Decision Maths

Want to get better grades?

Nope I Sure Do

Nope, I’m not ready yet

Get free, full access to:

Flashcards
Notes
Explanations
Study Planner
Textbook solutions

Decision Maths

Calculus

Decision Maths

Geometry

Mechanics Maths

Probability and Statistics

Pure Maths

Statistics

TABLE OF CONTENTS :

TABLE OF CONTENTS

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmelden

Nie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmelden

Have you ever wondered how the post office decides what order to deliver mail in? Or how a delivery service can bring everyone on the list a meal without having to backtrack a lot? Have you ever used a flow chart? Then you have already delved into decision maths!

Applications of decision mathematics

Before looking at the official definition of decision mathematics, let's look at some of the topics included in it.

Computer algorithm design: Even the basic bubble sort is an example of decision mathematics in computer science.
Graph theory: You have probably already heard of the classic travelling salesperson problem which involves finding the shortest path between all of the nodes on a graph.
Linear Programming: Here you will encounter things like the Simplex algorithm, which is used to find a basic solution that satisfies a list of restraints.
Critical Path Analysis: Anyone who has worked in customer service has encountered critical path analysis, which is used to figure out the quickest way to complete a project given a list of tasks, the relationship between the tasks, and how long each task takes.
Game Theory: How should you bet on games at a casino to win the most money anyway?

There are plenty more as well!

Definition of decision maths

So what exactly is decision maths anyway?

Decision maths is a branch of applied probability theory that identifies values and uncertainties related to decisions.

The goal is to make the best decision where each factor in the problem is given a probability of occurring. This can be as simple as deciding not to take an umbrella on your walk because the weather report says there is a low probability of rain today. By not taking an umbrella, you have just made a decision based on probability.

Types of Decision Mathematics

Decision maths is generally broken down into four types.

Making route choices: Package delivery services are especially interested in this type.
Influencing outcomes: You do this when you cook a meal and need to decide when to start various dishes based on the goal of having them all finish cooking at the same time.
Placing bets: Every professional gambler looks at the odds and probabilities before they place bets. Otherwise, they don't stay professional very long. This isn't just for casino gamblers, stock brokers use this too.
Making strategic decisions: Every coach makes strategic decisions, from which players to field to how often to practice.

Importance and uses of decision mathematics

Decision mathematics is also called operations research. The idea is to use apply analytical methods to help solve problems and make better decisions. This can lead to managers making better decisions on how the business runs so it is more energy efficient.

Critical Path Analysis: Suppose that a new road is being constructed. People need to be hired, supplies brought in, permits gotten, etc. If you know how long each task should take, and how they are related, it can help you figure out when each task should start, leading to less congestion on alternate paths while the new road is built.
Bin Packing: Every shipping company deals with bin packing, whether it is container ships, airlines squashing your luggage into a cargo hold, or you moving to a new home. Being able to pack efficiently given the constraints of things like shape, weight, and size, can lead to enormous fuel savings and a cleaner environment.
Matching Problem: Have you taken a chemistry class? Then you know about covalent bonds. Matching problems are used for things like modelling the bonds in chemistry.
Route Inspection Problem: Over the river and through the woods, to Grandma's house you go! You have just inspected and chosen a route. If you use GPS apps to help you find directions, then you have worked with route inspection problems.

Decision mathematics examples

You have already seen several examples of the ways decision maths is used, but let's look at another.

Dynamic programming might sound like a way to have a really exciting event, but it is really a way to optimize things like recursion. Recursive solutions to problems involve repeated calls in a program using the same inputs and are used in things like calculating Fibonacci numbers. Using dynamic programming to decide when and how to save the results of these recursive calls can drastically reduce the time required for a program to run.

This isn't just a programming question though, this is a maths one as well. Have you ever needed to multiply two matrices together? How about three? What if you had to multiply together 150 matrices that were each 3000 by 3000 square? You would certainly want the fastest way to do that! Matrix chain multiplication is another example of the kinds of problems that dynamic programming can help you solve.

Decision Maths - Key takeaways

Decision maths is a branch of applied probability theory.
Applications of decision maths include computer algorithm design, route analysis, bin packing, and critical path analysis.
Decision maths can be broken down into 4 main types:
- making route choices;
- influencing outcomes;
- making bets; and
- making strategic decisions.

Frequently Asked Questions about Decision Maths

Dijkstra's algorithm calculates the shortest path between nodes in a graph.

The are graphs where the parts of the tree represent decisions, rules, and outcomes.

Decision maths, also known as operations research, is the study of using analytical models to make better decisions.

Decision maths is used to determine the decidability of a problem. For example, what is the most efficient route to deliver packages, and how to find it quickly.

No. Discrete math applies to a wide variety of topics, including difference equations. Decision math, also known as operations research, is the study of using analytical models to make better decisions.

