The Numerical Methods for Real Options

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According to Mun(2006), Real Options(RO) is a " systematic and comprehensive method" used to value real tangible assets. The term "Real Options," first used by Myers(1977), refers to the application of financial options theory to investment decisions made by firms (Krychowski and Que´lin, 2010). RO has been of growing interest to the academic community as a promising approach to supporting investment decisions under uncertainty. Pioneering scholars such as Trigeorgis(1996) and Copeland(2001), have contributed valuable work to topics on real options such as the RO value in resource allocation and capital budgeting. Generally speaking, there are three main methods which are used as the tools to value the embedded RO. They are the Black-Scholes Model, the Binomial Model and Monte Carlo Simulation. Each of the method requires certain assumptions and can be best applied under specific situations. On the empirical side, RO analysis has been applied widely in a range of industries such as pharmaceutical drug development, oil and gas exploration and production, and the like. Survey results of 4,000 CFOs published in 2001 by Graham and Harvey revealed that 27% of the respondents claimed they "always or almost always" used some kind of options approach to evaluating and deciding upon growth opportunities (Copeland & Tufano, 2004). Compared with the traditional discounted cash flow methods which assumed that the future cash flows can be discounted by a single fixed rate, RO analysis enjoys the merit of being highly flexible because it incorporates the managers' ability to actively respond to the unfolded uncertainties. It is noted by Hall (2005) that around 30 percent of the values of high-growth, high volatility firms come from the value of embedded options. Primarily motivated by the usefulness of RO, after doing a general research on its background, I did further reading on approaches employed to value the embedded options. In this paper, my work can be divided into three parts. In the first section, a background on the real option analysis is presented. This includes an overview of typical categories of RO, the classical methods employed to value the RO and also the industrial practices of applying the RO Analysis. Additionally, a brief comparison between the traditional methods and RO is also presented. In the second section, I demonstrate the three methods in detail with elaboration on their assumptions and steps of analysis, as well as examples of application. In the final section, I conclude with a discussion of these numerical methods, including their merits and limitations as well as responses to some of the critics that RO analysis has incurred. I hope that this paper could serve as a motivator for further research.

