Coordinated scheduling of the transfer lots in an assembly-type supply chain: a genetic algorithm approach

Abstract

In this study, we consider coordinated scheduling of the transfer lots in an assembly-type supply chain. An assembly-type supply chain consists of at least two stages, where the upstream stages manufacture the components for several products to be assembled at the downstream stages. In order to enable faster flow of products through the supply chain and to decrease the work-in-process inventory, the concept of lot streaming is used as a means of supply chain coordination. We introduce a mathematical model, which finds the optimal transfer lot sizes in the supply chain. The objective is the minimization of the sum of weighted flow and inventory costs. We develop genetic algorithm (GA) based heuristics to solve the proposed model efficiently. The experimental results show that the proposed GA-based approaches provide acceptable results in reasonable amount of time. We also show that coordination with lot streaming provides improvements in the supply chain performance.

Appendix

• a. The weighted flow time cost: The products are delivered to the customers in transfer lots, and the portion of the demand delivered in each sublot is important. For this reason, the flow time cost is weighted by the portion of the demand delivered to the customer in Eq. (15).

  $$\begin{aligned} A=\sum _{k} {\alpha _{k}\sum _{l=1}^{n_{k}} {\left( {{X_{0,1,kl}}/{D_{k}}}\right) W_{0,1,kl}}} \end{aligned}$$

  (15)

• b. The inventory cost: We consider two types of inventory costs.

WIP inventory cost of components For the manufacturer, an example trajectory of a component’s WIP inventory is presented in Fig. 8. The total inventory costs for all the components are provided in Eq. (16). The first term of Eq. (16) denotes the waiting time of the components after receiving the sublot from supplier till starting to process it in the manufacturer. Note that $M_{1,j,k,l}$ denotes the inventory level of the component needed to manufacture product $k$ after the receipt of $l\hbox {th}$ sublot from $j\hbox {th}$ supplier. The second term of Eq. (16) is the WIP inventory of the component when the $l\hbox {th}$ sublot of product $k$ starts processing. Raw material inventory for the suppliers is not considered in the objective function.

$$\begin{aligned} B&= \sum \limits _{j=1}^{2} \sum \limits _{k} h_{1,jk} \sum \limits _{l=1}^{n_{k}} \left( M_{1,jkl} \left( {C_{0,1,kl} -\left( {W_{1,jkl} +T_{1,j}}\right) }\right) \right. \nonumber \\&\left. +\left( {2M_{1,j,k,l} -X_{0,1,kl} a_{1,jk}}\right) \left( {p_{0,1,k} {X_{0,1,kl}}/2}\right) \right) \end{aligned}$$

(16)

Inventory cost of finished goods For a supplier, an example trajectory of the finished good’s inventory is presented in Fig. 8. In Eq. (17) the first term denotes the inventory cost during the process of the sublot. The second term shows the inventory cost between sublot completion and delivery. We consider the inventory of finished goods for the suppliers and the manufacturer.

$$\begin{aligned} C&= \sum \limits _{i=0}^{1} \sum \limits _{j=1}^{2^{i}} \sum _{k} \gamma _{ijk} \sum \limits _{l=1}^{n_{k}} X_{ijkl} \left( \left( {{p_{ijk} X_{ijkl}}/2}\right) +\left( W_{ijkl}\right. \right. \nonumber \\&\left. \left. +\,T_{ij}\right) -\left( {C_{ijkl} +p_{ijk} X_{ijkl}}\right) \right) \end{aligned}$$

(17)

Fig. 8

The trajectories of the inventory levels for the manufacturer and supplier