Safety stock placement for serial systems under supply process uncertainty

Abstract

In this study, we address safety stock positioning when demand per period is a known constant but supply is uncertain. The supply is either available or not available, while the setting is that of a periodically reviewed, serial system following a base stock policy. Each stage is allowed to operate according to the guaranteed or stochastic service model. We use a Discrete Time Markov Chain model for expressing the expected on-hand inventories for each stage, along with other terms of interest, as a function of policy parameters determined by a given service level requirement for the end product. Exact models are constructed for single-stage and two-stage systems. As the number of states for a two-stage system grows exponentially, we propose an approximation for expressing the effect of the input stage using a single parameter. A generalization for the approximation is provided for a multi-stage problem. Computational evaluations of the approximation, as well as numerical comparisons of different cases, are presented.

Appendices

Appendix 1: Probability transition matrix for single-stage problem under the SSM

Appendix 2: Probability transition matrix for single-stage problem under the GSM

Appendix 3: Probability transition matrix for two-stage problem under the GSM

Appendix 4: Two-stage model: expressions for stage 1 for the SSM

	SSM
Replenishment Time ($\tau$)	$\tau =\begin{Bmatrix} i&with&probability&\pi ^{S_2}(1-p_1)(\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}]) \end{Bmatrix}$
	where i= 0,1,2,3,...
Demand during $\tau$	$D=\begin{Bmatrix}iQ&with&probability&\pi ^{S_2}(1-p_1)(\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}])\end{Bmatrix}$
	where i= 0,1,2,3,...
$E[D_\tau ]$	$Q\pi ^{S_2}(1-p_1)(\sum _{i=1}^{\infty }\sum _{j=0}^{i}[i(1-\pi ^{S_2})^{i-j}p_{1}^{j}])$
$\sigma ^2_\tau$	$Q^{2}\pi ^{S_2}(1-p_1)(\sum _{i=1}^{\infty }\sum _{j=0}^{i}[i^{2}(1-\pi ^{S_2})^{i-j}p_{1}^{j}])$
	$-[Q\pi ^{S_2}(1-p_1)(\sum _{i=1}^{\infty }\sum _{j=0}^{i}[i(1-\pi ^{S_2})^{i-j}p_{1}^{j}])]^2$

Appendix 5: Two-stage model: probability transition matrix for stage 1 under the SSM

Appendix 6: Two-stage model: expressions for Stage 1 for the GSM

	GSM
Replenishment Time	$\tau =\begin{Bmatrix} i&with&probability&\pi ^{S_2}(1-p_1)(\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}])\\ MQ&with&probability&1-\pi ^{S_2}(1-p_1)(\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}])\end{Bmatrix}$
	where $i=1,..,M_1-1$
Demand during $\tau$	$D=\begin{Bmatrix} iQ&with&probability&\pi ^{S_2}(1-p_1)(\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}])\\ MQ&with&probability&1-\pi ^{S_2}(1-p_1)(\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}])\end{Bmatrix}$
	where $i=1,..,M_1-1$
$E[D_\tau ]$	$Q\pi ^{S_2}(1-p_1)(\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[i(1-\pi ^{S_2})^{i-j}p_{1}^{j}]) +$
	$MQ[1-\pi ^{S_2}(1-p_1)\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}]]$
$E[D_{\tau }^{2}]$	$Q^{2}\pi ^{S_2}(1-p_1)(\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[i^{2}(1-\pi ^{S_2})^{i-j}p_{1}^{j}]) +$
	$M^{2}Q^{2}[1-\pi ^{S_2}(1-p_1)\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[(1-\pi ^{S_2})^{i-j}p_{1}^{j}]]$
$(E[D_\tau ])^2$	$[Q\pi ^{S_2}(1-p_1)(\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}[i(1-\pi ^{S_2})^{i-j}p_{1}^{j}]) +$
	$MQ(1-\pi ^{S_2}(1-p_1)\sum _{i=1}^{M_1-1}\sum _{j=0}^{i}((1-\pi ^{S_2})^{i-j}p_{1}^{j}))]^2$
$\sigma _{\tau }^{2}$	$E[D_{\tau }^2]-(E[D_\tau ])^2$

Appendix 7: Two-stage model: probability transition matrix for Stage 1 under the GSM

Appendix 8: Two-stage models: ranking according to expected total on-hand inventory costs

See Tables 6 and 7.

Table 6 Service level is at least 95.3% (range: [95.3% , 96.8%])

Table 7 Service level is at least 99% (range: [99%, 100%])