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The decision situations with flat uncertainty have the largest risk. For simplicity, consider a case where there are only two outcomes, with one having a probability of p. Thus, the variation in the states of nature is p 1-p. In such a case, the quality of information is at its lowest level. Remember from your Statistics course that the quality of information and variation are inversely related. That is, larger variation in data implies lower quality data i.

Relevant information and knowledge used to solve a decision problem sharpens our flat probability. Useful information moves the location of a problem from the pure uncertain "pole" towards the deterministic "pole". Probability assessment is nothing more than the quantification of uncertainty. In other words, quantification of uncertainty allows for the communication of uncertainty between persons. There can be uncertainties regarding events, states of the world, beliefs, and so on. Probability is the tool for both communicating uncertainty and managing it taming chance.

There are different types of decision models that help to analyze the different scenarios. Depending on the amount and degree of knowledge we have, the three most widely used types are: Decision-making under pure uncertainty Decision-making under risk Decision-making by buying information pushing the problem towards the deterministic "pole" In decision-making under pure uncertainty, the decision maker has absolutely no knowledge, not even about the likelihood of occurrence for any state of nature.

Some of these behaviors are optimistic, pessimistic, and least regret, among others. The most optimistic person I ever met was undoubtedly a young artist in Paris who, without a franc in his pocket, went into a swanky restaurant and ate dozens of oysters in hopes of finding a pearl to pay the bill. Optimist: The glass is half-full.

Pessimist: The glass is half-empty. Manager: The glass is twice as large as it needs to be. Or, as in the follwoing metaphor of a captain in a rough sea: The pessimist complains about the wind; the optimist expects it to change; the realist adjusts the sails. Optimists are right; so are the pessimists. It is up to you to choose which you will be.

The optimist sees opportunity in every problem; the pessimist sees problem in every opportunity. Both optimists and pessimists contribute to our society. The optimist invents the airplane and the pessimist the parachute. By doing so, the problem is then classified as decision making under risk. In such a case, the decision-maker may buy the expert's relevant knowledge in order to make a better decision. The procedure used to incorporate the expert's advice with the decision maker's probabilities assessment is known as the Bayesian approach.

For example, in an investment decision-making situation, one is faced with the following question: What will the state of the economy be next year? Then, a typical representation of our uncertainty could be depicted as follows: Further Readings: Howson C. Gheorghe A. Kouvelis P. Provides a comprehensive discussion of motivation for sources of uncertainty in decision process, and a good discussion on minmax regret and its advantages over other criteria. In such cases, the decision making depends merely on the decision-maker's personality type.

Worse case scenario. Bad things always happen to me. B 3 a Write min in each action row, S -2 b Choose max and do that action. Good things always happen to me. My decision should be made so that it is worth repeating. I should only do those things that I feel I could happily repeat. This reduces the chance that the outcome will make me feel regretful, or disappointed, or that it will be an unpleasant surprise. Regret is the payoff on what would have been the best decision in the circumstances minus the payoff for the actual decision in the circumstances.

Therefore, the first step is to setup the regret table: a Take the largest number in each states of nature column say, L. L - Xi,j. Limitations of Decision Making under Pure Uncertainty Decision analysis in general assumes that the decision-maker faces a decision problem where he or she must choose at least and at most one option from a set of options.

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In some cases this limitation can be overcome by formulating the decision making under uncertainty as a zero-sum two-person game. In decision making under pure uncertainty, the decision-maker has no knowledge regarding which state of nature is "most likely" to happen. He or she is probabilistically ignorant concerning the state of nature therefore he or she cannot be optimistic or pessimistic.

In such a case, the decision-maker invokes consideration of security. Notice that any technique used in decision making under pure uncertainties, is appropriate only for the private life decisions. Moreover, the public person i. Otherwise, the decision-maker is not capable of making a reasonable and defensible decision. You might try to use Decision Making Under Uncertainty JavaScript E-lab for checking your computation, performing numerical experimentation for a deeper understanding, and stability analysis of your decision by altering the problem's parameters.

Further Readings: Biswas T. Martin's Press, Driver M. Brousseau, and Ph. Eiser J. Flin R. Ghemawat P. Decision Making Under Risk Risk implies a degree of uncertainty and an inability to fully control the outcomes or consequences of such an action. Risk or the elimination of risk is an effort that managers employ. However, in some instances the elimination of one risk may increase some other risks. Effective handling of a risk requires its assessment and its subsequent impact on the decision process. The decision process allows the decision-maker to evaluate alternative strategies prior to making any decision.

