Inventory pinch gasoline blend scheduling algorithm combining discrete- and continuous-time models

Highlights

• MPIP-C inventory-pinch based three-level scheduling algorithm for nonlinear processes.

• Approximate scheduling of coarse time scale decisions via discrete-time model.

• Detailed scheduling via continuous-time model.

• MPIP-C solves linear problems to less than 0.5% optimality gap.

• MPIP-C solves problems 2–3 orders of magnitude faster than prior methods.

Abstract

This work introduces multi-period inventory pinch-based algorithm to solve continuous-time scheduling models (MPIP-C algorithm), a three level method which combines discrete-time approximate scheduling with continuous-time detailed scheduling and with inventory pinch-based optimization of operating states. When applied to gasoline blending, the top level computes optimal recipes for aggregated blends over periods initially delineated by inventory pinch points. Discrete-time middle level uses fixed blend recipes to compute an approximate schedule, i.e. what, when, and how much to produce; it also allocates swing storage and associated product shipments with specific storage. Continuous-time model at the third level computes when exactly to start/stop an operation (blend, tank transfer, shipment). MPIP-C algorithm solves linear or nonlinear problems 2–3 orders of magnitude faster than full-space models.

Introduction

Process industries’ supply chains are comprised of facilities and activities which procure raw materials to the plants, and store and distribute finished products to the customers. Structure of the supply chains (location, capacity, and type of each plant and storage) is decided upon by optimizing returns over long time horizons, e.g. five or ten years. Once the structure of the supply chain is fixed, one needs to decide the best way to operate it. Since supply chains are large systems, and there are many decisions to be made, it is customary to optimize supply chain operations over different lengths of the time horizon and at different levels of model accuracy. When optimizing operation over long time horizons, e.g. over one year, it is important to determine how much of each product will be produced from one season to another, while it is not important to decide at what specific time some equipment will change from one operation mode to another. The latter is a detailed decision which can be made e.g. one or two weeks prior to that day.

Scheduling is an activity that determines the operating states and their sequence for each equipment, amount of feed to be processed by each instance of an operating state, amount and time of shipment of each product, allocation of multipurpose storage to a given service, and others. Scheduling models for oil refinery operations are usually mixed-integer linear programs (e.g. Gothe-Lundgren et al., 2002, Jia and Ierapetritou, 2003, Jia and Ierapetritou, 2004, Li and Karimi, 2011) even if the underlying processes are nonlinear by nature, in order to decrease the computational burden. Some MINLP models have been published recently for the detailed scheduling of crude oil operations (Li et al., 2012) and the pooling problem (Kolodziej et al., 2013); they have been solved via the latest generation of specialized MINLP solvers. Currently, most scheduling models are usually formulated using a continuous-time representation (i.e. the horizon is divided into several time slots which duration is a variable to determine) since it requires a smaller number of discrete variables compared with the corresponding discrete-time model. One example of commercial scheduling software is Aspen Petroleum Scheduler (Aspen Technology).

Inclusion of sequencing terms in the model formulation makes scheduling a very challenging computational problem. Due to the inherent discrete decisions involved in the scheduling problem, it is at least NP-complete (Birewar and Grossmann, 1990, Pinto et al., 2000) and frequently NP-hard (Terrazas-Moreno and Grossmann, 2011); this means that there are no polynomial time bounded algorithms to solve this type of problems. Within this current paradigm, it is not possible to guarantee computation of optimal solutions in a reasonable amount of time by a general algorithm when scheduling problems grow beyond a certain model size (which depends on the problem type and instance). Most successful solution strategies are those that employ algorithms tailored to a specific class of scheduling problems and scale up to solve large problems within acceptable computational times. Necessity for such approaches has been presented as “no free lunch theorems for optimization” by Wolpert and Macready (1997).

Decomposition strategies are usually employed to handle large-scale scheduling problems. Bassett et al. (1996) presented various heuristic methods for discrete-time formulations of scheduling problems, most of them using time-based decompositions. Wu and Ierapetritou (2003) reviewed several heuristic approaches for scheduling problems with continuous-time models, including methods using time- and resource-based decompositions. They also included a review of rigorous mathematical approaches such as Lagrangean relaxation and Lagrangean decomposition.

