Data-Driven Decision-Making under Uncertainty - Revision history

S222572: /* Limitations */

2023-05-09T16:56:38Z

Limitations

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	While the described approaches can be applied in practice for different use cases in Project, Program and Portfolio Management, they also have their limitations: First of all, the assumption that the given data is reliable can be criticized. Since scenarios with certain outcomes are mostly developed based on conjectures rather than on facts, they always bring along inaccuracies. Secondly, the fact that every scenario is considered equally likely can cause distortions regarding extreme cases. For example, the worst- and the best-case scenario are usually less probable compared to a scenario with an outcome closer to the expected value. If the probabilities were known, tools for decision-making under risk could be applied to obtain a more accurate result. Another point of criticism is that with the classical approaches, only one criterion can be examined at a time. To compare multiple features like costs together with duration, the criteria can be assigned a percentage weight before applying the strategies to all the criteria and calculating the weighted result to determine a preferred alternative. However, this procedure comes with even more inaccuracy since setting a weight between different features comes with subjectivity. <ref name="Renou2010"/>		While the described approaches can be applied in practice for different use cases in Project, Program and Portfolio Management, they also have their limitations: First of all, the assumption that the given data is reliable can be criticized. Since scenarios with certain outcomes are mostly developed based on conjectures rather than on facts, they always bring along inaccuracies. Secondly, the fact that every scenario is considered equally likely can cause distortions regarding extreme cases. For example, the worst- and the best-case scenario are usually less probable compared to a scenario with an outcome closer to the expected value. If the probabilities were known, tools for decision-making under risk could be applied to obtain a more accurate result. Another point of criticism is that with the classical approaches, only one criterion can be examined at a time. To compare multiple features like costs together with duration, the criteria can be assigned a percentage weight before applying the strategies to all the criteria and calculating the weighted result to determine a preferred alternative. However, this procedure comes with even more inaccuracy since setting a weight between different features comes with subjectivity. <ref name="Renou2010"/>
	However, to provide a quantitative foundation for decision-making, these approaches are very useful and considering the required effort fairly easy to implement. <ref name="Mohanty2014"/>		However, to provide a quantitative foundation for decision-making, these approaches are very useful and considering the required effort, fairly easy to implement. <ref name="Mohanty2014"/>

	==Annotated bibliography==		==Annotated bibliography==

S222572: /* Application to outsourcing contracts within Project Management */

2023-05-09T16:56:27Z

Application to outsourcing contracts within Project Management

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	<math> \lambda _{B,C} = 0.778 </math> <br />		<math> \lambda _{B,C} = 0.778 </math> <br />

	Figure 2 shows a graphical representation of these functions. It can be seen that there are three sections in the diagram depending on the value of λ. From the Minimax approach (λ=0) until the intersection of the function for supplier A and C supplier A will be chosen since it shows the lowest value in this region. After that, supplier C becomes more attractive and will thus be chosen. However, after the intersection between the functions for supplier B and C, it is superseded by supplier B, with supplier B staying the most attractive option until the Maxmimax approach (λ=1). While it was already evident after the Minimax and the Maximax approach, how the project manager would decide in the extreme cases (λ=0 and λ=1), a new insight could be gained by examining the Hurwicz criterion using a Sensitivity Analysis: Picking supplier C can also be an option if the Optimism parameter lies between 0.600 and 0.778, which indicates a required above-average risk attitude by the project manager.		Figure 2 shows a graphical representation of these functions. It can be seen that there are three sections in the diagram depending on the value of λ. From the Minimax approach (λ=0) until the intersection of the function for supplier A and C supplier A will be chosen since it shows the lowest value in this region. After that, supplier C becomes more attractive and will thus be chosen. However, after the intersection between the functions for supplier B and C, it is superseded by supplier B, with supplier B staying the most attractive option until the Maxmimax approach (λ=1). While it was already evident after the Minimax and the Maximax approach how the project manager would decide in the extreme cases (λ=0 and λ=1), a new insight could be gained by examining the Hurwicz criterion using a Sensitivity Analysis: Picking supplier C can also be an option if the Optimism parameter lies between 0.600 and 0.778, which indicates a required above-average risk attitude by the project manager.