The process is as follows: The problem is defined and all feasible alternatives are considered. The possible outcomes for each alternative are evaluated. Outcomes are discussed based on their monetary payoffs or net gain in reference to assets or time. Various uncertainties are quantified in terms of probabilities. The quality of the optimal strategy depends upon the quality of the judgments. The decision-maker should identify and examine the sensitivity of the optimal strategy with respect to the crucial factors. In such cases, the problem is classified as decision making under risk.

The decision-maker is able to assign probabilities based on the occurrence of the states of nature. Expected Payoff: The actual outcome will not equal the expected value. What you get is not what you expect, i. Value B 0.

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Expected Opportunity Loss EOL : a Setup a loss payoff matrix by taking largest number in each state of nature column say L , and subtract all numbers in that column from it, L - Xij, b For each action, multiply the probability and loss then add up for each action, c Choose the action with smallest EOL. Loss Payoff Matrix G 0. Since I don't know anything about the nature, every state of nature is equally likely to occur: a For each state of nature, use an equal probability i.

Payoff Bonds 0. One important factor is the emotion of regret. This occurs when a decision outcome is compared to the outcome that would have taken place had a different decision been made. This is in contrast to disappointment, which results from comparing one outcome to another as a result of the same decision. Accordingly, large contrasts with counterfactual results have a disproportionate influence on decision making. Regret results compare a decision outcome with what might have been. Therefore, it depends upon the feedback available to decision makers as to which outcome the alternative option would have yielded.

Altering the potential for regret by manipulating uncertainty resolution reveals that the decision-making behavior that appears to be risk averse can actually be attributed to regret aversion. There is some indication that regret may be related to the distinction between acts and omissions. Some studies have found that regret is more intense following an action, than an omission. For example, in one study, participants concluded that a decision maker who switched stock funds from one company to another and lost money, would feel more regret than another decision maker who decided against switching the stock funds but also lost money.

People usually assigned a higher value to an inferior outcome when it resulted from an act rather than from an omission. Presumably, this is as a way of counteracting the regret that could have resulted from the act. You might like to use Making Risky Decisions JavaScript E-lab for checking your computation, performing numerical experimentation for a deeper understanding, and stability analysis of your decision by altering the problem's parameters. Further Readings: Beroggi G.

George Ch. Rowe W. Krieger Pub. Suijs J. For example, consider the following decision problem a company is facing concerning the development of a new product: States of Nature High Sales Med. Sales Low Sales A 0. We will refer to these subjective probability assessments as 'prior' probabilities. However, the manager is hesitant about this decision. Based on "nothing ventured, nothing gained" the company is thinking about seeking help from a marketing research firm. The marketing research firm will assess the size of the product's market by means of a survey.

The manager has to make a decision as to how 'reliable' the consulting firm is. By sampling and then reviewing the past performance of the consultant, we can develop the following reliability matrix : 1. What the Ap 0. These records are available to their clients free of charge.

To construct a reliability matrix, you must consider the marketing research firm's performance records for similar products with high sales. Then, find the percentage of which products the marketing research firm correctly predicted would have high sales A , medium sales B , and little C or almost no sales. Similar analysis should be conducted to construct the remaining columns of the reliability matrix. Note that for consistency, the entries in each column of the above reliability matrix should add up to one. In this example, what is the numerical value of P A A p? That is, what is the chance that the marketing firm predicts A is going to happen, and A actually will happen?

This important information can be obtained by applying the Bayes Law from your probability and statistics course as follows: a Take probabilities and multiply them "down" in the above matrix, b Add the rows across to get the sum, c Normalize the values i. Many managerial problems, such as this example, involve a sequence of decisions. When a decision situation requires a series of decisions, the payoff table cannot accommodate the multiple layers of decision-making. Thus, a decision tree is needed. Do not gather useless information that cannot change a decision: A question for you: In a game a player is presented two envelopes containing money.

He is told that one envelope contains twice as much money as the other envelope, but he does not know which one contains the larger amount. The player then may pick one envelope at will, and after he has made a decision, he is offered to exchange his envelope with the other envelope. If the player is allowed to see what's inside the envelope he has selected at first, should the player swap, that is, exchange the envelopes?

