PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA ELÉTRICA E INFORMÁTICA INDUSTRIAL
INSTITUTO SUPERIOR TÉCNICO DA UNIVERSIDADE DE LISBOA PROGRAMA DOUTORAL EM ENGENHARIA E GESTÃO
WILLIAM HITOSHI TSUNODA MEIRA
SCHEDULING OF STRAIGHT MULTIPRODUCT PIPELINE
SYSTEMS WITH MULTIPLE-SOURCES AND
MULTIPLE-DESTINATIONS
TESE
CURITIBA 2020
SCHEDULING OF STRAIGHT MULTIPRODUCT PIPELINE
SYSTEMS WITH MULTIPLE-SOURCES AND
MULTIPLE-DESTINATIONS
Tese apresentada ao Programa de Pós-Graduação em Engenharia Elétrica e Infor-mática Industrial (CPGEI) da Universidade Tecnológica Federal do Paraná (UTFPR) como requisito à obtenção do título de “Doutor em Ciências”. Área de Concentração: Engenharia de Computação.
Orientador: Prof. Dr. Leandro Magatão
Tese apresentada à Commissão de Acom-panhamento de Tese (CAT) do Programa Doutoral em Engenharia e Gestão do Instituto Superior Técnico da Universidade de Lisboa (IST-UL) como requisito à obtenção de grau de doutor no ramo de conhecimento em Engen-haria e Gestão.
Orientadora: Prof. Dra. Ana Paula Ferreira Dias Barbosa Póvoa
CURITIBA 2020
Meira, William Hitoshi Tsunoda
Scheduling of straight multiproduct pipeline systems with multiple-sources and multiple-destinations [recurso eletrônico] / William Hitoshi Tsunoda Meira.-- 2020.
1 arquivo texto (210 f.): PDF; 34,5 MB. Modo de acesso: World Wide Web
Título extraído da tela de título (visualizado em 2 abr. 2020) Texto em inglês, com resumo em português
Tese (Doutorado) - Universidade Tecnológica Federal do Paraná. Programa de Pós-Graduação em Engenharia Elétrica e Informática Industrial, Curitiba, 2020
Bibliografia: p. 187-193.
1. Engenharia elétrica - Teses. 2. Pesquisa operacional. 3. Programação linear. 4. Programação inteira. 5. Agenda de execução (Administração). 6. Método de de-composição. 7. Heurística. 8. Petróleo - Refinarias - Tubulações. 9. Modelos ma-temáticos. I. Magatão, Leandro. II. Póvoa, Ana Paula Ferreira Dias Barbosa. III. Universidade Tecnológica Federal do Paraná - Programa de Pós-graduação em En-genharia Elétrica e Informática Industrial. IV. Título.
CDD: Ed. 23 – 621.3 Biblioteca Central da UTFPR, Câmpus Curitiba
Universidade Tecnológica Federal do Paraná Diretoria de Pesquisa e Pós-Graduação
TERMO DE APROVAÇÃO DE TESE
A Tese de Doutorado intitulada SCHEDULING OF STRAIGHT MULTIPRODUCT
PIPELINE SYSTEMS WITH MULTIPLE-SOURCES AND MULTIPLE-DESTINATIONS,
defendida em sessão pública pelo(a) candidato(a) William Hitoshi Tsunoda Meira, no dia 25 de março de 2020, foi julgada aprovada em sua forma final para obtenção do título de
Doutor em Ciências, Área de Concentração – Engenharia de Computação, Linha de Pesquisa – Sistemas Inteligentes, pelo Programa de Pós-Graduação em Engenharia Elétrica e Informática Industrial – CPGEI. A referida Tese foi realizada em cotutela internacional, de acordo com convênio firmado em outubro de 2017 entre a UTFPR e a Universidade de Lisboa, sendo também lavrada ata específica da Universidade de Lisboa.
BANCA EXAMINADORA:
Prof. Dr. Lucia Valeria Ramos de Arruda – (UTFPR) – Presidente Prof. Dr. José Fernando da Costa Oliveira – (Universidade do Porto)
Prof. Dr. Ana Paula Ferreira Dias Barbosa Póvoa – (Universidade de Lisboa) Prof. Dr. Eduardo Camponogara – (UFSC)
Prof. Dr. Ricardo Lüders – (UTFPR)
A via original deste documento encontra-se arquivada na Secretaria do Programa, contendo a assinatura da Coordenação após a entrega da versão corrigida do trabalho.
Curitiba, 25 de março de 2020.
_______________________________________ Carimbo e Assinatura do(a) Coordenador(a) do Programa
I am grateful to all the incredible people that I had the pleasure of sharing inspiring moments, knowledge, conversations, and laughs over these 4 years’ adventure that, undoubtedly, made the path to the conclusion of this thesis possible and enjoyable.
First and foremost, I am extremely grateful to my supervisor, prof. Dr. Leandro Magatão (UTFPR). I am not able to express with enough words how much I appreciate everything I have learned from your experience, knowledge, and professionalism, which is an inspiration for me and for everyone around you. I am certain that I couldn’t have chosen a better person to guide me along my academic path. Once more, I thank you for everything.
Special thanks to my supervisor from IST, prof. Dra. Ana Paula Barbosa-Póvoa. I am profoundly grateful and honored for accepting me as your PhD student and for guiding me over these years. Also, I am thankful for welcoming me with such warmth and hospitality during my period at IST. Your experience, interest, and dedication to your students are extremely remarkable. Your guidance was invaluable to the conclusion of this thesis. Thank you for everything.
To my PhD committee, prof. Dr. José Fernando Oliveira (Universidade do Porto), prof. Dr. Eduardo Camponogara (UFSC), prof. Dr. Ricardo Lüders (UTFPR), and prof. Dra. Valéria Ar-ruda (UTFPR), I would like to thank you for all the relevant suggestions and comments on my research.
My sincere thanks to prof. Dr. Flávio Neves for the opportunity of being part of the LASCA laboratory and all the support over these years. To my co-authors, prof. Dra. Valéria Arruda and prof. Dra. Susana Relvas, for the interest and attention on the reviews of the produced articles from this research.
I would like to acknowledge the financial support from the Brazilian Oil Company PETROBRAS, CAPES, and Erasmus Mundus SMART2for sponsoring this research.
To all the researchers and colleagues from the CPGEI/UTFPR and LASCA laboratory, thank you for being part of this journey. Special thanks to my co-workers from the SConsuelo project: Bettoni, Bueno, Débora, Geraldo, Matheus, and all the previous members with whom I had the opportunity to work with.
Part of my research was developed at the Instituto Superior Técnico of Universidade de Lis-boa. For this, I am glad to the incredible memories and experiences shared with my colleagues and professors from the Department of Engineering and Management. Thank you to my office mates: Felipe, Jochem, and Diego for the interesting conversations while sharing the N8.22 at the end of the corridor. To the memorable talks and laughs at lunchtime with my dear colleagues
rique, Raul, for the events, beers, and adventures around Lisbon.
To all my friends, thank you for the support and company throughout these years. Special thanks to my closest friends, Conan and Johnny, who are like brothers to me. The social events would not be the same without my friends Bill, Fontoura, Edmar, Fraga, Leandro, Longen, Murilo, Tarzan.
I would like to thank my father, grandmother, sisters, and my entire family for the support, comprehension, and unconditional love along these years.
Gostaria de agradecer ao meu pai, vó, irmãs e toda minha família pelo suporte, compreensão e amor incondicional ao longo desses anos.
My sincere thanks and love to Marcelly. You are immensely special in my life, and I lack the words to describe how much I love being with you. Your presence always makes me feel better. Thank you for your constant support, comprehension, and love.
William Hitoshi Tsunoda Meira Curitiba, March 2020
MEIRA, W. H. T.. Scheduling of Straight Multiproduct Pipeline Systems with Multiple-Sources and Multiple-Destinations. 2020. 200 f. Tese (Duplo doutorado em Engenharia Elétrica e Informática Industrial e Engenharia e Gestão) – Universidade Tecnológica Federal do Paraná e Instituto Superior Técnico da Universidade de Lisboa. Curitiba, 2020.
O interesse no estudo de técnicas de otimização com o objetivo de auxiliar especialistas na tomada de melhores decisões tem recebido significante atenção nos últimos anos, pois qualquer aprimoramento no uso dos recursos disponíveis pode aumentar consideravelmente a margem de lucro das companhias, como no caso do transporte de derivados de petróleo por redes de dutos. Na cadeia de abastecimento do petróleo, o transporte de produtos por dutos é um dos mais adotados modais de distribuição, justificado pela sua alta capacidade volumétrica, confi-abilidade e segurança comparado com outros meios de transporte. Neste contexto, o objetivo desta tese é desenvolver uma ferramenta baseada em métodos de otimização para o scheduling de operações de transporte e o gerenciamento de inventário em sistemas de único duto com múltiplas origens e múltiplos destinos. Primeiramente é proposta uma estrutura de solução que integra heurísticas e modelos de Programação Linear Inteira Mista (PLIM), uma matheuristic, para resolver o scheduling de longo prazo dos sistemas de polidutos com apenas uma origem. Em seguida, a estrutura de solução é ampliada para resolver sistemas de polidutos com múlti-plas origens e múltiplos destinos. Essa estrutura é decomposta em dois módulos: Módulo de Alocação e Sequenciamento (ASM) e Módulo de Programação (SM). O ASM é responsável por agendar as operações de bombeamento da origem inicial, o que envolve determinar o pro-duto, volume, vazão e sequência de bateladas a serem bombeados durante o horizonte de tempo. O SM programa as operações de entrega em cada nó e as operações de bombeamento das ori-gens intermediárias durante o horizonte. Diversos aspectos operacionais da rede de polidutos estudada foram levados em consideração, como operações de entrega e bombeamentos simultâ-neos, tratamento rigoroso de sequências proibidas, períodos de manutenção do duto, períodos de manutenção dos tanques, controle da vazão do bombeamento, gerenciamento rigoroso do inventário. O desenvolvimento desta tese é um projeto colaborativo com a empresa petrolífera brasileira (Petrobras) que permitiu a validação dos resultados e do desempenho da estrutura de solução desenvolvida usando dados pautados em valores reais. Os resultados obtidos apre-sentam soluções viáveis com gerenciamento adequado de inventário que, certamente, podem auxiliar os especialistas do sistema em seu processo de tomada de decisão.
