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Assignment problem in linear programming : introduction and assignment model.

represent assignment problem in lpp form

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Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

job of Work

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

Assignment Model

The total assignment cost will be given by

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The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Assignment Model

Subjected to constraints

Assignment Model

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Related Articles:

  • Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
  • Linear Programming: Applications, Definitions and Problems

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Assignment Problem

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

  • First, prefix name of what this variable represents.
  • Second is the list of all the variables.
  • Third is the lower bound on this variable.
  • Fourth variable is the upper bound.
  • Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

  • Solving Blending Problem in Python using Gurobi
  • Transshipment Problem in Python Using PuLP

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Assignment Problem: Linear Programming

The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.

In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.

It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.

"A mathematician is a device for turning coffee into theorems." -Paul Erdos

Formulation of an assignment problem

Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:

The structure of an assignment problem is identical to that of a transportation problem.

To formulate the assignment problem in mathematical programming terms , we define the activity variables as

for i = 1, 2, ..., n and j = 1, 2, ..., n

In the above table, c ij is the cost of performing jth job by ith worker.

Generalized Form of an Assignment Problem

The optimization model is

Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn

subject to x i1 + x i2 +..........+ x in = 1          i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1          j = 1, 2,......., n

x ij = 0 or 1

In Σ Sigma notation

x ij = 0 or 1 for all i and j

An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.

Assumptions in Assignment Problem

  • Number of jobs is equal to the number of machines or persons.
  • Each man or machine is assigned only one job.
  • Each man or machine is independently capable of handling any job to be done.
  • Assigning criteria is clearly specified (minimizing cost or maximizing profit).

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Operations Research Simplified Back Next

Goal programming Linear programming Simplex Method Transportation Problem

Linear Programming

Linear programming is a process that is used to determine the best outcome of a linear function. It is the best method to perform linear optimization by making a few simple assumptions. The linear function is known as the objective function. Real-world relationships can be extremely complicated. However, linear programming can be used to depict such relationships, thus, making it easier to analyze them.

Linear programming is used in many industries such as energy, telecommunication, transportation, and manufacturing. This article sheds light on the various aspects of linear programming such as the definition, formula, methods to solve problems using this technique, and associated linear programming examples.

What is Linear Programming?

Linear programming, also abbreviated as LP, is a simple method that is used to depict complicated real-world relationships by using a linear function . The elements in the mathematical model so obtained have a linear relationship with each other. Linear programming is used to perform linear optimization so as to achieve the best outcome.

Linear Programming Definition

Linear programming can be defined as a technique that is used for optimizing a linear function in order to reach the best outcome. This linear function or objective function consists of linear equality and inequality constraints. We obtain the best outcome by minimizing or maximizing the objective function .

Linear Programming Examples

Suppose a postman has to deliver 6 letters in a day from the post office (located at A) to different houses (U, V, W, Y, Z). The distance between the houses is indicated on the lines as given in the image. If the postman wants to find the shortest route that will enable him to deliver the letters as well as save on fuel then it becomes a linear programming problem. Thus, LP will be used to get the optimal solution which will be the shortest route in this example.

Example of Linear Programming

Linear Programming Formula

A linear programming problem will consist of decision variables , an objective function, constraints, and non-negative restrictions. The decision variables, x, and y, decide the output of the LP problem and represent the final solution. The objective function, Z, is the linear function that needs to be optimized (maximized or minimized) to get the solution. The constraints are the restrictions that are imposed on the decision variables to limit their value. The decision variables must always have a non-negative value which is given by the non-negative restrictions. The general formula of a linear programming problem is given below:

  • Objective Function: Z = ax + by
  • Constraints: cx + dy ≤ e, fx + gy ≤ h. The inequalities can also be "≥"
  • Non-negative restrictions: x ≥ 0, y ≥ 0

How to Solve Linear Programming Problems?

The most important part of solving linear programming problem is to first formulate the problem using the given data. The steps to solve linear programming problems are given below:

  • Step 1: Identify the decision variables.
  • Step 2: Formulate the objective function. Check whether the function needs to be minimized or maximized.
  • Step 3: Write down the constraints.
  • Step 4: Ensure that the decision variables are greater than or equal to 0. (Non-negative restraint)
  • Step 5: Solve the linear programming problem using either the simplex or graphical method.

Let us study about these methods in detail in the following sections.

Linear Programming Methods

There are two main methods available for solving linear programming problem. These are the simplex method and the graphical method. Given below are the steps to solve a linear programming problem using both methods.

Linear Programming by Simplex Method

The simplex method in lpp can be applied to problems with two or more decision variables. Suppose the objective function Z = 40\(x_{1}\) + 30\(x_{2}\) needs to be maximized and the constraints are given as follows:

\(x_{1}\) + \(x_{2}\) ≤ 12

2\(x_{1}\) + \(x_{2}\) ≤ 16

\(x_{1}\) ≥ 0, \(x_{2}\) ≥ 0

Step 1: Add another variable, known as the slack variable, to convert the inequalities into equations. Also, rewrite the objective function as an equation .

- 40\(x_{1}\) - 30\(x_{2}\) + Z = 0

\(x_{1}\) + \(x_{2}\) + \(y_{1}\) =12

2\(x_{1}\) + \(x_{2}\) + \(y_{2}\) =16

\(y_{1}\) and \(y_{2}\) are the slack variables.

Step 2: Construct the initial simplex matrix as follows:

\(\begin{bmatrix} x_{1} & x_{2} &y_{1} & y_{2} & Z & \\ 1&1 &1 &0 &0 &12 \\ 2& 1 & 0& 1 & 0 & 16 \\ -40&-30&0&0&1&0 \end{bmatrix}\)

Step 3: Identify the column with the highest negative entry. This is called the pivot column. As -40 is the highest negative entry, thus, column 1 will be the pivot column.

Step 4: Divide the entries in the rightmost column by the entries in the pivot column. We exclude the entries in the bottom-most row.