The outcome of a good decision may not be good, therefor one must not confuse the quality of the outcome with the quality of the decision. As Seneca put it "When the words are clear, then the thought will be also". It utilizes a network of two types of nodes: decision choice nodes represented by square shapes , and states of nature chance nodes represented by circles.

Construct a decision tree utilizing the logic of the problem. For the chance nodes, ensure that the probabilities along any outgoing branch sum to one. Calculate the expected payoffs by rolling the tree backward i. You may imagine driving your car; starting at the foot of the decision tree and moving to the right along the branches. At each square you have control, to make a decision and then turn the wheel of your car.

At each circle , Lady Fortuna takes over the wheel and you are powerless. Here is a step-by-step description of how to build a decision tree: Draw the decision tree using squares to represent decisions and circles to represent uncertainty, Evaluate the decision tree to make sure all possible outcomes are included, Calculate the tree values working from the right side back to the left, Calculate the values of uncertain outcome nodes by multiplying the value of the outcomes by their probability i.

On the tree, the value of a node can be calculated when we have the values for all the nodes following it. The value for a choice node is the largest value of all nodes immediately following it. The value of a chance node is the expected value of the nodes following that node, using the probability of the arcs. By rolling the tree backward, from its branches toward its root, you can compute the value of all nodes including the root of the tree. Putting these numerical results on the decision tree results in the following graph: A Typical Decision Tree Click on the image to enlarge it Determine the best decision for the tree by starting at its root and going forward.

Based on proceeding decision tree, our decision is as follows: Hire the consultant, and then wait for the consultant's report. If the report predicts either high or medium sales, then go ahead and manufacture the product. Otherwise, do not manufacture the product. Clearly the manufacturer is concerned with measuring the risk of the above decision, based on decision tree.

Coefficient of Variation as Risk Measuring Tool and Decision Procedure: Based on the above decision, and its decision-tree, one might develop a coefficient of variation C. V risk-tree, as depicted below: Coefficient of Variation as a Risk Measuring Tool and Decision Procedure Click on the image to enlarge it Notice that the above risk-tree is extracted from the decision tree, with C.

For example the consultant fee is already subtracted from the payoffs. From the above risk-tree, we notice that this consulting firm is likely with probability 0. Clearly one must not consider only one consulting firm, rather one must consider several potential consulting during decision-making planning stage.

The risk decision tree then is a necessary tool to construct for each consulting firm in order to measure and compare to arrive at the final decision for implementation. You may start with the following extreme and interesting cases by using this JavaScript for the needed computation: Consider a flat prior, without changing the reliability matrix. Consider a perfect reliability matrix i. Consider a perfect prior, without changing the reliability matrix. Consider a flat reliability matrix i.

Consider the consultant prediction probabilities as your own prior, without changing the reliability matrix. Influence diagrams: As can be seen in the decision tree examples, the branch and node description of sequential decision problems often become very complicated. At times it is downright difficult to draw the tree in such a manner that preserves the relationships that actually drive the decision.

The need to maintain validation, and the rapid increase in complexity that often arises from the liberal use of recursive structures, have rendered the decision process difficult to describe to others. The reason for this complexity is that the actual computational mechanism used to analyze the tree, is embodied directly within the trees and branches.

The probabilities and values required to calculate the expected value of the following branch are explicitly defined at each node. Influence diagrams are also used for the development of decision models and as an alternate graphical representations of decision trees. The following figure depicts an influence diagram for our numerical example. In the influence diagram above, the decision nodes and chance nodes are similarly illustrated with squares and circles. Arcs arrows imply relationships, including probabilistic ones.

Finally, decision tree and influence diagram provide effective methods of decision-making because they: Clearly lay out the problem so that all options can be challenged Allow us to analyze fully the possible consequences of a decision Provide a framework to quantify the values of outcomes and the probabilities of achieving them Help us to make the best decisions on the basis of existing information and best guesses Visit also: Decision Theory and Decision Trees Further Readings Bazerman M.

Connolly T. Arkes, and K. Cooke R. Describes much of the history of the expert judgment problem. It also includes many of the methods that have been suggested to do numerical combination of expert uncertainties. Furthermore, it promotes a method that has been used extensively by us and many others, in which experts are given a weighting that judge their performance on calibration questions.

