# Optimization-Based Scheduling for the Process Industries: From Theory to Real-Life Industrial Applications

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## Abstract

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## 1. Introduction

- A large number of time periods is required to capture all significant events and extract a high quality solution—this usually results to extremely large models;
- Operations in which the processing time is dependent on the batch size are difficult to be modelled;
- The modelling of continuous and semi-continuous operations must be approximately modelled.

## 2. Theoretical Aspects of Optimization-Based Process Scheduling

#### 2.1. Classification of Scheduling Problems

- What tasks must be executed to satisfy the given demand (batching/lot-sizing)?
- How should the given resources be utilized (task-resource assignment)?
- In what order are batches/lots processed (sequencing and/or timing)?

- Facility data; e.g., processing stages and units, storage vessels, processing rates, unit to task compatibility.
- Production targets that need to be satisfied.
- Availability of raw materials and resource limitations; e.g., maintenance of units, availability of utilities.

#### 2.1.1. The Production Facility

#### Process Type

#### Production Environment

- Single stage: Production facility that consists of just one processing stage, which may consist of a single unit or multiple parallel units. The product to unit compatibility may be fixed (batch can be processed in a single unit) or flexible (batch can be processed in multiple units), but in all cases each batch must be processed in a single unit.
- Multistage: Each batch must be processed in more than one processing stages, each consisting of a single unit or multiple parallel units. The multistage environment can be further categorized into multiproduct and multipurpose, depending on the imposed routing restrictions. Multiproduct facilities are equivalent to flowshop environments in discrete manufacturing, where all products go through the same sequence of processing stages. In contrast, a facility is characterized as multipurpose when the routings are product-specific, or when a processing unit belongs to different processing stages depending on the product, thus being equivalent to jobshop environments in discrete manufacturing.

#### 2.1.2. Interaction with Other Planning Functions

#### 2.1.3. Processing Characteristics and Constraints

#### 2.2. Classification of Modelling Approaches

#### 2.2.1. Optimization Decisions

#### 2.2.2. Modelling Elements

#### 2.2.3. Time Representations

#### 2.3. Alternative MILP Models for Process Scheduling

#### 2.3.1. Models for Network Production Environments

#### 2.3.2. Models for Sequential Production Environments

## 3. Real-Life Process Systems Industrial Applications

#### 3.1. Chemical Industries

#### 3.2. Pharmaceutical Industries

#### 3.3. Petrochemical Industries

_{2}emissions. Castro and Mostafaei [90] motivated by the scheduling problem of liquid transportation, proposed an event-point MILP formulation for treelike transportation systems, where a single input node leads to multiple outputs. A continuous time representation was utilized and novel constraints for ensuring the avoidance of forbidden product sequences were adapted. A real-life study case from the Iranian Oil Pipelines and Telecommunication Company network was considered and the optimal schedules could lead to even a 6.2% capacity increase, as the given demand can be efficiently covered fourteen hours earlier. A comparison with previous methods, proposed from one of the co-authors [91], indicates the efficiency of the approach. A time termination criterion of 5 h has been used for the proposed formulation. The number of the event points has been identified as key parameter with high impact on the computational time and solution quality. Nearly optimal solutions can be generated in less than 1 h by reducing the available event points. A similar problem, referring to the scheduling of a transportation system of petroleum products, produced from a single oil refinery industry was tackled by Cafaro and Cerdá [92]. They proposed an MILP continuous time model in order to define the optimal lot size, the batch sequence, as well as the delivery time of batch order. A variety of constraints were taken into account, such as tank availability and quality control operations. A real-life study case consisting of six different oil derivatives produced by a unique oil refinery to a single distribution center was scheduled, and the results indicated that better solutions were produced in comparison with other approaches for the same problem, in less than 60 s CPU time. The same problem was also addressed by Cafaro et al. [93], but now allowing simultaneous product deliveries, thus providing more realistic solutions. The proposed two-level MILP-based solution technique aimed for the minimization of the total number of operations in order to reduce the number of restarts and stoppages of the pipeline. On the upper level, the feasibility of the problem was ensured, as more detailed decisions such as lot sizing, lot sequencing and timing decisions were defined on the lower level. A study case related to REPLAN refinery industry, consisting of five distribution centers at Brazil, is used to illustrate the applicability of the model. Significant savings were noticed in CPU time using the multiple delivery policy, as the illustrative examples under consideration can be solved in less than 125 s CPU time. Rejowski and Pinto [94], inspired also from the REPLAN refinery, proposed two discrete-time, MILP-based models, to solve a real-life problem, including the distribution of various petroleum products to five depots. An indicative instance of a 75-h time horizon that is discretized in 5-h intervals was presented. A good quality solution with integrality gap of 5.8% was returned within the time limit of 10,000 CPU s. Boschetto et al. [95], proposed an MILP-based solution algorithm for solving a large scale real-life pipeline network problem, by determining the delivery and the pumping times of 14 different oil products and ethanol, to a number of distribution centers. Efficient heuristic rules were utilized in parallel with a continuous time representation, for tackling the daily scheduling problem, in reasonable computational times within 3–5 min. The generated solution for various studied cases, have been also validated by the planners.

