Optimization inBusiness

EconomicsLife Sciences

Lesson 4.7

Maximizing Profits

• Profit = Revenue – Costs

• Profits are maximized when: '( ) '( )R x C x

( )R x

( )C x

( ) ( ) ( )P x R x C x

Maximizing Profits

• Try this:

• How many items should we make and sell to make the most profit?

22( ) 3 10

51

( ) (45 )5

C x x x

R x x x

Minimum Average Cost

• Average cost :

• This is minimized when marginal cost is equal to average cost.

( )( )

C xA x

x

( )'( )

C xC x

x

Minimum Average Cost

• Try for our previous example

22( ) 3 10

5C x x x

Application

• Consider balancing shipping versus inventory costs

• Total cost = storage cost + ordering cost + item cost

Shipping Inventory

•ordering•freight•handling

•storage

We seek to minimize this functionWe seek to minimize this function

Example

• Selling 600 cans of hairspray per year – cost $4 each– shipping $30 per shipment– storage $0.90 per can per year

• Assume – uniform sales throughout year– each shipment arrives as preceding used

• How frequently should shipments be ordered to minimize costs?

Hairspray Orders

• Let x = number of cans ordered each time

• Storage cost

• Ordering cost

• Item cost

0.92

60030

600 4

x

x

Optimal Holding Time

• Consider an asset that increases in value• There may be a point in time when better to sell• This is when present value …

– at current rate of interest

– is maximized

• Given V(t) = price t years from now– Present value P(t) = V(t) e-r*t

– Where r = annual interest rate t = years

Optimal Holding Time

• Given :

r = 10% = 0.10

• When is the best time to sell

• P(t) = ?

• P`(t) = ?

• Maximum when x = ?

( ) 200ln 2 (in thousands)V t t

Optimal Holding Time

Assignment

• Lesson 4.7

• Page 259

• 1 – 39 EOO