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Effectiveness of consignment stock policy in a three-level supply chain

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Abstract

The paper studies a three-level supply chain having one supplier, one vendor and multiple buyers under multi-shipment consignment stock policy which is applied by the supplier when he delivers semi-finished products to the vendor. The vendor who produces fully finished products applies the same policy when he delivers to buyers. The integrated system cost which consists of production cost, shipment cost and stock-holding cost is minimized subject to no-shortage in any part of the chain. Optimal decisions are obtained for a numerical example. It is found that the consignment stock policy performs better than the usual shipment policy when the stocking component of holding cost of the item is less in the downstream. The effectiveness of the consignment stock policy under different situations is also investigated.

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Correspondence to T. Maiti.

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Appendices

Appendix 1

To find the supplier’s holding area, we use the method adopted by Huang (2004):

$$\begin{aligned} \hbox {Holding area} = \hbox {bold area} - \hbox {shaded area}. \end{aligned}$$

However, the bold area in our model is different from that of Huang’s (2004) model because of synchronizing delivery policy. We have

$$\begin{aligned} \hbox {Bold area}&= kq_{vw} \left( \frac{k q_{vw}}{P_2} \right) - k q_{vw} \left( \frac{kq_{vw}}{2P_1} \right) \\&= \frac{k^2 q_{vw}^2}{P_2} - \frac{k^2q_{vw}^2}{2P_1}\\ \hbox {Therefore, the holding area}&= \frac{k^2 q_{vw}^2}{P_2} - \frac{k^2q_{vw}^2}{2P_1} - \frac{k(k-1)q_{vw}^2}{2P_2}\\&= \frac{k q_{vw}^2}{2 P_2} \left( k + 1 - \frac{k P_2}{P_1} \right) . \end{aligned}$$

Appendix 2

For the consignment stock policy, the shipment size is \(q_{s}\) and the length between two consecutive shipments is \(\frac{q_{s}}{P_{1}}\). Let \(X_{i}\) be the inventory level at the vendor’s warehouse at the time of ith shipment, \(i=1,2,3,\ldots .\). Then we have

$$\begin{aligned} X_{1}=q_{s}, \quad X_{2}=2q_{s}-\frac{q_{s}}{P_{1}}P_{2}, \quad X_{i}=iq_{s}-(i-1)\frac{q_{s}}{P_{1}}P_{2} \end{aligned}$$

The holding area includes \((k-1)\) trapeziums and one triangle. The sum of \((k-1)\) trapeziums is given by

$$\begin{aligned} \Delta =\sum _{i=1}^{k-1}\frac{q_{s}}{2P_{1}}\left[ X_{i} + (X_{i+1}- q_s)\right] = \frac{q_s}{2P_1} \left[ k(k -1)q_s - \frac{(k-1)^2 q_s P_2}{P_1}\right] \end{aligned}$$

and the area of the triangle is \(\frac{X_{k}^{2}}{2P_{2}}\).

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Giri, B.C., Chakraborty, A. & Maiti, T. Effectiveness of consignment stock policy in a three-level supply chain. Oper Res Int J 17, 39–66 (2017). https://doi.org/10.1007/s12351-015-0215-8

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