#### 3.4. Food Industries

#### 3.5. Consumer Goods Industries

#### 3.6. Steel Plants

#### 3.7. Paper Industries

## 4. Industrial Applications of Optimization-Based Scheduling—Challenges

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$i\in I$ | processing tasks |

$o\in O$ | orders to be processed |

$j\in J$ | units |

$t,t\prime \in T$ | time events |

$l\in L$ | processing stages |

$r\in R$ | resources |

$s\in S$ | states |

${i}^{ST}$ | storage tasks |

${I}_{r}$ | tasks requiring resource r |

${I}_{j}$ | tasks that can be executed in unit j |

${I}_{s}^{ST}$ | storage tasks for state s |

${I}_{s}^{P}$ | tasks that produce state s |

${I}_{s}^{C}$ | tasks that consume state s |

${R}^{J}$ | resources corresponding to unit j |

${R}_{i}^{ST}$ | resources corresponding to storage that can be used for task i |

${J}_{i}$ | units that can perform task i |

${J}_{o,o\prime ,l}$ | units that can execute both order i and order i’ at stage l |

${J}_{o,l}$ | units that can execute order i at stage l |

${L}_{o,o\prime}$ | stage required for the production of order o and o’ |

${\alpha}_{i}$ | constant term for the processing time of task i |

${\beta}_{i}$ | proportional term for the processing time of task i |

$H$ | time horizon |

${V}_{i}^{\mathrm{min}}$ | minimum batch size of task i |

${V}_{i}^{\mathrm{max}}$ | maximum batch size of task i |

${R}_{r}^{\mathrm{min}}$ | minimum available resource r |

${R}_{r}^{\mathrm{max}}$ | maximum available resource r |

${\mu}_{r,i}^{p}$ | fixed term for the production of resource r at the end of task i |

${\mu}_{r,i}^{c}$ | fixed term for the consumption of resource r at the beginning of task i |

${v}_{r,i}^{p}$ | variable term for the production of resource r at the end of task i |

${v}_{r,i}^{c}$ | variable term for the consumption of resource r at the beginning of task i |

${\rho}_{i,s}$ | proportion of state s consumed/produced by task i |

$p{t}_{o,j}$ | processing time of order o in unit j |

$s{u}_{o,j}$ | setup time for order o in unit j |

${\tau}_{o,o\prime ,j}$ | changeover time between orders o and o’ processed in unit j |

${T}_{t}$ | exact time of event point t |

${\overline{N}}_{i,t,t\prime}$ | defines a task I that starts at event point t and ends at time point t’ |

${\overline{\xi}}_{i,t,t\prime}$ | amount of material processed by task I, that starts at t and ends at t’ |

