Newspaper boy model

The Newspaper Boy Model is a mathematical model in production management and applied economics that is used to determine optimal inventory levels. It is typically used when there is a fixed price and uncertain demand for a perishable product. If you add up the stock quantity , then every unit demanded that is above the stock quantity is a unit that is lost in sales. The model is called "Newspaper Boy Model" or "Newspaper Boy Problem" as it is reminiscent of the situation of a newspaper boy deciding how many newspapers to buy in the face of fluctuating demand and the certainty that unsold copies will end up are worth nothing of the day. ${\ displaystyle q}$

history

The mathematical problem seems to have appeared for the first time in 1888. That year, Edgeworth used the Central Threshold Theorem to determine the optimal level of cash needed to satisfy infrequent withdrawals by savers. The modern formulation originated in a 1951 paper by Kenneth Arrow , T. Harris, and Jacob Marschak in Econometrica magazine.

Profit function

The classic newsboy profit function is

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

${\ displaystyle D}$ , a random variable with probability distribution , represents the demand. Each unit is sold at price and bought at price . is the stock quantity, and is the expectation operator . The solution for the optimal inventory of the newsboy that maximizes the expected profit: ${\ displaystyle F}$ ${\ displaystyle p}$ ${\ displaystyle c}$ ${\ displaystyle q}$ ${\ displaystyle E}$

Critical fractile formula

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

In this represents the inverse function of the cumulative distribution function of . ${\ displaystyle F ^ {- 1}}$ ${\ displaystyle D}$

This ratio, which is also known as the critical fractile, sets the costs of having too little inventory (loss of sales of ) in relation to the total costs of having either too little or too much inventory (here the costs of the excess inventory are the inventory costs or so the total cost is). ${\ displaystyle (pc)}$ ${\ displaystyle c}$ ${\ displaystyle p}$

The formula for the critical fractile is also known as Littlewood's rule in yield management .

Numerical examples

equal distribution

Assume that the selling price is [€ / unit] and the buying price is [€ / unit]. Furthermore, the demand should follow an equal distribution between the minimum demand and the maximum demand . ${\ displaystyle p = 7}$ ${\ displaystyle c = 5}$ ${\ displaystyle D _ {\ min} = 50}$ ${\ displaystyle D _ {\ max} = 80}$

{\ displaystyle 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 \ approx 59}

.

The optimal inventory quantity would be 59 units.

Normal distribution

Let us assume that the selling price is [€ / unit] and the buying price is [€ / unit]. Furthermore, let the demand be normally distributed with a mean of 50 and a standard deviation of 20. Then is ${\ displaystyle p = 7}$ ${\ displaystyle c = 5}$ ${\ displaystyle D}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ sigma}$

{\ displaystyle q _ {\ text {opt}} = F ^ {- 1} \ left ({\ frac {7-5} {7}} \ right) = \ mu + \ sigma Z ^ {- 1} \ left (0 {,} 285 \ right) = 50 + 20 \ cdot (-0 {,} 56595) = 38 {,} 68 \ approx 39}

.

The optimal inventory quantity is therefore 39 units.

Logarithmic normal distribution

Assume that the selling price is [€ / unit] and the buying price is [€ / unit]. Now let the demand be logarithmically normally distributed with a mean of 50 and a standard deviation of 0.2. ${\ displaystyle p = 7}$ ${\ displaystyle c = 5}$ ${\ displaystyle D}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ sigma}$

{\ displaystyle q _ {\ text {opt}} = F ^ {- 1} \ left ({\ frac {7-5} {7}} \ right) = \ mu e ^ {Z ^ {- 1} \ left (0 {,} 285 \ right) \ sigma} = 50e ^ {\ left (0 {,} 2 \ cdot (-0 {,} 56595) \ right)} = 44 {,} 64 \ approx 45}

.

The optimal inventory quantity is therefore 45 units.

Extreme case

If the selling price , i.e. the selling price , is below the buying price, the numerator becomes negative, so it does not make sense to keep any units in inventory, that is . ${\ displaystyle p <c}$ ${\ displaystyle q _ {\ text {opt}} = 0}$

Cost-oriented optimization of the inventory

Let's assume the 'newsboy' is a small company that wants to produce goods in an uncertain market. In this general situation, the function of the newsboy (company) can be formulated in the following way:

{\ displaystyle K (q) = c_ {f} + c_ {v} (qx) + pE \ left [\ max (Dq, 0) \ right] + hE \ left [\ max (qD, 0) \ right] }

the parameters represent the following:

${\ displaystyle c_ {f}}$ - the fixed costs. The costs that always arise when producing any series. [€ / production]
${\ displaystyle c_ {v}}$ - variable costs. The costs that only arise for the production of a product. [€ / product]
${\ displaystyle q}$ - the amount of product in stock after production. This is the sum of the initial stock and the additional quantity produced.
${\ displaystyle x}$ - Stock amount. We assume that the producer owns the amount of products at the beginning of the period. ${\ displaystyle x}$
${\ displaystyle p}$ - Penalty (or cost of recall). If there is not enough material in stock to meet the demand, the penalty for unsatisfied orders is [€ / product]. ${\ displaystyle p}$
${\ displaystyle E}$ - expected value operator.
${\ displaystyle D}$ - The demand of the end customer, modeled as a random variable. [Unit]
${\ displaystyle h}$ - Inventory and storage costs [€ / product]

When working on the basis of costs, finding the optimal inventory is a minimization problem. So in the long run the amount of the cost-optimal end product can be calculated in the following way:

{\ displaystyle q _ {\ text {opt}} = F ^ {- 1} \ left ({\ frac {p-c_ {v}} {p + h}} \ right)}

Individual evidence

^ FY Edgeworth, "The Mathematical Theory of Banking," Journal of the Royal Statistical Society, Issue 51 (1), 1888, pp. 113-127
↑ Guillermo Gallego: IEOR 4000 Production Management Lecture 7 . Columbia University . January 18, 2005. Retrieved May 30, 2012.
^ KJ Arrow, T. Harris, Jacob Marshak, Optimal Inventory Policy, Econometrica 1951
^ William J. Stevenson, Operations Management. 10th edition, 2009; Page 581