Since the first σ⁻¹(1) − 1 models of π and π′(i) are identical, it follows that we can readily construct a feasible mapping for π from σ′ by a simple relabeling of sequence positions for all remaining models with $\sigma(i)=\sigma'(i)+1 \quad \forall i=1,\ldots,T \land i\,{\neq}\,\sigma^{-1}(1)$. The feasibility of σ′ is thus sufficient for the feasibility of σ if $1 {\geq}\sigma^{-1}(1)-K$.

The same argument can now be applied to π′ and σ^'-1(1) and then repeatedly to any reduced sequence of positive length which after relabeling yields the required conditions $\sigma(i) {\geq} i - K \quad \forall i=1,\ldots,T$. $\square$