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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 in linear programming : introduction and assignment model.

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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

clip_image005

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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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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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Encyclopedia of Operations Research and Management Science pp 882–888 Cite as

Linear Programming

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Introduction

Linear programming is one of the most widely used techniques of operations research and management science. Its name means that planning (programming) is being done with a mathematical model (called a linear-programming model) where all the functions in the model are linear functions.

Linear Programming Models

Linear programming models come in a variety of forms. To illustrate one common form, consider the problem of determining the most profitable mix of products for a manufacturer. Let n be the number of possible products. For each product j ( j = 1, 2, …, n ), a decision variable x j is introduced to represent the decision on its production rate. Let c j be the profit per unit of product j produced, and let Z be the total rate of profit resulting from the choice of product mix. This choice is constrained by the limited capacities of the production facilities available for these products. Let m be the number of different types of facilities needed. For each type i ( i = 1,...

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Hillier, F.S. (2013). Linear Programming. In: Gass, S.I., Fu, M.C. (eds) Encyclopedia of Operations Research and Management Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1153-7_545

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Learning Objectives

In this chapter, you will:

  • Investigate real world applications of linear programming and related methods.
  • Solve linear programming maximization problems using the simplex method.
  • Solve linear programming minimization problems using the simplex method.
  • 4.1: Introduction to Linear Programming Applications in Business, Finance, Medicine, and Social Science In this section, you will learn about real world applications of linear programming and related methods.
  • 4.2.1: Maximization By The Simplex Method (Exercises)
  • 4.3.1: Minimization By The Simplex Method (Exercises)
  • 4.4: Chapter Review

Thumbnail: Polyhedron of simplex algorithm in 3D. (CC BY-SA 3.0; Sdo via Wikipedia)

Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

For example:

Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

The cost of performing each task by each resource is also known (shown in cells of matrix)

Fig 1-assigment model intro

  • In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

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, C i j \text {C}_{ij} C ij ​ denotes the cost of resources 'i' to the task 'j' ; such that

assignment method linear programming

→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

→ Subjected to constraint;

(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:

(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;

(iii) x i j x_{ij} x ij ​ is '0' or '1'.

Types of Assignment Problem:

(i) balanced assignment problem.

  • It consist of a suqare matrix (n x n).
  • Number of rows = Number of columns

(ii) Unbalanced Assignment Problem

  • It consist of a Non-square matrix.
  • Number of rows ≠ \not=  = Number of columns

Methods to solve Assignment Model:

(i) integer programming method:.

In assignment problem, either allocation is done to the cell or not.

So this can be formulated using 0 or 1 integer.

While using this method, we will have n x n decision varables, and n+n equalities.

So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

But, here in assignment problems, the matrix is a square matrix (m=n).

So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.

The assignments which gives minimum cost is selected as optimal solution.

But, such trail and error becomes very difficult and lengthy.

If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.

(iv) Hungarian Method:

It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

It is also know as Reduced matrix method or Flood's technique.

There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n). (2) Problem should be of minimization type.

Suggested Notes:

Unbalanced Transportation Problem Numerical

Unbalanced Transportation Problem Numerical

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Critical Path Method [CPM] - Steps and Introduction | Network Analysis | Operation Research

Critical Path Method [CPM] - Steps and Introduction | Network Analysis | Operation Research

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Indirect cost less than Crash Cost

Crashing Special Case - Indirect cost less than Crash Cost

Basics of Program Evaluation and Review Technique (PERT)

Basics of Program Evaluation and Review Technique (PERT)

Numerical on PERT (Program Evaluation and Review Technique)

Numerical on PERT (Program Evaluation and Review Technique)

Network Analysis - Dealing with Network Construction Basics

Network Analysis - Dealing with Network Construction Basics

Construct a project network with predecessor relationship | Operation Research | Numerical

Construct a project network with predecessor relationship | Operation Research | Numerical

Graphical Method | Methods to solve LPP | Linear Programming

Graphical Method | Methods to solve LPP | Linear Programming

Basics of Linear Programming

Basics of Linear Programming

Linear Programming Problem (LPP) Formulation with Numericals

Linear Programming Problem (LPP) Formulation with Numericals

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scipy.optimize.linear_sum_assignment #

Solve the linear sum assignment problem.

The cost matrix of the bipartite graph.

Calculates a maximum weight matching if true.

An array of row indices and one of corresponding column indices giving the optimal assignment. The cost of the assignment can be computed as cost_matrix[row_ind, col_ind].sum() . The row indices will be sorted; in the case of a square cost matrix they will be equal to numpy.arange(cost_matrix.shape[0]) .

for sparse inputs

The linear sum assignment problem [1] is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a ‘worker’) and vertex j of the second set (a ‘job’). The goal is to find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

where, in the case where the matrix X is square, each row is assigned to exactly one column, and each column to exactly one row.

