We have a set \u03a9<\/em><\/strong> of objects to classify according to a set C<\/em><\/strong>\nof criteria. The number of objects is finite. The partial evaluations of the objects\naccording to each criterion take values \u200b\u200bin easily identifiable sets. Approach assumption.<\/strong> \n\nThe objective associated with a criterion\n(information layer) will be described as a fuzzy set. The values \u200b\u200bof the\npixels for the layer located in the core will, therefore, be perfectly\ncompatible with the goal, while the values \u200b\u200blocated outside the support are, completely,\nincompatible. Even if the estimation of a mathematical function linking the depth to\nthe adequacy of the site for the oyster culture cannot be performed in an exact\nway, the shape of the curve makes it possible to express certain behaviours of\nthe decision-maker. This is why; in general, it is preferable to use a discrete\nnotation scale, usually comprising 5 levels, maximum 7, according to the\ndecision-maker’s perception threshold. Display items Search: <\/p>\n\n\n\n Showing 1\nto 5 of 5 entries <\/p>\n\n\n\n Previous\nNext \n\nRepresenting the criterion using a fuzzy\ninterval allows for a more convenient and abundant info representation. Indeed,\nthe decision-maker must provide a desired value, for example, of bathymetry. He\nmust establish an interval but the question necessarily arises: should he fix\nthis interval by being pessimistic and, thus, establishing distant boundaries,\nor being optimistic and thus tightening the limits? The criteria aggregation <\/strong><\/p>\n\n\n\n We will consider the case of a pair of criteria. The generalization in\nthe case of n criteria where n> 2 is presented in another document. <\/p>\n\n\n\n Two scenarios must be considered: <\/p>\n\n\n\n 1<\/strong>: Criteria<\/strong> of equal importance.<\/strong> <\/p>\n\n\n\n Two criteria of equal importance can be crossed according to the\nall-or-nothing principle or by introducing nuances. The principle of all or\nnothing excludes any compromise between the two criteria and results in two\naggregation operations: conjunction or disjunction. The conjunction is used in\nthe case where one wishes the simultaneous satisfaction of the two criteria\n(the \u00ab\u00a0and\u00a0\u00bb logic). That is, the overall assessment can only be better\nthan the worst of the partial evaluations. Disjunction is used in cases where the criteria are redundant (the logical \u00ab\u00a0or\u00a0\u00bb). That is, the overall assessment will be equal to the best of the partial evaluations. Procedure for determining the aggregation\noperation.<\/strong> <\/p>\n\n\n\n In the case where two objectives are aggregated, there is a simple\nprocedure for determining the type of operation to be performed. It consists in\nproposing to the decision-maker three typical situations and asking him to\nevaluate them. Considering the three answers given, we search in a functions catalogue\nthe one that better fits to the wishes of the decision maker. \n\n\n\n\n\nWe obtain three answers (RI, R2, R3) and we search\nfor the aggregation operation in the following table. \n\n\n\n<\/p>\n\n\n\n This table is not exhaustive and concerns only to the most common\nanswers. In reality, the set of possible answers has 50 triplets. These\ntriplets must however respect the following constraints: In the previous series of articles we have discussed examples of application for the Boolean and fuzzy logic, as well as two commands for ArcGis allowing the application of fuzzy logic to geographic information. We will…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"give_campaign_id":0,"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"sfsi_plus_gutenberg_text_before_share":"","sfsi_plus_gutenberg_show_text_before_share":"","sfsi_plus_gutenberg_icon_type":"","sfsi_plus_gutenberg_icon_alignemt":"","sfsi_plus_gutenburg_max_per_row":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1260],"tags":[],"class_list":["post-6628","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1IU","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6628","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/comments?post=6628"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6628\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=6628"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=6628"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=6628"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nA partial objective will be seen as a fuzzy set restricting the acceptable\nvalues \u200b\u200bof the associated criterion. Therefore, we accept the implicit hypothesis\nthat each objective defines a total order for \u03a9<\/em><\/strong> .
\nWe will use as an example the case of a set \u03a9<\/em><\/strong> representing the\npixels of a study area that we wish to classify according to their ability to\nreceive aquaculture breeding sites. The criteria set C<\/em><\/strong> is the dataset\nlayers available: bathymetry, slope, substrate, productivity, etc. Each of\nthese info layers adopts easily identifiable values : favourable, somehow favourable,\nunfavourable, and so on.
\nFor each layer of information we will set a goal, for example, for bathymetry\nthat is at least favourable, for productivity that is at least unfavourable,\nand so on. The goal is none other than the subset of the acceptable values \u200b\u200bof\nthe info layer.
