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)

• 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

→ 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.

Linear Programming Problem (LPP) Formulation with Numericals

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Assignment with Allowed Groups

This section describes an assignment problem in which only certain allowed groups of workers can be assigned to the tasks. In the example there are twelve workers, numbered 0 - 11. The allowed groups are combinations of the following pairs of workers. group1 = [[2, 3], # Subgroups of workers 0 - 3 [1, 3], [1, 2], [0, 1], [0, 2]]

group2 = [[6, 7], # Subgroups of workers 4 - 7 [5, 7], [5, 6], [4, 5], [4, 7]]

group3 = [[10, 11], # Subgroups of workers 8 - 11 [9, 11], [9, 10], [8, 10], [8, 11]]

An allowed group can be any combination of three pairs of workers, one pair from each of group1, group2, and group3. For example, combining [2, 3] , [6, 7] , and [10, 11] results in the allowed group [2, 3, 6, 7, 10, 11] . Since each of the three sets contains five elements, the total number of allowed groups is 5 * 5 * 5 = 125 .

Note that a group of workers can be a solution to the problem if it belongs to any one of the allowed groups. In other words, the feasible set consists of points for which any one of the constraints is satisfied. This is an example of a non-convex problem. By contrast, the MIP Example , described previously, is a convex problem: for a point to be feasible, all constraints must be satisfied.

For non-convex problems like this one, the CP-SAT solver is usually faster than a MIP solver. The following sections present solutions to the problem using the CP-SAT solver and the MIP solver, and compare the solution times for the two solvers.

CP-SAT solution

First, we'll describe a solution to the problem using the CP-SAT solver.

Import the libraries

The following code imports the required library.

Define the data

The following code creates the data for the program.

Create the allowed groups

To define the allowed groups of workers for the CP-SAT solver, you create binary arrays that indicate which workers belong to a group. For example, for group1 (workers 0 - 3), the binary vector [0, 0, 1, 1] specifies the group containing workers 2 and 3.

The following arrays define the allowed groups of workers.

For CP-SAT it is not necessary to create all 125 combinations of these vectors in a loop. The CP-SAT solver provides a method, AllowedAssignments , which enables you to specify the constraints for the allowed groups separately for each of the three sets of workers (0 - 3, 4 - 7, and 8 - 11). Here's how it works:

The variables work[i] are 0-1 variables that indicate the work status or each worker. That is, work[i] equals 1 if worker i is assigned to a task, and 0 otherwise. The line solver.Add(solver.AllowedAssignments([work[0], work[1], work[2], work[3]], group1)) defines the constraint that the work status of workers 0 - 3 must match one of the patterns in group1 . You can see the full details of the code in the next section.

Create the model

The following code creates the model.

Create the variables

The following code creates an array of variables for the problem.

The following code creates the constraints for the program.

Create the objective

The following code creates the objective function.

Invoke the solver

The following code invokes the solver and displays the results.

Display the results

Now, we can print the solution.

Here's the output of the program.

The entire program

Here is the entire program.

MIP solution

Next, we describe a solution to the problem using the MIP solver.

The following code creates the allowed groups, by looping through the three sets of subgroups shown above.

Declare the solver

The following code creates the solver.

Here's the output of the program:

Solutions times

The solution times for the two solvers are as follows:

• CP-SAT: 0.0113 seconds
• MIP: 0.3281 seconds

CP-SAT is significantly faster than MIP for this problem.

Last updated 2023-01-02 UTC.

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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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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

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• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.
Try it before moving to see the solution

Explanation for above simple example:

An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming.   The simplex method of Linear Programming Problems(LPP)   proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point.   Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres.     4 x 5 = 20 routes are possible.   Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs?   The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person?   The objective is minimizing total cost.   This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above.   Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

• To decide the transportation of new materials from various centres to different manufacturing plants.   In the case of multi-plant company this is highly useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.   These two are the uses of transportation model.   The objective is minimizing transportation cost.

Assignment model is used in the following:

• To decide the assignment of jobs to persons/machines, the assignment model is used.
• To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
• To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand.   Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available.   In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

Hungarian Method for Solving Assignment Problem:

The Hungarian method of assignment provides us with an efficient method of finding the optimal solution without having to make a-direct comparison of every solution. It works on the principle of reducing the given cost matrix to a matrix of opportunity costs.