12 / 1 = 12

The row containing the smallest quotient is identified to get the pivot row. As 8 is the smaller quotient as compared to 12 thus, row 2 becomes the pivot row. The intersection of the pivot row and the pivot column gives the pivot element.

Thus, pivot element = 2.

Step 5: With the help of the pivot element perform pivoting, using matrix properties , to make all other entries in the pivot column 0.

Using the elementary operations divide row 2 by 2 (\(R_{2}\) / 2)

\(\begin{bmatrix} x_{1} & x_{2} &y_{1} & y_{2} & Z & \\ 1&1 &1 &0 &0 &12 \\ 1& 1/2 & 0& 1/2 & 0 & 8 \\ -40&-30&0&0&1&0 \end{bmatrix}\)

Now apply \(R_{1}\) = \(R_{1}\) - \(R_{2}\)

\(\begin{bmatrix} x_{1} & x_{2} &y_{1} & y_{2} & Z & \\ 0&1/2 &1 &-1/2 &0 &4 \\ 1& 1/2 & 0& 1/2 & 0 & 8 \\ -40&-30&0&0&1&0 \end{bmatrix}\)

Finally \(R_{3}\) = \(R_{3}\) + 40\(R_{2}\) to get the required matrix.

\(\begin{bmatrix} x_{1} & x_{2} &y_{1} & y_{2} & Z & \\ 0&1/2 &1 &-1/2 &0 &4 \\ 1& 1/2 & 0& 1/2 & 0 & 8 \\ 0&-10&0&20&1&320 \end{bmatrix}\)

Step 6: Check if the bottom-most row has negative entries. If no, then the optimal solution has been determined. If yes, then go back to step 3 and repeat the process. -10 is a negative entry in the matrix thus, the process needs to be repeated. We get the following matrix.

\(\begin{bmatrix} x_{1} & x_{2} &y_{1} & y_{2} & Z & \\ 0&1 &2 &-1 &0 &8 \\ 1& 0 & -1& 1 & 0 & 4 \\ 0&0&20&10&1&400 \end{bmatrix}\)

Writing the bottom row in the form of an equation we get Z = 400 - 20\(y_{1}\) - 10\(y_{2}\). Thus, 400 is the highest value that Z can achieve when both \(y_{1}\) and \(y_{2}\) are 0.

Also, when \(x_{1}\) = 4 and \(x_{2}\) = 8 then value of Z = 400

Thus, \(x_{1}\) = 4 and \(x_{2}\) = 8 are the optimal points and the solution to our linear programming problem.

Linear Programming by Graphical Method

If there are two decision variables in a linear programming problem then the graphical method can be used to solve such a problem easily.

Suppose we have to maximize Z = 2x + 5y.

The constraints are x + 4y ≤ 24, 3x + y ≤ 21 and x + y ≤ 9

where, x ≥ 0 and y ≥ 0.

To solve this problem using the graphical method the steps are as follows.

Step 1: Write all inequality constraints in the form of equations.

x + 4y = 24

3x + y = 21

Step 2: Plot these lines on a graph by identifying test points.

x + 4y = 24 is a line passing through (0, 6) and (24, 0). [By substituting x = 0 the point (0, 6) is obtained. Similarly, when y = 0 the point (24, 0) is determined.]

3x + y = 21 passes through (0, 21) and (7, 0).

x + y = 9 passes through (9, 0) and (0, 9).

Step 3: Identify the feasible region. The feasible region can be defined as the area that is bounded by a set of coordinates that can satisfy some particular system of inequalities.

Any point that lies on or below the line x + 4y = 24 will satisfy the constraint x + 4y ≤ 24.

Similarly, a point that lies on or below 3x + y = 21 satisfies 3x + y ≤ 21.

Also, a point lying on or below the line x + y = 9 satisfies x + y ≤ 9.

The feasible region is represented by OABCD as it satisfies all the above-mentioned three restrictions.

Step 4: Determine the coordinates of the corner points. The corner points are the vertices of the feasible region.

B = (6, 3). B is the intersection of the two lines 3x + y = 21 and x + y = 9. Thus, by substituting y = 9 - x in 3x + y = 21 we can determine the point of intersection.

C = (4, 5) formed by the intersection of x + 4y = 24 and x + y = 9

Linear Programming by Graphical Method

Step 5: Substitute each corner point in the objective function. The point that gives the greatest (maximizing) or smallest (minimizing) value of the objective function will be the optimal point.

33 is the maximum value of Z and it occurs at C. Thus, the solution is x = 4 and y = 5.

Applications of Linear Programming

Linear programming is used in several real-world applications. It is used as the basis for creating mathematical models to denote real-world relationships. Some applications of LP are listed below:

  • Manufacturing companies make widespread use of linear programming to plan and schedule production.
  • Delivery services use linear programming to decide the shortest route in order to minimize time and fuel consumption.
  • Financial institutions use linear programming to determine the portfolio of financial products that can be offered to clients.

Related Articles:

  • Introduction to Graphing
  • Linear Equations in Two Variables
  • Solutions of a Linear Equation
  • Mathematical Induction

Important Notes on Linear Programming

  • Linear programming is a technique that is used to determine the optimal solution of a linear objective function.
  • The simplex method in lpp and the graphical method can be used to solve a linear programming problem.
  • In a linear programming problem, the variables will always be greater than or equal to 0.

