Quantitative Techniques and Models of Operations Research (OR Models)
Introduction: Quantitative Techniques / Brief Outlines of Operation Research Models
Operations Research (OR) helps organizations make better decisions by using mathematical models, quantitative techniques, and analytical methods.
Every problem cannot be solved by using the same technique or model. That is, there is no unique set of problems that can be solved by using Operation Research models or techniques.
The nature of the problem, the available data, and the objective of the study determine which quantitative technique or OR model should be used.
For this reason, Operations Research provides several quantitative techniques and mathematical models. Each model is designed to solve a particular type of decision-making problem, such as resource allocation, inventory control, project planning, waiting-line analysis, replacement decisions, or competitive situations.
Understanding these models helps managers and O.R. specialists choose the most appropriate method for obtaining the best possible solution. Some of the important models or techniques used in Operation Research are as follows:
Major Quantitative Techniques and Models of Operations Research
The following are some of the major quantitative techniques and models commonly used in Operations Research:
- Allocation Models
- Inventory Models
- Waiting Line Models
- Markov Models
- Game Theory
- Network Models
- Replacement Models
- Simulation Models
- Job Sequencing Models
1. Allocation (Distribution) Models
Resources or facilities such as money, manpower, machines, materials, and time are always limited, which does not allow each activity to be performed in the best possible way.
Therefore, managers must decide how these limited resources should be allocated among different activities to obtain the best possible results. These models are concerned with the allotment of available resources so as to minimize cost or maximize profit subject to prescribed restrictions.
Allocation Models help distribute available resources efficiently while satisfying specified constraints. Depending on the nature of the problem, the objective may be to maximize profit, minimize cost, maximize production, or achieve another organizational goal.
Methods for solving such types of problems are known as mathematical programming techniques. We distinguish between linear and non-linear programming problems on the basis of linearity and non-linearity of the objective function and/or constraints respectively.
Most allocation problems are solved using mathematical programming techniques.
The major allocation models include:
- Linear Programming
- Non-linear Programming
- Integer Programming
- Zero-One Programming
- Goal Programming
- Stochastic Programming
▪️Linear Programming (LP)
Linear Programming is used when both the objective function and all constraints are linear.
In linear programming problems (LPP), the objective function is linear and constraints are also linear inequalities/equations. Transportation and Assignment models are special cases of LPP.
Its purpose is to maximize or minimize an objective function while satisfying limited resources.
Example
A company manufactures two products using limited labour and raw materials. Linear Programming helps determine the most profitable production plan.
Exam Note: Linear Programming is applicable only when both the objective function and constraints are linear.
▪️Non-linear Programming (NLP)
Non-linear Programming is used when either the objective function or one or more constraints are non-linear.
This technique is suitable for problems where relationships between variables cannot be represented by straight-line equations.
Example
The production cost of a product increases at a varying rate rather than a constant rate.
Exam Note: Non-linear Programming is used whenever the assumptions of Linear Programming are not satisfied.
In case the decision variables in an LPP are restricted to either integer or zero-one value, it is known as Integer and Zero-One programming problems, respectively.
▪️Integer Programming (IP)
Some practical problems require decision variables to take only whole-number values.
Integer Programming is used when one or more decision variables must be integers.
Example
A company must decide the number of delivery trucks to purchase. Since fractional trucks are impossible, Integer Programming is used.
Exam Note: Integer Programming is a special form of Linear Programming in which decision variables are restricted to integer values.
▪️Zero-One Programming
In some decision-making problems, only two choices are possible: Yes or No, Accept or Reject, Select or Do Not Select.
Such problems are solved using Zero-One Programming, where decision variables can take only the values 0 or 1.
Example
A company must decide whether to open a new branch. The decision is either "Open" (1) or "Do Not Open" (0).
Exam Note: Zero-One Programming is also called Binary Programming because decision variables take only two values—0 or 1.
▪️Goal Programming
The problem having multiple, conflicting and incommensurable objective functions (goals) subject to linear constraints are called linear goal programming problems.
Many organizations pursue several objectives simultaneously.
Goal Programming helps management achieve multiple goals by assigning priorities to different objectives and finding the most satisfactory solution.