Palavras-chave: Poliduto. Scheduling. Abordagem por decomposição. Programação linear in-teira mista. Heurística.
MEIRA, W. H. T.. Scheduling of Straight Multiproduct Pipeline Systems with
Multiple-Sources and Multiple-Destinations. 2020. 200 p. Thesis (Double PhD in Electrical and Computer Engineering and Engineering and Management) – Universidade Tecnológica Federal do Paraná and Instituto Superior Técnico of Universidade de Lisboa. Curitiba, 2020. The interest on the optimization techniques with the objective to aid specialists on taking better decisions has received significant attention in the last years, since any improvement on the us-age of the available resources can considerably increase the profit of companies, as in the case of transport of oil derivatives through pipeline systems. In the oil supply chain, transportation of products through pipelines is one of the most used distribution modals, justified by its high volumetric capacity, reliability, and safety when compared to other transportation modes. In this context, the objective of this thesis is to develop an optimization-based decision supporting tool for the scheduling of transportation operations and the inventory management of straight multiproduct pipeline systems with multiple-sources and multiple-destinations. Firstly, a solu-tion framework was proposed; it integrates heuristics and Mixed Integer Linear Programming (MILP) models, a matheuristic, to solve the long-term scheduling of the straight pipeline sys-tems with just a single-source. Then, the developed solution framework is extended to solve straight pipeline systems with multiple-sources and multiple-destinations. This framework is divided into two modules: Allocation and Sequencing Module (ASM) and Scheduling Module (SM). The ASM is responsible for scheduling the pumping operations at the initial source that involves determining the product, volume, flow rate, and sequence of batches to be pumped during the time horizon. The SM schedules the delivery operations at each node and the pump-ing operations of the intermediate sources to be executed durpump-ing the considered horizon. Sev-eral operational aspects of the studied pipeline network have been taken into account, such as simultaneous deliveries and pumping operations, rigorous treatment of forbidden sequences, scheduling of pipeline interruptions, pipeline maintenance periods, tank maintenance periods, pumping flow rate control, rigorous inventory management. The development of this thesis is a collaborative project with the Brazilian oil company (Petrobras) that allowed the validation of the results and the performance of the developed solution framework using real-world data. The obtained results present viable solutions with proper inventory management that can, certainly, aid the system’s specialists in their decision-making process.
Keywords: Multiproduct pipeline. Scheduling. Decomposition approach. Mixed integer linear programming. Heuristic.
Algorithm 1 – Average segments flow rate heuristic . . . 50
Algorithm 2 – Mean flow rate per segment . . . 93
Algorithm 3 – Propagation of the initialization batches . . . 96
Algorithm 4 – Base heuristic: delivery strategy . . . 97
Algorithm 5 – Simulated Annealing Metaheuristic . . . 98
Algorithm 6 – Simulated Annealing: fitness function . . . 99
Algorithm 7 – Simulated Annealing: delivery strategy . . . 100
Figure 1 – Oil supply chain . . . 25
Figure 2 – Classification of pipeline topologies . . . 26
Figure 3 – Example of a straight pipeline network with a single source and multiple destinations . . . 30
Figure 4 – Example of a straight pipeline network with multiple sources and multiple destinations . . . 31
Figure 5 – Example of the inventory levels (physical, operational, and target ranges) as a percentage of the physical maximum capacity level . . . 32
Figure 6 – Example of the inventory levels with a tank maintenance period as a per-centage of the physical maximum capacity level . . . 33
Figure 7 – Example of a constant and weekday demand rate profile for a given pair node-product . . . 33
Figure 8 – Brazilian straight pipeline network . . . 34
Figure 9 – Examples of delivery operations types . . . 38
Figure 10 – Example of simultaneous delivery operations . . . 38
Figure 11 – Flow cases of two consecutive segments in a straight pipeline network . . . 39
Figure 12 – Example of Gantt chart of a pipeline scheduling solution . . . 40
Figure 13 – Example of inventory management for all products of the node . . . 41
Figure 14 – Typical inventory levels . . . 48
Figure 15 – Macro flowchart of the developed framework . . . 48
Figure 16 – Flowchart of the Allocation and Sequencing Module . . . 49
Figure 17 – ASM perspective of the network . . . 51
Figure 18 – Example of forward demand shifting heuristic . . . 51
Figure 19 – Example of the heuristic for an initialization batch delivery . . . 52
Figure 20 – Example of a rolling horizon execution . . . 53
Figure 21 – Flowchart of the Scheduling Module . . . 60
Figure 22 – Example of aggregate-DC . . . 61
Figure 23 – Cases to calculate the size of a batch at an event . . . 63
Figure 24 – Example of the propagation algorithm . . . 70
Figure 25 – CS1 - Illustrative pipeline network with 3 DCs . . . 70
Figure 26 – CS2 - Real-world pipeline network . . . 74
Figure 27 – CS2 - Gantt chart of the obtained solution for the base instance (example 1) 76 Figure 28 – CS2 - Inventory profile of product P1 in all DCs (example 1) . . . 77
Figure 29 – CS2 - Pumping flow rate profile for the base instance (example 1) . . . 78
Figure 30 – CS2 - Gantt chart of the obtained solution (example 2) . . . 79
Figure 31 – CS2 - Pumping flow rate profile for the extended instance (example 2) . . . 80
Figure 32 – CS2 - Inventory profile of product P1 in all DCs (example 2) . . . 81
Figure 33 – CS2 - Inventory profile comparison for 8-3 ℎ of uniform interval size in ASM (example 3) . . . 82
Figure 34 – Macro flowchart of the decomposition approach (ASM highlight) . . . 91
Figure 35 – Flowchart of the allocation and sequencing module. The highlighted blocks in red are improved in the proposed new solution . . . 91
Figure 36 – Example of violation curve for the inventory violation . . . 100
Figure 39 – Rolling horizon period division for the initial source . . . 103
Figure 40 – Inventory profile comparison between 𝑃 𝐼𝑜𝑓 𝑓 and 𝑃 𝐼𝑜𝑛 of product P2 in DC1-DC5 for the D40 instance . . . 118
Figure 41 – CS2 - Gantt chart with the base heuristic (BH) . . . 120
Figure 42 – CS2 - Gantt chart with the simulated annealing (SA) metaheuristic . . . 121
Figure 43 – Mean population fitness of the SA with different population size (𝑃 𝑆) for the CS1 (D100) instance . . . 122
Figure 44 – Mean population fitness of the SA with different population size (𝑃 𝑆) for the CS2 instance . . . 123
Figure 45 – Best and worst individual of the SA execution with 𝑃 𝑆 = 100 . . . 123
Figure 46 – Boxplots over 1,000 runs of CS1 (D100) instance with 𝑃 𝑆 = 20 varying the cooling rate 𝛼 . . . 124
Figure 47 – Mean population fitness of the SA with 𝑃 𝑆 = 20 varying the cooling rate 𝛼 parameter for the CS1 (D100) instance . . . 124
