The Continuous Joint Replenishment Problem is Strongly NP-Hard

TL;DR

This paper proves that the Continuous Periodic Joint Replenishment Problem (CPJRP), a key supply chain management problem, is strongly NP-hard, confirming its computational intractability despite existing approximation algorithms.

Contribution

It extends the NP-hardness proof from the discrete version to the continuous version of the CPJRP, establishing its strong NP-hardness.

Findings

CPJRP is strongly NP-hard.

The problem's complexity has been rigorously proven.

Existing approximation ratios do not imply polynomial solutions.

Abstract

The Continuous Periodic Joint Replenishment Problem (CPJRP) has been one of the core and most studied problems in supply chain management for the last half a century. Nonetheless, despite the vast effort put into studying the problem, its complexity has eluded researchers for years. Although the CPJRP has one of the tighter constant approximation ratio of 1.02, a polynomial optimal solution to it was never found. Recently, the discrete version of this problem was finally proved to be NP-hard. In this paper, we extend this result and finaly prove that the CPJRP problem is also strongly NP-hard.