	<br />		<br />

	After considering the Hurwicz Criterion, the following approach will determine the optimal alternative according to the '''Minimax Regret''' principle: Since there are no parameters to alter like the Optimism Parameter before, the result can directly be calculated. Supplier A will be chosen in this case since it has the lowest opportunity costs (which means the highest opportunity income), followed by Supplier C and then B as seen in the following table. This approach is an extension of the basic Minimax Criterion since it compares the alternatives with each other for each scenario instead of for only one outcome. Thus, it should be used for decisions where the risk of unexpected increased costs should be kept low.		After considering the Hurwicz Criterion, the following approach will determine the optimal alternative according to the '''Minimax Regret''' principle: Since there are no parameters to alter like the Optimism Parameter before, the result can be directly calculated. Supplier A will be chosen in this case since it has the lowest opportunity costs (which means the highest opportunity income), followed by Supplier C and then B as seen in the following table. This approach is an extension of the basic Minimax Criterion since it compares the alternatives with each other for each scenario instead of for only one outcome. Thus, it should be used for decisions where the risk of unexpected increased costs should be kept low.

	{\| class="wikitable" style="text-align:right;"		{\| class="wikitable" style="text-align:right;"
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	<br />		<br />

	[[File:Monte_Carlo_Simulation_Diagram.PNG\|thumb\|text-bottom\|right\|305px\|Figure 3: Histogram for a Monte Carlo Simulation for supplier selection]]Lastly, the '''Laplace Criterion''' will be calculated. Calculating the expected values for the suppliers yields the lowest cost for supplier A, followed by supplier B and finally C. However, this average value does not give an overview of the distribution of the possible outcomes, which can be additional information helping the project manager to make a decision. For example, two alternatives could have the same value for the Laplace criterion with ~~one~~ one featuring a broad distribution, while the values for the outcomes of the other alternative lie closer together. In order to display the distribution for the supplier selection, a Monte Carlo Simulation will be conducted. <ref name="Fang2021"/>		[[File:Monte_Carlo_Simulation_Diagram.PNG\|thumb\|text-bottom\|right\|305px\|Figure 3: Histogram for a Monte Carlo Simulation for supplier selection]]Lastly, the '''Laplace Criterion''' will be calculated. Calculating the expected values for the suppliers yields the lowest cost for supplier A, followed by supplier B and finally C. However, this average value does not give an overview of the distribution of the possible outcomes, which can be additional information helping the project manager to make a decision. For example, two alternatives could have the same value for the Laplace criterion, with one featuring a broad distribution, while the values for the outcomes of the other alternative lie closer together. In order to display the distribution for the supplier selection, a Monte Carlo Simulation will be conducted. <ref name="Fang2021"/>

	Since the scenario that will occur is uncertain, it can be modelled by generating random numbers. In this case, a random number from 1 to 5 was produced and then attributed to the respective costs for each supplier for the scenario matching the random number. This Simulation was run 1,000 times to ensure a comparably stable result. The results are illustrated in a histogram in Figure 3.		Since the scenario that will occur is uncertain, it can be modelled by generating random numbers. In this case, a random number from 1 to 5 was produced and then attributed to the respective costs for each supplier for the scenario matching the random number. This Simulation was run 1,000 times to ensure a comparably stable result. The results are illustrated in a histogram in Figure 3.

