Pareto curve

Figure 7.32 Nominal and robust Pareto curves for four-column SMB, Varicol, and PowerFeed processes ( ) the robust Pareto curves (reproduced from Mota, Araujo, and Rodrigues, 2007).

Each point in the Pareto curve represents an optimal combination of the objective functions. The selection of the operating conditions for a production is a matter of additional economic considerations. Any point above the Pareto curve has tvorse performance than the points constituting the Pareto curve, because there is aWays one element of the Pareto set which has a lower solvent consumption for the same productivity. The points located below the Pareto curve are not feasible because the purity requirements are not fulfilled. ... 

Figure 1231 Deterministic Pareto curve profit-customer satisfaction...

The solution of the optimization problem is depicted in a 2D plot of the involved objectives (figure 8.6). Each non-inferior attainable optimal solution is estimated at a given combination of objectives including constraint g xi,X2)- All these points define a curve of non-inferior solutions, normally referred to as Pareto curve. For the sake of transparency a linear relation between objectives is considered (. e. F = TjWi x fi), where Wi is the weighting factor of the objective function fi and Ylwi = 1. As expected, the utopia point is given by the coordinates... 

This curve is termed Pareto curve cf. explanatory note 8.1) and clearly shows that one objective function can only be improved at the expense of the other objective function. This trade-off leads to significant deviations between optimal designs and the utopia point (up to 60%) for the pair /4on /exergy The decision maker should have in mind the importance of compromising both objectives at early stages of the design cycle. 

The exergy loss (or entropy produced) can be extracted from the Pareto curve presented in figure 8.9. The relevant data for the classical and green designs and for the fully exergy-driven design i.e. w = [0 l]" ) are hsted in table 8.7. 

The shadow prices can be computed directly out from the data in table 8.7 or from the Pareto curve in figure 8.9. It was foimd that, when compared to the classical case,... 

Figure 4.24 The Pareto curve showing the tradeoff between total annualized cost (TAC) and the pollutant concentration at the final discharge location [52].

Figure 1 shows the Pareto curve for the deterministic case. This curve was obtained maximizing the NPV and constraining the consumer satisfaction. The curve shows that only above a 66% consumer satisfaction level some trade off between the objectives exists. Below 66% of requested consumer satisfaction the solution is the same as that of the model without constraining consumer satisfaction and therefore all the pareto solutions accumulate at the end point on the left. Figure 2 shows the same curve for the stochastic model. Figures 3 and 4 show the corresponding consumer satisfaction and financial risk curves of the pareto solutions of the multiobjective stochastic problem. Unsupported solutions are suspected to exist, but this could also be the effect of the small number of scenarios (100) used. In future work this matter will be resolved. 

Figure 7. Composite Risk Pareto Curves. Figure 8. Composite Risk Curves.

Constraints on regenerator temperature and coke content of regenerated catalyst were required in order to prevent catalyst deactivation due to thermal damage and excessive coke deposition. The Pareto optimal solutions obtained along with the decision variables corresponding to each point on the pareto curve is given in figure 3. 

Figure 3. Shifting the Pareto Curve through Solvent Substitution.

The formulation of the s-constraint technique is performed as one of the objectives is assigned as the objective function while the others are constrained within specified upper limits. The selected process parameters are assigned as the decision variables of the optimisation problem. The optimiser searches over the process variables, within the feasibility and constraints regions and feeds these selected variables to the model in HYSYS. Then, it waits for the process in HYSYS to converge and then recalculate the objectives and evaluate the optimisation results. This search loop between the optimiser in Excel and the model in HYSYS continues until a global optimum point is found which represents a point on the Pareto curve. The above optimisation process is repeated for different bounds of the constrained objectives to develop the entire Pareto curve. 

The E-constraint method was used to solve the multi-objective optimisation problem and obtain the Pareto curve. Here, the economic objective was optimised while the environmental objective was converted into a constraint with a specified upper bound as shown in Eq. (15). This multi-objective optimisation problem was performed for each designed HEN. 

Where x is the selected design variable of the recycled amount of HCl and each Pareto curve is generated by parametrically varying the upper bound (s) on the environmental objective over each entire range and solving the optimisation problem for each case. 

The optimization run was carried out with a population size of 200 during 200 generations. The parameter spac was fixed at a value of 20. The Pareto curve generated is presented in Fig. 9a where is compared with that of the e-constraint SRES method. Again, we can observe that the most economic designs have a very poor controllability. Really, for ISE values greater than 1, the design variables are essentially the same, and only the eontroller parameters are modified. On the other hand, this curve is quite similar to that of the s-constraint, except for a... 

In Fig. 9 we compare the different Pareto curves generated by the techniques considered in this work. It is possible to observe that the e-constraint SRES method that we presented here gives a very good approximation of the Pareto front, although it was not possible to obtain an even spread of points. 

With a product range of approximately 15,000 SKUs there is inevitably a long-tail on the sales pareto curve - for example 88 per cent of WDF s SKUs sell less... 

Figure 7.8 Pareto curve between the profit and environmental impact. Reproduced with permission from Martfnez-Cuido et al. (2014), 2014, American Chemical Society...

Fig. 2.6 Corporate value-profit Pareto curve, a Corporate value versus profit, b Corporate value versus NPV. c NPV versus profit...

Furthermore, there is a dependence of the SC structure on its total production. Other works related to SC design and environmental issues consider that demand must be completely fulfilled. This assumption leads to an invariable total production rate and suboptimal solutions. In Fig. 6.5 the iso-production curves correspond to solutions following this assumption. For these cases, minimum overall impact always leads to negative NPVs. These solutions are obviously dominated by the zero-production solution (origin). The actual Pareto curve is shown in Fig. 6.5 as a... 

Fig. 6.5 Overall environmental impact versus NPV Pareto curve (in gray iso-production curves)...

A maleic anhydride SC case study is presented where two potential technologies are available. Two problems were solved, a first approach that did not consider CO2 trading scheme and a second one (see Sect. 6.4.1), that took it into account. A SC for MA production based on butane was found to be more environmentally friendly than one based on benzene. Most works related to SC and environmental issues consider a fixed production/demand, but it was demonstrated that such constraint leads to dominated solutions. By allowing unsatisfied demand, the actual Pareto curve was obtained. 

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