Linear programming Example

As the minimum value of Z is 127, thus, B (3, 28) gives the optimal solution. Answer: The minimum value of Z is 127 and the optimal solution is (3, 28)

Linear Programming Problem

  • Example 3: Using the simplex method in lpp solve the linear programming problem Minimize Z = \(x_{1}\) + 2\(x_{2}\) + 3\(x_{3}\) \(x_{1}\) + \(x_{2}\) + \(x_{3}\) ≤ 12 2\(x_{1}\) + \(x_{2}\) + 3\(x_{3}\) ≤ 18 \(x_{1}\), \(x_{2}\), \(x_{3}\) ≥ 0 Solution: Convert all inequalities to equations by introducing slack variables. -\(x_{1}\) - 2\(x_{2}\) - 3\(x_{3}\) + Z = 0 \(x_{1}\) + \(x_{2}\) + \(x_{3}\) + \(y_{1}\) = 12 2\(x_{1}\) + \(x_{2}\) + 3\(x_{3}\) + \(y_{2}\) = 18 Expressing this as a matrix we get, \(\begin{bmatrix} x_{1} & x_{2} & x_{3} & y_{1} & y_{2} & Z & \\ 1 & 1 & 1 & 1 & 0 & 0 & 12\\ 2 & 1 & 3 & 0 & 1 & 0 & 18\\ -1 & -2 & -3 & 0 & 0 & 1 & 0 \end{bmatrix}\) As -3 is the greatest negative value thus, column 3 is the pivot column. 12 / 1 = 12 18 / 3 = 6 As 6 is the smaller quotient thus, row 2 is the pivot row and 3 is the pivot element. By applying matrix operations we get, \(\begin{bmatrix} x_{1} & x_{2} & x_{3} & y_{1} & y_{2} & Z & \\ 0.33 & 0.667 & 0 & 1 & -0.33 & 0 & 6\\ 0.667 & 0.33 & 1 & 0 & 0.33 & 0 & 6\\ 1 & -1 & 0 & 0 & 1 & 1 & 18 \end{bmatrix}\) Now -1 needs to be eliminated. Thus, by repreating the steps the matrix so obtained is as follows \(\begin{bmatrix} x_{1} & x_{2} & x_{3} & y_{1} & y_{2} & Z & \\ 0.5 & 1 & 0 & 1.5 & 0.5 & 0 & 9\\ 0.5 & 0 & 1 & -0.5 & 0.5 & 0 & 3\\ 1.5 & 0 & 0 & 1.5 & 0.5 & 1 & 27 \end{bmatrix}\) We get the maximum value of Z = 27 at \(x_{1}\) = 0, \(x_{2}\) = 9 \(x_{3}\) = 3 Answer: Maximum value of Z = 27 and optimal solution (0, 9, 3)

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represent assignment problem in lpp form

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Practice Questions on Linear Programming

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FAQs on Linear Programming

What is meant by linear programming.

Linear programming is a technique that is used to identify the optimal solution of a function wherein the elements have a linear relationship.

What is Linear Programming Formula?

The general formula for a linear programming problem is given as follows:

What is the Objective Function in Linear Programming Problems?

The objective function is the linear function that needs to be maximized or minimized and is subject to certain constraints. It is of the form Z = ax + by.

How to Formulate a Linear Programming Model?

The steps to formulate a linear programming model are given as follows:

  • Identify the decision variables.
  • Formulate the objective function.
  • Identify the constraints.
  • Solve the obtained model using the simplex or the graphical method.

How to Find Optimal Solution in Linear Programming?

We can find the optimal solution in a linear programming problem by using either the simplex method or the graphical method. The simplex method in lpp can be applied to problems with two or more variables while the graphical method can be applied to problems containing 2 variables only.

How to Find Feasible Region in Linear Programming?

To find the feasible region in a linear programming problem the steps are as follows:

  • Draw the straight lines of the linear inequalities of the constraints.
  • Use the "≤" and "≥" signs to denote the feasible region of each constraint.
  • The region common to all constraints will be the feasible region for the linear programming problem.

What are Linear Programming Uses?

Linear programming is widely used in many industries such as delivery services, transportation industries, manufacturing companies, and financial institutions. The linear program is solved through linear optimization method, and it is used to determine the best outcome in a given scenerio.

  • Math Article
  • Linear Programming Problem Lpp

Linear programming problem (LPP)

Linear Programming Problems (LPP): Linear programming or linear optimization is a process which takes into consideration certain linear relationships to obtain the best possible solution to a mathematical model. It is also denoted as LPP. It includes problems dealing with maximizing profits, minimizing costs, minimal usage of resources, etc. These problems can be solved through the simplex method or graphical method.

The Linear programming applications are present in broad disciplines such as commerce, industry, etc. In this section, we will discuss, how to do the mathematical formulation of the LPP.

Mathematical Formulation of Problem

Let x and y be the number of cabinets of types 1 and 2 respectively that he must manufacture. They are non-negative and known as non-negative constraints.

The company can invest a total of 540 hours of the labour force and is required to create up to 50 cabinets. Hence,

15x + 9y <= 540

x + y <= 50

The above two equations are known as linear constraints.

Let Z be the profit he earns from manufacturing x and y pieces of the cabinets of types 1 and 2. Thus,

Z = 5000x + 3000y

Our objective here is to maximize Z. Hence Z is known as the objective function. To find the answer to this question, we use graphs, which is known as the graphical method of solving LPP. We will cover this in the subsequent sections.

Graphical Method

The solution for problems based on linear programming is determined with the help of the feasible region, in case of graphical method. The feasible region is basically the common region determined by all constraints including non-negative constraints, say, x,y≥0, of an LPP. Each point in this feasible region represents the feasible solution of the constraints and therefore, is called the solution/feasible region for the problem. The region apart from (outside) the feasible region is called as the infeasible region .

The optimal value (maximum and minimum) obtained of an objective function in the feasible region at any point is called an optimal solution. To learn the graphical method to solve linear programming completely reach us.

Linear Programming Applications

Let us take a real-life problem to understand linear programming. A home décor company received an order to manufacture cabinets. The first consignment requires up to 50 cabinets. There are two types of cabinets. The first type requires 15 hours of the labour force (per piece) to be constructed and gives a profit of Rs 5000 per piece to the company. Whereas, the second type requires 9 hours of the labour force and makes a profit of Rs 3000 per piece. However, the company has only 540 hours of workforce available for the manufacture of the cabinets. With this information given, you are required to find a deal which gives the maximum profit to the décor company.

Linear Programming problem LPP

Given the situation, let us take up different scenarios to analyse how the profit can be maximized.