This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

This implementation is a modified Jonker-Volgenant algorithm with no initialization, described in ref. [2] .

New in version 0.17.0.

https://en.wikipedia.org/wiki/Assignment_problem

DF Crouse. On implementing 2D rectangular assignment algorithms. IEEE Transactions on Aerospace and Electronic Systems , 52(4):1679-1696, August 2016, DOI:10.1109/TAES.2016.140952

What Is Linear Programming? Meaning, Methods, and Examples

Linear programming helps determine how to arrive at the most optimized situation given a set of resource constraints.

Linear programming is defined as a technique in algebra that uses linear equations to figure out how to arrive at the optimal situation (maximum or minimum) as an answer to a mathematical problem, assuming the finiteness of resources and the quantifiable nature of the end optimization goal. This article explains how linear programming works with examples.

Table of Contents

What is linear programming, linear programming methods, examples of linear programming.

Linear programming is a technique in algebra that uses linear equations to determine how to arrive at the optimal situation (maximum or minimum) as an answer to a mathematical problem, assuming the finiteness of resources and the quantifiable nature of the end optimization goal. 

Linear programming (LP) uses many linear inequalities pertaining to a given scenario to determine the “optimal” value one can obtain under those constraints. A classic example would be calculating the “optimal” production levels to maximize profits, given the restrictions of supplies and personnel.

In the “real world,” linear programming is an essential subfield of mathematics known as optimization methods. This area of research (or at least its applicable findings) is used in resource allocation and management. These “real-world” systems may have dozens or even hundreds of variables. In algebra, however, you will only see the basic (and graphable) linear example with two variables.

Graphing the inequalities (called “constraints”) to construct a walled-off zone on the x,y plane is the typical method for solving linear programming problems (known as “feasibility region”). Then, you determine the dimensions of the extremities of this feasible zone (i.e., the intersection locations of the different pairs of lines) and evaluate these vertices in the equation (termed “optimization equation”) in which you’re attempting to get the maximum or minimum value.

Linear programming (LP) is among the most straightforward optimization techniques. It simplifies specific, complicated linear programming and optimization issues to help you reach a solution. Data analysts will always encounter applications and challenges requiring linear programming solutions.

Linear programming formula

A linear programming issue includes choice parameters, a nonlinear objective, constraints, and nonnegative limitations. The outcome of the LP model is determined by the choice variables x and y, which also reflect the ultimate answer.

Z is the function that must be optimized (maximized or minimized) to arrive at an answer. The constrictions are the limits placed on the variables to limit their values. According to the non-negative limitations, the variables must always possess a non-negative value.

The linear programming formula may be regarded as follows:

  • The function of the formula: ax + by = Z
  • The formula’s operating limitations: cx + dy ≤ e and fx + gy ≤ h
  • Other, non-negative restrictions: x ≥ 0, y ≥ 0

You need to know a few terms to understand the meaning of linear programming. First come the decision variables. These elements fight for limited resources, including products, services, etc. They are referred to as decision variables if they are connected with a linear connection capable of determining the most optimal option.

The next component would be the objective function. The challenge must have had a quantitatively measurable aim, such as maximizing profit, reducing costs, etc. These constraints also apply to the available resources, such as limited equipment, people, materials, etc. Some constraints are observably existent but do not hamper the process of the studied issue; they are referred to as redundant constraints.

A viable solution is the collection of all potential solutions that fulfill the constants in the format of variables. In addition, an optimum value is the best possible solution that effectively supports the problem’s aim.

The most crucial step in addressing a linear programming issue is formulating it using the provided data. The following are the steps for solving linear programming problems:

  • Determine the choice factors
  • Develop the objective function 
  • Determine whether the function should be decreased or maximized
  • Record the limitations
  • Verify that decision variables are either larger than or equal to 0. (Non-negative inhibition)
  • Utilize either the simplex or graphical method to resolve the linear programming issue

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Why is linear programming necessary?

We all encounter several target-based scenarios daily. Suppose a student has 15 days to finish an assignment, or a salesperson has one month to reach their sales quota, while another individual gets $600 to spend on an electronic device.

Let’s imagine that the student’s purpose for this endeavor is to get the highest possible grade. The salesman would strive for the highest possible monthly sales total. The purchaser of a device would want to decrease the price as much as feasible. They would attempt to purchase a device within their budget. The purpose of each situation described above is to maximize the benefits or reduce costs. Such issues are optimization challenges that may be resolved by linear programming.

