Nation’s Healthcare Inc (NHI) has collected historical data on the cost of operating a
large hospital. The operating cost turns out to be a nonlinear function of the number of
patient days per year, approximated by the function:
C  4,700,00  .00013x 2
where C is the total annual cost and x is the number of patient days per year.
a. Write the equation for cost per patient day.
b. Find the value of x (patient days) which minimizes cost per patient day.
c. Find the minimum cost per patient day.
Orion Outfitters is trying to price a new pair of ski goggles. They have estimates of the
relationship between price and the number of units sold as below:
p  50  .05 x
where p is the price and x is the number of units sold.
a. Write the equation for total revenue.
b. Find the number of units to sell in order to maximize revenue.
c. Find the revenue-maximizing price.
d. Find the maximum revenue.
SouthStar Inc. (SSI) produces lawn tractors at a single factory. Based on a number of
years of data, SSI has estimated a nonlinear cost function for the factory as below:
C  100,000  1500 x  .2 x 2
where C is the total annual cost in dollars and x is the number of units produced in a year.
a. Find the number of units to produce in order to minimize cost per unit.
b. What is the minimum cost per unit?
Nonlinear optimization, one variable, restricted interval
Find the minimum for the cost function:
C  2 x 3  12 x 2  100
where x is the production level in thousands of units and C is the total cost in millions of
dollars. Suppose that, due to other factors, the production level must be no lower than 1
thousand units and no more than 10 thousand units.
Nonlinear optimization, one variable, restricted interval
Consider the cost function:
C  2 x 3  6 x 2  10
where x is the production level in thousands of units and C is the total cost in millions of
dollars. Find the production level which yields the minimum cost. Assume that
production must be no lower than 1 thousand units and no higher than 5 thousand units.
Optimization with two variables (bivariate optimization), no constraints
A company is trying to construct an advertising plan. They can choose between TV
advertising and radio advertising. From previous experience they have found that the
following equation approximates the relationship between sales and advertising
expenditures:
f ( x, y)  50,000 x  40,000 y  10 x 2  20 y 2  10 xy
Where f(x,y) is unit sales, x is dollars spent on TV ads. And y is dollars spent on radio
ads. Find the advertising plan which will result in maximum sales.
A manufacturer sells two products. The demand functions for these two products are as
given below:
q1  150  2 p1  p 2
q 2  200  p1  3 p 2
where q1 is the number of units of product 1 sold, q2 is the number of units of product 2
sold, p1 is the price of product 1 in dollars and p2 is the price of product 2 in dollars.
Find the prices that the manufacturer should charge in order to maximize revenue.
A service company sells two products. Below is given the profit function of the company
as a function of the number of units of each product produced.
f ( x, y)  64 x  2 x 2  4 xy  4 y 2  32 y  14
where f(x,y) is profit, x is the number of units of product one sold, and y is the number of
units of product two sold. Find the number of units of each product that should be sold in
order to maximize profit.
Nonlinear Optimization, two variables with a constraint
Itech Cycle Company (ICC) has an order to produce 200 bicycles. ICC produces this
particular bicycle at two plants. The cost function for production at these two plants is:
Cost : f ( x1 , x 2 )  2 x1  x1 x 2  x 2  200
2
2
Where f(x1,x2) is the production cost in dollars, x1 is the number of bicycles produced at
plant 1 and x2 is the number of bicycles produced at plant 2. The company wants to split
the production between the two plants in such a way as to minimize production cost.
a. How many bicycles should ICC produce at each plant in order to meet the order at
minimum cost?
b. What is the minimum cost?
c. What would be the effect on cost of a one unit increase in the total production
requirement?
d. Now solve this problem on EXCEL.
A company has a requirement to produce 34 units of a new product. The order can be
filled by either product 1 or product 2 or a combination of the two. The company’s cost
function is:
C  6Q1  10Q2  Q1Q2  30
2
2
a. How many units of each product should be produced in order to minimize total cost?
b. What is the minimum cost?
c. What would be the effect on cost of a one unit increase in the total production
requirement?
d. Now solve this problem on EXCEL.
Nonlinear Product Mix Problem
A TV company produces two types of TV sets, the Astro and the Cosmo. There are two
production lines, one for each set, and there are two departments, both of which are used
in the production of each set. The capacity of the Astro production line is 70 sets per day.
The capacity of the Cosmo production line is 50 sets per day. In department A picture
tubes are produced. In this department the Astro set requires 1 labor hour and the Cosmo
set requires 2 labor hours. Presently in department A a maximum of 120 labor hours per
day can be assigned to production of these two types of sets. In department B the chassis
is constructed. In this department the Astro set requires 1 labor hour and the Cosmo set
requires 1 labor hours. Presently in department B a maximum of 90 labor hours per day
can be assigned to production of these two types of sets. The demand curve for each of
these two sets is downward sloping, meaning that more TV sets can be sold only if the
selling price is reduced. These demand curves are quantified by the following equations:
PA = .01A2 – 1.9A + 314
PC = -.14C + 243
Where:
A = daily production of Astros
PA = selling price of Astros
C = daily production of Cosmos
PA = selling price of Cosmos
Given:
Cosmos purchase cost = $220
Astros purchase cost = $210
What is the optimal mix for the Astros and Cosmos?
Practice Problems: Constrained Optimization
Problem 1. Min f ( x, y )  x 2  y 2
ST : x  2 y  10  0
Solution :
x*  2
y*  4
Problem 2. Max f ( x, y )  8 x  x 2  4 y  y 2
ST : x  y  10
Solution :
x*  6
y*  4
Review Problems: Nonlinear optimization
A company wants to maximize the profit from the production of two products. There is
one production department with 200 hours of production time per month. One thousand
units of product 1 requires 20 hours of production time and one thousand units of product
2 requires 40 hours of production time. The company wants to set up their production
plan so that there is no unused production time. The profit function is:
f (Q1 , Q2 )  60  140Q1  100Q2  10Q12  8Q22  6Q1Q2
where f(Q1,Q2) is monthly profit in millions $, Q1 is the number of units of product 1 in
thousands and Q2 is the number of units of product 2 in thousands.
a. Find the number of units of each product that should be produced in order to maximize
profit.
b. What would the maximum monthly profit be?
c. Suppose that the production capacity is to be increased by 1 hour. What would be the
approximate effect on profit?
Suppose a company wants to maximize revenue from production of a single product.
Due to limitations on the market, however, the production level must be between 100 and
1000 units. Given below is the revenue function
R  3x 4  x 3  2
where R is revenue in millions $, and x is the number of units produced in thousands.
a. Find the revenue-maximizing production point.
b. Find the maximum revenue.
A company produces two products, product 1 and product 2. The revenue function for
the company is:
R  3x 3  5 y 2  225 x  70 y  23
where R is revenue in millions $, x is the number of units of product 1 produced (in
thousands) and y is the number of units of product 2 produced (in thousands). How many
units of each product should be produced in order to maximize revenue?
A company wishes to maximize profit. The function which represents total revenue is:
TR  45 x  .5 x 2
where TR is revenue in millions $ and x is the number of units produced in thousands.
The total cost function is:
TC  x 3  39.5 x 2  120 x  125
where TC is total cost in millions $.
How many units should be produced in order to maximize profit?
A company produces two products, product 1 and product 2. The cost function for these
two products is:
C  6 x 2  9 x  3xy  7 y  5 y 2  200
where C is cost in millions $, x is units of product 1 in thousands, and y is units of
product 2 in thousands.
a. How many units of each product should be produced in order to minimize cost?
b. What is the minimum cost?