Decision making under risk - Revision history

Tkokotas at 19:28, 20 November 2018

2018-11-20T19:28:43Z

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			''Developed by Regitza Hansen''


	Decision making is one of the most important tasks in the management process and it is often a very difficult one. When having knowledge regarding the states of nature, subjective probability estimates for the occurrence of each state can be assigned. In such cases, the problem is classified as decision making under risk.		Decision making is one of the most important tasks in the management process and it is often a very difficult one. When having knowledge regarding the states of nature, subjective probability estimates for the occurrence of each state can be assigned. In such cases, the problem is classified as decision making under risk.
	In the decision making process, all relevant information is evaluated through decision analysis (DA). The decision analysis process consist of the use of a decison tool and a decsion theory. The decision tree is the most commonly applied decision tool in the decision analysis. The decision theory of interest in the decision analysis, regarding the decision making under risk, is the expected value of criterion also reffered to as the Bayesian principle. This is the only one of the four decision methods that incorporates the probabilities of the states of nature.		In the decision making process, all relevant information is evaluated through decision analysis (DA). The decision analysis process consist of the use of a decison tool and a decsion theory. The decision tree is the most commonly applied decision tool in the decision analysis. The decision theory of interest in the decision analysis, regarding the decision making under risk, is the expected value of criterion also reffered to as the Bayesian principle. This is the only one of the four decision methods that incorporates the probabilities of the states of nature.

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2017-07-13T08:37:22Z

Example

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	<br clear=all>		<br clear=all>




	== Key benefits ==		== Key benefits ==

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	E[C]=min{EVM <sub> a <sub> 1 <sub> <sub>;EVM <sub> a <sub> 2 <sub> <sub>}.		E[C]=min{EVM <sub> a <sub> 1 <sub> <sub>;EVM <sub> a <sub> 2 <sub> <sub>}.



			=== Example 2 - Prosterior analysis ===

			When additional information becomes available, the probability structure in the decision problem may be updated. Having updated the probability structure the decision analysis is unchanged in comparison to the situation with given prior information. The same condition and information provided in example 1 is used. Now acquired more information through a study about the chances of a bridge repairment. The
			study costs $5mio. Table 4 contains the estimation of the success rate (structure safe) of the bridge repairations in the study.
			Bridge repairment results
			𝑆 (structure safe) 𝑆̅(structure damaged)
			Indication 𝐼 90% 10%
			Table 4: Study indication of bridge repairment success.

			Given the result of the study, the updated probability structure or the posterior probability,now called P'(S) is evaluated by use of the Bayes' rule.
			P
			��[θi] = P[I\| θi]P[θi ]�j P[zk\| θj
			]P�[θj ]

			P[I]=

			The decision tree is now updated and is presented in figure 3.



			The cost for the repairment of the bridge is now changed to $25mio., since is is both the cost of the repairent and the study.


			This now give:


			Compared to the prioir analysis, the new
			The prerior analysis is related to the condition of known information. Using the same example as given previous under the construcion of the decision tree, following information is now provided:
			If the bridge structure is damaged, then a new bridge is reqiured which is asssociated with a cost of $100 mio. If the structure is safe then the cost is $0. It is estimated that there is a 10% probability that the bridge structure is safe. If the bridge is repaired then the probabaility of the bridge stucture having damaged is 20%. Reparing the bridge costs $20 mio.
			Table 1 provides the probabilities used in the construcion of the decision tree in figure 2.

			{\| class="wikitable" style="text-align:center"
			\|-
			! width="100" align="center"\|
			! width="100" align="center"\|''a<sub>1</sub>''- Do nothing
			! width="100" align="center"\|''a<sub>2</sub>''- Repair bridge

			\|-
			\| P(S1) - Safe
			\| 0.1
			\| 0.8
			\|-
			\| P(S2) - Damaged
			\| 0.9
			\| 0.2

			\|}
			''Table 1: Probabilities related to the two alternatives''

			The cost for reparing the bridge is $20mio, which means that ''cost1=$0mio.'' and ''cost2=$20mio.''. Providing the given values in figure 1, the decison tree is now represented in figure 2.
			<gallery widths=450px heights=280px perrow=1>
			File:Prioir1.JPG\|Figure 2 - Decision tree
			</gallery>

			The pogram used is having the cost related to each branch, which means that value for the terminal nodes, are the sum of the cost for each branch. This values is also called the Net Path value or NPV. The values are shown in figure 3 and futhermore presented in table 2 below.

