Some linear programming problems and examples

  • Identification and formation of a specific problem

There is a way for establishing linear programming problems. The basic challenge is to form the mathematical problem out of a particular situation for which the optimum values are to be found, like knowing the maximum profits or maximization of cost cuts etc. Once, the problem statement is in place we have to form the problem by adopting following steps:

* Identify the quantity that is to be optimized.

* Identify the decision variables attached to this problem. For example while finding the maximum profits the production quantities.

*  May be decision variables and production limits can be the constraints.

* Now write the objective function and constraints, which has to be in terms of decision variables. The information given in the problem statement can be used for this purpose.

* Now this system of equation should be arranged in a suitable form to make the solution easier.

Linear Programming Problems offer complex solutions

  • Binding and non-binding constraints

In every situation concerning linear programming problems, there are certain limitations and restrictions that affect the variable values There can be less severe limitations and such constraints are considered “non-binding” constraints as they may not affect the ultimate solution for finding optimal values. A “binding constraint” is the one that changes optimal solution also, if there is a change in value of such constraint.

The constraints can be mainly termed as “equality and inequality constraints” As well as “duality constraints”. In view of the above, Linear Programming is the most useful technique for getting the optimum results in any Operation Research studies.

  • Some common terms

The following common terms are often used in linear programming problems.

* Solution Space Also known as feasible region, it is the area that satisfies all non-negative restrictions and other constraints of the LPP. Usually this is a convex type set and the optimum vale is found at its vertex.

* Feasible Solution When the solution of a LPP satisfies all constraints including its non-negativity restrictions, it is called a feasible solution.

* Basic Solution  Suppose there are ‘x’ constraints and ‘y’ decision variables in a LPP, and the solution of ‘x’ basic variables setting each of (y-x) non basic variable equals to Zero, then such solution will be known as basic solution. Therefore, a basic solution that satisfies the non-negativity constraint is called a basic feasible solution.

* Degenerate and Non-degenerate Solution…If one or more basic variables are zero, then the solution is “degenerate”. However if all the basic values are non-zero, then the solution is “non-degenerate”.

* Unbounded Solution. In this case, the values of decision variables can increase infinitely but they should not violate the feasibility conditions.

* Optimum Solution…When a basic feasible solution optimizes the objective function, it is said to be the optimum solution.

  • Example

Given below is an example of one of the linear programming problems with constraints:

Find the maximum value of

p = 3x – 2y + 4z    The value should be subject to the following………………..(1)

4x + 3y – z ? 3

x + 2y + z ? 4 …..and

x ? 0, y ? 0, z ? 0

In this example, objective function is the equation (1). The three mathematical statements, given below it, represent the constraints.

The above completes information about linear programming problems. Meanwhile readers should go through topics to gain knowledge on other issues.



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