Figure 48 – Mean population fitness of the SA with 𝑃 𝑆 = 20 varying the cooling rate 𝛼 parameter for the CS2 instance . . . 124
Figure 49 – Example of the inventory levels (physical, operational, and target ranges) as a percentage of the physical maximum capacity level . . . 133
Figure 50 – Example of a typical constant and weekday demand rate for a given pair node-product . . . 135
Figure 51 – Macro-flowchart of the proposed solution framework . . . 137
Figure 52 – Iterations of the rolling horizon process . . . 138
Figure 53 – Flowchart of the Scheduling Module . . . 139
Figure 54 – Example of aggregate-node for the second segment (𝑠2) iteration . . . 140
Figure 55 – Example of batch numbering with the proposed strategy . . . 142
Figure 56 – Straight pipeline flow cases . . . 143
Figure 57 – CS1 - Illustrative pipeline network and the initialization batches . . . 161
Figure 58 – CS1 - Inventory profile comparison between 𝐷𝑅 and 𝑅𝐻 run of product P1 at the nodes N1-N4 . . . 164
Figure 59 – CS1 - Inventory profile comparison between 𝐷𝑅 and 𝑅𝐻 run of product P2-P3 at the node N4 . . . 165
Figure 60 – CS2 - Pipeline network from Mostafaei et al. (2016) . . . 165
Figure 61 – CS2 - Inventory profile of all products at nodes N1-N4 . . . 169
Figure 62 – CS3 - Brazilian pipeline network and the initialization batches . . . 169
Figure 63 – CS3 - Demand rate during the 30-day horizon for the product P1 at node N2 170 Figure 64 – CS3 - Inventory profile of all products at nodes N1-N6 . . . 171 Figure 65 – CS3 - Inventory profile of product P1 at node N4 with tank maintenance period172
Table 1 – Cost factors adopted for the case studies 1 and 2 . . . 71
Table 2 – CS1 - Initialization batches . . . 71
Table 3 – CS1 - Minimum and maximum batch size . . . 72
Table 4 – CS1 - Product incompatibility matrix . . . 72
Table 5 – CS1 - DC-product demand for 21 days . . . 72
Table 6 – CS1 - DCs characteristics . . . 72
Table 7 – CS1 - CPU time in seconds for each of the 24 different instances . . . 73
Table 8 – CS2 - Initialization batches . . . 74
Table 9 – CS2 - Minimum and maximum batch size . . . 75
Table 10 – CS2 - Product incompatibility matrix . . . 75
Table 11 – CS2 - DC-product demand for 30 days . . . 75
Table 12 – CS2 - DCs characteristics . . . 75
Table 13 – CS2 - MILP models statistics for the base instance run (example 1) . . . 76
Table 14 – CS2 - MILP models statistics for the extended instance run (example 2) . . . 79
Table 15 – CS2 - CPU time in seconds of each model execution for 8-3 hours of interval size (example 3) . . . 82
Table 16 – CS1 - ASM MILP statistics for D100, D50, and D40 instances for the 𝑃 𝐼𝑜𝑓 𝑓 and 𝑃 𝐼𝑜𝑛runs . . . 115
Table 17 – CS1 - CPU time in seconds of D100, D50, and D40 instances for the 𝑃 𝐼𝑜𝑓 𝑓 and 𝑃 𝐼𝑜𝑛runs . . . 116
Table 18 – CS1 - Objective function value of D100, D50, and D40 instances for the 𝑃 𝐼𝑜𝑓 𝑓 and 𝑃 𝐼𝑜𝑛runs . . . 116
Table 19 – CS1 - Statistics for D100, D50, and D40 instances for the 𝑃 𝐼𝑜𝑓 𝑓 and 𝑃 𝐼𝑜𝑛runs 117 Table 20 – CS2 - Initialization batches . . . 118
Table 21 – CS2 - Batch constraints: product incompatibility matrix and batch volume limits119 Table 22 – CS2 - DCs characteristics . . . 119
Table 23 – CS2 - DC-product demand for 30 days . . . 119
Table 24 – CS2 - Statistics for CS2 solution with BH and SA strategy . . . 121
Table 25 – CS1 - Node-product characteristics: inventory information, total demand and production forecasts (15-day horizon) . . . 162
Table 26 – CS1 - SM model statistics for the DR and RH execution procedures . . . 163
Table 27 – CS1 - Scheduled batches to be pumped by source N1 in the DR and RH solutions . . . 163
Table 28 – CS2 - Initial volume (𝑚3) and inventory maximum capacity (𝑚3) for each pair of node-product . . . 166
Table 29 – CS2 - SM model statistics for the rolling horizon execution . . . 167
Table 30 – CS2 - Scheduled batches to be pumped by sources N1 and N3 . . . 168
ACO Ant Colony Optimization
ASM Allocation and Sequencing Module DC distribution center
GUI Graphical User Interface IST Instituto Superior Técnico
LP Linear Programming
MILP Mixed Integer Linear Programming MINLP Mixed Integer Non-Linear Programming MIP Mixed Integer Programming
PIG Pipeline Injection Gauge RTN Resource-Task Network
SA Simulated Annealing
SM Scheduling Module
UL Universidade de Lisboa
The list of symbols contains the entire nomenclature presented in this thesis. When the nomenclature is limited to a specific chapter or model, the reference is mentioned between parenthesis. As a notation pattern, all parameters and sets start with an uppercase letter, and all continuous and binary variables start with a lowercase letter.
INDICES AND SETS
𝑏,𝑏′ ∈ 𝐵 Set of batches indexes. The first index is the closest batch to the last node, increasing a unit for each following initialization batch and then the allocated batches as each one enters the network
𝑏,𝑏′ ∈ 𝐵𝑡⊂ 𝐵 Set of in-transit batches. The in-transit batches are already inside the pipeline
segment or reaching the upstream node of the SM iteration (used in Chap-ter 5)
𝑏,𝑏′ ∈ 𝐵𝑖𝑛𝑖𝑡⊂ 𝐵𝑡 Set of only initialization batches indexes. The first element is the closest
batch to the last node, increasing a unit for each following batch. This set is a subset of all batches (𝐵𝑖𝑛𝑖𝑡 ⊂ 𝐵)
𝑏,𝑏′ ∈ 𝐵𝑛⊂ 𝐵 Set of new pumped batches, where the 𝑏 = |𝐵𝑡| + 1 is the first element of
𝐵𝑛
{𝑏,𝑏′} ∈ 𝐵𝐵 Sparse set containing tuple {𝑏,𝑏′}, which associates all the possible
prece-dence order between two batches 𝑏 ∈ 𝐵 and 𝑏′ ∈ 𝐵 {𝑏,𝑏′,𝑒} ∈ 𝐵𝐵𝐸
Sparse set containing tuple {𝑏,𝑏′,𝑒}, which associates all possible precedence orders between two batches {𝑏,𝑏′} ∈ 𝐵𝐵 and each event 𝑒 ∈ 𝐸
{𝑏,𝑒} ∈ 𝐵𝐸 Set containing the tuple {𝑏,𝑒}, which combines each batch index 𝑏 with an event 𝑒
{𝑏,𝑒,𝑛} ∈ 𝐵𝐸𝑁 Set containing the tuple {𝑏,𝑒,𝑛}, which associates each batch index 𝑏 with an event 𝑒 and a node 𝑛
{𝑏,𝑖} ∈ 𝐵𝐼 Set containing the tuple {𝑏,𝑖}, which associates each batch index 𝑏 with an interval 𝑖
{𝑏,𝑖,𝑛} ∈ 𝐵𝐼𝑁 Set containing the tuple {𝑏,𝑖,𝑛}, which associates each batch index 𝑏 with an interval 𝑖 and a node 𝑛
𝑒,𝑒′ ∈ 𝐸 Set of events
𝑓 ∈ 𝐹 Set of inventory levels, where 𝐹 = {𝑐𝑎𝑝, 𝑒𝑚𝑝, 𝑚𝑎𝑥, 𝑚𝑖𝑛, 𝑚𝑎𝑥𝑡𝑔, 𝑚𝑖𝑛𝑡𝑔} 𝑖,𝑖′,𝑖′′ ∈ 𝐼 Set of intervals. The interval 𝑖 starts at 𝑒 = 𝑖 − 1 and ends at 𝑒 = 𝑖. The
elements are indexed from 1 to |𝐸| 𝑖 ∈ 𝐼𝑘
𝑛 Set of intervals in the interation 𝑘 associated with node 𝑛
ization batches intervals, where 𝑖′ 6 𝑖 and 𝑖′ 6 (max{𝑛,𝑏,𝑖′′}∈𝑁 𝐵𝐼𝑖′′) (higher interval index of 𝑁 𝐵𝐼
𝑘 ∈ 𝐾 Set of iterations of the rolling horizon process 𝑛 ∈ 𝑁 Set of nodes (e.g., refinery, distribution centers)
{𝑛,𝑏,𝑖} ∈ 𝑁 𝐵𝐼 Sparse set containing the tuple {𝑛,𝑏,𝑖}, which associates each initialization batch 𝑏 ∈ 𝐵𝑖𝑛𝑖𝑡with interval 𝑖 and node 𝑛
{𝑛,𝑖,𝑖′} ∈ 𝑁 𝐼𝐼 Sparse set of intervals 𝑖′ (𝑖′ ∈ 𝐼
𝑛) related to the pumping interval 𝑖 (𝑖 ∈ 𝐼0)
from the initial source to node 𝑛 > 0. The set considers the mean transit time to 𝑛 (𝑇𝑛) and estimates the intervals when the pumped volume will be
passing by the node 𝑛
{𝑛,𝑝} ∈ 𝑁 𝑃 Sparse set containing the tuple {𝑛,𝑝}; node 𝑛 where product 𝑝 can be deliv-ered. Not necessarily all products can be delivered to all nodes
{𝑛,𝑝,𝑒} ∈ 𝑁 𝑃 𝐸 Sparse set containing the tuple {𝑛,𝑝,𝑒}, which associates every element of {𝑛,𝑝} ∈ 𝑁 𝑃 to every 𝑒 ∈ 𝐸
