Optimizing constant pricing and inventory decisions for a periodic review system with batch ordering

Abstract

In business practice the misalignment between frequent inventory adjustments and a sticky price is common. Motivated by Pipi Milk, a diary manufacturer who commits its customer at a fixed price for a whole year meanwhile weekly replenishes its cheese product stock at a specific batch size, this paper develops an integrated constant pricing and inventory control model in a periodic-review system with batch order consideration. For such a system, our focus is on an (nQ, r)-p policy: the price p is determined and committed at the beginning of the planning horizon; inventories are managed based on an (nQ, r) policy with reorder level r and order quantities at non-negative integer multiples of batch size Q. With a fixed ordering cost included, we develop an efficient algorithm to compute the optimal policy parameters with the purpose to maximize the expected long-run average profit. We also numerically investigate the policy performance and verify its advantages by comparing to a sequential optimization of price and inventory decisions. Finally, managerial insights are generated.

Appendix

1.1 Proof of Proposition 1

Proof

Given an (nQ, r)-p policy, if we can show $\sum _{i=1}^{Q}a_{ij}(p)=1$ for any $j\in \{1, 2,\ldots ,Q\}$, a uniform distribution, i.e., $\theta (y)=$ constant, will then satisfy (2). Combining with $\sum _{y=r+1}^{r+Q}\theta _t(y)=1$, the steady state distribution is $\theta (y)=\frac{1}{Q}$, $y \in \{r+1, \ldots ,r+Q\}$.

The following indicates $\sum _{i=1}^{Q}a_{ij}(p)=1$ for any $j\in \{1, 2,\ldots ,Q\}$. The one-step transition probability matrix is seen to be

$$\begin{aligned} a_{ij(p)} =\left\{ \begin{array} {r l} \sum \nolimits _{z=0}^{\infty }\phi _{z\cdot Q+i-j}(p), &{}\quad \text{ when } i \ge j; \\ \sum \nolimits _{z=1}^{\infty }\phi _{z\cdot Q+i-j}(p), &{}\quad \text{ when } i < j. \end{array} \right. \end{aligned}$$

The matrix $[a_{ij}(p)]$ is then doubly stochastic, since for any $j\in I$,

$$\begin{aligned} \sum _{i=1}^{Q}a_{ij}(p)= & {} \sum _{i=1}^{j}\sum _{z=1}^{\infty }\phi _{z\cdot Q+i-j}(p) + \sum _{i=j+1}^{Q}\sum _{z=0}^{\infty }\phi _{z\cdot Q+i-j}(p) \nonumber \\= & {} [\phi _{1}(p)+\phi _{2}(p)+\cdots +\phi _{Q-j}(p)] \nonumber \\&+\,[\phi _{Q-j+1}(p_{1})+\cdots +\phi _{Q}(p)] \nonumber \\&+\, [\phi _{Q+1}(p)+\phi _{Q+2}(p_{j+2})+\cdots +\phi _{Q+Q-j}(p)]\nonumber \\&+\,[\phi _{2Q-j+1}(p_{1})+\cdots +\phi _{2Q}(p)] +\cdots \nonumber \\= & {} 1 \end{aligned}$$

(5)

This, in align with our previous argument, completes the proof. $\square$

1.2 Proof of Proposition 2

Proof

For a given dummy profit $\beta <\pi ^*(nQ,r,p)$ and a given price $p^i$, we first fix the maximal order-up-to level $S_{\beta }^i$ and compare the policy $(r_{\beta }^i, S_{\beta }^i, p^i)$ to policy $(r_{\beta }^i+1, S_{\beta }^i, p^i)$.

$$\begin{aligned}&l_{\beta } (r_{\beta }^i,S_{\beta }^i, p^i) - l_{\beta } ( r_{\beta }^i+1, S_{\beta }^i,p^i) \nonumber \\&\quad = \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta Q_{\beta } - \left\{ \sum _{y=r_{\beta }^i+2}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) \right. \nonumber \\&\qquad \left. -K \bar{\varPhi }_{y-r_{\beta }^i-1}(p^i)]-\beta (Q_{\beta }-1)\right\} \nonumber \\&\quad = B(r_{\beta }^i+1,p^i) -K \bar{\varPhi }_{Q_{\beta }}(p^i)]- \beta \ge 0. \end{aligned}$$

(6)

The last inequality comes from the definition of the optimal parameters of $r_{\beta }^i$ and $S_{\beta }^i$. We then compare policy $(r_{\beta }^i, S_{\beta }^i, p^i)$ to policy $(r_{\beta }^i-1, S_{\beta }^i, p^i)$.

$$\begin{aligned}&l_{\beta }(r_{\beta }^i,S_{\beta }^i, p^i) - l_{\beta } ( r_{\beta }^i-1, S_{\beta }^i,p^i) \nonumber \\&\quad = \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta Q_{\beta } - \left\{ \sum _{y=r_{\beta }^i}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) \right. \nonumber \\&\qquad \left. -K \bar{\varPhi }_{y-r_{\beta }^i+1}(p^i)]-\beta (Q_{\beta }+1)\right\} \nonumber \\&\quad = -B(r_{\beta }^i,p^i) +K \bar{\varPhi }_{Q_{\beta }+1}(p^i)]+ \beta \ge 0. \end{aligned}$$

