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## OR methodologies

In section 3.2.2, analytical method for getting the solution was chosen to be implemented in the sustainable Phase III BioRSC model to be developed in this project. Then, in section 3.2.3, the sustainable Phase III BioRSC model was described as MO-BMIP optimization model. Therefore, in this section some of the techniques to solve it will be briefly described.

Table 3.2 show the principals methodologies for solve OR models describing briefly their features in the second column (Hillier and Lieberman 2001; A. Ravi Ravindran 2008; P. Rama Murthy 2008; Poler et al. 2014). Linear Programming, Integer Programming, Non-linear Programming, Queueing Theory, Inventory Theory, Simulation and Forecasting, were excluded because these are out of the scope of the model. In the third column, suitability of each methodology for solve MO-BMIP models is analyzed.

In Table 3.2, it can be noted that dynamic programming is not suitable to sustainable Phase III BioRSC conception due to the large amount of decisions related to all the SC decision-making levels. Secondly, game theory and decision analysis have difficulty to be applied due multi-objective nature of sustainability. Then, Markov chains can help to build the probability distribution for the model uncertain parameters. Also, Markov chains and Markov decision process are not suitable for sustainable Phase III BioRSC conception optimization model, as dynamic programming, due to the large amount of decisions related to all the SC decision-making levels and its interrelationship. While, multiple criteria decision making can handle multiple objective functions simultaneously, it does not consider the dynamism and stochasticity of the MO-BMIP for the sustainable Phase III BioRSC conception. Instead, stochastic programming and robust optimization can handle dynamism and stochasticity, and they can be developed as multi-objective models.

Therefore, as conclusion, there are mainly two OR methodologies to conceive the sustainable Phase III BioRSC: stochastic programming and robust optimization. They should ideally be integrated with multiple criteria decision making to include afterwards DM preferences. In section 3.3 two general methodologies are proposed and described.

**General methodology proposition for decision-making on sustainable Phase III BioRSC projects**

In this section, two general methodologies considering the integration of MCDM, Stochastic programming multistage and Robust Optimization for the sustainable Phase III BioRSC conception under uncertainty are proposed.

At first step, for any methodology, it should be analyzed the characteristics associated to the Phase III BioRSC, the sustainability dimensions and the SC decision-levels to identify the system elements and develop the correspondent model. Whereby, three different models can be noted, the design, management and scheduling models, related to strategical, tactical and operational SC decisionmaking level, respectively. Then, each model construction can be described by Figure 3.3. The design model described decisions that must to be taken here and now without information. In the other hand, the management and operational models describe decisions that are made before receive information about the random parameters, known as wait and see decisions.

**Model construction proposition for sustainable Phase III BioRSC design**

As presented in Chapter III the model to design a Phase III BioRSC is a BMILP (Binary Mixed Integer Linear Programming) optimization model, regarding the presence of mixed decision variables, for example decision variables for the production plants location, binary in nature. Thus, efforts will focus on continue developing a linear model to avoid possible discontinuities in space solution (Hamidian et al. 2008; Chinneck 2016) to design a sustainable Phase III BioRSC. Moreover, this model will be developed as deterministic to permit the development whether MSP or RO depending on available information about uncertain parameters.

Additionally, sustainability assessment should be multi-objective, because its framework is constituted by principles, criteria and indicators (Bautista et al. 2016), which could translate into more than one objective function. They are defined as:

Principles: The premises, bases or universal principles that define the sustainability of a biorefinery supply chain.

Criteria: Those measurable conditions (qualitative or quantitative) that establish the level of application of the principles of a sustainable biorefinery supply chain.

Indicators: There are observable qualitative or quantitative expressions, which can describe the characteristics, behaviours or phenomena of reality through the development of one or more variables.

The first level in the framework, the principles, represents the interaction between the five dimensions of the sustainability and the biorefinery supply chain stages. The second level is made up of a set of sustainability assessment criteria linked to each principle. These criteria were identified as a measurable condition (qualitative or quantitative) aiming to assess how the sustainability principle was applied to the BioRSC. The first and the second level in the framework were defined in order to make a general sustainability assessment. Therefore, the principles and criteria can be applied regardless of the economic, social, political or biogeographic context, the technological conditions or the raw materials used, among other aspects.

Finally, in the third level, indicators were established to evaluate the characteristics or behaviours of each criterion. Besides principles, and criteria, the indicators must refer to particular conditions of the biorefinery production system, or the assessment scale (national, regional, local).

Therefore the challenge on model construction is the required analysis to determine and integrate the decision variables, constraints and objective functions related to the sustainable Phase III BioRSC characteristics, the SC strategic decisions and the sustainability dimensions (principles, criteria and indicators). This combination results in a highly complex problem due to the amount of components to analyze. Then, for the model construction it is proposed a progressive development, adding elements one at a time. This working-way will permit to starts from a simply model to reach a very complex one. Enabling test the model in each element addition.