Literature Review

Managers nowadays are facing a rather volatile environment because of the mixed effects of globalization, deregulation and technology break through (Krychowski and Que´lin, 2010). RO helps them to make use the advantages of uncertainty and their flexibility. It has been crucial in the way that it helps the firm to identify, understand, value, prioritize, select, time, optimize and manage strategic investment and capital budgeting decisions (Mun, 2006). Similar to the financial option models, RO is useful both to evaluate an investment project and to determine the optimal investment timing (Krychowski and Que´lin, 2010). The most common forms of RO, based on the division given by Copeland and Antikarov(2001) and Mun (2006), are : option to abandon, option to expand, option to switch, option to defer and sequential compound options. In detail, for example, an option to expand enables the management to expand into different markets, products, and strategies or to expand its current operations under the right conditions (Mun, 2006, p19). Multiple methodologies and approaches are used in RO to calculate the embedded option's value. These range from using closed-form equations like the Black-Scholes model and its modifications, Binomial Models (for example, binomial lattices and binomial trees) and Monte Carlo simulations and other numerical techniques. Since this will be the main part of this paper, it will be illustrated in detail in the next section. Primarily used as a tool for strategic decision making in natural resource companies, in the recent decade, RO has been applied in a broader ranges of industries, including private equity, oil and gas exploration and production, manufacturing, IT infrastructure, pharmaceutical drug development, e-commerce and e-business, technology development, and the like ( Mun,2005). The following are some of the industry examples of applying RO. Equity According to Berger, Ofek and Swary (1996), a considerable proportion of equity value should be attributed to the equity holder's abandonment option. They prove a difference of 11.5 percent between equity market value and the present value of cash flows for more than seven thousands firm over a 6-year period horizon. By running a series of regressions the authors find an interaction between the market value to present value premium and variables which should be linked to higher values for the abandonment option. Natural resource While evaluating the investment projects for natural resource, Brennan and Schwartz (1985) isolate the disadvantages and inadequacies of the traditional DCF approach. Particularly, they point out that obvious deficiencies are due to the overlook of the stochastic nature of output prices and of potential managerial reactions to price changes. Price uncertainty is of central concern in many natural resource industries where price fluctuates around 30 percent per year are usual. Under such circumstances the evaluation results obtained through substituting the expected values of future prices by their distributions is likely to be misleading. Oil By extending the financial option theory, Paddock, Siegel and Smith (1988) develop a new approach to valuating options on a real asset, an offshore oil lease. The authors show us how to utilize an explicit model of equilibrium in the market for the underlying real asset, i.e. the developed petroleum reserves, with option-pricing technique to derive the value of a real option. At the same time, they examine a valuation problem in sufficient detail by using the oil leases as an example. This allows close reviewing of the many theoretical and practical issues involved in applying financial option valuation theory to RO. Gold In 1998, Kelly adopts an eight period binomial option approach to estimate the value of a discovered but yet unexploited gold mine, Lihir Gold Limited. In particular, she compares the value calculated from the option model to what was obtained from the traditional method. The option approach appears to provide a more useful and accurate technique to evaluate the value of the gold mine . In 2002, Moel and Tufano conduct a research on the opening and closing events of 285 gold mines in North America from the year of 1988 to 1997. Strong evidence is found to support the conclusion that, compared with other methods, real (switching) options provide better explanations for the decisions on openings and closings of the gold mines. Manufacturing Newbhard, Shi and Park (2000) use a case-study to stimulate the academic and practical research needed to support a real option framework for system changes in four major manufacturing transitions which are launch of new product, commercialization of R&D product, site selection of new plant and restarting production of existing commodities. By presenting a framework, they quantify manufacturing changes, develop a real option model for these activities, value the options, identify the best scenarios and integrate these elements in order to monitor and manage the overall process. They also propose a general model for optimizing real option valuation based on typical RO models such as the Black-Scholes Option Model, the Binomial Option Model. They conclude that a model that incorporates flexibility and economic factors could effectively enhance companies' manufacturing strategy. Unlike the traditional valuation approaches, such as the discounted cash flow (DCF) method which bases itself on a static environment, real option analysis takes into account the potential for possible future gains and incorporates active decision making. Thus it tackles uncertainty in a better way. Specifically, deterministic models such as DCF method bases itself on some rather flawed assumptions. It assumes that all the future outcomes are fixed and can be evaluated as individual cash flows. Even more unreasonable, it seems to give a "Once for All" solution which assumes once initiated, all projects are passively managed. RO, on the contrary, accepts the facts that projects are correlated and can be actively managed through its life path. By taking the fluid environment and managerial flexibility into account, RO provides value-added insights to decision making (Mun, 2006).

Numerical Methods

As I have mentioned in the previous section, multiple approaches have been employed by researchers and practitioners in RO. This part will introduce the readers to three common types of methods in RO, namely, the Binomial methods, the Black-Scholes Model and Monte Carlo Simulation, from the origins of them to present application examples. More specifically, a step-by-step binomial approach is used to analyzing a compound option problem in the case study with two different techniques, so as to offer a deeper understanding for the readers. As for the Black-Scholes and Monte Carlo Simulation, due to the word limit and time constraint, I used two simplified examples from Newbhard et al (2000) and Damodaran( 2005) in the hope that the readers could have a general sense of how the two methods work.

