---
# System prepended metadata

title: Short Term Decisions - Week 3
tags: [MGT8803]

---

# Short Term Decisions - Week 3 

## 🧭 Emoji Legend (Note Formatting Guide)
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---

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</details>

## 🧭 Table of Contents
[TOC]

# 🖊️ Video Lessons

## 🧩 Bullwhip Effect - M10L1

**Learning Objectives:**
* Discuss the supply chain phenomena known as the Bullwhip Effect
* Explain the causes of the Bullwhip Effect
* List some solutions to mitigate the Bullwhip Effect

### 🔸 Bullwhip Effect

* Hau Lee investigated the Bullwhip Effect after observing multi-echelon supply chain inventory problems
* First documented the problem with Pampers
* Found similar phenomenon in other industries

### 🔹 Magnification of Orders

![image](https://hackmd.io/_uploads/BkKYLbpTgg.png)

**💡 Note**: An amplification of quantities as we move upstream.

### 🔹 Causes

* Price Fluctuations (placing items on sale)
    * Sell out due to 'artificial' demand
    * Upstream perceives this as 'actual' demand and ramps up production
* Order Batching (Company ordering infrequently and in large amounts)
    * Upstream cannot distinguish change in batch size from change in demand
* Shortage Gaming (Ordering more than they need)
    * Suppliers ration orders
    * Buyers overcompensate to ensure they have product
* Forecast Inaccuracies 

### 🔹 How to Mitigate the Bullwhip Effect

* Increase information sharing of data through the supply chain! (Share data upstream)
* Reduce order costs (Reduces desire to order in larger batches)
* Eliminate discounts and promotions (Reduces 'artificial' demand)

---

## 🧩 Newsvendor Model - M10L2

**Learning Objectives:**
* Discuss the Newsvendor Model

### 🔹 Newsvendor Model Scenario Set-Up (How Many Newspapers Should He Get Today?)

A newsboy would sell newspapers. He buys them for $0.80 and sells them for $1.00. He faces two dilemnas:
* If he buys too few papers, some customers would not be able to purchase a paper and he would loss profits on lost sales
* If he buys too many papers, he carries excess stock and will lose profit

### 🔹 The Newsvendor Framework

* One chance to decide the stocking quantity for the product you are selling
* Demand for the product is uncertain BUT you have a *probability distribution*
* Known marginal profit for each unit sold and known marginal loss for ones that are bought and not sold
* **🎯 Goal:** Maximize ++expected++ profit

**💡 Note:** When you see the work ++expected++ that probably means you will see something to do with probabilities.

### 🔹 Where Is the Newsvendor Framework Used?

* Perishable Goods
    * Meals in a cafeteria
    * Dairy foods
* Short Selling Season
    * Christmas Trees
    * Flowers on Valentines Day
    * Fashion Clothes
    * Newspapers
    * Event Related Goods

### 🔸 Variables and Critical Fractile

**Underage Cost** = opportunity cost of underestimating demand

$$
c_u = \text{Underage Cost} = p - c \:\: \text{or retail price minus cost}
$$

**Overage Cost** = cost of overestimating demand

$$
c_o = \text{Overage cost} = c - s \:\: \text{or cost minus salvage value}
$$

**c:** Cost of each item
**p:** Retail selling price for each item
**s:** Salvage value for unsold items
**x:** Number of items you buy
**P(x):** Probability that the xth item item is not sold 

**💡 Note:** As **x** grows, so will the **P(x)**

* The likelihood of selling $x=1$ papers is HIGH, so **P(x)** is very low.
* The likelihood of selling $x=1000$ papers is LOW, so **P(x)** is very high.

**Critical Fractile:**
$Q = \text{Quantity Ordered}$
$D = \text{Demand}$

$$
F(Q) = P(D≤Q)≤\frac{c_u}{c_u+c_o}
$$

### 🔹 Normal Distribution of Demand

![image](https://hackmd.io/_uploads/ry_0uf6Txg.png)

We want to choose a quantity (Q) where the probability of not selling is equal to our Critical Fractile. This will maximize our expected profit.

![image](https://hackmd.io/_uploads/HkCHKzppxx.png)

$G(z)$ is called the Cumulative Distribution Function: Allows us to calculate area under the curve at any point. 

