10. A retail store in Des Moines, Iowa, receives shipments of a particular product from
Kansas City and Minneapolis. Let x 5 units of product received from Kansas City y 5 units
of product received from Minneapolis
a. Write an expression for the total units of product received by the retail store in Des
Moines.
The expression for the total units is:
Total units= X +Y
b. Shipments from Kansas City cost $0.20 per unit, and shipments from Minneapolis
cost $0.25 per unit. Develop an objective function representing the total cost of
shipments to Des Moines.
The expression for the shipments to Des Moines total cost is:
X*(.20) + Y*(.25) = Total shipping cost
b. Assuming the monthly demand at the retail store is 5000 units, develop a
constraint that requires 5000 units to be shipped to Des Moines.
The constraint of 5000 units shipped to Des Moines is:
X+Y=5000
d. No more than 4000 units can be shipped from Kansas City and no more than 3000 units
can be shipped from Minneapolis in a month. Develop constraints to model this situation.
Constraints model for this situation is:
X<=4000 Y<=3000
e. Of course, negative amounts cannot be shipped. Combine the objective function and
constraints developed to state a mathematical model for satisfying the demand at the Des
Moines retail store at minimum cost.
Mathematical model which can satisfy the request of the Des Moines store to minimum cost is:
Min .20 xs+.25Y
, X+Y=5000
X<=4000
Y<=3000
X, Y>=0
11. For most products, higher prices result in a decreased demand, whereas lower prices
result in an increased demand (economists refer to such products as normal goods). Let d 5
annual demand for a product in units p 5 price per unit Assume that a firm accepts the
following price–demand relationship as being a real- istic representation of its market: d 5
800 2 10p where p must be between $20 and $70.
a. How many units can the firm sell at the $20 per-unit price? At the $70 per-unit price?
firm can trade at the 800-10 (20) = 600 units. $70 /unit the value is 800-10 (70) = 100 units
b. What happens to annual units demanded for the product if the firm increases the per-
unit price from $26 to $27? From $42 to $43? From $68 to $69? What is the suggested
relationship between per-unit price and annual demand for the product in units?
The firm increased price per unit from $26 to $27 so the annual demand has decreased by 10
units.
800 – 10 (26) = 540 to 800 – 10 (27) = 530
The firm increased the price per unit from $42 to $43 so the annual units demand has decrease by
10 units.
$42, D=800-10*42=380
$43, D=800-10*43=370
Annual demand decreases from 800 – 10 (68) = 120 to 800 – 10 (69) 110. $1 increase in price
decrease in demand by 10 units.
c. Show the mathematical model for the total revenue (TR), which is the annual
demand multiplied by the unit price.
Kansas City and Minneapolis. Let x 5 units of product received from Kansas City y 5 units
of product received from Minneapolis
a. Write an expression for the total units of product received by the retail store in Des
Moines.
The expression for the total units is:
Total units= X +Y
b. Shipments from Kansas City cost $0.20 per unit, and shipments from Minneapolis
cost $0.25 per unit. Develop an objective function representing the total cost of
shipments to Des Moines.
The expression for the shipments to Des Moines total cost is:
X*(.20) + Y*(.25) = Total shipping cost
b. Assuming the monthly demand at the retail store is 5000 units, develop a
constraint that requires 5000 units to be shipped to Des Moines.
The constraint of 5000 units shipped to Des Moines is:
X+Y=5000
d. No more than 4000 units can be shipped from Kansas City and no more than 3000 units
can be shipped from Minneapolis in a month. Develop constraints to model this situation.
Constraints model for this situation is:
X<=4000 Y<=3000
e. Of course, negative amounts cannot be shipped. Combine the objective function and
constraints developed to state a mathematical model for satisfying the demand at the Des
Moines retail store at minimum cost.
Mathematical model which can satisfy the request of the Des Moines store to minimum cost is:
Min .20 xs+.25Y
, X+Y=5000
X<=4000
Y<=3000
X, Y>=0
11. For most products, higher prices result in a decreased demand, whereas lower prices
result in an increased demand (economists refer to such products as normal goods). Let d 5
annual demand for a product in units p 5 price per unit Assume that a firm accepts the
following price–demand relationship as being a real- istic representation of its market: d 5
800 2 10p where p must be between $20 and $70.
a. How many units can the firm sell at the $20 per-unit price? At the $70 per-unit price?
firm can trade at the 800-10 (20) = 600 units. $70 /unit the value is 800-10 (70) = 100 units
b. What happens to annual units demanded for the product if the firm increases the per-
unit price from $26 to $27? From $42 to $43? From $68 to $69? What is the suggested
relationship between per-unit price and annual demand for the product in units?
The firm increased price per unit from $26 to $27 so the annual demand has decreased by 10
units.
800 – 10 (26) = 540 to 800 – 10 (27) = 530
The firm increased the price per unit from $42 to $43 so the annual units demand has decrease by
10 units.
$42, D=800-10*42=380
$43, D=800-10*43=370
Annual demand decreases from 800 – 10 (68) = 120 to 800 – 10 (69) 110. $1 increase in price
decrease in demand by 10 units.
c. Show the mathematical model for the total revenue (TR), which is the annual
demand multiplied by the unit price.
