Question:

Mr. Mistry owns a small shop called "Mistry Furniture Shop" and deals with only two items—tables and chairs.

Total Budget: Rs 50,000

Storage Limit: 60 pieces

Cost & Profit per Item:

• Table: Cost = Rs 2,500, Profit = Rs 250
• Chair: Cost = Rs 500, Profit = Rs 75

Based on this information, answer the following:

1. Define the decision variables clearly.
2. Formulate the objective function to maximize profit.
3. Write down the constraints based on the given budget and storage limits.

(Do not solve the problem; just formulate the Linear Programming Problem.)

Decision Variables:

Let:

x = Number of tables purchased

y = Number of chairs purchased

Objective Function (Maximize Profit):

Maximize Z = 250x + 75y

Constraints:

1. Budget Constraint: 2500x + 500y ≤ 50000

2. Storage Constraint: x + y ≤ 60

3. Non-Negativity Constraint: x ≥ 0, y ≥ 0

Question (5 Marks)

A company produces two products, A and B. The production is limited by available resources. The company wants to determine the feasible region based on the given constraints.

Let:

x = Number of units of Product A

y = Number of units of Product B

The production is limited by the following constraints:

1. Labor Constraint: 2x + 4y ≤ 40

2. Material Constraint: 3x + 2y ≤ 30

3. Non-Negativity Constraint: x ≥ 0, y ≥ 0

Instructions:

- Plot the given constraint equations on a graph.

- Identify and shade the feasible region that satisfies all constraints.

(Do not solve the problem; only graph the feasible region.)

Hint:

To plot the constraint equations, follow these steps:

1. Convert each inequality into an equation (replace ≤ with =).

2. Find the x-intercept and y-intercept for each equation by setting one variable to zero and solving for the other.

3. Draw the lines for each equation on a graph.

4. Identify the region that satisfies all inequalities and shade it as the feasible region.

Example for 2x + 4y ≤ 40:

- Set x = 0 → 4y = 40 → y = 10 (y-intercept)

- Set y = 0 → 2x = 40 → x = 20 (x-intercept)

- Plot the points (0,10) and (20,0) and draw the line.

Repeat this process for the second equation, then shade the common feasible region.

Total cost after following all the steps will be 95 + 60 + 180 + 120 + 280 + 280 = 1015. 

Q. Find the feasible solution for the given transportation problem using the North-West Corner Method.

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Q. Find the initial basic feasible solution using the North-West Corner Method.

Q. Find the initial basic feasible solution for the given transportation problem using the Least-cost Method.

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Find the initial basic feasible solution for the given transportation problem using Vogel's Approximation Method.

A Payoff Table is a decision analysis tool that helps evaluate different decision alternatives by displaying potential outcomes under various conditions. It presents decision options in rows and possible states of nature or scenarios in columns, with each cell showing the payoff (profit, cost, or utility) for a specific combination of decision and scenario.

For example, consider a business deciding whether to launch Product A or Product B, with market conditions being either favorable or unfavorable.

This table helps decision-makers compare potential gains and losses under different scenarios. If the company is optimistic, it may choose Product A due to its higher maximum payoff. If risk-averse, it might prefer Product B, which has a better minimum outcome. By structuring choices clearly, a Payoff Table allows for better strategic decision-making based on data rather than intuition.

Decision-making under uncertainty occurs when a decision-maker lacks complete information about future outcomes, probabilities, or external factors influencing the decision. Unlike decision-making under risk, where probabilities are known, uncertainty involves unpredictable variables, making it difficult to assess potential results accurately.

One major challenge in decision-making under uncertainty is the lack of reliable data. Without historical trends or probabilities, decision-makers must rely on intuition, experience, or assumptions, increasing the risk of poor outcomes. Another challenge is the complexity of evaluating multiple unknown factors, which can lead to indecisiveness or biased judgments. Additionally, external factors such as market fluctuations, technological advancements, and regulatory changes can further complicate the decision-making process.

