 ## Introduction, Descriptive and Predictive Analytics Part – 1

Part – 1: Descriptive Analytics

• An Operational Decision Problem
• Forecasting with Past Historical Data
• Moving Averages
• Exponential Smoothing
• Thinking about Trends and Seasonality
• forecasting for new Products
• Fitting distributions

Descriptive Analytics

• Before we dive into analyzing data, let us a look at a fundamental problem that firms face
• An Operations problem:
• How much to produce?
• We need to know or estimate the cost of the product, price of the product, and some data on the demand of the product.
• Let us explore a problem to get started.

A Fundamental Operations Problem: An example

• Suppose that you are making operations decisions for a retailer who orders a product from a supplier and sells it to customers.
• The ordered product items are received and placed on store shelf.
• There is a large customer population
• Each customer may choose to buy or not buy the product.
• If the customer chooses to buy, he arrives at the store to buy the product.
• He buys it as long as it is available on the shelf.
• However, you have to order the product before you see the customer demand, since you have to have the items available on shelf.
• You get only one chance to order (i.e., you cannot change your purchase order after your decision).

An Operations Problem: Costs

• You order the product from the supplier at cost = 3 talers/item. (Talersare the currency units).
• After your order is received and placed on shelves, demand occurs.
• The product on the shelf sells at price = 12 talers/item.
• All unsold items are salvaged. Salvage value =0 talers/item.
• Let us look at timeline of events.

Timeline of Events Demand is uncertain. Suppose you bought 10 items

• A High Demand Scenario: Demand is 100. You will sell all 10 items, and make a profit of 10*(12-3) = 90 talers.
• A Low demand Scenario: No demand (i.e., demand = 0). You sell nothing and lose 10*3=30 talers.

Problem Recap

You dont know what the demand is going to be
You have to decide on the number of units to order from supplier before seeing the customer demand.

What could help?

• Past demand data
• Fortunately, we have the demand data from past 100 periods. The chart shows the demands (y-axis) observed in past 100 periods (x-axis).

Past Demand Data

• From the observations over the past 100 such periods.
– Maximum Demand observed was 81.
– Minimum Demand observed was 15.
– The arithmetic average of those 100 observations is 52.8
• Based on the data, I am going to ask you to go through an exercise
– On deciding how much to order.

• There is no penalty for a wrong answer, or conversely, no extra course credit for the right answer.
• You get one attempt at making your decision.
• The objective of the exercise is not to test or grade you, but to set a baseline initial thinking as we start the course.
• Write down your answer on a sheet of paper and keep the sheet through the course.
• We will see the best answer and you will then get a chance to compare your answers and calibrate learning progress.

The problem you just saw is called a Newsvendor problem.

Its characteristics are:

• You have an objective (usually maximize profits, minimize costs, improve market share, etc.)
• You have to make one decision (usually, how much to buy, or plan for).
• before you see the future demand
• Demand occurs, and profits and costs are realized.

This is called the newsvendor problem:

Because it is similar to a vendor who sells newspapers:

1. Buy too much and you may be left with unsold newspapers,
2. Or buy too little, and you will forgo revenue opportunity.

In this course, we will show you how to think about and analyze this problem

A Business Application at Time Inc.

Time Magazine Supply chain:

Stores were either selling out inventories (too little inventory)

Or sold only a small fraction of allocation (too much inventory).

Time Magazine evaluated and adjusted for every issue:

National print order (total number of copies printed and shipped),

Wholesale allotment structure (How those copies are allotted to wholesalers).

Store distribution (Final distribution to stores).

Note: above three decisions are made before the actual demand is realized

Need to analyze past data

Forecast future demand.

Time Magazine reports saving \$3.5M annually from tackling the newsvendor problem.

Broader applications of the Newsvendor problem

Governments order flu vaccines before the flu season begins, and before the extent or the nature of the flu strain

Is known

1. How many vaccines to order?

Smart phone users buy mobile data plans before they know their actual future usage

1. What is the right plan for you?

Consumers buy health insurance plans, before they know their actual health expenditures

1. How to think about the right plans?

For all the above examples: some forecast of future demand is essential

Introduction to Forecasting

What is forecasting?

• Primary Function is to predict the Future

Why are we interested?

• Dictates the decisions we make today

Examples: who uses forecasting in their jobs?

• forecast demand for products and services
• forecast inventory and capacity needs daily

What makes a good forecast?

• It should be timely, reliable.
• It should be as accurate as possible, and
• It should be in meaningful units
• The method should be easy to use and be understood in practice.

Characteristics of Forecasts

• Point forecasts are usually wrong! Why?

1. Examples: In December 2015, there will be 37cms of snow.

2. We will sell 314 umbrellas during the rains next Part.

3. Demand could be a random variable.

• Therefore, a good forecast should be more than a single number

1. Mean and standard deviation

2. range (high and low) (e.g. weather forecasts).

Modeling Uncertain Future: Probability Distributions

• We often do not control purchasing behavior as a result, we cannot predict future demand with certainty
• How do we describe uncertain future demand?
• We can try to decide what future demand scenarios are possible, for each scenario, estimate the likelihood of its realization
• Where do scenarios come from?
1. Past data
2. Expert estimates

An Example of a Model of Future Demand

• Lets start by looking at a small number of scenarios, say, three: high demand, ordinary demand and low demand.
• Lets say that high demand scenario corresponds to the demand value of 80, ordinary demand scenario to the value of 50, and low demand scenario – to a value of 20
• For each scenario, a likelihood of its occurring must be estimated

Example of a Model of Future Demand: How likely is Each Scenario?

