Quantitative Techniques – Question & Answers

Quick Revision Notes for Commerce & Management Students

Linear Programming (LP):

Linear Programming is a mathematical optimization technique used for finding the best outcome in a mathematical model with linear relationships. It involves maximizing or minimizing a linear objective function subject to a set of linear equality and inequality constraints. The basic components of linear programming include:

Decision Variables:

Definition: The variables that represent the quantities to be determined in the problem.

Example: In a production problem, decision variables might represent the quantities of different products to be manufactured.

Objective Function:

Definition: The linear equation representing the goal of optimization, whether to maximize profit or minimize cost.

Assumptions of Linear Programming:

Linearity: Assumes that the relationships between decision variables and the objective function, as well as constraints, are linear.

Additivity: Assumes that the total contribution of all decision variables is the sum of their individual contributions.

Divisibility: Assumes that decision variables can take any fractional value.

Certainty: Assumes that all parameters in the model are known with certainty.

Independence: Assumes that the decision variables are independent of each other.

Applications of Linear Programming:

Supply Chain Management: Optimal allocation of resources in the production and distribution process.

Finance: Portfolio optimization for maximizing returns with constraints on risk.

Operations Research: Resource allocation, production scheduling, and project management.

Marketing: Media planning, advertising budget allocation, and product mix optimization.

Agriculture: Crop planning, land use optimization, and resource allocation.

Manufacturing: Production planning, workforce scheduling, and inventory management.

Telecommunications: Network optimization and capacity planning.

Environmental Management: Waste disposal, pollution control, and resource conservation.

Linear programming provides a powerful tool for decision-makers to optimize resource allocation and make informed choices in a wide range of industries and applications.

In game theory, a saddle point (or sometimes called a minimax point) is a solution to a two-player zero-sum game. A zero-sum game is one in which the total amount of utility or payoff remains constant, meaning one player’s gain is exactly offset by the other player’s loss.

In the context of a payoff matrix representing a game, a saddle point is a specific entry where the player whose turn it is to move can guarantee at least a certain payoff, regardless of the opponent’s strategy. At the same time, the opponent can ensure that the moving player doesn’t achieve more than a certain payoff.

Mathematically, for a payoff matrix M in a two-player game:

1. A saddle point exists at (i, j) if M[i, j] is the minimum element in the i-th row and the maximum element in the j-th column.

1. The player whose turn it is to move can choose the strategy corresponding to row i with the guarantee of getting at least M[i, j].

1. The opponent can choose the strategy corresponding to the column j to ensure that the moving player gets no more than M[i, j].

The concept of a saddle point is closely related to the minimax strategy in game theory, where each player minimizes their maximum possible loss. Saddle points are important in determining optimal strategies for players in certain types of games. However, it’s worth noting that not all games have saddle points, and more complex games may require different solution concepts, such as mixed strategies.

Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM) are both project management techniques used for planning, scheduling, and controlling projects. Here’s a chart highlighting the key differences between PERT and CPM:

It’s important to note that in practice, PERT and CPM techniques are often used together or in conjunction, and the choice between them depends on the specific characteristics and requirements of the project.

A two-person zero-sum game is a type of strategic interaction in game theory where there are two players, and the total amount of utility or payoff in the game is constant, meaning that one player’s gain is exactly balanced by the other player’s loss. In other words, the sum of the players’ payoffs is always zero.

Here are key characteristics of a two-person zero-sum game:

1. Two Players: There are only two players involved in the game, often referred to as Player 1 and Player 2.

1. Zero-Sum Property: The total payoff or utility in the game is constant, and any gain by one player is offset by an equal loss for the other player. Mathematically, the sum of the payoffs for both players is always zero.

1. Competitive Nature: The interests of the two players are in direct opposition. When one player wins, the other loses.

1. Fixed Total Payoff: The total payoff does not change regardless of the strategies chosen by the players. The gains and losses are redistributed among the players.

1. Payoff Matrix: The game is often represented by a payoff matrix, where each player’s strategy choice corresponds to a specific cell in the matrix, indicating the payoffs associated with the interaction.

Two-person zero-sum games are commonly analyzed using various solution concepts, such as the minimax strategy, where each player aims to minimize the maximum possible loss. These games find applications in various fields, including economics, operations research, and decision analysis, where conflicts of interest can be modeled as competitive interactions.