			{\| class="wikitable" style="text-align:center"
			\|-
			! width="100" align="center"\| NPV
			! width="100" align="center"\|''Cost''-


			\|-
			\| NPV <sub> 1 <sub>
			\| $0mio.
			\|-
			\| NPV <sub> 2 <sub>
			\| $100mio.
			\|-
			\| NPV <sub> 3 <sub>
			\| $20mio.
			\|-
			\| NPV <sub> 4 <sub>
			\| $120mio.
			\|-

			\|}
			''Table 1: Net path values (NPV) for example 1.''

			The decision tree analysis is now preformed, starting with first chance node, associated with alternative ''a <sub> 1 <sub>''.
			''EVM=P(s1)0 + P(s2)$100mio=$90mio''
			Chance node 2
			''EVM=P(s1)$120mio + P(s2)$120mio=$40mio''

			The most benifical solution, is the one associated with the lowest cost, so:

			E[C]=min{EVM <sub> a <sub> 1 <sub> <sub>;EVM <sub> a <sub> 2 <sub> <sub>}.

S113782 at 16:44, 23 June 2017

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@@ Line 91: / Line 91: @@
 {| class="wikitable" style="text-align:center"
 |-
-! width="100" align="center"|Do nothing
+! width="100" align="center"|
-! width="100" align="center"|Repair bridge
+! width="100" align="center"|''a<sub>1</sub>''- Do nothing
 |-
-| P(S1)
+| P(S1) - Safe
 | 0.9
 | 0.2
 |-
 |}
-''Table 1: Exemplary assessment of EVA data''
+''Table 1: Net path values (NPV) for example 1.''

S113782 at 16:18, 23 June 2017

2017-06-23T16:18:47Z

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	* Terminal (consequence) Node – triangles		* Terminal (consequence) Node – triangles

	For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:		For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. If the bridge is damaged, it is unfit for service and a new one needs to be constructed. Your company is hired to find the best solution of two alternatives:
	<ol>		<ol>
	<li> Alternative ''a ~~&sub2;~~'' - Do nothing</li>		<li> Alternative ''a<sub>2</sub>'': ''Do nothing'' </li>
	<li> Alternative ''a<sub>2</sub>''- Repair the bridge</li>		<li> Alternative ''a<sub>2</sub>'': ''Repair the bridge'' </li>
	</ol>		</ol>
			The decision analysis is therefore to evaluate weither is most beneficial to do nothing or repair the bridge, so which alternative is associated with the lowest risk, in this case the lowest risk is the alternative with the lowest expected cost.

	The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.		In figure 1, the first node drawn is the decision node, a branch emanating from a decision node corresponds to a decision alternative, which in this case are either to repair the bridge or do nothing. It includes a cost or benefit value, which in this example is refered to as ''cost 1'' and ''cost 2''.
	In figure 1, the first node drawn is the decision node, a branch emanating from a decision node corresponds to a decision alternative, which in this case are either to ~~modify~~ the bridge or do nothing. It includes a cost or benefit value.


	<gallery widths=450px heights=280px perrow=1>		<gallery widths=450px heights=280px perrow=1>
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	</gallery>		</gallery>

			The next node is a chance node, since both alternatives does not lead to a new the decision. A branch emanating from a state of nature (chance) node corresponds to a particular state of nature, and includes the probability of this state of nature. For both alternatives the branches has the same outcome, either the bridge is safe and fit for service or the bridge is damaged. The probability associtaed with the bridge being safe or damaged, are denoted P(S1) and P(S2), respectively.
	The next node is a chance node, since both alternatives does not lead to a new the decision. A branch emanating from a state of nature (chance) node corresponds to a particular state of nature, and includes the probability of this state of nature. For both alternatives the branches has the same outcome, either ~~a new bridge needs to be constructed or~~ the bridge is fit for service.		The final node is the terminal node, this represent the cost consequens related to the choice of decision branch.

	The ~~next step in~~ the ~~decision making process~~, is the ~~decision tree analysis (DTA)~~, ~~which depends on~~ the ~~chose~~ of decision ~~theory, which under risk~~

	=== Decision theory ===		=== Decision theory ===
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	The prioir decicison analysis with known information, it quantify the beliefs before any evidince is taking into account.		The prioir decicison analysis with known information, it quantify the beliefs before any evidince is taking into account.



