## Formulation of Linear Programming Problem(LPP)

The construction of objective function as well as the constraints is known as formulation of **Linear Programming Problem(LPP) **.

The following are the basic steps in formulation of LPP.

- Identify the variables to be determined and then express these by some algebraic symbols.
- Locate the various constraints/restrictions present in the problem ans express these linear equations/inequalities which are some linear functions of the variables identified in step (i).
- Determine the objective of the problem and express it as linear function of the decision variable involved in the phenomenon.

**Example 1**

The Omni Furniture Manufacturing Company produces two products: Table and Chair. Both products require time in two processing departments. Assembly Department and Finishing Department. Data on the two products are as follows:

Processing |
Desk |
Table |
Available Hours |

Assembly | 2 | 4 | 100 |

Finishing | 3 | 2 | 90 |

Profit per unit | $20 | $35 |

The company wants to find the most profitable mix of these two products. Define the decision variables as follows :

__Example 2__

__Example 2__

A resourceful home decorator manufactures two types of lamps A and B. Both the lamps go through two technicians first a cutter and then a finisher. Lamp A requires 2 hours of cutter time and one hour of finisher’s time. Lamp B requires one hour of cutter time and two hours of finisher’s time. The cutter has 104 hours and finisher 76 hours available time. Profit on one lamp of A is Rs. 6.00 and one lamp of B is Rs. 11.00. Assuming that the decorator can sell all that he produces, how many of each type of lamps should he manufacture to obtain the best return?

**Solution **

It is typical LPP. Here the profit will depend or the output or Lamps A and B. SO we are to find the optimum combination of the outputs of A and B .Let the manufacturer produce X units of A and Y units of B. Then profit from X units of A will be 6X and profit from Y units of B will be 11Y and the total profit from X units of A and Y units of B will be

Z = 6X + 11Y

Now we want to find that combination of X, Y for which Z is maximum. Hence our objective function is

Maximize Z = 6X + 11Y

Again X units of A will require 2X hours of cutter time and Y hours of finisher time. Similarly Y units of B will require Y hours of cutter time and 2Y hours of finisher time.

Thus cutter time required for producing X units of A and Y units of

B = 2X + Y

And the corresponding finisher time will be = X + 2 Y

But the total time available for cutter and finisher is respectively 104 hours and 76 hours. Hence we get the constraints as

2X + Y ≤ 104 (Cutter time)

X + 2Y ≤ 76 ( Finisher Time )

And X, Y ≥ 0 is the non- negativity restriction. Thus the mathematical model for the formulated LPP can be written as

Max. Z = 6X + 11Y (Objective Function)

2X + Y ≤ 104

X + 2Y ≤ 76 (Constraints)

X, Y ≥ 0 (non-negativity restriction)