Decomposition and Nondifferentiable Optimization with the Projective Algorithm

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

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“…The material balances on the three nodes in Figure 5 for finished and intermediate products are expressed in constraints (25) - (27).…”

Section: Materials and Inventory Balancesmentioning

confidence: 99%

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

Calfa

Agarwal

Grossmann

et al. 2013

Ind. Eng. Chem. Res.

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“…Since a standard column generation method for solving the linear relaxation of the formulation (3) suffers from very slow convergence due to high degeneracy, two strategies for stabilizing column generation [16] were used and compared in [15]. That one for which the linear relaxation is solved by an interior-point algorithm, i.e., the weighted version of the analytic center cutting plane method (ACCPM) of Goffin, Haurie, and Vial [20], was found to be the best.…”

Section: Column Generation Algorithm Revisitedmentioning

confidence: 99%

An improved column generation algorithm for minimum sum-of-squares clustering

2010

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Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consist in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given in du Merle et al. [15]. The bottleneck of that algorithm is resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2300 entities, i.e., more than 10 times as much as previously done.

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“…ACCPM, initially developed as a nondifferentiable optimization algorithm [16], permits the solution of generalized monotone variational inequalities [17]. The key idea is that under the assumptions that F is a monotone and continuous mapping and that Y is a closed, convex and nonempty set, VI(F,Y) can be formulated as a convex feasibility problem:…”

Section: Accpm For Variational Inequalitiesmentioning

confidence: 99%

Using ACCPM in a simplicial decomposition algorithm for the traffic assignment problem

2007

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The purpose of the traffic assignment problem is to obtain a traffic flow pattern given a set of origindestination travel demands and flow dependent link performance functions of a road network. In the general case, the traffic assignment problem can be formulated as a variational inequality, and several algorithms have been devised for its efficient solution. In this work we propose a new approach that combines two existing procedures: the master problem of a simplicial decomposition algorithm is solved through the analytic center cutting plane method. Four variants are considered for solving the master problem. The third and fourth ones, which heuristically compute an appropriate initial point, provided the best results. The computational experience reported in the solution of real large-scale diagonal and difficult asymmetric problems-including a subset of the transportation networks of Madrid and Barcelona-show the effectiveness of the approach.

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Decomposition and Nondifferentiable Optimization with the Projective Algorithm

Cited by 165 publications

References 12 publications

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

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

An improved column generation algorithm for minimum sum-of-squares clustering

Using ACCPM in a simplicial decomposition algorithm for the traffic assignment problem

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