Capital flow constrained lot sizing problem with loss of goodwill and loan

We introduce capital flow constraints, loss of good will and loan to the lot sizing problem. Capital flow constraint is different from traditional capacity constraints: when a manufacturer launches production, its present capital should not be less than its present total production cost; otherwise, it must decrease production quantity or suspend production. Unsatisfied demand in one period may cause customer’s demand to shrink in the next period considering loss of goodwill. Fixed loan can be adopted in the starting period for production. A mixed integer model for a deterministic single-item problem is constructed. Based on the analysis about the structure of optimal solutions, we approximate it to a traveling salesman problem, and divide it into sub-linear programming problems without integer variables. A forward recursive algorithm with heuristic adjustments is proposed to solve it. When unit variable production costs are equal and goodwill loss rate is zero, the algorithm can obtain optimal solutions. Under other situations, numerical comparisons with CPLEX 12.6.2 show our algorithm can reach optimal in most cases and has computation time advantage for large-size problems. Numerical tests also demonstrate that initial capital availability as well as loan interest rate can substantially affect the manufacturer’s optimal lot sizing decisions.

💡 Research Summary

This paper presents a significant extension to the classic deterministic single-item lot-sizing problem by integrating critical real-world financial and marketing considerations: capital flow constraints, loss of customer goodwill, and the option of obtaining a loan.

The core innovation is the introduction of a capital flow constraint. Unlike traditional capacity constraints (e.g., machine hours), this constraint mandates that at the start of any production period, the manufacturer’s available capital must be sufficient to cover the total production cost (fixed setup cost + variable cost). If capital is insufficient, production must be reduced or suspended. This directly models the liquidity constraints faced by many firms, especially SMEs. Secondly, the model incorporates a dynamic loss of goodwill mechanism. Unmet demand in a period does not merely incur a penalty but actively shrinks the effective demand in the subsequent period, reflecting customer attrition and long-term revenue loss. Thirdly, the model allows for a fixed loan to be taken at the beginning of the planning horizon, which must be repaid with interest at a specified future period (T_L), adding a financial decision layer to the production planning problem.

The authors formulate this integrated problem as a mixed-integer linear programming (MILP) model, termed Model P, with the objective of maximizing the net capital increment at the end of the planning horizon. The model includes binary variables for production setup and effective demand positivity, continuous variables for production quantity, inventory, capital, and lost sales. The authors establish that the problem is NP-hard, as the capital constraint can be viewed as a specific type of capacity constraint.

To tackle this complexity, the authors perform a structural analysis of optimal solutions. A key property (Lemma 1) is proven: when unit variable production costs are equal across periods and the goodwill loss rate is zero, an optimal solution satisfies the Zero-Inventory-Ordering (ZIO) property, meaning production only occurs when inventory is depleted. Leveraging these properties, the problem is approximated to a Traveling Salesman Problem (TSP) framework by conceptualizing “production rounds” (sequences of production cycles from zero inventory to zero inventory). This approximation allows the problem to be decomposed into sub-problems that are linear programs without integer variables.

Based on this analysis, the authors propose a forward recursive algorithm enhanced with heuristic adjustments. The algorithm progresses chronologically from the first to the final period, evaluating feasible production and sales fulfillment decisions at each stage while pruning non-promising states for efficiency.

Numerical experiments demonstrate the algorithm’s effectiveness. Under the special case of equal variable costs and no goodwill loss, the algorithm guarantees optimality. For the general case with varying costs and positive goodwill loss, the algorithm is tested against the commercial solver CPLEX 12.6.2 on large-scale randomly generated instances. The results show that the proposed algorithm reaches optimal or near-optimal solutions in most cases and exhibits a significant computational time advantage over CPLEX for large-size problems, highlighting its practical utility.

Furthermore, the numerical analysis provides crucial managerial insights. It demonstrates that initial capital availability (B0) and the loan interest rate (r) substantially impact optimal lot-sizing decisions and final profitability. Scenarios with severe capital shortage limit production and profit, while abundant capital allows unconstrained planning. In intermediate cases, the decision to take a loan and the interest rate become critical factors, creating a complex interplay between production scheduling and financial cost management.

In conclusion, this paper successfully bridges operations (lot-sizing) and finance (capital constraints, loans) within a unified model. It provides both a formal modeling framework and an efficient solution algorithm for a more realistic production planning problem, emphasizing the necessity of integrated operational-financial decision-making in capital-constrained environments. Future research may extend this work to stochastic settings, multi-item problems, or more sophisticated financial and goodwill loss models.