Decision matrix

<h2 id="multiple-criteria-decision-analysis">Multiple-criteria decision analysis</h2>
The term decision matrix is used to describe a <a href="/facts/Multiple-criteria_decision_analysis/IXQLJgBx">multiple-criteria decision analysis</a> (MCDA) problem. An MCDA problem, where there are M alternative options and each needs to be assessed on N criteria, can be described by the decision matrix which has N rows and M columns, or M × N elements, as shown in the following table. Each element, such as Xij, is either a single numerical value or a single grade, representing the performance of alternative i on criterion j. For example, if alternative i is "car i", criterion j is "engine quality" assessed by five grades {Exceptional, Good, Average, Below Average, Poor}, and "Car i" is assessed to be "Good" on "engine quality", then Xij = "Good". These assessments may be replaced by scores, from 1 to 5. Sums of scores may then be compared and ranked, to show the winning proposal.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

Example of Comparison<table><tbody><tr><th></th><th>Alternative 1</th><th>Alternative 2</th><th>...</th><th>Alternative M</th></tr><tr><td>Criterion 1</td><td>x11</td><td>x12</td><td>...</td><td>x1M</td></tr><tr><td>Criterion 2</td><td>x21</td><td>x22</td><td>...</td><td>x2M</td></tr><tr><td>...</td><td>...</td><td>...</td><td>Xij = Good</td><td>...</td></tr><tr><td>Criterion N</td><td>xN1</td><td>xN2</td><td>...</td><td>xNM</td></tr><tr><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Sum</th><td></td><td></td><td></td><td></td></tr><tr><th>Rank</th><td></td><td></td><td></td><td></td></tr><tr><th>Status</th><td></td><td>No</td><td></td><td>No</td></tr></tbody></table>
<h2 id="belief-decision-matrix">Belief decision matrix</h2>
Further information: <a href="/facts/Belief_structure/VGXupB3F">Belief structure</a>
Similar to a decision matrix, a belief decision matrix is used to describe a multiple criteria decision analysis (<a href="/facts/MCDA/IXQLJgBx">MCDA</a>) problem in the <a href="/facts/Evidential_Reasoning_Approach/HuvpLQ84">Evidential Reasoning Approach</a>. Instead of being a single numerical value or a single grade as in a decision matrix, each element in a belief decision matrix is a <a href="/facts/Belief_distribution/VGXupB3F">belief distribution</a>.
For example, suppose Alternative i is "Car i", Criterion j is "Engine Quality" assessed by five grades {Excellent, Good, Average, Below Average, Poor}, and "Car i" is assessed to be “Excellent” on "Engine Quality" with a high degree of belief (e.g. 0.6) due to its low fuel consumption, low vibration and high responsiveness. At the same time, the quality is also assessed to be only “Good” with a lower degree of belief (e.g. 0.4 or less) because its quietness and starting can still be improved. If this is the case, then we have Xij={ (Excellent, 0.6), (Good, 0.4)}, or Xij={ (Excellent, 0.6), (Good, 0.4), (Average, 0), (Below Average, 0), (Poor, 0)}.
A conventional decision matrix is a special case of belief decision matrix when only one belief degree in a belief structure is 1 and the others are 0.

<table><tbody><tr><th></th><th>Criterion 1</th><th>Criterion 2</th><th>...</th><th>Criterion N</th></tr><tr><td>Alternative 1</td><td>x11</td><td>x12</td><td>...</td><td>x1N</td></tr><tr><td>Alternative 2</td><td>x21</td><td>x22</td><td>...</td><td>x2N</td></tr><tr><td>...</td><td>...</td><td>...</td><td>Xij={ (Excellent, 0.6), (Good, 0.4)}</td><td>...</td></tr><tr><td>Alternative M</td><td>xM1</td><td>xM2</td><td>...</td><td>xMN</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Belief_structure/VGXupB3F">Belief structure</a></li>
<li><a href="/facts/Decisional_balance_sheet/eWzXDkFn">Decisional balance sheet</a></li>
<li><a href="/facts/Decision-matrix_method/xC6kAS9u">Decision-matrix method</a></li></ul>

<h3>Sources</h3>

<ul><li>Shafer, G.A. (1976). <a href="https://archive.org/details/mathematicaltheo0000shaf">Mathematical Theory of Evidence</a>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08175-1.</li>
<li>Yang J.B., Singh M.G. (1994). "An evidential reasoning approach for multiple attribute decision making with uncertainty". IEEE Transactions on Systems, Man, and Cybernetics. 24: 1–18. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F21.259681">10.1109/21.259681</a>.</li>
<li>Yang J.B., Xu D.L. (2002). "On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty". IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans. 32 (3): 289–304. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FTSMCA.2002.802746">10.1109/TSMCA.2002.802746</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"When to use a weighted decision matrix". Weighted Decision. 15 October 2013. Retrieved 25 May 2022. <a href="https://www.weighteddecision.com/when-to-use-a-weighted-decision-matrix/" target="_blank">https://www.weighteddecision.com/when-to-use-a-weighted-decision-matrix/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Enz, Cathy A.; Thompson, Gary (June 2013). "The Options Matrix Tool (OMT): A Strategic Decision-making Tool to Evaluate Decision Alternatives". ecommons.cornell.edu. Retrieved 20 December 2022. <a href="https://ecommons.cornell.edu/handle/1813/71278" target="_blank">https://ecommons.cornell.edu/handle/1813/71278</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Decision matrix open-in-new

Decision matrix