			== Baysian decision analysis ==

			The theory and principles of decision making under risk, has been represented. An example with is now presented with the applied theory.

			=== Example 1 - Prior analysis ===

			The prioir analysis is related to the condition of known information. Using the same example as given previous under the construcion of the decision tree, following information is now provided:
			If the bridge structure is damaged, then a new bridge is reqiured which is asssociated with a cost of $100 mio. If the structure is safe then the cost is $0. It is estimated that there is a 10% probability that the bridge structure is safe. If the bridge is repaired then the probabaility of the bridge stucture having damaged is 20%. Reparing the bridge costs $20 mio.
			Table 1 provides the probabilities used in the construcion of the decision tree in figure 2.

	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"
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	\|}		\|}
	''Table 1: Exemplary assessment of EVA data''		''Table 1: Exemplary assessment of EVA data''



	~~== Baysian decision analysis ==~~

	~~Having the same sceneriao as~~
	~~Using the model constructed in figure 1,~~

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	For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:		For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:
	1) Alternative ''a&sub2;'' - Do nothing		<ol>
	2) Alternative ''a<sub>2</sub>''- Repair the bridge		<li> Alternative ''a &sub2;'' - Do nothing</li>
			<li> Alternative ''a<sub>2</sub>''- Repair the bridge</li>
	~~(²) to~~		</ol>
	~~1 + x2 + y3 (~~<~~sup~~>2</~~sup~~>~~) or~~
	~~1 + x2 + y3 ({{sup\|2}}~~

	The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.		The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.

S113782 at 15:17, 23 June 2017

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	For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:		For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:
	1) Alternative a~~{{sub\|1}}~~ - Do nothing		1) Alternative ''a&sub2;'' - Do nothing
	2) Alternative a{{sub\|2}}- Repair the bridge		2) Alternative ''a<sub>2</sub>''- Repair the bridge

	(²) to		(²) to

S113782 at 15:16, 23 June 2017

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	For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:		For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:
	1) Alternative ''a{{sub\|1}}'' - Do nothing		1) Alternative a{{sub\|1}} - Do nothing
	2) Alternative ''a{{sub\|2}}''- Repair the bridge		2) Alternative a{{sub\|2}}- Repair the bridge

			(²) to
			1 + x2 + y3 (<sup>2</sup>) or
			1 + x2 + y3 ({{sup\|2}}

	The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.		The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.

S113782 at 15:13, 23 June 2017

2017-06-23T15:13:25Z

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	* Terminal (consequence) Node – triangles		* Terminal (consequence) Node – triangles

	For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge ~~exhibits abnormally fast deterioration. If this~~ deterioration ~~is not stopped~~, ~~it is likely~~ that the bridge ~~has to~~ be ~~replaced by a new bridge soon~~. However, ~~it is known that a within the next year, the bridge has to~~ be ~~turn down anyway, cause of new city-planning~~. Your company is hired to find the best solution of two alternatives:		For the purpose of illustration, the decision tree in figure 1 considers the following very simple decision problem. An old bridge has been subject to deterioration, control data reveal that the bridge structure may be damaged. However, this cannot be indicated with certainty. Your company is hired to find the best solution of two alternatives:
	1) ~~Modify the bridge in order to stop the deterioration.~~		1) Alternative ''a{{sub\|1}}'' - Do nothing
	2) ~~Do nothing.~~		2) Alternative ''a{{sub\|2}}''- Repair the bridge

	The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed. In figure 1, the first node drawn is the decision node, a branch emanating from a decision node corresponds to a decision alternative, which in this case are either to modify the bridge or do nothing. It includes a cost or benefit value.
			The bridge modification does not garanti that the detoration will stop, and if the deterotaion continuous it will lead to building a new bridge. So it benficial to spent money on the modifacation or do nothing and replace the bridge when needed.
			In figure 1, the first node drawn is the decision node, a branch emanating from a decision node corresponds to a decision alternative, which in this case are either to modify the bridge or do nothing. It includes a cost or benefit value.

S113782 at 14:33, 23 June 2017

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	<gallery widths=450px heights=280px perrow=1>		<gallery widths=450px heights=280px perrow=1>
	File:Simple.~~jpg~~\|Figure 1 - Decision tree		File:Simple.JPG\|Figure 1 - Decision tree
	</gallery>		</gallery>