applications of derivatives 10.pdf

7/23/2019 Applications of Derivatives 10.pdf

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How Derivatives Help Us Make BusinessDecisions

Natalie Rovetto

Kennesaw State University



Introduction

! In this presentation, the applications of calculus inbusiness are considered.

! Examples of total revenue, marginal revenue,average revenue, price elasticity of demand,

discrete future and present value, and optimizationare shown.

! In addition, optimization examples using calculuswithin supply chain management are shown.

2



Total Revenue Defined

! Total Revenue (TR) is the total amount received

by selling x items of the product at a price P perunit. (Morey, 2002)

! It is represented by the equation :

3



Total Revenue Example

Example 1.

Based on sales data from 2000 to 2009, therelationship between the price per barrel of beer

(P) at the Boston Beer Company and the Number

of Barrels sold annually (Q) can be modeled bythe power function

P=209.7204Q-0.0209

where Q is in the thousands of barrels.

Find the revenue function TR(Q).

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Total Revenue Example Continued

Solution:

Total Revenue= Price*Quantity

TR(Q)=Q*209.7204Q-0.0209

TR(Q)=209.7204Q0.0791

thousands of dollars

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Marginal Revenue Defined

! Marginal Revenue (MR) is

the rate of change of totalrevenue with respect to thequantity demanded. (Morey,2002)

! It is represented by theequation:

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Marginal Revenue Example

Example 1 (continued).

Using the first example, approximate the marginalrevenue when 1,500,000 barrels are sold.

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Price Elasticity of Demand

! Price Elasticity of Demand is a concept that helps see

what the responsiveness of a consumer would be for aproduct if the price changed.

Example 2.

Using Data from Market years 2000 through 2010, therelationship between the price per bushel of oats and

the number of bushels of oats sold is given byQ=152.07P-0.543, where P is in dollars and Q is in

millions of bushels. Find the price elasticity of demandwhen the price per bushel is $1.75 and determine if the

demand is elastic or inelastic.

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Calculations Continued

The Price Elasticity of Demand is

represented by the equation

First, find the derivative

and then insert it into the elasticityfunction.

If the elasticity is less than -1, thedemand is elastic and if it is

greater than -1, but less than 0, itis inelastic

Since the elasticity is between

-1 and 0, it is inelastic.

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Example 2 (continued).



Discrete Future and Present Value

!

Future Value (FV) of a payment (P) is theamount that the payment would have grownif deposited today in an interest bearingaccount.

! Present Value (PV) of a future payment (P)is the amount that would have to bedeposited in a bank account today toproduce exactly FV in the account at therelevant time future.

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Discrete Future and Present Value Cont.

!

The relationshipbetween the values canbe illustrated afterinterest is compoundedn times a year at anannual rate for t years.

! In the case ofcontinuous compoundinterest:

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Solution:

FV= $10,000, r =0.08, t =3PV=?

PV is approximately$7,866.28

Discrete Future and Present Value Examples

! You need $10,000 inyour account 3 yearsfrom now and theinterest rate is 8% per

year, compoundedcontinuously. How muchshould you depositnow?

Example 3.



Discrete Future and Present Value in Excel

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! Delivery performance is measurement of thefulfillment of a customers demand to the wish date

! Delivery performance is a key customer serviceinitiative within a supply chain that must be

proactively managed (Min and Zhou, 2002).

! Delivery performance is a strategic level supplychain performance measure (Gunasekaran et al .,

2007, 2001).

! Delivery performance is a key factor in supplierselection decisions (Shin et al . 2009, Tracy, 2001).

Importance of Delivery Performance in

Supply Chain Management

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Illustration of Delivery Time in a Two-StageSupply Chain

Let X = delivery time to the buyer in a serial supplychain.

Buyer

Delivery

TimeSupplier

where f(x) is the probability density function ofdelivery time X

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Definition of Delivery Window

Legend:

! X = random variable defining delivery time

! c 1 , c 1+ !c = milestone times defining early, on-time

and late delivery

!

!

c = the width of the on time delivery window 16



!

Model expected penalty cost for uniformlydistribution delivery time

! Assess the effect of delivery time

distribution parameters on the optimalposition of the delivery window and theexpected penalty cost

17

In the Presentation We Consider:



Expected Penalty Cost as a Function of thePosition of the Delivery Window

(based Guiffrida and Nagi, 2006)

! Y = expected penalty cost

! c1 = beginning of on time delivery

! !c = width of the delivery window

! Q = constant delivery lot size

! H = inventory holding cost per unit per unit time

! K = penalty cost per time unit late

! f(x) = pdf of delivery time X 18



Uniform Distribution

! pdf

! Optimal position

! Expected cost

! or

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Proposition 1. Increasing parameter a ofuniform distribution will increase theoptimal position of the delivery window and

reduce the expected penalty cost.

Proposition 2. Increasing parameter b ofuniform distribution will increase the

optimal position of the delivery window andincrease the expected penalty cost.

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! The expected penalty cost is found foruniformly distribution delivery time

!

The results allow developing strategies forimproving delivery performance and answerthe question what supplier should do todecrease the expected penalty cost

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22

!

Increasing the optimal delivery window by increasingparameter a will decrease the expected penalty cost.

! Because of this, proposition 1 is the best option

! Increasing the optimal delivery window by increasing

parameter b will increase the expected penalty cost.

! Because of this, proposition 2 in not in the bestinterest of a company.

!

Decreasing parameter b is a better



Conclusion

In conclusion, there is a wide array ofsituations in which derivatives could assistbusiness people within the real world. Fromfinding the rate of change of total revenue to

finding optimal delivery time, derivatives willalways be useful in a business setting.

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References

Finan, M. (2003). 52 Applications to economics. Reform calculus: part 2 (pp. 99-105). Arkansas:Arkansas Tech University.

Tesler, G. (2012). Continuous distributions.

Guiffrida A.L. and Nagi R., 2006. Cost characterizations of supply chain delivery performance.

International Journal of Production Economics, 102(1), 22-36.

Morey, E. (2007). Economic Application of derivatives . Boulder: University of Colorado Boulder.

Shin H., Benton W.C., and Jun M., 2009. Quantifying supplier’s product quality and delivery

performance: A sourcing policy decision model. Computers and Operations Research, 36,

2462-2471.

Tatiana Rudchenko “Supply chain delivery performance improvement for uniformly distributed

delivery time”.Conference Proceedings, of Southeast Decision Sciences Institute Wilmington2014 Conference

Vickers, J. (2004). The Uniform Distribution. In Work Book 38 .

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