{𝑛,𝑝,𝑖} ∈ 𝑁 𝑃 𝐼 Sparse set containing the tuple {𝑛,𝑝,𝑖}, which associates every element of {𝑛,𝑝} ∈ 𝑁 𝑃 to every 𝑖 ∈ 𝐼 or 𝑖 ∈ 𝐼𝑛(ASM model from Chapter 4)
𝑝,𝑝′ ∈ 𝑃 Set of products (oil derivatives)
𝑝 ∈ 𝑃𝑝𝑢𝑚𝑝 Set of products that can be pumped by the current intermediate source
𝑠 ∈ 𝑆 Set of pipeline segments (𝑆 = 𝑁 −{0}). Segment 𝑠 connects node 𝑛 = 𝑠−1 to 𝑛 = 𝑠
PARAMETERS
𝛼 Multiplier of the geometric cooling rate for the simulated annealing heuristic 𝜀 Small constant value used to avoid equalities (e.g. 𝜖 = 10−4)
𝐿 ≪ 0 Lower bound value (e.g. 𝐿 = −𝑈 ) 𝑈 ≫ 0 Upper bound value (e.g. 𝑈 = 106)
𝑈𝑓 𝑟 Upper bound for the flow rate (𝑚3/ℎ). The highest value among all 𝐹 𝑠𝑚𝑎𝑥𝑝 , 𝐹 𝑝𝑚𝑎𝑥
𝑝 , 𝐹 𝑑𝑚𝑎𝑥𝑝 , and maximum in-transit flow rate (i.e., 𝑚𝑎𝑥𝑖∈𝐼(𝑉𝑖𝑡/𝑇𝑖))
(𝑚3/ℎ)
𝑈𝑝𝑢𝑚𝑝 Upper bound for the active pumping period of the initial station, disconsiders
the pipeline maintenance periods (ℎ)
𝑈𝑣 Upper bound for the volume equals to the highest batch size between the in-transit batches and maximum new batch size 𝐿𝑏𝑚𝑎𝑥
𝑝 (𝑚3)
𝐵𝑙𝑎𝑠𝑡
𝑒 Indicates the index of the closest batch 𝑏 to the upstream node at the event 𝑒,
which also means the last batch pumped into the segment
𝐵𝑖𝑝𝑎𝑠𝑠 Indicates the index of batch 𝑏 passing along the upstream node during inter-val 𝑖
𝐵𝑡
𝐶𝑓 𝑟𝑚𝑖𝑛 Penalty cost per unit of the minimum flow rate violation at an event
𝐶𝑓 𝑟𝑙𝑑𝑖𝑓 𝑓 Penalty cost per unit of the lower difference in flow rate compared to the last
interval
𝐶𝑓 𝑟𝑢𝑑𝑖𝑓 𝑓 Penalty cost per unit of the upper difference in flow rate compared to the last
interval
𝐶ℎ𝑠 Penalty cost per hour of pumping interruption 𝐶𝑖𝑑𝑐𝑎𝑝 Penalty cost per unit of inventory capacity violation 𝐶𝑖𝑑𝑒𝑚𝑝 Penalty cost per unit of inventory empty violation
𝐶𝑖𝑑𝑚𝑎𝑥 Penalty cost per unit of inventory maximum operational level violation 𝐶𝑖𝑑𝑚𝑖𝑛 Penalty cost per unit of inventory minimum operational level violation
𝐶𝑖𝑑𝑚𝑎𝑥𝑡𝑔 Penalty cost per unit of inventory maximum target level violation
𝐶𝑖𝑑𝑚𝑖𝑛𝑡𝑔 Penalty cost per unit of inventory minimum target level violation
𝐶𝑖𝑑𝑓 Penalty cost of violating the inventory level 𝑓 . A simplified way of
express-ing 𝐶𝑖𝑑𝑐𝑎𝑝, 𝐶𝑖𝑑𝑒𝑚𝑝, 𝐶𝑖𝑑𝑚𝑎𝑥, 𝐶𝑖𝑑𝑚𝑖𝑛, 𝐶𝑖𝑑𝑚𝑎𝑥𝑡𝑔, 𝐶𝑖𝑑𝑚𝑖𝑛𝑡𝑔
𝐶𝑖𝑛𝑡 Penalty cost associated with the changeover cost per product pumping change and the contamination volume cost generated in the interface be-tween two batches of different products
𝐶𝑚𝑓 𝑟 Penalty cost per unit of mean flow rate violation
𝐶𝑚𝑓 𝑟𝑙𝑜𝑤𝑒𝑟 Penalty cost per unit of the lower violation of the mean flow rate
𝐶𝑚𝑓 𝑟𝑢𝑝𝑝𝑒𝑟 Penalty cost per unit of the upper violation of the mean flow rate 𝐶𝑝𝑠 Penalty cost of an hour of pipeline stoppage
𝐶𝑝𝑢𝑚𝑝 Penalty cost of an hour of active pumping operation 𝐶𝑠 Penalty cost per pumping interruption
𝐷𝐵 1 if the first in-transit batch (𝐵1𝑡) has been delivered to the upstream node at some interval at the previous rolling horizon iteration; 0 otherwise
𝐷𝑒𝑚𝑛,𝑝,𝑖 Demand of product 𝑝 in node 𝑛 during the interval 𝑖 (𝑚3)
𝐸𝑛𝑘 End of iteration 𝑘 in node 𝑛 (ℎ)
𝐹 𝑑𝑚𝑎𝑥𝑛,𝑝 Maximum delivery flow rate of product 𝑝 at the node 𝑛 (𝑚3/ℎ) 𝐹 𝑑𝑚𝑖𝑛𝑛,𝑝 Minimum delivery flow rate of product 𝑝 at the node 𝑛 (𝑚3/ℎ 𝐹 𝑝𝑚𝑎𝑥
𝑝 Maximum pumping flow rate of product 𝑝 for the upstream source of the SM
iteration (𝑚3/ℎ)
𝐹 𝑝𝑚𝑎𝑥
𝑝,𝑖 Maximum pumping flow rate of product 𝑝 during the interval 𝑖 (𝑚3/ℎ)
𝐹 𝑝𝑚𝑖𝑛
𝑝 Minimum pumping flow rate of product 𝑝 for the upstream source of the SM
iteration (𝑚3/ℎ) 𝐹 𝑝𝑚𝑖𝑛
𝑝,𝑖 Minimum pumping flow rate of product 𝑝 during the interval 𝑖 (ASM model)
(𝑚3/ℎ)
𝐹𝑖𝑝𝑎𝑠𝑠 Flow rate of the batch part 𝐵𝑖𝑝𝑎𝑠𝑠 passing along the upstream node during interval 𝑖 (𝑚3/ℎ)
the SM iteration (𝑚3/ℎ)
𝐹𝑠𝑚𝑒𝑎𝑛 Estimated mean flow rate per pipeline segment 𝑠 (𝑚3/ℎ) 𝐹 𝑠𝑚𝑖𝑛
𝑖 Minimum passage flow rate during the interval 𝑖 for the pipeline segment of
the SM iteration (𝑚3/ℎ)
𝐻𝑃 𝑜𝑠𝑏 Volumetric coordinate of the head of the initialization batch 𝑏 at the
begin-ning of the horizon (𝑚3)
𝐼𝐷𝑛,𝑝 Initial inventory volume of product 𝑝 in node 𝑛 (𝑚3)
𝐼𝐷𝑛,𝑝,𝑒𝑐𝑎𝑝 Volume capacity of the aggregate inventory storage of product 𝑝 in node 𝑛 at
the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝐷𝑒𝑚𝑝
𝑛,𝑝,𝑒 Volume considered empty for the aggregate inventory storage of product 𝑝
in node 𝑛 at the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝐷𝑛,𝑝,𝑖𝑓 Volume of inventory level 𝑓 of product 𝑝 in node 𝑛 during interval 𝑖 (𝑚3)
𝐼𝐷𝑚𝑎𝑥
𝑛,𝑝,𝑒 Maximum operational level for the aggregate inventory storage of product 𝑝
in node 𝑛 at the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝐷𝑚𝑎𝑥𝑡𝑔
𝑛,𝑝,𝑒 Maximum target level for the aggregate inventory storage of product 𝑝 in
node 𝑛 at the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝐷𝑛,𝑝,𝑒𝑚𝑖𝑛 Minimum operational level for the aggregate inventory storage of product 𝑝
in node 𝑛 at the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝐷𝑚𝑖𝑛𝑡𝑔
𝑛,𝑝,𝑒 Minimum target level for the aggregate inventory storage of product 𝑝 in
node 𝑛 at the event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝐼𝑚𝑎𝑥 Maximum duration of an interval (ℎ) in the SM model
𝐼𝑁𝑏𝑒𝑓 𝑜𝑟𝑒 1 if the first in-transit batch (𝑏 = 𝐵𝑡
1) has already entered in the pipeline
segment in the previous rolling horizon iteration and it is not the last initial-ization batch; 0 otherwise
𝐼𝑃𝑝 Binary parameter indicating if product 𝑝 is being pumped from the initial
source (refinery) at the initial event (𝑒 = 0)
𝐼𝑃 𝐻 Hours of continuous pumping operation until the beginning of the horizon. If 𝐼𝑆 = 0, then 𝐼𝑃 𝐻 > 0; 0 otherwise
𝐼𝑆 Binary parameter indicating if the pumping operation is interrupted (stopped) at the beginning of the horizon. The value is 1 if the pipeline is interrupted; 0 otherwise
𝐼𝑆𝐻 Hours of continuous pumping interruption until the beginning of the horizon. If 𝐼𝑆 = 1, then 𝐼𝑆𝐻 > 0; 0 otherwise
𝐼𝑉𝑝 Initial volume of product 𝑝 being pumped from the initial source at the
begin-ning of the horizon. It is greater than 0 only for the product 𝑝 where 𝐼𝑃𝑝 = 1
previous rolling horizon iteration; 0 otherwise 𝐿𝑏𝑚𝑖𝑛𝑝 Minimum batch pumping size (𝑚3)
𝐿𝑏𝑚𝑎𝑥
𝑝 Maximum batch pumping size (𝑚3)
𝐿𝑑𝑚𝑖𝑛
𝑝 Minimum delivery size of product 𝑝 (𝑚3)
𝑁 𝐵𝑚𝑎𝑥 Maximum number of new batches allocated to be pumped by an intermediate source
𝑃𝑏 Product 𝑝 of the batch 𝑏
𝑃 𝐻𝑚𝑎𝑥 Maximum length of continuous active pumping operation (ℎ) 𝑃 𝐻𝑚𝑖𝑛 Minimum length of continuous active pumping operation (ℎ)
𝑃 𝑀𝑖 Binary parameter indicating if there is a pipeline maintenance period during
interval 𝑖
𝑃 𝑃𝑝,𝑝′ Matrix of forbidden sequences between two consecutive products 𝑝 and 𝑝′. The value is 1 if the sequence is forbidden; 0 otherwise
𝑃 𝑟𝑜𝑑𝑛,𝑝,𝑖 Production of product 𝑝 in node 𝑛 during interval 𝑖 (𝑚3)
𝑃 𝑆𝑚𝑎𝑥 Maximum number of pumping interruptions 𝑃𝑡
𝑏 Product of the in-transit batch 𝑏
𝑅𝑛,𝑖,𝑖′ Ratio (proportion) of the pumped volume during interval 𝑖 passing by node
𝑛 during interval 𝑖′, where {𝑛,𝑖,𝑖′} ∈ 𝑁 𝐼𝐼 (0..1)
𝑅𝑒𝑐𝑛,𝑝,𝑖 Volume of product 𝑝 being received in node 𝑛 during the interval 𝑖. The
val-ues are obtained during the initialization batches simplification in the ASM and also in the SM (𝑚3)
𝑆𝐻𝑚𝑎𝑥 Maximum length of continuous pumping interruption (ℎ)
𝑆𝐻𝑚𝑖𝑛 Minimum length of continuous pumping interruption (ℎ)
𝑆𝑘
𝑛 Start of iteration 𝑘 in node 𝑛 (ℎ)
𝑇𝑖 Duration of the interval 𝑖 (h)
