Inventory Control and Pricing with Alternative Delivery Times

Abstract

In this chapter, we consider joint inventory and pricing decisions. The seller can adjust the price to control the arrival rates directly. We derive the optimal joint policies and then compare the two mechanisms using a hybrid model in which the seller must choose between the two control means in each period. Our results indicate that the inventory decision is meaningful in the high-inventory case, and the pricing decision is advantageous when the inventory level is low. We further demonstrate the equivalence of the two models and show that the optimal policy in the hybrid model is a switching policy characterized by a switching threshold.

Appendix

5.1.1 Proof of Lemma 5.1

Proof

We prove this result by induction. It is obvious that the terminal function V(T, n, m) satisfies both properties. Assuming that the two properties hold for $V(t+1,n,m)$, we prove that they also hold for V(t, n, m).

Rewrite the dynamic programming equation as follows:

$$\begin{aligned} V(t,n,m_n)= & {} \max \limits _{\pi _s\in [\underline{\pi _s},\overline{\pi _s}]}\{\pi _sp_s+\pi _s \max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1) \\&+\pi _f p_f+(1-\pi _0-\pi _s) \max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1)\} \\&+\pi _0 \max _{u_0\in \mathscr {U}(n,m_n)}V(t+1,n-u_0,m_n) - hn \\= & {} \max \limits _{\pi _s\in [\underline{\pi _s},\overline{\pi _s}]}\{\pi _sp_s+\pi _f p_f \\&+\pi _s [\max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1) \\&-\max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1)]\} \\&+(1-\pi _0) \max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1) \\&+\pi _0 \max _{u_0\in \mathscr {U}(n,m_n)}V(t+1,n-u_0,m_n) - hn. \end{aligned}$$

The optimal $\pi _s$ depends on

$$\begin{aligned} \max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1) -\max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1), \end{aligned}$$

and the one-period revenue function. It equals

$$\begin{aligned} \arg \limits _{\pi _s}&\{\frac{\partial (\pi _sp_s+\pi _f p_f)}{\partial \pi _s}+ \max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1) \\&-\max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1) = 0 \}, \end{aligned}$$

if this value falls within $[\underline{\pi _s},\overline{\pi _s}]$; otherwise it is set to $\underline{\pi _s}$ or $\overline{\pi _s}$. Given the integer concavity of $V(t+1,n,m_n)$ in n, if $\pi _sp_s+\pi _f p_f$ is strictly concave in $\pi _s$, then for given values of n(t) and $m_n(t)$, the optimal $\pi _s$ is uniquely determined. Based on the integer concavity of $V(t+1,n,m_n)$ in n, we have the following result.

Lemma 5.4

For a given $m_n$, $\max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1) -\max _{u_f\in \mathscr {U}(n,m_n-1)} V(t+1,n-u_f, m_n-1)$ is non-decreasing in n.

We discuss two possible subcases as follows.

Subcase 1: given the concavity of $V(t+1,n,m_n)$, when n(t) is larger than the optimal on-hand inventory level for $m_n(t)-1$, $\pi _s$ is set to the value of $\arg \max \limits _{\pi _s}(\pi _sp_s+\pi _f p_f)$ to maximize the one-period revenue.

Subcase 2: if n(t) equals or is smaller than the optimal on-hand inventory level for $m(t)-1$, we know the following:

$$\begin{aligned} \max _{u_s\in \mathscr {U}(n-1,m_n-1)}V(t+1,n-1-u_s,m_n-1)= & {} V(t+1,n-1,m_n-1), \\ \max _{u_f\in \mathscr {U}(n,m_n-1)}V(t+1,n-u_f,m_n-1)= & {} V(t+1,n,m_n-1). \end{aligned}$$

Let $\widetilde{V}(t+1,x,m_n)$ represent the continuous counterpart of $V(t+1,n,m_n)$ by linear interpolation. For $\forall x\in \mathbb {R}^+$,

$$\begin{aligned} \widetilde{V}(t+1,x,m_n) = (\lceil x\rceil -x)V(t,\lfloor x\rfloor ,m_n) + (x-\lfloor x\rfloor )V(t,\lceil x\rceil ,m_n), \end{aligned}$$

where $\lceil x\rceil$ is the smallest integer that is larger than x and $\lfloor x\rfloor$ is the largest integer that is smaller than x. $\lceil x\rceil =\lfloor x\rfloor +1$. Given the integer concavity of $V(t+1,n,m_n)$ in n, we can obtain the following two results.

