# The Newsvendor Model

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The Newsvendor Model

In this lesson, you’re expected to understand how a mathematical model can be used in operations management to determine optimal inventory levels.
The newsvendor type of problem occurs in situations in which you plan a single-period inventory, to meet uncertain demandwith a perishable product.

Suppose that you have just started a business that serves an uncertain demand, and the product is highly perishable, such as holiday trees (which are supposed to be used on 24th and 25th of December, according to Western tradition).

You start by ordering your initial amount of trees from your supplier. You plan to sell the trees at full price, e.g. \$100.

Then, two scenarios may occur:

1) You order too many: Then, by the end of the period you have many leftover, which you may still sell at a marked-down price, e.g. \$5

2) You order too few: Then, before the end of the period, you are stocked-out, missing many sales opportunities.

This represents a common problem in many businesses, such as a restaurant, high-tech equipment, fashion, newspapers etc.

Assumptions of the Newsvendor Problem

• It is a one-time decision.

• Leftover items from the previous seasons are not used to satisfy demand for the current season.

Overview of the Newsvendor Problem

• There is a short selling season with a well-defined beginning and end.
• Demand during the season is uncertain.
• Buyers or producers have to determine how much to order or produce prior to the start of the selling season.
• If you order too much:
– inventory is left over at the end of the period.
– sometimes it is possible to salvage something (e.g. mark-down and sale at a loss).
• If you order too little:
– not all demand is served
– losing out on revenue
• When total demand in the season exceeds the stock made available, there are associated underage costs.
• When total demand in the season is less than the stock made available, there are associated overage costs.
[Optional] What is the Newsvendor Model?
Newsvendor Model Setup: Expected Profits

In the newsvendor model, we want to find the order quantity that maximizes total profits.

Enlarged version: http://bit.ly/2pVKqry
Notation
Costs of ordering too much and too little

The newsvendor model can be thought of simply in terms of costs of overage and underage.

1) Co: Cost of Overage
– The cost of ordering too much. That is, the consequence of ordering one more unit than what you would have ordered if you had known the demand. For example: suppose you had left over inventory (you over-ordered). Co is the increase in profit you would have enjoyed had you ordered one fewer unit.

– In this case, the cost of overage is simply the acquisition cost minus any eventual salvage value: Co = c – s

2) Cu: Cost of Underage

– The cost of ordering too little. That is, the consequence of ordering one fewer unit than what you would have ordered if you had known demand. For example: suppose you had lost sales (you stocked-out). Cu is the increase in profit you would have enjoyed had you ordered one more unit.

– In this case, the cost of underage is simply the selling price minus the acquisition cost, plus the goodwill that you would have avoided losing: Cu = p – c + g

Balancing the risk and benefit of ordering a unit

As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases.

Ordering one more unit increases the chance of overage:
Expected loss on the Qth unit = Co x F(Q)
– where F(Q) is the distribution function of demand, i.e. the probability that demand is less than or equal to Q.

But the benefit / gain of ordering one more unit is the reduction in the chance of underage:
Expected gain on the Qth unit = Cu x [1 – F(Q)]

To maximize expected profit, we order Q units so that the expected loss on the Qth unit equals its expected gain, which is given by:

Co x F(Q) = Cu x [1 – F(Q)]

Rearrange the terms in the above equation to obtain:

* The ratio Cu / Cu + Co is called the critical ratio.
Example of Newsvendor Problem with Normally Distributed Demand

Problem Setup
The local newsstand purchases copies of the weekly paper Extraat the start of every week. The observed sales (demands) during each of the last 52 weeks are given in the figure below.

Demand: Normal Distribution, Mean (μ) = 101, Standard Deviation (σ) = 18

p = 1, c = 0.5, s = 0.05, g= 0.15

Solution

Co = c – s = 0.5 – 0.05 = 0.45
Cu = p – c + g = 1 – 0.5 + 0.15 = 0.65
Critical Ratio: F(Q*) = 0.59

We must find quantity Q* in the demand distribution function that makes F(Q) = 0.59.

Since this is a normal distribution, we look at the cumulative normal distribution table to find z-score = 0.59, which is 0.23.

Then we find our quantity Q* for a normal distribution,
Q* = μ + Zσ
= 101 + (0.23 x 18)
= 105.14

[Optional] Newsvendor Model using Excel