### MATHEMATICAL TECHNIQUES – FUNCTIONS?

- Functions, equations and graphs: Linear, quadratic, cubic, exponential and logarithmic - Application of mathematical functions in solving business problems

### MATHEMATICAL TECHNIQUES – MATRIX ALGEBRA?

- Types and operations (addition, subtraction, multiplication, transposition and inversion) - Application of matrices: statistical modelling, Markov analysis, inputoutput analysis and general applications

### CALCULUS – DIFFERENTIATION?

- Rules of differentiation (general rule, chain, product, quotient) - Differentiation of exponential and logarithmic functions - Higher order derivatives: turning points (maxima and minima) - Ordinary derivatives and their applications - Partial derivatives and their applications - Constrained optimisation; lagrangian multiplier

### CALCULUS INTEGRATION?

- Rules of integration - Applications of integration to business problems

### PROBABILITY-SET THEORY?

- Types of sets - Set description: enumeration and descriptive properties of sets - Operations of sets: union, intersection, complement and difference - Venn diagrams

### PROBABILITY THEORY?

- Definitions: event, outcome, experiment, sample space - Types of events: elementary, compound, dependent, independent, mutually exclusive, exhaustive, mutually inclusive - Laws of probability: additive and multiplicative rules - Baye’s Theorem - Probability trees - Expected value, variance, standard deviation and coefficient of variation using frequency and probability

### PROBABILITY DISTRIBUTION?

- Discrete and continuous probability distributions (uniform, normal, binomial, poisson and exponential) - Application of probability to business problems

### HYPOTHESIS TESTING AND ESTIMATION?

- Hypothesis tests on the mean (when population standard deviation is unknown) - Hypothesis tests on proportions - Hypothesis tests on the difference between means (independent samples) - Hypothesis tests on the difference between means (matched pairs) - Hypothesis tests on the difference between two proportions

### CORRELATION AND REGRESSION ANALYSIS?

- Scatter diagrams - Measures of correlation –product moment and rank correlation coefficients (Pearson and Spearman) - Regression analysis - Simple and multiple linear regression analysis - Assumptions of linear regression analysis - Coefficient of determination, standard error of the estimate, standard error of the slope, t and F statistics - Computer output of linear regression - T-ratios and confidence interval of the coefficients - Analysis of Variances (ANOVA)

### TIME SERIES?

- Definition of time series - Components of time series (circular, seasonal, cyclical, irregular/ random, trend) - Application of time series - Methods of fitting trend: free hand, semi-averages, moving averages, least squares methods - Models - additive and multiplicative models - Measurement of seasonal variation using additive and multiplicative models - Forecasting time series value using moving averages, ordinary least squares method and exponential smoothing - Comparison and application of forecasts for different techniques - Trend projection methods: linear, quadratic and exponential

### LINEAR PROGRAMMING?

- Definition of decision variables, objective function and constraints - Assumptions of linear programming - Solving linear programming using graphical method - Solving linear programming using simplex method - Sensitivity analysis and economic meaning of shadow prices in business situations - Interpretation of computer assisted solutions - Transportation and assignment problems

### DECISION THEORY?

- Decision making process - Decision making environment: deterministic situation (certainty) - Decision making under risk - expected monetary value, expected opportunity loss, risk using coefficient of variation, expected value of perfect information - Decision trees - sequential decision, expected value of sample information - Decision making under uncertainty - maximin, maximax, minimax regret, Hurwicz decision rule, Laplace decision rule

### GAME THEORY?

- Assumptions of game theory - Zero sum games - Pure strategy games (saddle point) - Mixed strategy games (joint probability approach) - Dominance, graphical reduction of a game - Value of the game - Non zero sum games - Limitations of game theory

### NETWORK PLANNING AND ANALYSIS?

- Basic concepts – network, activity, event - Activity sequencing and network diagram - Critical path analysis (CPA) - Float and its importance - Crashing of activity/project completion time - Project evaluation and review technique (PERT) - Resource scheduling (leveling) and Gantt charts - Advantages and limitations of CPA and PERT

