Assignment Code:MMPO-001 / TMA / July / 2022 / January 23

Course Code: MMPO-001

Assignment Name: Operations Research

Year: 2022 - 2023

Verification Status: Verified by Professor

Note: Attempt all the questions

Q 1. What is Operations Research? Describe some of the tools of Operations Research.

Ans) Operations Research is a distinct discipline in its own way but also become

an important tool in all areas of study. Operations research (OR) is a scientific and systematic approach to decision-making and problem-solving. It is a field of study that uses mathematical models, statistical analysis, and optimization techniques to help organizations make better decisions and solve complex problems. The goal of OR is to identify the best possible solution for a given problem by analysing and optimizing the available resources. OR can be applied to a wide range of problems in many different industries, such as transportation, logistics, manufacturing, finance, healthcare, and the military.

Tools of Operations Research

Operations Research (OR) involves the application of mathematical and analytical methods to solve complex decision-making problems. OR involves a wide range of tools and techniques that help in decision-making, optimization, and problem-solving.

1. Linear Programming (LP): Linear Programming is a mathematical method used to determine the optimal solution to a linear problem. LP is used to solve optimization problems where the objective is to maximize or minimize a linear function subject to a set of linear constraints.

2. Nonlinear Programming (NLP): Nonlinear Programming is a mathematical method used to solve optimization problems where the objective function and constraints are nonlinear. NLP is used to solve complex problems that cannot be solved using LP.

3. Dynamic Programming (DP): Dynamic Programming is a method used to solve complex problems that involve making a sequence of decisions over time. DP is used to solve optimization problems where the solution evolves over time.

4. Integer Programming (IP): Integer Programming is a method used to solve optimization problems where some or all of the decision variables are restricted to integer values. IP is used to solve problems where the decision variables represent discrete choices, such as selecting the number of units of a product to produce.

5. Queuing Theory: Queuing Theory is a method used to analyze waiting lines or queues. Queuing Theory is used to optimize the performance of service systems by minimizing the average waiting time or the average queue length.

6. Simulation: Simulation is a method used to model and analyze complex systems by using a computer to imitate the system's behavior over time. Simulation is used to optimize the performance of systems by predicting their behavior under different scenarios.

7. Network Analysis: Network Analysis is a method used to analyze complex systems that can be represented as a network. Network Analysis is used to optimize the performance of systems by identifying the critical paths or bottlenecks in the system.

8. Symbolic Logic: Symbolic logic deals with substituting symbols for words, classes of things or functional systems. It incorporates rules, algebra of logic and propositions. There have been only limited attempts to apply this technique to business problems; however, it has had extensive application in the design of computing machinery.

9. Information Theory: Information Theory is an analytical process transferred from the electrical communication field to operations research. It seeks to evaluate the effectiveness of information flow within a given system. Despite its application mainly to communications networks, it has had an indirect influence in simulating the examination of business organizational structures with a view to improving information or communication flow.

10. Decision Analysis: Decision Analysis is a method used to analyze complex decision-making problems that involve uncertainty and risk. Decision Analysis is used to optimize the decision-making process by identifying the optimal decision based on the available information.

11. Game Theory: Game Theory is a method used to analyze strategic decision-making in situations where the outcome depends on the decisions of multiple players. Game Theory is used to optimize the performance of systems by predicting the behavior of multiple players and identifying the optimal strategy.

12. Utility/Value Theory: Utility/Value theory deals with assigning numerical significance to the worth of alternative choices. To date this has been only a concept and is in the stage of elementary model information and experimentation. When developed this may be very helpful in the decision-making process in assessing the worth of various possible outcomes.

13. Inventory Control Models: When to buy, how much to keep buying and how much to keep in stores are some of the questions which production managers, purchase managers and material managers address themselves to. Inventory control models provide rational answer to these questions in different situation of supply and demand for different kind of materials. Inventory control models help managers to decide reordering time, reordering level and optimal ordering quantity. The approach is to prepare a mathematical model of the situation that expresses total inventory costs in terms of demand, size of order, possible over or under stocking and other relevant factors and then to determine optimal order size, optimal order level etc. using calculus or some other technique.

14. Sequencing Theory: Sequencing Theory is related to Waiting Line Theory. It is applicable when the facilities are fixed, but the order of servicing may be controlled. The scheduling of service or the sequencing of jobs is done to minimize the relevant costs.