∆𝑡 Duration of each time interval (ℎ) (ASM model from Chapter 4) 𝑇𝑚𝑎𝑥 Initial (maximum) temperature for the simulated annealing heuristic 𝑇𝑚𝑖𝑛 Minimum temperature for the simulated annealing heuristic
𝑇 𝑁𝑛 Estimated mean transit time to arrive in node 𝑛 from the initial source (ℎ)
𝑇 𝑆𝑠 Estimated mean transit time of segment 𝑠 (ℎ)
𝑉 𝐶 Volumetric coordinate of the aggregate-node in relation to the upstream node (𝑚3), i.e., the volume of the pipeline segment of the current SM iteration 𝑉 𝐶𝑛 Volumetric coordinate of node 𝑛 from zero coordinate, 𝑛 = 0 (𝑚3)
𝑉 𝐷 Delivered volume until first in-transit batch until the end of the previous rolling horizon iteration. It is greater than o only if 𝐷𝐵 = 1 (𝑚3)
𝑉𝑖𝑛
𝑏 Volume of the initialization batch 𝑏 at the beginning of the horizon (𝑚3)
𝑉𝑖𝑛𝑖𝑡
interval 𝑖 (𝑚3). Also referenced in the algorithms as 𝑉𝑛,𝑏,𝑖𝑐𝑢𝑟𝑟𝑒𝑛𝑡, 𝑉𝑛,𝑏,𝑖𝑏𝑒𝑠𝑡, and 𝑉𝑛,𝑏,𝑖𝑔𝑏𝑒𝑠𝑡
𝑉𝑖𝑝𝑎𝑠𝑠 Volume of the batch part 𝐵𝑖𝑝𝑎𝑠𝑠 passing along the upstream node during in-terval 𝑖 (𝑚3)
𝑉𝑝𝑢𝑚𝑝 Batch size pumped until the end of the previous rolling horizon iteration. It
is greater than 0 only if 𝐿𝑎𝑠𝑡𝑝𝑢𝑚𝑝 = 1 (𝑚3)
𝑉 𝑆𝑏,𝑠 Volume of the initialization batch 𝑏 in pipeline segment 𝑠 (𝑚3)
CONTINUOS VARIABLES
𝑐𝑑ℎ𝑒𝑎𝑑
𝑏,𝑒 Volumetric coordinate of the batch’s head 𝑏 at event 𝑒 (𝑚3)
𝑐𝑑𝑡𝑎𝑖𝑙𝑏,𝑒 Volumetric coordinate of the batch’s tail 𝑏 at event 𝑒 (𝑚3)
𝑓 𝑟𝑝,𝑖 Pumping flow rate allocated to the product 𝑝 during the interval 𝑖 (𝑚3/ℎ)
𝑓 𝑟𝑑
𝑏,𝑖,𝑛 Flow rate of the delivery operation of batch 𝑏 to node 𝑛 during interval 𝑖
(𝑚3/ℎ)
𝑓 𝑟𝑠
𝑏,𝑖 Pipeline flow rate of the batch 𝑏 that enters into the pipeline segment being
considered in SM, which discounts the flow rate of the delivery operation (𝑚3/ℎ)
𝑓 𝑟𝑠
𝑖 Pipeline flow rate during interval 𝑖 (𝑚3/ℎ)
𝑖𝑛𝑣𝑛,𝑝,𝑒 Inventory level of product 𝑝 in the node 𝑛 at the event 𝑒. When interval 𝑖 is
used, it refers to the interval’s end event (𝑚3)
𝑙𝑚𝑓 𝑟𝑖 Lower difference between the flow rate during interval 𝑖 and the 𝑚𝑓 𝑟 (𝑚3/ℎ)
𝑚𝑓 𝑟 Mean pumping flow rate of the initial source (refinery) (𝑚3/ℎ)
𝑝ℎ𝑖 Length of continuous active pumping operation at the end of interval 𝑖 (ℎ)
𝑠ℎ𝑖 Length of continuous pumping interruption at the end of interval 𝑖 (ℎ)
𝑠𝑚𝑓 𝑟𝑖 Mean pumping flow rate of the initial source during each interval 𝑖 ∈ 𝐼0. If
𝑝𝑠𝑖 = 1, then 𝑠𝑚𝑓 𝑟𝑖 = 0; else 𝑠𝑚𝑓 𝑟𝑖 = 𝑚𝑓 𝑟 (𝑚3/ℎ)
𝑢𝑚𝑓 𝑟𝑖 Upper difference between the flow rate during interval 𝑖 and the 𝑚𝑓 𝑟 (𝑚3/ℎ)
𝑣𝑑
𝑏,𝑖,𝑛 Volume of batch 𝑏 delivered to node 𝑛 during interval 𝑖 (𝑚3)
𝑣𝑝,𝑖 Volume of product 𝑝 allocated to be pumped during the interval 𝑖 (𝑚3)
𝑣𝑏𝑎𝑡𝑐ℎ
𝑝,𝑖 Batch size of product 𝑝 being pumped during the interval 𝑖 (𝑚3)
𝑣𝑑𝑐
𝑛,𝑝,𝑖 Volume of product 𝑝 expected to be delivered to node 𝑛 during interval 𝑖
(𝑚3)
𝑣𝑓 𝑟𝑖𝑙𝑑𝑖𝑓 𝑓 Lower difference between the flow rate that passes the upstream node during
interval 𝑖 and 𝑖 − 1 (𝑚3/ℎ)
𝑣𝑓 𝑟𝑖𝑢𝑑𝑖𝑓 𝑓 Upper difference between the flow rate that passes the upstream node during
interval 𝑖 and 𝑖 − 1 (𝑚3/ℎ)
𝑣𝑓 𝑟𝑏,𝑖𝑚𝑎𝑥 Maximum flow rate violation of the batch 𝑏 during interval 𝑖 for the pipeline segment being considered in the SM iteration (𝑚3/ℎ)
𝑣𝑓 𝑟𝑚𝑖𝑛
𝑏,𝑖 Minimum flow rate violation of the batch 𝑏 during interval 𝑖 for the pipeline
segment being considered in the SM iteration (𝑚3/ℎ)
𝑣𝑓 𝑟𝑚𝑖𝑛
𝑖 Violation of the minimum pipeline flow rate during interval 𝑖 (𝑚3/ℎ)
𝑣𝑖𝑑𝑐𝑎𝑝
𝑛,𝑝,𝑒 Violated volume with respect to capacity level of product 𝑝 in node 𝑛 at event
𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑒𝑚𝑝
𝑛,𝑝,𝑒 Violated volume with respect to empty storage of product 𝑝 in node 𝑛 at
event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑚𝑎𝑥
𝑛,𝑝,𝑒 Violated volume with respect to the maximum operational level of product
𝑝 in node 𝑛 at event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑚𝑖𝑛𝑛,𝑝,𝑒 Violated volume with respect to the minimum operational level of product 𝑝
in node 𝑛 at event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑚𝑎𝑥𝑡𝑔
𝑛,𝑝,𝑒 Violated volume with respect to the maximum target level of product 𝑝 in
node 𝑛 at event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑚𝑖𝑛𝑡𝑔𝑛,𝑝,𝑒 Violated volume with respect to the minimum target level of product 𝑝 in
node 𝑛 at event 𝑒. When interval 𝑖 is used, it refers to the interval’s end event (𝑚3)
𝑣𝑖𝑑𝑓𝑛,𝑝,𝑖 Violated volume of inventory level 𝑓 with respect to product 𝑝 in node 𝑛 at
the end of interval 𝑖 (𝑚3)
𝑣𝑖𝑛
𝑏,𝑒 Volume of batch 𝑏 in the considered pipeline segment at event 𝑒 (𝑚3)
𝑣𝑝𝑑
𝑏,𝑝,𝑖 Volume of new batch 𝑏 associated with product 𝑝 delivered during interval 𝑖
to the downstream node (𝑚3)
𝑣𝑝𝑝𝑢𝑚𝑝𝑏,𝑝,𝑖 Volume of new batch 𝑏 associated with product 𝑝 pumped during interval 𝑖
(𝑚3)
𝑣𝑏,𝑖𝑝𝑢𝑚𝑝 Volume pumped of new batch 𝑏 during interval 𝑖 (𝑚3)
BINARY VARIABLES
𝑎𝑝,𝑖 1 if the product 𝑝 is allocated to be pumped in the interval 𝑖; 0 otherwise
𝑎𝑠𝑝,𝑖 1 if the product 𝑝 is allocated in interval 𝑖 (𝑎𝑝,𝑖= 1) and the pumping
opera-tion is interrupted (𝑝𝑠𝑖); 0 otherwise
𝑐 1 if the first new batch is the continuation of a batch that started being pumped in the previous iteration; 0 otherwise
𝑑𝑏𝑖,𝑖′ 1 when the initialization batches estimated to be delivered during interval 𝑖′ (𝑉𝑖𝑛𝑖𝑡
𝑛,𝑝,𝑖′) are already effectively considered in interval 𝑖 ({𝑖,𝑖′} ∈ 𝐼𝐼); 0 otherwise. The delay in the delivery operation is due to early pumping inter-ruptions
𝑓𝑏,𝑏′ 1 if batch 𝑏′ follows batch 𝑏, i.e., 𝑏′ entered in the pipeline segment just after 𝑏; 0 otherwise
𝑏 at event 𝑒 or in a previous event; 0 otherwise
ℎ𝑏𝑏,𝑒 1 if the head coordinate of batch 𝑏 has reached the downstream node of the
SM iteration at event 𝑒 or in a previous event; 0 otherwise
ℎ𝑏𝑏,𝑒,𝑛 1 if the head of batch 𝑏 has reached the node 𝑛 at the event 𝑒; 0 otherwise
𝑖𝑛𝑏,𝑒 1 if batch 𝑏 has entered in the pipeline segment at event 𝑒 or in a previous
event; 0 otherwise
𝑙𝑎𝑠𝑡𝑏,𝑒 1 if batch 𝑏 is the last batch (closest to the upstream node) at the event 𝑒; 0
otherwise
𝑚𝑖 1 if the upstream node is pumping a new batch during interval 𝑖; 0 otherwise
𝑝𝑐𝑝,𝑖 1 if the product 𝑝 being pumped during interval 𝑖 − 1 (𝑎𝑝,𝑖−1 = 1) changes
to other product at the beginning of interval 𝑖 (𝑎𝑝,𝑖= 0); 0 otherwise
𝑝𝑑𝑏,𝑖,𝑛 1 if there is the possibility to deliver batch 𝑏 to the node 𝑛 during interval 𝑖;
0 otherwise
𝑝𝑚𝑏,𝑖 1 if new batch 𝑏 is being pumped by an intermediate node during interval 𝑖;
0 otherwise
𝑝𝑠𝑖 1 if the pumping operation is interrupted during interval 𝑖; 0 otherwise
𝑠𝑐𝑖 1 if a pumping interruption starts at the beginning of the interval 𝑖 (𝑠𝑐𝑖 =
¬𝑝𝑠𝑖−1∧ 𝑝𝑠𝑖); 0 otherwise
𝑠𝑠𝑏,𝑖,𝑛 1 if batch 𝑏 has been scheduled to be delivered in node 𝑛 during interval 𝑖; 0
otherwise
𝑠𝑠𝑑𝑏 1 if batch 𝑏 has been delivered to the upstream node of the SM iteration; 0 otherwise
𝑠𝑠𝑑
𝑏,𝑛 1 if at least one delivery operation is scheduled for batch 𝑏 to the node 𝑛; 0
otherwise
𝑡𝑏𝑏,𝑒 1 if the tail coordinate of batch 𝑏 has reached the downstream node of the
SM iteration at event 𝑒 or in a previous event; 0 otherwise
1 INTRODUCTION . . . 24 1.1 OBJECTIVES . . . 27 1.2 PUBLICATIONS . . . 27 1.2.1 Journal articles . . . 27 1.2.2 Co-authored journal articles . . . 28 1.2.3 Conference proceedings . . . 28 1.3 DOCUMENT OUTLINE . . . 29 2 THE PIPELINE SCHEDULING PROBLEM . . . 30 2.1 STRAIGHT MULTIPRODUCT PIPELINE NETWORK . . . 30 2.1.1 Tankage characteristics . . . 31 2.1.2 Pipeline characteristics . . . 33 2.2 PIPELINE OPERATIONS . . . 35 2.2.1 Pumping operation . . . 35 2.2.2 Delivery operation . . . 37 2.3 PROBLEM INSTANCE AND SOLUTION . . . 39 2.4 CONSIDERATIONS . . . 41