Opportunity cost show the relative penalties associated with assigning resources to an activity as opposed to making the best or least cost assignment. If we can reduce the cost matrix to the extent of having at least one zero in each row and column, it will be possible to make optimal assignment.

The Hungarian method can be summarized in the following steps:

Step 1: Develop the Cost Table from the given Problem:

If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero.

Step 2:  Find the Opportunity Cost Table:

(a) Locate the smallest element in each row of the given cost table and then subtract that from each element of that row, and

(b) In the reduced matrix obtained from 2 (a) locate the smallest element in each column and then subtract that from each element. Each row and column now have at least one zero value.

Step 3: Make Assignment in the Opportunity Cost Matrix:

The procedure of making assignment is as follows:

(a) Examine rows successively until a row with exactly one unmarked zero is obtained. Make an assignment single zero by making a square around it.

(b) For each zero value that becomes assigned, eliminate (Strike off) all other zeros in the same row and/ or column

(c) Repeat step 3 (a) and 3 (b) for each column also with exactly single zero value all that has not been assigned.

(d) If a row and/or column has two or more unmarked zeros and one cannot be chosen by inspection, then choose the assigned zero cell arbitrarily.

(e) Continue this process until all zeros in row column are either enclosed (Assigned) or struck off (x)

Step 4: Optimality Criterion:

If the member of assigned cells is equal to the numbers of rows column then it is optimal solution. The total cost associated with this solution is obtained by adding original cost figures in the occupied cells.

If a zero cell was chosen arbitrarily in step (3), there exists an alternative optimal solution. But if no optimal solution is found, then go to step (5).

Step 5: Revise the Opportunity Cost Table:

Draw a set of horizontal and vertical lines to cover all the zeros in the revised cost table obtained from step (3), by using the following procedure:

(a) For each row in which no assignment was made, mark a tick (√)

(b) Examine the marked rows. If any zero occurs in those columns, tick the respective rows that contain those assigned zeros.

(c) Repeat this process until no more rows or columns can be marked.

(d) Draw a straight line through each marked column and each unmarked row.

If a no of lines drawn is equal to the no of (or columns) the current solution is the optimal solution, otherwise go to step 6.

Step 6: Develop the New Revised Opportunity Cost Table:

(a) From among the cells not covered by any line, choose the smallest element, call this value K

(b) Subtract K from every element in the cell not covered by line.

(c) Add K to very element in the cell covered by the two lines, i.e., intersection of two lines.

(d) Elements in cells covered by one line remain unchanged.

Step 7:  Repeat Step 3 to 6 Unlit an Optimal Solution is Obtained:

The flow chart of steps in the Hungarian method for solving an assignment problem is shown in following figures:

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.

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.

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The Assignment Model

The linear programming formulation of the assignment model is similar to the formulation of the transportation model, except all the supply values for each source equal one, and all the demand values at each destination equal one. Thus, our example is formulated as follows :

This is a balanced assignment model. An unbalanced model exists when supply exceeds demand or demand exceeds supply.

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Operation Research Models and Modelling

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Introduction to Models and Modelling

Models and modelling are fundamental to operations research and have broad application in organisational systems. A model may be defined as a representation of a system or phenomenon.

According to Perumpalath (2005), a manageable model of reality is necessary to understand an organisation’s components and their interrelationships. Models provide descriptions and/or simplifications of complex phenomena (Rouse & Morris, 1986). They can provide the basis for both scientific explanations of and predictions about phenomena (Gilbert, Boulter & Rutherford, 1998).

Models have two basic purposes : to convey the current understanding of a system or to generate a new understanding of the system or both. The aim of the model is to provide a way to assess the behaviour of a system to improve its performance. In the case of an anticipated system, models help to define its ideal structure by showing the functional relationships between its various components.

Table of Content

• 1 Introduction to Models and Modelling
• 2.1 What is Model in Operation Research?
• 2.2 What is Model in Operation Research?
• 2.3 Principles of Modelling
• 3 Advantages of Good Model
• 4 Characteristics of Model
• 5.1.1 Mathematical models
• 5.2.1 Descriptive models
• 5.2.2 Predictive models
• 5.3.1 Analogue or schematic models
• 5.3.2 Symbolic or mathematical models
• 5.4.1 Deterministic model
• 5.4.2 Probabilistic model
• 5.5.1 Static models
• 5.5.2 Dynamic models
• 6.1 Allocation Models (Distribution Models)
• 6.2 Replacement Models
• 6.3 Queueing Models (Waiting Line Models)
• 6.4 Network Models
• 6.5 Game Theory Models
• 6.6 Inventory Models
• 6.7 Simulation Models
• 6.8 Job Sequencing Models
• 6.9 Markovian Models

Modelling is the process of developing a model, conducting experiments using the model and evaluating the alternative management policies as well as decision-making practices. Managers can use modelling for a real-life system or phenomenon to understand the difference in the behaviour of the problem as compared with the description of the problem.