This is a good way of getting around the problem of assessing the "quality" of an expert, and lends a degree of objectivity to the results that is not obtained by other methods. Bouyssou D. Daellenbach H. Klein D. Thierauf R. Work they do not want to do themselves. Work they do not have time to do themselves. All such work falls under the broad umbrella of consulting service. Regardless of why managers pay others to advise them, they typically have high expectations concerning the quality of the recommendations, measured in terms of reliability and cost.

The following figure depicts the process of the optimal information determination. The Determination of the Optimal Information Deciding about the Consulting Firm: Each time you are thinking of hiring a consultant you may face the danger of looking foolish, not to mention losing thousands or even millions of dollars. To make matters worse, most of the consulting industry's tried-and-true firms have recently merged, split, disappeared, reappeared, or reconfigured at least once. How can you be sure to choose the right consultants?

Test the consultant's knowledge of your product. It is imperative to find out the depth of a prospective consultant's knowledge about your particular product and its potential market. Ask the consultant to provide a generic project plan, task list, or other documentation about your product. Is there an approved budget and duration? What potential customers' involvement is expected?

Who is expected to provide the final advice and provide sign-off? Even the best consultants are likely to have some less-than-successful moments in their work history. Conducting the reliability analysis process is essential. Ask specific questions about the consultants' past projects, proud moments, and failed efforts. Of course it's important to check a potential consultant's references. Ask for specific referrals from as many previous clients or firms with similar businesses to yours.

Get a clearly written contract, accurate cost estimates, the survey statistical sample size, and the commitment on the completion and written advice on time. Further Reading Holtz H. Weinberg G. Revising Your Expectation and its Risk In our example, we saw how to make decision based on objective payoff matrix by computing the expected value and the risk expressed as coefficient of variation as our decision criteria. You may like using Revising the Mean and Variance JavaScript to performing some numerical experimentation. You may apply it for validating the above example and for a deeper understanding of the concept where more than 2-sources of information are to be combined.

Determination of the Decision-Maker's Utility Function We have worked with payoff tables expressed in terms of expected monetary value. Expected monetary value, however, is not always the best criterion to use in decision making. The value of money varies from situation to situation and from one decision maker to another. Generally, too, the value of money is not a linear function of the amount of money.

In such situations, the analyst should determine the decision-maker's utility for money and select the alternative course of action that yields the highest expected utility, rather than the highest expected monetary value. Individuals pay insurance premiums to avoid the possibility of financial loss associated with an undesirable event occurring.

However, utilities of different outcomes are not directly proportional to their monetary consequences.

Statistical Decision Making: An Example

If the loss is considered to be relatively large, an individual is more likely to opt to pay an associated premium. If an individual considers the loss inconsequential, it is less likely the individual will choose to pay the associated premium. Individuals differ in their attitudes towards risk and these differences will influence their choices.

Therefore, individuals should make the same decision each time relative to the perceived risk in similar situations. This does not mean that all individuals would assess the same amount of risk to similar situations. Further, due to the financial stability of an individual, two individuals facing the same situation may react differently but still behave rationally.

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An individual's differences of opinion and interpretation of policies can also produce differences. The expected monetary reward associated with various decisions may be unreasonable for the following two important reasons: 1. Dollar value may not truly express the personal value of the outcome. Expected monetary values may not accurately reflect risk aversion. The gamble's outcome depends on the toss of a fair coin. Clearly, the second choice is preferred to the first if expected monetary reward were a reasonable criterion. Why do some people buy insurance and others do not?

The decision-making process involves psychological and economical factors, among others. The utility concept is an attempt to measure the usefulness of money for the individual decision maker. It is measured in 'Utile'. The utility concept enables us to explain why, for example, some people buy one dollar lotto tickets to win a million dollars. Therefore, in order to make a sound decision considering the decision-maker's attitude towards risk, one must translate the monetary payoff matrix into the utility matrix. The main question is: how do we measure the utility function for a specific decision maker?

Consider our Investment Decision Problem. By changing the value of p and repeating a similar question, there exists a value for p at which the decision maker is indifferent between the two scenarios. Suppose we find the following utility matrix: Monetary Payoff Matrix Utility Payoff Matrix A B C D A B C D 12 8 7 3 58 28 20 13 15 9 5 -2 30 18 0 7 7 7 7 20 20 20 20 At this point, you may apply any of the previously discussed techniques to this utility matrix instead of monetary in order to make a satisfactory decision.

