Progressive hedging for multi-stage stochastic lot sizing problems with setup carry-over under uncertain demand

TL;DR

The paper addresses multi-stage stochastic lot sizing with setup carry-over and external component demand by developing a Progressive Hedging (PH) algorithm that decomposes by scenario path and enforces non-anticipativity via a penalty parameter $\rho$. It investigates penalty parameter tuning with cost-proportional rules and introduces metaheuristic global/local adjustments, plus alternative consensus mechanisms (averaging vs. majority voting). Computational results show that a tuned PH approach attains solutions within about 1% of the compact model's cost while achieving shorter runtimes, and that majority voting can further reduce runtime at a small cost to accuracy. The work demonstrates PH's scalability to larger scenario trees and positions it as a practical tool for 24/7 multi-stage production planning with spare-parts demand, while offering partial-implicit heuristics as a complementary approach.

Abstract

We investigate multi-stage demand uncertainty for the multi-item multi-echelon capacitated lot sizing problem with setup carry-over. Considering a multi-stage decision framework helps to quantify the benefits of being able to adapt decisions to newly available information. The drawback is that multi-stage stochastic optimization approaches lead to very challenging formulations. This is because they usually rely on scenario tree representations of the uncertainty, which grow exponentially in the number of decision stages. Thus, even for a moderate number of decision stages it becomes difficult to solve the problem by means of a compact optimization model. To address this issue, we propose a progressive hedging algorithm and we investigate and tune the crucial penalty parameter that influences the conflicting goals of fast convergence and solution quality. While low penalty parameters usually lead to high quality solutions, this comes at the cost of slow convergence. To tackle this problem, we adapt metaheuristic adjustment strategies to guide the algorithm towards a consensus more efficiently. Furthermore, we consider several options to compute the consensus solution. While averaging the subproblem decisions is a common choice, we also apply a majority voting procedure. We test different algorithm configurations and compare the results of progressive hedging to the solutions obtained by solving a compact optimization model on well-known benchmark instances. For several problem instances the progressive hedging algorithm converges to solutions within 1% of the cost of the compact model's solution, while requiring shorter runtimes.

Figures (4)

• Figure 1: Static-dynamic decision framework
• Figure 2: Example scenario tree for two periods
• Figure 3: A partial scenario tree with adjusted scenario probabilities.
• Figure 4: BOM for assembly (a) and general (b) production structures including resource assignments ($i / k$ - item $i$ is assigned to resource $k$)