Applied mathematics is that branch of mathematics which deals with the application of mathematics in various other fields, like science, economics, finance, management problems, assigning problems, transportation problems, along with the probabilistic and statistical analysis for taking management decisions.

Applied mathematics as the name says, is the application of mathematics from the theoretical part to the real world practical part and its real life applications in the field of decision making.
Applied mathematics can be termed as that part of the mathematics, which is applied in reality in various departments like finance and budgeting:  investments, the process of purchasing, procurement and exploration.
Applied mathematics uses various tools and techniques of mathematics which is helpful in the field of research and development, in the determination of time cost trade off and control of development projects, reliability and alternative design.

A brief account of some of the important techniques or models used in applied mathematics is mentioned below:
1). Distribution (Allocation) Models
2). Production or Inventory Models
3). Waiting Line or Queuing Models
4). Markovian Models
5). Competitive Strategy Models (Games Theory)
6). Network Models
7). Job Sequencing Models
8). Replacement Models
9). Simulation Models
Out of the above mentioned models, Simulation is a very useful method to find solutions of various complicated problems, which are very difficult to solve otherwise, thus simulation is used generally for solving various problems, where we know that there are many number of constraints and variables, because its not possible to find the solution by manual calculations.

 Applied mathematics, gives stress on the analysis of mathematics on its application to real world problems as a whole. For this purpose, applied mathematics uses any suitable techniques or tools available from the fields of mathematics, statistics, cost analysis or numerical calculations.
Some of the techniques which are used in the field of applied mathematics are listed below:
1). Linear programming
2). Non linear programming
3). Integer programming
4). Dynamic programming
5). Goal programming
6). Games Theory
7). Inventory Control
8). PERT and CPM techniques
9). Simulation
10). Queuing Theory and many more to mention.

There are various methods used in the field of applied mathematics, some of which are mentioned below:
Method Number 1:
Using Linear Programming Approach:

Linear implies expression of the form a1x1 + a2b2 + …+ anxn, where a1, a2,  …, an are constants and x1, x2, ....., x are variables.
Programming implies the process of finding out a particular plan of action.
Hence, a linear programming problem optimizes (either maximize or minimize) any linear function of variables that is called as objective function and this objective function has to be optimized, subject to some pair of linear equations and or inequalities, that is called as constraints or restrictions.
This method is used for finding various assignment problems, transportation problems, the optimum estimation of the executive compensation in an industrial concern, allocation of some limited resources like labour, water, supply, working capital etc, so that to maximize the net revenue.
For determining the product mix, product smoothing and assembly time balancing in the field of production management and many more.

Method Number 2:
Using Integer Programming Approach:

A general integer programming problem is as follows:
maximize cT x
subject to $Ax \leq b$
$x\geq 0$
and $x\epsilon Z$
Note that here x belongs to the set of integers and therefore, in this method we have to find the optimal solution taking into consideration that all the solutions must satisfy the integer constraints.
There can be two ways to solve Integer programming problem which are either to use Gomory’s Cutting Plane Method or to use Branch and Bound Method.
 
Method Number 3:
Using Goal Programming Approach

When management has a multiple goals, they will most likely have some priority scale for the goals. Goal programming provides for the preferential ordering of goals through the use of priority coefficient (P’s). All goals (deviational variables) that have a top or first priority are assigned an objective function value of P1, the goals considered to be second in priority are assigned a P2 value and this process is continued until all goals have been ranked.
The coefficient P1, P2, etc are not parameters or variables. They do not, in general, assume a numerical value; they simply represent levels for the priorities.

Method Number 4:
Analytic Method

In the study of applied mathematics, if any problem is solved by using all the techniques of theoretical mathematics like calculus (integral and differential) and probability and statistics for solving the problem, then this kind of answers obtained are known as analytic solutions.

Method Number 5:
 Iterative Method

If the analytic method does not work for any problem solving because of the complicated complex constraints or variables, then we are usually forced to adopt an iterative method, where the method begins with an initial trial solution followed by some sets of rule for improving it.
This solution is then exchanged by a solution got after the improved iteration and this process is repeated until we get all the correct improved solutions.

Method Number 6:
The Monte – Carlo Method

The basis of this method is random sampling of variable’s values from a distribution of that variable.
Monte Carlo refers to the use of sampling methods to estimate the value of non stochastic variables.
Some solved examples of applied mathematics are mentioned below:
Example 1:
Determine if the following transition matrix is ergodic Markov chain.

Applied Mathematics Examples

                            
Solution:
Here we must check that it is possible to go from every present state to all other states. We observe that from state 1, it is possible to go directly to every other state except state 3. For state 3, it is possible to go from state 1 to state 2 to state 3. Therefore, it is possible to go from state 1 to any other state.
Similarly, from state 2, it is possible to go to state 3 or state 4, then from state 3 to state 1, or from state 4 to state 3 to state 1.
Also from state 3 it is possible to go directly to state 1. Finally from state 4, it is possible to go to state 3, then from state 3 to state 1.
Hence, the above transition matrix is an ergodic Markov chain.

Example 2: Find the range of the values of p and q which will render the entry (2, 2) a saddle point for the game:

Applied Mathematics Problem

Solution:
First ignoring the values of p and q we will determine the maximin and the minimax values of the payoff matrix as below:
Since the entry (2, 2) is a saddle point, thus maximin value v = 7, minimax value v- = 7. This imposes the condition on p as p $\leq$ 7 and on q as q $\geq$ 7. Hence, the range of p and q will be p $\leq$ 7 and q $\geq$ 7.

Example 3: Solve the following minimization problem:
Minimize p = 3x + y
subject to
2x - y $\leq$ 4
2x + 3y  $\leq$ 12
y  $\leq$ 3
x $\geq$ 0
y  $\geq$ 0
Solution: After solving the lines, taking the vertex and the lines through the vertex, along with the value of their objective function, we will get the following table:

 Vertex  Lines Through Vertex   Value of Objective
 (3, 2)  2x - y = 4; 2x + 3y = 12  11
 (2, 0)  2x - y = 4; y = 0  6
 (1.5, 3)  2x + 3y = 12; y = 3  7.5
 (0, 3)  y = 3; x = 0  3 Minimum
 (0, 0)   x = 0; y = 0  0 Minimum

Thus, we got the minimum value of the function at x = 0 and y = 3, hence, the minimum value is 3). However, if we can take the value (0, 0), then the minimum value of any objective function will always be zero.
The graph of the lines drawn can be shown as below:
Graph of Lines
Here, note that the region shown in the white part is the feasible region only, not the red shaded part.