An optimization issue in mathematics may include maximizing profit, minimizing costs, or minimizing resource use. We have previously described the aims of the three presented circumstances; we can now examine their respective limiting considerations. What does it imply?

In every instance, resources are scarce. In the first scenario, the deadline for completing the assignment is tight. Similarly, in example two, the individual must sell the greatest amount of things within a given time frame. In the third scenario, the individual must purchase the device within a defined budget; hence, the quantity of money is the restricting element. The lack of available resources hinders the search for optimal solutions to the presented challenges.

One cannot use the typical calculus and marginal analysis techniques in these circumstances. Calculus approaches can only handle precisely equal constraints, a limitation that linear programming does not have.

Numerous real-world applications make use of linear programming. It serves as the foundation for mathematical models that represent real-world connections. To organize and schedule production, manufacturing businesses employ linear programming extensively. To decrease travel time and fuel consumption, delivery services employ linear programming to determine the shortest route. Financial institutions establish the spectrum of investment instruments that may be supplied to customers using linear programming.

Linear programming delivers vital insights into business challenges by facilitating the identification of the ideal solution in each given circumstance.

Traits of a linear programming task

A problem being solved through linear programming will have the following traits or characteristics:

  • Subject to constraints : Regarding the resource, one should represent the restrictions in mathematical form.
  • Geared towards an objective function : The objective function of an issue should be described quantitatively.
  • A linear relationship : The function’s connection across two or more independent variables must be linear. It indicates that the variable’s degree is one.
  • Includes only finite numbers : There should be output and input numbers that are both finite and infinite. The optimum solution is not implementable if the function contains an unlimited number of elements.
  • Does not include negative values : The variable’s value must be zero or positive. The value should not be negative.
  • Hinges on decision variables : The result is determined by the decision variable. It provides the final solution to the issue. The first step in solving any issue is to determine the decision factors.

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There are several approaches to solving linear programming problems. The four most important approaches are:

1. The simplex method 

The simplex method is a typical methodology for tackling optimization problems in linear programming. Typically, it consists of a function and some restrictions written as inequalities. The inequality defines a polygonal area, with the solution often located at a vertex. This approach is a method for systematically examining the vertices as potential solutions.

The technique approaches and finally achieves the maximum or lowest value of the goal function via an iterative procedure. In addition, the strategy aids the decision-maker in identifying duplicate restrictions, a complete solution, several alternatives, and an infeasible issue, thereby providing a thorough grasp of the business situation.

A twin problem accompanies every linear programming issue. One may easily derive the answer to this issue using the simplex approach from resolving the initial problem.

George Dantzig created the simplex technique for linear programming. Dantzig developed planning systems for the United States Air Defense during World War II using a desk calculator. In 1946, a coworker challenged him to automate the planning procedure to prevent him from accepting another position. Dantzig defined the issue as linear inequalities, although he did not include an aim in his formulation at the time. Without a goal, a wide variety of plausible options exist. Therefore, military-specific “ground rules” should be applied to identify the best possible alternative.

Dantzig’s key realization was that most such ground rules might be expressed as a linear function of objectives that must be maximized.

2. Solving linear programming problems using R

Linear programming is an excellent tool for decision-making optimization. Several R programs, such as the lpSolve R package, enable the solution of linear programming difficulties. lpSolve is an R extension that allows links to a C-based framework for linear programming problem-solving. With only a few bits of open-source code, you may get statistically significant information (sensitivity analysis).

Whereas other free optimization solutions are available, having a linear programming R code in one’s individual code repository may save a considerable amount of time by eliminating the need to start writing the formula from scratch and requiring only the modification of the coefficients and signs of the respective matrices. This is helpful since R is regularly used for data science and statistical analysis. 

3. Graphical linear programming

The technique of resolving a linear equation system by generating a graph is often referred to as the graphical method. The same holds true for linear programming issues.

Using graphical approaches, it is simple to solve any optimization programming issue with just two variables. These variables may be referred to as x1 and x2, and most of the analysis can be performed on a two-dimensional graph using these variables. The graphical approach for solving linear programming employs the extreme or corner points method and the iso-profit (cost) efficiency line method.

The iso-cost or iso-profit approach identifies the point combination that yields the same costs/profits as any other point combinations on the same line. This is accomplished by drawing parallel lines to the gradient of the equation.

4. Linear programming using OpenSolver

OpenSolver is a tool designed to solve models of linear and integer programming. OpenSolver is an Excel VBA add-on that expands the capabilities of Excel’s built-in Solver. Andrew Mason developed and updated it with students at the University of Auckland’s Engineering Science department. In addition, it permits you to resolve linear and mixed-integer optimization methods in Google Sheets without arbitrary size restrictions.

