The Newsvendor Model
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.
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.
• Leftover items from the previous seasons are not used to satisfy demand for the current season.
• 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.
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
– 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
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)]
Co x F(Q) = Cu x [1 – F(Q)]
Rearrange the terms in the above equation to obtain:
Demand: Normal Distribution, Mean (μ) = 101, Standard Deviation (σ) = 18
p = 1, c = 0.5, s = 0.05, g= 0.15
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)