3 SINGLE-SOURCE AND MULTIPLE-DESTINATIONS PIPELINE
SYSTEMS . . . 42 3.1 INTRODUCTION . . . 43 3.2 PROBLEM DEFINITION . . . 46 3.3 DECOMPOSITION FRAMEWORK . . . 48 3.3.1 Allocation and sequencing module (ASM) . . . 49 3.3.1.1 Allocation and sequencing preprocessing . . . 50 3.3.1.2 MILP rolling horizon processing . . . 52 3.3.1.3 Allocation and sequencing model formulation . . . 53 3.3.1.3.1 Batch allocation . . . 54 3.3.1.3.2 Volume allocation . . . 54 3.3.1.3.3 Product change . . . 55 3.3.1.3.4 Batch size . . . 56 3.3.1.3.5 Forbidden sequences of products . . . 56 3.3.1.3.6 Flow rate control. . . 57 3.3.1.3.7 Inventory control . . . 57 3.3.1.3.8 ASM objective function . . . 58 3.3.2 Scheduling module (SM) . . . 59 3.3.2.1 Scheduling preprocessing . . . 60 3.3.2.2 Scheduling model formulation . . . 62 3.3.2.2.1 Batch size . . . 62 3.3.2.2.2 Batch volumetric coordinate . . . 64 3.3.2.2.3 Possibility of delivery . . . 65 3.3.2.2.4 Delivery operation . . . 65 3.3.2.2.5 Pipeline segment . . . 67 3.3.2.2.6 Inventory control . . . 67 3.3.2.2.7 SM objective function . . . 68
3.4 RESULTS AND DISCUSSION . . . 70 3.4.1 Case study 1: illustrative network . . . 70 3.5 CASE STUDY 2: REAL-WORLD NETWORK . . . 73 3.5.0.1 Example 1: base instance . . . 74 3.5.0.2 Example 2: extended instance . . . 78 3.5.0.3 Example 3: model performance tests . . . 80 3.6 CONSIDERATIONS . . . 82
4 SINGLE-SOURCE AND MULTIPLE-DESTINATIONS PIPELINE
SYSTEMS: EXTENDED FRAMEWORK . . . 84 4.1 INTRODUCTION . . . 85 4.1.1 Related work: single-source and multiple-destinations . . . 85 4.2 PROBLEM DEFINITION . . . 88 4.3 DECOMPOSITION FRAMEWORK . . . 90 4.4 NEW ALLOCATION AND SEQUENCING MODULE . . . 90 4.5 ASM PREPROCESSING . . . 92 4.5.1 Mean flow rate heuristic . . . 92 4.5.2 Initialization batches: estimation of delivery operations . . . 94 4.5.2.1 Propagation algorithm . . . 95 4.5.2.2 Base heuristic strategy . . . 95 4.5.2.3 Simulated annealing metaheuristic strategy . . . 97 4.5.2.3.1 Fitness function . . . 98 4.5.2.3.2 Initial solution . . . 99 4.5.2.3.3 Solution improvement local search . . . 100
4.6 ALLOCATION AND SEQUENCING MILP MODEL FORMULATION103
4.6.1 Rolling horizon process . . . 103 4.6.2 ASM MILP model formulation . . . 104 4.6.2.1 Product allocation . . . 105 4.6.2.2 DC volume allocation . . . 105 4.6.2.3 Product change . . . 105 4.6.2.4 Batch Size . . . 106 4.6.2.5 Forbidden sequences of products . . . 106 4.6.2.6 Flow rate control . . . 107 4.6.2.7 Pumping interruptions . . . 107 4.6.2.7.1 Modified flow rate control constraints . . . 109 4.6.2.8 Delivery delay of initialization batches . . . 110 4.6.2.9 Inventory control . . . 111 4.6.2.10 ASM objective function . . . 112 4.7 RESULTS AND DISCUSSION . . . 113 4.7.1 Case Study 1: instance with lower demand rate . . . 114 4.7.2 Case Study 2: base heuristic vs simulate annealing metaheuristic . . . 117 4.7.3 Analysis of the simulated annealing metaheuristic . . . 121 4.7.3.1 Analysis of the SA population . . . 122 4.7.3.2 Analysis of the cooling rate factor . . . 123 4.8 CONSIDERATIONS . . . 124
5 MULTIPLE-SOURCES AND MULTIPLE-DESTINATIONS
5.1.1 Multiple-sources and multiple-destinations straight pipeline operation . . . . 129 5.1.2 Related work: multiple-sources and multiple-destinations . . . 130 5.1.3 Contributions . . . 132 5.2 PROBLEM DEFINITION . . . 134 5.3 SOLUTION FRAMEWORK . . . 136 5.3.1 Rolling horizon process . . . 138 5.4 SCHEDULING MODULE (SM) . . . 138 5.4.1 Time representation . . . 140 5.4.2 Batch numbering strategy . . . 141 5.4.2.1 Maximum number of new batches . . . 143 5.4.3 SM MILP model formulation . . . 145 5.4.3.1 Pumping mode . . . 145 5.4.3.2 Batch has entered into the pipeline segment . . . 146 5.4.3.3 Last batch . . . 147 5.4.3.4 Pipeline stoppage . . . 149 5.4.3.5 Batch precedence . . . 149 5.4.3.6 Batch tracking . . . 150 5.4.3.7 Batch volume . . . 151 5.4.3.8 Pipeline flow rate . . . 152 5.4.3.9 Pumping operation . . . 152 5.4.3.10 Delivery operation . . . 153 5.4.3.11 Forbidden sequence . . . 156 5.4.3.12 Inventory management . . . 156 5.4.3.13 Rolling horizon . . . 157 5.4.3.14 Objective Function . . . 159 5.5 CASE STUDIES SOLUTION AND DISCUSSION . . . 160 5.5.1 Case study 1 . . . 161 5.5.2 Case study 2 . . . 165 5.5.3 Case study 3 . . . 168 5.6 CONSIDERATIONS . . . 172 6 FINAL CONSIDERATIONS AND FUTURE WORK . . . 174 6.1 FINAL CONSIDERATIONS . . . 174 6.2 FUTURE WORK . . . 176
REFERENCES . . . 178 APPENDIX A – SUPPLEMENTARY INFORMATION OF
1 INTRODUCTION
Companies are facing increasing challenges to maintain their competitiveness in a mar-ket that requires seamless integration of different supply chain activities, where higher service levels and lower prices are expected. As a consequence, industries are being forced to focus on a joint effort to improve communication, cooperation, integration of all entities (i.e., suppliers, manufacturers, distributors, and retailers), and activities throughout their entire business pro-cess. Thus, the supply chain deep analysis becomes a fundamental issue. Sahebi (2013) defines supply chain as the integrated process of all entities to acquire raw materials, convert them into finished goods and distribute these to the final customers. A supply chain is then a complex and dynamic network, within a collaborative or competitive environment, whose entities may cooperate in attending requests for products or services (LIMA et al., 2016). In order to secure adequate financial, material, and information flow along with the system, supply chain manage-ment is essential. In this context, the objective of the supply chain managemanage-ment is to plan and coordinate the integrated activities of each supply chain entity with the objective of minimizing overall costs, providing products with correct specifications at the correct time in the right place. One important type of supply chain is the oil supply chain in view of its enormous economic value, size, and importance. Besides the huge complexity of the petroleum or oil supply chain, it suffers from an uncertain environment due to global competition, geopolitical conflicts and interests, and price fluctuation (LIMA et al., 2016). Such complex system as stated by Sahebi (2013) may be divided into three segments of activities: upstream, midstream, and downstream. The upstream segment is responsible for the exploration, production, and crude oil transportation until the refineries and petrochemicals. Midstream segments concern with the refineries and petrochemicals operations, which transform crude oil into oil derivatives, such as gasoline, different grades of diesel, naphtha. The downstream segment includes the storage, pri-mary and secondary distribution of the oil products to the final customers. In the pripri-mary distri-bution, the oil products are transported to distribution centers (DCs), where products are stored in tanks. Afterwards, the secondary distribution step is responsible for attending the whole-sale and retail market demand. Figure 1 shows the division of each segment in the described structure.
