126 The UMAP Journal 30.2(2009) 10° We apply expert system and group decision theory to determine the weights of the species in Population 2. We have a multi-attribute deci on problem, where the aim is to select the optimal solution from many lternatives or to sort the available alternatives. Assume that the finite solution set is y=fy1 ., yny, the attribute set is C=fcl,., cab, and the decision expert set is E=el,., em] Let S=[s1,.., g be a predefined set consisting of odd-chain elements Expert ek selects one element from S as the value of solution yi under attribute c; i let it be denoted as pi E S, and let p=(ping denote the judgment matrix of expert ek on all the solutions for all the attributes. The attribute weight vector in evaluating information given by ert ek is W=wi, where wf is the weight of attribute c selected by expert ek from set S, v∈S This theory can be actualized through the Analytical Hierarchy Process (AHP), first put forward by American operational researcher T.L. Saaty in the 1970s AhP is a method for decision-making analysis that combines qualitative and quantitative methods. Using this method, decision-makers
126 The UMAP Journal 30.2 (2009) 12..xle 0 5 10 Figure 2. Numexical solutions for xi(t). We apply expert system and group decision theory to determine the weights of the species in Population 2. We have a multi-attribute decision problem, where the aim is to select the optimal solution from many alternatives or to sort the available alternatives. Assume that the finite solution set is Y = {yi, .. . ,y,}, the attribute set is C = {ci, . . . , cq}, and the decision expert set is E = {ei, ... , el}. Let S = {si,..., s.} be a predefined set consisting of odd-chain elements. Expert ek selects one element from S as the value of solution yi under attribute cj; let it be denoted as pikj E S, and let = (Pkj)n×q denote the judgment matrix of expert ek on all the solutions for all the attributes. The attribute weight vector in evaluating information given by expert ek is where wk is the weight of attribute cj selected by expert ek from set S, Wk ES. This theory can be actualized through the Analytical Hierarchy Process (AHP), first put forward by American operational researcher T.L. Saaty in the 1970s. AHP is a method for decision-making analysis that combines qualitative and quantitative methods. Using this method, decision-makers -X 0 ------- - --- -- - -- -- -- -- -- ----------- 4-- ------------ 48----- -- ------------------------------------------- --- L--------------------------- 6------------------------------ -------------- 24---- ------------------------------------------------------------------- /I SI I - - -I I 2------------------I----------------I---------------- I
Rebalancing 127 can separate complex problemsinto severallevels and factors, and compare and find the weights for different solutions, and provide the basis for the optimum solution AHP first classifies the problem into different levels based on the nature and the purpose of the problem, constructing a multilevel structure model ranked as the lowest level (program for decision making, measures etc ) compared with the highest level(the highest purpose). Based on AHP, we can establish the stratification diagram shown in Figure 3 Competitive relationship teen the decision Excreta quality adapt to the degree Oxygen s crustacea fish Figure 3. AHP stratification diagram. At last, we make consistency check of the result, finding that the consis- tency ratio of each expert's judgment matrix is below 1, so the consistency of the judgment matrix is acceptable. Finally we figure out the weight of the numbers of all the species in Population 2, as shown in Table 1 Table 1 Weight of each species in Population 2 as measured by AHP. Herbivorous fish Crustaceans Echinoderms Here we adopt population competition model to confirm the weight of es in Population 2: N1=N1(e1+nN2), N2=N2(e2+mM)
Rebalancing 127 can separate complex problems into several levels and factors, and compare and find the weights for different solutions, and provide the basis for the optimum solution. AHP first classifies the problem into different levels based on the nature and the purpose of the problem, constructing a multilevel structure model ranked as the lowest level (program for decision making, measures etc.), compared with the highest level (the highest purpose). Based on AHF, we can establish the stratification diagram shown in Figure 3. ................ .Competitive relationship .. . . . . . . . . . . .. . between the decision .. . . . . . . . . . . . . . Food requirements . Environment to . . . of the algae Excxeta quality, • adapt to the degree . . . . ............... .......... T O Figure 3. AHP stratification diagram. At last, we make consistency check of the result, finding that the consistency ratio of each expert's judgment matrix is below 1, so the consistency of the judgment matrix is acceptable. Finally we figure out the weight of the numbers of all the species in Population 2, as shown in Table 1: Table 1. Weight of each species in Population 2 as measured by AH-RP Species Weight Herbivorous fish .21 Crustaceans .23 Molluscs .31 Echinodermis .24 Here we adopt population competition model to confirm the weight of each species in Population 2: N1 = Ni(e1 +y71N2), N2. N2.(e2 + 'yNi)