**💡 Note:** To find $z$ using Excel: `Norm.S.Inv('Critical Fractile')`

---

## 🧩 Newsvendor Model Example - M10L3

**Learning Objective:**
* Use the Critical Fractile and Newsvendor Model in an example problem

### 🔸 Merchandise Buyer

You work for the University Bookstore:
* Georgia Tech Football is playing in the Sun Bowl
* Need to decide on a T-Shirt Order from supplier
    * Order too many and the costs go up
    * Order too few and you will miss out on sales

This is an example of a product with a limited selling season. 

**Data**

* Vendor charges $6.50 per shirt
* You sell the T-Shirts for $8.95 per shirt
* After the bowl game, no demand but Big Lots will buy any leftover shirt for $1.00 per shirt
* From past orders, you estimate demand to be normally distributed with a mean ($µ$) of 20,000 T-Shirts and a Standard Deviation ($σ$) of 1,000.

How many T-Shirts should you order?

**Solution**

p = $8.95, c = $6.50, s = $1.00
$c_u = 8.95 - 6.50 = 2.45$
$c_o = 6.50 - 1.00 = 5.50$
$µ=20,000 \;, σ=1,000$

$$
F(Q) = P(D≤Q)=\frac{c_u}{c_u+c_o} → \frac{2.45}{2.45+5.50} = .308
$$

Given $F(Q)=.308$, find that value on a z-table or use `Norm.S.Inv(.308)`: $z=-0.5$

$$
Q = µ \: + \: zσ = 20,000 - (0.5)(1,000) = 19,500 \: \text{T-Shirts}
$$

**💡Note:** Since $c_o > c_u$ our Quantity ($Q$) is less than the mean. If $c_o < c_u$ then our $Q$ would be higher than our mean.

---

## 🧩 Forecasting 1 - M10L4

**Learning Objectives:**
* Discuss Forecasting in the context of supply chain managements
* Discuss patterns of demand
* Identify qualitative and quantitative forecasting methods

### 🔸 What is Forecasting?

++Forecasting++: prediction of future events used for planning purposes

Used for:
* Strategic Planning
* Finance and Accounting
* Marketing
* Production and Operations

**General Characteristics:**
* Forecasts are almost always wrong!
* Forecasts are more accurate from groups or families of items
* Forecasts are more accurate for shorter periods of time
* Every forecast should include an error estimate
* Forecasts are no substitute for actual demand

### 🔹 Patterns of Demand

**Trends**

![image](https://hackmd.io/_uploads/B1e8DBaTxg.png)

**Seasonality**

![image](https://hackmd.io/_uploads/HkZDwSp6xl.png)

**Cylical Elements**

![image](https://hackmd.io/_uploads/ryYDPSTaeg.png)

**Autocorrelation**

![image](https://hackmd.io/_uploads/SkZ_PSppxl.png)

**Random Variation**

![image](https://hackmd.io/_uploads/HyYdPHaaxl.png)

**Data Can Exhibit Multiple Patterns**

![image](https://hackmd.io/_uploads/SyYoDH6Txx.png)

### 🔸 Types of Forecasting Methods

**Qualitative**
Rely on subjective opinions from one or more experts. More long-term.

**Quantitative**
Rely on data and analytical techniques. More medium to short-term.

Some Qualitatative Methods include:
* **Time Series**: models that predict future demand based on past history trends
* **Causal Relationships**: models that use statistical techniques to establish relationships between various items and demand (ex: linear regression)
* **Simulation**: models that can incorporate some randomness and non-linear effects

---

## 🧩 Forecasting 2: Simple Moving Average - M10L5

**Learning Objectives:**
* Explain time series forecasting methods
* Outline simple moving average

### 🔸 Time Series: Simple Moving Average

$$
F_{t+1} = \frac{A_{t} + A_{t+1} + ... + A_{t-n+1}}{n}
$$

**$t$**: current period
**$F_{t+1}$**: forecast for next period
**$n$**: number of periods
**$A$**: actual demand for a given period

#### 🔹 Example: 4-Period Simple Moving Average

![image](https://hackmd.io/_uploads/rkUdqBppxl.png)

What is the forecast for week 13 using a 4-period Simple Moving Average?

$$
F_{13} = \frac{A_{12} + A_{11} + A_{10} + A_{9}}{4} → \frac{844 + 789 + 920 + 892}{4} = 861.25
$$

