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6.2 Methods for Optimum Assignment

Note: if you cannot remember how to represent matching problems in graphs and matrices, revise notes for 6.1 Introduction to Matching Problems.

Optimum Assignment by Inspection

  • In smaller scale matching problems, it may be possible to determine the optimum assignment by inspection.
  • One method is to find the lowest weight assignment for each task, object, etc. being assigned to. If there is any conflict, then consider other tasks, objects, etc.

Example

Teacher

Banjo

Violin

Piano

Dave

10

25

35

Jill

11

15

40

Steve

10

18

46

 

Above is a matrix representing the time in days required for each teacher at a school to teach a beginner’s course in certain instruments. Only one teacher is required for each course and each teacher must be assigned to a course. As this is a smaller-scale problem, we will use inspection to find the optimum assignment.

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2.3 Extended Matrix Recurrence Relations

Note: if you cannot remember how to use simple matrix recurrence relations, revise notes for 2.1 Simple Matrix Recurrence Relations.

The Extended Matrix Recurrence Formula

  • The simple matrix recurrence formula we have analysed previously (in 2.1 Simple Matrix Recurrence Relations and 2.2 Applications of Matrix Recurrence Relations) only allows the values for the next state to be calculated based on proportional values of the current state. In some systems, this is not enough to produce an accurate model. The extended matrix recurrence formula allows us to model systems which have fixed values added or removed between states.
  • The general form for the extended recurrence formula is:

S_{n+1}=T S_{n}+B

Where S_n is the state matrix for the nth state, T is the transition matrix and B is a matrix containing constant values.

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2.2 Applications of Matrix Recurrence Relations

Note: if you cannot remember how to use a matrix recurrence relation, revise notes for 2.1 Simple Matrix Recurrence Relations.

Transition Diagrams

  • Transition diagrams provide a visual method for modelling how a system changes between states.
  • They consist of a series of nodes (dots) representing the variables in the system, with arrows drawn between them showing the movement of individuals/objects in the system between variables (e.g. the percentage of people who go from playing soccer one month to playing baseball the next).
  • By convention numerical values should be shown as percentages in transition diagrams.
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2.1 Simple Matrix Recurrence Relations

The Simple Matrix Recurrence Relation Formula

  • The simplest type of matrix recurrence relation formula we will analyse in Further Maths models a system where the next “state”; S_{n+1}, can be reached by multiplying the current state; S_{n}, by a transition matrix; T, in the form:

S_{n+1}=T S_{n}

  • The state matrices; S_{n} (where n is a positive whole number representing the state of the system), are column matrix listing the value of each of the system’s variables in the corresponding state.
  • The transition matrix; T, is a square matrix.
  • As with a linear recurrence relation, it is important to state the initial state of a system; S_{0}.
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1.8 Matrix Applications: Solving Systems of Equations

Representing Systems of Equations in Matrix Form

  • Matrices provide a concise way of representing systems of linear equations (i.e. multiple related linear equations). The equations are represented in 3 matrices:
    • A square coefficient matrix, generally denoted by a capital A, where each column lists the coefficients of a corresponding variable, while each row corresponds to a different equation.
    • A variable column matrix which lists the variables. It is denoted by a capital X. This is multiplied by the coefficient matrix to form the left-hand side of the matrix equation.
    • A column matrix of constants, which lists the constant in each equation. It is denoted by a lower-case b. This forms the right-hand side of the matrix equation.
    • Equation for a system of linear equations in matrix form:

AX=b

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1.7 Matrix Applications: Dominance Matrices

Dominance Matrices

  • A dominance matrix is a binary matrix which has individuals listed along the rows and columns. A 1 element indicates the individual corresponding to the row is “dominant to” or “beat” the individual corresponding to the column. A 0 element indicates this is not the case.
  • This type of matrix is useful when modelling systems such as round robin competitions, where multiple people play against each other in rounds, with each round producing a winner.
  • This type of matrix is also known as a one-step dominance matrix.
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1.6 Matrix Applications: Communication Matrices

Communication Matrices

  • Communication matrices provide a numerical method for visualising communication networks. Each individual in the network is listed along the rows and columns of the matrix.
  • A “1” element indicates the sender corresponding to the row can communicate directly to the receiver corresponding to the column. A “0” means they cannot.
  • By convention all elements on the main diagonal are “0”s, as it is generally redundant to analyse an individual’s ability to communicate with itself.
  • A basic communication matrix is also referred to as a one-step communication matrix as it shows the communication allowed via a single “step” (i.e. direct communication).
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1.4 Basic Matrix Applications

Using Matrices to Present Information

  • Matrices can be used to present information in a tabular form.
  • Similar to 2-way frequency tables, two categorical datasets are listed as the rows and columns, respectively, and each element represents a numerical value corresponding to the category in its row and column.
  • When presenting data in a matrix, the categories should be listed at the top of each column, and to the left of each row.
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1.3 The Inverse of a Matrix

The Determinant of a 2×2 Matrix

  • The determinant of a 2 x 2 matrix can be calculated as:

\text { det }=A_{1,1} A_{2,2}-A_{1,2} A_{2,1}

  • The determinant of an identity matrix is equal to 1.

Example

A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]

The determinant of the above matrix is:

\operatorname{det}=1 * 4-2 * 3=-2

Requirements for a Matrix to be Invertible

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