Hybrid Bilevel-Lagrangean Decomposition Scheme for the Integration of Planning and Scheduling of a Network of Batch Plants

Abstract

Motivated by a real-world industrial problem, this work deals with the integration of planning and scheduling in the operation of a network of batch plants. The network consists of single-stage, multiproduct batch plants located in different sites, which can exchange intermediate products in order to blend them to obtain finished products. The time horizon is given and divided into multiple time periods, at the end of which the customer demands have to be exactly satisfied. The planning model is a simplified and aggregate formulation derived from the detailed precedence-based scheduling formulation. Traveling Salesman Problem (TSP) constraints are incorporated at the planning level in order to predict the sequence-dependent changeovers between groups of products, within and across time periods, without requiring the detailed timing of operations, which is performed at the scheduling level. In an effort to avoid solving the full-space, rigorous scheduling model, especially for large problem sizes, two decomposition strategies are investigated: Bilevel and Temporal Lagrangean. We demonstrate that Bilevel Decomposition is efficient for small to medium problem instances and that further decomposition of the planning problem, yielding a hybrid decomposition scheme, is advantageous for tackling a large-scale industrial test case. ysl ill t 0-1 variable to force only one plant l to ship product i to customer l at time period t Notation Planning Model Variablesyg gult 0-1 variable to denote the assignment of group g to unit u of plant l at time period t ygf gult 0-1 variable to denote the first group g assigned to unit u of plant l at time period t ygl gult 0-1 variable to denote the last group g assigned to unit u of plant l at time period t yp iult 0-1 variable to denote the assignment of product i to unit u of plant l at time period t zg gg ult 0-1 variable to denote if group g is followed by group g in unit u of plant l within time period t zzg gg ult 0-1 variable to denote if link between group g and group g in unit u of plant l within time period t is broken zzzg gg ult 0-1 variable to denote a changeover between group g and group g in unit u of plant l across time period t Unit-Specific General Precedence Model VariablesT e iult End time of product i in unit u of plant l at time period t T s iult Start time of product i in unit u of plant l at time period t 4 wp ii ult 0-1 variable to denote changeover between product i and product i in unit u of plant l across time period t xp ii ult 0-1 variable to denote local precedence between product i and product i in unit u of plant l within time period t yp iult 0-1 variable to denote the assignment of product ito unit u of plant l at time period t ypf iult 0-1 variable to denote the first product i assigned to unit u of plant l at time period t ypl iult 0-1 variable to denote the last product i assigned to unit u of plant l at time period t zp ii ult 0-1 variable to denote global precedence between product i and product i in unit u of plant l within time period t

A General State-Space Formulation for Online Scheduling

“…To tackle the computational challenge of MILP scheduling models, several solution methods have been proposed: (1) tightening methods based on preprocessing algorithms and valid inequalities [32][33][34][35][36][37][38]; (2) reformulations [5,37,[39][40][41]; (3) decomposition methods [42][43][44][45][46][47]; (4) heuristics [19,[48][49][50]; and (5) hybrid methods [51][52][53][54][55]. Finally, parallel computing has been utilized to obtain faster solutions [56][57][58].…”

Integrated short‐term scheduling and production planning in an ethylene plant based on Lagrangian decomposition

“…You et al proposed a strategy using both bi‐level and Lagrangian decomposition methods to solve a multi‐period MILP problem for a multisite system . Calfa et al proposed a hybrid method with bi‐level and Temporal Lagrangian decomposition strategies which dealt with the integration of planning and scheduling problems in the operation of a network of batch plants …”

“…Material balances of production and consumption in continuous time t are presented in Equations (30,31). Equation (32) calculates the amount of leftover materials during the operation.…”

On the solution of large-scale mixed integer programming scheduling models

“…(1) tightening methods including preprocessing algorithms for fixing binary variables (Pinto and Grossmann, 1995;Blomer and Gunther, 2000) and generating valid inequalities , as well as the solution of auxiliary LP and MIP models for the generation of valid inequalities (Burkard and Hatzl 2005;Janak and Floudas, 2008); (2) reformulations including variable disaggregation (Sahinidis and Grossmann, 1991;Yee and Shah, 1998) and reformulation-linearization (Janak and Floudas, 2008) techniques; (3) decomposition methods relying on the structure of the network , the hierarchy of decisions Kelly and Zyngier, 2008), the iterative solution of a simpler MIP model , Lagrangean relaxation and decomposition (Wu and Ierapetritou, 2003;Calfa et al, 2013), and rolling horizon approaches (Dimitriadis et al, 1997;Lin et al, 2002); and (4) algorithmic enhancements including preprocessing algorithms to generate strong valid inequalities , and the use of heuristics (Mendez and Cerda, 2003;Roslof et al, 2001;Kopanos et al, 2010). Furthermore, researchers have proposed decomposition methods that rely on the integration of different solution methods, both for sequential (Jain and Grossmann, 2001;Harjunkoski and Grossmann, 2002;Maravelias, 2006) and network (Maravelias and Grossmann, 2004;Roe et al, 2005) environments.…”