  • He decides to construct all the cabinets of the first type. In this case, he can create 540/15 = 36 cabinets. This would give him a profit of Rs 5000 × 36 = Rs 180,000.
  • He decides to construct all the cabinets of the second type. In this case, he can create 540/9 = 60 cabinets. But the first consignment requires only up to 50 cabinets. Hence, he can make profit of Rs 3000 × 50 = Rs 150,000.
  • He decides to make 15 cabinets of type 1 and 35 of type 2. In this case, his profit is (5000 × 15 + 3000 × 35) Rs 180,000.

Similarly, there can be many strategies which he can devise to maximize his profit by allocating the different amount of labour force to the two types of cabinets. We do a mathematical formulation of the discussed LPP to find out the strategy which would lead to maximum profit.

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  • Linear Programming Problem and Its Mathematical Formulation

Sometimes one seeks to optimize (maximize or minimize) a known function (could be profit/loss or any output), subject to a set of linear constraints on the function. Linear Programming Problems (LPP) provide the method of finding such an optimized function along with/or the values which would optimize the required function accordingly.

Browse more Topics under Linear Programming

  • Different Types of Linear Programming Problems
  • Graphical Method of Solving Linear Programming Problems

It is one of the most important Operations Research tools. It is widely used as a decision making aid in almost all industries. There can be various fields of application of LPP, in the areas of Economics, Computer Sciences, Mathematics , etc. Since it is a very important topic with numerous practical applications , we will proceed slowly in building up this topic and making it very clear to you. So, let’s begin!

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Mathematical Formulation

Formulation of an LPP refers to translating the real-world problem into the form of mathematical equations which could be solved. It usually requires a thorough understanding of the problem.

LPP

Steps towards formulating a Linear Programming problem:

  • Step 1:   Identify the ‘n’ number of decision variables which govern the behaviour of the objective function (which needs to be optimized).
  • Step 2:  Identify the set of constraints on the decision variables and express them in the form of linear equations / inequations . This will set up our region in the n- dimensional space within which the objective function needs to be optimized. Don’t forget to impose the condition of non-negativity on the decision variables i.e. all of them must be positive since the problem might represent a physical scenario, and such variables can’t be negative.
  • Step 3:  Express the objective function in the form of a linear equation in the decision variables.
  • Step 4:  Optimize the objective function either graphically or mathematically.

Now let us look at an example aimed at enabling you to learn how to formulate a problem in Linear Programming!

Download Linear Programming Problem Cheat Sheet

linear programming problem cheat sheet

Solved Examples for You

Question 1: A calculator company produces a handheld calculator and a scientific calculator. Long-term projections indicate an expected demand of at least 150 scientific and 100 handheld calculators each day. Because of limitations on production capacity, no more than 250 scientific and 200 handheld calculators can be made daily.

To satisfy a shipping contract, a minimum of 250 calculators must be shipped each day. If each scientific calculator sold, results in a 20 rupees loss, but each handheld calculator produces a 50 rupees profit; then how many of each type should be manufactured daily to maximize the net profit?

Such word problems can be very tricky to solve if not formulated properly. Hence, let us approach it in a step by step manner as discussed above.

Step 1: The decision variables : Since the question has asked for an optimum number of calculators, that’s what our decision variables in this problem would be. Let,

x = number of scientific calculators produced y =  number of handheld calculators produced

Therefore, we have 2 decision variables in this problem, namely ‘ x’  and ‘ y’.

Step 2: The constraints: Since the company can’t produce a negative number of calculators in a day, a natural constraint would be:

x ≥ 0 y ≥ 0

However, a lower bound for the company to sell calculators is already supplied in the problem. We can note it down as:

x ≥ 150 y ≥ 100

We have also been given an upper bound for these variables, owing to the limitations on production by the company. We can write as follows:

x ≤ 250 y ≤ 200

Besides, we also have a joint constraint on the values of  ‘x’ and  ‘y’ due to the minimum order on a shipping consignment; given as:

x + y ≥ 250

Step 3:  Objective Function: Clearly, here we need to optimize the Net Profit function. The Net Profit Function is given as:

P = -20x + 50y

Step 4:  Solving the problem: the system here –

Maximization of  P = -20x + 50y, subject to: 150 ≤ x ≤ 250 100 ≤ y ≤ 200 x + y ≥ 250

We can solve it graphically or mathematically as per convenience. We will discuss it in later sections. This completes the discussion on the mathematical formulation of a Linear Programming problem !

Question 2: What is meant by LPP?

Answer: The full form of LPP is Linear Programming Problems. This method helps in achieving the best outcome in a mathematical model. The best outcome could be maximum profit or the lowest cost or the best possible price. The representation of this model’s requirements is by linear relationships.

Question 3: Explain how one can calculate LPP?

Answer: In order to calculate LPP, one must follow the following steps:

  • Formulate the LP problem.
  • Construct a graph and then plot the various constraint lines.
  • Ascertain the valid side of all constraint lines.
  • Identify the region of feasible solution.
  • Plot the objective function.
  • Finally, find out the optimum point.

Question 4: What are the various advantages of LPP?

Answer: The various advantages of LPP are as follows:

  • Utilized for analyzing numerous military, social, economic social, and industrial problems.
  • Linear programming is appropriate for solving complex problems.
  • It assists in productive management of an organization for better outcomes.

Question 5: Who is credited with the development of LPP?

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Linear programming is the simplest way of optimizing a problem. Through this method, we can formulate a real-world problem into a mathematical model. There are various methods for solving Linear Programming Problems and one of the easiest and most important methods for solving LPP is the graphical method. In Graphical Solution of Linear Programming, we use graphs to solve LPP.

We can solve a wide variety of problems using Linear programming in different sectors, but it is generally used for problems in which we have to maximize profit, minimize cost, or minimize the use of resources. In this article, we will learn about Solutions of Graphical solutions of linear programming problems, their types, examples, and others in detail.

