Linear Programing Problem LPP Examples

Q1. A Company produces two types of watches one for gents and another for ladies, both are to be processed on three machines, the processing time required and the total time available per week on each machine are as follows:

MACHINE          MODEL                 AVAILABLE TIME
                    GENTS    LADIES

M1                 3                3                    36
M2                 5                2                    50
M3                 2                6                    60

The contribution of profit for each unit of gents watch is Rs.20 and each unit of ladies watch is Rs.30. How should the company schedule the production to the maximum profit formulate the problem as LPP.

Soln:
Maximize z = 20x1+30x2
Subjected to :
3x1+3x2 <= 36
5x1+2x2 <= 50
2x1+6x2 <= 60
and decision variables x1, x2 >=0.
Explanation:
As per the problem given we have two variables
(i)Gents watch--x1 (ii)Ladies watch --x2
They are processed in three machines therefore we will have 3 sets of constraints.
We have to maximize the profit, hence the problem should maximize the profit therefore the objective function should be maximize z.

Q2. A leading leather good company manufactures two types of cricket ball A and B. Each type of ball requires work by both skilled and semi skilled employees , the available time per month and time required for each ball is given below.

TYPES OF EMPLOYES             MANUF. TIME                AVAILABLE TIME
                                               A                    B
SEMI SKILLED                      2                     3                       320
SKILLED                                4                     6                       600

The cost of the hour of semi skilled labor is Rs.10 & skilled labor is  Rs.15. To meet the monthly demands for at least 60 balls of type A and at least 40 balls of type B must be manufactured. Formulate LPP to minimize the cost of production.

Soln:
Minimize z = (2hrs)(10Rs)x1+(4hrs)(15Rs)x1+(3hrs)(10Rs)x2+(6hrs)(15Rs)x2
                 =80x1+120x2
Subjected to
2x1+3x2 <= 320
4x1+6x2 <= 600
x1>=60
x2>=40

and decision variables x1,x2 >=0

Explanation:
As per the problem given we havve two types of variables
(i) Ball A--x1 (ii)Ball B --x2
They are processed by two types of employees Skilled & semi skilled hence two sets of constraints.
We have to minimize the cost of production hence the objective function Minimize z.
Cost of labor and number of balls to be manufactured should be calculated as the objective functions.
Here the number of balls to be manufactured is provided as at least therefore the decision variables are given as >=

Linear Programing Problem(LPP)

Linear Programming Problem(LPP)

An equation with same degree/variable/power is a Ist order equation.

LP model has few basic elements which are described below.

Decision Variable  - The variable whose values determine the solution of a problem.

Objective function - the generalized format of an objective function is given as

maximize or minimize z =c1x1+c2x2 +.........+cnxn

where c1,c2,...cn are cost variables and x1,x2 ...xn are decision variables.

Technical Coefficient : (aij)

aij is the amount of resource i required for the activity j where i varies from 1 to m and j varies from 1 to n.

The generalized format of technical coefficient is 

[ a11 a12 .....a1n

  a21 a22 .....a2n 

  .

  .

  am1 am2 ... amn ]

Resource Availability: (bi)

The constant bi is the amount of resource i available during the planning period.

The general format is given as,

[b1

 b2

 .

 .

 bm]

Set of Constraints:

A constraint is a kind of restriction on the total amount of a particular resource required to carry out activities atvarious levels.

The generalized format is given as:

a11x1+a12x2+...+a1nxn <=,= or  >= b1

a21x1+a22x2+...+a2nxn <=, = or >= b2

.

.

am1x1+am2x2+...+amnxn<= ,= or >=bm

Non negativity constraint:

Each and every decision variable in LP model is a non negative variable.

The general format is

x1,x2..xn >=0

Mathematical formulation of LPP:

The general format is given by

Maximize/Minimize z = c1x1+c2x2+...+cnxn  --- (1 )

subjected to constraints

a11x1+a12x2+...+a1nxn <=,= or >= b1

a21x1+a22x2+...+a2nxn <=,= or >= b2

.                                                                        ---(2)

.

an1xn+anx2+...+amnxn<= , = or >=bm

and x1,x2,...xn >=0                                         ---(3)

variables are involved .

Definition of LPP :

Linear Programming problem deals with optimization of functions of decision variables known as objective functions subject to a set of simultaneous linear eqution known as constraints.

Applications:

LP technique is used in many indusrial and economic problems , airlines , railways, food processing etc.,