Newsvendor model

Newsvendor model

The newsvendor (or newsboy or single-period[1]) 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. 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.

Contents

Profit function

The standard newsvendor profit function is

\pi =E\left[p\min (q,D)\right]-cq

where D is a random variable with probability distribution F representing demand, each unit is sold for price p and purchased for price c, and E is the expectation operator. The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:

q=F^{-1}\left( \frac{p-c}{p}\right)

where F − 1 denotes the inverse cumulative distribution function of D.

Intuitively, this ratio, referred to as the critical fractile, balances the cost of being understocked (a lost sale worth (pc)) and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or c so total cost is simply p).

Numerical example

Assume that: retail price is p = 7 [$/unit] and purchase price is c = 5 [$/unit]. Furthermore the D demand follows a uniform distribution (continuous) between Dmin  = 50 and Dmax  = 80.

q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=F^{-1}\left( 0.285 \right) = D_\min+(D_\max-D_\min) \cdot 0.285 = 58.55\approx59.

Therefore optimal inventory level is approximately 59 units.

Extreme situation

If p < c (i.e. the retail price is less than the purchase price), the numerator becomes negative. In this situation, it isn't worth keeping any item in the inventory.

Cost based optimization of inventory level

Assuming that the 'newsvendor' is in fact a small company who wants to produce goods to an uncertain market. In this more general situation the cost function of the newsvendor (company) can be formulated in the following manner:

K(q) = c_f + c_v (q-x) + p E\left[\max(D-q,0)\right] + h E\left[\max(q-D,0)\right]

where the individual parameters are the following:

  • cf – fixed cost. This cost always exists when the production of a series is started. [$/production]
  • cv – variable cost. This cost type expresses the production cost of one product. [$/product]
  • q – The product quantity in the inventory. The decision of the inventory control policy concerns the product quantity in the inventory after the product decision. This parameter includes the initial inventory as well. If nothing is produced, then this quantity is equal to the initial quantity, i.e. concerning the existing inventory.
  • x – Initial inventory level. We assume that the supplier possesses x products in the inventory at the beginning of the demand of the delivery period.
  • p – penalty cost (or back order cost). If there is less raw material in the inventory than needed to satisfy the demands, this is the penalty cost of the unsatisfied orders. [$/product]
  • E[D] – Expected value of the D stochastic variable.
  • D – This means the demand from the receiver for the product, which is an optional probability variable. [unit]
  • h – inventory and stock holding cost. [$ / product]

On the basis of the cost function the determination of the optimal inventory level is a minimization problem. So in the long run the amount of cost-optimal end-product can be calculated on the basis of the following relation:[1]

q_\text{opt} = F^{-1}\left( \frac{p-c_v}{p+h}\right)

See also

References

  1. ^ a b William J. Stevenson, Operations Management. 10th edition, 2009; page 581

Further reading

  • Ayhan, Hayriye, Dai, Jim, Foley, R. D., Wu, Joe, 2004: Newsvendor Notes, ISyE 3232 Stochastic Manufacturing & Service Systems.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Newsvendor — The newsvendor (or newsboy) 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. If the inventory level is q …   Wikipedia

  • Extended Newsvendor models — are some variations of the classic Newsvendor problem involving production capacity constraints, multiple products, multiple production cycles, demand dependent selling price, etc. in the literature. Further reading * R.L. Zhan, Z.J.M. Shen:… …   Wikipedia

  • Economic order quantity — is that level of inventory that minimizes the total of inventory holding cost and ordering cost. The framework used to determine this order quantity is also known as Wilson EOQ Model. The model was developed by F. W. Harris in 1913. But still R.… …   Wikipedia

  • Jack C. Hayya — is professor emeritus of management science at the Pennsylvania State University.Education*B.S., Civil Enginering, University of Illinois at Champaign Urbana, 1952 *M.S., Management, California State University, Northridge, 1961 [Hayya, Jack C.… …   Wikipedia

  • Channel coordination — (or supply chain coordination) aims at improving supply chain performance by aligning the plans and the objectives of individual enterprises. It usually focuses on inventory management and ordering decisions in distributed inter company settings …   Wikipedia

  • Newsboy — may refer to: Contents 1 Occupations 2 Personal nicknames 3 Other uses 4 See also Occupations A …   Wikipedia

  • Inventory control system — An inventory control system is an integrated package of software and hardware used in warehouse operations, and elsewhere, to monitor the quantity, location and status of inventory as well as the related shipping, receiving, picking and putaway… …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia

  • Economic production quantity — model (also known as the EPQ model) is an extension of the Economic Order Quantity model. The EPQ model was developed by E.W. Taft in 1918. The difference being that the EPQ model assumes orders are received incrementally during the production… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”