Test15 - Revision history

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:51:32Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:51, 13 July 2017
Line 444:		Line 444:


	<center> <math> \textstyle EVM_{e_{2}}=P'[z_1] \cdot EVM_{e_2-z_1}+P'[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 \cdot $80 mio.=$61.5 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}}=P'[z_1] \cdot EVM_{e_2-z_1}+P'[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 \cdot $80 mio.=$61.6 mio. </math> </center>


Line 451:		Line 451:


	<center> <math> \textstyle E[C]=min \{EVM_{e_1};EVM_{e_2} \} = min \{$70 mio.;$61.5 mio. \} =$61.5 mio. </math> </center>		<center> <math> \textstyle E[C]=min \{EVM_{e_1};EVM_{e_2} \} = min \{$70 mio.;$61.6 mio. \} =$61.6 mio. </math> </center>


	It is seen that the lowest expected cost, largest utility, is obtained by installing the ~~strutural~~ health ~~monetering~~ system. Figure 8 show the ~~reults~~ of the decision analysis preformed in ''DPL9''. The figure ~~show~~ the net path values at the terminal nodes and the expected monetary values for both the chance and the decision nodes. The decision tree gives a ~~convinenet~~ overview of the paths and the related costs and probabilities. Example 3 gives a better presentation of the benefits regarding using a ~~decsion~~ tree, compared to the two previous examples.		It is seen that the lowest expected cost, largest utility, is obtained by installing the structural health monitoring system. Figure 8 show the results of the decision analysis preformed in ''DPL9''. The figure shows the net path values at the terminal nodes and the expected monetary values for both the chance and the decision nodes. The decision tree gives a convenient overview of the paths and the related costs and probabilities. Example 3 gives a better presentation of the benefits regarding using a decision tree, compared to the two previous examples.
	[[File:Preposterior21.JPG\|800px\|thumb\|center\|Figure 8: Decision analysis example 3 - Pre-posterior analysis (click to zoom)]]		[[File:Preposterior21.JPG\|800px\|thumb\|center\|Figure 8: Decision analysis example 3 - Pre-posterior analysis (click to zoom)]]

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:48:44Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:48, 13 July 2017
Line 444:		Line 444:


	<center> <math> \textstyle EVM_{~~e_2~~}=P´[z_1] \cdot EVM_{e_2-z_1}+P´[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 \cdot $80 mio.=$61.5 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}}=P'[z_1] \cdot EVM_{e_2-z_1}+P'[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 \cdot $80 mio.=$61.5 mio. </math> </center>

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:47:30Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:47, 13 July 2017
Line 438:		Line 438:


	The last ~~expexted~~ monetary value that needs to be evaluated is related to the last chance node. The chance node associated with the two indications ''z <sub> 1 </sub>'' and ''z <sub> 2 </sub>''. The two quantified expected monetary values <math> \scriptstyle EVM_{e_2-z_1} </math> and <math> \scriptstyle EVM_{e_2-z_2} </math> together with the ~~probabilitise~~ <math> \scriptstyle P'[z_1] </math> and <math> \scriptstyle P'[z_2] </math> are therefore used, this gives:		The last expected monetary value that needs to be evaluated is related to the last chance node. The chance node associated with the two indications ''z <sub> 1 </sub>'' and ''z <sub> 2 </sub>''. The two quantified expected monetary values <math> \scriptstyle EVM_{e_2-z_1} </math> and <math> \scriptstyle EVM_{e_2-z_2} </math> together with the probabilities <math> \scriptstyle P'[z_1] </math> and <math> \scriptstyle P'[z_2] </math> are therefore used, this gives:


Line 444:		Line 444:


	<center> <math> \textstyle EVM_{e_2}=P´[z_1] \cdot EVM_{e_2-z_1}+P´[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 cdot $80 mio.=$61.5 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2}=P´[z_1] \cdot EVM_{e_2-z_1}+P´[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 \cdot $80 mio.=$61.5 mio. </math> </center>

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:43:31Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:43, 13 July 2017
Line 435:		Line 435:


	<center> <math> \textstyle EVM_{e_2-~~z_1~~}=min \{ EVM_{e_2-~~z_1~~-a_1}; EVM_{e_2-~~z_1~~-a_2}\}=min \{$26 mio.;$80 mio.\}= $26 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2-z_2}=min \{ EVM_{e_2-z_2-a_1}; EVM_{e_2-z_2-a_2}\}=min \{$100 mio.;$80 mio.\}= $80 mio. </math> </center>


