{"id":141,"date":"2025-01-17T17:24:14","date_gmt":"2025-01-17T16:24:14","guid":{"rendered":"https:\/\/home.agh.edu.pl\/~kkulak\/?page_id=141"},"modified":"2026-08-23T15:44:52","modified_gmt":"2026-08-23T13:44:52","slug":"book-erratum","status":"publish","type":"page","link":"https:\/\/home.agh.edu.pl\/~kkulak\/book-erratum\/","title":{"rendered":"Book erratum"},"content":{"rendered":"\n<style>\n\/* LyX Provided Styles *\/\ndiv.bibtexentry { margin-left: 2em; text-indent: -2em; }\nspan.bibtexlabel:before{ content: \"[\"; }\nspan.bibtexlabel:after{ content: \"] \"; }\n\n\/* Layout-provided Styles *\/\nh1.title {\nfont-size: x-large;\nmargin-bottom: 1ex;\ntext-align: center;\n\n}\ndiv.author {\nfont-size: large;\nmargin-top: 1.3ex;\nmargin-bottom: 0.7ex;\ntext-align: center;\n\n}\ndiv.abstract {\nfont-size: small;\nmargin-top: 0.7ex;\nmargin-bottom: 0.7ex;\nmargin-left: 3ex;\nmargin-right: 3ex;\ntext-align: left;\n\n}\nspan.abstract_label {\nfont-weight: bold;\nfont-size: small;\ntext-align: center;\n\n}\n\ndiv.abstract {\n\tmargin: 4ex;\n}\ndiv.abstract_item {\n\tfont-size: small;\n\tpadding-top: 1ex;\n}\ndiv.abstract_label {\n\tfont-weight: bold;\n}\nh4.subsubsection_ {\nfont-weight: bold;\nfont-size: medium;\nmargin-top: 0.7ex;\nmargin-bottom: 0.4ex;\ntext-align: left;\n\n}\ndiv.standard {\n\tmargin-bottom: 2ex;\n}\ndiv.example {\ntext-align: left;\n\n}\n\n\n<\/style>\n<\/head>\n<body dir=\"auto\">\n<h1 class='title' id='magicparlabel-31247'>Book Erratum\n<br \/>\nUnderstanding The Analytic Hierarchy Process<\/h1>\n<div class='author' id='magicparlabel-31248'>Konrad Ku\u0142akowski<\/div>\n<div class='abstract'>\n<span class=\"abstract_label\">Abstract<\/span>\n<div class=\"abstract_item\" id='magicparlabel-31249'>Errors found after publishing Understanding The Analytic Hierarchy Process [<a href='#LyXCite-Kulakowski2020utahp'><span class=\"bib-label\">1<\/span><\/a>] <\/div>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31250'>Page 40, line 3 from the top<\/h4>\n<div class='standard' id='magicparlabel-31251'><b><div style='height:5ex'><\/div><em>is<\/em><\/b> <\/div>\n\n<div class='standard' id='magicparlabel-31252'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>n<\/mi>\n  <mrow>\n   <mi>i<\/mi>\n   <mi>j<\/mi>\n  <\/mrow>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <mrow>\n  <mo form='prefix' fence='true' stretchy='true' symmetric='true'>{<\/mo>\n  <mtable class=\"grid\">\n   <mtr>\n    <mtd>\n     <mn>0<\/mn>\n    <\/mtd>\n    <mtd>\n     <mrow>\n      <mtext>if<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mi>i<\/mi>\n      <mo stretchy='false'>=<\/mo>\n      <mi>j<\/mi>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mtext>or<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>{\n      <msub>\n       <mi>a<\/mi>\n       <mi>i<\/mi>\n      <\/msub>\n      <mo stretchy='false'>,<\/mo>\n      <msub>\n       <mi>a<\/mi>\n       <mi>j<\/mi>\n      <\/msub>}\n      <mo>&#x2208;<\/mo>\n      <mi>E<\/mi>\n     <\/mrow>\n    <\/mtd>\n   <\/mtr>\n   <mtr>\n    <mtd>\n     <mn>1<\/mn>\n    <\/mtd>\n    <mtd>\n     <mrow>\n      <mtext>if<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mi>i<\/mi>\n      <mo>&#x2260;<\/mo>\n      <mi>j<\/mi>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mtext>and<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>{\n      <msub>\n       <mi>a<\/mi>\n       <mi>i<\/mi>\n      <\/msub>\n      <mo stretchy='false'>,<\/mo>\n      <msub>\n       <mi>a<\/mi>\n       <mi>j<\/mi>\n      <\/msub>}\n      <mo>&#x2209;<\/mo>\n      <mi>E<\/mi>\n     <\/mrow>\n    <\/mtd>\n   <\/mtr>\n  <\/mtable>\n <\/mrow>\n<\/mrow>\n<\/math>\n<em><b>should be<\/b> <\/em><\/div>\n\n<div class='standard' id='magicparlabel-31253'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>n<\/mi>\n  <mrow>\n   <mi>i<\/mi>\n   <mi>j<\/mi>\n  <\/mrow>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <mrow>\n  <mo form='prefix' fence='true' stretchy='true' symmetric='true'>{<\/mo>\n  <mtable class=\"grid\">\n   <mtr>\n    <mtd>\n     <mn>0<\/mn>\n    <\/mtd>\n    <mtd>\n     <mrow>\n      <mtext>if<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mi>i<\/mi>\n      <mo stretchy='false'>=<\/mo>\n      <mi>j<\/mi>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mtext>or<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>{\n      <msub>\n       <mi>a<\/mi>\n       <mi>i<\/mi>\n      <\/msub>\n      <mo stretchy='false'>,<\/mo>\n      <msub>\n       <mi>a<\/mi>\n       <mi>j<\/mi>\n      <\/msub>}\n      <mo>&#x2209;<\/mo>\n      <mi>E<\/mi>\n     <\/mrow>\n    <\/mtd>\n   <\/mtr>\n   <mtr>\n    <mtd>\n     <mn>1<\/mn>\n    <\/mtd>\n    <mtd>\n     <mrow>\n      <mtext>if<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mi>i<\/mi>\n      <mo>&#x2260;<\/mo>\n      <mi>j<\/mi>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>\n      <mtext>and<\/mtext>\n      <mspace width=\"6px\"\/>\n      <mspace width=\"6px\"\/>{\n      <msub>\n       <mi>a<\/mi>\n       <mi>i<\/mi>\n      <\/msub>\n      <mo stretchy='false'>,<\/mo>\n      <msub>\n       <mi>a<\/mi>\n       <mi>j<\/mi>\n      <\/msub>}\n      <mo>&#x2208;<\/mo>\n      <mi>E<\/mi>\n     <\/mrow>\n    <\/mtd>\n   <\/mtr>\n  <\/mtable>\n <\/mrow>\n<\/mrow>\n<\/math>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31254'>Page 75, the numerical example is faulty. The correct version is: <\/h4>\n<div class='standard' id='magicparlabel-31255'><b><div style='height:5ex'><\/div><\/b><\/div>\n\n<div class='example' id='magicparlabel-31256'>and the constant term vector <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mi>r<\/mi>\n<\/math>\n is given as\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>r<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo>ln<\/mo>\n     <mn>6<\/mn>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo>ln<\/mo>\n     <mfrac>\n      <mn>21<\/mn>\n      <mn>8<\/mn>\n     <\/mfrac>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo stretchy='false'>&#x2212;<\/mo>\n     <mo>ln<\/mo>\n     <mn>3<\/mn>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo>ln<\/mo>\n     <mfrac>\n      <mn>36<\/mn>\n      <mn>7<\/mn>\n     <\/mfrac>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo stretchy='false'>&#x2212;<\/mo>\n     <mo>ln<\/mo>\n     <mn>27<\/mn>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mn>.<\/mn>\n<\/mrow>\n<\/math>\n<\/div>\n\n<div class='example' id='magicparlabel-31257'>Solving <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>G<\/mi>\n <mover>\n  <mi>w<\/mi>\n  <mo stretchy='true'>&#x005E;<\/mo>\n <\/mover>\n <mo stretchy='false'>=<\/mo>\n <mi>r<\/mi>\n<\/mrow>\n<\/math>\n leads to the following logarithmized ranking vector<\/div>\n\n\n\n<div class='example' id='magicparlabel-31264'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mover>\n  <mi>w<\/mi>\n  <mo stretchy='true'>&#x005E;<\/mo>\n <\/mover>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mrow>\n     <mfrac>\n      <mn>1<\/mn>\n      <mn>100<\/mn>\n     <\/mfrac>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <mn>22<\/mn>\n       <mo>ln<\/mo>\n       <mn>3<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>43<\/mn>\n       <mo>ln<\/mo>\n       <mn>6<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>2<\/mn>\n       <mo>ln<\/mo>\n       <mn>27<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>7<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>36<\/mn>\n        <mn>7<\/mn>\n       <\/mfrac>\n       <mo stretchy='false'>+<\/mo>\n       <mn>8<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>21<\/mn>\n        <mn>8<\/mn>\n       <\/mfrac>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mfrac>\n      <mn>1<\/mn>\n      <mn>25<\/mn>\n     <\/mfrac>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <mn>8<\/mn>\n       <mo>ln<\/mo>\n       <mn>3<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>2<\/mn>\n       <mo>ln<\/mo>\n       <mn>6<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>3<\/mn>\n       <mo>ln<\/mo>\n       <mn>27<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>2<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>36<\/mn>\n        <mn>7<\/mn>\n       <\/mfrac>\n       <mo stretchy='false'>+<\/mo>\n       <mn>12<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>21<\/mn>\n        <mn>8<\/mn>\n       <\/mfrac>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mfrac>\n      <mn>1<\/mn>\n      <mn>100<\/mn>\n     <\/mfrac>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <mn>22<\/mn>\n       <mo>ln<\/mo>\n       <mn>3<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>7<\/mn>\n       <mo>ln<\/mo>\n       <mn>6<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>2<\/mn>\n       <mo>ln<\/mo>\n       <mn>27<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>43<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>36<\/mn>\n        <mn>7<\/mn>\n       <\/mfrac>\n       <mo stretchy='false'>+<\/mo>\n       <mn>8<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>21<\/mn>\n        <mn>8<\/mn>\n       <\/mfrac>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mfrac>\n      <mn>1<\/mn>\n      <mn>100<\/mn>\n     <\/mfrac>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <mn>22<\/mn>\n       <mo>ln<\/mo>\n       <mn>3<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>7<\/mn>\n       <mo>ln<\/mo>\n       <mn>6<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>2<\/mn>\n       <mo>ln<\/mo>\n       <mn>27<\/mn>\n       <mo stretchy='false'>+<\/mo>\n       <mn>43<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>36<\/mn>\n        <mn>7<\/mn>\n       <\/mfrac>\n       <mo stretchy='false'>+<\/mo>\n       <mn>8<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>21<\/mn>\n        <mn>8<\/mn>\n       <\/mfrac>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mfrac>\n      <mn>1<\/mn>\n      <mn>50<\/mn>\n     <\/mfrac>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>4<\/mn>\n       <mo>ln<\/mo>\n       <mn>3<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mo>ln<\/mo>\n       <mn>6<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>14<\/mn>\n       <mo>ln<\/mo>\n       <mn>27<\/mn>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>36<\/mn>\n        <mn>7<\/mn>\n       <\/mfrac>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>6<\/mn>\n       <mo>ln<\/mo>\n       <mfrac>\n        <mn>21<\/mn>\n        <mn>8<\/mn>\n       <\/mfrac>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>1.04<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>1.484<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo stretchy='false'>&#x2212;<\/mo>\n     <mn>2.29<\/mn>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.963<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mo stretchy='false'>&#x2212;<\/mo>\n     <mn>1.19<\/mn>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mn>.<\/mn>\n<\/mrow>\n<\/math>\n<\/div>\n\n\n\n<div class='example' id='magicparlabel-31270'>Hence, the (unscaled) ranking vector is \n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>w<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <msup>\n     <mi>e<\/mi>\n     <mn>1.04064<\/mn>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mi>e<\/mi>\n     <mn>1.48464<\/mn>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mi>e<\/mi>\n     <mrow>\n      <mo stretchy='false'>&#x2212;<\/mo>\n      <mn>2.2937<\/mn>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mi>e<\/mi>\n     <mn>0.96356<\/mn>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mi>e<\/mi>\n     <mrow>\n      <mo stretchy='false'>&#x2212;<\/mo>\n      <mn>1.19512<\/mn>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>2.83103<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>4.4134<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.100889<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>2.62103<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.302668<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mn>.<\/mn>\n<\/mrow>\n<\/math>\n<\/div>\n\n<div class='example' id='magicparlabel-31271'>The last step to receive the ranking in the usual form is scaling so that the entries of the ranking vector sum up to <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mn>1<\/mn>\n<\/math>\n. The final form of the ranking vector is as follows: \n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>w<\/mi>\n  <mrow>\n   <mi>g<\/mi>\n   <mi>m<\/mi>\n  <\/mrow>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>0.275<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.429<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.0098<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.255<\/mn>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mn>0.0294<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mn>.