(7)

Next we fix the reorder point $r_{\beta }^i$ and compare the policy $(r_{\beta }^i, S_{\beta }^i, p^i)$ to policy $(r_{\beta }^i, S_{\beta }^i+1, p^i)$.

$$\begin{aligned}&l_{\beta } (r_{\beta }^i,S_{\beta }^i, p^i) - l_{\beta } (r_{\beta }^i,S_{\beta }^i+1, p^i) \nonumber \\&\quad = \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta Q_{\beta } - \left\{ \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }+1}[B(y,p^i) \right. \nonumber \\&\qquad \left. -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta (Q_{\beta }+1)\right\} \nonumber \\&\quad = -B(r_{\beta }^i+Q_{\beta }+1,p^i) +K \bar{\varPhi }_{Q_{\beta }+1}(p^i)]+ \beta \ge 0. \end{aligned}$$

(8)

We then compare policy $(r_{\beta }^i, S_{\beta }^i, p^i)$ to policy $(r_{\beta }^i, S_{\beta }^i-1, p^i)$.

$$\begin{aligned}&l_{\beta } (r_{\beta }^i,S_{\beta }^i, p^i) - l_{\beta } (r_{\beta }^i,S_{\beta }^i-1, p^i) \nonumber \\&\quad = \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }}[B(y,p^i) -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta Q_{\beta } - \left\{ \sum _{y=r_{\beta }^i+1}^{r_{\beta }^i+Q_{\beta }-1}[B(y,p^i) \right. \nonumber \\&\qquad \left. -K \bar{\varPhi }_{y-r_{\beta }^i}(p^i)]-\beta (Q_{\beta }-1)\right\} \nonumber \\&\quad = B(r_{\beta }^i+Q_{\beta },p^i) -K \bar{\varPhi }_{Q_{\beta }}(p^i)]- \beta \ge 0. \end{aligned}$$

(9)

Replacing $Q_{\beta }$ with $S_{\beta }^i -r_{\beta }^i$ and substituting it back into the above inequalities, we then complete the proof. $\square$

1.3 Proof of Corollary 1

Proof

Considering $B(y,p^i)$ is unimodal in y given $p^i$ and $K \bar{\varPhi }_{S_{\beta }^i-r_{\beta }^i}(p^i) > 0$, the interval between the crossing points shrinks by increasing $\beta$ to $\beta +K \bar{\varPhi }_{S_{\beta }^i-r_{\beta }^i}(p^i)$. Thus, $\hat{Q}^i_{\beta } \ge S_{\beta }^i-r_{\beta }^i$. We also have $\bar{\varPhi }_{S_{\beta }^i-r_{\beta }^i}(p^i) \ge \bar{\varPhi }_{\hat{Q}_{\beta }}(p^i)$.

By definition of $r_{\beta }^i$, $B(r_{\beta }^i+1,p^i) \ge \beta +K \bar{\varPhi }_{S_{\beta }^i-r_{\beta }^i}(p^i) \ge \beta +K \bar{\varPhi }_{\hat{Q}_{\beta }}(p^i)$. Considering $B(y,p^i)$ is increasing for $y<y^i_0$ where $y^i_0$ is the maximal y such that $B(y,p^i)$ archives maximum, and $\underline{r}_{\beta }^i+1$ is the minimal y such that $B(y,p^i) \ge \beta +K \bar{\varPhi }_{\hat{Q}_{\beta }}(p^i)$, we thus have $r_{\beta }^i \ge \underline{r}_{\beta }^i$.

Similarly, By definiton of $S_{\beta }^i$, $B(S_{\beta }^i,p^i) \ge \beta +K \bar{\varPhi }_{S_{\beta }^i-r_{\beta }^i}(p^i) \ge \beta +K \bar{\varPhi }_{\hat{Q}_{\beta }}(p^i)$. Considering $B(y,p^i)$ is decreasing for $y\ge y^i_0$ and $\bar{S}_{\beta }$ is the maximal y such that $B(y,p^i) \ge \beta +K \bar{\varPhi }_{\hat{Q}_{\beta }}(p^i)$, we thus have $S_{\beta }^i \le \bar{S}_{\beta }^i$. $\square$

1.4 Proof of Lemma 3

Proof

By definition, both $\underline{r}_{\beta _0}^i \le y_0$ and $\underline{r}_{\beta }^i \le y_0$. Considering the increasing monotonicity of $B(y,p^i)$ for $y\le y_0^i$, $\underline{r}_{\beta }^i \ge \underline{r}_{\beta _0}^i$ is straightforward since $\beta \ge \beta _0$. Similarly, $\bar{S}_{\beta }^i \le \bar{S}_{\beta _0}^i$ due to the monotonicity of $B(y,p^i)$, in addition, $y\ge y_0^i$ and $\beta \ge \beta _0$. $\square$

1.5 Proof of Lemma 4

Proof

$l_{\beta _1}(r,S,p^i) = \sum _{y=r+1}^{S}[B(y,p^i)-K\bar{\varPhi }_{y-r}(p^i)] - \beta Q +\beta Q-\beta _1Q = l_{\beta }(r,S,p^i) -(\beta _1-\beta )Q <0$. Thus, proved. $\square$