Payoffs, alternatives, and expected monetary values are terms associated with:

9.3.7 Decision analysis and decision trees

Decision making has a binary nature—we go for it or not. The primary objective of decision analysis is to identify the decision alternative that maximizes expected utility or expected monetary value with probability of occurrence as the outcome consequence weight factors. As mentioned in the introduction to this section, decision trees are a means to structure decision making taking account of the various aspects or components and motivate gathering the needed information. In computer science much use is made of binary decision trees. Binary refers to a Boolean basis: an aspect must be reckoned with or not, it is true or false, the value is one or zero [as in a truth table]. In fact, it is embodying the “if-then-else” rule. Binary decision trees are very useful in development of digital systems. The tree that branches up to the final decision is a directed acyclic graph.

Here, we are more interested in decision trees that include uncertainty as required for a system approach. Ian Jordaan,27 Memorial University of Newfoundland, described the field. Basic is the distinction of a [binary] choice node, which Jordaan calls a decision fork, followed at each branch by a probability node, or chance fork. In Figure 9.6, a simple example using point values is given of risk-based decision making use of the module PrecisionTree of [email protected] MS Excel-based decision analysis software [easily found on the Internet]. In a process under normal condition a light protective measure is adequate, but one needs a heavy protective measure if a coincidental process condition materializes. From the choice of protection four end states arise, called consequences or utilities because they represent values to the decision maker. Two of these end states can be classified as adequate protection, the other two as under- and over-protection. Hence, basically, the choice depends on how the decision maker perceives the chance or probability that the condition will occur. Here, an increase in occurrence probability of 0.08 to a value of 0.1 will change the preference from light to heavy protection.

An additional possibility is collecting more information about circumstances influencing the emergence of the coincidence. A value of information calculation can be cost-effective to lower uncertainty and reduce the risk of decision under uncertainty. Developing the knowledge by, for example, testing, requires funding, but the value of this information must be balanced against the gain in knowledge and reduction in uncertainty. The larger the uncertainty, the higher the value of the information and at lower cost to obtain than when the uncertainty is at lower levels. This calculation will form a pre-decision node. In case testing would be very costly or not possible, improved information could be gained through estimation by employing so-called pre-posterior analysis. This is by simulating a posterior distribution by taking the prior and estimate the conditional probabilities of what you would observe in a test [this is a kind of contingency analysis]. Another value of information is the value of control to reduce uncertainty of outcomes. Prior to finalizing a decision, both approaches can be simply added. The bad state probability may be strongly reduced. For this simple case, overview can be kept easily, but evaluating for example a complete bow-tie will be different. Apart from enabling to cope with complexity, the result of the calculation with the software offers clarity in team communication and later review.

By combining PrecisionTree with @Risk, or by calculation in Bayesian nets, uncertainties in the probabilities of occurrence and of consequences can be included. The Precision Tree software allows the tree to be converted to the appearance of an influence diagram. The GeNIe Bayesian Net can do this too, while it is at the same time more versatile and able to calculate a result using distributions of all relevant information.

7.7.3 Decision Trees and Utility Theory

Decision trees are a useful way to describe alternative scenarios and select the decision that maximizes the NPV or whatever the decision maker is trying to optimize. In a subsequent section, we will see that “utils” can express the relative desirability of various outcomes. In decision theory, the most desired outcome is based on the goals and preferences of the decision maker. The reservoir engineer can use decision trees to describe complex scenarios with multiple decisions and multiple probabilities. This discussion can be considered only a brief introduction. In constructing a decision tree, we use rectangles to represent decision nodes and circles to represent probability nodes. Two or more decisions can be associated with each decision node, and multiple nodes can be associated with a probability node. A probability node representing betting $1000 on number “30” at a roulette wheel11 in Las Vegas is shown in Figure 7.11

The single bet on number 30 can easily be evaluated as to its expected value as follows:

EV=−1000$+[138]×$36,000+[3738]×0=−$52.63

In other words, a single bet of $1000 on number 30 [or any other number] has a negative expected value of $52.63. Similar analyses will show negative expectations for each of the gambling games explaining the fabulous hotels and inexpensive “all you can eat” buffets in Las Vegas. But is it crazy to play roulette or make other decisions selecting lower expected values than other alternatives? No, the decider may have a different use for $35,000 than $1000. Maybe he owes a debt that is immediately due and has a major negative result if he is unable to generate $35,000 right away. This particular preference for risk is actually unusual; most people have less utility for expected outcomes that have large negative impacts. This analysis does not mean that every player will lose money playing roulette. It is a relatively straightforward exercise to model a roulette wheel with various strategies in which a significant fraction of the players win.12 It is the aggregate EMV of all players over the long run that is negative.

Suppose someone gives you the chance to play a game in which a fair coin is flipped. In the case of heads,13 you receive $2 and for tails you get nothing. You will no doubt be happy to play this game as it has an expected monetary value [EMV] of $1. How much would you be willing to sell your ticket for? It is unlikely anyone will pay you much more than $1, and if you sell it for much less you are “giving away” EMV. Now consider another game. In this game you have to buy a ticket. In this game a heads pays $3 and a tail pays $1. How much would you be willing to pay for this ticket? The EMV of this game is $2, and if you pay any less than that you are [on an expected value basis] gaining money. Would you pay more than $1? If you paid $1, the second game becomes equivalent to the first with the net result of a head being $3−1=$2 and the result of a tail would be $1−1=$0. Is there a difference in how much you are willing to sell your ticket for in the first game and what you are willing to pay for it in the second game? Decision makers often make decisions on other than an expected value basis based on how much investment exposure is necessary.

Let us consider another set of decisions. In the first option, you pay €1000 by investing in a very small percentage [0.1%] of a drilling well that you anticipate has a 50% chance of success [or a coin flip for heads if you prefer]. In the case of a discovery you win a series of cash flows with an NPV of €4000, while a dry hole pays nothing. The EMV is 0.5×€4000−1000=1000. Are you interested in this investment? If you believe these numbers and have €1000 to invest, it is an obvious decision to participate in the project. Now let us look at the 100% working interest position. In this case, you need to invest €1,000,000 and have a 50% chance of €4,000,000. Assume that your net worth is just enough that you could come up with the money by mortgaging your house, cashing in your retirement, and borrowing all of the money that you can; it is unlikely that you would accept such an investment opportunity. A single investment or a series of investments that has the potential to bankrupt an investor is known as “gambler’s ruin.” Your utility for a positive €1,000,000 is considerably less than 1000 times greater than it is for €1000. By analyzing your responses to a series of similarly constructed alternatives, an individual with game theory expertise could construct your “indifference curve.” Your personal utility and indifference curves and those of the decision maker are not as important as are the utility functions of the corporation. For our purposes, we will assume that the corporation has a unit slope linear utility function and makes its decisions entirely on EMV. Exceptions to this would only occur for massive investments.

In the drill vs. farmout example, we had a decision tree, see Fig. 7.12.

There were only two decisions: drill and farmout. The probability nodes were only dry hole or discovery. The analysis of a decision tree proceeds from right to left as the EMV is calculated for each probability node. The expected value of each probability node is replaced with its expected value, and the highest EMV decision node is selected. There can be multiple probabilities at each probability, and the probability node can be replaced by Monte Carlo simulations. In fact, the entire decision tree can be replaced by Monte Carlo simulations with a distribution of decisions being made and the corresponding variability in results conveyed to decision makers.

2.3.3 Decision Under Risk

In this section, we revisit the decision under risk, in which the probabilities that affect the outcomes usually are assumed to be known. We introduce more rigorous treatment to the decision-making model, in which we revisit the car-buying example for illustration.

To facilitate our discussion, we assign N and U to the options of new and used car, respectively; and W and F to the events of car works well without major repairs and car fails due to major problems, respectively. The buyer also consulted with the dealership, and the historical data suggest that there is an 80% chance that a new car does not encounter major problems in 10 years. The historical data also suggest that the probability is 50% for a used car. At the time, this was the best possible estimate that the buyer was able to attain. Next, we construct a decision tree to aid the buyer in making a decision. The payoff table of Table 2.4 is expanded by incorporating the probabilities and expected values. In Table 2.6, the notation P[W|N] stands for the probability of new car [N] that works [W] well in 10 years. Similarly, P[F|U] stands for the probability of used car [U] that fails [F] in 10 years [i.e., encountering major problems], and so on.