The construction and use of models are fundamental to operations research. Models are generally used in OR to make simplified depictions of complex systems.

What is Model in Operation Research?

A model may be defined as a representation of a system (a unit or process that exists and operates through the interaction of its parts) or a phenomenon. These representations can be in the form of diagrams, maps, flowcharts, mathematical equations, graphs, computer simulations, or physical replicas of the system.

Models can be physical, abstract or somewhere in the middle. Models have two basic purposes: to convey the current understanding of a system or to generate a new understanding of the system or both.

The aim of a model is to offer ways for examining the performance of the system for further improvement. Models are created so that they remain dynamic and this is why they do not only serve the purpose of system representation but also enable managers to forecast future events based on past and present factors, draw up explanations, expose gaps in understanding, and present new questions for further analysis.

The reliability of the conclusion/s derived using a model depends on the validity of the model or the basic assumptions on which the model was constructed.

Modelling refers to the process of developing a model for an existing or anticipated system, conducting experiments using the model to gain knowledge about the performance of the system under various operating conditions, and evaluating of the alternative management policies and decision-making practices.

Modelling forms the core of operation research. Modelling a real-life system or phenomenon enables managers to study the difference in the behaviour of the problem in relation to the description of the problem. By developing a model of a decision-making problem, the associated complexities and uncertainties are transformed into a logical structure that is more open to formal analysis.

This model is able to identify the different alternatives available along with their expected outcomes for all probable events. It also helps to specify the relevant data for assessing the alternatives and allows managers to arrive at meaningful and informative conclusions. Modelling thus provides a means to obtain a well-defined structural framework of the given problem.

Modelling involves the follow- ing steps, regardless of the type of model used:

• Defining the problem
• Collecting data
• Building a model of the system
• Deriving a solution
• Testing to validate the model and the solution
• Implementing the solution

Principles of Modelling

When building a model, the following principles should always be kept in mind:

• If a simple model will be adequate, never choose a complicated model
• Remember, models never replace decision-makers
• The deduction phase of modelling must be performed diligently
• Validation of models is necessary before implementation
• A model is only as good as the information that it is given

A good model has many advantages as follows:

• The model is easier to study than the whole system itself.
• It offers a logical and systematic approach to the problem.
• It specifies the scope and limitations of the problem.
• It reveals the nature of quantifiable factors (that can be measured in numeric terms) in a problem.
• It includes useful tools that help in removing duplication of methods used to solve a particular problem.
• A model can help to identify gaps in understanding, areas for further analysis and opportunities for system improvement.
• A model offers an economic explanation of the operations of the system it represents.

Characteristics of Model

The following characteristics are important for a good OR model:

• A good model should have few and simple assumptions.
• The number of variables should be as less as possible for a model to be simple and easy to understand.
• In the previous chapter, you learnt that models can be formulated and reformulated over the course of the OR process. A good model should be able to take into account new formulations without permitting any major change in its frame.
• A good model is open to the parametric type of approach.
• A model should not take too much time in its construction for any problem.
• A model should be flexible and open to adjustments.
• A model should be able to show the associations and interrelations of cause and effect in operational conditions.

Types of Models in Operation Research

Various factors can form the basis on which OR models can be categorised.

Based on the Degree of Abstraction

Mathematical models.

A mathematical model is a set of equations that are used to represent a real-life situation or problem. For example, linear programming problems for maximising profits or transportation problem.

Concrete models: These are least abstract models and a viewer can observe the shape and characteristics of the modelled entity immediately. For example, a globe of the earth or 3-D model of human DNA.

Language models: Language models are more abstract than concrete models but less abstract than mathematical models. For example, language model for speech recognition.

Based on the Function

Descriptive models.

A descriptive model is one in which all the operations involved in a system are represented using non-mathematical language. Also, the relationships and interactions among different operations is also defined using non-mathematical language. Such models are used to define and represent a system but they cannot predict their behaviour. For example, a layout plan.