Clearly, the decision could be different. Notice that any technique used in decision making with utility matrix is indeed very subjective ; therefore it is more appropriate only for the private life decisions. You may like to check your computations using Determination of Utility Function JavaScript, and then perform some numerical experimentation for a deeper understanding of the concepts.

Utility Function Representations with Applications Introduction: A utility function transforms the usefulness of an outcome into a numerical value that measures the personal worth of the outcome. The utility of an outcome may be scaled between 0, and , as we did in our numerical example, converting the monetary matrix into the utility matrix. This utility function may be a simple table, a smooth continuously increasing graph, or a mathematical expression of the graph. The aim is to represent the functional relationship between the entries of monetary matrix and the utility matrix outcome obtained earlier.

You may ask what is a function? What is a function? A function is a thing that does something. For example, a coffee grinding machine is a function that transforms the coffee beans into powder. A utility function translates converts the input domain monetary values into output range, with the two end-values of 0 and utiles.

In other words, a utility function determines the degrees of the decision-maker sensible preferences. This chapter presents a general process for determining utility function. The presentation is in the context of the previous chapter's numerical results, although there are repeated data therein. Utility Function Representations with Applications: There are three different methods of representing a function: The Tabular, Graphical, and Mathematical representation. The selection of one method over another depends on the mathematical skill of the decision-maker to understand and use it easily.

The three methods are evolutionary in their construction process, respectively; therefore, one may proceed to the next method if needed. The utility function is often used to predict the utility of the decision-maker for a given monetary value. The prediction scope and precision increases form the tabular method to the mathematical method. Tabular Representation of the Utility Function: We can tabulate the pair of data D, U using the entries of the matrix representing the monetary values D and their corresponding utiles U from the utility matrix obtained already.

The Tabular Form of the utility function for our numerical example is given by the following paired D, U table: Utility Function U of the Monetary Variable D in Tabular Form D 12 8 7 3 15 9 5 -2 7 7 7 7 U 58 28 20 13 30 18 0 20 20 20 20 Tabular Representation of the Utility Function for the Numerical Example As you see, the tabular representation is limited to the numerical values within the table.

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One may apply an interpolation method: however since the utility function is almost always non-linear; the interpolated result does not represent the utility of the decision maker accurately. To overcome this difficulty, one may use the graphical method.


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Graphical Representation of the Utility Function: We can draw a curve using a scatter diagram obtained by plotting the Tabular Form on a graph paper. Having the scatter diagram, first we need to decide on the shape of the utility function. The utility graph is characterized by its properties of being smooth, continuous, and an increasing curve. Often a parabola shape function fits well for relatively narrow domain values of D variable.

For wider domains, one may fit few piece-wise parabola functions, one for each appropriate sub-domain. For our numerical example, the following is a graph of the function over the interval used in modeling the utility function, plotted with its associated utility U-axis and the associated Dollar values D-axis.

Note that in the scatter diagram the multiple points are depicted by small circles. Reading a value from a graph is not convenient; therefore, for prediction proposes, a mathematical model serves best. Mathematical Representation of the Utility Function: We can construct a mathematical model for the utility function using the shape of utility function obtained by its representation by Graphical Method. For wider domains, one may fit a few piece-wise parabola functions, one for each appropriate sub-domain.

We know that we want a quadratic function that best fits the scatter diagram that has already been constructed. Therefore, we use a regression analysis to estimate the coefficients in the function that is the best fit to the pairs of data D, U. The above mathematical representation provides more useful information than the other two methods. For example, by taking the derivative of the function provides the marginal value of the utility; i. Notice that for this numerical example, the marginal utility is an increasing function, because variable D has a positive coefficient; therefore, one is able to classify this decision- maker as a mild risk-taker.

You might like to use Quadratic Regression JavaScript to check your hand computation. For higher degrees than quadratic, you may like to use the Polynomial Regressions JavaScript. It allows the decision maker to examine the risk within its expected return, and identify the critical issues in assessing, limiting, and mitigating risk.

This process involves both the qualitative and quantitative aspects of assessing the impact of risk. Decision theory does not describe what people actually do since there are difficulties with both computations of probability and the utility of an outcome. Decisions can also be affected by people's subjective rationality and by the way in which a decision problem is perceived. Traditionally, the expected value of random variables has been used as a major aid to quantify the amount of risk.