Table of Content

Graphical Solution of a Linear Programming Problems

Corner point methods, iso-cost methods.

  • Solved Examples

Linear programming is a mathematical technique employed to determine the most favorable solution for a problem characterized by linear relationships. It is a valuable tool in fields such as operations research, economics, and engineering, where efficient resource allocation and optimization are critical.

Now let’s learn about types of linear programming problems

Types of Linear Programming Problems

There are mainly three types of problems based on Linear programming they are,

Manufacturing Problem: In this type of problem, some constraints like manpower, output units/hour, and machine hours are given in the form of a linear equation. And we have to find an optimal solution to make a maximum profit or minimum cost.

Diet Problem: These problems are generally easy to understand and have fewer variables. Our main objective in this kind of problem is to minimize the cost of diet and to keep a minimum amount of every constituent in the diet. 

Transportation Problem: In these problems, we have to find the cheapest way of transportation by choosing the shortest route/optimized path.

Some commonly used terms in linear programming problems are,

Objective function: The direct function of form Z = ax + by, where a and b are constant, which is reduced or enlarged is called the objective function. For example, if Z = 10x + 7y. The variables x and y are called the decision variable.

Constraints: The restrictions that are applied to a linear inequality are called constraints.

  • Non-Negative Constraints: x > 0, y > 0 etc.
  • General Constraints: x + y > 40, 2x + 9y ≥ 40 etc.

Optimization problem: A problem that seeks to maximization or minimization of variables of linear inequality problem is called optimization problems.

Feasible Region: A common region determined by all given issues including the non-negative (x ≥ 0, y ≥ 0) constrain is called the feasible region (or solution area) of the problem. The region other than the feasible region is known as the infeasible region.

Feasible Solutions: These points within or on the boundary region represent feasible solutions of the problem. Any point outside the scenario is called an infeasible solution.

Optimal(Most Feasible) Solution: Any point in the emerging region that provides the right amount (maximum or minimum) of the objective function is called the optimal solution.

  • If we have to find maximum output, we have to consider the innermost intersecting points of all equations.
  • If we have to find minimum output, we consider the outermost intersecting points of all equations.
  • If there is no point in common in the linear inequality, then there is no feasible solution.

We can solve linear programming problems using two different methods are,

To solve the problem using the corner point method you need to follow the following steps:

Step 1: Create mathematical formulation from the given problem. If not given.

Step 2: Now plot the graph using the given constraints and find the feasible region.

Step 3: Find the coordinates of the feasible region(vertices) that we get from step 2.

Step 4: Now evaluate the objective function at each corner point of the feasible region. Assume N and n denotes the largest and smallest values of these points.

Step 5: If the feasible region is bounded then N and n are the maximum and minimum value of the objective function. Or if the feasible region is unbounded then:

  • N is the maximum value of the objective function if the open half plan is got by the ax + by > N has no common point to the feasible region. Otherwise, the objective function has no solution.
  • n is the minimum value of the objective function if the open half plan is got by the ax + by < n has no common point to the feasible region. Otherwise, the objective function has no solution.

Examples on LPP using Corner Point Methods

Example 1: Solve the given linear programming problems graphically: 

Maximize: Z = 8x + y

Constraints are,

  • 2x + y ≤ 60
  • x ≥ 0, y ≥ 0

Step 1: Constraints are,

Step 2: Draw the graph using these constraints. 

Graph for Z = 8x + y

Here both the constraints are less than or equal to, so they satisfy the below region (towards origin). You can find the vertex of feasible region by graph, or you can calculate using the given constraints:

 x + y = 40 …(i)

2x + y = 60 …(ii)

Now multiply eq(i) by 2 and then subtract both eq(i) and (ii), we get

Now put the value of y in any of the equations, we get

x = 20 

So the third point of the feasible region is (20, 20)

Step 3: To find the maximum value of Z = 8x + y. Compare each intersection point of the graph to find the maximum value

So the maximum value of Z = 240 at point x = 30, y = 0.

Example 2: One kind of cake requires 200 g of flour and 25g of fat, and another kind of cake requires 100 g of flour and 50 g of fat Find the maximum number of cakes that can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients, used in making the cakes.

Solution: 

Step 1: Create a table like this for easy understanding (not necessary).

Step 2: Create linear equation using inequality

  • 200x + 100y ≤ 5000 or 2x + y ≤ 50
  • 25x + 50y ≤ 1000 or x + 2y ≤ 40
  • Also, x > 0 and y > 0

Step 3: Create a graph using the inequality (remember only to take positive x and y-axis)

Corner Point Method Example 2

Step 4: To find the maximum number of cakes (Z) = x + y. Compare each intersection point of the graph to find the maximum number of cakes that can be baked.

Clearly, Z is maximum at co-ordinate (20, 10). So the maximum number of cakes that can be baked is Z = 20 + 10 = 30.

The term iso-cost or iso-profit method provides the combination of points that produces the same cost/profit as any other combination on the same line. This is done by plotting lines parallel to the slope of the equation.

To solve the problem using Iso-cost method you need to follow the following steps:

Step 3: Now find the coordinates of the feasible region that we get from step 2.

Step 4: Find the convenient value of Z(objective function) and draw the line of this objective function.

Step 5: If the objective function is maximum type then draw a line which is parallel to the objective function line and this line is farthest from the origin and only has one common point to the feasible region. Or if the objective function is minimum type then draw a line which is parallel to the objective function line and this line is nearest from the origin and has at least one common point to the feasible region.

Step 6: Now get the coordinates of the common point that we find in step 5. Now, this point is used to find the optimal solution and the value of the objective function.

Linear Programming Graphical Solution of Linear Inequalities

Solved Examples of Graphical Solution of LPP

Maximize: Z = 50x + 15y

  • 5x + y ≤ 100

Step 1: Finding points 

We can also write as 

5x + y = 100….(i)

x + y = 50….(ii)

Now we find the points 

so we take eq(i), now in this equation

When x = 0, y = 100

When y = 0, x = 20

So, points are (0, 100) and (20, 0)

Similarly, in eq(ii)

When x = 0, y = 50

When y = 0, x = 50

So, points are (0, 50) and (50, 0)

Step 2: Now plot these points in the graph and find the feasible region.