Predictive models

A predictive model is developed and validated using known results and is used to predict future events or outcomes. These models are used to increase the probability of forecasting future outcomes and risks by incorporating historical information.

Unlike mathematical models explained under the next heading, predictive models are not easy to explain in the equation form, and simulation techniques are often needed to generate a prediction. Curve and surface fitting, time series regression, or machine learning techniques may be used to construct predictive models.

Normative or prescriptive models

Such models are used for recurring problems. In such models, decision rules or criteria are developed for finding optimal solution. The solution pro- cess can be programmed easily without the management’s involvement. For example, linear programming is a prescriptive model.

Based on the Structure

Physical or iconic models

A physical model is similar to the system it represents. Different physical models are used to predict and understand systems that cannot be directly observed. Examples are the double helix model of the DNA, the model of an atom, a map, a model of the universe, etc. There are always limitations to each physical model and it is not always possible to get a complete representation with a physical model.

An iconic model is one that is usually a scaled-down or scaled-up physical replica of the system it represents. An iconic model is exactly or extremely similar to the system it represents. An aeroplane model and a globe are examples of iconic models.

Analogue or schematic models

Analogue models are used more frequently than iconic models and are used to represent dynamic situations as they represent the characteristics of a system under study.

For example, graphs. When the elements of an overall system are represented using abstract symbols, graphical symbols or hierarchy, then, it is called as schematic modelling. For example, organisational chart is a schematic model.

Symbolic or mathematical models

A mathematical or abstract model is one in which mathematical symbols and statements constitute the model. An abstract model is needed in systems where the complexity of relationships is not possible to represent physically or the physical representation is difficult and time-consuming.

A lot of operational science analysis is performed using abstract/mathematical models that utilise mathematical symbols. These types of models are not specific but quite general instead and may be used to represent diverse situations. They can moreover be readily manipulated for ex- perimentation and forecasting purposes.

In a mathematical model, the information about the significance of variables vis- à-vis their influence on the solution is important. A mathematical model depicts relationships and interrelationships among the variables and other factors relevant to solving the problem.

Based on the Nature of Environment

Deterministic model.

A deterministic model is one that allows you to precisely calculate an outcome, without the involvement of uncertainty or unpredictability or randomness. With a deterministic model, one has all the data required to predict the out- come with certainty. A deterministic model always provides the same output for a certain set of input variables.

Deterministic models are simple and easy to understand. The outcome of the model is completely decided by the parameter values and the initial values. These models can be fundamentally flawed since they are not able to take into account the different variables that will have an effect.

Probabilistic model

A probabilistic (also called stochastic) model can handle uncertainties or randomness in the applied inputs. Stochastic refers to the property of having a random probability distribution that can be statistically analysed but may not be predicted accurately. Probabilistic models have some innate randomness so that the same set of parameter values and initial conditions will result in a combination of different outputs.

Unlike deterministic models, probabilistic models may not always provide the same output for a certain set of input variables because of the randomness it includes. These models are more sophisticated than deterministic as they incorporate historical data to demonstrate the probability of the occurrence of an event. Probabilistic models can replicate real-world scenarios better and offer a range of possible outcomes for a problem.

Based on Time Horizon

Static models.

A static model is one that describes relationships that do not change with respect to time. Such a model remains at an equilibrium or a steady state. A defining feature is that static models do not contain an internal memory of previous input values, internal variables or output values. A static model gives an idea of a system’s response to a given set of input conditions.

This type of model does not account for the effect of factors that vary over time but it is still useful in offering an initial analysis of the given problem. A static model is rather structural than behavioural representation of a system.

A key feature is that it only uses a set of algebraic equations. Stat- ic modelling comprises class diagrams as well as object dia- grams and allows the representation of static constituents of the system. In the static model, if the same set of input values are entered, the result is the same set of output values.

A static model is the model of the system not during runtime. Being a time-independent representation a system, static modelling is quite rigid and cannot be modified in real-time.

Dynamic models

In contrast, a dynamic model describes time-varying relationships. Dynamic models offer a way of modelling the time-dependent behaviour of a given system. They keep changing according to time. Unlike a static model, a dynamic model is a behavioural representation of the static components. Dynamic models characteristically maintain an internal memory of previous input values, internal variables or output values.