Despite these challenges, decision-makers can use techniques like scenario analysis, sensitivity analysis, and decision trees to structure their approach. By investigating possible alternatives and assessing their potential impacts, they can make more informed choices, even in uncertain environments.

Decision analysis is crucial for businesses as it provides a structured approach to evaluating alternatives, reducing uncertainty, and improving decision-making efficiency. It enables organizations to assess potential risks, maximize opportunities, and allocate resources effectively, leading to better strategic and operational outcomes.

The Payoff Table helps businesses compare different decision alternatives under various scenarios by displaying potential outcomes. By analyzing this table, decision-makers can identify the most profitable or least risky option, depending on their risk appetite. For example, if a company is considering launching a new product, a Payoff Table can outline expected profits under different market conditions, guiding them toward the best choice.

The Opportunity Loss Table, on the other hand, highlights the potential losses incurred by not choosing the best alternative. It helps businesses recognize the cost of missed opportunities and make decisions that minimize regret. For instance, if a company must decide between two investment projects, the Opportunity Loss Table can show how much profit would be lost if the less optimal choice is made, ensuring a more informed decision.

By using these tools, businesses can justify their decisions with data-driven insights, avoid guesswork, and improve overall decision quality, leading to long-term success and sustainability.

Decision analysis is crucial for businesses as it provides a structured approach to evaluating alternatives, reducing uncertainty, and improving decision-making efficiency. It enables organizations to assess potential risks, maximize opportunities, and allocate resources effectively, leading to better strategic and operational outcomes.

The Payoff Table helps businesses compare different decision alternatives under various scenarios by displaying potential outcomes. By analyzing this table, decision-makers can identify the most profitable or least risky option, depending on their risk appetite. For example, if a company is considering launching a new product, a Payoff Table can outline expected profits under different market conditions, guiding them toward the best choice.

The Opportunity Loss Table, on the other hand, highlights the potential losses incurred by not choosing the best alternative. It helps businesses recognize the cost of missed opportunities and make decisions that minimize regret. For instance, if a company must decide between two investment projects, the Opportunity Loss Table can show how much profit would be lost if the less optimal choice is made, ensuring a more informed decision.

By using these tools, businesses can justify their decisions with data-driven insights, avoid guesswork, and improve overall decision quality, leading to long-term success and sustainability.

An Opportunity Loss Table, also known as a Regret Table, helps decision-makers assess the potential loss from not choosing the best alternative. It compares each decision’s outcome to the highest possible payoff in that scenario, highlighting missed opportunities.

Unlike a Payoff Table, which shows actual gains for each decision, an Opportunity Loss Table focuses on the difference between the chosen option and the best possible result. This allows businesses to minimize regret and make more informed choices.

For example, if a company chooses between two investment options and one yields lower profits than the other under certain market conditions, the difference in earnings represents the opportunity loss. By using this approach, decision-makers can evaluate risks more effectively and select the option that reduces potential regret.

Game Theory is a mathematical framework used to analyze strategic interactions where the outcome for each participant depends on the actions of others. The basic elements of Game Theory include players, strategies, payoffs, and outcomes.

Players are the decision-makers in the game. Strategies refer to the different choices available to each player. Payoffs represent the rewards or consequences of a chosen strategy, and outcomes are the final results based on the strategies chosen by all players.

For example, in a pricing competition between two companies, each company (player) can choose to lower or maintain its prices (strategy). If both lower prices, profits decrease (payoff), but if only one lowers prices, that company gains a competitive advantage while the other loses customers (outcome). Understanding these elements helps businesses, governments, and individuals make optimal strategic decisions.

The basic elements of Game Theory are players, strategies, payoffs, and outcomes, all of which shape decision-making in competitive situations.

Players are the individuals or entities making strategic choices. Strategies represent the possible actions each player can take. Payoffs are the rewards or penalties associated with different strategy combinations. Outcomes result from the interaction of strategies chosen by all players.

