Safety stock management in a supply chain model with waiting time and price discount dependent backlogging rate in stochastic environment

Abstract

In today’s dynamic global economy, with highly oscillated demand for any product, safety stock and lead-time management have become even more challenging. In this paper, we develop an integrated supply chain model under continuous-review inventory policy where the lead-time demand is uncertain and the lead-time consists of two components: production time and transportation time; the production time being dependent on ordered quantity and the transportation time being in a range between minimum and normal durations which can be crashed to minimum duration with some additional investment. The unsatisfied demands are partially backlogged with the backlogging parameter being dependent on the time the customers wait before receiving the item. The buyer provides a certain range of price discount to increase the backorder rate. Unlike the previous research, this study considers that the safety factor for the first shipment is different from the rest of the shipments. The model is formulated to find the optimal solution for order quantity, safety factors, price discount, transportation time, and the number of shipments from the vendor to the buyer so that the joint total cost attains the minimum value. Some theoretical results are derived to demonstrate the existence and uniqueness of the optimal solution. It is seen from the numerical study that lead-time reduction is more profitable when the backorder rate depends on both price discount and lead-time.

Appendix: Notation

	Description
Decision variables
Q	size of a shipment (units)
$\kappa _{1}$	Safety factor of the first shipment
$\kappa _{2}$	Safety factor of the r-th shipment, $r = 2,3,\ldots ,n$
$b_{x}$	bBackorder price discount
T	Transportation time
n	Number of deliveries from the vendor to the buyer
Parameters
D	Demand rate at the buyer (units/year)
P	Production rate at the vendor (units/year)
$A_{s}$	Vendor’s setup cost per setup $(\$/$setup)
$A_{b}$	Buyer’s ordering cost per order $(\$/$order)
$H_{b}$	Unit holding cost at the buyer ($\$/$/unit/year)
$H_{v}$	Unit holding cost at the vendor ($\$/$/unit/year)
F	Transportation cost per shipment $(\$/$shipment)
$b_{0}$	Buyer’s marginal profit $(\$/$unit)
R(T)	Transportation time crashing cost
$p_{t}$	Time required to produce Q units by the vendor,
	i.e., $p_{t}=\frac{Q}{P}$ (time unit)
$S^{'}$	Safety stock of the first shipment
$S^{''}$	Safety stock of the r-th shipment, $r = 2,3,\ldots ,n$
$r^{'}$	Reorder point of the first shipment
$r^{''}$	Reorder point of the r-th shipment, $r = 2,3,\ldots ,n$
$\alpha _{1}$	Lead-time demand of the first shipment, a random variable
$\alpha _{2}$	Lead-time demand of the r-th shipment, $r = 2,3,\ldots ,n$,
	a random variable
$\beta _{0}^{'}$	Maximum backorder rate of the first shipment
$\beta _{0}^{''}$	Maximum backorder rate of the r-th shipment, $r = 2,3,\ldots ,n$
$f(\alpha _{1})$	Probability density function of $\alpha _{1}$
$f(\alpha _{2})$	Probability density function of $\alpha _{2}$
$\sigma$	Lead-time demand deviation
$ETC_{b}$	Buyer’s expected total cost per unit time
$ETC_{v}$	Vendor’s expected total cost per unit time
JETC	Joint expected total cost per unit time