A major part of an oil refinery's profit comes from gasoline (Mendez et al., 2006, Li and Karimi, 2011). Therefore, determining the best possible way to mix the refinery's intermediate products (blend components) to produce the different gasoline grades is an important task. For that reason, inclusion of recipe optimization in gasoline blend scheduling models has become the norm in the last ten years. Both discrete- and continuous-time formulations have been developed to solve the gasoline blend scheduling problem. In order to develop a mixed-integer linear model, quality properties are transformed into blend indices that blend linearly on a volumetric or weight basis (e.g. Li et al., 2010, Li and Karimi, 2011). A different approach is to solve a sequence of MILP models in order to converge to the proper quality values (e.g. Mendez et al., 2006). As well, stochastic methods have been proposed to solve nonlinear recipe optimization problems (e.g. Chen and Wang, 2010, Zhao and Wang, 2011); however, no operational features nor logistic constraints are considered in such cases.

Commercial MILP solvers may require several hours to obtain the optimal solution of detailed scheduling models for medium- and large-scale problems (e.g. Li and Karimi, 2011, Shah and Ierapetritou, 2011). Castillo-Castillo and Mahalec (2016) presented a detailed continuous-time scheduling model with reduced number of discrete variables, and although such model was able to solve some large-scale examples to optimality in less time than previously published models, execution times still can reach more than 12 h.

Glismann and Gruhn (2001) proposed a two-level approach to solve the blend scheduling problem: at the top level, a discrete-time NLP model computes blend recipes and production targets, and at the lower level, a discrete-time MILP model solves the short term scheduling problem using the recipes and targets from the top level. An iterative procedure is required to handle possible infeasibilities. The scheduling model is based on a resource-task-network representation. The scheduling model does not consider multipurpose tanks (i.e. swing tanks) nor the delivery scheduling problem (i.e. distribution or shipping problem).

Castillo and Mahalec (2014b) presented a three-level decomposition approach to solve the gasoline blending problem using discrete-time models at each level. They included recipe optimization using linear or nonlinear models, blend size threshold constraints, and most of the operational features described by Li and Karimi (2011). The discrete-time formulation for scheduling horizons of 1 or 2 weeks with 1-h time periods resulted in a large size scheduling model at the 3rd level that required to be solved in subintervals. These subintervals were solved sequentially, first in a forward direction and then in a reverse direction. Solutions computed by this approach were better and the execution times for large problems were two orders of magnitude shorter than those from previous works (Li et al., 2010, Li and Karimi, 2011). For most of their nonlinear examples, their algorithm computed better solutions than MINLP solvers BARON, ANTIGONE, and GloMIQO, with execution times an order of magnitude shorter. Analogously to Li and Karimi (2011), Castillo and Mahalec (2014b) did not penalize the delivery of the same product order from different tanks.

This work introduces Multi-Period Inventory Pinch-Continuous time algorithm (MPIP-C) for scheduling of processes described by linear and nonlinear models. MPIP-C algorithm decomposes the original scheduling problem into (i) blend recipe optimization, (ii) approximate scheduling, and (iii) detailed scheduling. The 1st level determines the blend recipes (by solving a discrete-time LP or NLP model), and the 2nd level computes an approximate schedule via discrete-time MILP. The time periods at the 1st level are initially delineated by inventory pinch points (Castillo et al., 2013, Castillo and Mahalec, 2014a). Discrete-time model for approximate scheduling is a modification of a model used in our previous work (Castillo and Mahalec, 2014a). The 3rd level model (i.e. the detailed scheduling problem) is a continuous-time MILP which includes additional constraints arising from the approximate scheduling solution. If there are infeasibilities encountered at the 2nd or the 3rd level, they are resolved by subdividing the corresponding periods at the 1st level. The problem does not have a feasible solution if the 1st level is infeasible.

The rest of this article is structured as follows. Section 2 presents the problem statement. Section 3 presents an overview of the algorithm and of the models used at each level. Section 4 describes the examples used in this work. Section 5 discusses the results obtained using the MPIP-C algorithm. As summarized in Section 6, the computational results show that the proposed MPIP-C algorithm computes optimal or near-optimal solutions with execution times which are much smaller (for both linear and nonlinear models) than those required by the full-space model.