<\/mn>\n<\/mrow>\n<\/math>\nAccording to the computed ranking, the most preferred alternative is <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>a<\/mi>\n <mn>1<\/mn>\n<\/msub>\n<\/math>\n with the ranking value <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>w<\/mi>\n <mrow>\n  <mo stretchy='false'>(<\/mo>\n  <msub>\n   <mi>a<\/mi>\n   <mn>2<\/mn>\n  <\/msub>\n  <mo stretchy='false'>)<\/mo>\n <\/mrow>\n <mo stretchy='false'>=<\/mo>\n <mn>0.429<\/mn>\n<\/mrow>\n<\/math>\n. The second place is taken by <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>a<\/mi>\n <mn>4<\/mn>\n<\/msub>\n<\/math>\n with <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>w<\/mi>\n <mrow>\n  <mo stretchy='false'>(<\/mo>\n  <msub>\n   <mi>a<\/mi>\n   <mn>1<\/mn>\n  <\/msub>\n  <mo stretchy='false'>)<\/mo>\n <\/mrow>\n <mo stretchy='false'>=<\/mo>\n <mn>0.275<\/mn>\n<\/mrow>\n<\/math>\n, then <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <msub>\n  <mi>a<\/mi>\n  <mn>4<\/mn>\n <\/msub>\n <mo stretchy='false'>,<\/mo>\n <msub>\n  <mi>a<\/mi>\n  <mn>5<\/mn>\n <\/msub>\n<\/mrow>\n<\/math>\n and <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>a<\/mi>\n <mn>3<\/mn>\n<\/msub>\n<\/math>\n.<\/div>\n\n<div class='example' id='magicparlabel-31272'>Of course, one may verify that GMM applied to the following matrix \n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n<mtable class=\"matrix\">\n <mtr>\n  <mtd>\n   <mn>1<\/mn>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>2<\/mn>\n    <mn>3<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.275<\/mn>\n    <mn>0.0098<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.275<\/mn>\n    <mn>0.255<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>9<\/mn>\n  <\/mtd>\n <\/mtr>\n <mtr>\n  <mtd>\n   <mfrac>\n    <mn>3<\/mn>\n    <mn>2<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>1<\/mn>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.429<\/mn>\n    <mn>0.0098<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>7<\/mn>\n    <mn>4<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.429<\/mn>\n    <mn>0.0294<\/mn>\n   <\/mfrac>\n  <\/mtd>\n <\/mtr>\n <mtr>\n  <mtd>\n   <mfrac>\n    <mn>0.0098<\/mn>\n    <mn>0.275<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.0098<\/mn>\n    <mn>0.429<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>1<\/mn>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.0098<\/mn>\n    <mn>0.255<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>1<\/mn>\n    <mn>3<\/mn>\n   <\/mfrac>\n  <\/mtd>\n <\/mtr>\n <mtr>\n  <mtd>\n   <mfrac>\n    <mn>0.255<\/mn>\n    <mn>0.275<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>4<\/mn>\n    <mn>7<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.255<\/mn>\n    <mn>0.0098<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>1<\/mn>\n  <\/mtd>\n  <mtd>\n   <mn>9<\/mn>\n  <\/mtd>\n <\/mtr>\n <mtr>\n  <mtd>\n   <mfrac>\n    <mn>1<\/mn>\n    <mn>9<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>0.0294<\/mn>\n    <mn>0.429<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>3<\/mn>\n  <\/mtd>\n  <mtd>\n   <mfrac>\n    <mn>1<\/mn>\n    <mn>9<\/mn>\n   <\/mfrac>\n  <\/mtd>\n  <mtd>\n   <mn>1<\/mn>\n  <\/mtd>\n <\/mtr>\n<\/mtable>\n<mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n<\/math>\nresults in <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>w<\/mi>\n <mrow>\n  <mi>g<\/mi>\n  <mi>m<\/mi>\n <\/mrow>\n<\/msub>\n<\/math>\n.<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31273'>Page 80, line 1 from the top <\/h4>\n<div class='standard' id='magicparlabel-31274'><em><b>is<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31275'>where <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>p<\/mi>\n <mi>i<\/mi>\n<\/msub>\n<\/math>\n is the number of existing comparisons in the i-th row of <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mi>C<\/mi>\n<\/math>\n,<\/div>\n\n<div class='standard' id='magicparlabel-31276'><em><b>should be <\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31277'>where <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<msub>\n <mi>p<\/mi>\n <mi>i<\/mi>\n<\/msub>\n<\/math>\n is the number of existing comparisons in the i-th row of <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mi>C<\/mi>\n<\/math>\n except