# Newsvendor (newsboy) Model: A Brief Introduction

Essay by Raghava Bar • October 31, 2018 • Research Paper • 2,213 Words (9 Pages) • 790 Views

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Newsvendor (Newsboy) Model: A Brief Introduction

From sweatshirts (in the EOQ model introduction) to summer dresses. The big difference is that while sweatshirts were continuous selling items, the demand for summer dresses is limited to summer months. Once the summer season is over, the unsold dresses must be heavily discounted. You are a local design firm that designs northwest-accented summer dresses, sources them from China and sells them through retailers here.

The problem is that for a particular summer dress, total demand during the summer season is hard to predict. All you can do is to make a guess, that is, develop a probability distribution of demand. Let us generate our demand with the throw of a regular dice; it can be any number from 1 to 6, each with probability 1/6.

On the supply side, the lead time from your Chinese supplier is long. There is no possibility of making multiple orders. You make one order before the summer season starts, sell as many as you can during the season and then whatever is left is discounted. Let us say that per unit purchase cost c is $80. For any units that you are able to sell per unit revenue r is $100. For the units you are not able to sell during the season, let us say that you can discount them and are able to sell them at a per unit salvage value s of $30.

The big decision is the order quantity S of dresses you should order from your Chinese supplier at the beginning of the summer season.

The Trade-off

If you order a very large quantity, there is a bigger chance that you will not be able to sell all of them. There will be excess units at the end of the season that you will have to discount. You will lose money on them. On the other hand, if you order a small quantity, there is a bigger chance that you will be short. That is, there will be some demand you will not be able to satisfy. You will not be able to make as much money as you could have.

The order quantity decision resolves this trade-off between the expected cost of having excess inventory and the expected cost of falling short. We will call the sum of these two costs as Mismatch cost. The optimal order quantity will minimize mismatch cost.

Marginal Costs

To resolve this trade-off, we start with defining marginal costs of excess and shortage.

Marginal Cost of excess Ce is defined as the cost of having one unit excess. You bought this unit for purchase cost of c=$80, were not able to sell it during the season and then had to discount it down to the salvage value of s=$30. The cost to you is $80-$30=$50. That is, in this setting, Ce=c-s.

Marginal Cost of shortage Cs is defined as the cost of having one unit short. Had you bought this unit for purchase cost of c=$80, you would have been able to sell it during the season for a revenue of r=$100. We say, that the cost to you for being one unit short is $100-$80=$20. That is, in this setting, Cs=r-c.

Service Level

Service level is the chance that you will be able to meet all the demand in a single period (summer season). Suppose you bought an order quantity S=3 units. Recall that demand is any number between 1 and 6 with equal probability 1/6. In this case, you will be able to meet all the demand only if demand is either 1 unit, 2 units or 3 units. That is, the probability that demand is less than or equal to S=3 units. This probability is known as cumulative probability and is given by the sum of the probabilities that demand is 1, demand is 2, and demand is 3 = (1/6)+(1/6)+(1/6)=3/6=1/2. That is, if you buy S=3 units, you will provide a service level of 50%.

Here is a quick table to provide cumulative probabilities in our case:

Demand | 1 | 2 | 3 | 4 | 5 | 6 |

Probability | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

Cumulative Probability | 1/6 | 1/6+1/6 =2/6 | 2/6+1/6 =3/6 | 3/6+1/6 =4/6 | 4/6+1/6 =5/6 | 5/6+1/6 =6/6=1 |

Optimal Service Level and Optimal Order Quantity

Single-period model tells us that, given the marginal costs of excess and shortage, Ce and Cs, the optimal service level is given by (Cs/(Cs+Ce). In our case, the optimal service level is equal to 20/(20+50)= 0.2857.

Optimal order quantity S* is the minimum size of the order that will be able to provide the optimal service level. Going by the above table, if you buy, for example, S=1, you will be able to provide a service level of 1/6=0.1667 which is less than the optimal service level we wish to provide. If we buy 2, service level is 2/6=0.3333 and we will be able to satisfy the optimal service level requirement of 0.2857. Therefore S*=2.

Rule: compute optimal service level and find the minimum value of demand for which cumulative probability, for the first time, equals or exceeds optimal service level. That is the optimal order quantity.

Summary of Formulas for Continuous Demand: Normal Distribution

The demand distribution we considered above is a discrete distribution because demand can only take a limited number of values. In some real settings, it is easier to work with the assumption that the demand follows Normal distribution with a given mean and standard deviation. Normal is a continuous distribution because demand can take any value. In this case, we can use the following formulas:

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