A continuous-time integrated model for discrete control of production ¯ows in a multi-level bills of material environment Konstantin Kogan Tel-Aviv University, Department of Industrial Engineering and Holon Center for Technological Education, Tel Aviv, Israel Abstract An integrated dynamic model for discrete production scheduling and continuous capacity expansion is presented in this paper. The modeled manufacturing system, based on multi-level bills of materials, is characterized by ¯exible machines with negligible setups and production rates ®xed for the current capacity. Make-to-stock formulation of the problem is studied and optimal behavior of the system is determined with the help of the maximum principle. A fast time-decomposition algorithm is suggested to locate the optimal solution. # 1999 IFORS. Published by Elsevier Science Ltd. All rights reserved. Keywords: Continuous-time production scheduling and planning; Optimal control 1. Introduction Production ¯ow control is a complex industrial problem while scheduling production throughout available capacities. The problem becomes especially complicated when dealing with multi-level product structures and capacity expansion. While classical static approaches to such scheduling problems commonly lead to combinatorial explosion, the optimal control approach has con®rmed its applicability to cope with dierent production ¯ow control problems (Akella and Kumar, 1986; Akella et al., 1990; Sharifnia, 1988; Khmelnitsky et al., 1995a, b). Beginning from Kimemia and Gershwin (1983) who ®rst presented production process by dierential equations, various extensions of continuous-time optimal control models for production planning and scheduling have been formulated. A failure-prone environment consideration (Akella and Kumar, 1986) and setup eects (Khmelnitsky et al., 1995, 1996) are Intl. Trans. in Op. Res. 6 (1999) 263±273 0969-6016/99/$20.00 # 1999 IFORS. Published by Elsevier Science Ltd. All rights reserved. PII: S0969-6016(98)00054-9