${R}_{r,t}$ | amount of resource r consumed at event point t |

${W}_{i,t}^{s}$ | denotes that a task i starts at event point t |

${W}_{i,t}^{f}$ | denotes that a task i ends at event point t |

${W}_{i,t}$ | denotes that a task i is active at event point t |

${B}_{i,j,t}^{s}$ | batch size of the task i in unit j started at event point t |

${B}_{i,j,t}$ | batch size of the task i in unit j being processed at event point t |

${B}_{i,j,t}^{f}$ | batch size of the task i in unit j finished at event point t |

${B}_{{i}^{ST},j,t}^{ST}$ | batch size of storage task i^{st} at event point t |

${T}_{i,j,t}^{s}$ | time at which the execution of task i by unit j at event point t starts |

${T}_{i,j,t}^{f}$ | time at which the execution of task i by unit j at event point t ends |

${Y}_{o,j}$ | binary variable denoting that order o is allocated to unit j |

${C}_{o,l}$ | completion time of order o in unit j |

${X}_{o,o\prime ,l}$ | binary variable that is activated when order o is processed before order o′ at stage l |

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Author | Industrial Sector | Main Research Features |
---|---|---|

Lin and Floudas [75] | Chemical industry | • Continuous time event-based mixed-integer linear programming (MILP) • Decomposition methodology |

Janak et al. [129] | Chemical industry | • Graphical user interface development • Rolling horizon approach |

Westerlund et al. [77] | Chemical industry | • Planning tool connected with the plant’s ERP system |

Velez, Merchan and Maravelias [78] | Chemical industry | • Multiple discrete-time grids • A real case study from Dow company |

Moniz et al. [81] | Pharmaceutical industry | • A Visio-based decision-making tool development |

Stefansson et al. [82] | Pharmaceutical industry | • Discrete and continuous time representations • Stage decomposition |

Castro, Harjunkoski and Grossmann [83] | Pharmaceutical industry | • Decomposition-based algorithm |

Kopanos, Méndez and Puigjaner [39] | Pharmaceutical industry | • Precedence-based MILP models • Decomposition-based solution strategy |

Liu et al. [84] | Pharmaceutical industry | • Maintenance planning • 1 h CPU time |

Kabra et al. [85] | Pharmaceutical industry | • State-task network (STN) representation • Computational time in the range of 1–2 min. |

Shah, Sahay and Ierapetritou [86] | Oil refineries | • Six-step MILP based heuristic algorithm • A case study provided by Honeywell Process Solutions (HPS). |

Zhang and Hua [87] | Oil refineries | • Integration of the plant processes and the utility system |

Iyer et al. [88] | Oil refineries | • Decomposition algorithm • Feasible solutions within 6600 s are obtained |

Assis et al. [89] | Oil refineries | • Scheduling of a crude oil terminal • A case study by the national refinery of Uruguay |

Casrto and Mostafei [90] | Pipeline systems | • A case study from the Iranian Oil Pipelines and Telecommunication Company • 6.2% capacity increase |

Cafaro et al. [93] | Pipeline systems | • Simultaneous product deliveries are allowed • A case study, related to REPLAN refinery |

Rejowski and Pinto [94] | Pipeline systems | • Integrality gap of 5.8% in 10,000 CPU s • A case study, related to REPLAN refinery |

Boschetto et al. [95] | Pipeline systems | • Heuristic rules • Computational times within 3–5 min. |

Baldo et al. [96] | Food industries | • A novel MILP-based relax and fix heuristic algorithm • A case study from a brewery industry |

Kopanos, Puigjaner and Maravelias [29] | Food industries | • An immediate precedence-based MILP formulation • A case study from a brewery industry |

Abakarov andSimpson [97] | Food industries | • A food cannery case study • Scheduling of the sterilization stage |

Georgiadis et al. [98] | Food industries | • A case study from a large-scale canned fish industry case study • MILP based decomposition algorithm |

Liu, Pinto and Papageorgiou [99] | Food industries | • An edible-oil deodorized industry case study • Mixed discrete and continuous MILP mathematical |