It’s important to note that almost all linear programming and mixed-integer linear programming libraries that are widely used are authored in Fortran, C, or C++ and are native to those languages. This is because linear programming requires a lot of work with (often sizable) matrices, which is hard to do in a language like Python . This kind of library is called a solver.

5. Mixed-integer linear programming

Linear programming can be made even more robust with mixed-integer linear programming. It can solve problems in which at least one variable has a discrete integer value instead of a continuous value. At first glance, mixed-integer problems look like continuous variable problems, but they are much better in flexibility and accuracy.

Integer variables are essential for accurately representing numbers expressed with integers, like the number of airplanes made or the number of customers served. The binary variable is a type of integer variable that is very important. It can only have the values 0 or 1 and helps make yes-or-no decisions, like whether or not to build a plant or turn on a machine. You can also use them to imitate logical constraints.

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The Linear Programming Problem (LPP) involves finding the best value of a given linear function. The ideal value may either be the largest or smallest one. Here, the linear function provided is regarded as an objective function. The objective function may include several variables dependent on the situation and must fulfill the linear constraints, a collection of linear inequalities. One may utilize linear programming issues to determine the ideal solution for manufacturing, diet, transportation, and allocation problems, among others.

Listed below are a few illustrations of the kind of issues commonly addressed by linear programming techniques:

Example 1: Optimizing dietary needs and cost constraints

A doctor wants to combine two food kinds such that the mixture’s vitamin content includes a minimum of 8 elements of vitamin A and ten elements of vitamin C. Food ‘I’ includes 2 vitamin A units per kilogram and 1 vitamin C unit per kilogram. Food ‘II’ has 1 vitamin A unit per kilogram and 2 vitamin C units per kilogram. Food ‘I’ is priced at $5 per kilogram, whereas Food ‘II’ costs $7 per kilogram. To minimize the price of such a combination, this may be expressed as a problem of linear programming.

Example 2: Optimizing food ingredients and food volume

One type of cake calls for 200g of flour and 25g of fat, but another one calls for 100g of flour and 50g of fat. This issue may be expressed as a linear programming problem to determine the highest proportion of cake that can be baked using 5kg of wheat and 1kg of fat. It also implies that there are sufficient quantities of the other cake-making components.

Example 3: Optimizing goods transportation costs

Consider a manufacturing business with two plants in cities F1 & F2 and three retail outlets in cities C1, C2, or C3. Monthly demand at retail locations is 8, 5, and 2 units, whereas monthly supply at manufacturers is 6 and 9, accordingly. Notice that the entire supply and demand are equal. We are also provided with the cost of transporting one unit from manufacturing to retail outlets. This linear programming challenge aims to estimate the amount that must be shipped from each manufacturer to each retail area to satisfy demand at the lowest possible total shipping cost.

Example 4: Optimizing product sales to arrive at maximum profit

A bakery produces two types of cookies: chocolate chip and caramel. The bakery anticipates daily demand for a minimum of 80 caramelized & 120 chocolate chip cookies. Due to a lack of raw materials and labor, the bakery can produce 120 caramel cookies and 140 chocolate chip cookies daily. For the bakery to be viable, it must sell a minimum of 240 cookies each day. Every chocolate chip cookie served generates $0.75 in profit, whereas each caramel biscuit generates $0.88. The solution to the number of chocolate chip and caramel cookies that the bakery must produce each day to maximize profit may be determined using linear programming.

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Linear programming, like decision trees and fuzzy logic , is essential for computing algorithms. It states that, given a set of fixed resource constraints, an optimal or best solution exists. This has myriad applications in cognitive technologies such as AI or machine learning , which try and apply mathematical formulae and statistical models to real-world problems. 

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  5. assigning students to schools--Assignment Problem Ch 5-12

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COMMENTS

  1. Assignment problem

    The formal definition of the assignment problem (or linear assignment problem) is Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function : is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:

  2. Solving Assignment Problem using Linear Programming in Python

    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.

  3. PDF Lecture 5 1 Linear Programming

    A linear program is an optimization problem in which we have a collection of variables, which can take real values, and we want to nd an assignment of values to the variables that satis es a given collection of linear inequalities and that maximizes or minimizes a given linear function.

  4. A linear Programming Formulation of Assignment Problems

    1. Introduction Linear programming (LP) has been successfully applied to a wide range of problems, such as capital budgeting, maintenance, production scheduling and traveling salesman problems. LP has in the last decade been shown to be a flexible, efficient and commercially successful technique for scheduling, planning and allocation.

