## Linear Programming Problems (LLP): Preface

In this article, we will learn about the different types of linear programming problems such as manufacturing problems, diet problems, optimal assignment problems, and transportation problems. Finding the most efficient resource allocation is often done using linear programming.

A one-dimensional relationship between numerous variables is referred to as "linear" in this context. Programming is selecting the best option from a list of options. When a set of linear inequalities, known as linear constraints, are satisfied and all of the variables are non-negative, linear programming seeks to determine the optimal value of a linear function of many variables.

## Diet Problems

Diet problems, as the name suggests, involves maximising the consumption of particular foods high in particular nutrients that can help with the application of a particular diet plan. Finding a collection of meals that will satisfy a set of daily nutritional needs while costing the least is the goal of a diet problem.

Limitation – Dietary requirements must be fulfilled, such as a specific calorie intake or a required amount of sugar or cholesterol.

The objective function is the price of food consumption.

## Manufacturing Problems

The goal of manufacturing is to increase production rates or net profits, which may depend on a variety of factors, including the amount of space available, the number of workers, the number of machine hours used, the type of packaging used, the amount of raw materials needed, the market value of the product, and others. These are employed in the industrial sector and can be utilised to predict a company's potential future capital growth.

Limitations – Factors like labour hours, the price of packaging supplies, and so forth.

The objective function is the production rate.

## Optimal Assignment Problems

The optimal assignment problems deal with a corporation finishing a particular task or assignment by choosing a particular number of employees to complete the assignment within the given deadline, provided that each person works on just one task inside the assignment. These problems can be found, among other places, in huge organisations’ event management and planning.

Limitations – The size of the workforce, the hours each person works, etc.

The objective function is represented by the total number of jobs finished.

## Transportation Problems

The study of efficient transportation routes, or how effectively goods from diverse sources of production are transported to various markets in such a way that the overall transportation cost is minimised, is related to the study of transportation problems. Analysing such problems is crucial for large firms with numerous production units and a sizable client base.

Limitations – Specific supply and demand trends.

The objective function is the cost of transportation.

## Applications of Linear Programming

Numerous sectors, including delivery services, the transportation sector, manufacturing firms, and financial institutions, mainly depend on linear programming. Let’s discuss some applications of Linear Programming problems in this section.

Production management can use linear programming to decide on the right product mix, smooth out products, and balance assembly times.

The people manager can analyse personnel policies using linear programming.

Based on the available advertising medium, linear programming aids in measuring the success of marketing campaigns and timing.

For physical distribution, the best place to site warehouses, manufacturing facilities, and distribution hubs are determined using the linear programming method.

## Linear Programming Problems

1. Purchase some filing cabinets. You are aware that Cabinet X costs $10$ per unit, takes up six square feet of floor space, and has a file capacity of eight cubic feet. Cabinet Y costs $20$ each, takes up eight square feet of floor area, and has a file capacity of twelve cubic feet. Although you don't have to spend that much money, you have been granted $140$ to use toward this purchase. There is only enough room in the office for $72$ square feet of cupboards. Which model should you purchase the most in order to maximise the storage capacity? Put your maximum storage capacity in the box provided.

Ans:

Let the number of Y models be y and the number of X models be x, them

For floor 6x + 8y = 72 or 3x + 4y = 36

10x +20y ≤ 140

Volume is 8x + 12y

Now x + 2y ≤ 14

Therefore, 2x + 4y ≤ 14

Volume = 8x + 12y

= 8x +108 – 9x

= 108 – x

Now, 2x + 4y ≤ 28

2x + 36 – 3x ≤ 28

x ≥ 8

So, when x is minimum x = 8, the volume is maximum.

Therefore, maximum volume = 108 – 8 = 100.

2. Use the graphical approach to solve the linear programming problem below:

Maximise Z = 2x + 3y

x + y ≤ 30,

x ≤ 20, y ≤ 12

x, y ≥ 0

Ans. Equations resulting from the inequalities are as follows: x + y = 30 passing through (0, 30) and (30, 0). Points on or under this line will satisfy x + y ≤ 30

The y-axis and the line at x = 20 are parallel. Any point along this line or to its left will satisfy the condition x ≤ 20.

Line y = 12 is parallel to the x-axis. Any point above or below this line will meet the condition y ≤ 12. According to the graph,

Hence, Z has a maximum value of 72 and occurs at C. (18, 12)

## Practice Questions

1. In a linear programming problem involving the variables x and y, what constraints are placed on the variables?

x, y ≥ 0

x, y ≤ 0

x, y = 0

x, y ≤ 0

Ans: Option (a)

2. Find the optimal(best) solution using linear programming.

Maximise Z = 50x + 120y

x + 2y ≤ 100

x + 3y ≤ 120

x + y ≤ 110

x, y ≥ 0

(60, 20)

(20, 10)

(5, 7)

(-12, -8)

Ans: Option (a)

## Conclusion

Linear programming, commonly known as LP, is a straightforward technique for representing complex real-world interactions by means of a linear function. The components of the resulting mathematical model are linearly related to one another. In order to reach the optimal result, linear optimisation is carried out using linear programming. Example problems have also been solved in this article to boost your concept and clear your doubt about the topic.

## FAQs on Types of Linear Programming Problems

**1. What is linear programming?**

Constrained problems can be solved in some way with the help of linear programming. It involves maximising or minimising linear functions while adhering to a linear inequality's constraints. It is said to be the easiest task to solve a linear programming problem.

**2. What are the two forms of LPP?**

**The two forms of LPP are:**

Canonical form of linear programming problem

Standard form of linear programming problem

**3. What are the two limitations of LPP?**

**The limitations or constraints of linear programming problems are:**

They are only applicable when using the objective function and all constraints represented by linear inequalities.

They are only applied when a problem's entirety can be quantified.