Optimal Policies and Bounds for Stochastic Inventory Systems with Lost Sales

Abstract

This paper studies the classical discrete-time, single-location inventory model with stochastic demand, lost sales, and positive lead time. We transform the problem into an equivalent problem of the Markovian demand inventory model with zero lead time and zero initial inventory, which provides a better understanding to the original problem. Based on this transformation, we introduce a key concept—effectual demand, which determines both the performance in the current period and the evolution into future periods. In this way, we provide a bridge of two research streams in the literature: Markovian demand inventory model and lost sales inventory model. We believe this Markovian approach is more straightforward to work with various applications. In this way, we easily show the existence of optimal policies in discounted and average cost, and finite and infinite horizon cases, when cost functions are of polynomial growth. The polynomial growth cost functions virtually cover all practical scenarios in real business settings. We then present simpler proofs and examples for the linear order cost case. The analytical solutions are first. We also derive bounds analytically on the optimal policy; these bounds are equivalent to those of the myopic policy but tighter than the popular bound in the literature.

Appendix: Proofs

1.1 Proof of Lemma 5.1a

• (1) Obviously $\alpha ^2cu_1$ is convex in $u_1$.

• (2) $\alpha ^2E\left[ {g_1 \left( {{x_1 ,u_1 }}\right) } \right]$ is convex in $u_1$because $\alpha ^2E\left[ {g_1 \left( {{x_1 ,u_1 }}\right) } \right]$ is the expected cost of the newsvendor problem facing effectual demand, $D_1 ( {x_1 ,u_1 })=u_1 -z_2 =-( {{x_1} -{d_1}}){^+}+d_2$.

1.2 Proof of Lemma 5.1b.

• (1) We try to show the optimal expected single-period cost

$$\begin{aligned} \alpha ^2E\left[ {g_1 ( {x_1 ,u_1 })} \right] =\alpha ^2E\left[ {g_1 ( {z_2 })} \right] =\alpha ^2E( {h[z_2 ]{^+}+b[z_2 ]^-})=\alpha ^2E\\ \left( {{h\left[ {[z_1 ]{^+}+u_1 -d_2 } \right] {^+}+b\left[ {[z_1 ]{^+}+u_1 -d_2 } \right] ^-}}\right) \end{aligned}$$

is convex and nondecreasing in $\left[ {z_1 } \right] ^+$.

$\alpha ^2E\left[ {g_1 ( {x_1 ,u_1 })} \right] \hbox {is convex in} [z_1 ]^+\hbox { because}$

$$\begin{aligned} \begin{array}{lll} &{}&{}\alpha ^2E\left[ {g_1 ( {x_1 ,u_1 })} \right] =\alpha ^2E\left[ {g_1 ( {z_2 })} \right] =\alpha ^2E\left[ {( {h[z_2 ]{^+}+b[z_2 ]^-})} \right] \\ &{}&{}=\alpha ^2E({h\left[ {[z_1 ]{^+}+u_1 -d_2 } \right] {^+}+b\left[ {[z_1 ]^++u_1 -d_2 } \right] ^-}) \\ &{}&{}=\alpha ^2({h\int \limits _{-\infty }^{[z_1 ]{^+}+u_1 } {F_{d_2 } (y)\mathrm{d}y} +b\int \limits _{[z_1 ]^++u}^\infty {F_{d_2^{\prime }} (y)dy} })\\ &{}&{}=\alpha ^2({(h+b)\int \limits _{-\infty }^{[z_1 ]{^+}+u} {F_{d_2 } (y)\mathrm{d}y}-b([z_1 ]{^+}+u_1 -\mu )}) \\ &{}&{}=\alpha ^2( {(h+b)\int \limits _0^{[z_1 ]{^+}+u} {F_{d_2 } (y)dy} -bu_1 +b(\mu -[z_1 ]^+)}) \\ \end{array} \end{aligned}$$

is the single-period cost of newsvendor problem with $[z_1 ]{^+}+u_1$.

To show nondecreasing in $\left[ {z_1 } \right] ^+$, for any given $\left[ {z_1 } \right] {^{+{\prime }}}$ let $\left[ {z_1 } \right] {^{+{\prime \prime }}}=\left[ {z_1 } \right] {^{+{\prime }}}-\delta$, where $\delta >0$ is very small; we compare optimal cost between $\left[ {z_1 } \right] {^{+{\prime }}}$ and $\left[ {z_1 } \right] {^{+{\prime \prime }}}$. Let the optimal order size be $u_1^{{\prime }*}$ for

$\left[ {z_1 } \right] {^{+{\prime }}}$. The optimal expected single-period cost for a given $\left[ {z_1 } \right] {^{+{\prime }}}$ thus is

$$\begin{aligned} \begin{array}{l} \alpha ^2E\left[ {{g_1^*} ( {[z_1 ]{^+{\prime }}})} \right] =\alpha ^2E\left[ {cu_1^{{\prime }*} +( {h[z_2 ]{^+}+b[z_2 ]^-})} \right] =\alpha ^2cu_1^{{\prime }*} +\alpha ^2E\\ ( {h\left[ {[z_1 ]{^+{\prime }}+u_1^{{\prime }*} -d_2} \right] {^+}+b\left[ {[z_1 ]{^+{\prime }}+u_1^{^{\prime }*} -d_2 } \right] ^-}) \\ =\alpha ^2( {(h+b)\int \limits _0^{u_1^{{\prime }*} +[z_1 ]{^+{\prime }}} {F_{d_2} (y)\mathrm{d}y} -(b-c)u_1^{{\prime }*} +b(\mu -[z_1 ]{^{+{\prime }}})}), \\ \end{array} \end{aligned}$$

(9)

where $u_1^{\prime *} =\left[ {F_{^{d_2 -[z_1 ]{^+{\prime }}}}^{-1} ( \theta )} \right] {^+}$that means $(h+b)F_{d_2 -[z_1 ]{^{+\prime }}} (u_1^{{\prime }*} )-b=(h+b)F_{d_2 } (u_1^{\prime *} +[z_1 ]{^{+\prime }})-b\ge 0$.