Example
A company wants to maximize profit, reduce overtime, and improve product quality at the same time.
Exam Note: Goal Programming is used for problems involving multiple objectives.
▪️Stochastic Programming
In some real-life problems, decision variables or parameters are uncertain and depend on probability.
If the decision variables in an LPP depend on chance, then such problems are called stochastic linear programming problems.
Stochastic Programming helps solve such problems by considering uncertainty in the mathematical model.
Example
A company prepares its production plan without knowing the exact future demand.
Exam Note: Stochastic Programming deals with optimization problems involving uncertainty or probabilistic data.
2. Inventory Models (Production Models)
It is used in determining when and how much production or purchase. Every organization must maintain an adequate level of inventory to ensure smooth production and customer service.
However, keeping excessive inventory increases costs, while insufficient inventory may interrupt production or lead to customer dissatisfaction.
Inventory Models of Operation Research help managers determine the optimum inventory policy by balancing different inventory-related costs.
The major costs considered in inventory models are:
- Shortage Cost
- Ordering Cost
- Holding (Storage) Cost
- Interest or Carrying Cost
This study of Inventory Models helps in making decisions about:
- How much to purchase.
- When to order.
- Whether to manufacture or to purchase i.e., make and buy decisions.
The most well-known use is in the form of the Economic Order Quantity equation for finding economic lot size.
The Economic Order Quantity (EOQ) model determines the most economical order quantity that minimizes the total inventory cost.
Example
A supermarket must decide how many grocery items should be ordered every month. Ordering too frequently increases ordering costs, while ordering large quantities increases storage costs. Inventory Models help determine the optimum order quantity.
Exam Note: Inventory Models determine how much to order or produce and when to order or produce while minimizing total inventory costs.
3. Waiting Line (Queuing) Models
In many organizations, customers have to wait before receiving a service. Banks, hospitals, railway reservation counters, airports, supermarkets, and call centres are common examples of waiting-line systems.
Queuing Models study these waiting-line situations and help managers provide efficient service while controlling operating costs.
These models help estimate:
- The average waiting time of customers (how much average time will be spent by the customer in a queue?)
- The average number of customers in the queue.
- The average queue length.
- The utilization of service facilities.
- The traffic intensity of a queuing system.
- The optimum number of service counters. etc.
The study of waiting line problems provides us to minimize the sum of costs of providing service and cost of obtaining service which is primarily associated with the value of time spent by the customer in a queue.
The main objective is to balance the cost of providing service with the cost of customer waiting.
Example
A bank uses a Queuing Model to determine whether two or three cash counters should remain open during peak business hours.
Exam Note: Queuing Models help determine the optimum level of service facilities by analysing waiting-line systems.
4. Markovian Models
Many real-life systems change from one condition to another over time. These conditions are known as states, and the movement from one state to another depends on probability.
That is, these models are used in situations where various states of the system are defined and the system moves from one state to another on a probability basis.
Markovian Models are used to study such systems where future behaviour depends only on the present state and not on previous states.
These models are widely used in production planning, inventory systems, marketing analysis, equipment maintenance, and customer behaviour studies.
Example
A company studies customer brand loyalty. Based on past purchasing behaviour, it estimates the probability that customers will continue buying the same brand or switch to a competing brand.
Exam Note: Markovian Models analyse systems that move from one state to another according to probability.
5. Game Theory (Competitive Strategy Models)
Game theory is used for decision making under conflicting situations where there are one or more players. In the game theory, we consider two or more persons with different objectives, each of whose actions influence the outcomes of the game.
Business organizations often operate in competitive environments where the decisions of one competitor influence the results of others.
Game Theory provides a mathematical framework for analysing such competitive situations. The game theory provides solutions to such games, assuming that each of the players wants to maximize his profits and minimize his losses.
That is, in Game Theory, two or more decision-makers (called players) choose strategies to maximize their own gains while minimizing possible losses.
This technique is useful for studying competitive decision-making, pricing policies, advertising strategies, bidding problems, and market competition.
However, many practical business situations are more complex than the assumptions made in basic Game Theory models. Therefore, its direct application in industrial decision-making may sometimes be limited.