However, the expected value is not necessarily a good measure alone by which to make decisions since it blurs the distinction between probability and severity. Of course, this is a subjective assessment. The following charts depict the complexity of probability of an event and the impact of the occurrence of the event, and its related risk indicator, respectively: From the previous section, you may recall that the certainty equivalent is the risk free payoff.

Dymocks - (ebook) Introduction to Decision Making Support Using Statistics, eBook ()

Moreover, the difference between a decision maker's certainty equivalent and the expected monetary value EMV is called the risk premium. We may use the sign and the magnitude of the risk premium in classification of a decision maker's relative attitude toward risk as follows: If the risk premium is positive, then the decision maker is willing to take the risk and the decision maker is said to be a risk seeker. Clearly, some people are more risk-accepting than others: the larger is the risk premium, the more risk-accepting the decision-maker.

If the risk premium is negative, then the decision-maker would avoid taking the risk and the decision maker is said to be risk averse. If the risk premium is zero, then the decision maker is said to be risk neutral. Buying Insurance: As we have noticed, often it is not probability, but expectation that acts a measuring tool and decision-guide. Many decision cases are similar to the following: The probability of a fire in your neighborhood may be very small.

But, if it occurred, the cost to you could be very great. Not only property but also your "dear ones", so the negative expectation of not ensuring against fire is so much greater than the cost of premium than ensuring is the best. Further Readings Christensen C. Hammond J. Keeney, and H. Richter M. Wong, Computable preference and utility, Journal of Mathematical Economics , 32 3 , , Tummala V.

The Discovery and Management of Losses In discovery and management of losses expressed in the monetary terms perception and measuring the chance of events is crucial. Losses might have various sources. It uses various scientific research -based principles, strategies , and analytical methods including mathematical modeling , statistics and numerical algorithms to improve an organization's ability to enact rational and meaningful management decisions by arriving at optimal or near optimal solutions to complex decision problems.

Management scientists help businesses to achieve their goals using the scientific methods of operational research. The management scientist's mandate is to use rational, systematic, science-based techniques to inform and improve decisions of all kinds. Of course, the techniques of management science are not restricted to business applications but may be applied to military, medical, public administration, charitable groups, political groups or community groups.

Management science is concerned with developing and applying models and concepts that may prove useful in helping to illuminate management issues and solve managerial problems, as well as designing and developing new and better models of organizational excellence. The application of these models within the corporate sector became known as management science. Some of the fields that have considerable overlap with Operations Research and Management Science include [32] :. Applications are abundant such as in airlines, manufacturing companies, service organizations , military branches, and government.

The range of problems and issues to which it has contributed insights and solutions is vast. It includes: [30]. Management is also concerned with so-called 'soft-operational analysis' which concerns methods for strategic planning , strategic decision support , problem structuring methods. In dealing with these sorts of challenges, mathematical modeling and simulation may not be appropriate or may not suffice.

Therefore, during the past 30 years [ vague ] , a number of non-quantified modeling methods have been developed. These include: [ citation needed ]. From Wikipedia, the free encyclopedia. For the academic journal, see Operations Research. This section needs expansion. You can help by adding to it. March Main article: Management science. West Churchman William W.

Cooper Robert Dorfman Richard M. Karp Ramayya Krishnan Frederick W. Lanchester Thomas L. Magnanti Alvin E. Roth Peter Whittle Related fields Behavioral operations research Big data Business engineering Business process management Database normalization Engineering management Geographic information systems Industrial engineering Industrial organization Managerial economics Military simulation Power system simulation Project Production Management Reliability engineering Scientific management Search-based software engineering Simulation modeling System safety Wargaming.

Retrieved 7 January American Mathematical Society. Retrieved 13 November Retrieved 27 January Archived from the original on 27 May Sodhi, "What about the 'O' in O. Dunnigan Dirty Little Secrets of the Twentieth Century. Harper Paperbacks. Allen Journal of the American Statistical Association. May United States Naval Institute Proceedings. Pound, J.

Bell, and M. Shenoy and T. Retrieved 5 June The University of Tennessee, Omega - International Journal of Management Science. Archived from the original on 24 April Retrieved 31 March The Science of Better. Learn about OR. Retrieved 19 March IGI Global. Systems science. Doubling time Leverage points Limiting factor Negative feedback Positive feedback.

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The Importance of Statistics in Management Decision Making

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