Economic production quantity with quantity discounts and interruptions in a multi-item single-machine environment: A fuzzy–probabilistic chance-constrained programming framework

Identifying the optimal economic production quantity (EPQ) under practical operating conditions is a key issue in production planning. This paper develops an extended EPQ formulation that simultaneously considers rework, scrap, and supplier quantity discounts under uncertainty. We model uncertainty through a hybrid fuzzy–probabilistic setting: interruption/interruptions are represented as fuzzy parameters elicited from expert judgment, whereas machine setup times are treated as probabilistic parameters estimated from historical records. Fuzzy and probabilistic chance-constrained programming (FCCP and PCCP) are adopted to control constraint satisfaction for the fuzzy and stochastic terms, respectively, and fuzzy parameters appearing in the objective are handled via their expected values. The resulting bi-objective model minimizes (i) the expected total cost and (ii) the variance of total cost as a measure of risk/stability. We solve the model using three established multi-objective approaches (i.e., LP-metric, goal attainment (GA), and Selim–Ozkarahan (SO)) and evaluate them on test problems of different sizes. Sensitivity analyses show that the SO approach yields the lowest expected costs, and the LP-metric approach is most computationally efficient, especially for larger instances. Interestingly, cost variance can dramatically increase with increased allowed interruption time (up to 1,100% in the complex case). In comparison, the expected cost can significantly drop with increased allowed interruption time (up to 72% in the complex case). A key operational trade-off to note is that the cost variance could be very high by increasing the allowed interruption time (up to 1,100% when the interruption time is allowed to be high and complicated), on the other hand, many times the expected cost could drop significantly (up to 72% in the complex case). In most experimental scenarios, the TOPSIS-based ranking prioritizes the goal attainment (GA) approach since it reflects the best compromise between the three competing criteria: expected cost, cost variance, and CPU time.