assignment model in operation research examples

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:

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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:

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

Column 3 contains no zero. Go to Step 2.

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

Step 3 (Assignment) :

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ₹ 35

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How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

Assignment Problem: Maximization

There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.

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 produces the same assignment solution as the original maximization problem.

• Unbalanced Assignment Problem
• Multiple Optimal Solutions

Example: Maximization In An Assignment Problem

At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.

How should the counters be assigned to persons so as to maximize the profit ?

Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.

On small screens, scroll horizontally to view full calculation

Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.

Select the smallest element from all the uncovered elements, i.e., 4. Subtract this element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment. Repeating step 3, we obtain a solution which is shown in the following table.

Final Table: Maximization Problem

Use Horizontal Scrollbar to View Full Table Calculation

The total cost of assignment = 1C + 2E + 3A + 4D + 5B

Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.

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Assignment Problem

5.1  introduction.

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.

5.2  GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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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.

Related posts:

• Operations Research approach of problem solving
• Introduction to Transportation Problem
• Procedure for finding an optimum solution for transportation problem
• Initial basic feasible solution of a transportation problem
• Top 7 Best Ways of Getting MBA Assignment Writing Help
• Introduction to Decision Models
• Transportation Cost Elements
• Modes of Transportation in Logistics
• Factors Affecting Transportation in Logistics
• Export/Import Transportation Systems

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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.

Suggested Notes:

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Stepping Stone | Transportation Problem | Transportation Model

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

Transportation Model - Introduction

North West Corner 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

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Operation Research Models – 8 Common Models Explained in Detail | Operations Management

Operation research models.

Table of Contents

Operational Research (OR) Models, also known as Management Science Models and Decision Science Models, are mathematical and analytical methods used to answer complex questions and make informed decisions in many fields, including business, engineering, healthcare, logistics, and finance.

By formulating real-world problems as mathematical equations or algorithms, OR models allow decision-makers to find the best solutions under given constraints, optimizing processes, resources, and outcomes.

It is the main objective of OR models to maximize profits, minimize costs, improve efficiency, and maximize overall performance. Decision-making situations involving multiple variables, uncertainties, and constraints need to be considered simultaneously using these models. There are several types of OR models, each suited for a different type of problem. Here are some of the most common types of OR models:

1. Linear Programming (LP) Model:

Linear Programming (LP) is one of the most widely used and prominent OR models. A linear equation represents the relationship between a decision variable and an objective/constraint when the objective function and constraints are all linear.

Profit, cost, utility, or any other relevant metric is typically represented by a linear function, and the objective of LP is to maximize or minimize it. Constraints limit the possible values of these variables, reflecting real-world limitations on resources and capacity, while decision variables represent the quantities to be determined.

A variety of fields utilize LP, including production planning, supply chain optimization, portfolio optimization, resource allocation, and transportation planning. In 1947, George Dantzig developed the Simplex Method, a popular algorithm for solving linear programming problems.

2. Integer Programming (IP) Model:

The concept of integer programming is an extension of linear programming that deals with problems where the decision variables must have integer values, i.e., solutions must be whole numbers, not fractions.

If a decision involves discrete choices, such as selecting a facility location, assigning workers tasks, or determining the number of units to be produced, IP models are particularly useful. Among the applications are project selection, workforce scheduling, and routing.

Since IP problems have discrete variables, solving them is more challenging and computationally intensive than solving LP problems. Algorithms such as Branch and Bound and Cutting Plane are often used to find optimal or near-optimal solutions.

3. Non-Linear Programming Model:

The concept of nonlinear programming refers to problems in which the objective function or constraints are nonlinear. Unlike linear relationships, these problems involve nonlinear equations that may not be easily solved analytically.

There are many applications for non-linear programming models, including engineering design, portfolio optimization, financial planning, and resource management. Iterative methods like Gradient Descent or Newton’s method are often used to solve non-linear programming problems, where successive approximations lead to the optimal solution.

4. Network Models:

A network model is a type of OR model that focuses on problems involving interconnected elements or networks. These models are widely used in the transportation industry, project scheduling, and supply chain logistics, among other applications.

The following are common network models:

a. Shortest Path Problem:

The shortest path problem aims to find the shortest path between two nodes in a network, taking into account distances, costs, or transit times.

b. Max Flow-Min Cut Problem:

This is a problem that determines the maximum flow that can be sent through a network from a source node to a sink node while minimizing the cut (the minimum capacity of edges to disconnect source and sink).

c. Critical Path Method (CPM):

The CPM method is used to determine the critical path, i.e. the sequence of tasks that must be completed in order to avoid project delays.

It is possible to optimize resource utilization, routing, and scheduling in complex systems by using network models.