For a $\left[ {z_1}\right] ^+{\prime \prime }=\left[ {z_1}\right] ^+{\prime }-\delta$, the single-period cost with same order size $u_1^{\prime *}$ is

$$\begin{aligned} \begin{array}{l} \alpha ^2E\left[ {g_1 ( {[z_1 ]^{\prime \prime },u_1^{\prime *} })} \right] =\alpha ^2cu_1^{\prime *} +\alpha ^2E( {h[z_2 ]{^+}+b[z_2 ]^-})\\ =\alpha ^2cu_1^{\prime *} +\alpha ^2E( {h\left[ {[z_1 ]{^{\prime \prime }}+u_1^{\prime *} -d_2 } \right] {^+}+b\left[ {[z_1 ]^{+{\prime \prime }}+u_1^{\prime *} -d_2 } \right] ^-}) \\ =\alpha ^2( {(h+b)\int \limits _0^{u_1^{{\prime }*} +[z_1 ]^+\prime \prime } {F_{d_2 } (y)\mathrm{d}y} -(b-c)u_1^{\prime *} +b(\mu -[z_1 ]{^{+\prime \prime }})}) \\ \end{array} \end{aligned}$$

(10)

Applying (9) and (10), we have

$$\begin{aligned} \alpha ^2E\left[ {cu_1^{\prime *} +g_1 ( {[z_1 ]^{\prime },u_1^{\prime *}})} \right] -\alpha ^2E\left[ {cu_1^{\prime *} +g_1 ( {[z_1 ]^{\prime \prime },u_1^{\prime *}})} \right] \\ =\alpha ^2( {(h+b)F_{d_2 } (u*+[z_1 ]^+{\prime })\delta -b\delta })\ge 0 \end{aligned}$$

We thus show $\alpha ^2E\left[ {cu_1^{\prime *} +g_{_1 }^*( {[z_1 ]^{\prime }})} \right] \ge \alpha ^2E\left[ {cu_1^{\prime *} +g{_{1}}{^*}} \left( [z_1]^{\prime \prime }\right) \right]$. In other words, the optimal expected single-period cost is nondecreasing in $\left[ {z_1 } \right] ^+$.

(2) $\left[ {z_1 } \right] ^+=\left[ {x_1 -d_1 } \right] ^+$is convex and nondecreasing in $x_1$ because $\left[ {z_1 } \right] ^+=\left[ {x_1 -d_1 } \right] ^+=\int \limits _0^{x_1 } {F_d (y)\mathrm{d}y}$.

Combining (1) and (2) together, we prove the result.

1.3 Proof of Lemma 5.2.

The proof is done by induction.

• (1) From Lemma 5.1, we know above results hold for the single-period cost, $t=T$.

• (2) We assume that these results hold for $t+1$. We need to show that they are true for $t$.

• 6a. (i) The one-period cost $\alpha ^2cu_t +\alpha ^2E\left[ {g_t (x_t ,u_t ;u_t )} \right]$ is convex in $u_t$. (ii) $v_{\alpha ,t+1} ( {x_{t+1} ,\text{ u }_{t+1} })$ is nondecreasing and convex in $x_{t+1}$ and$x_{t+1} =( {x_t -d_t }){^+}+u_t$ is convex in $u_t$, so $\alpha E\left[ {v_{\alpha ,t+1} ( {x_{t+1} ,u_{t+1} })} \right]$is convex in $u_t$. Combining (i) and (ii), we have $v_{\alpha ,t} (x_t ,u_t )=\mathop {\inf }\limits _{u_t \ge 0}$ $\left\{ {\alpha ^2cu_t +\alpha ^2E\left[ {g_t (x_t ,u_t ;u_t )} \right] +\alpha E\left[ {v_{\alpha ,t+1} ( {x_{t+1} ,u_{t+1}})} \right] } \right\} ,$ is convex in $u_t$.

• 6b. (i) The one-period cost $\alpha ^2cu_t +\alpha ^2E\left[ {g_t\left( {x_t ,u_t ;u_t}\right) }\right]$ is convex in $x_t$. (ii) To show nondecreasing, the proof is in a same manner as for the single-period case. For any given $x_t$, the optimal order size ${u_t}{^*}$ is the smallest number such that the decrease rate of the one-period cost, $\alpha ^2cu_t +\alpha ^2E\left[ {g_t (x_t ,u_t ;u_t )} \right]$, is not smaller than the increase rate of the optimal expected cost for $t+1$, $\alpha v_{\alpha ,t+1} ( {x_{t+1} ,u_{t+1} })$ because $\alpha ^2cu_t +\alpha ^2E\left[ {g_t (x_t ,{{{\varvec{u}}}_t} ;u_t)} \right]$ is convex in $u_t$ and $\alpha v_{\alpha ,t+1} ( {x_{t+1} ,u_{t+1} })$ is always nondecreasing in $x_{t+1}$. Following the proof of Lemma 5.1b, we thus show the optimal expected cost of $x_t^{\prime }$ and $u_t^{\prime *}$ is not smaller than that of $x_t^{\prime \prime }$ and$u_t^{\prime *}$, where again $x_t^{\prime \prime } =x_t^{\prime } -\delta$ and $\delta >0$ are very small. $\square$