Binomial approach

Work by John Cox, Steve Ross, and Mark Rubinstein has led to the creation of binomial, or lattice, models that are built around decision trees and are ideally suited to real-option valuation. As it noted by Copeland and Tufano (2004), "RO don't have to be a black box." Binomial methods, with its advantage of easy math and apparent illustration has make Real option analysis a more practical tool for manager in the new era. According to Brandao et al (2005), a binomial lattice may be viewed as a probability tree with binary chance branches, with the unique feature that the outcome resulting from moving up(u) and then down (d) in value is the same as the outcome from moving down and then up. This probability tree, also referred as decision tree, can be used in modeling managerial flexibility by incorporating the decision nodes which represent decisions the managers can make to optimize the value of the project. A three-step binomial tree is illustrated below in figure1. Before entering details about how to use the binomial method, it is worthwhile to make certain clarifications on the assumptions behind this approach. In their book RO, Copeland and Antikarov (2001) made the marketed asset disclaimer assumption (henceforth MAD) that the market value of a project is best estimated by the present value of the project without options. Additionally, if the movements in the value of the project without options are then assumed to change over time according to a geometric Brownian motion (GBM), then the value of options can be obtained through traditional option pricing methods. Generally, there are three essential steps that need to be gone through when a binomial approach is adopted. Step1 Calculating the expected present value of the project at Time0 Step2 Obtaining estimates of the standard deviation of returns (or volatility of the project) by using a Monte Carlo simulation. Step3 Constructing a binomial tree to model the dynamics of the project value using the estimated parameters of the second step and add the decision nodes to model the project's RO. No matter what real option model is of interest, the basic structure almost always exists, taking the form: Inputs: S, X,, T, rf, b u= and d== P= Source: Mun, 2006 The basic inputs are the present value of the underlying asset(S), present value of implementation cost of the option( X), volatility of the natural logarithm of the underlying free cash flow returns in percent(,time to expiration in years(T), risk-free rate or the rate of return on a riskless asset(rf), and continuous dividend outflows in percent(b). In addition, the binomial lattice approach requires two additional sets of calculations, the up and down factors ( u and d). The up factor is simply the exponential function of the cash flow returns volatility multiplied by the square root of time-steps or stepping time (.The volatility measure is an annualized value; multiplying it by the square root of time steps breaks it down into the time step's equivalent volatility. The down factor is simply the reciprocal of the up factor. In addition the higher the volatility measure, the higher the up and down factors. This reciprocal magnitude ensures that the lattices are recombining signs. The second required calculation is that of the risk-neutral probability, defined simply as the ratio of the exponential function of the difference between risk-free rate and dividend, multiplied by the stepping time less the down factor, to the difference between the up and down factors. In order to give the readers a more clear understanding on this, below is a case study of a sequential compound option problem, in which the execution and value of future strategic options depend on previous options. It represents a simplification of the business decision-making case and my purpose is to illustrate how a RO valuation is implemented using binomial approaches.

Case Study

A chemical company is considering a phased investment in a plant. There are three periods. In the beginning of year one, an initial outlay of $50 million is required to cover the cost of permits and preparation. At the end of that year, the firm has the choice to pay a commitment of $200 million to enter into the design phase. Once the design is finished one year later, the firm is believed to have a two year window during which to make the final investment in constructing the plant for $400 million. If the firm chooses not to make any investments during these two years, it can no longer to build the plant. For managers who think from the real-options perspective, this phased investment opportunity is a sequential compound option. Clearly, the initial payment of $50 million allows the firm to have the option to go on with the project for one year. At the end of year one, it again faces the choice of whether or not enter the stage of design by investing an additional $200 million. As the result, the execution of the design phase gives the firm the option to construct the plant at the end of year three or at the end of year four for $400 million. The firm estimates that if the plant existed today it would be worth $550 million by using non-option valuation techniques such as the DCF. In applying Binomial method, basically there are two techniques. The one is the decision tree approach the other is the replicating portfolio technique. I will use both of them to analyze the above case and give some comments on these two techniques.

Decision Tree Analysis

Prior to analyzing this problem, we must make some assumptions concerning the uncertainty in the future value of the project. We assume that current prices of the project have already incorporated all relevant information available at this point, known as part of the efficient market theory. At the same time, future changes are modeled as a random walk. This facilitates the use of a Geometric Brownian Motion (GBM) to model the dynamic uncertainty associated with prices movements (Hull, 2003). The key parameters required here are the estimated initial value, $550 million, the annual risk-free interest rate r, assumed to be 6% and the volatility, represented by, which is the annualized percentage standard deviation of the returns and is given as 18.23% here. By using the calculating structure given previously, we can calculate the corresponding values of u and d, and values for each branch of the binomial approximation. We then obtain the risk-neutral probability p ==. In this case study, the model have three periods and choose time interval to be t = 1. Therefore, u = 1.2, d =0.83, and p =0.673. For details about this binomial approximation, see Hull (2003). The value of the project is calculated via Vi,j =V0ui-jdj.For example in the right top scenario, the value of the project is $798 million which equates $550 million multiplied by 1.23. (Note: Values shown at each node in the tree are discounted Year 3 values, instead of the actual values at each point.) After approximate the project value according to the GBM, now we are going to value the Value of the Option to invest in this project. Here we used the same parameters in a decision tree with binary chance nodes to yield an equivalent binomial tree for the project value, as shown in Figure 3 below. Decision tree analysis works in the way that it models managerial flexibility in discrete time by constructing a tree with decision nodes. These nodes represent choices the manager can make to optimize the value of the project as uncertainties are resolved over the project's life. Note: represents a chance node in which the project can either move up or down with the probabilities of up=0.673, down=0.327 represents a decision node in which the firm can chose to invest or not denotes the termination of one possible case the line in bold shows the optimal investment strategy in different cases Let's suppose that at the end of year3, we arrive at the best scenario in which the project value is $798 million (See Figure2). If we choose to invest the extra $400 million, we will have an income of $223 million. Otherwise we will lose what we have paid for the preparation and design phase, say, $239 million. Rational managers will of course choose to invest further. The same calculation applies in the scenario with the increases in first two years and a decline in year three. By multiplying the values obtained from the decision nodes with their up and down probabilities, we arrive at the option value in year2. Using this rollback method, finally we obtain the value of option at year0, which is $31million.