These are just a few examples of the many tools and techniques used in Operation Research. The choice of which tools to use depends on the specific problem being solved and the available data and resources. By using these tools, organizations can make more informed decisions, optimize their operations, and improve their overall performance

Q 2. What do you mean by Goal Programming? Describe the specialities of goal programming.

Ans) Goal Programming is a powerful optimization technique that is used to solve decision-making problems where there are multiple objectives or goals to be achieved. It is a technique that is widely used in Operations Research, Management Science, and Decision Analysis. In Goal Programming, the decision-maker defines a set of goals, or objectives, and then seeks to find the best possible solution that satisfies as many of these goals as possible. The basic idea behind Goal Programming is to assign weights to each goal, indicating the relative importance of that goal, and then to minimize the deviations from each goal subject to the constraints. The deviations from each goal are measured by the so-called "deviation variables", which are added to the objective function. The constraints are expressed as linear equations or inequalities that must be satisfied.

There are different types of Goal Programming, depending on the number and nature of the goals. These include:

1. Pre-emptive Goal Programming: In this type of Goal Programming, the decision-maker seeks to achieve the most important goals first, and then move on to the less important goals. The less important goals are only considered if they do not conflict with the more important goals.

2. Lexicographic Goal Programming: In this type of Goal Programming, the goals are ranked in order of importance, and the decision-maker seeks to satisfy the most important goal completely, and then move on to the next most important goal, and so on.

3. Interactive Goal Programming: In this type of Goal Programming, the decision-maker interacts with the model to adjust the goals and constraints, and to evaluate the trade-offs between different goals.

4. Multi-Objective Linear Programming: In this type of Goal Programming, there are multiple objectives to be optimized simultaneously, and the decision-maker seeks to find the best compromise solution that satisfies all the objectives.

Goal Programming has many applications in real-world decision-making problems. For example, it can be used in production planning to balance the conflicting objectives of minimizing costs, maximizing profits, and meeting customer demand. It can also be used in financial planning to balance the conflicting objectives of maximizing returns, minimizing risk, and satisfying regulatory requirements.

Specialities of Goal Programming

Goal Programming is a powerful optimization technique that is used to solve decision-making problems where there are multiple objectives or goals to be achieved. The technique has several specialities that make it an effective tool for solving complex decision-making problems.

1. Handles Multiple Objectives: One of the primary specialities of Goal Programming is its ability to handle multiple objectives. Decision-making problems often involve several goals that need to be achieved simultaneously. Goal Programming can be used to find the best possible solution that satisfies as many of these goals as possible, even if they conflict with each other.

2. Allows for Trade-offs: Another speciality of Goal Programming is its ability to allow for trade-offs between different goals. In many decision-making problems, it is not possible to achieve all the goals at the same time. Goal Programming can be used to find the best possible compromise solution that balances the different goals.

3. Incorporates Constraints: Goal Programming can also incorporate constraints into the decision-making process. Constraints are conditions that must be satisfied in order to achieve the goals. These constraints can be expressed as linear equations or inequalities that must be satisfied.

4. Assigns Weights to Objectives: In Goal Programming, weights are assigned to each objective, indicating the relative importance of that objective. The weights can be adjusted to reflect the decision-maker's priorities. This allows for the decision-maker to make subjective judgments about the relative importance of each goal.

5. Uses Deviation Variables: Deviation variables are used in Goal Programming to measure the deviations from each goal. These variables are added to the objective function, and the goal is to minimize these deviations subject to the constraints.

6. Handles Nonlinear Objectives: Goal Programming can also handle nonlinear objectives. Nonlinear objectives are those that cannot be expressed as a linear function. This makes Goal Programming more flexible than other optimization techniques, which are limited to linear objectives.

7. Flexible Formulations: Goal Programming can be formulated in many different ways, depending on the nature of the problem and the decision-maker's preferences. This allows for a great deal of flexibility in the decision-making process.

In summary, Goal Programming is a powerful optimization technique that has several specialities that make it an effective tool for solving complex decision-making problems. It can handle multiple objectives, allow for trade-offs, incorporate constraints, assign weights to objectives, use deviation variables, handle nonlinear objectives, and be formulated in many different ways. These specialities make Goal Programming a valuable tool for decision-makers in a wide range of fields.

Q 3. Describe the unique features of goal programming.

