{"id":349,"date":"2015-11-13T23:30:02","date_gmt":"2015-11-14T04:30:02","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=349"},"modified":"2021-11-25T20:49:36","modified_gmt":"2021-11-26T01:49:36","slug":"making-rational-decisions","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=349","title":{"rendered":"Making Rational Decisions"},"content":{"rendered":"<p><style>\ntable, th, td {<br \/>\nborder-collapse: collapse;<br \/>\nbackground: transparent;<br \/>\nborder: none;<br \/>\n}<br \/>\nth {<br \/>\ntext-align: center;<br \/>\ncolor: red;<br \/>\n}<br \/>\ntd {<br \/>\nborder: 1px solid black;<br \/>\ntext-align: center;<br \/>\n}<br \/>\ntd.left {<br \/>\nborder: none;<br \/>\ntext-align: center;<br \/>\ncolor: red;<br \/>\nfont-weight: bold;<br \/>\n}<br \/>\n<\/style><\/p>\n<p>I recently came across an interesting method for combining qualitative and quantitative data on a common footing to allow for a mathematically supported framework for make complicated decisions where many criteria are involved.\u00a0 The method is called the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Analytic_hierarchy_process\">Analytic Hierarchy Process<\/a>.<\/p>\n<p>The Analytic Hierarchy Process (AHP), which was invented Thomas L. Saaty in the 1970s, uses a technique based on matrices and eigenvectors to structure complex decision making when large sets of alternatives and criteria are involved and\/or when some of the criteria are described by attributes that cannot be assigned objective rankings are in play.\u00a0 It is especially useful in group-based decision making since it allows the judgements of disparate stake-holders, often with quite different points-of-view, to be considered in a dispassionate way.<\/p>\n<p>In a nut-shell, the AHP consists of three parts: the objective, the criteria, and the alternatives.\u00a0 Criteria can be sub-divided as finely as desired, with the obvious, concomitant cost of more complexity in the decision making process.\u00a0 Each alternative is then assigned a value in each criterion and each criteria is given a weighting.\u00a0 The assessments are normalized and matrix methods are used to link the relative values and weightings to give a ranking.\u00a0 Graphically, these parts are usually presented in hierarchical chart that looks something like:<\/p>\n<p><a href=\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-351\" src=\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure.jpg\" alt=\"AHP_structure\" width=\"1454\" height=\"724\" srcset=\"https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure.jpg 1454w, https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure-300x149.jpg 300w, https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure-1024x510.jpg 1024w, https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2015\/11\/AHP_structure-810x403.jpg 810w\" sizes=\"auto, (max-width: 1454px) 100vw, 1454px\" \/><\/a><\/p>\n<p>A nice tutorial exists by Haas and Meixner entitled <a href=\"https:\/\/mi.boku.ac.at\/ahp\/ahptutorial.pdf\">An Illustrated Guide to the Analytic Hierarchy Process<\/a> and this posting is patterned closely after their slides.\u00a0 The decision-making process that they address is buying a car.\u00a0 This is the objective (\u2018the what\u2019) that we seek to accomplish.\u00a0 We will use three criteria when selecting the car to buy:\u00a0 Style, Reliability, and Fuel Economy.<\/p>\n<p>Two of these criteria, Style and Reliability, are qualitative or, at least, semi-qualitative, whereas the Fuel Economy is quantitative.\u00a0 Our alternatives\/selections for the cars will be AutoFine, BigMotors, CoolCar, and Dynamix.<\/p>\n<p>The first step is to make assign numerical labels to the qualitative criteria.