(423f) Planning and Long-Term Scheduling of a Single-Stage Multi-Product Continuous Process with Complex Recycling Structure

This paper addresses the planning and scheduling of a real-world process for manufacturing tinted glass in a continuous multi-product single stage process with recycling of the waste glass (cullet). The aim of this work is the development and solution of a large scale mixed integer linear programming (MILP) model that has enough accuracy to contribute to real decisions as part of a decision supporting tool. The model and solution approach are designed for long-term scheduling, planning, and cullet recycling in a rolling horizon strategy. The optimization of the production schedule is driven by high transition costs due to long transition times, in the order of days, and their impact on the profit of the process.

The products are characterized by color, which depends on the color of the substrate and the color of a coating if added. The changeover from one substrate to another substrate implies a transition time in the order of days, while the changeover from a specific substrate color to a product with the same substrate with a coating requires a comparatively short changeover. During changeovers, the process produces cullet at the same conditions (energy, raw materials, and production rate). The process scheduling is restricted by: 1) sequence-dependent changeovers between a subset of the products and no changeovers between another subset of products; 2) impositions on the sequence of production and production times between products without changeovers; 3) minimum run lengths due to process control and stability; and 4) by a complex compatibility and recycling profile of cullet.

In this paper, we use as a basis the continuous time slot MILP model proposed by Erdirik-Dogan and Grossmann (2008). Due to the particular characteristics of the process described, several new features are proposed that are motivated by this application but are also generic. The main features are: a) carry-over changeovers across the due dates; b) minimum run lengths across the due dates; c) a rigorous aggregation of the products based on the type of changeovers; d) definition of minimum inventory levels at the end of the time horizon, and e) recycling of material (cullet). The two first extensions are motivated by long changeovers and minimum run lengths that use considerable time of the time period, therefore reducing the flexibility of the model to fit production times and changeovers within the time periods, especially for short time periods. The cost of adding the first extension is only given by the addition of linear equations, while the second extension may require the addition of binary variables depending on the lengths of the time periods set, and the formulation used, i.e. if the the minimum run length is considerably greater than the length of the time period, then binary variables are required. The third extension aggregates the products without changeovers into pseudo-products for the scheduling part of the model, while in the inventory balances and objective function the products are disaggregated to consider the different selling prices, costs, and production rates. This aggregation eliminates a large number of equations, continuous and binary variables, which would be used to formulate the restrictions on the sequence of production and production times between the products without changeovers. Each product has a minimum requirement for cullet feed to the process in order to achieve glass quality standards, and to minimize energy costs. An interesting feature is the bidirectional relation between the scheduling, and the cullet generated and consumed, since a given sequence of production will generate specific amounts of cullet, but also the quality and quantity of cullet available in storage can restrict the products to be produced. The cullet is generated due to the process yield and during the transitions, creating two major types of cullet, transition cullet and product cullet. The recycling of cullet is characterized by a compatibility matrix between cullet recycled and product to produce, and by specific rates of cullet consumption as a function of the production run. The cullet recycling is modeled using mass balances for the transition cullet and product cullet that is produced and consumed, and inventory balances, since there is a maximum storage capacity. Cullet generated over the storage capacity has to be sold. These balances are applied at the end of each slot because the dilution cullet generated in slot l of a specific changeover may be used for a different product in slot l+1.

Two solution approaches are proposed. The first is based on the solution of the MILP model using an off-the-shelf MILP solver, and the second is based on a rolling horizon strategy. The latter is motivated by the large size of the model and associated computer burden to achieve good solutions. The rolling horizon strategy involves the solution of a set of subproblems where the scheduling model is applied to an increasing time horizon, while the time horizon for the planning problem is also moving forward in the time. The time horizon for the planning model is two times the length of the time horizon for the scheduling model in order to provide feedback information to the scheduling sub-models of the future demand. In addition, constraints over the inventory levels at the end of the planning time horizon also feedback the information that the demand is not over after the planning time horizon. Due to the long transition times and minimum run lengths, these are also modeled over the due date at the interface between the scheduling and the planning model.

Different case studies are presented to demonstrate a) the importance of the extensions proposed to the model in terms of inventory levels, and potential profit; b) the influence of the length of the time period in the inventory levels, and sequence of production; c) the impact of different final inventory level constraints at the end of the planning horizon, d) capability of handling recycled material in a scheduling model. The two solution approaches are compared in terms of size of the models, and computational efficiency in terms of integrality gaps and CPU time. In addition, the inventory cullet profiles represent a contribution to the real-world process operation decision makers.

References

Erdirik-Dogan M, Grossmann IE, Simultaneous planning and scheduling of single-stage multi-product continuous plants with parallel lines, Comp. Chem. Eng. (2008), 32, 11, pg 2664-2683.