Approximation algorithms for the joint replenishment problem with deadlines

Abstract

The Joint Replenishment Problem (${\hbox {JRP}}$) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers’ waiting costs. We study the approximability of ${\hbox {JRP-D}}$, the version of ${\hbox {JRP}}$ with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of $1.207$, a stronger, computer-assisted lower bound of $1.245$, as well as an upper bound and approximation ratio of $1.574$. The best previous upper bound and approximation ratio was $1.667$; no lower bound was previously published. For the special case when all demand periods are of equal length, we give an upper bound of $1.5$, a lower bound of $1.2$, and show APX-hardness.

Appendix: Wald’s lemma

Here is the variant of Wald’s Lemma (also known as Wald’s identity, and a consequence of standard “optional stopping” theorems) that we use in Sect. 2. The proof is standard; we present it for completeness.

Lemma 10

(Wald’s Lemma) Consider a random experiment that, starting from a fixed start state $S_0$, produces a random sequence of states $S_1,S_2,S_3,\ldots$ Let random index $T\in \{0,1,2,\ldots \}$ be a stopping time for the sequence (that is, for each positive integer $t$, the event “$T \le t$” is determined by state $S_t$). Let function $\phi :\{S_t\} \rightarrow {\mathbb R}$ map the states to ${\mathbb R}$. Suppose that, for some fixed constants $\xi$ and $F$,

• (i) $(\forall t<T)~ \mathbf{{E}}[\phi (S_{t+1})~|~ S_t] \ge \phi (S_t)+\xi$,

• (ii) either $(\forall t<T)~ \phi (S_{t+1})-\phi (S_t) \ge F$ in all outcomes, or $(\forall t<T)~ \phi (S_{t+1})-\phi (S_t) \le F$ in all outcomes, and

• (iii) $T$ has finite expectation.

Then, $\mathbf{{E}}[\phi (S_T)]\, \ge \,\phi (S_0) + \xi \, \mathbf{{E}}[T]$.

Proof

For each $t\ge 0$, define random variable $\delta _t = \phi (S_{t+1})-\phi (S_{t})$. By assumption (i), $\mathbf{{E}}[\delta _{t}~|~S_t] \ge \xi$ for $t<T$. Since the event “$T > t$” is determined by $S_t$, this implies that $\mathbf{{E}}[\delta _{t}~|~T>t] \ge \xi$. Then the inequality in the lemma can be derived as follows:

$$\begin{aligned}&\textstyle \mathbf{{E}}[\phi (S_T) - \phi (S_0)] ~=~ \mathbf{{E}}[ \sum _{t<T} \delta _t ]\\&\quad =~ \textstyle \sum _{\tau \ge 0} \mathrm Pr [T=\tau ]\cdot \mathbf{{E}}[\sum _{t<\tau } \delta _t ~|~ T=\tau ] \\&\quad =~ \textstyle \sum _{\tau \ge 0} \sum _{t<\tau } \mathrm Pr [T=\tau ]\cdot \mathbf{{E}}[ \delta _t ~|~ T=\tau ] \\&\quad =~ \textstyle \sum _{t \ge 0} \sum _{\tau >t} \mathrm Pr [T=\tau ]\cdot \mathbf{{E}}[\delta _t ~|~ T=\tau ] \\&\quad =~ \textstyle \sum _{t\ge 0} \mathrm Pr [T>t]\cdot \mathbf{{E}}[ \delta _t ~|~ T> t] \\&\quad \ge ~ \textstyle \sum _{t\ge 0} \mathrm Pr [T>t]\cdot \xi \\&\quad =~ \xi \, \mathbf{{E}}[T], \end{aligned}$$

Exchanging the order of summation in the third step above does not change the value of the sum, because (by assumptions (ii) and (iii)) either the sum of all negative terms is at least $\sum _{\tau \ge 0} \sum _{t<\tau } \Pr [T=\tau ] F ~=~ F \sum _{\tau \ge 0} \tau \Pr [T=\tau ] ~=~ F\,\mathbf{{E}}[T]$, which is finite, or (likewise) the sum of all positive terms is finite.

Each application in Sect. 2 has $\xi \ge \mathcal{Z}(p)> 0$ and $\phi (S_T) - \phi (S_0) \le U$ for some fixed $U$. In this case Wald’s Lemma implies $\mathbf{{E}}[T] \le U/\xi \le U/\mathcal{Z}(p)$.