\nFinally, we accept the hypothesis that each layer of information can be\nclassified in its entirety by the set goal, that is to say that we are able for\neach pixel to determine the corresponding value of the layer. <\/p>\n\n\n\n\n\n\n\n
\nIf we use only two values categories, for example favourable and unfavourable,\nwe will have for Bathymetry the following representation of the goal. \n\n\n\n<\/p>\n\n\n\n![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/f1.png?resize=630%2C216&ssl=1\")
<\/figure>\n\n\n\n
<\/p>\n\n\n\n
\nA simple way is to, linguistically, express compatibility levels between goal\nand evaluation, and then project them to [0 , 1 ] using the following table: <\/p>\n\n\n\n <\/td><\/tr> \n Linguistic Appreciation<\/strong> \n <\/td> \n Level of compatibility consequence – objective<\/strong> \n <\/td> \n Digital convention in [0,1]<\/strong> \n <\/td> \n Ordinary Convention<\/strong> \n <\/td><\/tr><\/thead> \n Very\n well \n <\/td> \n Fully\n compatible \n <\/td> \n 1 \n <\/td> \n AT \n <\/td><\/tr> \n Good\n <\/td> \n Rather\n compatible \n <\/td> \n 0.75 \n <\/td> \n B \n <\/td><\/tr> \n Pretty\n good \n <\/td> \n Moderately\n compatible \n <\/td> \n 0.5 \n <\/td> \n C \n <\/td><\/tr> \n Poor \n <\/td> \n Weakly\n compatible \n <\/td> \n 0.25 \n <\/td> \n D \n <\/td><\/tr> \n Very\n bad \n <\/td> \n Incompatible\n \n <\/td> \n 0 \n <\/td> \n E \n <\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\nThe fuzzy interval makes it possible to have both representations at once: the\npessimistic interval will be the support and the optimistic interval the\nnucleus.
\nFor example: if the decision maker considers that it is impossible to raise\noysters at a depth less than 4m and more\nthan 25m, but the optimum depths are between 8 and 12m, we will have as\nobjective the following fuzzy interval: \n\n\n\n<\/p>\n\n\n\n![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/f2.png?resize=618%2C320&ssl=1\")
<\/figure>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n\n
\nExample: aggregation of the substrate and productivity criteria. If the\ndecision-maker’s attitude implies the simultaneous satisfaction of the two\ncriteria, this means that if the substrate is moderately favourable and the\nproductivity is very favourable, the result of the aggregation of the two\ncriteria will be the most unfavourable of the two, that is to say moderately favourable.\n<\/p>\n\n\n\n
Example: aggregation of \u201cwater quality\u201d and \u201cproductivity\u201d criteria. If the decision-maker’s attitude implies a redundancy of these two criteria, it means that if the quality of the water is average and the productivity is very good, the result of the aggregation will be the most favourable of the two, that is to say \u00ab\u00a0very good\u00a0\u00bb.
A third attitude of the decision maker leaves aside all or none to introduce nuances into the aggregation. If the objectives become nuanced, the compromise between the two criteria becomes one of the natural attitudes of the decision maker.
The compromise results in the fact that the overall assessment is at an intermediate level between the partial evaluations. Taking the example of water quality and productivity, if one has average quality and excellent productivity, the result will be; for example, \u00ab\u00a0good\u00a0\u00bb.
On fuzzy sets, this type of set-up operation is performed using two families of aggregation operations: symmetric sums and parametric medians.<\/p>\n\n\n\n
\nThe three typical situations (Si, S2, S3) are chosen according to the two\ncriteria (C1, C2) so that: <\/p>\n\n\n\n\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/f3-1.png?resize=437%2C512&ssl=1\")
<\/figure>\n\n\n\n
<\/p>\n\n\n\n
\n1) R3 \u2265 max (R1, R2), the evaluation of a situation that completely satisfies\ncriterion 2 and moderately criterion 1 must be at least equal to the best\nevaluation of the other two situations (R1 and R2), where number one does not\nsatisfy the first criterion at all and the other satisfies only moderately the\ntwo criteria;
\n2) R3\u2265 note C or 0.5, the total satisfaction of the second criterion cannot\nbring down the overall satisfaction below the level of satisfaction of the\nfirst criterion;
\n3) the aggregation function must be symmetrical, ie the objectives are of equal\nimportance and can therefore be interchanged in the aggregation process. (Caution:\nto say that the objectives are of equal importance does not imply that they are\nof the same critical nature, see below: objectives of unequal importance). \n\nTo be discussed in the following article … \n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"