	Since supplier A shows no deviation depending on the scenario, all 1,000 results yielded a cost of 100 monetary units. Comparing suppliers B and C makes clear that supplier B shows a wide spread around the estimated costs, while supplier C stays closer to the average. The reason that supplier C shows a comparably high ~~occurence~~ of cases with a cost between 88.75 and 96.25 is that both scenarios 1 and 2 fall into this range. Another insightful piece of information can be the comparison of the average values for the Monte Carlo Analysis per supplier with the respective Laplace Criteria, which the following table shows:		Since supplier A shows no deviation depending on the scenario, all 1,000 results yielded a cost of 100 monetary units. Comparing suppliers B and C makes clear that supplier B shows a wide spread around the estimated costs, while supplier C stays closer to the average. The reason that supplier C shows a comparably high occurrence of cases with a cost between 88.75 and 96.25 is that both scenarios 1 and 2 fall into this range. Another insightful piece of information can be the comparison of the average values for the Monte Carlo Analysis per supplier with the respective Laplace Criteria, which the following table shows:

	{\| class="wikitable" style="text-align:right;"		{\| class="wikitable" style="text-align:right;"

S222572: /* Application to outsourcing contracts within Project Management */

2023-05-09T16:53:42Z

Application to outsourcing contracts within Project Management

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	<br />		<br />

	For the '''Maximax Criterion''', the costs for scenario 1 are taken into account. Supplier B as the tenderer with the highest price deviations, shows the lowest costs for the best-case scenario, which is why this supplier would be selected in this case. The fact that the costs for supplier A don't change with the scenarios shows the opposite effect to the Minimax Criterion: Since there are also no deviations towards lower prices, this supplier will not be the most attractive in case the best possible outcome occurs. The Maximax criterion can hence be used if the decision's impact is minor and taking a higher risk can be afforded.		For the '''Maximax Criterion''', the costs for scenario 1 are taken into account. Supplier B, as the tenderer with the highest price deviations, shows the lowest costs for the best-case scenario, which is why this supplier would be selected in this case. The fact that the costs for supplier A don't change with the scenarios shows the opposite effect to the Minimax Criterion: Since there are also no deviations towards lower prices, this supplier will not be the most attractive in case the best possible outcome occurs. The Maximax criterion can hence be used if the decision's impact is minor and taking a higher risk can be afforded.

	<br />		<br />

S222572: /* Monte Carlo Simulation */

2023-05-09T16:52:54Z

Monte Carlo Simulation

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	====Monte Carlo Simulation====		====Monte Carlo Simulation====

	A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, skewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation, and it can even be combined with the first concept. In this way, the user can get a better understanding ~~about~~ the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>		A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, skewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation, and it can even be combined with the first concept. In this way, the user can get a better understanding of the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>

	====Sensitivity Analysis====		====Sensitivity Analysis====

S222572: /* Monte Carlo Simulation */

2023-05-09T16:52:27Z

Monte Carlo Simulation

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	====Monte Carlo Simulation====		====Monte Carlo Simulation====

	A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, skewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>		A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, skewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation, and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>

	====Sensitivity Analysis====		====Sensitivity Analysis====

S222572: /* Monte Carlo Simulation */

2023-05-09T16:51:55Z

Monte Carlo Simulation

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	====Monte Carlo Simulation====		====Monte Carlo Simulation====

	A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, ~~scewness~~ or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>		A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, skewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>

	====Sensitivity Analysis====		====Sensitivity Analysis====

S222572: /* Monte Carlo Simulation */

2023-05-09T16:51:10Z

Monte Carlo Simulation

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	====Monte Carlo Simulation====		====Monte Carlo Simulation====

	A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, box plot or analyzed further by determining statistical parameters such as the mean, variance, scewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>		A Monte Carlo Simulation is a probabilistic simulation method used to model randomness in certain parameters to analyze how results change depending on these parameters. It is a very versatile tool and can provide valuable insights even for problems with high complexity. It is usually run a high number of times to generate multiple values to obtain a more precise end result and minimize statistical deviations. <ref name="Brandimarte2014"/> The end result can then be either illustrated in a histogram, or box plot or analyzed further by determining statistical parameters such as the mean, variance, scewness or kurtosis. With regard to decision-making problems, the Monte Carlo Simulation can be used to model the occurrence of a scenario. Since it is assumed that all scenarios are equally probable, a decision-making basis can be the mean outcome for each alternative after generating a random scenario for a defined number of times using a Monte Carlo Simulation. Another application of the Monte Carlo Simulation is modelling a deviation of the outcome values <math>x_{i,s}</math>. This can be either an absolute or relative deviation and it can even be combined with the first concept. In this way, the user can get a better understanding about the influence of different parameters and the consequences of choosing a certain alternative. <ref name="Raychaudhuri2008"/>