Table 2.6. Cost of Different Options and Events [Payoff Table]

States of Nature [Events]Courses of ActionNew Car [N]Used Car [U]ProbabilityPayoffExpected ValueProbabilityPayoffExpected ValueWithout major problems [W]P[W|N] = 0.8$25,000$20,000P[W|U] = 0.5$15,000$7500With major problems [F]P[F|N] = 0.2$35,000$7000P[F|U] = 0.5$30,000$15,000$27,000$22,500

Assuming these historical data are reliable, the payoff table and the probability estimates can be combined to arrive at the expected payoff of individual decisions. The expected payoff is also called the expected monetary value [EMV] in decision theory. The calculations are summarized as follows:

[2.3]E[ai]=∑jP[ϕj]v[ai,ϕj]

where E[ai] is the EMV of event ai.

Hence, for the event of buying a new car, the expected payoff is

E[N]=∑jP[ϕj]v[N,ϕj]=P[W|N]v[N,W]+P[F|N]v[N,F]=0.8[$25,000]+0.2[$35,000]=$27,000

Similarly, for the event of buying a used car, the expected payoff is E[U] = $22,500, as shown in Table 2.6. Therefore, based on the expected payoff, buying a used car presents a better option.

The car-buying example can be represented in a decision tree similar to that of Section 2.2.2. The decision tree for the car-buying example is shown in Figure 2.4.

As discussed in Section 2.2.2, the general approach to solving a decision tree is to move backward through the tree [from right to left] until we reach the originating decision node. We select a payoff node and move left to trace the branch to encounter the next node. If the next node is a chance node, we calculate the expected value [E] of all nodes connected immediately to the right of the encountered node by using Eq. 2.3. In this example, we calculate the expected values for the options of buying a new car E[N] and buying a used car E[U], respectively.

We enter these values to their respective chance nodes, and then move left to encounter a decision node—in this case, the originating decision node, as shown in Figure 2.4b. At the decision node, we select the branch that leads to the best value. In this case, $22,500 is the best value, representing a lesser overall cost. At this point, we reach a decision of buying a used car.

One of the difficulties in using the decision tree method is coming up with the probabilities of the uncertain event occurring. In the car-buying example, how certain is the event that the probability of a used car without major problem is P[W|U] = 50%? In this case, we do not have to agonize too much over the accuracy of the estimate because we can easily test to see if the decision to buy a used car is sensitive to the probability estimated. We first let the probability be P[W|U] = x. Then the probability of a used car having major problems is P[F|U] = 1 − x. The expected value of buying a used car is

E[U]=x[$15k]+[1−x][$30k]=$30k−$15kx

If we equate the two expected values of used and new car E[U] = E[N] = $27k, we have x = 0.2. In other words, as long as the probability of a used car without major problems is greater than 20%, buying a used car is still a good choice.

25.2 CCS-related acronyms

7.2 Objective functions

Based on the type of optimization [i.e., nominal vs. robust], the objective function can accordingly be either deterministic or probabilistic. While deterministic objective functions are evaluated for a single model only, probabilistic objective functions are evaluated across multiple models. Deterministic objective functions can be financial or production/injection indicators such as NPV, revenue, recovery factor, cumulative fluid production [oil, gas or water], cumulative fluid injection [water, gas, solvent or CO2], displacement efficiency at water breakthrough [Sudaryanto and Yortsos, 2011], sweep efficiency through equalizing the arrival times of water/gas front at all producers [Alhuthali et al., 2007, 2008, 2009; Elfeel et al., 2018], and water breakthrough time [Bagherinezhad et al., 2017]. Probabilistic objective functions are usually described as expected indicators such as EMV, expected recovery factor, expected cumulative fluid production [oil, gas or water], and expected cumulative fluid injection [water, gas, solvent or CO2]; sometimes combined with a risk measure.

