QUME 507 Module 1: Quantitative Models and Probability

1.1 Quantitative Methods Managers and the Approach to Statistical Analysis Managers face many decision-making dilemmas as they struggle to make the best decision for their organizations. An excellent decision can result in millions of additional dollars for the company or promotions and/or bonuses for the decision maker. Thus, a bad decision can lead to huge financial losses for the company and possible job losses for the decision maker. People make any decision at a personal level or even for their companies based on their instincts. However, especially in business, decision-making tools can provide excellent guidance in illustrating the advantages and disadvantages of various alternatives. This is the essence of decision modeling (Balakrishnan et al. 2017). A model can be defined as a denotative and concise representation of the structure or function of selected aspects of our world to one or more observers for the purpose of communicating a belief about some relationships, expressing a guess, making a prediction, or specifying the design of something or a set of events. It could be assumed that rational people accept a model or belief because the preponderance of evidence supports that model or belief. However, people are always free to choose, and in companies it is the managers who have the task of deciding (Sheridan, 2017). In companies, decision modeling is a scientific approach to business decision making. It is defined as the development of a (usually mathematical) model of a real-world problem scenario or environment. Decision modeling also refers to a quantitative analysis (Balakrishnan, et al. 2017). The approach to quantitative analysis is to follow a structured and rational process in making decisions to obtain the solution to problems in the organization. According to Anderson et al. (2016) problem solving is defined as the process of identifying a difference between the current state of affairs and the desired state and then taking action to reduce or eliminate the difference.

Render et al. (2018) the approach of quantitative analysis is to define a problem, develop a model, obtain input data, develop a solution, test the solution, analyze the results, and implement them. The Model presented in Figure 1. Figure 1. Quantitative Analysis Approach Defining the problem Developing the model Acquiring input data Developing the solution Testing the solution Analyzing the results Implementation Adapted by Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson.

The seven steps that are indispensable for the development of quantitative models and the solution of problems affecting companies are discussed below. First Step: Define the Problem The first step in the quantitative approach is to develop a clear and concise statement about the problem. This statement will give direction and meaning to the next steps. In many cases, defining the problem is the most important and difficult step. It is essential to go beyond the symptoms of the problem and identify the real causes. Besides, you shouldn't pick and choose all the problems at once. You must take into consideration what the main problem is. Similarly, when a problem is difficult to quantify, it may be necessary to develop specific objectives that can be measured. Second Step: Develop the Model The second step is to develop a model. In quantitative problems, the problem is expressed mathematically through a model. According to Sheridan (2017) any meaningful mathematical equation or statement in symbolic form is a model. In the case of a mathematical model, it is a set of mathematical relationships that are expressed by inequalities and equations. These models should be simple and easy to apply with the resources available. In mathematical models there are variables and parameters. A variable is a measurable quantity that can vary or is subject to change. Variables can be controllable or uncontrollable. A controllable variable is also known as a decision variable. An example would be how many inventory items to order. A parameter is a measurable quantity that is inherent in the problem. Therefore, the variables are not known in advance, but the parameters are. Third Step: Data Collection Once a model is developed, the data is collected. Obtaining accurate data for the model is critical; even if the model is a perfect representation of reality, inadequate data will lead to wrong results. This situation is known as garbage-in, garbage-out. Measurement errors should be avoided. Therefore,

one must be cautious about using data that represents approximations, as these may be measuring other things that do not necessarily indicate what is needed to evaluate the citation correctly. Fourth Step: Develop a solution The fourth step is the development of a solution. It involves manipulating the model to arrive at the best (optimal) solution to the problem. In some cases, this requires solving an equation to achieve the best decision. When a solution is found, the model can be modified, and different scenarios can be evaluated. In other cases, trial and error may be used, trying various approaches, and choosing the one that result in the best decision. Fifth Step: Testing the Solution The fifth step is to test the solution. Testing the input data and the model includes determining the accuracy and completeness of the data used by the model. Inaccurate data will lead to an inaccurate solution; the solutions provided by the model and the data are converted into predictions. Proposals based on the predictions must be approved by management. It should be noted that models that offer complex solutions face the challenge of being rejected by managers because they are not intuitive. Sixth Step: Analyze the Results The sixth step is the analysis of the results. Once a solution has been tested, the results must be analyzed in terms of how they will affect the organization as a whole. Be aware that even small changes in organizations are often difficult to make. If the results indicate major changes in organizational policy, the analyst can expect resistance. In analyzing the results, one should identify who should change and how much, and who is the person who has the power to lead the change