One interesting problem of the oil supply chain is the scheduling of transportation op-erations at the primary distribution of the defined downstream segment. Refined oil products are transported from a production source (e.g., refineries) to destinations (e.g., DCs) using var-ious modals, such as tank-trucks, trains, vessels, and pipelines. Oil pipelines are used by many companies as their main transportation choice for the primary distribution, justified because of high volume capacity, reliability, economy, and safety of pipelines when compared to other means (SASIKUMAR et al., 1997). Currently, most of the planning and scheduling operations in pipeline systems are decided by specialists, where their decision-making process is usually
Figure 1 – Oil supply chain.
Source: Lima et al. (2016).
based on manual calculations, spreadsheets, and past experiences. In order to aid these special-ists in performing better decisions, optimization techniques have received significant interest in the last years since any improvement in the usage of resources may considerably increase the oil company profit (POLLI et al., 2017).
Pipeline networks may be organized in a range of topologies with differences in com-plexity depending on how one or more pipelines are combined to connect one or more sources and destinations, as presented in Figure 2(a-e). The simplest possible pipeline network is com-posed of a single-source connecting a single-destination (Figure 2(a)), and transporting just one type of product, where simple planning is required for the scheduling of operations. These pipelines networks can be found, for example, in the transportation of crude oil from coastal ports to inland refineries (SASIKUMAR et al., 1997). The complexity increases considerably when a multiproduct pipeline is considered, which means that more than one type of product is transported through the same pipeline and the sequencing of pumping operations is required. In the doctoral thesis of Jittamai (2004), it was proved that a single-source oil pipeline distributing multiple products subject to delivery time-windows is an NP-complete1problem. Next, there is the network configuration of Figure 2(b), single-source connected to multiple-destinations by a straight multiproduct pipeline. Figure 2(c) shows a more complex straight pipeline system in-cluding multiple-sources (CAFARO; CERDÁ, 2009), or DCs (outputs) that can act also as input (known as dual purpose terminals), which are approached in the works of Cafaro et al. (2015b), Mostafaei et al. (2016), Mostafaei and Castro (2017). This particular class of topologies from Figure 2(a-c) account for the straight pipeline systems that will be the focus of this thesis.
For the multi-pipeline systems, Figure 2(d) shows an example of tree-like pipeline topology, where a mainline (trunk) with high-volume capacity covers a longer path and sec-ondary lines (branches) transport smaller quantities over shorter distances (MOSTAFAEI et al., 2015b). Another and more rare topology is mesh-like systems as shown in Figure 2(e), mainly approached in the works of (MAGATÃO et al., 2012; MAGATÃO et al., 2015; POLLI et al., 2017).
1 NP is a complexity definition used to classify decision problems, where a solution can be verified in polynomial
time. An NP problem is considered complete if every problem in NP is reducible to this problem in polynomial time. For more details on the NP-completeness theory, see Garey and Johnson (1979).
Figure 2 – Classification of pipeline topologies. Straight pipeline: (a) source and single-destination; (b) single-source and destinations; (c) sources and multiple-destinations. Multi-pipeline: (d) tree-like structure; (e) mesh-like structure. The nodes with an white “S” are sources
Source: Own authorship.
As mentioned before, this thesis focuses on the development of a solution framework to aid specialists on the decision-making procedure of scheduling operations to straight mul-tiproduct pipeline systems connecting a single or sources to a single or multiple-destinations. In the literature, several approaches to this problem have been proposed using different techniques, such as heuristics, meta-heuristics, Mixed Integer Linear Programming (MILP), Mixed Integer Non-Linear Programming (MINLP). Moreover, solution approaches proposed in the literature tend to include real-world aspects and constraints of the problem, or long-term scheduling plans, combining different techniques with structural, temporal, or both decomposition structures to achieve solutions in a reasonable computational time.
The fundamental study for the development of the proposed innovative solution frame-work to solve straight pipeline systems with multiple-sources and multiple destinations comes from Kira (2011) (bachelor’s thesis), then Meira (2016) (master’s thesis). Both studies focused on a solution to a specific pipeline system with a single-source and multiple-destinations. The motivation of this thesis also comes from the Department of Logistics of Petrobras, the Brazil-ian oil company, that supports the development of this research project in scientific collabora-tion with Universidade Tecnológica Federal do Paraná (UTFPR) and Instituto Superior Técnico (IST) of Universidade de Lisboa (UL).
1.1 OBJECTIVES
The general objective of this thesis is to propose a new solution framework combin-ing mathematical programmcombin-ing and heuristics to solve the schedulcombin-ing of pumpcombin-ing and deliv-ery operations in straight multiproduct pipeline systems with sources and multiple-destinations. In order to achieve such objective, secondary objectives have been defined, which are:
• Implement a new solution framework based on Meira (2016) to solve the scheduling of straight multiproduct pipeline systems with single-source and multiple-destinations; • Expand this new proposed framework to solve the scheduling of a generic straight
multi-product pipeline system with multiple-sources and multiple-destinations;
• Propose mathematical models and heuristics to better treat operational constraints and real-world aspects of the considered problem, such as tank and pipeline maintenance periods, scheduling of pipeline interruptions, inventory management, flow rate control; • Analyze and compare the obtained results from real instances with different operational
aspects and configuration parameters.
1.2 PUBLICATIONS
This thesis and the following references are the results of this PhD research. The con-tent of the journal articles from Subsection 1.2.1 are the basis for the Chapters 3, 4, and 5 of this thesis. My contribution roles to each journal article publication are defined by the terms of the Contributor Roles Taxonomy (CRediT) (BRAND et al., 2015).
1.2.1 Journal articles
• MEIRA, W. H. T.; MAGATÃO, L.; RELVAS, S.; BARBOSA-PÓVOA, A. P. F. D.; NEVES-JR, F.; ARRUDA, L. V. R. A matheuristic decomposition approach for the scheduling of a single-source and multiple destinations pipeline system. European Jour-nal of OperatioJour-nal Research, v. 268, n. 2, p. 665–687, 2018.
– Roles: conceptualization; methodology; software; validation; formal analysis; in-vestigation; data curation; writing - original draft; visualization.
• MEIRA, W. H. T.; MAGATÃO, L.; NEVES-JR, F.; ARRUDA, L. V. R.; VAQUEIRO, J. P.; RELVAS, S.; BARBOSA-PÓVOA, A. P. F. D. Scheduling of a single-source multi-product pipeline system by a matheuristic approach: combining simulated annealing and MILP. Computers and Chemical Engineering, v. 136, p. 106784, 2020.
– Roles: conceptualization; methodology; software; validation; formal analysis; in-vestigation; data curation; writing - original draft; visualization.
• MEIRA, W. H. T.; MAGATÃO, L.; NEVES-JR, F.; ARRUDA, L. V. R.; VAQUEIRO, J. P.; RELVAS, S.; BARBOSA-PÓVOA, A. P. F. D. A solution framework for the long-term scheduling and inventory management of straight pipeline systems with multiple-sources. Computers and Operations Research, 2020. Submitted on February, 2020.
– Roles: conceptualization; methodology; software; validation; formal analysis; in-vestigation; data curation; writing - original draft; visualization.
