{"id":6614,"date":"2018-10-10T04:01:10","date_gmt":"2018-10-10T02:01:10","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=6614"},"modified":"2018-10-17T12:00:47","modified_gmt":"2018-10-17T10:00:47","slug":"gis-and-decision-support-8-aggregation-of-uneven-relevance-criteria","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/gis-and-decision-support-8-aggregation-of-uneven-relevance-criteria\/","title":{"rendered":"GIS and decision support (8): aggregation of uneven relevance criteria"},"content":{"rendered":"\n<p>We have discussed how to aggregate two or more criteria as part of a\ngeographical objects ranking ( <a href=\"https:\/\/translate.google.com\/translate?hl=en&amp;prev=_t&amp;sl=auto&amp;tl=en&amp;u=http:\/\/www.sigterritoires.fr\/index.php\/sig-et-aide-a-la-decision-4-un-outil-pour-agreger-des-criteres-flous-avec-arcmap\/\">GIS and decision support (4): a tool for aggregating fuzzy criteria<\/a>\n<a href=\"https:\/\/translate.google.com\/translate?hl=en&amp;prev=_t&amp;sl=auto&amp;tl=en&amp;u=http:\/\/www.sigterritoires.fr\/index.php\/sig-et-aide-a-la-decision-4-un-outil-pour-agreger-des-criteres-flous-avec-arcmap\/\">\u00a0with ArcMap<\/a>\n). The method employee requires the criteria be perceived by the decision maker\nas being of equal relevance. In this article we will discuss the theoretical\nbases to meet the uneven relevance criteria. <\/p>\n\n\n\n<p>To determine the aggregation method to be used across two criteria, we\npropose to the user to evaluate three situations: <\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The first question (S1) proposes a very poor value for the first criterion and an\nexcellent value for the second criterion. <\/li><li>The second question (S2) proposes two average\nvalues for both criteria. <\/li><li>The third question (S3) proposes an average value for the first criterion and an excellent value for the second . <\/li><\/ul>\n\n\n\n<!--more-->\n\n\n\n<p>The three replies (triplets) are, then, used as key input for a table of\naggregation methods including 50 possible replies (triplets). <\/p>\n\n\n\n<p>One of the constraints of this method is that both criteria have equal\nrelevance, ie that the answers to the three questions are the same whichever\nthe criteria order. This is called Replies \u201csymmetry\u201d. <\/p>\n\n\n\n<p>Two criteria have the same importance if the aggregation function is symmetrical,\ni.e. if the answer to the three assessment questions is the same if we reverse\nthe order of the criteria. <br\/>\nFor example, in the case of choosing a car, the consideration of the criteria\n\u00ab\u00a0colour and price\u201d, we can build the first question in two different ways:\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>a) a totally incompatible colour (E) and a fully compatible price (A) if we consider C1 = colour and C2 = price, or so <\/li><li>(b) a completely incompatible price (E) and a completely compatible colour(A) if we consider C1 = price and C2 = colour . <\/li><\/ul>\n\n\n\n<p>If both criteria have the same relevance, the answer to this question\nwill be the same in both cases . Through this reply, the subjective way of\naggregating the two criteria ( conjunction or disjunction) or the underlying compromise\nmechanism that the decision maker uses, will be considered. <br\/>\nBy cons, if one of the two criteria is more relevant than the other, the\nsymmetry is not verified. In our example it would not be unusual to get\n\u00ab\u00a0good\u00a0\u00bb as reply, if the price is our concern, and not the colour, and\n\u201cpoor\u201d if the colour satisfies us but not the price. <br\/>\nIn this scenario, the aggregation operations table is no longer valid. <br\/>\nThe concept of relevance for a criterion in relation to another has not been,\nyet, solved. The meaning we give to this word is very variable according to the\ndecision makers or the situations. <br\/>\nUnlike equal relevance aggregation criteria, for which we can find the\ndevelopment of the calculations in the current literature, a method to treat in\nthe case of uneven relevance criteria aggregation has to be developed. <\/p>\n\n\n\n<p><strong>Problem statement<\/strong> <\/p>\n\n\n\n<p>How to enhance the list of Questions SI, S2, S3 with the lowest number\nof new questions to determine: <\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u00a0 if\nthe function aggregation is symmetrical or not, and therefore if we can use the\naggregation operations table of equal relevance goals; <\/li><li>if the\nfunction is not symmetrical , what is the weight relative of each criterion C1\nand C2? <\/li><\/ol>\n\n\n\n<p><strong>Suggested<\/strong> <strong>solution<\/strong>. <\/p>\n\n\n\n<p>We have S1 (E, A), S2 (C, C), S3 (C, A). We suggest to add S4 (A, E),\nthat is to say the symmetrical question to S1 comprising a proposal fully\ncompatible with criterion C1 and another totally incompatible with criterion\nC2. <\/p>\n\n\n\n<p>All responses forming a doublet S1, S4 (AA, BB, CC, DD, EE) refers to equal\nrelevance criteria treatment. <\/p>\n\n\n\n<p>The doublets (A, E) and (E, A) correspond to a particular case where the\nweight of a criterion is equal to 0, the aggregation is not necessary because\nthe result