	The last expexted monetary value that needs to be evaluated is related to the last chance node. The chance node associated with the two indications ''z <sub> 1 </sub>'' and ''z <sub> 2 </sub>''. The two quantified expected monetary values ~~EVM~~ and ~~EVM~~ together with the probabilitise P ~~(z)~~ and P~~(z)~~ are therefore used, this gives:		The last expexted monetary value that needs to be evaluated is related to the last chance node. The chance node associated with the two indications ''z <sub> 1 </sub>'' and ''z <sub> 2 </sub>''. The two quantified expected monetary values <math> \scriptstyle EVM_{e_2-z_1} </math> and <math> \scriptstyle EVM_{e_2-z_2} </math> together with the probabilitise <math> \scriptstyle P'[z_1] </math> and <math> \scriptstyle P'[z_2] </math> are therefore used, this gives:


	''''7. chance node - associated with alternative e <sub> 2 </sub>'''''		''''7. chance node - associated with alternative e<sub>2</sub>'''''


	<center> <math> \textstyle ~~EVM_1~~=P´[z_1] \cdot EVM_{}+P´[z_2] \cdot EVM_{} = $70 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2}=P´[z_1] \cdot EVM_{e_2-z_1}+P´[z_2] \cdot EVM_{e_2-z_2} = 0.34 \cdot $26 mio. + 0.66 cdot $80 mio.=$61.5 mio. </math> </center>


Line 451:		Line 451:


	<center> <math> \textstyle E[C]=min \{EVM_{};EVM_{} \} = $70 mio. </math> </center>		<center> <math> \textstyle E[C]=min \{EVM_{e_1};EVM_{e_2} \} = min \{$70 mio.;$61.5 mio. \} =$61.5 mio. </math> </center>

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:38:14Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:38, 13 July 2017
Line 423:		Line 423:


	<center> <math> \textstyle EVM_{e_1}=min \{ EVM_{e_1-a_1}; EVM_{e_1-a_2}\}= = $70 mio. </math> </center>		<center> <math> \textstyle EVM_{e_1}=min \{ EVM_{e_1-a_1}; EVM_{e_1-a_2}\}=min \{$70 mio.;$75 mio.\} = $70 mio. </math> </center>


Line 429:		Line 429:


	<center> <math> \textstyle EVM_{e_2-z_1}=min \{ ~~EVM_1~~; ~~EVM_2~~\}=~~0.65~~ \~~cdot~~ $40 mio. ~~+ 0.35 \cdot~~ $~~140~~ mio.= $75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2-z_1}=min \{ EVM_{e_2-z_1-a_1}; EVM_{e_2-z_1-a_2}\}=min \{$26 mio.;$80 mio.\}= $26 mio. </math> </center>


Line 435:		Line 435:


	<center> <math> \textstyle EVM_{e_2-~~z_2~~}=min \{ ~~EVM_1~~; ~~EVM_2~~\}=~~0.65~~ \~~cdot~~ $40 mio. ~~+ 0.35 \cdot~~ $~~140~~ mio.= $75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2-z_1}=min \{ EVM_{e_2-z_1-a_1}; EVM_{e_2-z_1-a_2}\}=min \{$26 mio.;$80 mio.\}= $26 mio. </math> </center>

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:35:41Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:35, 13 July 2017
Line 423:		Line 423:


	<center> <math> \textstyle EVM_{e_1}=min \{ ~~EVM_1~~; ~~EVM_2~~\}= = $70 mio. </math> </center>		<center> <math> \textstyle EVM_{e_1}=min \{ EVM_{e_1-a_1}; EVM_{e_1-a_2}\}= = $70 mio. </math> </center>

S113782 at 09:34, 13 July 2017

2017-07-13T09:34:20Z

← Older revision		Revision as of 09:34, 13 July 2017
Line 410:		Line 410:


	<center> <math> \textstyle EVM_{e_{2}-z_2-a_1}=P''[\theta_1 \| z_2] \cdot NPV_1 + P''[\theta_2 \| z_2] \cdot NPV_2= 0.05 \cdot $5 mio. + 0.95 \cdot $105 mio. =$75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}-z_2-a_1}=P''[\theta_1 \| z_2] \cdot NPV_1 + P''[\theta_2 \| z_2] \cdot NPV_2= 0.05 \cdot $5 mio. + 0.95 \cdot $105 mio. =$100 mio. </math> </center>

	'''''6. chance node - associated with alternative e <sub> 2 </sub> - z <sub> 2 </sub> - a <sub> 2 </sub>'''''		'''''6. chance node - associated with alternative e <sub> 2 </sub> - z <sub> 2 </sub> - a <sub> 2 </sub>'''''

