{"id":6754,"date":"2018-10-22T05:30:10","date_gmt":"2018-10-22T03:30:10","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=6754"},"modified":"2018-10-22T08:09:47","modified_gmt":"2018-10-22T06:09:47","slug":"gis-and-decision-support-1-classifying-with-fuzzy-numbers","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/gis-and-decision-support-1-classifying-with-fuzzy-numbers\/","title":{"rendered":"GIS and decision support (1): classifying with fuzzy numbers"},"content":{"rendered":"\n

Decision processes are based on information from very diverse source and\ntype. This information is used by the decision-makers to perform   choices,\ni.e. to retain a certain number of entities and to exclude others. <\/p>\n\n\n\n

Let’s discuss the following example:
\nAn action must be performed on municipalities, but this action depends on: 1. the area of \u200b\u200bthe communes, between 2500 and 3000 hectares
\n2. the number of inhabitants of the municipality, between 2500 and 5000
\nThe purpose of the operation is to perform a classification of objects (municipalities)\naccording to two criteria (population and area). <\/p>\n\n\n\n\n\n\n\n

Using traditional GIS queries.<\/strong> <\/p>\n\n\n\n

The tools offered by GIS work as follows:
\nA selection of the municipalities which have a population between the two limits\n(in this example   between 2500 and 5000 inhabitants) is performed.
\nA selection of the municipalities which have a surface between the two desired\nbounds (in this example   between 2500 and 3000 hectares) is performed.
\nThe final result is those municipalities that appear in the   two previous\nselection results, eliminating those that only appear   in only one of the\nselections. <\/p>\n\n\n\n

Let’s use as example all   municipalities for the Finist\u00e8re region.\n<\/p>\n\n\n\n

To determine which municipalities meet the criterion Population<\/em><\/strong>\nwe apply the selection request: <\/p>\n\n\n\n

\u00ab\u00a0<\/em><\/strong> POPULATION<\/em><\/strong> \n\u00bb> = 2500 AND\u00ab<\/em><\/strong> POPULATION<\/em><\/strong> \n\u00bb<= 5000<\/em><\/strong> \n\nThe 54 municipalities with a population between\n2500 and 5000 are the following  \n\n\n\n<\/p>\n\n\n\n

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

To determine the municipalities that meet the criteria Area<\/em><\/strong>\nwe apply the following selection request: <\/p>\n\n\n\n

\u00ab\u00a0<\/em><\/strong> AREA<\/em><\/strong> \n\u00bb> = 2500 AND\u00ab<\/em><\/strong> AREA<\/em><\/strong> \n\u00bb<= 3000<\/em><\/strong> \n\nThe 26 municipalities with an area between 2500\nand 3000 ha are the following: \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\n10 municipalities that meet both conditions are:\n\n\n\n<\/p>\n\n\n\n

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

If we are a technical service, the result suits us and we pass it on to\nour dear elected officials. <\/p>\n\n\n\n

If we are the elected officials, our problems begin: <\/p>\n\n\n\n

Why the commune of Plouenan is not in the result? Because it has a\npopulation of 2451 inhabitants and an area of \u200b\u200b3077 ha. <\/p>\n\n\n\n

And the commune of R\u00e9den\u00e9? Because it has 2464 ha, but 2870 inhabitants.\n<\/p>\n\n\n\n

And the commune of … In short, the list will be more or less long, but\nat each classification performed using our all or none logic (Boolean), classic\nfor GIS, we will have more or less limited situations that will cause problems.\n<\/p>\n\n\n\n

Let’s understand then that for a large part of elected officials, the\nfact that they are told that our tool is a \u201cdecision  help\u201d is far from convincing them! <\/p>\n\n\n\n

Another logic, another result<\/strong> <\/p>\n\n\n\n

A municipality having 2499 inhabitants and not 2500 will be eliminated\nfrom the result, as any municipality having 3001 ha and not 3000.
\nIn the decision processes, the values \u200b\u200bof the variables used are always tainted\nwith some uncertainty. In order for the GIS to be a decision help tool, it is\nessential to give the user tools that match his method of reasoning.
\nThe value of 2500 inhabitants is used by the GIS as a strict value. Besides,\n  in the decision-maker’s mind, this value is only a representative value\nof a municipality \u00ab\u00a0Size\u00a0\u00bb (eg \u00ab\u00a0average municipality\u00a0\u00bb). <\/p>\n\n\n\n

The use of  \u201cfuzzy numbers\u201d is\nanother possibility to classify objects. <\/p>\n\n\n\n

How do we define an average municipality with fuzzy logic? Instead of\nusing two values \u200b\u200bas minimum-maximum limits, we will use four values: <\/p>\n\n\n\n