• For each scenario, a likelihood of its coming true must be estimated
• where do estimates of likelihood come from?
1. Statistical analysis of past data
• Suppose that after analyzing the past data and using subjective inputs, we estimate that scenarios have the following likelihoods of being realized in the next selling season:
1. Likelihood of high demand is 20%
2. Likelihood of normal demand is 70%
3. Likelihood of low demand is 10%

Three Scenarios and Probability Distribution

In other words, we project that the demand is not equal to a certain number with probability 1, but, rather can take one of three values with those probabilities

We have just created a probability distribution for the future demand:

1. D1= 80 with probability 1=0.2
2. D2= 50 with probability 2=0.7
3. D3= 20 with probability 3=0.1

Probability distributions like that one, described by a number of distinct scenarios with attached probabilities, are called discrete

Note that the probabilities are
1. greater than zero, and
2. they sum up to 1.

In other words, we project that the demand is not equal to a certain number with probability 1, but, rather can take one of three values with those probabilities

Three Scenarios Probability Distribution: Scenarios and Their Probabilities Describing Probability Distribution: Mean and Standard Deviation

• For any probability distribution, including a simple one reflecting three demand scenarios, two useful descriptive quantities are often calculated: mean (also called expected value) and standard deviation
• For a discrete probability distribution, the mean is defined as a sum of the products of scenario values and their probabilities
• For our demand distribution, the mean D=1D1 + 2D2 + 3D3 = 0.2∗80+0.7∗50+0.1∗20=53.
• Mean reflects the demand value that we will get, on average, in a selling season, if we keep observing the demand realizations over infinite number of selling seasons

Three Scenarios Probability Distribution: Mean Describing Probability Distribution: Mean and Standard Deviation

• Standard deviation describes, roughly speaking, how far away actual random variable values are from the mean, on average. In other words, it describes how, in a colloquial sense, spread out the distribution is around its mean
• Standard deviation is defined as a square root of the sum of products of scenario probabilities and the squares of the difference between scenario value and the mean value
• For example, for the three-scenario demand probability distribution we consider, the standard deviation is calculated as Three Scenarios Probability Distribution: Mean and Standard Deviation

Knowledge of mean and standard deviation values helps to support a general intuition about the nature of a random variable Mean and Standard Deviation: More than three scenarios

• What if we have more than three scenarios?

– D1 with probability 1

– D2with probability 2

– D3with probability 3

————————–

– Dn with probability

And 1+2+3+⋯+=1

• What about mean and standard deviation of this demand distribution for n scenarios? Discrete vs. Continuous Probability Distributions

• So far, we have looked at a discrete probability distributions with a number of future scenarios with attached probability for each scenario
• But what will happen to a discrete probability picture when

a) Random variable being modeled has a really large number of scenarios on any small interval of possible                          interval of values and

b) The probability that any one scenario is realized is really small

• Think of examples such as stock prices, or the amount of rainfall in a region.
• In such cases, it makes sense to describe such probability distribution using groups of scenarios rather than focusing on individual scenarios

Continuous Distribution: Random Variable X Normal Distribution

• One of the most popular examples of a continuous probability distribution is normal distribution

Normal distribution:

• Allows the underlying random variable to take any value from negative infinity to positive infinity, and
• Is completely characterized by two parameters mean and standard deviation s. There exist statistical formulas (also implemented in Excel) that calculate a probability that a normal random variable X with given mean and standard deviation s produces a value within a specified interval of values

[Xmin, Xmax] Other Continuous Probability Distributions

• There exist a large number of other popular continuous probability distribution: exponential, beta, etc. with easily computablemean and variance/standard deviation
• Each of those distributions is often used to describe specific uncertain setting/quantity
• For example, normal distribution is used to describe a distribution of a future relative (percentage) changes in the values of stocks, FX rates
• Another example: exponential distribution can be used in characterizing time between successive arrivals of customers in service systems (e.g. call centers).

Returning back: Characteristics of Forecasts

• Point forecasts are usually wrong! Why?
1. Demand could be a random variable
• Therefore, a good forecast should be more than a single number
• Forecasts should include some distribution information
1. Mean and standard deviation
2. Range (high and low)
• Aggregate forecasts are usually more accurate
• Accuracy of forecasts erodes as we go further into the future
• Don’t exclude known information

Subjective Forecasting Methods

• Composites
1. Sales Force Composites: Aggregation of sales personnel estimates.
2. Election Polling Composites: sites that aggregate polls.
• Customer Surveys
• Jury of Executive Opinion
• The Delphi Method
1. Individual opinions are compiled and reconsidered. Repeat until overall group consensus is (hopefully)               reached.
• We will return to subjective forecasting methods at the end of the Part 1 (Last Session).

How to forecast with past data, objectively?

We can leverage past data to come up with forecasts:

Two primary methods:

1. Causal models

2. Time series methods

Causal Models

– Let D be the demand or future outcome to be predicted and assume that there

– Are n variables (or root causes) that influence the demand.

– A causal model is one which demand D is formulated as a theoretical function of all those n causes.

– Causal models are generally intricate and complex, and need advanced tools in addition to domain expertise.

– In this course, we will focus mainly on time series based models.

Time Series Methods

• A time series is just collection of past values of the variable being predicted.
• Can be considered as a nave method. Goal is to isolate patterns in past data.
• Past data might have characteristics such as:
1. Trend
2. Seasonality/Cycles
3. Randomness

Next….

Forecasting with past historical data
Moving averages
Exponential smoothing