Seventh Step: Implementation The seventh and final step is implementation. In this step the organization can face resistance from employees, managers and consumers. Necessary measures must be taken. In addition, after implementation, the success or failure of the measures implemented should be measured and monitored. Finally, it should be clear that implementation is not just another step that takes place when the modeling process is over. Each of these steps significantly affects the possibility of implementing the results obtained from a quantitative study. The Break-even Point and Decision Making As we have discussed, the development of a model is important in the approach to quantitative analysis. Decision makers are often interested in the breakeven point (BP). It provides the point where costs are satisfied with revenues. It sets the minimum that must be produced so that at least the profit is zero and does not assume negative values. In the following mathematical model, we can see the representation that profit equals income minus expenses: Profit = Income - Expenses In addition, expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs by the number of units. Here is an example. Profit = Income - (Fixed Cost + Variable Cost) Profit = (Selling price per unit) (Number of units sold) - [Fixed cost + (Variable cost per unit) (Number of units sold) Profit = sX - [f + vX] Profit = sX - f – vX where s = selling price per unit v = variable cost per unit f = fixed cost X = number of units sold

Application: Pritchett Fine Watches The company buys, sells, and repairs antique watches. Rebuilt springs are sold at $10 per unit. The fixed cost of the spring building equipment is $1,000. The variable cost of spring material is $5 per unit. s = 10 f = 1,000 v = 5 Number of springs sold = X Profits = sX - f – vX When we solve the equation, we see: If sales = 0, profit = -$1,000 If sales = 1,000, profit = [(10) (1,000) - 1,000 - (5) (1,000)] = $4,000 For additional information on the use of the break-even point in decision making you can access the video: Break-even 101 - Calculate break-even for buy or make decision and process selection in Supply Chain https://youtu.be/5XOVUPGPcbg Application: Use Excel QM for Balance Point exercises First Step: When you open the program, choose the Excel Qm window. Choose from the Break-Even-Analysis/Break-even Cost vs Revenue menu as shown in Figure 1.

Image 1 Select Select Select Second step Check Volume Analysis and Graph in the window. Press OK, as shown in image 2. Image 2

Third Step When entering the data, you will get the result and the graph as shown in image 3. In this case the break-even point is in 200 units. Image 3 To see a demonstration of the use of Excel QM in a PE problem you can access the following video: Break-Even Point Analysis at https://youtu.be/jzjnfv9ITWI To see articles and research where the topic of break-even point analysis as a tool in decision making is deepened, you can access the following articles that are in the Folder with Complementary Material. Alnasser, N., Shaban, O. S. & Al-Zubi, Z. (2014). The Effect of Using Break-Even-Point in Planning, Controlling, and Decision Making in the Industrial Jordanian Companies. International Journal of Academic Research in Business and Social Sciences, 4(5), 626-636.

Jakupi, K., Statovci, B. & Havrizi, B. (2017). Break-Even Analysis as a powerful tool in Decision Making. International Journal of Management Excellence, 9(3), 1169-1171.

