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Chapters

Introduction
Exercise 1
Exercise 2
Exercise 3
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3

Introduction

Optimization of resources (cost and time) is required in every aspect of our lives. We need the optimization because we have limited time and cost resources, and we need to take maximum out of them. From manufacturing to resolving supply chain issues, every aspect of the business world today requires optimization to stay competitive.

Linear programming offers the most easiest way to do optimization as it simplifies the constraints and helps to reach a viable solution to a complex problem. In this article, we will solve some of the linear programming problems through graphing method.

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Exercise 1

A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of $20 m^#$ and a non-refrigerated capacity of $40 m^3$ while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of $3,000 m^3$ of refrigerated stock and $4, 000 m^#$ of non-refrigerated stock. The cost per kilometer of a Type A is $30, and $40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

Exercise 2

A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is $800 and $600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

Exercise 3

A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for $30. Offer B is a package of three shirts and a pair of pants, which will sell for $50. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to maximize the money generated from the promotion?

Solution of exercise 1

A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of $20 m^3$ and a non-refrigerated capacity of $40 m^3$ while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of $3000 m ^3$ of refrigerated stock and $4,000 m^3$ of non-refrigerated stock. The cost per kilometer of a Type A is $30, and $40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

a) Choose the unknowns.

x = Type A trucks

y = Type B trucks

b) Write the objective function.

$f(x,y) = 30x + 40y$

c) Write the constraints as a system of inequalities.

$40x + 30y \geq 4000$

$20x + 30y \geq 3000$

$x \geq 0$

$y \geq 0$

d) Find the set of feasible solutions that graphically represent the constraints.

e) Calculate the coordinates of the vertices from the compound of feasible solutions.

f) Calculate the value of the objective function at each of the vertices to determine which of them has the maximum or minimum values.

$f(0, 400/3) = 30 \cdot 0 + 40 \cdot \frac {400}{3} = 5,333.33^ 2$

$f(150,0) = 30 \cdot 150 + 40 \cdot 0 = 4500$

As x and y must be natural numbers round the value of y.

$f(50,67) = 30 \cdot 50 + 40 \cdot 67 = 4180$

By default, we see what takes the value x to y = 66 in the equation $20x + 30y = 3000 \cdot x = 51$ which it is within the feasible solutions.

$f(51,66) = 30 \cdot 51 + 40 \cdot 66 = 4170$

The minimum cost is $4170. To achieve this 51 trucks of Type A and 66 trucks of Type B are needed.

Solution of exercise 2

a)Choose the unknowns.

x = small buses

y = big buses

b) Write the objective function.

$f(x,y) = 600x + 800y$

c) Write the constraints as a system of inequalities.

$40x + 50y \geq 400$

$x + y \leq 9$

$x \geq 0$

$y \geq 0$

d) Find the set of feasible solutions that graphically represent the constraints.

e) Calculate the coordinates of the vertices from the compound of feasible solutions.

f) Calculate the value of the objective function at each of the vertices to determine which of them has the maximum or minimum values.

$f(0,8) = 600 \cdot 0 + 800 \cdot 8 = 6400$

$(0,9) = 600 \cdot 0 + 800\cdot 9 = 72000$

$f(5,4) = 600 \cdot 5 + 800 \cdot 4 = 6200$

Hence, the minimum cost is $6200. This is achieved with 4 large and 5 small buses.

We substituted the points (0,9), (0,8), and (5,4) in the equation to determine the minimum cost. However, you can tell this by directly looking at the graph. The coordinate (5,4) comes under the feasible region and is the minimum point of it.

Solution of exercise 3

a) Choose the unknowns.

x = number of packages of Offer A

y = number of packages of Offer B

b) Write the objective function.

$f(x,y) = 30x + 50y$

c) Write the constraints as a system of inequalities.

$x + y \leq 100$ $x + 3y \leq 200$

$x \geq 20$

$y \geq 10$

d) Find the set of feasible solutions that graphically represent the constraints.

e) Calculate the coordinates of the vertices from the compound of feasible solutions.

f) Calculate the value of the objective function at each of the vertices to determine which of them has the maximum or minimum values.

$f(x,y) = 20 \cdot 20 + 50 \cdot 10 = 1100$

$f(x,y) = 30 \cdot 90 + 50 \cdot 10 = 3200$

$f(x,y) = 30 \cdot 20 + 50 \cdot 60 = 3600$

$f(x,y) = 30 \cdot 50 + 50 \cdot 50 = 4000$

50 packages of each offer generates a maximum amount of $4000 in sales.

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arcee

April 2022

can you help me?
You are deciding what vehicles to buy for a very large rental car company. You have decided that the company will rent 5 different types of cars. They are listed here, along with the most important qualities of each car. You wish to minimize the total cost of purchase subject to the following constraints:
• At least 30% of the cars are fuel-efficient
• At least 20% of the cars are fast
• At least 50% of the cars appeal to low-cost customers
• At least 40% of the cars are spacious
• At least 10% of the cars are stylish

Harold dave Rago

April 2022

Can someone help me?

3. A store has a sale for two packs of candies: one pack contains 2 brand A and 6 brand B whereas the other pack contains 3 brand A and 3 brand B. Near closing time one day, only 24 of brand A and 48 of brand B are available. If brand A sells for Php 1.20 and brand Bsells for Php 1.50, then how many of each should be made to maximize the total revenue?

Mary Anthonette Rose Valdez

April 2022

2. A company packages and sells 16-ounce containers of mixed nuts. Brand A contains 12 o unces of peanuts and 4 ounces of cashews. Brand B contains 8 ounces peanuts and 8 ounces of of cashews . Suppose 120 pounds of anda 96 pounds of cashews are available. Find nthe number of containers of each brand that can be packaged.

Mary Anthonette Rose Valdez

April 2022

HELP ME SOLVE THIS PLEASEEE

1. A garden requires at least 10,12 , and 12 units of chemicals A, B, and C, respectively. A jar of liquid fertilizer contains 5, 2, and 1 unit of A, B, and C , respectively. A bag of granular fertilizer contain 1, 2, and 4 units of A, B, and C respectively. Find the combinations of jars and bags that will meet the requirements of the garden.

Lula

March 2022

Who can solve this equation
Domestic cows can be bought for $200 each but Hybrid cows cost$ 500 each. The domestic cows produce 30 gallons of milk per week and the Hybrid ones produce 50 gallons of milk per week. A gallon of milk can be sold for $3. A cow costs$ 100 per week to feed. If the financial constraint is to spend $8000 for cows and the capacity constraint is that total number of cows to be bought cannot exceed 20 cows. Formulate a linear programming model for this problem, and find the optimal combination of domestic and hybrid cows that maximize the total profit earned using the graphic.

gina

February 2022

please help me to solve this problem
A communication system has to satisfy certain specifications. The bit error rate (BER)
at the receiver has to be smaller than 10-2
and the transmitted power has to be between
0.5 to 1 watt. The noise power at the receiver is 0.1 watt. An attenuation factor between
the transmitter and receiver is 0.7. The relation that relates link the BER and the signal
to noise ratio is 𝐵𝐸𝑅 = 𝑄(√𝑆/𝑁). (𝑄-function table is given to get values form it).
Formulate the optimization problem and classify its type and find the optimal solution

Linear Programming Problems and Solutions

Introduction

Exercise 1

Exercise 2

Exercise 3

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Linear Programming Examples

Steps to Solve a Linear Programming Problem