The Price of Simplicity: Analyzing Decoupled Policies for Multi-Location Inventory Control

TL;DR

This work investigates the cost of using simple, decoupled control policies in inherently coupled multi-location inventory systems with nonlinear cross-location ordering costs, motivated by energy-efficient coordination in refrigeration. It derives an equivalence result showing that, within the infinite-horizon stationary base-stock policy class, optimal base-stock levels are unchanged by coupling, and it provides tight worst-case bounds on the performance gap between decoupled and optimally coupled policies that depend on cost nonlinearity and the number of locations. The study also analyzes an online randomized cost-balancing algorithm that achieves competitive ratios independent of full dynamic programming, making it practical for real-time operation. Numerical simulations reveal that decoupled policies typically outperform their worst-case guarantees but remain suboptimal to fully coordinated strategies, offering actionable guidance for balancing control simplicity and operational efficiency in coupled systems.

Abstract

What is the performance cost of using simple, decoupled control policies in inherently coupled systems? Motivated by industrial refrigeration systems, where centralized compressors exhibit economies of scale yet traditional control employs decoupled room-by-room temperature regulation, we address this question through the lens of multi-location inventory control. Here, a planner manages multiple inventories to meet stochastic demand while minimizing costs that are coupled through nonlinear ordering functions reflecting economies of scale. Our main contributions are: (i) a surprising equivalence result showing that optimal stationary base-stock levels for individual locations remain unchanged despite the coupling when restricting attention to decoupled strategies; (ii) tight performance bounds for simple decoupled policies relative to optimal coupled policies, revealing that the worst-case ratio depends primarily on the degree of nonlinearity in the cost function and scales with the number of locations for systems with fixed costs; and (iii) analysis of practical online algorithms that achieve competitive performance without solving complex dynamic programs. Numerical simulations demonstrate that while decoupled policies significantly outperform their worst-case guarantees in typical scenarios, they still exhibit meaningful suboptimality compared to fully coordinated strategies. These results provide actionable guidance for system operators navigating the trade-off between control complexity and operational efficiency in coupled systems.

Figures (2)

• Figure 1: We consider $M = 2$ inventories, integer-valued parameters, a finite horizon $N = 2$, holding/backlog cost $r(z) = \max \{0,z\} + 10 \max \{0,-z\}$, and stochastic demand $w_k$ that takes values zero or one with equal probability. (a) shows the optimal policy for the linear ordering cost $c(z) = 2z$. (b) shows the optimal policy for the nonlinear ordering cost function \ref{['eq:c_nonlinear']}. (c) shows the expected cost from each initial condition for the nonlinear ordering cost function \ref{['eq:c_nonlinear']} under the policies in (a),(b).
• Figure 2: For all simulations we consider $M = 2$ inventories, state space $x \in [-2,8]$, discretization $\Delta x = 0.5$, a finite horizon $N = 20$, and stochastic demand $w_k \in \{0,0.5,1,1.5\}$ where $w_k \sim \mathrm{Bin}(3,0.5)$. (a) shows average cost ratios for the decoupled policy $\pi_$ and online policy $\pi_\vartriangle$ from each initial state under sector-bounded ordering cost. We set holding/backlog cost $r(z) = 0.1 \max\{0,z\} + 10 \max \{0,-z\}$. (b) shows average cost ratios for the decoupled policy $\pi_s$ and online policy $\pi_\vartriangle$ from each initial state under affine-bounded ordering cost. We set holding/backlog cost $r(z) = 0.2 \max\{0,z\} + 10 \max \{0,-z\}$.

Theorems & Definitions (11)

• Definition 1
• Theorem 1: Equivalence of stationary base-stock policies
• proof
• Definition 2
• Definition 3
• Definition 4
• Theorem 2: Suboptimality of base-stock and $(s,S)$ policies
• proof
• Proposition 3: Suboptimality of online algorithm
• proof