Solved Example 1: Z = 50x + 15y

Step 3: Now we find the convenient value of Z(objective function) 

So, to find the convenient value of Z, we have to take the lcm of coefficient of 50x + 15y, i.e., 150. So, the value of Z is the multiple of 150, i.e., 300. Hence, 

50x + 15y = 300

Put x = 0, y = 20

Put y = 0, x = 6

draw the line of this objective function on the graph:

Solved Example 1 Z = 50x + 15y Step 3

Step 4: As we know that the objective function is maximum type then we draw a line which is parallel to the objective function line and farthest from the origin and only has one common point to the feasible region.

Solved Example 1: Z = 50x + 15y Step 4

Step 5: We have a common point that is (12.5, 37.5) with the feasible region. So, now we find the optimal solution of the objective function:

Z = 50x + 15y

Z = 50(12.5) + 15(37.5)

Z = 625 + 562.5

Thus, maximum value of Z with given constraint is, 1187

Example 2: Solve the given linear programming problems graphically: 

Minimize: Z = 20x + 10y

  • x + 2y ≤ 40
  • 3x + y ≥ 30
  • 4x + 3y ≥ 60

l 1 = x + 2y = 40 ….(i)

l 2 = 3x + y = 30 ….(ii)

l 3 = 4x + 3y = 60 ….(iii)

So we take eq(i), now in this equation

When x = 0, y = 20

When y = 0, x = 40

So, points are (0, 20) and (40, 0)

When x = 0, y = 30

When y = 0, x = 10

So, points are (0, 30) and (10, 0)

Similarly, in eq(iii)

When y = 0, x = 15

So, points are (0, 20) and (15, 0)

Solved Example 2: Z = 20x + 10y Step 2

So let us assume z = 0

20x + 10y = 0

Solved Example 2: Z = 20x + 10y Step 3

Step 4: As we know that the objective function is minimum type then we draw a line which is parallel to the objective function line and nearest from the origin and has at least one common point to the feasible region.

Solved Exanple 2: Z = 20x + 10y Step 4

This parallel line touch the feasible region at point A. So now we find the coordinates of point A:

As you can see from the graph at point A l2 and l3 line intersect so we find the coordinate of point A by solving these equations:

l 2 = 3x + y = 30 ….(iv)

l 3 = 4x + 3y = 60 ….(v)

Now multiply eq(iv) with 4 and eq(v) with 3, we get

12x + 4y = 120 

12x + 9y = 180 

Now subtract both the equation we get coordinates (6, 12)

Step 5: We have a common point that is (6, 12) with the feasible region. So, now we find the optimal solution of the objective function:

Z = 20x + 10y

Z = 20(6) + 10(12)

Z = 120 + 120

Thus, the minimum value of Z with the given constraint are 240

FAQs on Graphical Solution of LPP

1. what is linear programming problems(lpp).

Linear Programming Problems are the mathematical problems that are used to solve or optimize the mathematical problems. We can solve Linear Programming Problems to maximize and minimize the special linear condition.

2. What are Types of Linear Programming Problems(LPP) Solution?

There are various types of Solution to Linear Programming Problems that are, Linear Programming Problems Solution by Simplex Method Linear Programming Problems Solution by R Method Linear Programming Problems Solution by Graphical Method

3. What are Types of Graphical Solution of Linear Programming Problems(LPP)?

There are various types of graphical solution of linear programming problems that are, Corner Points Methods Iso-Cost Methods

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Linear Programming Problem (LPP) Formulation with Numericals

To formulate Linear Programming Problem (LPP) for given statement type problems(numerical) is easy if we go through its Mathematical Model.

General Mathematical Modelling of LP

As explained in the video, mathematical model of LP consists of the following:

i) Objective Function (denote with “Z”) ii) Decision variables (represented in terms of “x”) iii) Constraints

Note: To know more about above terms you can go through the notes of the Basics of Linear programming, link to the notes- click here

Now, let us generate the mathematical model of the LPP:

Let, Decision variables be x 1 , x 2 , x 3 , x 4 , … , x n x_1, x_2, x_3, x_4,…, x_n x 1 ​ , x 2 ​ , x 3 ​ , x 4 ​ , … , x n ​ with total ‘m’ constraints. Now, LP can be formulate as:

Thus, we will have any of the three conditions applied on constraints as per the provided problem ( ≤ , = , ≥ ) (≤ , = , ≥) ( ≤ , = , ≥ ) .

Steps for LPP Formulation

Identify the decision variables: It is the most important step on LPP Formulation, because based on the decision variables only, the whole problem governs. We will learn about, how to identify decision variables from given numerical/problem in this note only. (Also you can go through video for the same. Link provided in cover page)

Identify Objective Function (Z) and express it as a linear function of decision variables: As we have seen above in mathematical model, objective of any problem can be maximize or minimize. Based on that, we will have to define the objective function in terms of decision variables. (For eg: Maximize Z = 2 x 1 + 5 x 2 + 9 x 3 Z= 2x_1 + 5x_2 + 9x_3 Z = 2 x 1 ​ + 5 x 2 ​ + 9 x 3 ​ )

Identify all constraints and express it as linear inequalities or equalities in terms of decision variables: As we know that, in the real world all the resources are limited and so in each particular problem you will find some limitation/constraint on the use of available resources.