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:20:52Z

Example 3 - Pre-Posterior analysis

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:15:39Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:15, 13 July 2017
Line 289:		Line 289:
	=== Example 3 - Pre-Posterior analysis ===		=== Example 3 - Pre-Posterior analysis ===

	Often the decision maker has the possibility to ‘buy’ additional information through an experiment before making the choice of action. If the cost of this information is small in comparison to the potential value of the information, the decision maker should perform the experiment. If several different types of experiments are possible, the decision maker must choose the experiment yielding the overall largest expected value of utility. The conditional probability is used in the pre-posterior analysis. Let the prior condition be the same as the one presented in example 1. The alternative actions have however change, since there now is the option to install a SHM (Structural health monitoring) system. This gives the action alternatives no do the experiment, <math> \scriptstyle {e_1}</math> or do it <math> \scriptstyle {e_2} </math>, before chosen between doing nothing or repair the bridge. The SHM system indicates either the structure is safe <math> \scriptstyle {z_1} </math> or damaged <math> \scriptstyle {z_2} </math>. The cost of the SHM system is $5 mio. For to the finite precision, the system indications are probabilities depending the states. Table 5 present the indications of the SHM system.		Often the decision maker has the possibility to ‘buy’ additional information through an experiment before making the choice of action. If the cost of this information is small in comparison to the potential value of the information, the decision maker should perform the experiment. If several different types of experiments are possible, the decision maker must choose the experiment yielding the overall largest expected value of utility. The conditional probability is used in the pre-posterior analysis. Let the prior condition be the same as the one presented in example 1. The alternative actions have however change, since there now is the option to install a SHM (Structural health monitoring) system. This gives the action alternatives no do the experiment ''<math> \scriptstyle {e_1}</math>'' or do the experiment ''<math> \scriptstyle {e_2} </math>'', before chosen between doing nothing or repair the bridge. The SHM system indicates either the structure is safe ''<math> \scriptstyle {z_1} </math>'' or damaged ''<math> \scriptstyle {z_2} </math>''. The cost of the SHM system is $5 mio. For to the finite precision, the system indications are probabilities depending the states. Table 5 present the indications of the SHM system.
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"
	\|+''Table 5: Structural health monitoring indications''		\|+''Table 5: Structural health monitoring indications''
Line 392:		Line 392:


	<center> <math> \textstyle ~~EVM_2~~=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{1}-a_2}=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>


Line 398:		Line 398:


	<center> <math> \textstyle ~~EVM_3~~=P''[\theta_1 \| z_1] \cdot NPV_1 + P''[\theta_2 \| z_1] \cdot NPV_2= 0.79 \cdot $5 mio. + 0.21 \cdot $105 mio. =$26 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}-z_1-a_1}=P''[\theta_1 \| z_1] \cdot NPV_1 + P''[\theta_2 \| z_1] \cdot NPV_2= 0.79 \cdot $5 mio. + 0.21 \cdot $105 mio. =$26 mio. </math> </center>

S113782: /* Example 3 - Pre-Posterior analysis */

2017-07-13T09:11:48Z

Example 3 - Pre-Posterior analysis

← Older revision		Revision as of 09:11, 13 July 2017
Line 289:		Line 289:
	=== Example 3 - Pre-Posterior analysis ===		=== Example 3 - Pre-Posterior analysis ===

	Often the decision maker has the possibility to ‘buy’ additional information through an experiment before making the choice of action. If the cost of this information is small in comparison to the potential value of the information, the decision maker should perform the experiment. If several different types of experiments are possible, the decision maker must choose the experiment yielding the overall largest expected value of utility. The conditional probability is used in the pre-posterior analysis. Let the prior condition be the same as the one presented in example 1. The alternative actions have however change, since there now is the option to install a SHM (Structural health monitoring) system. This gives the action alternatives to do the experiment or ~~not~~, before chosen between doing nothing or repair the bridge. The SHM system indicates either the structure is safe <math> \scriptstyle {z_1} </math> or damaged <math> \scriptstyle {z_2} </math>. The cost of the SHM system is $5 mio. For to the finite precision, the system indications are probabilities depending the states. Table 5 present the indications of the SHM system.		Often the decision maker has the possibility to ‘buy’ additional information through an experiment before making the choice of action. If the cost of this information is small in comparison to the potential value of the information, the decision maker should perform the experiment. If several different types of experiments are possible, the decision maker must choose the experiment yielding the overall largest expected value of utility. The conditional probability is used in the pre-posterior analysis. Let the prior condition be the same as the one presented in example 1. The alternative actions have however change, since there now is the option to install a SHM (Structural health monitoring) system. This gives the action alternatives no do the experiment, <math> \scriptstyle {e_1}</math> or do it <math> \scriptstyle {e_2} </math>, before chosen between doing nothing or repair the bridge. The SHM system indicates either the structure is safe <math> \scriptstyle {z_1} </math> or damaged <math> \scriptstyle {z_2} </math>. The cost of the SHM system is $5 mio. For to the finite precision, the system indications are probabilities depending the states. Table 5 present the indications of the SHM system.
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"
	\|+''Table 5: Structural health monitoring indications''		\|+''Table 5: Structural health monitoring indications''
Line 298:		Line 298:

	\|-		\|-
	\|''z <sub> 1 </sub>'' - Safe		\|''z <sub>1</sub>'' - Safe
	\| 0.9		\| 0.9
	\| 0.1		\| 0.1
	\|-		\|-
	\|''z <sub> 2 </sub>''-Damaged		\|''z <sub>2</sub>''-Damaged
	\| 0.1		\| 0.1
	\| 0.9		\| 0.9

← Older revision		Revision as of 09:20, 13 July 2017
Line 404:		Line 404:


	<center> <math> \textstyle ~~EVM_4~~=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2= 0.65 \cdot $45 mio. + 0.35 \cdot $145 mio. =$80 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}-z_1-a_2}=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2= 0.65 \cdot $45 mio. + 0.35 \cdot $145 mio. =$80 mio. </math> </center>


Line 410:		Line 410:


	<center> <math> \textstyle ~~EVM_5~~=P''[\theta_1 \| z_2] \cdot NPV_1 + P''[\theta_2 \| z_2] \cdot NPV_2= 0.05 \cdot $5 mio. + 0.95 \cdot $105 mio. =$75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}-z_2-a_1}=P''[\theta_1 \| z_2] \cdot NPV_1 + P''[\theta_2 \| z_2] \cdot NPV_2= 0.05 \cdot $5 mio. + 0.95 \cdot $105 mio. =$75 mio. </math> </center>

	'''''6. chance node - associated with alternative e <sub> 2 </sub> - z <sub> 2 </sub> - a <sub> 2 </sub>'''''		'''''6. chance node - associated with alternative e <sub> 2 </sub> - z <sub> 2 </sub> - a <sub> 2 </sub>'''''


	<center> <math> \textstyle ~~EVM_6~~=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2= 0.65 \cdot $45 mio. + 0.35 \cdot $145 mio. =$80 mio. </math> </center>		<center> <math> \textstyle EVM_{e_{2}-z_2-a_2}=P'[\theta_1] \cdot NPV_1 + P'[\theta_2] \cdot NPV_2= 0.65 \cdot $45 mio. + 0.35 \cdot $145 mio. =$80 mio. </math> </center>

			The expected monetary values for the three decision nodes related to alternatives ''a<sub>1</sub>'' and ''a<sub>2</sub>'', is then established, the value is determined as the minimum cost, which is equivalent to the largest utility. The expected monetary values are then:

	The expexted monetary values for the three decision nodes related to alternatives ''a<sub>1</sub>'' and ''a<sub>2</sub>'', the value is determined as the minimum cost, which is equliant to the largest utility. The expected monetary values are then:

	'''''1. decision node - associated with alternative e <sub> 1 </sub>'''''		'''''1. decision node - associated with alternative e<sub>1</sub>'''''


	<center> <math> \textstyle ~~EVM_1~~=min \{ EVM_1; EVM_2\}= = $70 mio. </math> </center>		<center> <math> \textstyle EVM_{e_1}=min \{ EVM_1; EVM_2\}= = $70 mio. </math> </center>


	'''''2. decision node - associated with alternative e <sub> 1 </sub> - z <sub> 1 </sub>'''''		'''''2. decision node - associated with alternative e<sub>2</sub> - z<sub>1</sub>'''''


	<center> <math> \textstyle ~~EVM_2~~=min \{ EVM_1; EVM_2\}=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2-z_1}=min \{ EVM_1; EVM_2\}=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>


	'''''3. decision node - associated with alternative e <sub> 2 </sub> - z <sub> 2 </sub>'''''		'''''3. decision node - associated with alternative e<sub>2</sub> - z<sub>2</sub>'''''


	<center> <math> \textstyle ~~EVM_2~~=min \{ EVM_1; EVM_2\}=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>		<center> <math> \textstyle EVM_{e_2-z_2}=min \{ EVM_1; EVM_2\}=0.65 \cdot $40 mio. + 0.35 \cdot $140 mio.= $75 mio. </math> </center>