{"id":6742,"date":"2018-10-19T04:40:19","date_gmt":"2018-10-19T02:40:19","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=6742"},"modified":"2018-10-19T08:14:07","modified_gmt":"2018-10-19T06:14:07","slug":"gis-and-decision-support-2-the-crossing-of-two-fuzzy-criteria","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/gis-and-decision-support-2-the-crossing-of-two-fuzzy-criteria\/","title":{"rendered":"GIS and decision support (2): the crossing of two fuzzy criteria"},"content":{"rendered":"\n

Let\u2019s reconsider our example regarding the towns ranking\nfor the Finist\u00e8re region according to the following two criteria: population\nand area. <\/p>\n\n\n\n

How to rank entities according to<\/strong> a single<\/strong> criterion<\/strong> \n\nTo establish the fuzzy number allowing the\nclassification of the towns according a single criterion, we rely on our assessment\nof two qualities: the complete satisfaction of the criterion (1) or the complete\ndissatisfaction (0).\n\n\n\n<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

By accomplishing this task, we have generated a complete\nseries of intermediate values, between 0 and 1. Working with numbers is not\neasy but, above all, it is not natural. Let\u2019s say that a town meets the 0.356\nsurface criterions and that the town surface is rather unsatisfactory, it does\nnot affect the classification process but it sure does the operator. <\/p>\n\n\n\n\n\n\n\n

Usually, we use a 5 satisfaction values scale:\nexcellent, good, average, mediocre, bad. Our brain can, sometimes, to manage a\nscale of 7 classes, adding two further values, very good and very mediocre. But\nbeyond that, it becomes metaphysical. <\/p>\n\n\n\n

Therefore we will rely on a classification with 5\nclasses for the rest of our example. <\/p>\n\n\n\n

As you could have already noticed, classifying the towns\naccording to a single criterion whatever the population or area, it is not too\ncomplicated with a fuzzy number. But you will, also, observe that it involves a\nsubjective aspect that is not unique. Just as much as we have only one classification\nin Boolean logic when we define a range, for example 2500 to 5000 inhabitants, we\ncould have different low and high limits according to each operator to define\nwhen an average town is completely excluded. In our example we used 1500 and\n7500 but this is fully subjective. For someone else it could be 1500 and 6000,\netc. <\/p>\n\n\n\n

Here we hit an essential point when working with\nfuzzy logic: there is not ONE possible result, there are as many as possible actors.\nAnd this is not a question of knowing which the REAL result is. They are all\ntrue because they correspond to the vision of each concerned actor concerned. <\/p>\n\n\n\n

How to rank entities according to<\/strong> two<\/strong> criteria<\/strong> <\/p>\n\n\n\n

Once you have performed both rankings according to\neach of the chosen criteria, it is time to classify the towns according to the\ntwo associated criteria: those who fully satisfy both criteria will be at the\ntop of the ranking, those who do not satisfy both criteria, at the bottom. And those\nthat partially satisfy one of the criteria will be in the middle.
\nIn the middle, but, where and in which order? <\/p>\n\n\n\n

Let\u2019s forget the numbers and use our five classes.\nA town that wholly satisfies one of the two criteria and that does not satisfy\nat all the other criterion; we\u2019ll consider that it satisfies both as excellent,\ngood, average, poor or bad? <\/p>\n\n\n\n

Well, it depends! Depending on the actor it will adopt\nany value. Therefore the goal is not to know which is the right way to cross\nboth criteria, but to determine how our actor realizes this crossing in his\nhead. <\/p>\n\n\n\n

To find the formula underlying mathematical the reasoning\nof our actor, it is enough to propose him three crossings <\/p>\n\n\n\n

  • How does he classify a town that totally satisfies the population criterion and not at all the area one? <\/li>
  • How does he classify a town that moderately satisfies both\ncriteria?<\/li>
  • How does he classify a town that totally satisfies the population criterion and moderately the surface\none? <\/li><\/ul>\n\n\n\n

    In our example for the towns of the Finist\u00e8re area\n, the ranking results from the crossing of both parameters taken into account. The\ndefinition of the crossing function is performed by the response to the three\ncases: <\/p>\n\n\n\n
    \n Population<\/strong> \n <\/td>\n Area<\/strong> \n <\/td>\n Expected Result<\/strong>\n <\/td><\/tr>
    \n 3750 (\n very good) \n <\/td>\n 1500 (\n bad ) \n <\/td>\n average\n <\/td><\/tr>
    \n 2000 ( average\n ) \n <\/td>\n 2250 ( average\n ) \n <\/td>\n average\n <\/td><\/tr>
    \n 3750 (\n very good) \n <\/td>\n 2250 ( average\n ) \n <\/td>\n Rather\n good \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