\u00a0 We will use a 1-10 scale for Style and Reliability.\u00a0 Since we are weighing judgements, the absolute values of these scores are meaningless.\u00a0 Instead the labels indicate the relative ranking. \u00a0For example, we can assume that the 1-10 scale can be interpreted as:<\/p>\n<ul>\n<li>1 \u2013 perfectly equal<\/li>\n<li>3 \u2013 moderately more important\/moderately better<\/li>\n<li>5 \u2013 strongly more important\/strongly better<\/li>\n<li>7 \u2013 very strongly more important\/very strongly better<\/li>\n<li>9 \u2013 extremely more important\/extremely better<\/li>\n<\/ul>\n<p>with the even-labeled values slightly greater in shading than the odd labels that precede them.\u00a0 This ranking scheme can be used to assign weightings to the criteria relative to each other (for example style is almost strongly more important than reliability \u2013 4\/1) and to weigh the alternatives against each other in a particular criteria (for example AutoFine is moderately better than CoolCar in reliability).<\/p>\n<p>To be concrete, let\u2019s suppose our friend Michael is looking to buy a car.\u00a0 We interview Michael and find that he feels that:<\/p>\n<ul>\n<li>Style is half as important as Reliability<\/li>\n<li>Style is 3 times more important as Fuel Economy<\/li>\n<li>Reliability is 4 times more important as Fuel Economy<\/li>\n<\/ul>\n<p>Based on these responses, we construct a weighting table<\/p>\n<table>\n<thead>\n<tr>\n<th width=\"160\">\u00a0<\/th>\n<th width=\"160\">Style<\/th>\n<th width=\"160\">Reliability<\/th>\n<th width=\"160\">Fuel Economy<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"left\" width=\"160\">Style<\/td>\n<td width=\"160\">1\/1<\/td>\n<td width=\"160\">1\/2<\/td>\n<td width=\"160\">3\/1<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"160\">Reliability<\/td>\n<td width=\"160\">\u00a0<\/td>\n<td width=\"160\">1\/1<\/td>\n<td width=\"160\">4\/1<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"160\">Fuel Economy<\/td>\n<td width=\"160\">\u00a0<\/td>\n<td width=\"160\">\u00a0<\/td>\n<td width=\"160\">1\/1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>where the first number in the entry corresponds to the row and the second to the column.\u00a0 So the \u20184\/1\u2019 entry encodes the statement that Reliability is 4 times more important as Fuel Economy. The omitted entries below the diagonal as simply the reverses of the one above (e.g. 1\/2 goes to 2\/1).<\/p>\n<p>This table is converted to a weighting matrix $\\mathbf{W}$ which numerically looks<\/p>\n<p>\\[ {\\mathbf W} = \\left[ \\begin{array}{ccc} 1.000 &amp; 0.5000 &amp; 3.000 \\\\ 2.000 &amp; 1.000 &amp; 4.000 \\\\ 0.3333 &amp; 0.2500 &amp; 1.000 \\end{array} \\right] \\; . \\]<\/p>\n<p>In a similar fashion, we interview Michael for his judgments of each automobile model with each criteria and find corresponding weighting matrices for Style and Reliability:<\/p>\n<p>\\[ {\\mathbf S} = \\left[ \\begin{array}{ccc} 1.000 &amp; 0.2500 &amp; 4.000 &amp; 0.1667 \\\\ 4.000 &amp; 1.000 &amp; 4.000 &amp; 0.2500 \\\\ 0.2500 &amp; 0.2500 &amp; 1.000 &amp; 0.2000 \\\\ 6.000 &amp; 4.000 &amp; 5.000 &amp; 1.000 \\end{array} \\right] \\; \\]<\/p>\n<p>and<\/p>\n<p>\\[ {\\mathbf R} = \\left[ \\begin{array}{ccc} 1.000 &amp; 2.000 &amp; 5.000 &amp; 1.000 \\\\ 0.5000 &amp; 1.000 &amp; 3.000 &amp; 2.000 \\\\ 0.2000 &amp; 0.3333 &amp; 1.000 &amp; 0.2500 \\\\ 1.000 &amp; 0.5000 &amp; 4.000 &amp; 1.000 \\end{array} \\right] \\; . \\]<\/p>\n<p>Finally, we rank the fuel economy for each alternative.