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A graph-based algorithm for the non-stationary lot-sizing problem with penalty scheme

Xiyuan Ma, Roberto Rossi, Thomas Archibald

Abstract

This paper introduces a graph-based algorithm for solving single-item, single-location inventory lot-sizing problems under non-stationary stochastic demand using the $(R_t, S_t)$ policy and a penalty cost scheme. The proposed method relaxes the original mixed-integer linear programming (MILP) model by eliminating non-negative order quantity constraints and formulating it as a shortest-path problem on a weighted directed acyclic graph. A repetitive augmentation procedure is proposed to resolve any infeasibility in the solution. This procedure consists of three stages: (1) filtration, (2) repeated augmentation by redirecting, reconnecting, and duplicating between newly introduced and existing nodes to adjust the graph and eliminate negative replenishment orders, and (3) re-optimising. The effectiveness and computational efficiency of the proposed approach are assessed through extensive experiments on 1,620 test instances across various demand patterns and parameter settings. The results show that 195 instances required augmentation, mainly those with high penalty costs, low fixed ordering costs, large demand variability, and extended planning horizons. The efficiency of the algorithm for instances with extended planning horizon scenarios demonstrates its suitability for use in real-world scenarios.

A graph-based algorithm for the non-stationary lot-sizing problem with penalty scheme

Abstract

This paper introduces a graph-based algorithm for solving single-item, single-location inventory lot-sizing problems under non-stationary stochastic demand using the policy and a penalty cost scheme. The proposed method relaxes the original mixed-integer linear programming (MILP) model by eliminating non-negative order quantity constraints and formulating it as a shortest-path problem on a weighted directed acyclic graph. A repetitive augmentation procedure is proposed to resolve any infeasibility in the solution. This procedure consists of three stages: (1) filtration, (2) repeated augmentation by redirecting, reconnecting, and duplicating between newly introduced and existing nodes to adjust the graph and eliminate negative replenishment orders, and (3) re-optimising. The effectiveness and computational efficiency of the proposed approach are assessed through extensive experiments on 1,620 test instances across various demand patterns and parameter settings. The results show that 195 instances required augmentation, mainly those with high penalty costs, low fixed ordering costs, large demand variability, and extended planning horizons. The efficiency of the algorithm for instances with extended planning horizon scenarios demonstrates its suitability for use in real-world scenarios.

Paper Structure

This paper contains 14 sections, 15 equations, 26 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description and preliminary formulations
  3. A shortest-path formulation of ($\bm{R_t,S_t}$) policy
  4. A relaxation of flow balance constraints
  5. An equivalent shortest path formulation
  6. A repetitive augmenting procedure for infeasibility
  7. Filtering by Principle of Optimality
  8. An augmenting procedure to resolve infeasibility
  9. A repetitive algorithm
  10. An illustrative example
  11. Computational study
  12. Conclusion
  13. Notations
  14. Implementing revised augmenting procedure on Example 1 by periods.

Figures (26)

  • Figure 1: A replenishment cycle $R(i,j)$ over period $i$ to $j$.
  • Figure 2: A negative order takes place in period $i$.
  • Figure 3: Shortest path problem:
  • Figure 4: The relation between replenishment cycles, arcs and cost associated with arcs.
  • Figure 5: The elimination of the negative order: inventory-level view.
  • ...and 21 more figures