  • The two limits of the number of\ninhabitants between which the municipalities totally correspond to his perception of an average municipality: by   example: 2500 and 5000; <\/li>
  • The lower limit from which the\nmunicipality t is completely excluded as average: for example 1500; <\/li>
  • The upper limit from which the\nmunicipality is excluded as average: for example 7500. <\/li><\/ul>\n\n\n\n

    \n\n\n\n\n\nThis makes it possible to build a\n\u00ab\u00a0belonging\u00a0\u00bb function that takes the following form\n\n\n\n<\/p>\n\n\n\n

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

    Here we measure the set membership \u201cAverage municipalities\u201d between 0\nand 1: the population values \u200b\u200bhaving a   belonging 0 are completely\n\u00ab\u00a0excluded\u00a0\u00bb from the classification, the values \u200b\u200bhaving   a\nmembership of 1 are \u00ab\u00a0completely included\u00a0\u00bb and the values \u200b\u200bbetween 0\nand 1   correspond to a \u00ab\u00a0more or less\u00a0\u00bb belonging (hence the term\n\u00a0\u00bb   fuzzy  \u00ab\u00a0). \n\nIf we apply this belonging function to our\nmunicipalities for the Finist\u00e8re region (we will discuss the tools in the next\narticle) to the category Municipalities, we obtain the following result: \n\n\n\n<\/p>\n\n\n\n

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

    Each municipality has a resultant value between 0 and 1. <\/p>\n\n\n\n

    We have grouped the municipality into 5 classes: <\/p>\n\n\n\n

    • those that correspond very well\nto the criterion: resultant values \u200b\u200bbetween 0.8 and 1.0: 63 municipalities (in\nred) <\/li>
    • those that match   rather well to the criterion: resulting values \u200b\u200bbetween 0.6 and 0.8   : 16   common (in dark orange) <\/li>
    • those that correspond moderately   to the criterion: resultant values \u200b\u200bbetween 0.4 and 0.6: 16   common (in light orange) <\/li>
    • those that match   rather badly   the criterion: result values \u200b\u200bbetween 0.2 and 0.4:\n18 common (in yellow   dark) <\/li>
    • those that do not match the criterion: Resulting values \u200b\u200bbetween 0.0 and 0.20:\n170 communes (in light yellow) <\/li><\/ul>\n\n\n\n

      \n\n\n\n\n\n\n\n\n\nIf we apply the   belonging function to our\nFinist\u00e8re municipalities to the category of average surface municipalities with\nlimits between 2000 ha – 2500 ha – 3000 ha and 4000 ha, we obtain the following\nresult: \n\n\n\n<\/p>\n\n\n\n

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

      Each municipality has a resultant value between 0 and 1. <\/p>\n\n\n\n

      We have grouped the municipalities into 5 classes according to the population:\n<\/p>\n\n\n\n

      • those which correspond very well\nto the criterion: resultant values \u200b\u200bbetween 0.8 and 1.0: 46   common (in red) <\/li>
      • those that match   rather well to the criterion: resulting values \u200b\u200bbetween 0.6 and 0.8   : 14 common (in dark orange) <\/li>
      • those that correspond moderately the criterion: resulting values \u200b\u200bbetween 0.4 and\n0.6: 10   common (in light orange) <\/li>
      • those that match   rather badly the criterion: resulting values \u200b\u200bbetween 0.2 and 0.4: 6   common (in yellow   dark) <\/li>
      • those that do not match the criterion:\nresulting values \u200b\u200bbetween 0.0 and 0.20: 207   common (in light yellow) <\/li><\/ul>\n\n\n\n

        Now let’s cross the two fuzzy numbers (again, we’ll discuss the tools in\nthe following article): \n\nThe aggregation results of the two fuzzy\nnumbers: area and population is as follows \n\n\n\n<\/p>\n\n\n\n

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

        Each municipality has a resultant value between 0 and 1. <\/p>\n\n\n\n

        We have grouped the communes into 5 classes: <\/p>\n\n\n\n

        • those which correspond very well\nto both criteria: resultant values \u200b\u200bbetween 0.8 and 1.0: 15   common (in red) <\/li>
        • those that match   rather well to both criteria: resulting values \u200b\u200bbetween 0.6 and 0.8   : 6   common (in dark orange) <\/li>
        • those that correspond moderately\nto both criteria: resultant values \u200b\u200bbetween 0.4 and 0.6: 113 common (in light\norange) <\/li>
        • those that match   rather poorly to both criteria: resultant values \u200b\u200bbetween 0.2 and 0.4:\n19   common (in yellow   dark) <\/li>
        • those that do not match both\ncriteria: resulting values \u200b\u200bbetween 0.0 and 0.28   : 130   common (in light yellow) <\/li><\/ul>\n\n\n\n