	====Sensitivity Analysis====		====Sensitivity Analysis====

S222572: /* Laplace Criterion */

2023-05-09T16:50:19Z

Laplace Criterion

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	====Laplace Criterion====		====Laplace Criterion====
	The Laplace Criterion (also called Equal Likelihood Criterion) compares the expected value of each alternative with each other. Thereby, every outcome is considered, which means that outliers that would act as extreme values for the Minimax or Maximax Criterion will have a considerably lower importance with the Laplace Criterion. Furthermore, since there will be no consideration of only minimum or maximum values, the formula for the criterion per alternative is the same for both benefits and costs, while for benefits the maximum and for costs the minimum value is considered:		The Laplace Criterion (also called Equal Likelihood Criterion) compares the expected value of each alternative with each other. Thereby, every outcome is considered, which means that outliers that would act as extreme values for the Minimax or Maximax Criterion will have a considerably lower importance with the Laplace Criterion. Furthermore, since there will be no consideration of only minimum or maximum values, the formula for the criterion per alternative is the same for both benefits and costs, while for benefits, the maximum and for costs, the minimum value is considered:

	<math>Laplace_B = max((\sum_{s=1}^{S} x_{i,s})/S ~\|~ i \in I) </math> <br />		<math>Laplace_B = max((\sum_{s=1}^{S} x_{i,s})/S ~\|~ i \in I) </math> <br />

S222572: /* Maximax Criterion */

2023-05-09T16:44:06Z

Maximax Criterion

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	====Maximax Criterion====		====Maximax Criterion====
	The Maximax Criterion (also called Minimin Criterion) works opposite to the Minimax approach: Here, the best possible outcome is considered, which means that all other outcomes are ignored. Therefore, it selects an alternative based on the maximum benefit possible (or the minimum cost possible), while disregarding more negative consequences for decision alternatives:		The Maximax Criterion (also called Minimin Criterion) works opposite to the Minimax approach: Here, the best possible outcome is considered, which means that all other outcomes are ignored. Therefore, it selects an alternative based on the maximum benefit possible (or the minimum cost possible) while disregarding more negative consequences for decision alternatives:

	<math>B:Maximax(i) = max(x_{i,max}~\|~i \in I)</math> <br />		<math>B:Maximax(i) = max(x_{i,max}~\|~i \in I)</math> <br />

S222572: /* Differences between Decision-Making under uncertainty and risk */

2023-05-09T16:43:30Z

Differences between Decision-Making under uncertainty and risk

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	==Differences between Decision-Making under uncertainty and risk==		==Differences between Decision-Making under uncertainty and risk==

	While decision-making under risk implies a certain probability for each possible outcome, decision-making under uncertainty assumes that these probabilities are unknown, and thus, each scenario is considered equally important. This subsequently leads to a higher inaccuracy of uncertainty problems than risk problems. An example of that would be a problem with three possible outcomes: best case, most probable case and worst case. While for decision-making under risk both extreme scenarios would be considered with a presumable low probability and thus a low weight, for decision-making under uncertainty, all of the outcomes would be regarded in the same way, which means that case outliers have a considerably higher impact on the decision. <ref name="Kerzner2017"/>		While decision-making under risk implies a certain probability for each possible outcome, decision-making under uncertainty assumes that these probabilities are unknown, and thus, each scenario is considered equally important. This subsequently leads to a higher inaccuracy of uncertainty problems than risk problems. An example of that would be a problem with three possible outcomes: best case, most probable case and worst case. While for decision-making under risk, both extreme scenarios would be considered with a presumable low probability and thus a low weight, for decision-making under uncertainty, all of the outcomes would be regarded in the same way, which means that case outliers have a considerably higher impact on the decision. <ref name="Kerzner2017"/>

	==Decision-Making Approaches==		==Decision-Making Approaches==