The Replenishment Schedule to Minimize Peak Storage Problem: The Gap Between the Continuous and Discrete Versions of the Problem

Abstract

The replenishment storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system, where each item has a given reorder size and cycle length. We consider the discrete RSP, where reorders can only take place at an integer time unit within the cycle. Discrete RSP was shown to be NPhard for constant joint cycle length (the least common multiple of the length of all individual cycles). We show here that discrete RSP is weakly NP-hard for constant joint cycle length and prove that it is strongly NP-hard for nonconstant joint cycle length. For constant joint cycle-length discrete RSP, we further present a pseudopolynomial time algorithm that solves the problem optimally and the first known fully polynomial time approximation scheme (FPTAS) for the single-cycle RSP. The scheme is utilizing a new integer programming formulation of the problem that is introduced here. For the strongly NP-hard RSP with nonconstant joint cycle length, we provide a polynomial time approximation scheme (PTAS), which for any fixed , provides a linear time approximate solution. The continuous RSP, where reorders can take place at any time within a cycle, seems (with our results) to be easier than the respective discrete problem. We narrow the previously known complexity gap between the continuous and discrete versions of RSP for the multi-cycle RSP (with either constant or nonconstant cycle length) and the single-cycle RSP with constant cycle length and widen the gap for single-cycle RSP with nonconstant cycle length. For the multi-cycle case and constant joint cycle length, the complexity status of continuous RSP is open, whereas it is proved here that the discrete RSP is weakly NP-hard. Under our conjecture that the continuous RSP is easier than the discrete one, this implies that continuous RSP on multi-cycle and constant joint cycle length (currently of unknown complexity status) is at most weakly NP-hard.

A Fully Polynomial Time Approximation Scheme for the Replenishment Storage Problem

“…A weakly NP-hard problem can have an FPTAS and it was shown in [4] that for constant k the RSP problem is weakly NP-hard. For constant parameter k, Hochbaum and Rao [4] devised for the single-cycle RSP an FPTAS, which we observe here is fixed-parameter tractable (FPT). We devise here an FPTAS for the multi-cycle RSP with constant joint cycle length for the first time.…”

“…The RSP is an NP-hard problem [3,4], so there is no polynomial time optimization algorithm unless P = N P . But a polynomial time approximation scheme may exist for the problem.…”

“…If the running time is polynomial in the problem size for every fixed ǫ, then this scheme is a Polynomial Time Approximation Scheme (PTAS); furthermore, if the running time is polynomial in both the problem size and 1/ǫ, then it is a Fully Polynomial Time Approximation Scheme (FPTAS). Hochbaum and Rao [4] gave a Fully Polynomial Time Approximation Scheme (FPTAS) for the single-cycle RSP when k is a constant [4]. For the multicycle case however no FPTAS has been known to date.…”

“…Since the single-cycle RSP is a special case of the multi-cycle RSP, it implies that the multi-cycle RSP is also NP-hard, even when k is small. Hochbaum and Rao [4] investigated the complexity status of the single-cycle and the multi-cycle RSPs and showed that the problems are strongly NP-hard when k is not a constant, but weakly NP-hard when k is a constant. They further provided in [4] a pseudo-polynomial optimization algorithm for the two problems.…”

“…To that end several approximation results have been delivered for the single-cycle RSP. Hall [3] provided a linear time approximation algorithm for the single-cycle RSP, with an approximation factor of 1 + 2 k , even for non-constant k. Hochbaum and Rao [4] devised for the single-cycle RSP with constant k an FPTAS, and for the single-cycle RSP with non-constant k a Polynomial Time Approximation Scheme (PTAS). The complexity of the FPTAS for a (1 + ǫ)-approximation algorithm is O( n ǫ 2k ) 1 , and the complexity of the PTAS for non-constant cycle length is O(( 2 ǫ )!…”