Replicating portfolio technique

Using the binomial model which adopts replicating portfolio technique also requires two main steps. First, we need to figure out the full range of possible values for the underlying asset, in other words, draw the event tree, as shown in figure4. Figure4 (It has to be noted that, unlike the numbers for the binomial tree which have been discounted to present value, the numbers I used here are the value in that specific period.) Secondly, our task is to calculate the possible values of the project as an option at each stage. It is a backward working process and we have to begin from the end. If we abandon the project, its value is zero. Otherwise, the value at the end of that year, year three, for example, is the difference between the value of the plant at the end of year three and the expense of building it. As you can see from the figure5, we have got three potential scenarios in which the project's incremental value at the end of year three is positive and one in which the costs of the project exceed the plant's value, so the project value is zero. We now work back from the end of year three to determine the project's potential values at the end of year two. The decision rule is that in each scenario, the value will be the larger of the value of exercising the option by building the plant at that point for a cost of $550 million and the value of keeping the option window open-deferring the decision until the next period. The steps can be summarized in the followings and Figure5 serves as an illustration of the results. Step1: Calculate the potential final project values by subtracting the $400 million cost (from the event tree of Figure5). For the $314 million scenario at the bottom right of the event tree, the project's value is zero due to the cost is greater than the plant value. Step2: Obtain the potential end-of-year-two project values by comparing two calculation results. One is the value by exercising the project immediately, the other is the value if the project is kept alive by applying the replicating portfolio technique. Step3¼šSimilar to step2, yet the number used to be compared with the value derived from replicating portfolio technique is $200 million, since immediate exercise of the project is not possible. Step4: Calculate the starting project value of $81million. Since the initial required investment is $50 million, the project is profitable. The option value is the same as what is derived by the decision tree method, which is $31 million. Figure5 Some Comments: As we can see from the above, the results obtained from Binomial Decision Tree and Replicating portfolios Techniques are largely similar. It is worthwhile to compare them briefly. The binomial approach is suggested by Copeland and Antikarov(2001), they emphasize the use of binomial lattices and replicating portfolios while Brandao et al( 2005) believe that the use of binomial trees is more "intuitive appealing". The replicating approach bases itself on traditional option pricing methods, requiring that markets be complete. An important advantage of this approach to valuation is that the value of option can be calculated from market data. This avoids the estimation of the probability q of an up move in the stock price. However, for most projects involving real assets, there is no such a complete market. In this sense, using this approach appears to be complicated. Additionally, it is criticized for its computational cumbersomeness especially in a multi-stage project.

Black & Scholes model

With their article from 1973, Fisher Black, Myron Scholes, and Robert Merton were the first to give a closed form solution for the equilibrium price for a European call option, the Black &Scholes Model (B&S model). It has since been the basis for numerous studies and papers about the pricing of options and empirical testing hereof. In essence, the model is a special case of the binomial model where the underlying asset is assumed to follow a continuous stochastic process instead of a discrete. Otherwise, it is based on the same underlying assumptions of no arbitrage and market replicating portfolio and that the movement of the underlying asset follows a lognormal distribution (Copeland & Antikarov, 2001) It has to be noted that variants of the B&S model have been made, which relaxes some of these assumptions. B&S models are based on calculus of stochastic differential equation which is highly complex. So unless one can find a modified B&S model that fits one own specific situation, the process of deriving a B&S model that does is very cumbersome and complex. The Black & Scholes model is a so called closed form solution, meaning that a value can be found with an equation using a set of inputs. The inputs in the B&S model are the same as the binomial model, with dividend as the one exception. The value of a call option( C) is calculated as: Source: Copeland & Antikarov, 2001,p.106 Where and is the cumulative normal probability of unit normal variable and respectively. They are calculated as:

;

Source: Copeland & Antikarov, 2001, p.106 Other than the assumptions also applying to the binomial model mentioned above, the B&S model has several other restrictive assumptions embedded (Copeland & Antikarov, 2001 and Mun, 2002) which are: The option can only be exercised at maturity-it is a European option There is only one source of uncertainty It can only be used on a single underlying risky asset; ruling out compound options No dividends on the underlying asset The current market price and stochastic process of the underlying asset I known (observable) The variance of the underlying asset is constant over time The exercise price is known and constant over time No transaction costs Benaroch and Kauffman (1999) provide a formal theoretical grounding for the soundness of the B&S option pricing model in capital budgeting methods that might be employed to assess Information Technology (IT) investments. They also demonstrate why the assumptions of both the B&S and the binomial option pricing models impose constraints on the spectrum of IT investment situations that can be evaluated similarly by traditional capital budgeting methods such as discounted cash flow analysis. Most importantly, they demonstrate the first application of the B&S model that uses a real-world IT business case as its experimental area. To illustrate the application of B&S more clearly, I borrow an example of Brennan and Schwartz (1985) which uses option pricing theory to value a gold mine and present it with a simplified version below.

Option to delay for a Gold Mine

Consider a gold mine with an estimated reserve of 1 million ounces and a capacity output rate of 50,000 ounces annually. The firm maintains the ownership of this mine for the next 20 years. We expect the gold price will grow at 3% per year. It will cost $100 million to open the mine and the average variable cost is $250 per ounce; once the mine opened, the variable cost is expected to grow 5% a year. The current price of gold is $375 per ounce and its standard deviation is estimated as 20%. The riskless rate is given as 6%. The inputs to the model are summarized as follows: S = $ 47.24 million X= $100 million = 0.04 t=20 years rf= 6% Dividend Yield = = 1 / 20 = 5% Based upon these inputs, the Black-Scholes model provides the following value for the call: The value of the mine as an option is $ 3.19 million which is recognized as the mine's embedded option.

Monte Carlo Simulation :

Because of the difficulty in obtaining the needed parameters for analytical models such as the Black-Scholes model, researchers find an alternative way to value RO by using an approximate numerical method such as Monte Carlo simulation. Monte Carlo simulation, named for the famous gambling capital of Monaco, is a very powerful methodology (Mun, 2006). Monte Carlo, in its simplest form, is a random number generator that is useful for forecasting, estimation, and risk analysis. A simulation calculates numerous scenarios of a model by repeatedly picking values form a user-predefined probability distribution, such as the normal, uniform and lognormal distributions, for the uncertain variables and using those values for the model (Mun,2006,p317-318). Boyle (1977) was among the first to propose using Monte Carlo simulation to study option valuation. What distinguishes this approach is its generality in being able to model "imperfect" market conditions which are difficult to be captured in other models. The Monte Carlo method proves to be most effective in situations where it is difficult to proceed using a more accurate approach (Boyle, 1977). Researchers share a common emphasis on the need for investigating practical issues related to efficiently approximating various option models via Monte Carlo simulation and including sensitivity analysis and Quasi-Monte Carlo simulation approaches (Boyle,1977; Fu and Hu,1995; Birge 1994; Newbhard, Shi and Park ,2000 ).

Example:

For a manufacturing company A, market research revealed a demand for a new product. The unit price of the new product is $100. The initial monthly demand for this product is 1,000 units with a standard deviation of σ = 0.33. The product will be introduced over a four month period (T = 4) with monthly fixed interest rate of1%. Suppose we let S equal X, i.e. $100 multiplied by 1,000 which gives $100,000. To imitate the path followed by the state Variable S, we divide the phase of the variable into four time intervals. If dt is the length of one interval, then the relation between the S values is offered by Source: Newbhard, Shi and Park , 2000 Executing 1,000 Monte Carlo runs of this equation obtains an option value of $8,203, which is quite similar compared to the value of $8,155 derived from the B&S model.

