Table 2.

Algorithm for finding ${\mathrm{\Delta }}^{\ast }$ and $s^{\ast }$ in the modified (s, S) policy

Algorithm for finding ${\mathrm{\Delta }}^{\ast }$ and $s^{\ast }$ in the modified (s, S) policy
Step 0:	Initialization. $L^{\ast }\leftarrow inf$, ${\mathrm{\Delta }}^{\ast }\leftarrow inf$, $s^{\ast }\leftarrow inf$, ${\widehat{L}}^{\ast }(1:M_{\mathit{max}})=inf$, and ${\widehat{s}}^{\ast }(1:M_{\mathit{max}})=inf$.
Step 1.1:	For $\mathrm{\Delta }=1:{(M_{\mathit{min}}-D_{\mathit{max}}+1)}^{+}$, calculate $P_{i,j}$ by (EC.22) in the online appendix.
Step 1.2:	Given $\mathrm{\Delta }$, ${\widehat{s}}^{\ast }(\mathrm{\Delta })\leftarrow {\arg \min }_{s\in [y^{\ast },y^{\ast }+M_{\mathit{min}}]}L(\mathrm{\Delta },s)$ and ${\widehat{L}}^{\ast }(\mathrm{\Delta })\leftarrow L(\mathrm{\Delta },{\widehat{s}}^{\ast }(\mathrm{\Delta }))$, where
	$L(\mathrm{\Delta },s)$ is calculated as in Case 6. Then, go to Step 2.1.
Step 2.1:	For $\mathrm{\Delta }={(M_{\mathit{min}}-D_{\mathit{max}}+1)}^{+}+1:min\{M_{\mathit{min}},M_{\mathit{max}}-D_{\mathit{max}}+1\}-1$, calculate $P_{i,j}$ by
	(EC.14) in the online appendix.
Step 2.2:	Given $\mathrm{\Delta }$, ${\widehat{s}}^{\ast }(\mathrm{\Delta })\leftarrow {\arg \min }_{s\in [y^{\ast },y^{\ast }+M_{\mathit{min}}]}L(\mathrm{\Delta },s)$ and ${\widehat{L}}^{\ast }(\mathrm{\Delta })\leftarrow L(\mathrm{\Delta },{\widehat{s}}^{\ast }(\mathrm{\Delta }))$, where
	$L(\mathrm{\Delta },s)$ is calculated as in Case 4. Then, go to Step 3.0.
Step 3.0:	If $M_{\mathit{max}}-D_{\mathit{max}}+1\le M_{\mathit{min}}$, go to Sept 3.1;
	Else go to Step 3.3.
Step 3.1:	For $\mathrm{\Delta }=M_{\mathit{max}}-D_{\mathit{max}}+1:M_{\mathit{min}}$, calculate $P_{i,j}$ by (EC.19) in the online appendix.
Step 3.2:	Given $\mathrm{\Delta }$, ${\widehat{s}}^{\ast }(\mathrm{\Delta })\leftarrow {\arg \min }_{s\in [y^{\ast },y^{\ast }+M_{\mathit{min}}]}L(\mathrm{\Delta },s)$ and ${\widehat{L}}^{\ast }(\mathrm{\Delta })\leftarrow L(\mathrm{\Delta },{\widehat{s}}^{\ast }(\mathrm{\Delta }))$, where
	$L(\mathrm{\Delta },s)$ is calculated as in Case 5. Then, go to Step 4.1.
Step 3.3:	For $\mathrm{\Delta }=M_{\mathit{min}}:M_{\mathit{max}}-D_{\mathit{max}}+1$, calculate $P_{i,j}$ by (1).
Step 3.4:	Given $\mathrm{\Delta }$, ${\widehat{s}}^{\ast }(\mathrm{\Delta })\leftarrow {\arg \min }_{s\in [y^{\ast },y^{\ast }+\mathrm{\Delta }]}L(\mathrm{\Delta },s)$ and ${\widehat{L}}^{\ast }(\mathrm{\Delta })\leftarrow L(\mathrm{\Delta },{\widehat{s}}^{\ast }(\mathrm{\Delta }))$, where $L(\mathrm{\Delta },s)$
	is calculated as in Case 1. Then, go to Step 4.1.
Step 4.1:	For $\mathrm{\Delta }=max\{M_{\mathit{min}},M_{\mathit{max}}-D_{\mathit{max}}+1\}+1:M_{\mathit{max}}$, calculate $P_{i,j}$ by ((EC.8) in the
	online appendix.
Step 4.2:	Given $\mathrm{\Delta }$, ${\widehat{s}}^{\ast }(\mathrm{\Delta })\leftarrow {\arg \min }_{s\in [y^{\ast },y^{\ast }+\mathrm{\Delta }]}L(\mathrm{\Delta },s)$ and ${\widehat{L}}^{\ast }(\mathrm{\Delta })\leftarrow L(\mathrm{\Delta },{\widehat{s}}^{\ast }(\mathrm{\Delta }))$, where
	$L(\mathrm{\Delta },s)$ is calculated as in Case 2. Then, go to Step 5.
Step 5:	${\mathrm{\Delta }}^{\ast }\leftarrow {\arg \min }_{\mathrm{\Delta }\in [1,M_{\mathit{max}}]}{\widehat{L}}^{\ast }(\mathrm{\Delta })$, $L^{\ast }\leftarrow \widehat{L}({\mathrm{\Delta }}^{\ast })$, and $s^{\ast }\leftarrow {\widehat{s}}^{\ast }({\mathrm{\Delta }}^{\ast })$.