is equal to C1 in the case of (A, E), or C2 in the case of (E, A). <br\/>\nFor the other possible doublets it is necessary to determine what aggregation\noperation can be used, before determining the weights to be applied. <\/p>\n\n\n\n<p>Among the aggregation operations, min, max and the symmetrical sums, can\napply only on symmetrical criteria, and therefore they have to be eliminated ex\nofficio . <\/p>\n\n\n\n<p>Among the average operations, only the arithmetic average can give a\nresult different from 0 in the case when one of the criteria is 0 (\u221a xy = 0 and\n2xy \/ ( x + y ) = 0 if x = 0 or y = 0). <\/p>\n\n\n\n<p>Therefore we will withhold the arithmetic average as aggregation operation\nas follows <\/p>\n\n\n\n<p>( Px.x + Py.y ) \/ ( Px + Py ) <\/p>\n\n\n\n<p>Px and Py being the respective weights for C1 and C2 criteria. <\/p>\n\n\n\n<p>In the case of doublets (D, B) and (B, D) it is easy to demonstrate that\nthe weights have to be 3 and 1 for (D, B) and 1 and 3 for (B, D). <\/p>\n\n\n\n<p>There are no other possible doublets (DC , DA , &#8230;) if Px and Py are\nconstant. The other doublets assume that Px = f (x) and Py = f (y). <\/p>\n\n\n\n<p>We can conclude that the weighting of objectives is, only, necessary when\nthe number of classes is greater than three, and cannot apply, for example, to\na criterion that would be : good, average , poor . In this case it would always\nbe within the field of a symmetrical function . <\/p>\n\n\n\n<p>In the case of n = 5, \u00a0 only the weighting factor 3-1 is usable in\norder to keep on within the range of a human decision maker. <\/p>\n\n\n\n<p><strong>Practical<\/strong> <strong>solution<\/strong> <\/p>\n\n\n\n<p>There are 25 possible combinations answers to question S4, S1\nsymmetrical. We have divided them in 5 groups \u00a0 : <\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Replies confirming the equal relevance of both criteria: five combinations, the doublets AA, BB, CC, DD and EE. The aggregation formula is sought in the table of 50 triplets ignoring the S4 reply. <\/li><li>Replies demonstrating the lack of consideration of a whole criterion: two combinations, the doublets AE and EA. The result of the\naggregation is directly the value of the criterion taken in account, ignoring the value of the other criterion. <\/li><li>The replies where the difference between S1 and S4 is 0,5 and which imply a different weight from the two criteria ( weight 1 and 3); six combinations , doublets AC, BD, CA, CE, DB and EC. We use the weights of 1 and 3 for the input criteria and the arithmetic average. <\/li><li>Answers approaching an equal relevance criteria, but dubious. The difference between the answers is 0.25 \u00a0 : six combinations, the doublets AB, BA, BC, CB, DC and CD. The aggregation formula is sought in the table of 50 triples ignoring the S4 reply, then we shade the 0, 25 result obtained for the most relevant criterion. <\/li><li>The replies that come close to not taking into account one of the two criteria. The gap between the two responses is 0 , 75 \u00a0 : six combinations , the doublets AD, BE, ED, DA, DE and EB. The result of the\naggregation will be the criterion considered as relevant, shadowed by 0, 25 for the less important criterion. <\/li><\/ul>\n\n\n\n<p>\n\n\n\n\n\n\n\n\n\nThe following table details the 25 combinations\nand the aggregation formulas used. \n\n\n\n<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"283\" height=\"630\" data-attachment-id=\"6616\" data-permalink=\"https:\/\/www.sigterritoires.fr\/index.php\/en\/gis-and-decision-support-8-aggregation-of-uneven-relevance-criteria\/formu\/\" data-orig-file=\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/formu.png?fit=283%2C630&amp;ssl=1\" data-orig-size=\"283,630\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"formu\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/formu.png?fit=283%2C630&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/formu.png?resize=283%2C630&#038;ssl=1\" alt=\"\" class=\"wp-image-6616\" srcset=\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/formu.png?w=283&amp;ssl=1 283w, https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/10\/formu.png?resize=135%2C300&amp;ssl=1 135w\" sizes=\"auto, (max-width: 283px) 100vw, 283px\" \/><\/figure>\n\n\n\n<p>\u00a0In the next article we will discuss an ArcMap command that allows performing these uneven relevance aggregation criteria.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We have discussed how to aggregate two or more criteria as part of a geographical objects ranking ( GIS and decision support (4): a tool for aggregating fuzzy criteria \u00a0with ArcMap ). The method employee requires&hellip;<\/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-6614","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1IG","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6614","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=6614"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6614\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=6614"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=6614"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=6614"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}