5. Queuing Models:

A queueing model analyzes the lines or queues in various systems, including customer service centers, manufacturing facilities, and healthcare facilities. Using these models, service levels can be optimized, waiting times minimized, and resources allocated more efficiently.

When organizations understand the dynamics of queueing systems, they can enhance customer satisfaction and operational efficiency. Queuing models consider factors such as arrival rates, service rates, and the number of servers.

6. Simulation Models:

A simulation model is another group of OR models used to reproduce real-world processes through computer-based models. Simulations allow decision-makers to see how systems behave under different circumstances.

In product design, risk analysis, financial planning, and supply chain optimization, simulation models are particularly useful when real-world experiments would be either too expensive, risky, or time-consuming.

7. Markov Decision Process (MDP) Models:

The MDP model is used for decision-making in uncertain environments. In such situations, the outcomes are probabilistic, and the decision-maker aims to select actions that maximize long-term rewards or minimize long-term costs.

Artificial intelligence and reinforcement learning applications use MDPs to teach agents how to interact with environments and optimize their decisions.

8. Heuristic Models:

In a heuristic model, the solution is not guaranteed to be optimal, but it is good and efficient and can be completed in a reasonable amount of time. For large-scale and complex problems, where finding exact solutions is computationally infeasible, these models are particularly useful.

As a rule-of-thumb strategy, heuristics help narrow down the search space to find satisfactory solutions by guiding the search. Although heuristics do not guarantee optimality, they are an effective tool for tackling real-world problems and delivering practical results.

Operation Research (OR) Models have become indispensable tools for modern decision-making. In complex, dynamic environments, OR models assist organizations in optimizing resources, improving efficiency, and making informed choices by leveraging mathematical and analytical techniques.

In addition to linear programming and integer programming, non-linear programming, network models, queueing models, simulation models, and more, each type of OR model offers unique insights into specific types of problems.

A business, government, or individual seeking to navigate the complexity of today’s world will find OR models invaluable assets because of their versatility and capability to handle uncertainty, discrete choices, and complex interdependencies. OR models will continue to be integral to enhancing decision-making processes and advancing progress across a wide range of fields as technology advances and data availability increases.

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Operation Research Models and Modelling

• Post last modified: 21 July 2022
• Reading time: 23 mins read
• Post category: Operations Research

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 goal of this article is to help readers understand models and modelling in OR, the advantages and characteristics of a good model, model design and development and different models commonly used in OR.

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

Advantages of Good Model

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.

Business Ethics

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Corporate social responsibility (CSR)

• Theories of CSR
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Lean Six Sigma

• What is Six Sigma?
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• What is Process Audits?
• Six Sigma Implementation at Ford
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• Research Methodology
• What is Research?
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• Sampling Method
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Operations Research

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Duality in Linear Programming

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Project Network Analysis Methods

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

Operation Management

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Service Operations Management

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Procurement Management

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Strategic Management

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• Mission Statement
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• What is SWOT Analysis?
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• What is Ansoff Matrix?
• Prahalad and Gary Hammel
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• 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
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• What is Strategy Gap Analysis?
• Issues In Strategy Implementation
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• What is Strategic Management Process?

Supply Chain

• What is Supply Chain Management?
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• What is Packaging?
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Applications and Mathematical Modeling in Operations Research

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Theoretical understanding of the relevant problem structure and consistent mathematical modeling are necessary keys to formulating operations research models to be used for optimization of decisions in real applications. The numbers of alternative models, methods and applications of operations research are very large. This paper presents fundamental and general decision and information structures, theories and examples that can be expanded and modified in several directions. The discussed methods and examples are motivated from the points of view of empirical relevance and computability.

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Acknowledgements

My thanks go to Professor Hadi Nasseri for kind, rational and clever suggestions.

Recommender: 2016 International workshop on Mathematics and Decision Science, Dr. Hadi Nasseri of University of Mazandaran in Iran.

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Lohmander, P. (2018). Applications and Mathematical Modeling in Operations Research. In: Cao, BY. (eds) Fuzzy Information and Engineering and Decision. IWDS 2016. Advances in Intelligent Systems and Computing, vol 646. Springer, Cham. https://doi.org/10.1007/978-3-319-66514-6_5

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Reducing PIA by over 50% While Generating New Patient Flows: A Comprehensive Assessment of Emergency Department Redesign