So, methods for solving such models have not been found suitable for industrial applications, mainly because they are referred to an idealistic world neglecting many essential features of reality.
Example
Two competing companies are deciding whether to reduce the price of their products. The profit earned by one company depends on the pricing decision taken by the other company.
Exam Note: Game Theory is used for decision-making under competitive or conflicting situations.
6. Network Models
Many projects involve a large number of activities that must be completed in a proper sequence.
Network Models help managers plan, schedule, coordinate, and control such complex projects efficiently.
These models are an approach to planning, scheduling and controlling complex projects. Such large projects are very common in the field of construction, maintenance, computer system installation, research and development design, etc.
The two most widely used network techniques are:
- PERT (Program Evaluation and Review Technique)
- CPM (Critical Path Method)
PERT and CPM are used for planning, scheduling and controlling complex projects which can be characterized as networks.
These techniques identify the sequence of project activities, estimate completion time, and determine the critical path of the project.
Network Models are widely used in construction projects, research and development, software development, plant installation, maintenance work, and defence projects.
Example
A construction company uses PERT and CPM to prepare the activity schedule for building a new bridge.
Exam Note: Network Models use PERT and CPM for project planning, scheduling, and control.
7. Job Sequencing Models
In many organizations, several jobs have to be performed on one or more machines. The order in which these jobs are processed has a direct effect on the total production time and machine utilization.
Job Sequencing Models help determine the best sequence of performing jobs so that the overall performance of the system is improved.
The main objective is usually to:
- Minimize the total elapsed time.
- Reduce machine idle time.
- Improve production efficiency.
The term elapsed time refers to the total time from the beginning of the first job until the completion of the last job.
These models involve the selection of such a sequence of performing a series of jobs to be done on service facilities (machines) that optimize the efficiency measure of performance of the system. In other words, sequencing is concerned with such a problem in which efficiency measure depends upon the order or sequence of performing a series of jobs.
For example, consider a sequencing problem where n jobs are to be performed on m different machines. In such a case, our problem is to determine the sequence, which minimizes the total elapsed time. Here, the term elapsed time means the time from the start of the first job up to the completion of the last job.
Example
A workshop has five different jobs to be processed on two machines. Job Sequencing Models determine the order of processing that minimizes the total completion time.
Exam Note: Job Sequencing Models determine the best order of processing jobs to improve system efficiency.
8. Replacement Models
Machines, vehicles, and equipment gradually become less efficient because of wear and tear, increasing maintenance costs, or technological improvements.
Replacement Models help managers determine the most economical time to replace an existing asset with a new one. The objective is to minimize the total cost of operating, maintaining, and replacing equipment.
These models deal with the determination of optimum replacement policy in situations that arise when some items or machinery need replacement by a new one.
These models are commonly used for:
- Machinery replacement
- Vehicle replacement
- Equipment replacement
- Electronic system replacement
Example
A transport company decides whether an old delivery truck should be repaired or replaced with a new one.
Exam Note: Replacement Models determine the optimum replacement policy for equipment and machinery.
9. Simulation Models
Some real-life problems are so complex that obtaining an exact mathematical solution is difficult or even impossible.
Simulation Models provide an alternative approach by creating a mathematical or computer-based representation of the real system. Different situations are then tested on the model to study their possible outcomes before implementing decisions in practice.
Simulation is a very powerful technique for solving many complex models which cannot be solved otherwise and thus it is being extensively applied to solve a variety of problems. In fact, such models are solved by simulation techniques where no other method is available for its solution.
A Simulation Model is a mathematical business model which combines both mathematical and logical concepts that tries to emulate a real-life system through the use of computer software.
Simulation does not always provide the optimal solution. Instead, it helps managers understand the behaviour of complex systems and compare different alternatives.
Simulation Models are widely used in manufacturing, healthcare, transportation, banking, defence, and supply chain management.
Example
An airport uses a computer simulation model to analyse passenger flow before increasing the number of security counters.
Exam Note: Simulation is used when real-life systems are too complex for exact mathematical analysis.
Frequently Asked Questions (FAQs)
1. What are Quantitative Techniques in Operations Research?
Quantitative Techniques are mathematical and analytical methods used to analyse data and solve decision-making problems scientifically.