\u00a0 Here we don\u2019t need to depend on Michael\u2019s judgment and can simply look up the <a href=\"ohttp:\/\/www.nhtsa.gov\/fuel-economy\">CAFE<\/a> standards to find<\/p>\n<p>\\[ {\\mathbf F} = \\left[ \\begin{array}{c}34\\\\27\\\\24\\\\28 \\end{array} \\right] mpg \\; . \\]<\/p>\n<p>Saaty\u2019s method directs us to first find the eigenvectors of each of the $4 \\times 4$ criteria matrices and of the $3 \\times 3$ weighting matrix that correspond to largest eigenvalues for each.\u00a0 Note that the Fuel Economy is already in vector form.\u00a0 The L1 norm is used so that each vector is normalized by the sum of it elements.\u00a0 The resulting vectors are:<\/p>\n<p>\\[ {\\mathbf vW} = \\left[ \\begin{array}{c}0.3196\\\\0.5584\\\\0.1220 \\end{array} \\right] \\; , \\]<\/p>\n<p>\\[ {\\mathbf vS} = \\left[ \\begin{array}{c}0.1163\\\\0.2473\\\\0.0599\\\\0.5764 \\end{array} \\right] \\; , \\]<\/p>\n<p>\\[ {\\mathbf vR} = \\left[ \\begin{array}{c}0.3786\\\\0.2901\\\\0.0742\\\\0.2571 \\end{array} \\right] \\; , \\]<\/p>\n<p>and<\/p>\n<p>\\[ {\\mathbf vF} = \\left[ \\begin{array}{c}0.3009\\\\0.2389\\\\0.2124\\\\0.2479 \\end{array} \\right] \\; . \\]<\/p>\n<p>A $4 \\times 3$ matrix is formed whose columns are ${\\mathbf vS}$, ${\\mathbf vR}$, ${\\mathbf vF}$ which is then left multiplied into ${\\mathbf vW}$ to give a final ranking.\u00a0 Doing this gives:<\/p>\n<div align=\"center\">\n<table style=\"width: auto;\">\n<tbody>\n<tr>\n<td class=\"left\" width=\"108\">AutoFine<\/td>\n<td width=\"90\">0.2853<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"108\">BigMotors<\/td>\n<td width=\"90\">0.2702<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"108\">CoolCar<\/td>\n<td width=\"90\">0.0865<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"108\">Dynamix<\/td>\n<td width=\"90\">0.3580<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<\/p>\n<p>So from the point-of-view of our criteria, Dynamix is the way to go.\u00a0 Of course, we haven\u2019t figured in cost.\u00a0 To this end, Haas and Meixner recommend scaling these results by the cost to get a value.\u00a0 This is done straightforwardly as shown in the following table.<\/p>\n<table>\n<tbody>\n<tr>\n<th width=\"78\"><strong>\u00a0<\/strong><\/th>\n<th width=\"66\"><strong>Ranking<\/strong><\/th>\n<th width=\"90\"><strong>Cost<\/strong><\/th>\n<th width=\"85\"><strong>Normalized Cost<\/strong><\/th>\n<th width=\"278\"><strong>Value = Ranking\/Normalized Cost<\/strong><\/th>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td class=\"left\" width=\"78\">AutoFine<\/td>\n<td width=\"66\">0.2853<\/td>\n<td width=\"90\">$20,000<\/td>\n<td width=\"85\">0.2899<\/td>\n<td width=\"278\">1.016<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"78\">BigMotors<\/td>\n<td width=\"66\">0.2702<\/td>\n<td width=\"90\">$15,000<\/td>\n<td width=\"85\">0.2174<\/td>\n<td width=\"278\">1.243<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"78\">CoolCar<\/td>\n<td width=\"66\">0.0865<\/td>\n<td width=\"90\">$12,000<\/td>\n<td width=\"85\">0.1739<\/td>\n<td width=\"278\">0.497<\/td>\n<\/tr>\n<tr>\n<td class=\"left\" width=\"78\">Dynamix<\/td>\n<td width=\"66\">0.3580<\/td>\n<td width=\"90\">$22,000<\/td>\n<td width=\"85\">0.3188<\/td>\n<td width=\"278\">1.122<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0<\/p>\n<p>With this new data incorporated, we now decided that BigMotors gives us the best value for the money.\u00a0 Whether our friend Michael will follow either of these two recommendations, is, of course, only answerable by him but at least the AHP gives some rational way of weighing the facts.\u00a0 I suspect that Aristotle would have been pleased.<\/p>\n\n\n\n","protected":false},"excerpt":{"rendered":"<p>I recently came across an interesting method for combining qualitative and quantitative data on a common footing to allow for a mathematically supported framework for make complicated decisions where many&#8230; 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