---

## 🧩 Forecasting 2: Weighted Moving Average - M10L6

**Learning Objectives:**
* Outline weighted moving average

### 🔸 Time Series: Weighted Moving Average

$$
\begin{align}
F_{t+1} &= w_{1}A_{t} + w_{t-1}A_{t-1} + ... + w_{t-n}A_{t-n}
&w_{t} + w_{t-1} + ... + w_{t-n}=1
\end{align}
$$

**$t$**: current period
**$F_{t+1}$**: forecast for next period
**$n$**: number of periods
**$A$**: actual demand for a given period
**$w$**: importance (weight for a given period)

#### 🔹 Example: 3-Period Weighted Moving Average

![image](https://hackmd.io/_uploads/Syy-2S6Tge.png)

What is the forecast for week 10 using a 3-period Weighted Moving Average with weights 0.7, 0.2 and 0.1?

$$
F_{10} = 0.7(A_{9}) + 0.2(A_{8}) + 0.1(A_{7}) → 0.7(892) + 0.2(758) + 0.1(850) = 861
$$

### 🔹 Why do we need WMA Models?

Because of the ability to gie more importance to more recent data without losing the impact of the past.

![image](https://hackmd.io/_uploads/rJHhhSTpeg.png)

### 🔹 How do We Choose the Weights?

* Trial and Error
* Depends upon:
    * importance we feel past data has on future data
    * known seasonality

**💡 Note:** Simple Moving Average is actually Weighted Moving Average where all of the weights are the same.

![image](https://hackmd.io/_uploads/ryrzaSpTlg.png)

---

## 🧩 Forecasting 3: Exponential Smoothing - M10L7

**Learning Objectives:**
* Outline exponential smoothing

### 🔸 Time Series: Exponential Smoothing

The prediction of the future depends mostly on the most recent observation and on the error for the latest forecast.

Assume that we are currently in period $t$. We calculated the forecast for the last period $F_{t-1}$ and we know the actual demand from last period $A_{t-1}$.

$$
F_{t+1} = F_{t} + α(A_{t} - F_{t}), \: \text{where} \: 0≤α≤1
$$

The smoothing constant $α$ expresses how much our forecast will react to observed differences:
* if $α$ is **low**, little reaction to difference
* if $α$ is **high**, large reaction to difference

#### 🔹 Example

![image](https://hackmd.io/_uploads/ByYbk8ppgg.png)

![image](https://hackmd.io/_uploads/rkn718aagx.png)

### 🔹Why Exponential Smoothing?

* Uses less storage space for data - not a modern problem
* Extremely accurate
* Easy to understand
* Little calculation complexity

---

## 🧩 Forecasting 4: Error Measurements - M10L8

**Learning Objectives:**
* Explain the use of errors to evaluate accuracy of time series methods

### 🔹 How can we compare/evaluate different methods?

++Bias++: when a consistent mistake is made.
++Random++: errors that are not explained by the model being used.

### 🔸 Measure of Forecast Accuracy

$$
E_{t}=A_{t}-F_{t}
$$

**$E_{t}$**: error, can be positive or negative
* Positive means forecast was too low ($F_{t} < A_{t}$)
* Negative means forecast was too high ($F_{t} > A_{t}$)

### 🔸 Measures of Forecast Error

**RSFE - Running Sum of Forecast Error**

$$
RSFE = \sum (A_{i} - F_{i})
$$

**MFE - Mean Forecast Error (Bias)**

$$
MFE = \frac{\sum (A_{i}-F_{i})}{N} → \frac{RSFE}{N}
$$

**MAD - Mean Absolute Deviation**

$$
MAD = \frac{\sum |A_{i}-F_{i}|}{N}
$$

**TS - Tracking Signal (Bias)**
* Measure of how often our estimates have been above or below the actual value
* Used to decide when to re-evaluate the model

$$
TS = \frac{RSFE}{MAD}
$$

* Positive $TS$ - Generally, actuals > forecasted
* Negative $TS$ - Generally, actuals < forecasted
* <font color="#f00"> if $-4>TS$ ++OR++ $TS>4$ </font>

#### 🔹 Example

![image](https://hackmd.io/_uploads/H1blQUTTxe.png)

$$
RSFE = \sum (A_{i} - F_{i}) = 10,200 - 10,000 = 200
$$

$$
\text{Bias} = \text{MFE} = \frac{\sum (A_{i} - F_{i})}{N} = \frac{200}{10} = 20
$$

$$
\text{Mean Absolute Deviation (MAD)} = \frac{\sum |A_{i} - F_{i}|}{N} = \frac{1600}{10} = 160
$$

$$
TS = \frac{RSFE}{MAD} = \frac{200}{160} = 1.25
$$

---

## 🧩 Forecasting Example Problem - M10L9

**Learning Objectives:**
* Use Simple Moving Average, Weighted Moving Average and Exponential Smoothing to forecast a series of demand values
* Assess each method based on a variety of error values

