Simultaneous pricing and inventory management of deteriorating perishable products

Abstract

This study addresses the problem of combined pricing and inventory control in the context of perishable goods—specifically, when demand is uncertain and price-sensitive and the consumers are free to choose between new and old units, based on the relative affordability of prices. First, the problem is formulated as a periodic review problem for a product with finite arbitrary lifetime and a myopic policy, which is often followed in practice, is discussed. Analytically it is shown that the solution obtained by the policy is also the unique solution that maximizes the individual profits from the inventories at each age. Next, this study addresses a special case—that of products with two-period lifetime, which is a scenario of practical importance. It is proven that under the policy of discounts and sticky pricing, which is often practiced in the brick-and-mortar retail stores, the myopic policy is in fact the steady-state optimal policy of the infinite-horizon problem. The study concludes with a numerical example and directions for future research in the area of joint pricing and inventory management of deteriorating, perishable products.

Proofs of Lemmas and Theorems

Proof of Lemma 1

Let $x = \left\langle p_0^x, p_1^x, \ldots , p_{n-1}^x \right\rangle$ and $y = \left\langle p_0^y, p_1^y, \ldots , p_{n-1}^y \right\rangle$ be two points in ${\fancyscript{S}}$. Therefore,

$$\begin{aligned}&\overline{p}_0 \geqslant p_i^x \geqslant p_{i+1}^x \geqslant 0 \text { and } \overline{p}_0 \geqslant p_i^y \geqslant p_{i+1}^y \geqslant 0 \\&\quad \Rightarrow \overline{p}_0 \geqslant \lambda p_i^x + (1 - \lambda ) p_i^y \geqslant \lambda p_{i+1}^x + (1 - \lambda ) p_{i+1}^y \geqslant 0 \; \forall \lambda \in [0,1]\\&\quad \Rightarrow \lambda x + (1 - \lambda ) y \in {\fancyscript{S}} \end{aligned}$$

Hence, ${\fancyscript{S}}$ is convex. Further, ${\fancyscript{S}}$ is a closed and bounded set. Therefore, ${\fancyscript{S}}$ is compact. Because ${\fancyscript{R}}$ is a closed subset of ${\fancyscript{S}}$, ${\fancyscript{R}}$ is compact. The convexity of ${\fancyscript{R}}$ is proven as follows:

Let $x = \left\langle p_0^x, p_1^x, \ldots , p_{n-1}^x \right\rangle$ and $y = \left\langle p_0^y, p_1^y, \ldots , p_{n-1}^y\right\rangle$ be two points in ${\fancyscript{R}}$. Therefore, for each $i, \xi _i(x) \geqslant 0$ and $\xi _i(y) \geqslant 0$, which implies $\lambda \, \xi _i(x) + (1 - \lambda ) \, \xi _i(y) \geqslant 0$ for all $\lambda \in [0,1]$. Using the definition of $\xi _i(\cdot )$, it can be shown that $\lambda \, \xi _i(x) + (1 - \lambda ) \, \xi _i(y) = \xi _i(\lambda \, x + (1-\lambda ) \, y)$. $\square$

Proof of Lemma 2

The interior optimizer of (3) satisfies the FOCs with respect to the decision variables.

$$\begin{aligned}&\frac{\partial J}{\partial z_0^t} \left( z_0^t, {\mathbf {P}}^t; {\mathbf {X}}^t\right) = \left( p_0^t - c\right) - p_0^t F_0\left( z_0^t\right) = 0\\&\frac{\partial J}{\partial p_0^t} \left( z_0^t, {\mathbf {P}}^t; {\mathbf {X}}^t\right) = z_0^t + D_0\left( {\mathbf {P}}^t) + b_0(p_0^t - c\right) - E\left[ z_0^t - \epsilon _0\right] ^+\\ {}&\quad + \sum _{i=1}^{n-1} c_{i0} p_i^t F_i\left( x_i^t - D_i({\mathbf {P}}^t)\right) = 0 \\&\frac{\partial J}{\partial p_j^t} \left( z_0^t, {\mathbf {P}}^t; {\mathbf {X}}^t\right) = x_j^t - E\left[ x_j^t - \xi _j({\mathbf {P}}^t)\right] ^+\\ {}&\quad + \sum _{i=1}^{n-1} c_{ij} p_i^t F_i\left( x_i^t - D_i({\mathbf {P}}^t) \right) + c_{0j} p_0^t F_0\left( z_0^t\right) = 0 \end{aligned}$$

The FOCs listed above form the necessary conditions for interior optimality. If there is no interior maximizer, then the solution occurs on the boundary of the compact set ${\fancyscript{R}}$. $\square$

