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in Wiley InterScience (www.interscience.wiley.com).A novel approach is presented for the solution of production planning problems for multiproduct processes. A mixed-integer programming (MIP) scheduling model is analyzed off-line to obtain a convex approximation of feasible production levels and a convex underestimation of total production cost as a function of production levels. The two approximating functions are expressed via linear inequalities that involve only planning variables yet provide all the relevant scheduling information necessary to solve the planning problem with high quality. A rolling horizon algorithm is also presented for generation (if necessary) of detailed schedules. Keywords: production planning, scheduling, attainable region IntroductionIn 2005, the US chemical industry contributed over 2.3%, $540 billion, 1 to the gross domestic product (GDP). It is the second largest manufacturing sector and one of the largest private sector investors in R&D, with chemical patents accounting for 21% of total awarded patents in 2004.2 At the same time, however, the chemical industry faces a number of major challenges such as migration of customer industries, saturation of markets, increased global competition, continued environmental regulation, and energy price and availability. 3,4 To remain healthy in today's competitive environment, chemical companies must operate efficiently by simultaneously optimizing multiple levels of operation. 5,6 To achieve this, advanced modeling methods and optimization tools for the integration and solution of large-scale supply chain optimization models are necessary.In this article, we are specifically interested in mediumterm production planning of multiproduct processes. Production planning seeks to determine optimal production targets and product inventories for a given demand forecast. To effectively solve this problem, we should also account for the capacity of manufacturing facilities and production costs, which is currently achieved via the integration of planning and scheduling models. However, integrated models are notoriously difficult to solve despite improvements in computer hardware and optimization software. Current state of the art methods are insufficient to meet industrial needs, while ad hoc approaches are not adequate due to the complexity of the chemical manufacturing facilities (e.g., batch splitting/mixing, recycle streams, coproduction, different storage policies, utility constraints, etc.). To address this problem, we propose developing a set of constraints to accurately define the space of feasible production levels (amounts) and associated production costs. The strength of this framework is being able to locate feasible production levels without needing to on-line obtain a detailed schedule for achieving each target.This article is organized as follows. We begin by reviewing the area of supply chain management (SCM), defining the production planning problem, discussing integration of planning and scheduling, and reviewing previously proposed me...

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in Wiley InterScience (www.interscience.wiley.com).An algorithm is presented for identifying the projection of a scheduling model's feasible region onto the space of production targets. The projected feasible region is expressed using one of two mixed-integer programming formulations, which can be readily used to address integrated production planning and scheduling problems that were previously intractable. Production planning is solved in combination with a surrogate model representing the region of feasible production amounts to provide optimum production targets, while a detailed scheduling is solved in a rolling-horizon manner to define feasible schedules for meeting these targets. The proposed framework provides solutions of higher quality and yields tighter bounds than previously proposed approaches. V

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