总结 Summary

15 Jul 2024

库存定价问题总结 Summary for Joint Pricing and Inventory Management

Single Period Model

Objective: the Expected Profit $\pi(x, d)$

Decision Variable: the Stocking Quantity $x$, Sales Price $p=p(d)$

$(x, d)^* =\underset{x, d}{argmax}\enspace\pi(x, d)$

According to newsvendor problem, $x^=d^+F^{-1}(\frac{b-(1-\alpha)c}{h+b+\alpha c})$

$\pi(x,d)$ jointly concave, $d^=d_{min}, d_{max}, or R^\prime (d^)=c$

Multi Period Model

Base Model

Objection: the Value Function $V_t(I_t)$

T−period planning horizon: $ {T, T-1, T-2, …, 1}$

Definition of net inventory level: $I_t$ , $I_t=x_{t+1}-d_{t+1}-\epsilon_{t+1}$

$V_t(I_t)=cI_t+\underset{x_t, d_t}{max}\enspace[\pi(x_t, d_t)+\alpha V_{t-1}(I_{t-1})]=cI_t+\underset{x_t, d_t}{max}\enspace J_t(x_t, d_t)$

Boundary: $V_0(I_0)\equiv0$

Model with Fixed Ordering Cost

the cost of ordering the inventory from $I_t$ up-to $x_t$ $(I_t \leq x_t)$

Base Model $c(x_t-I_t)$

Model with Fixed Ordering Cost $k\delta (x_t-I_t)+c(x_t-I_t)$

Model with Positive Leadtime

the time duration $L$ between placing an order and receiving

Base Model $L\equiv0$

Model with Positive Leadtime $L>0$

the order quantity in period $t$ : $q_t$

the state of the system : $\mathbf {I_t}=(I_{0,t},I_{1,t},…,I_{L-1,t})$.

$I_{0,t} $ is the inventory level at the beginning of period $t$ after the order due in $t$ arrives and $I_{i,t}=q_{t-L+i}, i=1,2,…,L-1$

($I_{0,t}$ is similar to $x_t$ in base model, and the left are orders due after $t$)

the starting state of next period is $\mathbf {I_t^+}=(I_{0,t}-d_t-\epsilon_t,I_{1,t},…,I_{L-1,t})$ .

($I_{0,t}-d_t-\epsilon_t$ is similar to $I_t$ in base model)

Objection: the Value Function ${\hat{V}}^{l}_{t}(I_t)$

${\hat{V}}^l_t(I_t)=\underset{x_t, d_t}{max}\enspace{\hat{J}}^l_t(\mathbf{I_t},q_t, d_t)$

Model with Supply Uncertainty

the stochastically proportional yield model

the firm can receive $q_t\zeta_t$ units of inventory if he orders $q_t$ units, ${\zeta}^T_{t=1}$ are $i.i.d.$ nonnegative random variables supported on $[0,1]$ independent of $\epsilon_t$’s.

Base Model $\zeta_t\equiv 1$

Model with Supply Uncertainty $\zeta_t \in [0,1]$

Question

1. Page 4, Line 5, the range of $\alpha$ (Single Period Model)

This paper assumes $\alpha \in [0, 1]$, $\alpha c \leq c$.

It might exist the other situation that the incurring per-unit cost $c$ increases along with time.

Editing Problem

1. Page 4, Line 4, missing a comma

$x$ represents stocking quantities, according to Page 3, last Line.

It also shoud be “a per-unit holding cost h is charged, whereas if ……”

2. Page 4, Line 10, using the same symbol to define different quantities

$d(p, \epsilon) = d(p) + \epsilon$, $d(p,\epsilon)$ represents the stochastic price dependent demand, $\epsilon$ is a random perturbation, $d(p)$ represents the mean demand, $p(d)$ is the inverse function of $d(p)$, represents the price when the mean demand is $d$.

It could be better if the mean demand and the real demand could use dfferent symbol.

3. Page 5, Line 2, missing the definition of $F$

The definition of $F$ seems to be the cumulative distribution function of perturbation $\epsilon$.

4. Page 10, Line 5, spelling mistake

“$L^♮$-convcavity” should be $L^♮$-concavity.