          If we compare the two types of approach, considering that for the fuzzy\napproach an 80% membership can be considered very good, we get: <\/p>\n\n\n\n
          \n Approach\n \n <\/td>\n  \n Population \n <\/td>\n  \n Area \n <\/td>\n   Crossing\n \n <\/td><\/tr>
          \n Boolean\n \n <\/td>\n     54 \n <\/td>\n     26  \n <\/td>\n  \n 10 \n <\/td><\/tr>
          \n Fuzzy \n <\/td>\n  \n   63 \n <\/td>\n  \n   46 \n <\/td>\n  \n 15 \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

          Of course, both municipalities, R\u00e9den\u00e9 and Plouenan, which initially\nposed a problem for us, are included in the 15 municipalities selected through\nthe fuzzy treatment. R\u00e9den\u00e9 has a membership of 0.93 and Plouenan of 0.92. <\/p>\n\n\n\n

          The classification of geographical entities<\/strong> <\/p>\n\n\n\n

          In this example we have used two criteria to \u201cclassify\u201d our\nmunicipalities. <\/p>\n\n\n\n

          The classification of objects according to several criteria is a common\noperation in everyday life. When you buy a product you take into account the\ndegree of   satisfaction given by its price, its lifetime, its\n\u00ab\u00a0standing\u00a0\u00bb … <\/p>\n\n\n\n

          For each criterion we define ourselves \u00ab\u00a0fuzzy\u00a0\u00bb functions\n(price   between X and Y euros, up to Z maximum) or \u00ab\u00a0fuzzy\nclassifications \u00ab\u00a0, for example the\u00a0\u00bb standing \u00ab\u00a0(ie: bad, average,\ngood, high, very   above). <\/p>\n\n\n\n

          We make our choices by crossing the values \u200b\u200bof the different variables\ntaken into account and obtaining a ranking of the different products according\nto the degree of   overall satisfaction. <\/p>\n\n\n\n

          Consider the simple case of crossing two criteria to which five values\n\u200b\u200bare attributed   for satisfaction: bad, rather bad, average, rather\ngood, good. Each object will have   as  result a degree of satisfaction coded on these\nsame five values. <\/p>\n\n\n\n

          If we are looking for a vehicle based on its resistance and price characteristics,\nfor example, we will find very resistant vehicles, so maximum satisfaction\n for the first criterion, but whose price is a little above what we want,\n  therefore average satisfaction fort the second criterion. <\/p>\n\n\n\n

          What is the resulting value of the crossing? In fact there is not a\nsingle result value, but many, depending on who makes the choice. Some will do\na mental average of both and will give a \u00ab\u00a0pretty good\u00a0\u00bb rating, for\n  others the price   will prevail and classify this vehicle as\n\u00ab\u00a0medium\u00a0\u00bb, others   Finally, they will be more sensitive to the\nresistance criterion and will classify the vehicle as   \u00ab\u00a0Good\u00a0\u00bb. <\/p>\n\n\n\n

          The example becomes even clearer if we consider a very resistant vehicle\nbut   very expensive (complete satisfaction of one criterion and complete\ndissatisfaction of the other).   Will this scenario give an \u00ab\u00a0average\u00a0\u00bb,\n\u00ab\u00a0rather bad\u00a0\u00bb or \u00ab\u00a0bad\u00a0\u00bb value? <\/p>\n\n\n\n

          GIS tools based on classical logic work on the principle of minimum\nvalue. The result of the crossing is the smallest value of the two criteria,\nwhich will be coded only as 0 or 1. If one of the two criteria is not satisfied\nin a pair of values \u200b\u200b1-0, the resultant crossing will be 0. <\/p>\n\n\n\n

          The use of a flexible spatial analysis tool makes it possible to\ndetermine the function of   crossing used by the operator. This step is\nsimply to ask   the operator the result of three crosses: Very good –\nmedium, medium – medium, and   Very good – bad. <\/p>\n\n\n\n

          The result of this test allows you to choose a function among the 50\nfunctions of possible crossings  when taking into account 5 degrees of\nsatisfaction (Theory   of possibilities, Applications to the\nrepresentation of computer skills,   D. DUBOIS and H. PRADE, Masson 1988).\n<\/p>\n\n\n\n

          In the following articles we will discuss two tools developed for ArcGis\nthat will allow you to perform all these operations. <\/p>\n","protected":false},"excerpt":{"rendered":"

          Decision processes are based on information from very diverse source and type. This information is used by the decision-makers to perform   choices, i.e. to retain a certain number of entities and to exclude others. Let’s…<\/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-6754","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1KW","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6754","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=6754"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/6754\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=6754"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=6754"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=6754"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}