A physical model approach to order lot sizing

TL;DR

This work addresses the classical lot-sizing problem by introducing a physical-analogy framework that maps material supply to a one-dimensional elastic system. The key contribution is the analytical derivation of an exact solution governed by the parameter $γ=2C_O/C_H$, with $N_{op}=M$ for $γ\le1$ and $N_{op}=γ^{-1/2}M$ for $γ>1$, plus a cost function reformulated in terms of the intensive variable $θ=N/M$. The model yields either equal-sized orders or two-sized orders when integer constraints apply, enabling rapid, precise purchasing decisions without heuristics. Validation on a real-world pulp plant shows substantial cost savings compared to traditional strategies and to EOQ-based planning, highlighting practical relevance and computational efficiency. The findings suggest promising extensions to more complex demand and cost structures.

Abstract

The growing need for companies to reduce costs and maximize profits has led to an increased focus on logistics activities. Among these, inventory management plays a crucial role in minimizing organizational expenses by optimizing the storage and transportation of materials. In this context, this study introduces an optimization model for the lot-sizing problem based on a physical system approach. By establishing that the material supply problem is isomorphic to a one-dimensional mechanical system of point particles connected by elastic elements, we leverage this analogy to derive cost optimization conditions naturally and obtain an exact solution. This approach determines lot sizes that minimize the combined ordering and inventory holding costs in a significantly shorter time, eliminating the need for heuristic methods. The optimal lot sizes are defined in terms of the parameter $ γ= 2C_O / C_H $, which represents the relationship between the ordering cost per order ($ C_O $) and the holding cost per period for the material required in one period ($ C_H $). This parameter fully dictates the system's behavior: when $ γ\leq 1 $, the optimal strategy is to place one order per period, whereas for $ γ> 1 $, the number of orders $ N $ is reduced relative to the planning horizon $ M $, meaning $ N < M $. By formulating the total cost function in terms of the intensive variable $ N/M $, we consolidate the entire optimization problem into a single function of $ γ$. This eliminates the need for complex algorithms, enabling faster and more precise purchasing decisions. The proposed model was validated through a real-world case study and benchmarked against classical algorithms, demonstrating superior cost optimization and reduced execution time. These findings underscore the potential of this approach for improving material lot-sizing strategies.

Figures (6)

• Figure 1: Schematic representation of an array of ($N-1$) particles (solid circles) connected by $N$ linear springs.
• Figure 2: The minimum total cost $C_{T,0}$ (in $\log$ scale) as a function of the number of orders $N$ for a planning horizon of $M = 12$, with $C_O=1.0$ and different values of $\gamma$. The various values of $\gamma$ were achieved by adjusting $C_H$: $C_H=25/2$ ($\gamma = 0.16$); $C_H=2.0$ ($\gamma = 1.0$); $C_H=1/2$ ($\gamma = 4.0$); $C_H=2/9$ ($\gamma = 9.0$); $C_H=1/18$ ($\gamma = 36$) and $C_H=1/72$ ($\gamma = 144$).
• Figure 3: (a) The minimum total cost $C_{T,0}$ (in $\log$ scale) versus amount of orders $N$ for four typical cases: $\gamma=4$, $C_O=1$ and $C_H=1/2$, solid circles; $\gamma=4$, $C_O=2$ and $C_H=1$, circles with a plus inside; $\gamma=144$, $C_O=1$ and $C_H=1/72$, open triangles; and $\gamma=144$, $C_O=72$ and $C_H=1$, triangles with a plus inside. All curves were obtained for $M=12$. In part (b), the curves presented in part (a) have been normalized by the ordering cost $C_O$. With this rescaling, it is observed that all curves corresponding to the same values of $\gamma$ and $M$ collapse into a single curve.
• Figure 4: Data collapse of the minimum total cost $C_{T,0}/M$ (in $\log$ scale) versus $N/M$ for the case $\gamma = 4$. The plots were made using $C_O=1.0$, $C_H = 1/2$ and three different values of $M$: $M=12$, solid circles; $M=24$, open circles; and $M=36$, circles with a plus inside. The non-normalized curves ($C_{T,0}$ versus $N$) corresponding to the data in the main figure are displayed in the inset.
• Figure 5: $\widetilde{C}_{T,0}(\theta)$ as a function of $\theta$ for different values of the parameter $\gamma$: $10^{-2}$, 0.16, 4, 144 and $10^6$, as indicated. Solid lines represent the results obtained from equation \ref{['minctita']}. Symbols correspond to different sets of parameters $C_O-C_H-M$ that are compatible with the same $\gamma$ value. $\gamma=0.16$: $C_O=0.08$, $C_H=1$, $M=12$ (solid squares); $C_O=0.16$, $C_H=2$, $M=24$ (squares with a plus inside); $C_O=0.32$, $C_H=4$, $M=36$ (open squares); and $C_O=0.48$, $C_H=6$, $M=48$ (squares with a point inside). $\gamma=4$: $C_O=2$, $C_H=1$, $M=12$ (solid circles); $C_O=4$, $C_H=2$, $M=24$ (circles with a plus inside); $C_O=8$, $C_H=4$, $M=36$ (open circles); and $C_O=12$, $C_H=6$, $M=48$ (circles with a point inside). $\gamma=144$: $C_O=72$, $C_H=1$, $M=12$ (open triangles); $C_O=144$, $C_H=2$, $M=24$ (solid triangles); $C_O=288$, $C_H=4$, $M=36$ (triangles with a plus inside); and $C_O=432$, $C_H=6$, $M=48$ (triangles with a point inside).