2. What are Operations Research Models?
Operations Research Models are mathematical representations of real-life systems that help managers analyse problems and select the best possible solution.
3. What is the difference between Quantitative Techniques and OR Models?
Quantitative Techniques are the methods used for analysis, whereas OR Models represent the problem in mathematical form. Both work together to support scientific decision-making.
4. Which is the most widely used OR Model?
Linear Programming is one of the most widely used Operations Research models because it helps optimize the use of limited resources.
5. Why are Simulation Models important?
Simulation Models help analyse complex systems where exact mathematical solutions are difficult or impossible to obtain.
6. Which OR Model is used for project management?
Network Models, particularly PERT and CPM, are used for planning, scheduling, and controlling projects.
7. Which OR Model is used for waiting-line problems?
Queuing Models are used to analyse waiting-line systems and determine the optimum level of service facilities.
8. Comparison of Major Operations Research Models
| OR Model | Main Purpose |
|---|---|
| Allocation Models | Allocate limited resources efficiently |
| Inventory Models | Determine when and how much to order or produce |
| Queuing Models | Analyse waiting-line systems |
| Markovian Models | Study probabilistic state changes |
| Game Theory Models | Analyse competitive decision-making |
| Network Models | Plan, schedule, and control projects |
| Job Sequencing Models | Determine the best order of performing jobs |
| Replacement Models | Decide the optimum replacement time |
| Simulation Models | Analyse complex systems using computer models |
9. Applications of Operations Research Models
Operations Research models are widely used in both public and private organizations.
Some important applications include:
- Production planning and scheduling
- Inventory management
- Transportation and logistics
- Supply chain management
- Project management
- Hospital administration
- Banking and financial planning
- Marketing and sales planning
- Airline scheduling
- Defence and military operations
- Telecommunication systems
- Manufacturing industries
10. What are Quantitative Techniques in Operations Research?
Quantitative Techniques are mathematical, statistical, and analytical methods used to analyse problems and support decision-making. These techniques help convert real-life business problems into mathematical models so that they can be analysed scientifically.
In Operations Research, quantitative techniques are used to evaluate different alternatives and select the best solution based on available data and defined objectives.
Example — A manufacturing company wants to determine the best production plan using limited labour, machines, and raw materials. Quantitative techniques help analyse different production alternatives and identify the most profitable plan.
Quantitative techniques help managers make objective and scientific decisions by using numerical data and mathematical analysis.
11. What are Operations Research Models?
An Operations Research model is a simplified mathematical representation of a real-life problem or system. It shows the relationship between decision variables, objectives, and constraints.
Instead of studying the actual system directly, O.R. specialists analyse the model to understand the problem and identify the best possible solution.
Different models are developed for different types of managerial problems.
Example — A transportation company develops a mathematical model to determine the least-cost method of delivering goods from factories to warehouses.
An OR model represents a real-life system in mathematical form to simplify analysis and decision-making.
12. Relationship Between Quantitative Techniques and OR Models
Quantitative techniques and OR models are closely related.
Quantitative techniques provide the mathematical methods used for analysis, while OR models provide the mathematical representation of the problem to which those techniques are applied.
In many standard Operations Research textbooks, these two terms are often used together because both contribute to scientific decision-making.
Quantitative techniques provide the method, whereas OR models represent the problem to be solved.
Conclusion
Quantitative Techniques and Operations Research Models provide a scientific foundation for solving complex managerial problems. Each model is designed for a specific type of decision-making situation and helps organizations use their resources efficiently, reduce costs, improve productivity, and make better decisions.
Since no single model is suitable for every problem, managers and O.R. specialists must carefully study the nature of the problem before selecting the most appropriate quantitative technique or model. A proper understanding of these models enables organizations to achieve their objectives more effectively and supports sound managerial decision-making.
Read Next
References
- Hamdy A. Taha, Operations Research: An Introduction, Pearson.
- Kanti Swarup, P. K. Gupta & Man Mohan, Operations Research, Sultan Chand & Sons.
- J. K. Sharma, Operations Research: Theory and Applications, Macmillan India.
- Frederick S. Hillier & Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill.
- H. M. Wagner, Principles of Operations Research, Prentice Hall.
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