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## Operations Management

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**Operations Management**Session 10: Probability Concepts**Simulation Game**• Game codes due. • Please go to http://usc.responsive.net/lt/usc/start.html to register. • Course code: usc. • Individual code: what you purchased from the bookstore. • Case groups posted. Please double-check. Operations Management**Today’s Class**• Probability Concept Review • Basic Statistics Formula • Common Distribution Operations Management**Quote of the day**• Without the element of uncertainty, the bringing off of even, the greatest business triumph would be dull, routine, and eminently unsatisfying. J. Paul Getty Operations Management**Blackjack**• You have a 9 and 5, what will happen if you hit? Operations Management**Random Experiment**• Random Experiment: An experiment in which the precise outcome is not known ahead of time. The set of possibilities however is known • Examples: • Demand for blue blazers next month • The value of a rolled die • The waiting times of customers in the bank • The waiting time for an ATT service person • Tomorrow’s closing value of the NASDAQ • The temperature in Los Angeles tomorrow Operations Management**Random Variable**• A random variable is the numerical value determined by the outcome of a random experiment • A random variable can be discrete (i.e. takes on only a finite set of values) or continuous • Examples: • The value on a rolled die is a discrete random variable • The demand for blazers is a discrete random variable • The birth weight of a newborn baby is a continuous variable • The waiting time for the AT&T service person is a continuous random variable Operations Management**Sample Space**• Sample space is the list of possible outcomes of an experiment • Examples: • For a die, the sample space S is: {1,2,3,4,5,6} • For the demand for blue blazers it is all possible realizations of the demand. For example: {1000,1001,1002…,2000} • The waiting time in the bank is any number greater than or equal to 0. This is a continuous random variable • The waiting time for a bus at a bus stop is any number between 0 and 30 minutes. This is a continuous random variable that is bounded Operations Management**Event**• An event is a set of one or more outcomes of a random experiment • Examples: • Getting less than 5 by rolling the die: This event occurs if the values observed are {1,2,3, or 4} • The demand is smaller or equal to 1500. This event occurs if the values of the demand are {1000, 1001, … 1500} • The waiting time for a bus at the bus stop exceeds 10. This event occurs if the wait time is in the interval (10, 30) Operations Management**Probability**• The probability of an event is a number between 0 and 1 • 1 means that the event will always happen • 0 means that the event will never happen • The probability of an event A is denoted as either P(A) or Prob(A) • Example: Probability of Rolling die and observing a number less than 5 = • P(outcome< 5) = Prob(observing {1,2,3 or 4}) = 4/6 = 2/3 Operations Management**Probability**• Probability that A doesn’t occur: • P(not A) = 1 – P(A) • Thus, the probability you will roll a number larger or equal to 5 is or 6 is: 1 – Probability (Outcome <5) = 1 – 2/3 = 1/3 Operations Management**Probability**• Suppose all the outcomes that constitute the “waiting time” for an AT&T operator are equally likely. The minimum waiting time is 30 min and the maximum is 90 min. • Then the probability of waiting less than 45 min is: • P (waiting more than 45 min) is: Event Sample Space Operations Management**Probability Distribution for Discrete Random Variables**• Let us begin with discrete outcomes • A probability distribution is a list of: • All possible values for a random variable (Sample space); and • The corresponding probabilities • For a die, the probability distribution is: Operations Management**Probability Distribution for Discrete Random Variables**• The chart below depicts the probability distribution Operations Management**Cumulative Probability Distribution for Discrete Random**Variables • Probability that a random number will be less than or equal to some given number • For a die, the cumulative probability distribution is: Additional: What is the probability a die roll is less than 3.5? Operations Management**Continuous Random Variables and Probability Density**Functions (PDF) • The probability density function is the analog of the probability distribution (table 1) for discrete random numbers • Example: Suppose we have a computer program that can generate any number between 1 and 6 (not just the integers) • Assume that each number is equally likely to be generated. Then we have a continuous random number • This random number has a uniform distribution between 1 and 6 Operations Management**Continuous Random Variables and Probability Density**Functions (PDF) Probability Density 1/5 1 2 3 4 5 6 Outcome Operations Management**Properties of Probability Density Functions**• By convention the total area under the probability density function must equal 1 • The base of the rectangle in the figure is 6 – 1 = 5 units long, the probability density is 1/5 for all values between 1 and 6. This ensures that the total area is 1 • The probability of observing any value between two numbers is equal to the area under the probability density function between those numbers • The probability of observing any number between 4.0 and 5.0 will be (5.0 – 4.0)* 1/5 = 1/5 = 0.2 Operations Management**Properties of Probability Density Functions**Probability Density, f 1/5 1 2 3 4 5 6 Outcome Operations Management**Properties of Cumulative Distribution Functions**CDF, F Probability that the outcome is smaller than 5: is 4/5 1 4/5 Probability that the outcome is smaller than 2: is 1/5 1/5 0 1 2 3 4 5 6 Outcome Operations Management**Relationship between Density and Cumulative Distribution**Functions CDF 1 4/5 1/5 0 1 2 3 4 5 6 Outcome Operations Management**Other distributions: Triangular**Probability that the outcome is between 2 and 5 2 5 Operations