### Harry's Hardware Example

![image](https://hackmd.io/_uploads/ByVAUUTpxl.png)

**Evaluate the following models:**

1. SMA (2 period, 3 period, 4 period)

2. WMA (0.5, 0.3, 0.2 and 0.7, 0.2, 0.1)

3. ES (α = 0.9 and α = 0.2, where $F_{1}$ = 289)

**Using these Error Methods:**

1. RSFE

2. MFE

3. MAD

4. TS

![image](https://hackmd.io/_uploads/B17avI66lg.png)

---

## 🧩 Trends - M10L10

**Learning Objectives:**
* Discuss technology trends that will impact the supply chain of the future

### 🔹 Techonology

* 3D Printers
* Artificial Intelligence
* Robotics
* Autonomous Vehicles

---

# ❓ Self-Assessment Questions

1. Where would the newsvendor model be least useful?

items with a long selling season

2. What is the critical fractile for a newsboy who buys papers for $.75 and sells them on the street corner for $1.25. He has a standing agreement with the local animal boarding facility to sell papers for $.10 each.

$$
F(Q) = P(D≤Q) = \frac{1.25-0.75}{1.25-0.75+0.75-0.10} = 0.434
$$

3. Consider the following scenario for the ABC Widget Integrated Supply Chain: a multi-tiered supply chain consisting of a Wholesaler (W), a Retailer ( R), and a Manufacturer (M). The Wholesaler  supplies a product to a Retailer ( R), which, in turn, sells the product to end Customers ( C). C pays R $11.50 per unit. R pays W $4.50 per unit. W pays the Manufacturer (M) $2.25 per unit. The costs to M are $1.00/unit. Note that ABC Widget sales are seasonal and unsold widgets have no retail value after the end of the season.
The optimal service level for the integrated supply chain is:

$$
F(Q) = P(D≤Q) = \frac{11.50-1.00}{11.50-1.00+1.00-0.00} = 0.91
$$

4. Continuing with the ABC Widget Integrated Supply Chain scenario. What is R's optimal service level? 

$$
F(Q) = P(D≤Q) = \frac{11.50-4.50}{11.50-4.50+4.50-0.00} = 0.61
$$

5. Continuing with the Widget Integrated Supply Chain described in the question above. R finds a scrap dealer that will buy all unsold widgets for $.50. R’s new optimal service level will be?

$$
F(Q) = P(D≤Q) = \frac{11.50-4.50}{11.50-4.50+4.50-0.50} = 0.64
$$

6. What happens to the critical fractile if the salvage value for the item increases?

The critical fractile gets larger

7.  (True/False). In the newsvendor problem, the optimal order quantity depends on the relative cost of stocking too much and stocking too little. 

True.

8. (True/False). Consider a two-tier supply chain consisting of a Wholesaler (W) supplying a product to a Retailer ( R) which, in turn, sells the product to end Customers ( C). C pays R $2.50 per unit. R pays W$1.25 per unit. The unit cost to W is $0.75 per unit. (True/False). If W agrees to buy back unsold product from R at $.25 per unit, R’s optimal service level will increase. 

True.

9. The Dollar Store stocks Cadbury Chocolate Easter Eggs for a limited time each Spring. Of course the store only charges a dollar for each egg sold during the Easter season (since it is a dollar store). At the end of the season the remaining egg inventory is given to a children’s charity (assume no financial benefit). The Dollar Store advertises and is committed to a 95% service level on all products sold. What is the implied overage cost target The Dollar Store must hit that justifies the 95% fill rate for the Cadbury Eggs. (Assume that Cadbury does not buyback any unsold product). Choose the closest answer.