Unlike static models, dynamic models use both algebraic equations and differential equations. Dynamic modelling comprises a sequence of states, state transitions, events, actions, activities and memory. In a dynamic model, the input values provided at the time and the input values provided in the past both influence the output values at any given time.

A dynamic model is a runtime model of the system. The ability to change with time makes dynamic modelling flexible and it can provide a view of how an object deals with various possibilities that might come up in time.

Major OR Models

Allocation models (distribution models).

Allocation models are used to distribute the available resources amongst competing alternatives in a way that allows maximum total profit or minimum total cost, depending on existing and predicted limitations or constraints. These are applied to problems that contain the following elements:

• a set of resources available in fixed amounts
• a set of tasks, each of which utilises a particular amount of resources
• a set of costs or profit for every task and resource

The tools for solving allocation models are linear programming, assignment problem and transportation problem.

Replacement Models

Replacement models attempt to find the ideal time to replace equipment, machinery or its parts, an individual or capital assets due to various reasons such as scientific advancement, deterioration due to wear and tear, accidents, failure, etc. Replacement models focus on methods of evaluating alternative replacement policies.

Queueing Models (Waiting Line Models)

A queueing model is constructed to help predict:

• The average waiting time used up by the customer, job or item waiting in a line
• The standard length of the waiting line or queue
• The utilisation factor of a queue system

Limited resources for providing a service lead to the formation of a queue. This model helps to lower the sum of costs of providing and getting service and the value of the time used up by the customer, job or item in a waiting line.

Network Models

Network models are suitable for large-scale projects that have many interdependencies and complexities of activities. Critical Path Method (CPM) and Program Evaluation Review Technique (PERT) techniques are used to analyse, plan, schedule and control the various activities of complex projects which can be explained using a network diagram.

Game Theory Models

Game theory models are used to help with decision-making when there is a conflict or competition. Game theory models are useful for identifying the optimal outcome in case of multiple players (players can be cooperative or non-cooperative) and also the required trade-offs to be able to reach that outcome.

Inventory Models

Inventory models are a type of mathematical model that helps to determine the optimum level of inventory that needs to be maintained in a production process to ensure uninterrupted service to consumers without delivery delays.

Inventory models help in decision-making regarding order quantity and ordering production intervals, considering the various factors such as frequency of ordering, order placing costs, demand per unit time, amount of inventory to be stored, inventory flow, inventory holding costs, and cost owing to a scarcity of goods, etc.

For example, Economic Order Quantity (EOQ) model, production order quantity model, quantity discount model, single-period inventory model, etc.

Simulation Models

A simulation model is a mathematical model that uses both mathematical and logical concepts in an attempt to mimic a real-life scenario or system. This model in generally used to solve problems where the number of variables and constrained relationships is significantly high.

For example, Monte Carlo simulation model, risk analysis simulation model, agent-based modelling, discrete event simulation, system dynamics simulation, solutions models, etc.

Job Sequencing Models

Job sequencing models deal with the problem of determining the sequence in which a number of jobs should be performed on different machines in order to maximise the efficiency of available facilities and maximise system output.

For job sequencing, Johnson’s rule is used. Johnson’s rule states the procedure for minimising makespan for scheduling a group of jobs on two workstations.

• Step 1: Find the job with the shortest processing time from the jobs that have not been scheduled yet. If two or more jobs are tied, then, choose any job randomly.
• Step 2: If the shortest processing time is on workstation 1, then, allocate the job as early as possible. and, if the shortest processing time is on workstation 2, schedule the job as late as possible.
• Step 3: Now, consider only the jobs left after processing of the last job and repeat steps 1 and 2 until all jobs have been scheduled.

Ma rkovian Models

A Markovian model is a probabilistic model used for modelling randomly changing systems. A Markovian model assumes that future states depend only on the present state and not on previous events. These models are applicable in situations where the state of the system transforms from one to another on a probability basis.

( Click on Topic to Read )

• What is Ethics?
• Values, Norms, Beliefs and Standards in Business Ethics
• Indian Ethos in Management
• Ethical Issues in Marketing
• Ethical Issues in HRM
• Ethical Issues in IT
• Ethical Issues in Production and Operations Management
• Ethical Issues in Finance and Accounting
• What is Corporate Governance?
• What is Ownership Concentration?
• What is Ownership Composition?
• Types of Companies in India
• Internal Corporate Governance
• External Corporate Governance
• Corporate Governance in India
• What is Enterprise Risk Management (ERM)?
• What is Assessment of Risk?
• What is Risk Register?
• Risk Management Committee

Corporate social responsibility (CSR)

• Theories of CSR
• Arguments Against CSR
• Importance of CSR in India
• Drivers of Corporate Social Responsibility
• Developing a CSR Strategy
• Implement CSR Commitments
• CSR Marketplace
• CSR at Workplace
• Environmental CSR
• CSR with Communities and in Supply Chain
• Community Interventions
• CSR Monitoring
• CSR Reporting
• Voluntary Codes in CSR
• What is Corporate Ethics?