Management**0.80**0.70 Normal distribution #2 Normal distribution #1 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -4 -3 -2 -1 0 1 2 3 4 X Normal Distribution Operations Management**0.80**0.70 Normal distribution #2 Normal distribution #1 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -4 -3 -2 -1 0 1 2 3 4 X Continuous Random Variables and Probability Density Functions (PDF) Operations Management**Cumulative Density Function (CDF) for Continuous Random**Numbers • This is analogous to the cumulative distribution function for discrete random numbers • The cumulative density function gives the probability of the continuous random variable being equal to or smaller than a given number Operations Management**Cumulative Density Function (CDF) for Continuous Random**Numbers Cumulative Density Function 1 1/5 o 1 2 3 4 5 6 Operations Management**Mean or Expected Value of a Random Number**• Expected value can be thought of as the average value of a random variable • Let us denote by X the value of the random variable. • If the random variable is the value of a die, then X denotes the value rolled. If we roll a 6, then X = 6). We will use the notation E[X] to denote the expected value of X • If the random number is a discrete variable that can take on values between 1 and N then: • E[X] = Thus for the die, E[X] = 1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6 = 3.5 Operations Management**Mean or Expected Value of a Random Number**• What if the variable is a continuous random variable? • Let f(X) be the probability density function. • Example: for the uniform distribution, we have seen: • f(X) = 0.2 whenever X is between 1 and 6. f(X) = 0 if X is not between 1 and 6.] • Integration of continuous variables in lay terms is equivalent to summation for discrete variables. Operations Management**The Variance of X**• When X is a discrete random variable: • Var(X) = (X – E[X])2*Prob(X) • If X is the random number generated by the roll of a die then: • Var(X) = (1-3.5)2*1/6 + (2-3.5)2*1/6 +(3-3.5)2*1/6 +(4-3.5)2*1/6 +(5-3.5)2*1/6 +(6-3.5)2*1/6 = 2.9166 • Standard Deviation = square root of variance SD(X) = 1.708 in this example Operations Management**How to measure variability?**• A possible measure is variance, or standard deviation • Is this good enough? Operations Management**Which one has the larger variability?**Operations Management**Which one has the larger variability?**• The variation in the first set appears to be significantly higher than the second set. • Nevertheless, the standard deviation of the first graph is 5, the standard deviation of the second graph is 10. Operations Management**Coefficient of Variation**• A better measure of variability is the ratio of the standard deviation to the average. This ratio is called the coefficient of variation. • Coefficient of Variation = Standard Deviation / Average (expected value) • A similar measure is squared coefficient of variation: SCV = (CV)2= (SD/M)2 Operations Management**Sum of Random Numbers**• Often we have to analyze sum of random numbers. • Examples include: • The sum of the demand of different products processed by the same resource • The total demand for cars produced by GM • The total demand for knitwear at DD • The total completion time of a project • The sum of throughput times at two different stages of a service system (waiting time to place an order at a cafeteria and waiting time in the line to pay for the food) Operations Management**Sum of Random Numbers**• Let X and Y be two random variables. The sum of Xand Y is another random variable. Let S = X +Y • The distribution of S will be different from that of X and Y • Example: • Let Sbe the sum of the values when you roll 2 dice simultaneously. Let Xrepresent the value die #1 and Y represent the value of die #2 • S = X + Y Operations Management**Sum of Random Numbers**• The distribution of the sum S is given below: Operations Management**Sum of Random Numbers**• E[S] = 2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 + 6*5/36 + 7*6/36 + 8*5/36 + 9*4/36 + 10*3/36 + 11*2/36 + 12*1/36 = 7 • Var(S) = (2 - 7)2*1/36 + (3 - 7)2*2/36 +…….+ (12 - 7)2*1/36 = 5.83 • SD(S) = 5.83^1/2 = 2.42 Operations Management**Sum of Random Numbers**Operations Management**Expected Value and Standard Deviation of Sum of Random**Numbers • If a and b are 2 known constant and X and Y are random independent variables: • E[aX+bY] = aE[X] + bE[Y] • Var(aX+bY) = a2Var(X) + b2Var(Y) Operations Management**Specific Distributions Of Interest**• We will also utilize Uniform Distributions • Uniform Distribution: Whenever the likelihood of observing a set of numbers is equally likely • Continuous or discrete • We use notation U(a,b) to denote a uniform distribution • Example U(1,5) is uniform distribution between 1 and 5. • If it is a discrete distribution then outcomes 1,2,3,4, and 5 are equally likely (each with probability 1/5) • If it is a continuous distribution then all numbers between 1 and 5 are equally likely • The p.d.f. for U(1,5) (continuous) will be f(X) = 0.25 for X between 1 and 5 Operations Management**Exponential Distribution**• The exponential distribution is often used as a model for the distribution of time until the next arrival. • The probability density function for an Exponential distribution is: f(x) = e-x, x > 0 • is a parameter of the model (just as m and s are parameters of a Normal distribution) • E[X] = 1/ Var(X) = 1/2 • Coefficient of Variation = Standard deviation / Average = 1 Operations Management**Exponential Distribution**Shape of the Exponential Probability Density Function f(X) X Operations Management**Poisson Distribution**• The Poisson Distribution is often used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period • The probability of n events is: p(n) = ne-/n!, n = 0, 1, 2, 3, … • is a parameter of the model • E[N] = • Var(N) = Operations Management**Poisson Distribution**Operations Management**Next Class**• Waiting-line Management • How uncertainty/variability and utilization rate determines the system performance • Article Reading: “The Psychology of Waiting-lines” Operations Management