Polon et al. [100] | Food industries | • A case study from a sausage production industry • Scheduling of the production stage |

Doganis and Sarimveis [101] | Dairy industry | • A single yoghurt production line |

Sel, Bilgen and Bloenhof-Ruwaard [102] | Dairy industry | • Integrated planning and scheduling of a yoghurt facility |

Touil, Echchatbi and Charkaoui [103] | Dairy industry | • A case study from a milk industry |

Kopanos, Puigjaner and Georgiadis [104] | Dairy industry | • A case study from a yoghurt industry • Novel resource constraints |

Georgiadis et al. [105] | Dairy industry | • An integrated software tool connects the plant’s ERP system with the proposed MILP model • A total cost decrease of 20% is achieved |

Giannelos and Georgiadis [108] | Consumer goods industry | • STN continuous time formulation • Medium-size industrial consumer goods manufacturing process |

Baumann and Trautmann [110] | Consumer goods industry | • General precedence MILP hybrid method • 10 large-scale problem instances provided by The Procter and Gamble Company |

Elzakker et al. [106] | Fast-moving consumer goods (FMCG) industry | • A unit-specific, continuous time interval-based algorithm • Ice cream production process of Unilever |

Kopanos, Puigjaner and Georgiadis [107] | Fast-moving consumer goods (FMCG) industry | • Ice cream production process |

Elekidis, Corominas and Georgiadis [112] | Consumer goods industry | • An immediate-general precedence-based decomposition algorithm |

Georgiadis et al. [113] | Manufacturing industries | • Resource task network (RTN) and STN based models • A comparison with a PSE scheduling tool and an MILP model • Development of a middleware interface for data transfer |

Biondi, Saliba and Harjunkoski [114] | Steel industry | • Slot-based MILP formulation • communication with ERP and DCS |

Li et al. [116] | Steel industry | • A unit-specific event-based continuous-time MILP model |

Gajic et al. [117] | Steel industry | • Integrated scheduling and electricity optimization problem • A melt shop case study • 3% electricity cost reduction |

Hadera et al. [118] | Steel industry | • A melt shop case study • Integrated scheduling and electricity optimization problem • A general precedence MILP scheduling model |

Castro, Harjunkoski and Grossmann [120] | Steel industry | • Integrated scheduling and energy optimization problem • RTN-based MILP model • 20% electricity cost reduction |

Wang et al. [122] | glass company | • Bi-objective optimization problem • Makespan and the total energy cost minimization |

Westerlund, Isaksson and Harjunkoski [123] | Paper Industry | • Trim-loss problem of a paper converting mill |

Roslöf et al. [124] | Paper Industry | • MILP based decomposition algorithm |

Giannelos and Georgiadis [126] | Paper Industry | • A slot-based MILP model • A paper mill company case study |

Castro, Barbosa-Povoa and Matos [127] | Paper Industry | • Continuous and discrete time RTN representation • A case study from a pulp mill plant |

Castro, Westerlund and Forssell [128] | Paper Industry | • RTN-based formulation • Novel recycling policies |

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**MDPI and ACS Style**

Georgiadis, G.P.; Elekidis, A.P.; Georgiadis, M.C. Optimization-Based Scheduling for the Process Industries: From Theory to Real-Life Industrial Applications. *Processes* **2019**, *7*, 438.
https://doi.org/10.3390/pr7070438

**AMA Style**

Georgiadis GP, Elekidis AP, Georgiadis MC. Optimization-Based Scheduling for the Process Industries: From Theory to Real-Life Industrial Applications. *Processes*. 2019; 7(7):438.
https://doi.org/10.3390/pr7070438

**Chicago/Turabian Style**

Georgiadis, Georgios P., Apostolos P. Elekidis, and Michael C. Georgiadis. 2019. "Optimization-Based Scheduling for the Process Industries: From Theory to Real-Life Industrial Applications" *Processes* 7, no. 7: 438.
https://doi.org/10.3390/pr7070438