the diagonal<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31278'>Page 81, line 12 from the bottom<\/h4>\n<div class='standard' id='magicparlabel-31279'><b><div style='height:5ex'><\/div><em>is<\/em><\/b><\/div>\n\n<div class='standard' id='magicparlabel-31280'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>q<\/mi>\n  <mn>3<\/mn>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>i<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mrow>\n   <mi>n<\/mi>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mn>2<\/mn>\n  <\/mrow>\n <\/munderover>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>j<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mi>i<\/mi>\n   <mo stretchy='false'>+<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mrow>\n   <mi>n<\/mi>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n <\/munderover>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>k<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mi>j<\/mi>\n   <mo stretchy='false'>+<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mi>n<\/mi>\n <\/munderover>\n <mrow>\n  <mo>(<\/mo>\n  <mrow>\n   <mn>2<\/mn>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mfrac>\n    <msub>\n     <mi>a<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n    <mrow>\n     <msub>\n      <mi>a<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n     <msub>\n      <mi>a<\/mi>\n      <mrow>\n       <mi>j<\/mi>\n       <mi>k<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mfrac>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mfrac>\n    <mrow>\n     <msub>\n      <mi>a<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n     <msub>\n      <mi>a<\/mi>\n      <mrow>\n       <mi>j<\/mi>\n       <mi>k<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n    <msub>\n     <mi>a<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mfrac>\n  <\/mrow>\n  <mo>)<\/mo>\n <\/mrow>\n<\/mrow>\n<\/math>\n<em><b>should be<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31281'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>q<\/mi>\n  <mn>3<\/mn>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>i<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mrow>\n   <mi>n<\/mi>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mn>2<\/mn>\n  <\/mrow>\n <\/munderover>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>j<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mi>i<\/mi>\n   <mo stretchy='false'>+<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mrow>\n   <mi>n<\/mi>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n <\/munderover>\n <munderover>\n  <mo>&#x2211;<\/mo>\n  <mrow>\n   <mi>k<\/mi>\n   <mo stretchy='false'>=<\/mo>\n   <mi>j<\/mi>\n   <mo stretchy='false'>+<\/mo>\n   <mn>1<\/mn>\n  <\/mrow>\n  <mi>n<\/mi>\n <\/munderover>\n <mrow>\n  <mo>(<\/mo>\n  <mrow>\n   <mn>2<\/mn>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mfrac>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n    <mrow>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>j<\/mi>\n       <mi>k<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mfrac>\n   <mo stretchy='false'>&#x2212;<\/mo>\n   <mfrac>\n    <mrow>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>j<\/mi>\n       <mi>k<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mfrac>\n  <\/mrow>\n  <mo>)<\/mo>\n <\/mrow>\n<\/mrow>\n<\/math>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31282'>Page 113, line 14 from the bottom <\/h4>\n<div class='standard' id='magicparlabel-31283'><b><div style='height:5ex'><\/div><em>is<\/em><\/b><\/div>\n\n<div class='standard' id='magicparlabel-31284'>6.3.8.1 Effectiveness of the Koczkodaj index\n<br \/>\n<\/div>\n\n<div class='standard' style='text-align: left;' id='magicparlabel-31285'><em><b>should be<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31286'>6.3.8.1 Effectiveness and the Koczkodaj index\n<br \/>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31810'>Page 96, line 2 from the top <\/h4>\n<div class='standard' id='magicparlabel-31811'><b><div style='height:5ex'><\/div><em>is<\/em><\/b><\/div>\n\n<div class='standard' id='magicparlabel-31812'>Let <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>w<\/mi>\n <mo stretchy='false'>:<\/mo>\n <mi>A<\/mi>\n <mo>&#x2192;<\/mo>\n <msub>\n  <mi>&#x1d563;<\/mi>\n  <mo stretchy='false'>+<\/mo>\n <\/msub>\n<\/mrow>\n<\/math>\n be a priority function assigning priorities to alternatives and <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>C<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mrow>\n  <mo stretchy='false'>[<\/mo>\n  <msub>\n   <mi>c<\/mi>\n   <mrow>\n    <mi>i<\/mi>\n    <mi>j<\/mi>\n   <\/mrow>\n  <\/msub>\n  <mo stretchy='false'>]<\/mo>\n <\/mrow>\n<\/mrow>\n<\/math>\n be an <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>n<\/mi>\n <mo>&#x00D7;<\/mo>\n <mi>n<\/mi>\n<\/mrow>\n<\/math>\n PC matrix.