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 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.

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.

---

Operation Research Models and Modelling

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.

What is Model in Operation Research?

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:

1. Defining the problem
2. Collecting data
3. Building a model of the system
4. Deriving a solution
5. Testing to validate the model and the solution
6. 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.

Markovian 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.