**Multi-Objective Evolutionary Algorithm (MOEA) description**

Multi-Objective Evolutionary Algorithm (MOEA) is a stochastic search methodology to solve multi-objective problems, emulating the Darwinian principle of survival-of-the-fittest in natural selection and adaptation (Chi-Keong Goh and Kay Chen Tan 2009). The evolutionary algorithm is an iterative optimization process. The process starts with the initialization of the population of candidate-solution. Then, the evaluation stage considers the performance of each candidate-solution and the density (diversity) of candidate solutions group. Performance evaluation is calculated on the basis of the criteria optimization problem. After that, the performance of individuals is compared one by one, giving them a rating. Then a classification from highest to lowest is carried out, obtaining an update of candidate-solutions.

The selection of individuals can be performed in different ways. Some MOEAs maintain a fixed amount of the population, while others only keep individuals who are non-dominated, for the next stage of the process. Nonetheless, in most cases, a truncation process will be conducted based on some density assessment to restrict the number of achieved solutions.

The remaining individuals will be eliminated and replaced with new individuals. The objective is generating variation to explore and to exploit the selected individuals to generate a new population of solutions. The variation operators are two mechanisms:

Birth: two surviving individuals are selected randomly and the range of variation of its characteristics is defined. Births are accompanied by a performance test, and the new individual should have better performance to the last survivor of the population to be part of the new population.

Mutation: A predefined percentage of the population chromosomes are mutated at random within a range of calculation. A mutant is taken into account for the new population if their performance is better than the individual who replaced.

### Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II)

The NSGA-II procedure (Deb 2008a) attempts to find multiple Pareto-optimal solutions in a multi-objective optimization problem. It has the following three features:

Uses an elitist principle, i.e. it incorporates a mechanism for preserving the dominant solutions through several generations of a genetic algorithm.

Uses an explicit diversity preserving mechanism.

Emphasizes non-dominated solutions.

The optimization process that follows this algorithm is detailed below and represented schematically in Figure 4.8 and detailed following.

#### NSGA-II programming and optimization features

Related to the sustainability analysis and the different objective functions in the integrated model, particular attention should be paid to the amount of objective functions to compare by optimization. Because, in multiobjective problems graphical representation of the optimization results has a great importance in the analysis and decision making process (Blasco et al. 2017). In fact, depending on the number of objective functions to be optimized and the type of graphic to be performed there will be a number of possible combinations. For explain it, on table 4.4 it is presented the quantity of graphics that will be generated depending on the total amount of objective functions and on the graphic type.

**Table of contents :**

**Chapter I. Why develop a biorefinery? **

1.1. Introduction

1.2. Biorefinery background

1.3. Biorefinery integration degree

1.4. Special characteristics of biorefineries

1.5. Summary

1.6. References

**Chapter II. Key challenges and requirements for sustainable and industrialized biorefinery supply chain: A bibliographic analysis **

2.1. Introduction

2.2. Challenges and requirements for a sustainable biorefinery supply chain conception

2.3. Methods

2.4. Results

2.5. Discussion

2.6. Summary

2.7. References

**Chapter III. Conceptual framework: Decision-making on sustainable biorefinery supply chain **

3.1. Introduction

3.2. Operations Research

3.3. General methodology proposition for decision-making on sustainable Phase III BioRSC projects

3.4. Discussion

3.5. Summary

3.6. References

**Chapter IV. Methodology proposition: Modeling strategy methodology and bibliographic study for the selection of optimization techniques **

4.1. Introduction

4.2. Model construction proposition for sustainable Phase III BioRSC design

4.3. Model resolution methodology for a Multi-Objective Optimization Problem

4.4. NSGA-II programming and optimization features

4.5. Summary

4.6. References

**Chapter V. Model construction by sustainability dimensions analysis **

5.1. Introduction

5.2. Economic dimension analysis

5.3. Political dimension analysis

5.4. Technological dimension analysis

5.5. Social dimension analysis

5.6. Environmental dimension analysis

5.7. Summary

5.8. References

**Chapter VI. Case study parameter description **

6.1. Introduction

6.2. Parameter definition for strategic decisions in SC and the BioRSC characteristics

6.3. Parameter definition for the economic dimension equations

6.4. Political dimension analysis

6.5. Technological dimension analysis

6.6. Social dimension analysis

6.7. Environmental dimension analysis

6.8. Conclusions

6.9. References

**Chapter VII. Multiobjective algorithm and optimization results **

7.1. Introduction

7.2. NSGA-II parameters and parents production

7.3. Multiobjective optimization results

7.4. An example of optimal solutions with compromises between the sustainability dimensions

7.5. Sensitivity analysis

7.6. Model validation

7.7. Conclusions

7.8. References

**Chapter VIII. Conclusions and perspectives **

Appendix Chapter V

Appendix Chapter VI