Those previously described closed-loop studies that have used nominal optimization, RM nominal optimization or ensemble nominal optimization, represent the cases of using deterministic objective functions [with discounted or undiscounted NPV as a widely used objective function]. Furthermore, those working with RM robust optimization or robust optimization have used probabilistic objective functions [with expected discounted or undiscounted NPV as a widely used objective function]. Alhuthali et al. [2009] and TAMU in Peters et al. [2010] used expected arrival time of water fronts as the objective function in their water-flooding RM robust optimization problems.

Furthermore, regardless of deterministic or probabilistic optimization, the optimization problem could be either single or multi-objective, depending on the number of objective functions considered. Single-objective optimization aims to maximize or minimize a certain objective function [measures of production, injection, economic, risk, etc]. Another form of single-objective optimization also exists in which optimization is performed by lumping several objective functions into a single general balanced objective function, each objective function having its own weight [Marler and Arora, 2004]. Nevertheless, the difficulty is finding the suitable weighting factor corresponding to each objective function. As the weighting factors strongly govern the characteristics of the optimal solution, a vast number of trial and error runs with different weighting factors may be required to obtain a satisfactory solution [van Essen et al., 2011].

The practical optimization problems should normally consider multiple, possibly competitive and conflicting, objectives [Yasari et al., 2013; Moradi and Rasaei, 2017]. The multi-objective [or multi-criterion] optimization overcomes the difficulty of the single-objective optimization to address objectives with differing data types, to accommodate multiple objectives, and to handle the possible conflicts between objectives [Isebor and Durlofsky, 2014; Hutahaean et al., 2019]. For instance, in a water flood project, one may be interested in maximizing oil recovery while minimizing water injection, or maximizing produced oil while minimizing produced water.

In simultaneous multi-objective optimization, several objective functions are optimized simultaneously. Usually, the final optimal solution set [Pareto front] provides different solutions for decision-makers to select the production strategy by trade-off between objectives [Bagherinezhad et al., 2017; Hutahaean et al., 2019]. Yasari et al. [2013] performed a multi-objective robust optimization to optimize the different components of NPV under economical and geological uncertainty with the aim of omitting the relevancy of the optimization problem to the prices. Liu and Reynolds [2015, 2016], Isebor and Durlofsky [2014], Yasari and Pisvaie [2015] and Hutahaean et al. [2019] documented cases where the optimization objectives were to maximize the expected NPV while minimizing its associated uncertainty [standard deviation] over a set of models. Liu and Reynolds [2016] studied an optimization case where the objective was to maximize life-cycle NPV and to maximize the short-term NPV of production. In a work by Bagherinezhad et al. [2017], a procedure was applied for reservoir development optimization subject to maximization of the cumulative oil production and minimization of water front velocity [or respectively maximization of water breakthrough time]. Hasan et al. [2013] documented a case where short-term and life-cycle objective functions were optimized simultaneously.

Although production optimization studies normally focus on a life-cycle window, in practice short-term objectives usually dictate the course of the production strategy, especially in view of geological and economic uncertainties [van Essen et al., 2011; Chen et al., 2012]. Therefore, short-term objectives should also be incorporated into the life-cycle optimization problem [Pinto et al., 2015]. Following Jansen et al. [2009], who showed that a life-cycle performance could be optimized while maintaining freedom to perform short-term production optimization, van Essen et al. [2011], Chen et al. [2012] and Fonseca et al. [2014] utilized hierarchical optimization processes where maximization of the life-cycle NPV served as the primary objective and maximization of the short-term operational performance was the secondary objective [short-term in the context of reservoir engineering, in contrast to production engineering]. In their approach, optimality of the primary objective function constrains the secondary optimization problem. In other words, optimization of the second objection function is constrained by the requirement that the primary objective function must remain close to its optimal value.

To the best of our knowledge, all of the previous closed-loop studies have been based on optimization of a single-objective function.

Which document is a list of the components their description and the quantity of each required to make one unit of a product?

Which of the following is true concerning advantages of CAD?