Scheduling tank trucks at a fuel distribution terminal using max-plus model-based predictive control

https://doi.org/10.1016/j.jprocont.2021.05.005Get rights and content

Highlights

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    Model predictive control and max-plus algebra applied to a scheduling problem.

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    A pre-trained neural network provides the process times for the max-plus model.

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    Optimizing the sequencing and timing of entry through switching max-plus-linear systems.

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    Notes on the implementation and performance analysis of the control strategies.

Abstract

This study addresses the problem of scheduling tank trucks at a fuel distribution terminal. The plant was modeled in the max-plus algebra, applying machine learning to determine process times. With this model, and based on a just-in-time approach, we have developed a predictive controller with two operation modes. This control system aims to prevent the excess of tank trucks inside the loading yard, thus achieving a better flow, efficiency, and safety in the process. Next, we have investigated the case study of a realistic and representative fuel distribution terminal, developing a simulator to enable a performance comparison between the proposed algorithms and the current heuristic. There was a 42.7% reduction in the work-in-progress (WIP) and 41.4% in the lead time, while productivity suffered a 2.8% loss. Bearing in mind, however, that there is flexibility in parametrization to mitigate this loss of productivity. In doing so, the reductions in WIP and lead time are slightly lower, at 34.7% for both metrics. The results show that the proposed control system can contribute significantly to improving the company’s performance indicators.

Introduction

Fuel distribution is a relevant segment of the Brazilian market, responsible for taking products from refineries, fuel plants, and ports to points of consumption, such as gas stations and industries. In Brazil, the largest companies in the sector are BR Distribuidora, Raízen, and Ipiranga, holding together 63.48% of the market share of a total volume of 140 billion liters annually [1]. A large distribution terminal typically loads more than 300 tank trucks (TTs) in a single day. Because of this characteristic, companies generally use a system that schedules the entry of vehicles for loading.

Despite the high volume handled and the importance of the sector, we have noticed that few papers address intramural scheduling. Generally, the studies focus on the supply chain as a whole, as in [2], [3], [4]. Also, we have observed some in-depth papers in the context of fuel distribution, as in [5], [6], [7]. In these studies, the terminal is modeled in simulation software, used as a tool to support strategic decisions, such as investments and changes in operating hours.

In [5], there is an extension beyond modeling. It proposes a simplified scheduling algorithm that prioritizes a particular tank truck to prevent the loading bay from becoming idle. This paper suggests, as future work, the study of a method to optimize the allocation in bays, being in line with the theme of the present work.

Based on the described scenario, the lack of a systematic approach to the scheduling problem in fuel distribution terminals becomes evident. The present study addresses this issue, aiming to extend the results beyond the academic environment, developing a technique with applicability and direct return to the productive sector.

The case study is conducted at a fuel distribution corporation with operations in Brazil, where a simple scheduling heuristic is used to keep the loading yard full of TTs. The motivation for this strategy is to achieve higher productivity by preventing the loading equipment from being idle. However, this excess of vehicles hinders movement in the yard, causing a loss of efficiency. In addition, it represents a safety risk because of the fuel handling operations carried out in the loading yard.

This document is organized as follows. Section 2 describes the loading process of a fuel distribution terminal, also presents the problem under consideration and the proposed solution. The basic mathematical concepts for understanding this paper are exposed in Section 3. Based on these fundamentals, the case study plant is modeled in Section 4, the system dynamics are discussed in Section 5, and the optimization problem is established in Section 6. Then, we present in Section 7 the simulator developed to compare the current and proposed scenarios. Next, Section 8 provides the technical details of this implementation, aiming to expose relevant contributions to future work with similar applications. In Section 9, we conduct the verification process to demonstrate the correct functioning of the proposed algorithms. Subsequently, in Section 10, we carry out the validation, presenting the graphs and analyzing the impacts on the performance indicators. Finally, the last section summarizes the results of this study and suggests directions for future research.