    The mathematical translation of these responses is\n: R = average ( Population, Area ) \n\nThe tool we have developed allows performing this\noperation as well as the crossing: \n\n\n\n<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe\nresult is the following: \n\n\n\n<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    We find 20 towns located in the range of\nsatisfaction for both criteria 0.8-1.0 \n\nThis result has to be compared with the result when\nusing Boolean logic (see previous article): \n\n\n\n<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    where we found 10 towns satisfying both criteria . <\/p>\n\n\n\n

    With the same data, if the operator enters different\nanswers in the table: <\/p>\n\n\n\n
    \n Population<\/strong> \n <\/td>\n Area<\/strong> \n <\/td>\n Result<\/strong> wish<\/strong> \n <\/td><\/tr>
    \n 3750 (\n very good) \n <\/td>\n 1500 (\n bad ) \n <\/td>\n bad \n <\/td><\/tr>
    \n 2000 ( average\n ) \n <\/td>\n 2250 ( average\n ) \n <\/td>\n average\n <\/td><\/tr>
    \n 3750 (\n very good) \n <\/td>\n 2250 ( average\n ) \n <\/td>\n average\n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

    The mathematical translation of these responses is:\nR = minimum ( Population, Area )
    \nThis function corresponds to the one used by classic tools. \n\nWith our tool we enter the following parameters:\n\n\n\n\n<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAnd\nwe obtain the result of applying this function to the data   : \n\n\n\n<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Considering these answers, now we get only 15 towns\nin the 0.8 – 1.0 satisfaction range, while we had found 20 with the previous\nanswers. <\/p>\n\n\n\n

    What<\/strong> conclusions to<\/strong> draw<\/strong>?<\/strong> <\/p>\n\n\n\n

    The main conclusions are the following: <\/p>\n\n\n\n

    • The transformation of each criterion into a fuzzy function brings the notion of nuance into each criterion. We do not proceed by selection (which implies rejection and abandonment\nof some elements) but by ranking the objects according to each criterion. <\/li>
    • Whatever the number of criteria used at any time, the set of objects is present in the crossings. An object that is misclassified by the first cross can to be \u00a0\u00bb caught up \u00a0\u00bb by later crossings. <\/li>
    • In any ranking there is a part of subjectivity due to the operator. This subjectivity can be measured and modelled by simple mathematical functions. According to the operator (decision maker) the result will not be exactly the same. <\/li>
    • In any case, this method provides richer results and closer to the reality of the decision process. It allows, also, determining the differences of appreciation between actors (different crossing methods) and highlighting\nthe source of divergence rather than the discordant final result. <\/li><\/ul>\n\n\n\n

      The use of\nfuzzy sets blurred on GIS existing data, allows a range of answers that offer\nall the intermediate variations between completely satisfactory and completely unsatisfactory.\nTherefore, the user benefits with  a\nresponse closer to his reasoning – nuanced – and his decision criteria  is\nno longer bounded, exclusively, to the wholly satisfactory responses. A special\nadvantage can be observed in the case where no answer fully corresponds to the\nselection criteria: a traditional query gives 0 result while the answers come\nclosest to these criteria.
      \nBy setting mechanisms of the theory of possibility in the query tools and data\naggregation allows to introduce notions of imprecision and uncertainty that\nintervene both at knowledge and decision process. The information system is\ncompatible with poorly known or poorly defined information: information can\nbecome more certain or more   uncertain over time while  \nkeeping the same measured value. <\/p>\n\n\n\n

      In the next article we will introduce the two tools\nwe have developed for ArcGis to transform an attribute as fuzzy number and to\nperform the aggregation of two fuzzy criteria. You will have the chance to\ndownload and use them as you please. <\/p>\n","protected":false},"excerpt":{"rendered":"

      Let\u2019s reconsider our example regarding the towns ranking for the Finist\u00e8re region according to the following two criteria: population and area. How to rank entities according to a single criterion To establish the fuzzy number allowing…<\/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-6742","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1KK","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6742","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=6742"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6742\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=6742"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=6742"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=6742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}