\n<br \/>\n<\/div>\n\n<div class='standard' style='text-align: left;' id='magicparlabel-31813'><em><b>should be<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31814'>Let <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>w<\/mi>\n <mo stretchy='false'>:<\/mo>\n <mi>A<\/mi>\n <mo>&#x2192;<\/mo>\n <msub>\n  <mi>&#x1d563;<\/mi>\n  <mo stretchy='false'>+<\/mo>\n <\/msub>\n<\/mrow>\n<\/math>\n be a priority function assigning priorities to alternatives where <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>C<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mrow>\n  <mo stretchy='false'>[<\/mo>\n  <msub>\n   <mi>c<\/mi>\n   <mrow>\n    <mi>i<\/mi>\n    <mi>j<\/mi>\n   <\/mrow>\n  <\/msub>\n  <mo stretchy='false'>]<\/mo>\n <\/mrow>\n<\/mrow>\n<\/math>\n be an <math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'>\n<mrow>\n <mi>n<\/mi>\n <mo>&#x00D7;<\/mo>\n <mi>n<\/mi>\n<\/mrow>\n<\/math>\n PC matrix and the relative priority of the two alternatives corresponds to their direct comparison.\n<br \/>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31287'>Page 123, line 5 from the top<\/h4>\n<div class='standard' id='magicparlabel-31288'><em><b>is<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31289'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>T<\/mi>\n  <mrow>\n   <mi>i<\/mi>\n   <mi>j<\/mi>\n   <mi>k<\/mi>\n  <\/mrow>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>j<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mn>1<\/mn>\n     <mo stretchy='false'>\/<\/mo>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>k<\/mi>\n      <mi>j<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n   <mtd>\n    <mrow>\n     <mn>1<\/mn>\n     <mo stretchy='false'>\/<\/mo>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>k<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n<\/mrow>\n<\/math>\n<em><b>should be<\/b><\/em> \n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <msub>\n  <mi>T<\/mi>\n  <mrow>\n   <mi>i<\/mi>\n   <mi>j<\/mi>\n   <mi>k<\/mi>\n  <\/mrow>\n <\/msub>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>j<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <mn>1<\/mn>\n     <mo stretchy='false'>\/<\/mo>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>i<\/mi>\n       <mi>j<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>j<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msub>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mi>k<\/mi>\n     <\/mrow>\n    <\/msub>\n   <\/mtd>\n   <mtd>\n    <mrow>\n     <mn>1<\/mn>\n     <mo stretchy='false'>\/<\/mo>\n     <msub>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>j<\/mi>\n       <mi>k<\/mi>\n      <\/mrow>\n     <\/msub>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n<\/mrow>\n<\/math>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31290'>Page 150, line 11 from the bottom<\/h4>\n<div class='standard' id='magicparlabel-31291'><em><b>is<\/b><\/em> <\/div>\n\n<div class='standard' id='magicparlabel-31292'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>C<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mn>1<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mn>2<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mn>1<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mi>n<\/mi>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mn>2<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mn>1<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EE;<\/mi>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mi>&#x22EE;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22F1;<\/mi>\n   <\/mtd>\n   <mtd>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mi>n<\/mi>\n         <mo stretchy='false'>&#x2212;<\/mo>\n         <mn>1.