Lemma 5.5

$\widetilde{V}(t+1,x,m_n)$ is concave in x.

Lemma 5.6

There exists a value of $x\in [0,1]$, such that $(1-\pi _0)\widetilde{V}(t+1,n-1+x,m_n)=\pi _s V(t+1,n-1,m_n-1)+(1-\pi _0-\pi _s)V(t+1,n,m_n-1)$. Furthermore, $x=1-\pi _s$.

Let

$$ \phi (\pi _s)= \left\{ \begin{array}{ll} \pi _sp_s+\pi _fp_f, &{} \text { if }\pi _s\in [\underline{\pi _s},\overline{\pi _s}]; \\ 0, &{} \text {otherwise.} \\ \end{array} \right. $$

We can rewrite the value function as follows:

$$\begin{aligned} V(t,n,m_n)= & {} \max \limits _{x\in [\underline{\pi _s},\overline{\pi _s}]} [\phi (x) + (1-\pi _0)\widetilde{V}(t+1,n-x,m_n)] \\&+\pi _0 \max _{u_0\in \mathscr {U}(n,m_n)}V(t+1,n-u_0,m_n) - hn. \end{aligned}$$

We now prove that $\max \limits _{x\in [\underline{\pi _s},\overline{\pi _s}]} [\phi (x) + (1-\pi _0)\widetilde{V}(t+1,n-x,m_n)]$ preserves the integer concavity in n. Denote the optimal x for $V(t,n-1,m_n),V(t,n,m_n)$, and $V(t,n+1,m_n)$ as $x^*,{x^*}'$, and ${x^*}''$, respectively.

$$\begin{aligned}&\{[\phi (x^*) + (1-\pi _0)\widetilde{V}(t+1,n-1-x^*,m_n)]+[\phi ({x^*}'') \\&+(1-\pi _0)\widetilde{V}(t+1,n+1-{x^*}'',m_n)]\}/2 \\\le & {} \phi (\frac{x^*+{x^*}''}{2}) + (1-\pi _0)\widetilde{V}(t+1,n-\frac{x^*+{x^*}''}{2},m_n)] \\\le & {} \phi ({x^*}') + (1-\pi _0)\widetilde{V}(t+1,n-{x^*}',m_n). \end{aligned}$$

It is easy to see that the boundary condition at $n(t)=0$ and the conjunction at the optimum also satisfy the integer concavity. Hence, the integer concavity in n(t) holds for $V(t,n,m_n)$.

We skip the proof of modularity as it is similar to the proof of modularity in the non-capacitated upgrade model.

5.1.2 Proof of Corollary 5.1

Proof

Given the integer concavity of $V(t,n,m_n)$ in n(t), the first order condition can be written as follows:

$$ \left\{ \begin{array}{l} \displaystyle \frac{\partial (\pi _fp_f+\pi _sp_s)}{\partial \pi _s}, \\ \text { when }n(t) \text { is larger than the optimum for }m_n(t)-1; \\ \begin{aligned} \frac{\partial (\pi _fp_f+\pi _sp_s)}{\partial \pi _s}+V(t+1,n,m_n-1)\\ -V(t+1,n-1,m_n-1), \end{aligned} \\ \text { otherwise.} \\ \end{array} \right. $$

Hence, $\pi _s^*$ is non-decreasing in n(t). Correspondingly, $p_s^*$ is non-increasing in n(t).

5.1.3 Proof of Lemma 5.2

Proof

We use induction to prove this result. First, it is easy to see that the terminal function satisfies the integer concavity. Assume that the concavity holds for $V(t+1,n,m_n)$. We prove that it holds for $V(t,n,m_n)$ too.

From previous analysis, we already see that the pricing and upgrade controls depend on n(t), regardless of $m_n(t)$. The integer concavity thus can be carried over.

5.1.4 Proof of Theorem 5.4

Proof

First, it is obvious that the joint model dominates the hybrid model, as every strategy in the hybrid model can be replicated by the joint model. We now show that the optimal policy in the joint model can be achieved by the hybrid model. According to Theorem 5.1, when the inventory control , i.e., upgrade, is used, the short lead-time customer arrival rate is set as $\pi _s^*$. When there is no upgrade, the optimal rate is set according to the inventory level. The former case can be considered as pure inventory control , and the latter case corresponds to pure dynamic pricing . Hence, the optimal policy of the joint model is achievable by the hybrid model.