GPS model

Genetic programming [137] is an evolutionary technique which uses the concepts of Darwinian selection to generate and optimise a desired computational function or mathematical expression. It has been comprehensively studied theoretically over the past few years, but applications to real laboratory data as a practical modelling tool are still rather rare. Unlike many simpler modelling methods, GP model variations that require the interaction of several measured nonlinear variables, rather than requiring that these variables be orthogonal. 

In order to analyze the experimental Raman spectrum of the carbonized film, we simulated four kinds of line spectra for Chito, DLC, AC, and GP models, respectively, on the left in Fig. 27.3, while four kinds of broader simulated spectra... 

In Fig. 27.6, we showed the simulated valence X-ray photoelectron spectra of Chito (chitosan 2-mer, C12H24N2O9), DLC (CioHi2(CH3)4), AC (CioHie), and GP (CieHio) models in the upper part of (a) (d), respectively, as the four constitutional contributions for the carbonized film from calculations of the SAOP method in ADF software. The upper simulated spectra of Chito, ALC, and GP models in (a), (b), and (d) of the figure are in considerably good agreement with the experimental ones in lower part of (a), (b), and (d), respectively. By considering the four component... 

Hansen FF, Wagner Gp. Modeling genetic architecture A multilinear theory of gene interaction. Theor Popul Biol 2001 59 61-86. 

Tan P, Tong L, Steven GP. Modelling for predicting the mechanical properties of textile... 

Consider the general MCMP problem given in Equation 6.21. The assumption that there exists an optimal soluhon to the MCMP problem involving multiple criteria implies the existence of some preference ordering of the criteria by the DM. The goal programming (GP) formulation of the MCMP problem requires the DM to specify an acceptable level of achievement b for each criterion f, and specify a weight W (ordinal or cardinal) to be associated with the deviation between f, and bj Thus, the GP model of an MCMP problem becomes ... 

Equation 6.22 represents the objective function of the GP model, which minimizes the weighted sum of the deviational variables. The system of equations... 

In the non-preemptive GP model, the buyer sets goals to achieve for each objective and preferences in achieving those goals expressed as numerical weights. Here, the buyer has three goals as follows ... 

The weights Wy Wz, and W3 can be obtained using the methods discussed in Section 6.3. The non-preemptive GP model can be formulated as... 

In this GP model, the DM only specifies the goals/targets for each objective. The model minimizes the maximum deviation from the stated goals. For the supplier selection problem the Tchebycheff goal program becomes ... 

For additional readings on the variants of fuzzy GP models, the reader is referred to Ignizio and Cavalier (1994), Tiwari et al. (1986), Tiwari et al. (1987), Mohammed (1997), and Hu et al. (2007). An excellent source of reference for goal programming methods and applications is the textbook by Schniederjans (1995). 

Write the objective fimction of the GP model for each part clearly. Explain any new variable(s) and constraints you add to the GP model for each part. 

Based on Phase 1 results, price is the most important goal followed by MtT goal on quality, lead time, and VaR in that order, dn and dX are the deviation variables representing how far the solution deviates from each goal. In general, goal values are to be provided by the DMs. To determine the optimal solution for a preemptive GP model, a sequence of optimization problems has to be solved. 

The Tchebycheff GP model minimizes the maximum weighted deviation from the stated goals. The objective function of the Tchebycheff GP model is as follows ... 

Fuzzy GP uses the ideal values as targets and minimizes the maximum weighted normalized distance from fhe ideal solution for each objective. An ideal solution is the vector of best values of each objective obtained by optimizing each objective independently, ignoring other objectives. In this application, ideal solutions were obtained by minimizing price, lead time, MtT and VaR type risks independently. If M equals the maximum deviation from fhe ideal solufion, then the fuzzy GP model is as follows ... 

Zi, Z2, Z3 and Z4 in the aforementioned formulation represent individual minima, that is, the ideal values for fhe respective objectives. Weight values Wi, W2, W3, and W4 are the same that are used in the non-preemptive GP model. The advantage of fuzzy GP is fhaf no target values have to be specified by the DM and there are no deviational variables. 

Four GP models discussed in Section 7.14.4 were used to solve the multiobjective supplier selection problem. Ideal solutions, obtained by optimizing each objective separately in the model, given by Equations 7.23 through 7.32, are given in Table 7.24. Optimal solutions obtained by each GP model are discussed next. 

In the preemptive GP model, price is given the highest priority, followed by MtT risk of quality, lead time and VaR risk respectively. This is consistent with the criteria rankings obtained from the company staff (Section 7.14.2). 

Target values for each of the objectives are set at 110% of the ideal value. For example, the ideal (minimum) value for price objective is 508,680 hence the target value for price is 559,548 and the goal is to minimize the deviation above the target value. Table 7.25 illustrates the optimal solution using the preemptive GP model. Suppliers S8, Sll, and S19 are selected and target values for objectives Zy Z2, and Z3 are achieved in the optimal solution. The optimal solution also provides the optimal order quantities for each product and supplier. 

Same target values are used in the Tchebycheff GP model also. Objectives are scaled as in the non-preemptive GP. Using the Tchebycheff method, the optimal solution obtained is illustrated in Table 7.27. The same three suppliers S8, Sll, and S19 are chosen and only the price goal is achieved. 

In the fuzzy GP model, ideal values (Table 7.24) are used as targets for the four objectives. The optimal solution obtained using fuzzy GP is shown in Table 7.28. 

The four GP models provided different optimal solutions and goal achievements. In order to compare the four solutions and their trade-offs, we use the value path approach (VPA) discussed in Ghapter 6, Section 6.4.9. VPA is one of the most efficient ways to demonstrate the trade-offs among conflicting criteria. The display consists of a set of parallel scales, one for each criterion, on which is drawn the value path for each optimal solution. 

In both preemptive and non-preemptive GP models, the DM has to specify fhe targets or goals for each objective. In addition, in the preemptive GP models, the DM specifies a preemptive priority ranking on the goal achievements. In the non-preemptive case, the DM has to specify relative weights for goal achievements. 

Under the preemptive GP model, if the DM indicates that/, is mucii more important than/2, then the objective function will be... 

Under the non-preemptive GP model, the DM specifies relative weights on the goal achievements, say Wj and W2. Then the objective function becomes... 

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