Section snippets

Problem definition and approach

To clarify the scenario under consideration, we present in Fig. 1 a simplified layout of a fuel distribution terminal. The loading process starts when the driver arrives, parks the tank truck, and heads for the waiting area. There, invoices and transport documents are issued. Then, the driver watches the call monitor, waiting for his authorization to enter the loading yard. Once allowed, the TT is driven to the loading bay, where the filling process is carried out. Upon completion, the vehicle

Max-plus algebra

The max-plus algebra is built on the usual maximization and addition operations. Based on [15], [16], the max-plus-algebraic addition (⊕) and multiplication (⊗) are defined as follows: x⊕y=max(x,y),x⊗y=x+y, for x,y∈Rε=defR∪{−∞}. Let the notation (Z)i,j denote the entry zi,j in the ith row and jth column of the matrix Z. Regarding the max-plus matrix operations, the sum and product are defined as follows: (A⊕B)i,j=ai,j⊕bi,j,(A⊗C)i,j=⨁k=1nai,k⊗ck,j, for A,B∈Rεm×n and C∈Rεn×p. Also, the neutral

System modeling

Within the scope of max-plus algebra, a timed event graph (TEG) is typically used to represent the process. It is a subclass of Petri nets, in which places have an associated holding time that defines how long the token must wait before enabling the firing of the output transition [15].

Fig. 3 depicts the TEG of the fuel distribution terminal under analysis. It consists of n loading bays with their process times defined for each index k, being k the event counter. Considering the bays i=1,…,n,

Evolution of the system

For simplicity, the following notations are adopted: A(v(ρ))=defAρ,B(v(ρ))=defBρ,C(v(ρ))=defCρ.

Given an input u(k), a control input v(k), and an initial state x(0), Eqs. (1), (2) lead to the following expression for the evolution of the system: y(k)=Ck⊗⨂ℓ=0k−1Ak−ℓ⊗x(0)⊕Ck⊗⨁m=1k−1⨂ℓ=0k−m−1Ak−ℓ⊗Bm⊗u(m)⊕Ck⊗Bk⊗u(k), for k=1,2,….

Looking forward to the MPC implementation based on [14], we introduce the prediction horizon Np. It determines the number of future events considered in the current

Problem formulation

As described in Section 2.1, this study deals with a control system with two operation modes: (1) the Scheduling Algorithm that minimizes deviation from the planned schedule; and (2) the Maximization Algorithm that maximizes the utilization of the loading bays. In Section 10, we compare both algorithms with the current heuristic used for scheduling tank trucks, here called Current Strategy.

Based on the control diagram in Fig. 2, we define the reference input r(k)=r1(k),…,rn(k)T, used in the

The simulator of a fuel distribution terminal

To verify the proposed control system effectiveness, comparing it with the current heuristic, we have developed a simulator of the fuel distribution terminal under consideration. The application was titled RT2MP Scheduler, being an acronym for Real-Time Max-Plus Model Predictive Scheduler. It was developed with the following composition of tools: MATLAB and App Designer [23] for implementing the code and the graphical interface; YALMIP [24] for modeling the optimization problem; and Gurobi

Implementation aspects

In this section, we describe the technical details of the implementation. The objective is to present relevant aspects that can serve as a reference for future research.

A significant aspect of MPC is the time required to solve the optimization problem, representing a limiting factor for real-time control. From the first studies to the structure presented in this paper, we have established several different techniques. The early versions only worked for two loading bays, as they became

Verification of the control system

Verification and validation are processes that determine whether the specified system requirements have been fulfilled and whether the intended use has been satisfied [28]. In the verification process, we answer the question: “Does the system meet the specified requirements?”. While in the validation process, we answer: “Does the system solve the problem it was designed for?”. Therefore, this section presents the verification of the proposed control system and, subsequently, Section 10

Validation of the control system

Continuing the analysis started in the previous section, we now present the validation of the proposed control system. For this, we have performed simulations to analyze the impacts on the process performance indicators.

The simulations consist of performance comparisons between the Current Strategy, the Maximization Algorithm, and the Scheduling Algorithm (see definitions in Section 6). The simulation time was set to 720 h (30 days). Regarding the prediction horizon, we have set Np=3 since the

Conclusions and future work

In this paper, the problem of scheduling tank trucks at a fuel distribution terminal was analyzed, proposing the adoption of a technique to eliminate the excess of vehicles inside the loading yard. As this accumulation of TTs hinders movement in the yard, besides representing a safety risk because of the fuel handling operations, we have conducted the present study to improve the terminal efficiency and safety. The case study plant was modeled in the max-plus algebra, using machine learning to

CRediT authorship contribution statement

Marcos Vinícios Gonçalves: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Project administration. Antonio Eduardo Carrilho da Cunha: Conceptualization, Methodology, Validation, Resources, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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