<\/mn>\n         <mi>n<\/mi>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mi>n<\/mi>\n         <mo stretchy='false'>,<\/mo>\n         <mn>1<\/mn>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mo stretchy='false'>,<\/mo>\n<\/mrow>\n<\/math>\n<\/div>\n\n<div class='standard' id='magicparlabel-31293'><em><b>should be<\/b><\/em> <\/div>\n\n<div class='standard' id='magicparlabel-31294'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>C<\/mi>\n <mo stretchy='false'>=<\/mo>\n <mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>(<\/mo>\n <mtable class=\"matrix\">\n  <mtr>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <mrow>\n     <munderover>\n      <mo>&#x220F;<\/mo>\n      <mrow>\n       <mi>q<\/mi>\n       <mo stretchy='false'>=<\/mo>\n       <mn>1<\/mn>\n      <\/mrow>\n      <mi>r<\/mi>\n     <\/munderover>\n     <msubsup>\n      <mi>c<\/mi>\n      <mrow>\n       <mn>1<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mn>2<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mi>q<\/mi>\n      <\/mrow>\n      <msub>\n       <mi>&#x3B7;<\/mi>\n       <mi>q<\/mi>\n      <\/msub>\n     <\/msubsup>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mrow>\n     <munderover>\n      <mo>&#x220F;<\/mo>\n      <mrow>\n       <mi>q<\/mi>\n       <mo stretchy='false'>=<\/mo>\n       <mn>1<\/mn>\n      <\/mrow>\n      <mi>r<\/mi>\n     <\/munderover>\n     <msubsup>\n      <mi>c<\/mi>\n      <mrow>\n       <mn>1<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mi>n<\/mi>\n       <mo stretchy='false'>,<\/mo>\n       <mi>q<\/mi>\n      <\/mrow>\n      <msub>\n       <mi>&#x3B7;<\/mi>\n       <mi>q<\/mi>\n      <\/msub>\n     <\/msubsup>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <munderover>\n      <mo>&#x220F;<\/mo>\n      <mrow>\n       <mi>q<\/mi>\n       <mo stretchy='false'>=<\/mo>\n       <mn>1<\/mn>\n      <\/mrow>\n      <mi>r<\/mi>\n     <\/munderover>\n     <msubsup>\n      <mi>c<\/mi>\n      <mrow>\n       <mn>2<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mn>1<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mi>q<\/mi>\n      <\/mrow>\n      <msub>\n       <mi>&#x3B7;<\/mi>\n       <mi>q<\/mi>\n      <\/msub>\n     <\/msubsup>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EE;<\/mi>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mi>&#x22EE;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22F1;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mrow>\n     <munderover>\n      <mo>&#x220F;<\/mo>\n      <mrow>\n       <mi>q<\/mi>\n       <mo stretchy='false'>=<\/mo>\n       <mn>1<\/mn>\n      <\/mrow>\n      <mi>r<\/mi>\n     <\/munderover>\n     <msubsup>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>n<\/mi>\n       <mo stretchy='false'>&#x2212;<\/mo>\n       <mn>1.<\/mn>\n       <mi>n<\/mi>\n       <mo stretchy='false'>,<\/mo>\n       <mi>q<\/mi>\n      <\/mrow>\n      <msub>\n       <mi>&#x3B7;<\/mi>\n       <mi>q<\/mi>\n      <\/msub>\n     <\/msubsup>\n    <\/mrow>\n   <\/mtd>\n  <\/mtr>\n  <mtr>\n   <mtd>\n    <mrow>\n     <munderover>\n      <mo>&#x220F;<\/mo>\n      <mrow>\n       <mi>q<\/mi>\n       <mo stretchy='false'>=<\/mo>\n       <mn>1<\/mn>\n      <\/mrow>\n      <mi>r<\/mi>\n     <\/munderover>\n     <msubsup>\n      <mi>c<\/mi>\n      <mrow>\n       <mi>n<\/mi>\n       <mo stretchy='false'>,<\/mo>\n       <mn>1<\/mn>\n       <mo stretchy='false'>,<\/mo>\n       <mi>q<\/mi>\n      <\/mrow>\n      <msub>\n       <mi>&#x3B7;<\/mi>\n       <mi>q<\/mi>\n      <\/msub>\n     <\/msubsup>\n    <\/mrow>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mi>&#x22EF;<\/mi>\n   <\/mtd>\n   <mtd>\n    <mn>1<\/mn>\n   <\/mtd>\n  <\/mtr>\n <\/mtable>\n <mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='0.1666em'>)<\/mo>\n <mo stretchy='false'>,<\/mo>\n<\/mrow>\n<\/math>\n<\/div>\n<h4 class='subsubsection_' id='magicparlabel-31295'>Page 150, line 7 from the bottom <\/h4>\n<div class='standard' id='magicparlabel-31296'><em><b>is<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31297'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>w<\/mi>\n <mrow>\n  <mo stretchy='false'>(<\/mo>\n  <msub>\n   <mi>a<\/mi>\n   <mi>i<\/mi>\n  <\/msub>\n  <mo stretchy='false'>)<\/mo>\n <\/mrow>\n <mo stretchy='false'>=<\/mo>\n <msup>\n  <mrow>\n   <mo>(<\/mo>\n   <mrow>\n    <munderover>\n     <mo>&#x220F;<\/mo>\n     <mrow>\n      <mi>k<\/mi>\n      <mo stretchy='false'>=<\/mo>\n      <mn>1<\/mn>\n     <\/mrow>\n     <mi>n<\/mi>\n    <\/munderover>\n    <msup>\n     <mrow>\n      <mo>(<\/mo>\n      <mrow>\n       <munderover>\n        <mo>&#x220F;<\/mo>\n        <mrow>\n         <mi>q<\/mi>\n         <mo stretchy='false'>=<\/mo>\n         <mn>1<\/mn>\n        <\/mrow>\n        <mi>r<\/mi>\n       <\/munderover>\n       <msubsup>\n        <mi>c<\/mi>\n        <mrow>\n         <mi>i<\/mi>\n         <mo stretchy='false'>,<\/mo>\n         <mi>k<\/mi>\n         <mo stretchy='false'>,<\/mo>\n         <mi>q<\/mi>\n        <\/mrow>\n        <msub>\n         <mi>&#x3B7;<\/mi>\n         <mi>q<\/mi>\n        <\/msub>\n       <\/msubsup>\n      <\/mrow>\n      <mo>)<\/mo>\n     <\/mrow>\n     <mrow>\n      <mn>1<\/mn>\n      <mo stretchy='false'>\/<\/mo>\n      <mi>r<\/mi>\n     <\/mrow>\n    <\/msup>\n   <\/mrow>\n   <mo>)<\/mo>\n  <\/mrow>\n  <mrow>\n   <mn>1<\/mn>\n   <mo stretchy='false'>\/<\/mo>\n   <mi>n<\/mi>\n  <\/mrow>\n <\/msup>\n<\/mrow>\n<\/math>\n<\/div>\n\n<div class='standard' id='magicparlabel-31298'><em><b>should be<\/b><\/em><\/div>\n\n<div class='standard' id='magicparlabel-31299'>\n<math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' display='block'>\n<mrow>\n <mi>w<\/mi>\n <mrow>\n  <mo stretchy='false'>(<\/mo>\n  <msub>\n   <mi>a<\/mi>\n   <mi>i<\/mi>\n  <\/msub>\n  <mo stretchy='false'>)<\/mo>\n <\/mrow>\n <mo stretchy='false'>=<\/mo>\n <msup>\n  <mrow>\n   <mo>(<\/mo>\n   <mrow>\n    <munderover>\n     <mo>&#x220F;<\/mo>\n     <mrow>\n      <mi>k<\/mi>\n      <mo stretchy='false'>=<\/mo>\n      <mn>1<\/mn>\n     <\/mrow>\n     <mi>n<\/mi>\n    <\/munderover>\n    <munderover>\n     <mo>&#x220F;<\/mo>\n     <mrow>\n      <mi>q<\/mi>\n      <mo stretchy='false'>=<\/mo>\n      <mn>1<\/mn>\n     <\/mrow>\n     <mi>r<\/mi>\n    <\/munderover>\n    <msubsup>\n     <mi>c<\/mi>\n     <mrow>\n      <mi>i<\/mi>\n      <mo stretchy='false'>,<\/mo>\n      <mi>k<\/mi>\n      <mo stretchy='false'>,<\/mo>\n      <mi>q<\/mi>\n     <\/mrow>\n     <msub>\n      <mi>&#x3B7;<\/mi>\n      <mi>q<\/mi>\n     <\/msub>\n    <\/msubsup>\n   <\/mrow>\n   <mo>)<\/mo>\n  <\/mrow>\n  <mrow>\n   <mn>1<\/mn>\n   <mo stretchy='false'>\/<\/mo>\n   <mi>n<\/mi>\n  <\/mrow>\n <\/msup>\n <mn>.<\/mn>\n<\/mrow>\n<\/math>\n<\/div>\n\n<div class='standard' id='magicparlabel-31300'><h2 class='bibtex'>References<\/h2><div class='bibtex'><div class='bibtexentry' id='LyXCite-Kulakowski2020utahp'><span class='bibtexlabel'>1<\/span><span class='bibtexinfo'><span class=\"bib-fullnames:author\">Ku\u0142akowski, K.<\/span>, <i><span class=\"bib-title\">Understanding the Analytic Hierarchy Process<\/span><\/i> (<span class=\"bib-address\">6000 Broken Sound Parkway, Boca Raton, FL, 33487, USA<\/span>: <span class=\"bib-publisher\">Chapman and Hall \/ CRC Press<\/span>, <span class=\"bib-year\">2020<\/span>).<\/span><\/div>\n<\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Erratum can also be downloaded as the <a href=\"\/kkulak\/wp-content\/uploads\/2025\/01\/book_erratum.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">PDF file here<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Book Erratum Understanding The Analytic Hierarchy Process Konrad Ku\u0142akowski Abstract Errors found after publishing Understanding The Analytic Hierarchy Process [1] Page 40, line 3 from the top is n i j = { 0 if i = j or { a i , a j } &#x2208; E 1 if i &#x2260; j and { [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-templates\/full-width.php","meta":{"footnotes":""},"class_list":["post-141","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/pages\/141","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/comments?post=141"}],"version-history":[{"count":5,"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/pages\/141\/revisions"}],"predecessor-version":[{"id":594,"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/pages\/141\/revisions\/594"}],"wp:attachment":[{"href":"https:\/\/home.agh.edu.pl\/~kkulak\/wp-json\/wp\/v2\/media?parent=141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}