1.2 Probability A probability is a numerical expression of the possibility of an event occurring. In the following, we will discuss the concepts, definitions, and basic relationships of probability, as well as probability distributions that are useful for solving many quantitative analysis problems. There are two basic rules of mathematical probabilities: 1. The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal to 1: 2. 0 P (event) 1 3. The sum of the simple probabilities for all possible outcomes of an activity should equal 1. Example Diversey Paint The demand for white latex paint at Diversey Paint & Supply has always been 0, 1, 2, 3 or 4 gallons per day. Over the past 200 days, the owner has observed the following frequencies of demand shown in Figure 1. Figure 1. Diversey Paint & Supply Demand Frequencies Demand Days Probability Adapted de Render, B. Stair, R. M. & Hanna & Hale (2018). Quantitative Analysis for Management. (13 ed). Pearson. So, the probability of sales being 2 gallons of paint on any given day is P (2 gallons) 0.25 (25%). The probability of any level of sales must be greater than or equal to 0, and less than or equal to 1. Since 0, 1, 2, 3, and 4 gallons encompass all possible events or outcomes, the sum of their probabilities must equal 1.

Types of Probability There are two different ways to determine probability: the objective approach and the subjective approach. The objective approach is based on the relative frequency which is typically based on historical data or the logical or classical method. This is used to logically determine probabilities without testing. Frequency Method: P (event) = Number of occurrences of the event Total amount of trials or results Logical method: Number of opportunities to obtain something Number of results Event classification: Mutually exclusive and non-exclusive events Events are said to be mutually exclusive if only one of them can occur in any one trial. They are called collectively exhaustive if the list of results includes all possible outcomes. Most everyday experiences involve events that have both properties. In the sum of mutually exclusive events, what matters is whether an event or a second event will occur, which is known as the union of two events. When the two events are mutually exclusive, the law of addition is simple: P (event A or event B) = P (event A) + P (event B) – P (occurrences de among, event A and event B) Where P (A or B) = P (A) + P (B) – P (A and B) Figure 2: Difference between mutually exclusive and non-exclusive events

Events that are mutually Events that are not mutually exclusive exclusive Adapted by Render, B. Stair, R. M. & Hanna & Hale (2018). Quantitative Analysis for Management. (13 ed). Pearson. Probability: Dependence and Independence The three types of probability under statistical independence and dependence are: marginal, joint and conditional. 1. The marginal (or simple) probability is only the probability of an event occurring: P (A) 2. The joint probability is the probability of two or more events occurring and is the product of their marginal probabilities for independent events: P (AB) = P (A) x P (B) 3. The conditional probability is the probability of event B given that event A has occurred P (B | A) = P (B) 4. the probability of event A given that event B has occurred P (A | B) = P (A) Statistically independent events When events are statistically dependent, the occurrence of one event affects the probability of another event occurring. As we saw earlier, marginal, conditional and

joint probability exist with dependence as well as with independence, but the form of the last two changes. A marginal probability is calculated exactly as for independent events. Again, the marginal probability of event A occurring is denoted by P (A). Calculating a conditional probability with dependence is a bit more complicated than under independence. The formula for the conditional probability of A, given that event B occurs, is set as: In the case of the joint probability of two events the equation is: P (AB) = P (B | A) P (A) Bayes' Theorem Often, an analysis begins with estimates of initial or past probabilities for specific events of interest; these initial estimates are usually developed using the relative frequency approach (applied to historical data) or the subjective approach. Additional information about the events is then obtained from sources such as a sample, a special report, or a product test. With this new information, you can update the previous probability values by calculating revised probabilities, called subsequent probabilities. Bayes' theorem provides a means of performing these probability checks (Anderson et al., 2016). This means that we can take new or recent data and then revise and improve our previous probability estimates for an event as shown in Figure 3. Figure 3. Using the Bayes Process Previous Bayes Subsequent probabilities Process Probabilities New Information

Adapted by Render, B. Stair, R. M. & Hanna & Hale (2018). Quantitative Analysis for Management. (13 ed). Pearson. General form of Bayes' Theorem The revised probabilities are also calculated more directly using the general form of Bayes' theorem: ������(������|������)������(������) ������(������|������) = ������(������|������)������(������) + ������(������|������′)������(������′) where = the event's complement A Probability distributions The best-known probability distributions are: the normal, Poisson, binomial and exponential probability distributions. These help to save time and effort. Before explaining them it is necessary to address the issue of the random variable. A random variable assigns a real number to each possible result or event in an experiment. There are two types of random variables: 1. Discrete random variables may assume only a finite or limited set of values. When we have a discrete random variable, there is a probability value assigned to each event. These values must be between 0 and 1, and all must add up to 1. 2. Continuous random variables can assume any of an infinite set of values. As with discrete probability distributions, the sum of the probability values must equal 1. However, since there are an infinite number of values of the random variable, the probability of each value must be 0. If the probability values for the values of the random variable were greater than zero, the sum would be infinitely large.