$$
0.95 = \frac{1.00-0.05}{1.00-0.05+0.05-0.00}
$$

10. Jimmy sells a highly perishable product to dock workers at the Port of Miami. The product costs Jimmy $4.20 wholesale and he sells the product for $10. Ignore sales tax, Jimmy doesn’t pay it anyway. When Jimmy runs out of product he simply takes orders from the dock workers, batches them up and runs over to a local retail store and buys the exact amount of product needed to fill the orders. At the retail store Jimmy pay $8.90 for each product (instead of the $4.20 wholesale price he paid at the beginning of the day). Jimmy feels the need to do this to maintain high levels of customer service. If Jimmy instead of being under, is over, he has to discard the unsold product.  Jimmy believes that his demand follows a normal distribution with a mean of 75 and a standard deviation of 16. Considering this information, what is Jimmy’s optimum service level? (Round up to next highest full percent point)

**$c_{u}=(p-c_{initial}) - (p-c_{restock})$**
**$c_{u-initial} = 10.00-4.20 = 5.80$** and **$c_{u-restock}=10.00-8.90 = 1.10$**
**$c_{u} = 5.80-1.10 = 4.70$**
**$c_{o} = 4.20 - 0.00$**

$$
F(Q) = P(D≤Q) = \frac{4.70}{4.70+4.20} = 0.528
$$

11. Bob sells a highly perishable food product to construction workers at 777 Peachtree, a new luxury high rise under-construction in mid-town Atlanta.  The product costs Bob $4.50 wholesale and he sells the product for $11.00 to his construction worker clientele. Ignore sales tax, Bob doesn’t pay it anyway. When Bob runs out of product he simply takes orders from his remaining customers, walks to the Varsity Drive In , and buys the exact amount of product needed to fill the orders. Bob charges each customer exactly $1 more per order than his costs,  which will be variable based on the exact food orders placed by the individual workers. Bob only provides this "courier" service if he runs out of his own product and performs the service to maintain goodwill with his customers.  If Bob instead of being under (on his own product), is over, he has to discard the unsold product.  Bob believes that his demand follows a normal distribution with a mean of 175 and a standard deviation of 32. Considering this information, what is Bob's optimum service level? (Round up to next highest full percent point).

**$p_{initial}$** = 11.00,  **$p_{restock}$** = 1.00, **$c$** = 4.50
**$c_{u} = (11.00-4.50) - 1.00 = 5.50$**
**$c_{o} = 4.50-0.00 = 4.50$**

$$
F(Q) = P(D≤Q) = \frac{5.50}{5.50+4.50} = 0.55
$$

:::info
**🧠 Insight:** $c_{u}=p-c$ is just another way of saying profit. 
$c_{u} = \text{profit if I have the item}-\text{profit if I don't have the item}$
:::

12. Which of the following is not a cause of the bullwhip effect?

small batch sizes

13. A(n) _________ is a prediction of future events that is used for planning purposes.

forecast

14. Consider both statements: Statement 1. Forecasts are almost always wrong. Statement 2.  Demand can show random variation over time

Both statements are True

15. What types of patterns can Demand exhibit? (choose all that apply)

trends
randomness
autocorrelation
seasonality

16. Consider the following data to forecast sales for 2019 using a 3 period simple moving average.

![image](https://hackmd.io/_uploads/B1mFs_6pee.png)

$$
F_{2019} = \frac{518+563+584}{3} = 555
$$

17. Consider the following sales data. Forecast the sales for 2017 using a 3 period moving average.

$$
F_{2017} = \frac{450+495+518}{3} = 487.66
$$

18. Consider the following data to forecast sales for 2019 using a 2 period simple moving average.

$$
F_{2019} = \frac{563+584}{2} = 573.5
$$

19. Consider the following data to forecast sales for 2019 using a 3 period weighted moving average with weights of .15, .25, and .60. (NOTE that the .15 weight applied to 2018)

$$
F_{2019} = 0.15(584)+0.25(563)+0.60(518) = 539.15
$$

20. Consider the following data to forecast sales for 2019 using exponential smoothing with an $α = 0.4$.

![image](https://hackmd.io/_uploads/HykoaO6pge.png)

$$
F_{2018} = 488 + 0.4(563-488) = 518
$$

$$
F_{2019} = 518 + 0.4(584-518) = 544.4
$$

21. Consider the following data to forecast sales for 2019 using exponential smoothing with an $α = 0.8$.

$$
F_{2018} = 488 + 0.8(563-488) = 548
$$

$$
F_{2019} = 548 + 0.8(584-548) = 576.8
$$

22. Given the following data, what is the Tracking Signal and what is your assessment of the value?

![image](https://hackmd.io/_uploads/Hy1AC_6Tel.png)

$$
RSFE = \sum(A-F) = 0
$$

$$
MAD = \frac{\sum |A-F|}{N} = 2.5
$$

$$
RS = \frac{RSFE}{MAD} = \frac{0}{2.5} = 0
$$

23. Given the following data, what is the Mean Forecast Error?

$$
MFE = \frac{RSFE}{N} = 0
$$

24. What is the Mean Absolute Deviation?

$$
MAD = \frac{\sum |A-F|}{N} = \frac{10}{4} =2.5
$$

25. Which best describes the bullwhip effect?

The increasing fluctuations in demand as one moves upstream in the supply chain.