Ans) Goal programming is a multi-criteria decision-making approach that allows decision-makers to incorporate multiple conflicting objectives into a single optimization model. The goal programming model provides a flexible framework for dealing with multiple objectives and constraints, and it can be applied to a wide range of problems in various fields such as engineering, finance, and management. The unique features of goal programming are:

1. Multiple objectives: Goal programming is designed to handle multiple, often conflicting, objectives. It allows decision-makers to specify several goals that must be satisfied simultaneously. The model aims to find the best possible solution that minimizes the deviations from each of the goals.

2. Priority levels: In goal programming, objectives are often grouped by priority levels. Priority levels allow decision-makers to indicate the relative importance of each objective. Goals with higher priority levels are given more weight in the optimization process.

3. Deviation variables: Goal programming introduces deviation variables, which measure the degree of deviation from each objective. Deviation variables can be positive or negative, indicating whether the objective is over- or under-achieved. The model minimizes the sum of deviation variables across all objectives.

4. Target values: Goal programming requires decision-makers to specify target values for each objective. Target values represent the desired level of achievement for each objective. These values are often based on expert judgment, historical data, or market trends.

5. Constraints: Like other optimization models, goal programming also involves constraints. Constraints limit the feasible region of the solution space and ensure that the solution is practical and feasible. Constraints can be linear or nonlinear and can involve several variables.

6. Flexibility: Goal programming provides decision-makers with the flexibility to adjust objectives and constraints as the situation changes. The model can be easily modified to accommodate new goals or constraints, making it a powerful tool for dynamic decision-making. It can be applied to a range of problems in different fields, including finance, engineering, logistics, and agriculture. Moreover, it can be customized to meet the specific needs and requirements of a particular problem.

7. Sensitivity analysis: Goal programming also offers sensitivity analysis, which allows decision-makers to evaluate the effect of changes in the objectives, constraints, or parameters of the model. Sensitivity analysis can help decision-makers identify critical factors that affect the solution and make better-informed decisions.

8. Visualization: Goal programming can be visualized using graphs and charts, which make it easier for decision-makers to understand and interpret the results. Graphical representations can also help decision-makers identify trade-offs and make informed decisions.

9. Applications: Goal programming has a wide range of applications in various fields. It can be used to optimize investment portfolios, production planning, resource allocation, transportation planning, and many other decision-making problems.

10. Multiple Solutions: Goal programming may not always produce a unique solution. In some cases, the model may have multiple solutions that satisfy the constraints and objectives. This is due to the presence of multiple goals or conflicting objectives. However, the decision-maker can select the most appropriate solution based on their preferences and priorities.

11. Incorporating Risk: Goal programming can incorporate risk and uncertainty in the decision-making process. This is achieved by including risk constraints or objectives in the model. For instance, a company may want to minimize costs while also ensuring that it has a buffer stock of inventory to handle unexpected demand fluctuations. In this case, a risk objective could be included to account for the uncertainty in demand and supply.

12. Hierarchical Structure: Goal programming can be structured hierarchically, with goals and objectives organized in a hierarchy. This allows for a more organized and structured approach to decision-making. The model can be broken down into smaller sub-models, each with its own goals and objectives. This facilitates the modelling process and makes it easier to identify and address trade-offs and conflicts among different objectives.

13. Sensitivity Analysis: Goal programming can be used to conduct sensitivity analysis, which involves examining the impact of changes in parameters or constraints on the model's output. This can help decision-makers understand how changes in input data or assumptions can affect the optimal solution. Sensitivity analysis can also be used to identify critical constraints or parameters that may need to be monitored closely or adjusted in response to changing circumstances.

In conclusion, goal programming is a powerful optimization tool that can be used to solve complex decision-making problems involving multiple goals and objectives. Its unique features include the ability to handle conflicting goals, incorporate risk, and accommodate multiple solutions. Its flexibility and hierarchical structure make it a versatile tool that can be applied to a wide range of problems in various fields. Moreover, the use of sensitivity analysis can enhance the decision-making process by providing insights into the impact of changes in parameters or assumptions on the model's output.

Q 4. Determine which course of action Player B will not use in the following game. Obtain the best strategies for both players and the value of the game.

To determine the best strategies for both players, we will use the following steps:

Step 1: Find the maximum payoff for Player A in each row.

Max for Player A: 7, -1

Step 2: From the maximum payoffs, choose the minimum payoff for Player A.

Min of max for Player A: -1

Step 3: Find the minimum payoff for Player B in each column.

Min for Player B: -4, -1, -2

Step 4: From the minimum payoffs, choose the maximum payoff for Player B.