For instance, in a business rivalry, two competing firms may choose to advertise or not. If both advertise, profits remain stable. If only one advertises, it gains a larger market share. These elements interact to influence decisions, as each player must anticipate the response of others to maximize their payoff.

The basic elements of Game Theory are players, strategies, payoffs, and outcomes, all of which shape decision-making in competitive situations.

Players are the individuals or entities making strategic choices. Strategies represent the possible actions each player can take. Payoffs are the rewards or penalties associated with different strategy combinations. Outcomes result from the interaction of strategies chosen by all players.

For instance, in a business rivalry, two competing firms may choose to advertise or not. If both advertise, profits remain stable. If only one advertises, it gains a larger market share. These elements interact to influence decisions, as each player must anticipate the response of others to maximize their payoff.

The Payoff Matrix in Game Theory defines the potential outcomes of different strategies chosen by players in a competitive scenario.

It helps decision-makers evaluate various strategic options by presenting possible rewards or losses associated with each choice.

By analyzing the matrix, businesses and economists can predict competitor behavior, assess risks, and make informed decisions to maximize benefits.

A two-person zero-sum game is a situation where two players compete, and whatever one player gains, the other loses by the same amount. The total amount of rewards or losses in the game remains constant, meaning one player’s success comes at the expense of the other.

For example, in a simple game of chess, one player wins, and the other loses. There is no way for both to win or both to lose at the same time. If Player A wins, Player B automatically loses, making it a zero-sum situation.

This concept is used in real-life competitive situations like business, sports, and military strategies, where one party’s advantage often means another party’s disadvantage.

A pure strategy is when a player always chooses the same action in a game, without changing it. They have a fixed plan and do not switch strategies. For example, in rock-paper-scissors, if a player always chooses "rock," that is a pure strategy.

A mixed strategy is when a player changes their choices and does not stick to one action every time. They may use probabilities to decide their moves. For example, in rock-paper-scissors, if a player randomly picks rock, paper, or scissors each time, that is a mixed strategy.

The main difference is that pure strategy is predictable, while mixed strategy involves randomness, making it harder for the opponent to guess the next move.

A two-person zero-sum game is a situation where one player’s gain is exactly equal to the other player’s loss. This concept is commonly seen in competitive business environments and sports.

For example, consider two rival companies competing for market share in the same industry. If Company A launches a new product and gains customers, Company B loses those customers, reducing its sales. The total market remains the same, so one company’s success directly causes the other’s decline, making it a zero-sum game.

Similarly, in a sports match like tennis, if one player wins, the other must lose. There is no possibility for both players to win simultaneously. The total outcome remains constant, reinforcing the idea of a zero-sum game.

These situations justify how strategic decision-making in competitive environments follows the principles of a two-person zero-sum game, where each move directly impacts the opponent’s outcome.

One can argue that the assumption of rational decision-making in Game Theory does not always hold true in real life. While the theory assumes players act logically to maximize their benefits, human behavior is often influenced by emotions, biases, and external pressures.

For example, in a negotiation, a person might accept a less favorable deal due to fear of conflict or emotional attachment rather than making the most rational choice. Similarly, in financial markets, panic selling during a crisis often contradicts the rational goal of maximizing long-term profits.

These examples highlight that while rationality is a useful assumption in Game Theory, real-world decisions are frequently shaped by psychological and situational factors.

The Payoff Matrix is a useful tool for analyzing decisions where risks and rewards must be carefully considered. It helps individuals and businesses compare different choices and predict possible outcomes based on various scenarios.

For example, imagine you are deciding whether to take a new job with a higher salary but more workload or stay in your current job with a stable routine. The Payoff Matrix would list the possible outcomes of each choice, helping you weigh the benefits of increased income against the downside of added stress.

By using a Payoff Matrix, decision-makers can systematically evaluate their options and make informed choices. This structured approach defends the idea that decisions should not be made impulsively but rather through careful consideration of all possible outcomes.