Lean Six Sigma

• What is Six Sigma?
• What is Lean Six Sigma?
• Value and Waste in Lean Six Sigma
• Six Sigma Team
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• Six Sigma in Supply Chains
• What is Binomial, Poisson, Normal Distribution?
• What is Sigma Level?
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• What is DMADV in Six Sigma?
• Six Sigma Project Charter
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• Process Mapping Six Sigma
• Flowchart and SIPOC
• Gage Repeatability and Reproducibility
• Statistical Diagram
• Lean Techniques for Optimisation Flow
• Failure Modes and Effects Analysis (FMEA)
• What is Process Audits?
• Six Sigma Implementation at Ford
• IBM Uses Six Sigma to Drive Behaviour Change
• Research Methodology
• What is Research?
• What is Hypothesis?
• Sampling Method
• Research Methods
• Data Collection in Research
• Methods of Collecting Data
• Levels of Measurement
• What is Sampling?
• Hypothesis Testing
• Research Report
• What is Management?
• Planning in Management
• Decision Making in Management
• What is Controlling?
• What is Coordination?
• What is Staffing?
• Organization Structure
• What is Departmentation?
• Span of Control
• What is Authority?
• Centralization vs Decentralization
• Organizing in Management
• Schools of Management Thought
• Classical Management Approach
• Is Management an Art or Science?
• Who is a Manager?

Operations Research

• What is Operations Research?
• Operation Research Models
• Linear Programming
• Linear Programming Graphic Solution
• Linear Programming Simplex Method
• Linear Programming Artificial Variable Technique

Duality in Linear Programming

• Transportation Problem Initial Basic Feasible Solution
• Transportation Problem Finding Optimal Solution
• Project Network Analysis with Critical Path Method

Project Network Analysis Methods

Project evaluation and review technique (pert), simulation in operation research, replacement models in operation research.

Operation Management

• What is Strategy?
• What is Operations Strategy?
• Operations Competitive Dimensions
• Operations Strategy Formulation Process
• What is Strategic Fit?
• Strategic Design Process
• Focused Operations Strategy
• Corporate Level Strategy
• Expansion Strategies
• Stability Strategies
• Retrenchment Strategies
• Strategic Choice and Strategic Alternatives
• What is Production Process?
• What is Process Technology?
• What is Process Improvement?
• Strategic Capacity Management
• Production and Logistics Strategy
• Taxonomy of Supply Chain Strategies
• Factors Considered in Supply Chain Planning
• Operational and Strategic Issues in Global Logistics
• Logistics Outsourcing Strategy
• What is Supply Chain Mapping?
• Supply Chain Process Restructuring
• Points of Differentiation
• Re-engineering Improvement in SCM
• What is Supply Chain Drivers?
• Supply Chain Operations Reference (SCOR) Model
• Customer Service and Cost Trade Off
• Internal and External Performance Measures
• Netflix’s Niche Focused Strategy
• Disney and Pixar Merger
• Process Planning at Mcdonald’s

Service Operations Management

• What is Service?
• What is Service Operations Management?
• What is Service Design?
• Service Design Process
• Service Delivery
• What is Service Quality?
• Gap Model of Service Quality
• Juran Trilogy
• Service Performance Measurement
• Service Decoupling
• IT Service Operation
• Service Operations Management in Different Sector

Procurement Management

• What is Procurement Management?
• Procurement Negotiation
• Types of Requisition
• RFX in Procurement
• Vendor Managed Inventory
• Internal Conflict During Purchasing Operation
• Spend Analysis in Procurement
• Sourcing in Procurement
• Supplier Evaluation and Selection in Procurement
• Blacklisting of Suppliers in Procurement
• Total Cost of Ownership in Procurement
• Incoterms in Procurement
• Documents Used in International Procurement
• Transportation and Logistics Strategy
• What is Capital Equipment?
• Procurement Process of Capital Equipment
• Acquisition of Technology in Procurement
• What is E-Procurement?
• E-marketplace and Online Catalogues
• Fixed Price and Cost Reimbursement Contracts
• Contract Cancellation in Procurement
• Ethics in Procurement
• Legal Aspects of Procurement
• Global Sourcing in Procurement
• Intermediaries and Countertrade in Procurement