Max of min for Player B: -1

Therefore, the value of the game is -1, which is the outcome that both players will agree to.

To determine which course of action Player B will not use, we can identify the strategy that is dominated by another strategy. A strategy is said to be dominated if there exists another strategy that always gives a higher payoff regardless of what the opponent does. For Player B, strategy III is dominated by strategy II since choosing strategy II guarantees a payoff of at least -1, whereas choosing strategy III guarantees a payoff of -2. Therefore, Player B will not use strategy III.

The best strategies for both players are:

1. Player A: Choose strategy II

2. Player B: Choose strategy I

In this case, both players receive a payoff of -1.

Q 5. What is queueing system? Discuss the characteristics of a queueing model.

Ans) Queueing system, also known as queuing theory, is a branch of applied mathematics that deals with the study of waiting lines or queues. It involves the mathematical analysis of various types of queues and aims to optimize the performance of a queuing system.

A queuing system consists of three basic components:

1. Arrival Process: The arrival process is the sequence in which customers or entities arrive at the queue. The arrival of each entity can be independent or dependent on other entities.

2. Service Process: The service process refers to the process of providing service to the customers who are waiting in the queue. This process can be either single-phase or multi-phase, depending on the nature of the service.

3. Queue Discipline: Queue discipline is the set of rules that are used to determine the order in which customers are served. There are different types of queue disciplines, such as first-in, first-out (FIFO), last-in, first-out (LIFO), priority, and random.

Queuing theory employs various mathematical models and statistical techniques to study the behavior of the queuing system. It helps in analysing the performance of the system by evaluating key parameters such as waiting time, queue length, service time, and service utilization. Queuing theory can be applied to a wide range of real-world applications such as call centers, banks, hospitals, airports, and traffic systems.

There are several important features of queuing systems:

1. Arrival rate: This refers to the rate at which customers or entities arrive at the queue. It can be described by the Poisson distribution, which assumes that the arrival rate is constant over time.

2. Service rate: The service rate refers to the rate at which customers are served by the system. It can be described by the exponential distribution, which assumes that the service time is independent and follows a negative exponential distribution.

3. Queue length: Queue length refers to the number of customers waiting in the queue at any given time. It is an important measure of the performance of the system.

4. Waiting time: Waiting time is the amount of time that a customer spends waiting in the queue before being served.

5. Service time: Service time is the time it takes to serve a customer.

6. Utilization: Utilization refers to the proportion of time that the server is busy providing service.

There are several different types of queuing models, each with its own set of assumptions and parameters. Some of the most commonly used models include the M/M/1, M/M/c, M/G/1, and M/G/c models. These models assume different distributions for the arrival and service processes, and different queue discipline rules. Overall, queuing theory plays an important role in the design and optimization of systems that involve waiting lines, helping organizations to minimize wait times, reduce congestion, and improve overall efficiency.

Characteristics of a Queueing Model

A queueing model is a mathematical framework used to study queueing systems, which are systems characterized by the arrival of customers or requests to a service facility that can serve them, subject to certain constraints.

The following are some of the key characteristics of a queueing model:

1. Arrival process: This is the process by which customers arrive at the system. The arrival process is typically modelled using a probability distribution, such as a Poisson process or a negative exponential distribution.

2. Service process: This is the process by which the service facility serves customers. The service process is also typically modelled using a probability distribution, such as a normal distribution or an exponential distribution.

3. Queue discipline: This refers to the rule by which customers are served when they arrive at the system and find that the service facility is busy. Different queue disciplines can be modelled, such as first-come-first-served (FCFS), last-come-first-served (LCFS), or priority-based queues.

4. System capacity: This refers to the maximum number of customers that can be in the system at any given time. The capacity can be finite or infinite, and it can be modelled using various parameters, such as the number of servers or the size of the waiting area.

5. Performance measures: These are the metrics used to evaluate the performance of the queueing system. Common performance measures include the average waiting time, the average number of customers in the system, and the utilization of the service facility.

6. Queueing networks: In some cases, it is necessary to model more complex systems, such as those with multiple service facilities or those that are interconnected. Queueing network models can be used to capture such systems and analyze their performance.

Overall, queueing models are useful for understanding the behavior of complex systems that involve waiting lines, such as call centers, airports, and manufacturing facilities. By studying the characteristics of queueing models, researchers and practitioners can identify areas for improvement and optimize the performance of these systems.