Newsvendor Model

The newsvendor (or newsboy or single-period or perishable) model is a mathematical model in operations management and applied economics used to determine optimal inventory levels. It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is q, each unit of demand above q is lost in potential sales. This model is also known as the Newsvendor Problem or Newsboy Problem by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day’s paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

These are called Newsvendor Problems because they mimic the dilemma faced by the manager of a newsstand stocking newspapers each morning. They are quite prevalent in the stocking of perishable goods, such as milk and baked goods, which must be discarded after their expiration dates. They also apply to seasonal products, such as Christmas lights and Easter eggs, which tend to be salvaged at the end of the season. Fashion goods, such as designer clothes, also exhibit Newsvendor economics

The newsvendor problem is a one-time business decision that occurs in many different business contexts such as:

• Buying seasonal goods for a retailer – Retailers have to buy seasonal goods (sometimes called style goods) once per season. (A “season” can be a day, week, year, etc.) For example, most swimsuits can only be purchased seasonally. If a buyer orders too few swimsuits, the retailer will have lost sales and dissatisfied customers. If the buyer orders too many swimsuits, the retailer will have to sell them at a clearance price or even throw some away.
• Making the last buy or last production run decision – Manufacturers have to make a last buy (or last production run) for a product (or component) that is near the end of its life cycle. If the order size is too small, the firm will have stockouts and disappointed customers. If the order size is too large, the firm will only be able to sell the items for their salvage value.
• Setting safety stock levels – A distributor has to set the safety stock level for an item. If the safety stock is too low, stockouts will occur. If safety stock is too high, the firm has too much carrying cost. Nearly all safety stock models are newsvendor problems with the selling season being one order cycle or one review period.
• Setting target inventory levels – A salesperson carries inventor y in the trunk of a vehicle. The inventory is controlled by a target inventory level. If the target is too low, stockouts will occur. If the target is too high, the salesperson will have too much carrying cost.
• Selecting the right capacity for a facility or machine – If the capacity of a factory or a machine over the planning horizon is set too low, stockouts will occur. If capacity is set too high, the capital costs will be too high.
• Overbooking customers – If an airline overbooks too many passengers, it incurs the cost of giving away free tickets to inconvenienced passengers. If the airline does not overbook enough seats, it incurs an opportunity cost of lost revenue from flying with empty seats.

All of these newsvendor problem contexts share a common mathematical structure with the following four elements:

• A decision variable ( Q) – The newsvendor problem is to find the optimal Q for a one-time decision, where Q is the decision quantity (order quantity, safety stock level, overbooking level, etc.). Q* denotes the optimal (best) value for Q.
• Uncertain demand (D) – Demand is a random variable defined by the demand distribution (e.g., normal distribution, Poisson distribution, etc.) and estimates of the distribution parameters (e.g., mean, standard deviation). Demand may be either discrete (integer) or continuous. This paper develops the newsvendor models for both cases.
• Unit overage cost (co) – This is the cost of buying one un it more than the demand during the one-period selling season. In the standard retail context, the overage cost is the unit cost ( c ) less the unit salvage value ( s ), i.e., co = c – s . The salvage value is the salvage revenue less the salvage cost required to dispose of the unsold product.
• Unit underage cost (cu) – This is the cost of buying one unit less than the demand during the one-period selling season. This is also known as the stockout (or shortage) cost. In the retail context, the underage cost is computed as the lost contribution to profit, which is the unit price ( p) less the unit cost ( c ), i.e., cu = p – c . The lost customer goodwill ( g) associated with a lost sale can also be included (i.e., cu = p – c + g). However, it is difficult to estimate the g parameter because it is the net present value of all future lost profit from this customer and all other customers affected by this customer’s negative reports (negative “word of mouth”).

Since co and cu are both cost parameters, taxes should be considered for both or neither. Given that the newsvendor problem is in a single period, cash flows do not need to be discounted.

Discrete Demand Model

When demand only takes on integer (whole number) values, it is said to be “discrete.” With order quantity Q and specific demand D, the cost for the one-period selling season is

For discrete demand, the demand distribution is defined by the probability mass function p( D). The equation for the expected cost, therefore, is given by:

The first term in equation above is the expected overage (scrap) cost and the second term is the expected underage (shortage) cost. The optimal order quantity Q* can be found at the Q value where the expected cost function is flat. This is where the expected costs for Q and Q+1 units are approximately equal (i.e., ECost(Q) = ECost(Q+1) ). Therefore, Q* is the smallest value of Q such that the following relationship holds true:

The value R is called the “critical ratio” or “critical fractile” and is always between zero and one. The optimal Q is denoted as Q* and can be found with a simple search procedure starting at Q = 1 and increasing Q until the above relationship is satisfied. When cu = co, the critical ratio is R = 0.5, which is consistent with the intuition that suggests that Q* should be equal to the median demand when the costs are equal.

Continuous Demand Model

As with the discrete demand case, the cost for order quantity Q and specific demand D is

We assume that demand (D) is a continuous random variable with density function f(D) and cumulative distribution function F(D). The expected cost function is given by

This equation is analogous to equation above for the discrete demand problem. In order to find the optimal Q, we take the derivative of the expected cost function and set it to zero to find

Testing the second derivative proves that Q* is a global optimum.