Proof of Proposition 3

Consider a game of n players wherein each player tries to maximize its payoff. Let player i denote the manager who manages the inventory of class i to maximize his own profit. In such a game, the payoff of the players are:

$$\begin{aligned}&\Pi _0(p_0) = \Pi _0\left( p_0^t,{z_0^t}^*(p_0)\right) = \left( p_0^t - c\right) \left[ {z_0^t}^*(p_0) + D_0({\mathbf {P}}^t)\right] - p_0^t E\left[ {z_0^t}^*(p_0) - \epsilon _0\right] ^+\\&\Pi _i\left( p_i^t\right) = p_i^t \left[ x_i^t + E[x_i^t - \xi _i({\mathbf {P}}^t)]^+ \right] \;; 1 \leqslant i \leqslant n-1 \end{aligned}$$

The second-order conditions of optimality indicate that for all $1 \leqslant i \leqslant n-1$

$$\begin{aligned} \frac{\partial ^2 \Pi _i}{\partial {p_i^t}^2} = - 2 b_i F_j\left( x_j^t - D_j({\mathbf {P}}^t) \right) - \sum _{j=0}^{n-1} p_j^t c_{ij}^2 f_j\left( x_j^t - D_j({\mathbf {P}}^t) \right) <0 \end{aligned}$$

Petruzzi and Dada (1999) prove the unimodality of $\Pi _0(p_0)$ under the (technical) Assumption V. Hence, by Theorem 1.2 of Fudenberg and Tirole (1991), there exists a unique pure-strategy Nash equilibrium in the convex compact set ${\fancyscript{R}}$. Thus, the solution to the simultaneous Eq. (4) is the unique solution that maximizes the profits from inventory of each class (or age). $\square$

Proof of Proposition 1

According to Assumption II, whenever $p_0 = p_1 = p \geqslant 0$, it is required that $\tilde{a}_1 - b_1 p_1 + c_1 p_0 = 0$. The boundary condition $p = 0$ indicates that $\tilde{a}_1 = 0$. Hence, for non-zero values of p, it is true that $(b_1 - c_1) p = 0 \Rightarrow b_1 = c_1$. Therefore, using Assumption I, it can be established that $b_0 > c_{10} = b_1 > c_{01} > 0$.$\square$

Proof of Theorem 1

For each period, the state of the system is the initial on-hand inventory (x); the set of states is ${\mathcal {S}}= {\mathbb {R}}_{\geqslant 0}$; the action is the quantity of new units to be ordered ($y$); and the action set for each state $x \in {\mathcal {S}}$ is $A_x = {\mathbb {R}}_{\geqslant 0}$. The state-action space is ${\fancyscript{C}}\,\triangleq \,\lbrace (x,y): x \in {\mathcal {S}}, y \in A_s \rbrace$. Let $S(y)\,\triangleq \,\lbrace x : y \in A_x, x \in {\mathcal {S}} \rbrace$ be the set of states where action y is feasible.

The profit function defined in (1) satisfies the nine sufficient conditions listed in Heyman and Sobel (2004) and Porteus (2002), for existence of optimal myopic policy, as shown below. Condition I: Expected single-stage profit depends additively on the state x and action y: For a given price vector $(p_0,p_1)$, $J(y,p_0,p_1) = K(y) + L(x) \, \forall (x,y) \in {\fancyscript{C}}$ where $K(y) = (p_0 - c)y - p_0 E [y - \xi _0]^+$ and $L(x) = p_1 x - p_1 E[x - \xi _1]^+$.

Condition II: The future state depends only on the current action: The on-hand inventory at the start of period $t+1$ depends on the action taken in t as $x^{t+1} = [y^t - \xi _0^t]^+$. Therefore, $\forall \, y \in A, \; \exists \; \omega (y) = [y-\xi _0]^+ : x_{n+1} = [y_n-\xi _0^n]^+ = \omega (y_n)$. Therefore, $x_{n+1} \sim \omega (y_n)$.

Condition III: Let $\gamma (y) \triangleq K(y) + \beta E\lbrace L[\omega (y)] \rbrace ,\, y \in A$. A maximizer $y^*$ of $\gamma$ exists:

$$\begin{aligned} \gamma (y)&= K(y) + \beta E[L[\omega (y)]] \; ; \quad y \in A \\&= (p_0-c) y - (p_0 - \beta p_1) E[y-\xi _0]^+ - \beta p_1 E[[y-\xi _0]^+-\xi _1]^+ \end{aligned}$$

Note that

$$\begin{aligned} \left[ [y-\xi _0]^+-\xi _1\right] ^+= {\left\{ \begin{array}{ll} 0, &{} y \le \xi _0+\xi _1\\ y-\xi _0-\xi _1, &{} y > \xi _0+\xi _1 \end{array}\right. } \end{aligned}$$

is convex in y and hence, so is its expected value. Since, $\gamma (y)$ is concave in y, there exists a unique constrained maximizer, $y^*$ subject to the constraint $y \geqslant 0$. That is $y^* = \hbox {argmax}_{y \geqslant 0} \gamma (y)$.

