Solution Steps

modeFRONTIER® is a multi-disciplinary and multi-objective optimisation and design environment developed by Esteco SpA (ESTECO 2013). This study is the first case

Chapter Four

61

where LRPs are solved using this package. Extensive experimentation with the package reveals that the software is capable of handling LRPs by way of producing robust de- signs which have previously not been considered in the literature. However, the mode- FRONTIER® package is not designed particularly to handle LRPs. Therefore, extensive customisation of the robust design optimisation solver using multi-objective meta- heuristics has been made before implementing the two-layer MP-LRP. Design of Exper- iments (DoE) is coupled to the optimisers in such a manner that the solution of the two- layer MO-LRP is possible only if optimal feasible designs are obtained. A schematic description of the solution steps using modeFRONTIER® is presented in Figure 3.5:

Figure 4.5 The process of implementing the two-layer MO-LRP using mode-

FRONTIER®

4.5.1. Transforming the two-layer MO-LRP into modeFRONTIER®

The two-layer MO-LRP is transformed using modeFRONTIER®’s input components and then connected properly to create a full logical imitation of the mathematical model

Transforming the quantified two- layer MO-LRP into the modeFRONTIER® _language

Introduction of DoE to guide the chosen optimiser

Deployment of the optimisers

Setting-up the chosen optimiser followed by execution of the model

Analysing results  Scenario Analysis  Graphical Maps

DoE is used to define the initial population for the optimisers; using: - Design of Experiment

Sequence - Random - Sobol

- Uniform Latin Hypercube - Incremental Space Filler - Constraint Satisfaction

Using modeFRONTIER®

workflow components to transfer the mathematical model into mF format;

1. Following the logic of modeFRONTIER® _{in creating}

component of the model properly

2. Following the logic of mF in connecting the components of the model properly

Defined based on the type of optimiser;

Initial population table: guided by

DoE

Number of generations: For MOGA-II optimisers: Probability of crossover: 0.5 Crossover type: directional crossover Probability of selection: 0.05 Probability of mutation: 0.1 Mutation type: DNA string mutation

with ratio of 0.05

Elitism: Enable

Random generator Seed: 1 For NSGA-II optimiser: Max number of evaluations: 2000 Crossover probability: 0.9 Mutation probability in real-coded Vectors: 1.0

Mutation probability for binary Strings: 1.0

For MOPSO optimiser: Turbulence: 0.2 Random generator seed: 1

Genetic Algorithm-based optimisers: MOGA-II, NSGA-II, HYBRID Particle Swarm-based optimiser: MOPSO Simulated Annealing -

based optimiser: MOSA

 Refinement of results  Selection of results  Ranking the selected

results using TOPSIS

 Pareto frontier  ANOVA

Chapter Four

62

in modeFRONTIER®. The workflow of the model in modeFRONTIER® is presented in Figure 4.6:

Figure 4.6 The two-layer MO-LRP design in modeFRONTIER®

Figure 4.6, presents the logical design for the two-layer MO-LRP model in mode- FRONTIER® using the MOPSO optimiser. Graphically the design looks the same for all optimisers only the name of the optimiser is changed. The mathematical details of the developed model are all satisfied in this designed workflow using modeFRONTIER®.

4.5.2. Introducing DoE to the chosen optimiser

Logically in modeFRONTIER® the optimiser is connected to the main two-layer MO- LRP by the use of DoE. DoE generates the initial population sets for the optimisers to ensure the achievement of an optimum set of non-dominated solutions. The initial popu- lation table consists of 51 designs. The initial 51 DoE-guided designs consist of: (i) 10 ‘design of experiment sequence’, (ii) 10 ‘random’, (iii) 10 ‘sobol’, (iv) 10 ‘uniform Lat- in hypercube’, (v) 10 ‘incremental space filler’ and (vi) 1 ‘constraint satisfaction’ de- signs. The process of optimisation in modeFRONTIER® is DoE-guided by way of in- troducing the initial population table to the optimisers.

Constraint 3; AHP vector Decision variables with

Chapter Four

63 4.5.3. Deployment of optimisers

Five multi-objective evolutionary optimisers in modeFRONTIER® are initially selected to deploy the two-layer MO-LRP. The optimisers include three multi-objective Genetic Algorithm (GA)-based, one multi-objective Particle Swarm (PS)-based and one multi- objective Simulated Annealing (SA)-based. Based on extensive literature reviews in this area, it is believed that this is the first time that the modeFRONTIER® solver has been used to implement an LRP. Therefore, there is no report available on the performance of its multi-objective optimisers on LRPs. In this research disparate optimisers are used to compare the optimisers’ performance on the two-layer MO-LRP. Results reveal that two of these optimisers, viz., HYBRID and MOSA, do not perform efficiently in solv- ing the two-layer MO-LRP.

4.5.4. Setting-up the selected optimisers and executing the model

The chosen optimisers are set up separately. These optimisers have different requirements and distinctive specifications. Therefore they have disparate set up details. In order to compare the results obtained from these GA-based optimisers, the initial population and number of generations are kept the same in those optimisers. Table 4.11 presents the set up details for the GA-based optimisers:

Table 4.11 Set up details for GA-based optimisers in modeFRONTIER®

MOGA-II NSGA-II

Number of generations: 50 Initial population: 51 Probability of crossover: 0.5 Type of crossover: Directional Probability of mutation: 0.1 Type of mutation: DNA String DNA string mutation ratio: 0.05 Elitism: Enabled

Random generator seed: 1

Number of generations: 50 Initial population: 51 Crossover probability: 0.9 Mutation probability for real-coded vectors: 1.0

Mutation probability for binary strings: 1.0

Distribution index for real-coded crossover: 20.0

Distribution index for real-coded Mutation: 20.0

The set up specifications for MOPSO is presented in Table 4.12:

Table 4.12 Set up details for PS-based optimiser

MOPSO

Number of generations: 50 Initial population: 51 Turbulence: 0.2

Chapter Four

64

The workflow and the mathematical model in modeFRONTIER® remain the same during implementation of the LRP. In order to compare the optimisers’ performance, the initial population is kept the same in all generation-based optimisers. 50 generations with an initial population of 51, which generates 2,500 results. HYBRID in modeFRONTIER® works with number of iterations instead of generations. This optimiser is set at 2,500 evaluations with a DoE table that is identical to the other optimisers. Part-III of this chapter presents the categorised and analysed results.

Chapter Four

65