Discussion

For the practitioner, simulation opens the door for solving difficult and complex but practical problems with great ease. Monte Carlo creates artificial futures by generating thousands and even millions of sample paths of outcomes and looks at their prevalent characteristics (Mun, 2006). When modeled correctly, Monte Carlo simulation provides similar answers to the more mathematically elegant methods. Closed-form solutions can be obtained from models such as the Black-Scholes, given that a set of input parameters are estimated. B&S model is exact, quick, and easy to implement with the assistance of some basic programming knowledge but is difficult to explain because applying it requires high skills of technical stochastic calculus mathematics. They are also very specific in nature, with limited modeling flexibility. Closed-form solutions are mathematically elegant but very difficult to derive and are highly specific in nature. The use of algebra distinguishes binomial models and enables the models to be built using standard spreadsheet software such as EXCEL. Binomial models can also be easily customized to reflect changing volatility, early decision points, as well as multiple decisions (Copeland, 2004). Another practical advantage is that because the transparency of the model, it could be understood and used by managers without very strong mathematical background. Binomial lattices, compared with close-form solutions, are easy to implement and easy to explain. They are also highly flexible but require significant computing power and lattice steps to obtain good approximation. The process of working through lattices can be cumbersome and less intuitive, which is particularly true for more complex cases to real assets with several coinstantaneous and compound options. It is important to note, however, that in the limit, results obtained through the use of binomial lattices tend to approach those derived from closed-form solutions. The choice of models in practice is largely depending on the underlying assets' nature of changes. A possible process for deciding upon which model to use is presented below. Identify and Define RO (1) Quantify Activities Related to Changes Related to Changes (2, 3) Choose Solution Method

Binomial Model

Underlying change

is a binomial (discrete) process

Black-Scholes Model

Underlying change

is a lognormal

(continuous) process

Monte Carlo Simulation

When parameter

Estimation is needed

REAL OPTION ANALYSIS

Source: Newbhard H. B., Shi. L, Park .C (2000) Although managers today are facing a more volatile environment, most of them still rest their decisions on deterministic methods such as the discounted cash flow method, which is static in nature (Krychowski and Que´lin, 2010). RO have problems in the implementation sector and empirical evidence shows that it is little used in practice. Whereas about 75% to 85% of firms use NPV for their investment decisions, only about 6% to 27% of them use RO analysis. Empirical studies on the implementation of RO are still rare, and research remains relatively silent on how to concretely apply RO theory (Krychowski and Que´lin, 2010). Main Critics of options-based methods to valuing and managing growth opportunities often have two arguments. One is the huge difference between the highly complex real options and relatively simple financial options. The other is that the real option analysis relies itself on some simplified assumptions. However, just as Copeland (2004) noted, differences do exist but are not unsolvable. Let's take the availability of information as an example. Financial options can be valued based on their underlying assets price such as the stock price. For real options, in some cases the values of their underlying assets can be observed in the same way. A coal company could estimate the value of its reserve coal by checking the expected exploitation costs and the current price of coal. It is true that at most circumstances we don't know the exact value of the underlying asset. Yet by observing comparable assets or making educated guess based on certain assumptions, we can do the estimation. In the end, option models are not the only method that requires assumptions. The main alternative to real-option analysis-discounted cash flow method-does make simplified assumptions such as the pre-committed future cash flows.

Conclusion

Real Options, as it is remarked by Hall(2005), is one of the most crucial important decision-making tools introduced to managers in the last three or four decades. By utilizing financial theory, economic analysis, management science and econometric modeling, real option analysis can be very useful for investment decision making in the context of uncertain environment. Managers are allowed to make flexible midcourse corrections when adopting RO (Mun, 2006). I began paper by familiarizing the readers with the essential background of RO which includes the classifications of RO and methodologies adopted to value the embedded options. The merits of RO have been discussed via a comparison with the traditional DCF method. Meanwhile, they have also been introduced to several industry applications, ranging from the equity market to manufacturing industry. Then it comes the main part of this thesis-demonstration of the three RO methods, which are the Binomial Approach, the Black and Scholes Model and Monte Carlo Simulation. A case study has been worked out via using the two techniques of the Binomial approach with the details shown on the graphs. Two simplified examples of applications of the other two methods are also included to offer the readers a general sense. Furthermore, in the discussion section, I have compared the three methods with respect to their merits and limitations. A general process of model selection is presented in a flow chart as an illustration. As for the further research, it is suggested that there is a need to find a general approach which integrates at least two models discussed in this paper. At the same time, it might also be an interesting subject to compare the usefulness of RO in different industry settings.
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