Express decision variables as Feasible variables: It means that, we have lastly define all the decision variables are greater than or equal to zero (For eg:   x 1 , x 2 , x 3 ≥ 0 \ x_1, x_2, x_3 ≥ 0   x 1 ​ , x 2 ​ , x 3 ​ ≥ 0 )

Let’s move to numerical now,

represent assignment problem in lpp form

Numerical 1: Two products ‘A’ and ‘B’ are to be manufactured. Single unit of ‘A’ requires 2.4 minutes of punch press time and 5 minutes of assembly time, while single unit of ‘B’ requires 3 minutes of punch press time and 2.5 minutes of welding time. The capacity of punch press department, assembly department and welding department are 1200 min/week, 800 min/week and 600 min/week respectively. The profit from ‘A’ is ₹60 and from ‘B’ is ₹70 per unit. Formulate LPP such that, profit is maximized.

Tip: Always relate/treat LPP with real life problems for better understanding and ease of solution

Step 1: Identify the decision variables

Here, the decision has to be taken for how much quantities of product ‘A’ and ‘B’ to be manufactured in order to maximize the profit. (Quantities of product is governing this problem)

Let, Quantity of product ‘A’ manufactured = x 1 x_1 x 1 ​ Quantity of product ‘B’ manufactured = x 2 x_2 x 2 ​

Step 2: Identify Objective Function (Z)

Here, the objective is to maximize the profit from product ‘A’ and ‘B’. Each unit produced product ‘A’ gives profit of ₹60, while ‘B’ gives ₹70. So, the objective function can be defined in terms of decision variables as:

Step 3: Identify all the constraints

Product ‘A’ manufactured by: Punch press and Assembly Product ‘B’ manufactured by: Punch press and Welding

So, we have a total of three processes for the manufacturing of two products A and B. Thus, there are total three constraints.

Now, we will bifurcate this problem based on processes to get a clear idea of constraints on resources, as follows:

Product ‘A’ requires 2.4 min and product ‘B’ requires 3 min of Punch press time . Also, punch press time is limited to 1200 min/week .

Product ‘A’ requires 5 min of Assembly time . Also, assembly time is limited to 800 min/week .

Product ‘B’ requires 2.5 min of Welding time . Also, welding time is limited to 600 min/week .

Step 4: Express decision variables as feasible variables

We assume that the decision variables has feasible solution, that implies - decision variables are greater than or equal to zero.

So, complete LP can be written as,

Note that, here the solution provided (in above numerical) to you with steps is just for explanation. You can always skip this step wise solution to LP problems and can use direct method as provided in video of this topic (click here to watch video) .

Numerical 2: A tailor prepares two kinds of dresses. First kind of dress is having raw material cost of ₹150 and labour cost of ₹80, while the second kind of dress is raw material cost of ₹250 and labour cost of ₹100. He can sell the dresses of first and second kind at the rate of ₹300 and ₹500 respectively. First kind of dress takes 2 hours and second kind takes 2.5 hours of cutting. The total labour hours are restricted to 84 hours/week. Also, first kind of dress takes 1.5 hours of stitching and second requires 2 hours of stitching. The stitching hours are restricted to 60 hours/week. Formulate LPP, such that maximum profit can be earned by tailor.

The solution of this numerical will be such that, you can present this in your examination. This will save your time and gives complete idea of your explanation too in examination. All the explanation stuff will be provided in {"Explanation"} brackets for your understanding only, in the solution of this problem.

Let, Number of first kind of dress = x 1 x_1 x 1 ​ units Number of first kind of dress = x 2 x_2 x 2 ​ units

{The decision variables selected here are the units of dresses of both kind so produced, as we are going to maximize the profit of tailor. Also, by understanding the whole problem you can easily make out that, to increase the profit, tailor needs to produce more goods.}

Now, Net Profit earned by selling a unit of first kind of dress = ₹ ( 300 − 150 − 80 ) = ₹ 70 = ₹(300-150-80) = ₹70 = ₹ ( 300 − 150 − 80 ) = ₹70

Net Profit earned by selling a unit of first kind of dress = ₹ ( 500 − 250 − 100 ) = ₹ 150 = ₹(500-250-100) = ₹150 = ₹ ( 500 − 250 − 100 ) = ₹150

{As you can understand that the problem provided here is based on revenue so generated by selling different kinds of dresses. Now, revenue will be the profit generated after excluding the raw material and manufacturing cost on each of these dresses. So, for example you can see above we have excluded raw material cost of first kind of dress ₹150 and labour cost ₹80 from total selling cost of ₹300 . Thus revenue/net profit generated by selling each unit of first kind of dress is ₹ (300-150-80) = ₹70 and the same for the second kind of dress.}

∴ The objective function can be defined as;

Now, constraints are provided on labour hours for cutting and stitching. Thus, LP is subject to the following constraints;

{As we can find from problem provided, the cutting and stitching hour requirements for both kinds of dresses. Also total hours are restricted on cutting and stitching is 84 hours/week and 60 hours/week respectively.}

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Stay in touch, [notes] operation research, [notes] dynamics of machinery, [notes] maths, [notes] science, [notes] computer aided design.

IMAGES

  1. Maximization Assignment Problem LPP

    represent assignment problem in lpp form

  2. Linear Programming Problem LPP BASIC Answer of Assignment 1

    represent assignment problem in lpp form

  3. Formulating LPP from Word Problem

    represent assignment problem in lpp form

  4. Assignment LPP. & Assignment Model Doc

    represent assignment problem in lpp form

  5. Mathematical Formulation of Linear Programming Problem (LPP)

    represent assignment problem in lpp form

  6. Maximization LPP problem using Graphical Solution

    represent assignment problem in lpp form

VIDEO

  1. ANOUS & XXISIMON

  2. MALAYSIA TAMIL NEWS 23-08-2023

  3. Lightcodes Manifested and Anchored on Behalf of the Collective

  4. Assignment problem

  5. Assignment Three

  6. Linear Programming Problem

COMMENTS

  1. Assignment Model

    How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model. → Assignment Model is a special application of Linear Programming (LP). → The mathematical formulation for Assignment Model is given below: → Let, Cij denotes the cost of resources 'i' to the task 'j'; such that

  2. A linear Programming Formulation of Assignment Problems

    Mathemtical LP Model for assignment problem Some linear programming models for the assignment problem is presented .It is assumed that the cost (or time) for every machine is known denoting that: Cij=is the cost of machine(j). Xij=is the element machining job(i)on position in the job-machine matrix.

