Lot streaming for a two-stage assembly system in the presence of handling costs

Abstract

Lot streaming is a strategy of splitting a production lot into several sublots so that completed sublots can be transferred to a downstream machine before the entire lot has been completely processed at a current machine. Such a strategy increases the velocity of material flow through a system. However, an increase in the number of transfers between stages also increases material handling cost. In this paper, we address the problem of minimizing a weighted sum of the makespan and handling costs when multiple lots are produced in an assembly system consisting of s suppliers in the first stage and a single assembly machine in the second stage. We exploit a relationship between an optimal solution for this problem and that for a related single-lot, two-machine makespan minimization problem in developing a polynomial-time algorithm to obtain the optimal number and sizes of sublots for each lot when a sequence for processing the lots on the assembly machine is given. We also provide novel integer programming formulations to simultaneously determine an optimal production sequence for the lots and the number of sublots and sublot sizes for each lot, and present a polynomial-time method to obtain valid inequalities for this problem. Experimental results are presented to demonstrate the effectiveness of this formulation over the traditional linear-ordering-based formulations.

Appendices

Appendix

A Proof of Lemma 1

Proof

Consider a two-machine flow shop consisting of a supplier $v^{*}$ at the first stage and the assembly machine at the second stage, and a single-lot $i^{*}$ is produced and transferred in $q^{*}$ sublots. For the two-machine problem, $q^{*}$ is fixed. (Note that there may be some other value of $q^{*}$ for which the joint objective function is minimized for the two-machine problem, but that value is not relevant to any results presented in this paper.) Let sublot sizes $u_{1},\,u_{2},\ldots ,u_{q^{*}}$ minimize the makespan for the two-machine problem. Then we need to show that if supplier $v^{*}$ uses $q^{*}$ sublots to transfer lot $i^{*}$ in an optimal schedule for the ALSP, sublot sizes $u_{1},\,u_{2},\ldots ,u_{q*}$ are optimal for that lot.

We prove the result by construction. Consider an optimal schedule for the ALSP in which supplier $v^{*}$ uses $q^{*}$ sublots of sizes $u'_{1},\,u'_{2},\ldots ,u'_{q^{*}}$. In order to simplify the notation, assume, without loss of generality that the optimal sequence in which the lots are processed is $1,\ldots ,l$. The makespan for such a production sequence is given by

$$\begin{aligned} \mathop {max}\limits _{\{ k,v\} }\left( \sum _{i=1}^{k-1}p_{i}^{v}+\sum _{q=1}^{n_{k}^{v}}f_{kq}^{'v}\delta _{kq}^{v}+\sum _{i=k+1}^{l}p_{i}\right) , \end{aligned}$$

(49)

where $f_{kq}^{'v}$ is the makespan of the associated two-machine, single-lot flow shop problem when sublot sizes $u_{1}^{'},\,u_{2}^{'},\ldots ,u_{q}^{'}$ are used by the supplier for lot k, and other terms in (49) are independent of the sublot sizes used. By definition, $f_{k*q*}^{'v*}\ge f_{k*q*}^{v*}$ for all $k=1,\ldots ,l$ and $v=1,\ldots ,s$. Hence, replacing sublot sizes $u_{1}^{'},\,u_{2}^{'},\ldots ,u_{q^{*}}^{'}$ for lot $i^{*}$ from supplier $v^{*}$ with $u_{1},\,u{}_{2},\ldots ,u{}_{q^{*}}$, and hence, replacing $f_{i^{*}q^{*}}^{'\,v^{*}}$ with $f_{i^{*}q^{*}}^{v^{*}}\left( \le f_{i^{*}q^{*}}^{'\,v^{*}}\right)$ does not increase the makespan. Also, note that only the sublot sizes were altered; the number of sublots used remains unaltered as $q^{*}$ in both the initial and the final schedules. So the transportation cost is not altered since it is a function of the number of sublots alone. Hence, the schedule remains optimal. $\square$