Список курсовых работ по математике, методам оптимизации в экономике по программе MBA - Master of Business Administration - для обращения за помощью к репетитору - преподавателю МФТИ, МГЛУ и MIT

Список курсовых работ по математике, методам оптимизации в экономике по программе Master of Business Administration - MBA - для обращения за помощью к онлайн репетитору - дистанционному преподавателю МФТИ, МГЛУ и MIT

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:

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:

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:

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?

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Nonlinear optimization, one variable, restricted interval

Find the minimum for the cost function:

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:

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.

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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:
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:
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.
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.

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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:
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:

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.

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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?

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Practice Problems: Constrained Optimization
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:
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

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:

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:

where TR is revenue in millions $ and x is the number of units produced in thousands. The total cost function is:

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:

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?