Strategic Management

• What is Strategic Management?
• What is Value Chain Analysis?
• Mission Statement
• What is SWOT Analysis?
• What is Vision?
• What is Ansoff Matrix?
• Strategic Management In Global Environment
• Competitor Analysis Framework
• Competitive Rivalry Analysis
• Competitive Dynamics
• What is Competitive Rivalry?
• Five Competitive Forces That Shape Strategy
• What is PESTLE Analysis?
• Fragmentation and Consolidation Of Industries
• What is Technology Life Cycle?
• What is Diversification Strategy?
• What is Corporate Restructuring Strategy?
• Resources and Capabilities of Organization
• Role of Leaders In Functional-Level Strategic Management
• Functional Structure In Functional Level Strategy Formulation
• Information And Control System
• What is Strategy Gap Analysis?
• Issues In Strategy Implementation
• Matrix Organizational Structure
• What is Strategic Management Process?

Supply Chain

• What is Supply Chain Management?
• Supply Chain Planning and Measuring Strategy Performance
• What is Warehousing?
• What is Packaging?
• What is Inventory Management?
• What is Material Handling?
• What is Order Picking?
• Receiving and Dispatch, Processes
• What is Warehouse Design?
• What is Warehousing Costs?

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Forecasting Electricity price in India

Ash-0701/GNA_Test_Assignment

Folders and files, repository files navigation, gna-mle-assignment.

The Electricity Price Forecast Assignment is a machine learning project aimed at predicting electricity prices based on various factors such as time, generation sources, and other relevant parameters. The project involves data analysis, preprocessing, feature engineering, model training, and deployment of the trained models using a Flask API.

Project Structure:

The project consists of the following components:

Dataset : The dataset (final_energy_dataset.csv) contains information about electricity generation from different sources, forecasts, actual load, and prices. The data is preprocessed to handle missing values, and relevant features are selected for analysis and modeling.

Jupyter Notebook : The analysis and modeling tasks are performed in a Jupyter Notebook (electricity_price_forecast.ipynb). The notebook includes steps for data exploration, preprocessing, feature engineering, model training, and evaluation.

Flask API : The trained machine learning models (XGBoost, Random Forest) are deployed using a Flask API. The API provides a web interface for users to input relevant parameters and get predictions for electricity prices.

HTML Template : The HTML template (index.html) provides a user-friendly interface for interacting with the Flask API. Users can input details such as time, generation sources, and other parameters to obtain predictions for electricity prices.

Data Analysis and Preprocessing : The dataset is initially explored to understand its structure and identify any missing values. Missing values are handled using appropriate techniques such as imputation or removal of rows/columns. Feature engineering techniques are applied to extract relevant information from the dataset and create new features if necessary.

Model Training and Evaluation:

Two machine learning models, XGBoost and Random Forest, are trained using the preprocessed dataset. Hyperparameter tuning is performed using techniques like GridSearchCV to optimize the models' performance. The trained models are evaluated using metrics such as Mean Squared Error (MSE) and R-squared to assess their predictive accuracy.

The Flask API serves as the backend for making predictions using the trained models. It loads the serialized models and provides endpoints for prediction requests. The HTML template provides a user-friendly interface for users to input parameters and obtain predictions for electricity prices.

Conclusion:

The Electricity Price Forecast Assignment demonstrates the application of machine learning techniques to predict electricity prices accurately. By leveraging data analysis, preprocessing, and model training, the project provides valuable insights for stakeholders in the energy sector. The deployment of the trained models using a Flask API enables easy access to predictions, facilitating informed decision-making and planning.

• Jupyter Notebook 99.8%

1. Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver. Example. In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3).

2. Assignment Model

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).

3. Assignment problem

The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.

4. Assignment with Allowed Groups

Assignment with Allowed Groups. This section describes an assignment problem in which only certain allowed groups of workers can be assigned to the tasks. In the example there are twelve workers, numbered 0 - 11. The allowed groups are combinations of the following pairs of workers. An allowed group can be any combination of three pairs of ...