Decision maths is the study of using analytical models to make better decisions.

Final Decision Maths Quiz

Decision Maths Quiz - Teste dein Wissen

Question

In a two-player, zero-sum game, if player one's pay-off is 4, what is player two's pay-off?

Show answer

Answer

-4.

Show question

Question

In a two-player, zero-sum game, if player two's pay-off is 2, what is player ones's pay-off?

Show answer

Answer

-2.

Show question

Question

What is the value of a two-player, zero-sum game?

Show answer

Answer

The expected value of player one's pay-off, if both players act rationally.

Show question

Question

Given a pay-off matrix for player one, how would you find the corresponding pay-off matrix for player two?

Show answer

Answer

Flip the axis, and multiply all of the numbers in the matrix by -1.

Show question

Question

What is a stable solution of a two-player zero-sum game?

Show answer

Answer

An outcome in which neither player's pay-off would be increased by changing their strategy, whilst the other player's strategy stays the same.

Show question

Question

Do all zero-sum games have a stable solution?

Show answer

Answer

No.

Show question

Question

What do players do if there is no stable solution?

Show answer

Answer

Use a mixed strategy, meaning they play different strategies at random with a certain probability for each choice.

Show question

Question

Can a single game in a rugby tournament be modelled as a two-player, zero-sum game?

Show answer

Answer

Yes.

Show question

Question

Can the Prisoner's Dilemma be modelled as a zero-sum game?

Show answer

Answer

No.

Show question

Question

What is a play-safe strategy?

Show answer

Answer

The strategy with the highest minimum possible pay-off.

Show question

Question

In a two-player, zero-sum game, what is player one's play-safe strategy?

Show answer

Answer

The max(row minimum's), called the maximin.

Show question

Question

In a two-player, zero-sum game, what is player two's play-safe strategy?

Show answer

Answer

The min(row maximums), called the minimax.

Show question

Question

When does a stable solution exist?

Show answer

Answer

When the minimax and maximin values are equal.

Show question

Question

What is the value of a game with a stable solution?

Show answer

Answer

The same as the minimax and maximin values.

Show question

Question

If no stable solution exists, what do we know about the value of the game?

Show answer

Answer

It is bounded between the minimax and maximin values.

Show question

Question

What are algorithms?

Show answer

Answer

An algorithm shows the order in which a process should be followed for an event to occur or for a problem to be solved.

Show question

Question

List the properties of algorithms

Show answer

Answer

Input
Output
Finitness
Definiteness
Effectiveness

Show question

Question

How or where can algorithms be applied?

Show answer

Answer

Algorithms can be used for solving mathematical problems.
They can be used for solving scientific problems.
They can be applied in our everyday lives like using a recipe for a meal.
Computer programmers use algorithms to state out the instructions to follow before writing their codes.

Show question

Question

An algorithm can have more than one input.

TRUE OR FALSE

Show answer

Answer

True

Show question

Question

An algorithm can have more than one output.

TRUE OR FALSE

Show answer

Answer

False

Show question

Question

An algorithm can have zero input.

TRUE OR FALSE

Show answer

Answer

True

Show question

Question

An algorithm should be made to go on and on without stopping.

TRUE OR FALSE

Show answer

Answer

False

Show question

Question

An instruction in an algorithm should have a definite meaning.

TRUE OR FALSE

Show answer

Answer

True

Show question

Question

If an algorithm does not perform the task it is written for, we say the algorithm is...

Show answer

Answer

Ineffective.

Show question

Question

What does it mean if strategy P dominates strategy Q?

Show answer

Answer

Strategy P dominates strategy Q if for every possible strategy the other player would play, playing strategy P results in a higher pay-off than playing strategy Q.

Show question

Question

What can you do with strategy Q if it is dominated by another strategy?

Show answer

Answer

Remove strategy Q from the pay-off matrix.

Show question

Question

If player one has two strategies, A and B, and A dominates B, which of the following statements are true?

Show answer

Answer

All values in row A are greater than or equal to the corresponding values in row B.

Show question

Question

If player two has two strategies, X and Y, and X dominates Y, which of the following statements are true?

Show answer

Answer

The values in the pay-off matrix in column X are all less than or equal to the corresponding values in column Y.

Show question

Question

What is reducing a pay-off matrix?

Show answer

Answer

The action of removing all dominated strategies from the pay-off matrix.

Show question

Question

When can you find a mixed strategy graphically?

Show answer

Answer

When one of the players has only two strategies to use.

Show question

Question

What is a mixed strategy?

Show answer

Answer

A strategy that uses different strategies at random, with a set probability for each strategy.

Show question

Question

At what point is the optimal allocation for the probability p, of choosing strategy A?

Show answer

Answer

When the minimum possible pay-off is maximised.

Show question

Question

Which players perspective must you formulate the Linear Programming problem from?

Show answer

Answer

Player two.