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Background: On June 6, 2011 the Emergency Department (ED) at Southlake Regional Health Center, a very high-volume ED, initiated a comprehensive redesign project to improve patient waiting times. The primary initial goal of the project was to reduce Time to Physician Initial Assessment (TPIA) - one of the Key Performance Indicators (KPIs) tracked by the Ontario Ministry of Health and Long-Term Care. The objective was to achieve a significant improvement in TPIA without sacrificing performance on any other important KPIs such as Length of Stay (LOS), Left Without Being Seen (LWBS), or time to admission (T2A). The effect on TPIA was immediate and dramatic: the 90th percentile TPIA declining from 4 hrs to under 2.5 hrs, with further improvements seen over time. The patient in-flows also increased; anecdotally this increase was directly related to shorter wait time. However, like any other large-scale and ongoing system redesign project, the impacts are not limited to the listed KPIs, but are multi-dimensional, affecting patient inflows, flows within the ED, workloads, staffing levels, etc. Thus, teasing out the impact of system redesign requires from other concurrent factors (population changes, staffing changes, etc.) requires a comprehensive system assessment. The available data exhibits auto-correlations, heteroscedasticity, and interdependence among variables, rendering simple statistical analysis of individual KPIs inapplicable. We develop a novel methodology and conduct counterfactual analysis demonstrating that the decrease in TPIA, as well as new patient in-flows can indeed be attributed to the ED redesign. This suggests that a similar system redesign should be considered by other EDs looking to improve wait times. Objectives: To (1) statistically estimate the impacts of the redesign project on various performance measures over time, (2) examine whether the initial goal of improvement in TPIA without compromising other service performance measures was achieved, and (3) study whether the project impacted patient inflows. Methods: We (1) estimate simultaneous equations models to quantify interdependent and time-varying relations among variables, (2) conduct an iterative counterfactual analysis to estimate the mean-level impacts of the project, and (3) construct 95% confidence intervals for the estimated impacts using the Bootstrap method. Results: We study project impacts over 720 days after it was initiated. During this time, the 90th percentile of TPIA has been reduced by nearly 2.5 hours on average (translating into an over 50% improvement), with continuous improvement over the study period. This effect is statistically and operationally significant. The project also improved LOS for non-admitted patients (both acute and non-acute), and did not have statistically significant impact on LOS for admitted patients. There was also a decrease in LWBS, though it was not statistically significant. Thus the project achieved its stated primary goals. We also observed an increase in inflows of both acute and nonacute patients; our analysis confirms that this increase can be attributed to the project, indicating that improvements in TPIA attracted new patients to the ED. All of these effects have persisted over the 720-day post-project period. Conclusions: The redesign project has significantly reduced TPIA over time while also improving some LOS measures; none of the waiting time KPIs were compromised. The reduction in TPIA also attracted significant volumes of new patients. However, the redesigned process was able to deal with this volume without compromising performance. The redesign project involved a number of major changes in ED operations. We provide an overview of these changes, and while our analysis cannot attribute specific project impacts to specific changes, we believe that implementing similar changes should receive strong consideration by other EDs.

Competing Interest Statement

The authors have declared no competing interest.

Funding Statement

This study was funded by NSERC and MITACS.

Author Declarations

I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained.

The details of the IRB/oversight body that provided approval or exemption for the research described are given below:

The Research Ethics Board of The Southlake Regional Health Centre gave ethical approval for this work.

I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals.

I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance).

I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable.

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Subject Area

• Emergency Medicine
• Addiction Medicine (324)
• Allergy and Immunology (632)
• Anesthesia (168)
• Cardiovascular Medicine (2398)
• Dentistry and Oral Medicine (289)
• Dermatology (207)
• Emergency Medicine (381)
• Endocrinology (including Diabetes Mellitus and Metabolic Disease) (850)
• Epidemiology (11795)
• Forensic Medicine (10)
• Gastroenterology (705)
• Genetic and Genomic Medicine (3766)
• Geriatric Medicine (350)
• Health Economics (637)
• Health Informatics (2408)
• Health Policy (939)
• Health Systems and Quality Improvement (905)
• Hematology (342)
• HIV/AIDS (786)
• Infectious Diseases (except HIV/AIDS) (13346)
• Intensive Care and Critical Care Medicine (769)
• Medical Education (368)
• Medical Ethics (105)
• Nephrology (401)
• Neurology (3523)
• Nursing (199)
• Nutrition (528)
• Obstetrics and Gynecology (679)
• Occupational and Environmental Health (667)
• Oncology (1832)
• Ophthalmology (538)
• Orthopedics (221)
• Otolaryngology (287)
• Pain Medicine (234)
• Palliative Medicine (66)
• Pathology (447)
• Pediatrics (1037)
• Pharmacology and Therapeutics (426)
• Primary Care Research (424)
• Psychiatry and Clinical Psychology (3187)
• Public and Global Health (6178)
• Radiology and Imaging (1290)
• Rehabilitation Medicine and Physical Therapy (751)
• Respiratory Medicine (832)
• Rheumatology (380)
• Sexual and Reproductive Health (373)
• Sports Medicine (324)
• Surgery (403)
• Toxicology (50)
• Transplantation (172)
• Urology (147)