  5. PDF Linear programming 1 Basics

    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= b b . 2 The Simplex Method In 1947, George B. Dantzig developed a technique to solve linear programs | this technique is referred to as the simplex method. 2.1 Brief Review of Some ...

  6. 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.

  7. Assignment Problem, Linear Programming

    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.

  8. What is Assignment Problem

    Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows:

  9. Tutorial and Practice in Linear Programming

    The term linear programming arises from the fact that the objective function is a linear combination of decision variables and parameters that one seeks to maximize or minimize. For example, classic problems seek to maximize profits and flow and to minimize cost or time. The parameters in the linear combination of variables are fixed values ...

  10. PDF Lecture 7 1 Linear Programming Relaxations

    The vertex cover approximation algorithm based on linear programming is very ele-gant and simple, but it requires the solution of a linear program. Our previous vertex cover approximation algorithm, instead, had a very fast linear-time implementation. Can we get a fast linear-time algorithm that works in the weighted case and achieves

  11. Linear Programming

    Linear programming is one of the most widely used techniques of operations research and management science. Its name means that planning (programming) is being done with a mathematical model (called a linear-programming model) where all the functions in the model are linear functions. Linear Programming Models

  12. 4: Linear Programming

    4.3: Minimization By The Simplex Method. In this section, we will solve the standard linear programming minimization problems using the simplex method. The procedure to solve these problems involves solving an associated problem called the dual problem. The solution of the dual problem is used to find the solution of the original problem.

  13. 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.

  14. Assignment Model

    → Assignment model is a special application of Linear Programming Problem (LPP), in which the main objective is to assign the work or task to a group of individuals such that; i) There is only one assignment. ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

  15. Hands-On Linear Programming: Optimization With Python

    The basic method for solving linear programming problems is called the simplex method, which has several variants. Another popular approach is the interior-point method . Mixed-integer linear programming problems are solved with more complex and computationally intensive methods like the branch-and-bound method , which uses linear programming ...

  16. linear programming

    When trying to solve for assignments given a cost matrix, what is the difference between. using Scipy's linear_sum_assignment function (which I think uses the Hungarian method). describing the LP problem using a objective function with many boolean variables, add in the appropriate constraints and send it to a solver, such as through scipy.optimize.linprog?

  17. PDF Methods for Fast Linear Programming

    Finally, we also include a brief description of linear programming, descriptions of previous methods for solving linear programs, and Lee and Sidford' improvements to the path nding problem in the Appendix. 2 Interior Point Methods Interior point methods traverse the interior of the feasible region before nding an -approximate solution.

  18. PDF Modeling and Solving Linear Programming with R

    oped linear programming as a technique for planning expenditures and returns in order to optimize costs to the army and increase losses to the enemy. The method was kept secret until 1947, when George B. Dantzig published the simplex method for solving linear programming [2]. In this same year, John von Neumann developed the theory of

  19. Assignment-Method

    Assignment method is a particular model of linear programming in which the main objective is to assign the work or task to a group of individuals such that; i) There is only one assignment. ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

  20. scipy.optimize.linear_sum_assignment

    The linear sum assignment problem [1] is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C [i,j] is the cost of matching vertex i of the first partite set (a 'worker') and vertex j of the second set (a 'job'). The goal is to find a complete assignment of workers to ...

  21. Linear Programming Explained: Formulas and Examples

    The linear programming formula may be regarded as follows: The function of the formula: ax + by = Z. The formula's operating limitations: cx + dy ≤ e and fx + gy ≤ h. Other, non-negative restrictions: x ≥ 0, y ≥ 0. You need to know a few terms to understand the meaning of linear programming. First come the decision variables.

  22. Assigning Fastest Pick-Ups to Uber Drivers with Linear Programming

    Formulating the problem as a standard linear program above, we explore three methods using Munkres' Hungarian Algorithm and Google OR Tools' Linear Sum Assignment and Minimum Cost Flow. We then compare the runtime of the method for different sizes of networks i.e. different numbers of drivers and pick-ups.

  23. (PDF) Assignment model with multi-objective linear programming for

    The linear assignment method was first introduced as a programming model for selecting brands by consumers. The linear assignment model is a binary programming model and is used widely in solving ...

  24. A cutting plane algorithm for globally solving low ...

    The method builds on iteratively solving a small concave problem and a large linear programming problem. ... paper we consider the k-means problem for instances with low dimensional data and formulate it as a structured concave assignment problem. This allows us to exploit the low dimensional structure and solve the problem to global optimality ...