Show question

Question

What is $x_i$ defined as?

Show answer

Answer

\[x_i = \frac{p_i}{V'}\] where $p_i$ is the probability that player one will choose strategy $i$, and $V'$ is the value of the altered pay-off matrix.

Show question

Question

What is $y_i$ defined as?

Show answer

Answer

\[y_i = \frac{q_i}{V'}\] where $q_i$ is the probability that player two will choose strategy $i$, and $V'$ is the value of the altered pay-off matrix.

Show question

Question

How can you tell that a tableau does not need to be pivoted anymore?

Show answer

Answer

The top row is all non-negative.

Show question

Question

If no stable solution exists, what sort of strategy do the players use?

Show answer

Answer

A mixed strategy.

Show question

Question

Which of the following statements is true for the following pay-off matrix?

Show answer

Answer

Strategy X dominates strategy Y.

Show question

Question

If each player has 2 strategies and you have found the optimal mixed strategy for player one, how can you find the optimal mixed strategy for player two?

Show answer

Answer

Rewrite the pay-off matrix from player two's perspective and solve in the usual way.

Show question

Question

If player one has 2 strategies and player two has 3 strategies, and you have found player one's optimal mixed strategy, how can you find the optimal mixed strategy for player two?

Show answer

Answer

Create simultaneous equations using the probabilities that player one chooses each strategy, and solve for p, q and (1-p-q).

Show question

Question

How can you solve a game where both players have more than 2 strategies (assuming the pay-off matrix is already fully reduced)?

Show answer

Answer

Using Linear Programming.

Show question

Question

What is a linear programming problem?

Show answer

Answer

A linear programming problem deals with optimising (maximising or minimising) a function.

Show question

Question

What are the three constituents of a linear programming problem?

Show answer

Answer

Objective function, decision variable and constraints.

Show question

Question

Is happiness a valid quantity for a linear programming problem?

Show answer

Answer

No. Only quantifiable quantities elements can be included in a linear programming problem.

Show question

Question

Can the expression $x^2$ be an objective function in a linear programming problem?

Show answer

Answer

No. Objective function and constraints in a linear programming problem must be linear equations or inequalities.

Show question

Question

A firm manufactures two types of products, A and B and sells them at a profit of $2 on type A and $3 on type B. What would be the objective of a linear programming problem which has to maximise the profit?

Show answer

Answer

Maximise $2x_1+3x_2$

Here, $x_1$ denotes the units of product A and $x_2$ the units of product B.

Show question

Question

A firm manufactures two types of products, A and B and sells them at a profit of $2 on type A and $3 on type B. What are the decision variables of a linear programming problem which has to maximise the profit?

Show answer

Answer

The firm has to decide how many units of products A and B are to be manufactured to maximise its profit. So, they are the decision variables.

Show question

Question

A person wants to decide the constituents of a diet which will fulfill his daily requirements of proteins, fat, and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. What are the decision variables for this optimisation problem?

Show answer

Answer

The units of food of type 1, 2, 3, and 4 to be consumed to maintain the desired diet are the decision variables.

Show question

Question

What is the function to be optimised in a linear programming problem called?

Show answer

Answer

Objective function

Show question

Question

David wants to optimise his travel route from his house to work. There is only one roadway connecting his house and workplace. Can this be formulated as a linear programming problem?

Show answer

Answer

No. There must be alternative courses of the line of action to choose from.

Show question

More about Decision Maths

Select your language

Decision Maths

Want to get better grades?

Get free, full access to:

Decision Maths

Applications of decision mathematics

Definition of decision maths

Types of Decision Mathematics

Importance and uses of decision mathematics

Decision mathematics examples

Decision Maths - Key takeaways

Frequently Asked Questions about Decision Maths

What does Dijkstra's algorithm calculate?

What are the decision tree attributes?

What is decision making in mathematics?

What are the characteristics of decision mathematics?

Is discrete maths the same as decision?

How does maths help in decision making?

Final Decision Maths Quiz

Decision Maths Quiz - Teste dein Wissen

More explanations about Decision Maths

Discover the right content for your subjects

Biology

Business Studies

Chemistry

Combined Science

Economics

English

English Literature

Environmental Science

Geography

History

Human Geography

Macroeconomics

Marketing

Microeconomics

Physics

Politics

Psychology

Sociology

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Quizzes

Flashcards

Notes

Study Sets

Documents

Study Analytics

Weekly Goals

Smart Reminders

Rewards

Magic Marker

Smart Formatting

Join millions of people in learning anywhere, anytime - every day

This is still free to read, it's not a paywall.

You need to register to keep reading

This is still free to read, it's not a paywall.

You need to register to keep reading

Get FREE ACCESS to all of our study material, tailor-made!

StudySmarter bietet alles, was du für deinen Lernerfolg brauchst - in einer App!