26. (True/False). Trial and error can be used to determine the weights for a weighted moving average forecast. 

True.

27. Bills Bookstore places orders Wall Calendars once a year, approximately 6 months before the holiday season. Bill best estimate of demand is as follows:
![image](https://hackmd.io/_uploads/Hk3ueKp6lx.png)
Bills calendar supplier sells calendars in lots of 10. Each lot costs $27, but calendars are sold individually to customers at $5.79 (each). Calendars unsold after the new year starts are sold to a discounter for $.50 each. How many calendars should Bill order if he intends to deliver a service level at least equal to the optimum service level?
NOTE: To solve this problem use the logic of the Newsvendor model but substitute a discrete distribution for a normal distribution.

**Approach 1:**

$$
F(Q) = P(D≤Q) = \frac{5.79-2.70}{5.79-2.70+(2.70-0.50)} = 0.584
$$

| Demand | Probability | Midpoint(x) |
| -------- | -------- | -------- |
| 0-10 | 0.30 | 5 |
| 11-20 | 0.25 | 15.5 |
| 21-30 | 0.20 | 25.5 |
| 31-40 | 0.15 | 35.5 |
| 41-50 | 0.10 | 45.5 |

$$
µ = 0.30(5) + 0.25(15.5) + 0.20(25.5) + 0.15(35.5) + 0.10(45.5) = 20.35
$$
$$
σ = \sqrt{0.30(5-20.35)^2 + 0.25(15.5-20.35)^2 + 0.20(25.5-20.35)^2 + 0.15(35.5-20.35)^2 + 0.10(45.5-20.35)^2} = \sqrt{179.9} = 13.4
$$
$$
Q = 20.35 + (0.21)13.4 = 23.164
$$

**Approach 2:**

Given $F(Q) = P(D≤Q) = 0.584$, Cumulative Probability ($0.30 + 0.20 = 0.55$) means that we should order 30 calendars. Smallest $Q$ where $P(D≤Q)≥0.584$.

# Synchronous Class (10/15/2025)

Myers said that some of the example problems will be on the exam.

## 🧩 The Newsvendor Model

For perishable goods and seasonal/time-related items (Newspaper).

**$\text{Cost of Being Over} = c-s$** where $c$ is the cost of item and $s$ is the salvage value (marginal loss)
**$\text{Cost of Being Under} = p - c$** where $p$ is the retail price of each item (marginal profit)

**Critical Fractile → Critical Ratio → Service Level**
Equals the service level where expected profit is maximized.
$$
F(Q) = P(D≤Q) = \frac{c_{u}}{c_{u}+c_{o}} → \frac{p-c}{p-s} ,\;\;\;\; Q=µ + z(σ)
$$

### 🔹 Sample News Vendor Problem

**Question 1**

Pat sells papers. He sells $1.75 and pay $0.98 per paper. No salvage value.

$c_{u}=1.75-0.98=0.77$ and $c_{o}=0.98-0.00=0.98$

$$
F(Q) = P(D≤Q) = \frac{0.77}{0.98+0.77} = 0.44
$$

**Question 1, Part 2**

If Pat ++runs out++, he can buy more papers at $1.50 and sell at $1.75. Find the Critical Fractile.

$c_{u}= (1.75-0.98)-(1.75-1.50) = 0.52$ while $c_{o}$ remains the same.

$$
F(Q) = P(D≤Q) = \frac{0.52}{0.98+0.52} = 0.3467
$$

How many should Pat order? Given $µ = 110$ and $σ=17$.

$$
Q=µ + z(σ) → 110 + (-0.39*17)=103.37 → 104 \:\text{newspapers}
$$

### 🔹 The Newsvendor Model in an Integrated Supply Chain

:::warning
This was going to require a lot of notes for something that I was familiar with how to do. Lazy? Yes, I know.
:::