  3. PDF Linear programming 1 Basics

    A linear program in canonical form can be replaced by a linear program in standard form by just replacing Ax bby Ax+ Is= b, s 0 where sis a vector of slack variables and Iis the m m identity matrix. Similarly, a linear program in standard form can be replaced by a linear program in canonical form by replacing Ax= bby A0x b0where A0= A A and b0 ...

  4. Assignment problem

    The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. [1]

  5. Assignment Problem in Linear Programming : Introduction and Assignment

    Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum.

  6. PDF Formulating Linear Programming Models

    Assignment Steps for Developing an Algebraic LP Model What decisions need to be made? Define each decision variable. What is the goal of the problem? Write down the objective function as a function of the decision variables. What resources are in short supply and/or what requirements must be met?

  7. Solving Assignment Problem using Linear Programming in Python

    1 Assignment Problem 2 Problem Formulation 3 Initialize LP Model 4 Define Decision Variable 5 Define Objective Function 6 Define the Constraints 7 Solve Model 8 Summary Assignment Problem

  8. Assignment Problem, Linear Programming

    The assignment problem is a special type of transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

  9. Elements of a Linear Programming Problem (LPP)

    Elements of a basic LPP. Decision Variables: These are the unknown quantities that are expected to be estimated as an output of the LPP solution. Objective Function: All linear programming problems aim to either maximize or minimize some numerical value representing profit, cost, production quantity, etc.

  10. PDF Section 2.1

    A linear programming problem with a bounded set always has an optimal solution. This means that a bounded set has a maximum value as well as a minimum value. Example 1: Given the objective function P = 10 x − 3 y and the following feasible set, Find the maximum value and the point where the maximum occurs.

  11. Linear Programming

    Important Notes on Linear Programming. Linear programming is a technique that is used to determine the optimal solution of a linear objective function. The simplex method in lpp and the graphical method can be used to solve a linear programming problem. In a linear programming problem, the variables will always be greater than or equal to 0.

  12. PDF Lecture 7 1 Linear Programming Relaxations

    In which we show how to use linear programming to approximate the vertex cover problem. 1 Linear Programming Relaxations An integer linear program (abbreviated ILP) is a linear program (abbreviated LP) with the additional constraints that the variables must take integer values. For ex-ample, the following is an ILP: maximize x 1 x 2 + 2x 3 ...

  13. Applications of Linear Programming Problem (LPP)

    LPP applications may include production scheduling, inventory policies, investment portfolio, allocation of advertising budget, construction of warehouses, etc. In this article, we would focus on the different components of the output generated by Microsoft excel while solving a basic LPP model. We would solve and discuss four examples together ...

  14. Linear Programming Problem (LPP)- Simplex and Graphical method

    Math Article Linear Programming Problem Lpp Linear programming problem (LPP) Linear Programming Problems (LPP): Linear programming or linear optimization is a process which takes into consideration certain linear relationships to obtain the best possible solution to a mathematical model. It is also denoted as LPP.

  15. Linear Programming Problem and Its Mathematical Formulation

    Step 1: Identify the 'n' number of decision variables which govern the behaviour of the objective function (which needs to be optimized). Step 2: Identify the set of constraints on the decision variables and express them in the form of linear equations / inequations.

  16. PDF Linear Programming: Model Formulation and Solution

    Linear programming uses linear algebraic relationships to represent a firm's decisions, given a business objective, and resource constraints. Steps in application: 1. Identify problem as solvable by linear programming. 2. Formulate a mathematical model of the unstructured problem. 3. Solve the model. 4. Implementation Introduction

  17. Graphical Solution of Linear Programming Problems

    Step 2: Create linear equation using inequality. 200x + 100y ≤ 5000 or 2x + y ≤ 50. 25x + 50y ≤ 1000 or x + 2y ≤ 40. Also, x > 0 and y > 0. Step 3: Create a graph using the inequality (remember only to take positive x and y-axis) Step 4: To find the maximum number of cakes (Z) = x + y.

  18. PDF Exact extended formulation of the linear assignment problem (LAP

    tours. It can be solved optimally using any linear programming (LP) solver, hence o ering a new (incidental) proof of the equality of the computational complexity classes \P" and \NP." The extensions of the model to the time-dependent traveling salesman problem (TDTSP) as well as the quadratic assignment problem (QAP) are straightforward.

  19. PDF Tutorial 6: Converting a linear program to standard form

    transformed problem, then there is a feasible solution for the original problem with the same objective value. For example, if there is a feasible solution with y. 3 = 1, and w. 3 = 5, then there is a feasible solution for the original problem with the same objective value. In this case, let x. 3 = -4. But for every solution to the

  20. PDF Solving Assignment Problem, a Special Case of Transportation Problem by

    In this paper assignment problem represented as linear programming problem (LPP) and then different methods like two-phase method &Big-M methods are used to solve LPP.

  21. PDF UNIT 2 LINEAR PROGRAMMING PROBLEMS

    2.2 LINEAR PROGRAMMING PROBLEM (LPP) In Unit 1, you have learnt about the concept of optimisation. Usually, the requirements of any agency, industry or country far exceed their limited resources of land, workforce, capital, organisational facilities, etc.

  22. PDF Chapter 5: Linear Programming: Transportation and Assignment Models

    discussion of transportation problems by formulating a linear programming model. First we will proceed to identify the decision variables, constraints and objective functions as we do in general linear programming. In general, the following steps define a transportation problem: 1. A set of m supply points from which a good is shipped.

  23. Linear Programming Problem (LPP) Formulation with Numericals

    Solution: Tip: Always relate/treat LPP with real life problems for better understanding and ease of solution. Step 1: Identify the decision variables. Here, the decision has to be taken for how much quantities of product 'A' and 'B' to be manufactured in order to maximize the profit.