STEPS TO SOLVE LINEAR PROGRAMMING PROBLEMS

LINEAR PROGRAMMING: 
Linear programming (LP, or linear optimization) can be defined as a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
Generally, there are two type of problems:

1. MINIMIZATION PROBLEMS : ex cost minimization, operation time minimization etc.
2. MAXIMIZATION PROBLEMS: ex profit maximization, sales maximization etc.

LIMITATION:

       A LP problem with only 2 decision variables can be solved using graphical method.

STEP 1: Formulate the data given in the question.

A) List out the decision variables :
                                                  The variables for which the problems are asking solution for are called as decision variables.

B) Define the OBJECTIVE FUNCTION :
                                                             It is a linear function of the decision variables expressing the objective of the decision-maker.
C) List out the constraints :
                                            This are nothing but the limitations that a problem holds. Also define non- negativity constraints.

STEP 2:  Plot the graph according to the equations formed for the constraints.

STEP 3: Shade/ mark the regions based on the given conditions.
             
          case 1: equations with (>=  some value) condition are shaded away from the origin.
        case 2: equations with (<= some value) condition are shaded towards the origin.

STEP 4: Look for common region i.e the feasible region or region covered by all the constraint equations.

STEP 5: Find the corner points coordinates of the common/feasible region.

STEP 6: Put the corner point values in the objective function 

STEP 7: Compare values of objective function calculated using step 6.

STEP 8: 
IN MINIMIZATION PROBLEM: Consider the coordinates of the point which gives minimum objective function value.

IN MAXIMIZATION PROBLEM: Consider the coordinates of the point which gives maximum objective function value.

THE COORDINATE POINTS ARE THE SOLUTION TO THE PROBLEM.

EXAMPLE:
London Tourister is a manufacturer of travelling bags and its distributor has agreed to buy all the bags it produces for next 3 months. The bags have to undergo 4 processes as shown below.

Department	Standard bag (hrs/bag)	Deluxe bag (hrs/bag)
Cutting & dyeing	7/10	1
Sewing	½	5/6
Finishing	1	2/3
Inspection & Packaging	1/10	1/4

LT's manager has analyzed the constraints in production and estimates that 630hrs for cutting & dyeing, 600hrs for sewing, 708hrs for finishing, 135hrs for inspection & packaging will be available for production of traveling bags during the next 3 months. The profit contribution of each std. Bag is Rs.10 and rs.9 for each deluxe bag. Formulate the problem to determine the no. Of std. Bags and no. Of deluxe bags LT has to mfg in order to maximize the profit contribution.

Solution:
STEP 1: Formulate the data given in the question.
A) List out the decision variables :

1. S= No. Of standard bags 
2. D= No. Of deluxe bags

Lines from question: Formulate the problem to determine the no. Of std. Bags and no. Of deluxe bags LT has to mfg.

B) Define the OBJECTIVE FUNCTION : 
 _{10S +9D}

lines from question specifying objective function: In order to maximize the profit contribution.
                                                            

C) list out the constraints :

                                    

    7/10S + D <= 630

    1/2S + 5/6D <= 600

     S + 2/3D <= 708

    1/10S + 1/4D <= 135 ; 

    
    S,D >= 0 (NON NEGATIVITY CONSTRAINTS)

Lines from question : LT's manager has analyzed the constraints in production and estimates that 630hrs for cutting & dyeing, 600hrs for sewing, 708hrs for finishing, 135hrs for inspection & packaging 

AND THE VALUES FROM THE TABLE.


FOLLOW STEP 2 to STEP 8: 

ANSWER = (S=540, D=252)
TOTAL PROFIT = 7668 

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