Normal Distribution One of the most popular and useful continuous probability distributions is the normal distribution. The probability density function of this distribution is given by the formula, which is somewhat complex. The normal distribution is fully specified when we know the mean, µ and the standard deviation. Among the most important data of the normal distribution to point out is that it is symmetric, with the midpoint representing the mean. Moreover, changing the mean does not change the way of the distribution. The values on the X-axis are measured in the number of standard deviations outside the mean. Finally, as shown in Figure 4, as the standard deviation becomes larger, the curve flattens out. On the other hand, as the standard deviation becomes smaller, the curve becomes steeper. Figure 4. Normal Distribution Equal µ, smaller Equal µ,larger µ Adapted by Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. To see a video demonstration of the use of the Normal Distribution Calculator in Excel QM you can access the Excel OM/QM Tutorials: The Normal Distribution Calculator at https://youtu.be/3RhFN7itYK8 Binomial Distribution According to Render et al. (2018) many business experiments can be characterized by a Bernoulli process. The probability of obtaining specific results in a Bernoulli process is described by the binomial probability distribution. For a process to be considered Bernoulli, the experiment must have the following characteristics:

1. Each trial has only two possible outcomes. 2. The probability remains the same from one test to the next. 3. The trials are statistically independent. 4. The number of attempts is a positive integer. A common example is the process of flipping a coin. The binomial distribution is used to find the probability of a specific number of successes in n trials of a Bernoulli process. To determine this probability, it is necessary to know the following: n = number of tests p = the probability of success in a single trial are r = the number of successes q = 1 - p = probability of failure The formula for binomial distribution is as follows Probability of success in n trials The symbol ! means factorial and n! = n(n – 1)(n – 2)…(1) For example 4! = (4)(3)(2)(1) = 24 By definition 1! = 1 y 0! = 1 Poisson Distribution The Poisson probability distribution is used in many waiting line models to represent arrival patterns. The formula is as follows: ������������ ������ −������ ������(������) = ������!

where P (X) = probability of exactly X arrivals or incidents = average number of arrivals per unit of time (the average arrival rate) e = 2,718, the basis of natural logarithms X = specific value (0, 1, 2, 3 ...) of the random variable To see video demonstration of the use of Ms Excel 2016 in binomial and Poisson distribution you can access Excel Finding Binomial Distribution and Poisson Distribution in MS Excel 2016 at https://youtu.be/5kOG2P1fvzY Distribution F The F distribution is a continuous probability distribution useful in testing hypotheses about variances. The F distribution is often used when testing significance regression models. This will be discussed further in Module 4. Exponential Distribution The exponential distribution, also called negative exponential distribution, is used to calculate waiting line problems. This distribution often describes the time required to serve a customer. Exponential distribution is a continuous distribution. Its probability function is given by ������(������) = ������������−������������ Where X = random variable (operating times) µ = average number of units that the service facility can handle in a specific time period e = 2,718 (the basis of natural logarithms)

Final Considerations The basic concepts of probability and distributions are used for decision theory. Later we will see their application in inventory control, Markov analysis, project management, simulation and statistical quality control. Para To see an article where you can go deeper into the topic of probability, you can access the following article which can be found in the Folder with Complementary Material. Saraswathi, CH., Moulali, S. D. & Nagamani, A. (2017). The Real-Life Applications of Probability in Mathematics. International Journal of Management and Applied Science, 3(4), 62-64.

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