### QUEUING THEORY?

- Components/elements of a queue: arrival rate, service rate, departure, customer behaviour, service discipline, finite and infinite queues, traffic intensity - Elementary single server queuing systems - Finite capacity queuing systems - Multiple server queues

### SIMULATION?

- Types of simulation - Variables in a simulation model - Construction of a simulation model - Monte Carlo simulation - Random numbers selection - Simple queuing simulation: single server, single channel “first come first served” (FCFS) model - Application of simulation models

### EMERGING ISSUES AND TRENDS

Functions, equations and graphs: Linear, quadratic, cubic, exponential and logarithmic

FUNCTIONS

Definitions

1. Variables

A variable is any quantity that assumes different values in a particular analysis. Examples are:

• Production costs
• Material costs
• Sales revenue
1. Constant

This is any quantity whose value remains unchanged in a particular analysis Examples ar

• Fixed costs
• Rents
• Tuition fees:

Note: In a given analysis there are two types of variables namely:

• Independent variable/predictor variable – is that which influences the value of the other variables in a particular analysis.
• Dependent / response variable – is that whose value is influenced or changes when the value of other variables (independent) changes.

3. Functions

A function is a mathematical expression which describes a relationship between two or more variables in a particular analysis specifically one dependant variable and one or more independent variables.

Examples

If the price of the consumer product is Sh 40 per Kg, then the total sales revenue, S when Q units of the products are produced and sold is obtained as follows:

S = 40q

In this case S is the dependent variable, q the independent variable and 40 is a constant. In terms of number of variables in a function, functions can be classified into the following categories:

1.  Univariate function
2. Bivariate function
3. Multivariate function

A univariate function is that which involves two variables only, one dependent variable and one independent and is generally written as:

y = f (x) where y = dependent variable

x = independent variable

and f(x) = Function of x

Example of univariate function

The price of a house is dependent among other factors, on the size of the house. In functional form, this could be written as follows:

Price = f (size)

Where price is dependent variable Size is independent variable

A Bivariate function is that which involves three variables only, one dependent variable and two independent variables:

Example

A student’s performance or grade in an examination could be dependent upon the following factors

1. IQ
2. Time spent on studying in terms of Hours, H

In functional form, this is written as follows:

IQ, H Are independent variables

Multivariable function is that function which involves four or more variables, one dependent variable and three or more independent variables.

Example

The price of a house depends on the following factors:

1. Size
2. Location
3. Security
4. Nature of the house

In functional form this is written as follows; Price = f (size, location, security, nature of the house)

Where price – is dependent variable

Size, location, security, nature of the house are independent variables.

Graph of a function

A graph is a visual method of illustrating the behaviour of a particular function. It is easy to see from a graph how as x changes, the value of f(x) is changing.

The graph is thus much easier to understand and interpret than a table of values. For example by looking at a graph we can tell whether f(x) is increasing or decreasing as x increases or decreases. We can also tell whether the rate of change is slow or fast. Maximum and minimum values of the function can be seen at a glance. For particular values of x, it is easy to read the values of f(x) and vice versa i.e. graphs can be used for estimation purposes

Different functions create different shaped graphs and it is useful knowing the shapes of some of the most commonly encountered functions. Various types of equations such as linear, quadratic, trigonometric, exponential equations can be solved using graphical methods.

These include

1. Linear functions
3. Cubic functions
4. Exponential functions
5. Logarithmic functions
6. Hybrid functions
7. Linear functions

A linear function is a first degree polynomial function that takes the following general form.

y= a +bx

Where y is dependent variable

x is independent variable

a is y-intercept or the value of y when x = 0

b is the slope or gradient or the amount by which y changes in value when x changes by a unit

Properties/characteristics of linear functions

When plotted on an x-y coordinate system, the result is a straight line whose general direction is dependent on the slope, b of the function.

Specifically, if

1.
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