Condition IV: $y^*$ defined in Condition III is feasible in every period with probability 1: The action set for every state $x$ is $A_x = {\mathbb {R}}_{\geqslant 0}$. So, since $y^* \geqslant 0$, it is feasible for any state x. Therefore, $P \lbrace \omega (y^*) \in S(y^*) \rbrace = 1$.

Condition V: For every $x \in {\mathcal {S}}$ there is $b_x \in A_x$ such that $\gamma (b_x) \geqslant \gamma (y) \,\, \forall \,\, y \in A_x$ : Since $x \in S(y^*) \; \forall \; x \in {\mathcal {S}}$, the constrained maximizer $y^*$ is $b_x$ and $\gamma (b_x) = \gamma (y^*) \geqslant \gamma (y) \, \forall \, y \in A_x$ and $\forall \, x \in {\mathcal {S}}$, without loss of optimality.

Condition VI: $\gamma (\cdot )$ is non increasing on $[y^*,\infty )$: This is true since $\gamma (\cdot )$ is concave.

Condition VII: If y and $y'$ are such that $y^* \leqslant y \leqslant y'$, then $P \lbrace \omega (y) \leqslant \omega (y') \rbrace = 1$: Let $\delta = y' - y \geqslant 0$. By definition $\omega (y) = [y-\xi _0]^+$ and $\omega (y') = [y'-\xi _0]^+ = [y + \delta -\xi _0]^+$. So,

$$\begin{aligned} \omega (y) =\left\{ \begin{array}{ll} 0, &{} y \le \xi _0 \\ y-\xi _0, &{} y > \xi _0 \end{array}\right.&\quad {\mathrm {and}}\quad \omega (y') =\left\{ \begin{array}{ll} 0, &{} y \le \xi _0-\delta \\ y-\xi _0+\alpha , &{} y > \xi _0 - \delta \end{array}\right. \\ \Rightarrow \omega (y')-\omega (y) \ge 0 \end{aligned}$$

Condition VIII: There is a nondecreasing function $M(\cdot )$ such that $M(x) = \min \lbrace y : y \in A_x \rbrace \,\, \forall \, x \not \in S(y^*)$ : Because there does not exist a state x at which the action $y^*$ is not feasible, any definition of $M(\cdot )$ vacuously adheres to the definition mentioned above. Without loss of generality, the function $M(x)$ can be defined as the constant function $M(x) = 0$.

Condition IX: If $x \not \in S(y^*)$ and $y \in A_x$, then $y > y^*$: This condition holds true vacuously.

Since, for a given price vector $(p_0,p_1)$, the sufficient conditions I through IX are satisfied for the profit function, there exists a optimal myopic ordering policy for new units. $\square$

Proof of Lemma 4

Let $X_0$ and $X_1$ be two positive random variables, and let the positive random variable $Z$ be defined as $Z = \left[ [K_0-X_0]^+ - K_1 - X_1\right] ^+$ where $K_0$ and $K_1$ are positive real numbers. The expected value of $Z$ can be obtained as below:

$$\begin{aligned} E[Z]&= \int _0^\infty P(Z>z) dz \\ P(Z>z)&= P \left\{ \left[ [K_0-\epsilon _0]^+-K_1-\epsilon _1\right] ^+ > z \right\} \\&= P\left\{ \left[ [K_0-\epsilon _0]^+-K_1-\epsilon _1 \right] ^+ > z, K_0-\epsilon _0>0\right\} \\&\qquad + P\left\{ \left[ [K_0-\epsilon _0]^+-K_1-\epsilon _1\right] ^+ > z, K_0-\epsilon _0 \le 0\right\} \\&= \int _0^{K_0-K_1-z} P[\epsilon _0 < K_0 - K_1 -z-x_1] f_1(x_1)\,dx_1 \\&\qquad + \int _{K_0-K_1-z}^\infty P[\epsilon _0 < K_0 - K_1 -z-x_1] f_1(x_1)\,dx_1\\&= \int _0^{u-z} F_0(u-z-x_1) f_1(x_1) \,dx_1 \end{aligned}$$

$Z = \left[ [y - D_0 - \epsilon _0]^+ -D_1 - \epsilon _1 \right] ^+$ when $K_0 = y - D_0$ and $K_1 = D_1$, and $P(Z>z) = \int _0^{u-z} F_0(u-z-x_1) f_1(x_1) dx_1$. The, expected value of Z is:

$$\begin{aligned} E[Z]&= \int _0^u \int _0^{u-z} F_0(u-z-x_1) f_1(x_1)\, dx_1 \; dz = \int _0^u F_0(t) F_1(u-t) \end{aligned}$$

$\square$