1.2.2 Co-authored journal articles
• KONOWALENKO, F.; MEIRA, W. H. T.; MAGATÃO, L. Constraint logic programming applied to sequencing tasks in a pipeline network. IEEE Latin America Transactions, v. 17, n. 8, p. 1309–1317, 2019. ISSN 1548-0992.
– Roles: supporting conceptualization; supporting methodology; software; investiga-tion; writing – review and editing.
1.2.3 Conference proceedings
• MEIRA, W. H. T.; MAGATÃO, L.; RELVAS, S.; BARBOSA-PÓVOA, A. P. F. D.; NEVES-JR, F. A decomposition approach for the long-term scheduling of a single-source multiproduct pipeline network. In: VAZ, A. I. F.; ALMEIDA, J. P.; OLIVEIRA, J. F.; PINTO, A. A. (Ed.). Operational Research. Cham: Springer International Publishing, 2018. (Springer Proceedings in Mathematics and Statistics, v. 223), p. 235–248.
• MEIRA, W. H. T.; MAGATÃO, L.; RELVAS, S.; ARRUDA, L. V. R.; NEVES-JR, F.; BARBOSA-PÓVOA, A. P. F. D. Scheduling of a multiproduct and multiple destinations pipeline system with repumping operations. In: FRIEDL, Anton; KLEMEŠ, J. J.; RADL, Stefan; VARBANOV, P. S.; WALLEK, Thomas (Ed.). 28th European Symposium on Computer-Aided Process Engineering. Amsterdam: Elsevier, 2018. (Computer Aided Chemical Engineering, v. 43), p. 931–936.
• MEIRA, W. H. T.; MAGATÃO, L.; NEVES-JR, F.; ARRUDA, L. V. R.; VAQUEIRO, J. P.; RELVAS, S.; BARBOSA-PÓVOA, A. P. F. D. A matheuristic decomposition ap-proach for the long-term scheduling of a single-source multiproduct pipeline system with pumping interruptions. In: LI Simpósio Brasileiro de Pesquisa Operacional. Campinas: Anais Eletrônicos, Galoá, 2019. v. 2.
1.3 DOCUMENT OUTLINE
This chapter introduced the reader to the oil supply chain and the primary distribu-tion problem being studied. Chapter 2 presents the main terminology and operadistribu-tional aspects existing in the pipeline scheduling problem, especially the aspects concerning the straight mul-tiproduct pipeline systems. The goal is to initially present to the reader the main singularities involved in this pipeline-scheduling problem. Further operational aspects are exploited in Chap-ters 3 to 5.
Chapter 3 presents a solution framework that integrates heuristic procedures and MILP models, a matheuristic, to solve the long-term scheduling of single-source and multiple-destinations pipeline systems. The proposed approach decomposes the problem into two mod-ules: the Allocation and Sequencing Module (ASM), and the Scheduling Module (SM). The ASM is responsible for determining the pumping operations of the initial source (first node) during the scheduling horizon. The SM plans the detailed scheduling of the delivery operations to be executed at each node. The content of this chapter includes in full with slight adapta-tions the published paper Meira et al. (2018b): A matheuristic decomposition approach for the scheduling of a single-source and multiple destinations pipeline system.
Chapter 4 extends the previous proposed solution framework with a new ASM formu-lation to treat the scheduling of pumping interruptions. The pipeline restart is a high cost oper-ation; thus, the interruption of the pumping activity should be avoided. However, the pipeline operations may face situations of low consumption, for example, during an off-season period, to a point where the pipeline system cannot operate at the minimum operational pumping rate without increasing the overall inventory of the nodes, increasing the risk of an inventory prob-lem. In this case, the appropriate scheduling of pumping interruptions respecting certain oper-ational constraints are an important task. Moreover, a better solution methodology based on a Simulated Annealing (SA) heuristic has been developed to improve the solution quality. The content of this chapter derives from the published paper Meira et al. (2020a): Scheduling of a single-source multiproduct pipeline system by a matheuristic approach: combining simulated annealing and MILP.
Chapter 5 improves the solution framework to also solve straight pipeline systems with multiple-sources. This chapter includes the content of the submitted paper Meira et al. (2020b): A solution framework for the long-term scheduling and inventory management of straight pipeline systems with multiple-sources. The solution approach respects several oper-ational aspects, especially pipeline and tank maintenance periods, considering a rigorous inven-tory management at every node-product of the pipeline system.
Finally, Chapter 6 presents the final considerations and main contributions of this the-sis. Also, future research directions are suggested for the reader interested in continuing the developed work.
2 THE PIPELINE SCHEDULING PROBLEM
This chapter introduces the main concepts associated with the pipeline scheduling problem, which will serve as basis for the comprehension of the remaining chapters. The prob-lem consists of the scheduling of pumping and delivery operations in a pipeline system. The scope is limited to straight multiproduct pipeline network topologies with a focus on real char-acteristics of the oil industry and oil derivatives transportation. Sections 2.1 and 2.2 describe the main elements that constitute a straight oil pipeline network: physical characteristics; op-erational constraints; common assumptions; and typical operations. Section 2.3 describes the required data for an operational input instance and an example of an output solution. Thus, this chapter’s objective is to initially present to the reader the main characteristics involved in the studied problem. Further operational aspects are exploited in Chapters 3-5.
2.1 STRAIGHT MULTIPRODUCT PIPELINE NETWORK
Oil refinery operations are responsible for the transformation of crude oil into refined oil products (e.g., gasoline, diesel, jet fuel). The produced products are mainly stored in tanks with the intended purpose of attending local market consumption and the demand of down-stream distribution centers. Such tanks are often located in distribution centers (DCs) that are terrestrial terminals used as storage tank farms that receive and distribute products to the respec-tive local market, where none or almost no transformation of the received product is required. From the DCs to the local clients, the distance is usually short, and distribution flexibility is required, making tank-trucks the most viable option for the secondary distribution step.
In a straight multiproduct pipeline network, source and destination nodes are repre-sented by refineries and distribution centers; and the edges, which connect two nodes, are the pipeline segments. Figure 3 illustrates an example of a straight pipeline network with a single-source (refinery) and multiple-destinations (DCs).
Figure 3 – Example of a straight pipeline network with a single-source and multiple-destinations
Source: Meira et al. (2020a).
Regarding the multiple-sources and multiple-destinations topology, the system is com-posed of the initial source and downstream nodes, which in this case can be a destination, a
source, or a dual-purpose node. A dual-purpose node is capable of receiving products from the pipeline as a destination node and also pumping products into the pipeline as a source node. Figure 4 presents an example of multiple-sources and multiple-destinations pipeline network with N1 as the initial source, N3 as a dual-purpose node (notice downward and upward arrows), N2 and N4 as destination nodes.
Figure 4 – Example of a straight pipeline network with multiple-sources and multiple-destinations
Source: Meira et al. (2020b).
The main tankage and pipeline characteristics are described in Subsections 2.1.1 and 2.1.2, respectively.
2.1.1 Tankage characteristics
Tanks are present in the pipeline nodes for the storage of oil products. Tanks with a cylindrical shape are commonly used for liquid products (e.g., gasoline, diesel), and tanks with a spherical shape are used to store below ambient temperature or pressurized products (e.g., liquefied natural gas). The main tank characteristics and assumptions considered in this thesis are the following:
• Admissible products: a tank has a limited number of admissible products depending on the characteristic of the product and also the distribution center policy, where each tank is usually associated with, storing just one admissible product at a certain moment. The tank product swap to other product is not a frequent operation since it is costly and time demanding; thus, should be avoided. The tank swap aspect is addressed in Fabro et al. (2014);
• Physical storage levels: the levels are the maximum physical capacity of storage, the empty status level, which is the minimum possible level, the “ballast” level. Impurities of the stored product are usually sedimented below the ballast level, and this contaminated portion is not considered as part of the operational tank inventory;
• Inventory management: additional restrictive ranges to the physical levels may also be considered to achieve better inventory management and, consequently, protect against un-certain events during the scheduling horizon. In this thesis, two additional intermediate
ranges (operational range and target range) are introduced. The operational range rep-resents the maximum and minimum recommended inventory levels, which maintain a certain difference to the physical storage capacity or physical empty status. Inner the op-erational range, the target maximum/minimum levels frame the adequate inventory profile to be maintained for each pair node-product. Figure 5 presents an example of inventory profile and typical inventory levels during the scheduling horizon. Any inventory viola-tion of the upper or lower limit of each one of the three inventory ranges is penalized with an increasing cost from the target range to the physical range;
Figure 5 – Example of the inventory levels (physical, operational, and target ranges) as a percentage of the physical maximum capacity level
Source: Meira et al. (2020b).
• Aggregate tank: one common simplification in the literature is to consider as a single aggregate tank all the tanks associated with storing the same product at the same terminal, summing their physical levels and inventory levels;
• Tank maintenance period: due to technical problems, tank maintenance periods may exist, which means that a specific tank will become out of operation for a period to be examined or repaired. Each tank maintenance agenda is specified in advance by input pa-rameters to the problem. During a tank maintenance period, the volume of the unavailable tank is subtracted from the respective aggregate tank total capacity of the corresponding node-product. Figure 6 presents an example of a tank maintenance period that is repre-sented by the decrease in the inventory levels at the beginning of the horizon.
• Forecasted demand and production: for each noproduct, there is a forecasted de-mand and production profile for the scheduling horizon. The forecasted dede-mand and pro-duction values are input parameters of the system, and they are estimated by specialists of the oil company. The solution approach should anticipate the forecasted demand to avoid a shortage of products in the destination nodes and also avoid a surplus of the pro-duced products at production source nodes. The demand and production rates could be distributed in a constant rate along the horizon or may vary according to circumstances