5. Assignment model

In this video you will learn how to solve an assignment model problem

6. Assignment Problem in Linear Programming : Introduction and Assignment

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 ...

7. ASSIGNMENT PROBLEM (OPERATIONS RESEARCH) USING PYTHON

The Assignment Problem is a special type of Linear Programming Problem based on the following assumptions: However, solving this task for increasing number of jobs and/or resources calls for…

8. Assignment Problem, Linear Programming

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 ...

9. Operations Research with R

The assignment problem represents a special case of linear programming problem used for allocating resources (mostly workforce) in an optimal way; it is a highly useful tool for operation and project managers for optimizing costs. The lpSolve R package allows us to solve LP assignment problems with just very few lines of code.

10. 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 ...

11. Hungarian Algorithm for Assignment Problem

The Hungarian algorithm, aka Munkres assignment algorithm, utilizes the following theorem for polynomial runtime complexity (worst case O(n 3)) and guaranteed optimality: If a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an ...

12. Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming. The simplex method of Linear Programming Problems (LPP) proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like.

13. Assignment Problem: Meaning, Methods and Variations

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations. Meaning of Assignment Problem: An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total ...

14. Solving Assignment Problem using Linear Programming in Python

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.

15. Models

Matrix model of the assignment problem. The network model is in Fig. 13. It is very similar to the transportation model except the external flows are all +1 or -1. The only relevant parameter for the assignment model is arc cost (not shown in the figure for clarity) ; all other parameters should be set to default values.

16. Assignment Problem, Maximization Example, Hungarian Method

The Hungarian Method can also solve such assignment problems, as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss. The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss ...

17. Assignment Model in Operation Research

The task is to assign 1 job to 1 person so that the total number of hours are minimized. So, the first step in the assignment model would be to deduce all the numbers by the smallest number in the row. Hence, the smallest number becomes 0 and then we can target the zeroes to arrive at a conclusion.

18. The Assignment Model

The assignment model is a special form of a linear programming model that is similar to the transportation model. There are differences, however. In the assignment model, the supply at each source and the demand at each destination are each limited to one unit. An assignment model is for a special form of transportation problem in which all ...

19. PDF Transportation and Assignment Models

model file. Clearly we want to set up a general model to deal with this prob-lem. 3.2 An AMPL model for the transportation problem. Two fundamental sets of objects underlie the transportation problem: the sources or origins (mills, in our example) and the destinations (factories). Thus we begin the. AMPL. model with a declaration of these two sets:

20. (PDF) A New Method to Solve Assignment Models

models the source is connected to one or more of destination. The most common. method to solve assignment models is the Hungarian metho d. In this paper. introduced another method to solve ...

21. Assignment Model in Quantitative Techniques (QT)

The Assignment Model is a valuable tool in the field of Operations Research with several important applications. Here are some key reasons highlighting its importance: 1. Optimal resource allocation: The Assignment Model is used to solve allocation problems where a set of tasks or resources needs to be assigned to a set of individuals or entities.

22. Operation Research Models And Modelling

Introduction to Models and Modelling. Models and modelling are fundamental to operations research and have broad application in organisational systems. A model may be defined as a representation of a system or phenomenon. According to Perumpalath (2005), a manageable model of reality is necessary to understand an organisation's components and ...

23. Application of Assignment Model in PE Human Resources Allocation

The Assignment Model is a classic integer linear programming model of 0-1 and it is widely applied in dealing with assignment allocation, personnel selection, the programming of transport system and other practical issues. In the field of PE, the point of assignment gives full play in selecting the proper athletes, assigning tasks and even in ...

24. Estimating Markov Chain Mixing Times: Convergence Rate Towards

Network equilibrium models have been extensively used for decades. The rationale for using equilibrium as a predictor is essentially that (i) a unique and globally stable equilibrium point is guaranteed to exist and (ii) the transient period over which a system adapts to a change is sufficiently short in time that it can be neglected.

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The Online Writing Lab at Purdue University houses writing resources and instructional material, and we provide these as a free service of the Writing Lab at Purdue.

26. Ash-0701/GNA_Test_Assignment

The Electricity Price Forecast Assignment is a machine learning project aimed at predicting electricity prices based on various factors such as time, generation sources, and other relevant parameters. The project involves data analysis, preprocessing, feature engineering, model training, and deployment of the trained models using a Flask API.