A key aspect of solving real business problems is dealing appropriately with uncertainty. This involves recognizing explicitly that uncertainty exists and using quantitative methods to model uncertainty.

Review Section 5-1, and address the following:

• Explain how to use a flow chart for modeling uncertainty.
• Describe how flow charts work and how it is applied.

5' Introduction A. key aspect of solving real business problems is dealing appropriately with uncertainty. This involves recognizing explicitly that uncertainty exists and using quantitative methods to model uncertainty. if you want to develop realistic business models. you cannot simply act as if uncertainty doesn't exist. Fm example. if you don't know next month's demand. you shouldn't build a model that assumes next month's demand is a sure t units This is only wishful thinking. 1'I'ou sbtmld in stead incorporate demand uncertainty explicitly into your model.To do this. you need to knmv how to deal quantitatime with uncertainty. This involtes probability and probabilityr distributions. We introduce these topics in this chapter and then use them in a number of liner chapters. There are many sources of uncertainty. Demands for products are uncertain. times between arrivals to a supermarket are uncertain. stock price returns are uncertain. changes in interest rates are uncertain. and so on. in many situations. the uncertain quantity demand. time between an'ivals. stock price return. change in interest rateis a numerical quantity. In the language of probability. it is called a random variable- More formally. a random "variable associates a numerical value with each possible random outcome. Associated with each random variable is a probability distribution that lists all of the possible values of the random variable and their corresponding probabilities. A. proba- bility distribution provides very useful information. It not only indicates the possible val- ues of the random variable but it also indicates how likely they are. For ex ample. it is useful to know that the possible demands for a product are. say. Int], zoo. 300. and son, but it is even more useful to know that the probabilities of these four values are. say. {1. | . 0.2. 1.14. and 0.3-. This implies. for example. that there is a 7% chance that demand will be at least 3110. It is often useful to summarize the information from a probability distribution with numerical summary measures. These include the mean. variance. and standard deviation. As their names imply. these summary measures are much like the corresponding summary measures in Chapters 2 arid 3. However, they are not identical. The summary measures in this chapter are based on probability distributions. not an observed data set. We will use numerical examples to explain the difference between the tooand how they are related- The purpose of this chapter is to explain the basic concepts and tools necessary to work with probability distributions and their summary measures. The chapter then dis cusses several important prnhability distributions. particularly the normal distribution and the binomial distribution. in some detail. The purpose of this chapter is to explain the basic concepts and tools necessary to work with probability distributions and their summary measures. The chapter then dis- cusses several important probability distributions, particularly the normal distribution and the binomial distribution, in some detail. Modeling uncertainty, as we will be doing in the next chapter and later in Chapters 15 and 16. is sometimes difficult, depending on the complexity of the model, and it is easy to get so caught up in the details that you lose sight of the big picture. For this rea- son, the flow chart in Figure 5.1 is useful. (A colored version of this chart is available in the file Modeling Uncertainty Flow Chart.xIsx.) Take a close look at the middle row of this chart. You begin with inputs, some of which are uncertain quantities, you use Excel formulas to incorporate the logic of the model, and the result is probability distributions of important outputs. Finally, you use this information to make decisions. (The abbreviation EMV stands for expected monetary value. It is discussed extensively in Chapter 6.) The other boxes in the chart deal with implementation issues, particularly with the software you can use to perform the analysis. Study this chart carefully and return to it as you pro- ceed through the next few chapters and Chapters 15 and 16. Before proceeding, we discuss two terms you often hear in the business world: uncertainty and risk. They are sometimes used interchangeably, but they are not really the same. You typically have no control over uncertainty; it is something that simply exists. A good example is the uncertainty in exchange rates. You cannot be sure what the exchange rate between the U.S. dollar and the euro will be a year from now. All you can try to do is measure this uncertainty with a probability distribution.Figure 5.1 Flow Chart for Modeling Uncertainty Assess probability distributions of Two fundamental approaches: uncertain inputs: 1. Build an exact probability model that 1. If a lot of historical data is available. incorporates the rules of probability. find the distribution that best fits the Pros: It is exact and amenable to sensitivity analysis. Cons: It is often data. difficult mathematically, maybe not For simulation models, this can be done 2. Choose a probability distribution even possible.) "manually" with data tables and bull-in (normal? triangular?) that seems 2. Build a simulation model. (Pros; It is functions like AVERAGE, STDEV.S, clo. reasonable. Add-ins like But add-ins like DADM_Tools or @RISK Use decision trees, made easier with DADM_Tools or @ RISK are helpful typically much easier, especially with add-ins like DADM_Took or BRISK, take care of these bookkeeping details add-in like DADM_Tools or for exploring distributions and extremely versatile, Cons: I: is automaticaly. Precision Tree, if the number of possible 3. Gather relevant information, ask only approximate and runs can be decisions and the number of possible exports, and do the best you can. time consuming for complex models) outcomes are not too large Examine important outputs. Decide which inputs are important Model the problem. The result of these formulas should Make decisions based on this for the model. be probability distribution(s) of information. Use Excel formulas to relate inputs important output(s), Summarize 1. Which are known with certainty? Criterion is usually EMV, but it could to important outputs, i.e., enter the these probability distributions with 2. Which are uncertain? business logic. be something else, e.g. minimize (1) histograms (risk profiles). (2) means and standard deviations, the probability of losing money. (3) percentiles, (4) possibly others. For simulation models, random values This is an overview of for uncertain inputs are necessary. spreadsheet modeling with uncertainty. The 1. They can sometimes be generated main process is in red. with built-in Excel functions. This often The blue boxes deal with involves tricks and can be obscure. implementation issues, 2. Add-ins like DADM_Tools or @RISK provide functions that make it much easier.In contrast. risk depends on your position. Even though you don't know what the exchange rate will be. it makes no difference to youthere is no risk to youif you have no European investments. you aren't planning a trip to Europe. and you don't have to buy or sell anything in Europe. You have risk only when you stand to gain or lose money depending on the eventual exchange rate. Of course. the type of risk you face depends on your position. If you are holding euros in a money market account. you are hoping that euros gain value relative to the dollar. But if you are planning a European vacation. you are hoping Ihat euros lose value relative to the dollar. Uncertainty and risk are inherent in many of the examples in this book. By learning about probability. you will learn how to measure uncertainty. and you will also learn how to measure the risks involved in various decisions. One important topic you will not learn much about is risk mitigation by various types of hedging. For example. if you know you have to purchase a large quantity of some product from Europe a year from now. you face the risk that the value of the euro could increase dramatically. thus cost- ing you a lot of truancy. Fortunately. there are ways to hedge this risk. so that if the euro does increase relative to the dollar. your hedge minimizes your losses. Hedging risk is an extremely important topic. and it is practiced daily in the real world. but it is beyond the scope of this book