In game theory, players can adopt different types of strategies to maximize their payoffs. Two fundamental concepts are pure strategy and mixed strategy.

1. Pure Strategy:
• Definition: A pure strategy in game theory refers to a specific and well-defined course of action that a player will take in a game.

• Characteristics: Players using a pure strategy make a single, unambiguous choice. There is no randomness or probability involved.

• Example: In a simple game like rock-paper-scissors, if a player always chooses “rock,” “paper,” or “scissors,” that player is employing a pure strategy.

1. Mixed Strategy:
• Definition: A mixed strategy involves a player randomizing between different pure strategies with certain probabilities.

• Characteristics: Players using a mixed strategy introduce an element of randomness into their decision-making process. The probabilities assigned to each pure strategy reflect the player’s willingness to take certain actions.

• Example: In a game where a player can choose between “A” and “B,” adopting a mixed strategy might involve choosing “A” with a certain probability (p) and “B” with the complementary probability (1-p). The player randomizes between the two strategies.

Comparison:

• Pure Strategy:
• Involves a specific, deterministic choice.

• No randomness or probability is involved.

• Easy to analyze and understand.

• Mixed Strategy:
• Involves randomization between different pure strategies.

• Introduces an element of uncertainty and probability.

• Requires the use of probability distributions to represent the player’s choices.

Use in Game Theory:

• Pure Strategy:
• Often used when there is a dominant or optimal strategy for a player.

• Suitable for simpler games with clear and distinct choices.

• Mixed Strategy:
• Useful when no dominant strategy exists.

• Applicable in more complex games where players seek to introduce unpredictability to gain an advantage.

In summary, pure strategies involve unambiguous choices, while mixed strategies introduce an element of randomness and probability into a player’s decision-making process, allowing for a more flexible and unpredictable approach in certain game situations.

The Hurwitz Principle, named after economist Leonid Hurwicz, is a decision-making criterion used in decision theory and game theory. It seeks to strike a balance between optimism and pessimism when making decisions under uncertainty. The Hurwicz Principle is particularly applicable in situations where decision-makers face incomplete information about the probabilities of different outcomes.

The principle involves two key parameters:

1. Optimism Parameter (α): A value between 0 and 1 that reflects the decision-maker’s degree of optimism. A higher α indicates a more optimistic perspective.

1. Pessimism Parameter (1 – α): The complementary value to the optimism parameter. It reflects the decision-maker’s degree of pessimism.

The Hurwicz Rule combines the possible payoff under the best-case scenario (maximum) and the worst-case scenario (minimum) to determine the decision. The decision is made by calculating a weighted average of the best and worst outcomes based on the optimism parameter.

The Hurwicz Principle acknowledges the uncertainty in decision-making by allowing decision-makers to incorporate both optimistic and pessimistic perspectives. The choice of the optimism parameter reflects the decision-maker’s risk attitude. If the decision-maker is risk-averse, they might choose a lower α, while a risk-taker might opt for a higher α.

It’s important to note that the Hurwicz Principle is just one approach to decision-making under uncertainty, and the choice of the optimism parameter is subjective and context-dependent. In practice, decision-makers may use their judgment to determine the appropriate level of optimism or pessimism based on the specific circumstances and available information.

1. Strategy:
• Definition: In game theory, a strategy is a plan of action or a decision rule used by a player to choose their moves in a game. It represents a complete set of choices or actions available to a player at each decision point in the game.

• Types: Strategies can be pure (a specific, deterministic choice) or mixed (a randomized choice with probabilities assigned to different actions).

• Significance: Understanding and analyzing strategies is crucial in predicting and influencing the outcomes of strategic interactions.

1. Player:
• Definition: In game theory, a player is an individual or entity participating in a game. Players make decisions and choose strategies to maximize their payoffs or utility.

• Attributes: Each player has a set of possible strategies, preferences over outcomes, and a rational decision-making process.

• Roles: Players interact with each other through their strategies, and the outcomes depend on the combination of strategies chosen by all players.

1. Two-Person Zero-Sum Game:
• Definition: A two-person zero-sum game is a type of game in which there are only two players, and the total payoff remains constant. Any gain by one player is exactly offset by a loss to the other player, resulting in a zero-sum outcome.

• Mathematical Property: The sum of payoffs for both players is always zero, making the game inherently competitive.

• Common Examples: Poker, chess, and various economic scenarios can be modeled as two-person zero-sum games.