02 lp_graphical method simplex algorithm

7/28/2019 02 LP_Graphical Method Simplex Algorithm

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TTN DoMS, IIT Madras, 17-Apr-13

T.T. NarendranDepartment of Management Studies

Indian Institute of Technology Madras


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Example Problem No. 8 :

A furniture shop manufactures tables and

chairs. The operations take place

sequentially in two work centers.

The associated profits and the man hours

required by each product at each work centerare shown in the table below :

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Formulate a linear programme to determine theoptimal number of tables and chairs to be

manufactured so as to maximize the profit.

Furniture Shop Tables Chairs

Available

man

hours

Profit / unit 8 6

Work Center-I 4 2 60

Work Center-II 2 4 48

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Solution to Example Problem No.8

The following linear programme is formulated

This problem can be solved graphically.

The following figure shows the constraints ;

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The shaded region is the feasible solution set to the problem

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A(0,12)

X1

C(15,0)

B(12,6)

X2

O(0,0)


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Before we indicate the procedure to

solve this problem, a few concepts are

introduced.


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Convex Set:

If there exists a set in which a straight line

joining any two points in the set is also

contained in the set, then such a set is

called a convex set


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The extreme of a convex set will always lie

at a corner point. This property is used for

determining the solution to the given linear

programme.

The expression


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A series of parallel lines for various assumed

values of Z can be drawn. These are called

Isoprofit lines.

The profit Zalong each line is the same.

The value of Z is seen to increase as the

isoprofit lines move farther away from the origin.

The corner point through which the last isoprofit

line passes, gives the optimal solution.

In the figure OABC, the corner points are

evaluated as follows:


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Iso Profit Line

A(0,12)

X1

C(15,0)

B(12,6)

X2

O(0,0)


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Z0 = 0

ZA = 72

ZB = 132

ZC = 120

For the given problem, the optimal solution

occurs at corner point B (12,6)

i.e.,X1= 12,X2 = 6 with the corresponding profit

of 132 which is the maximum.


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Note: The graphical method, obviously, can

solve problems with just two variables.

The need, however, is for a method that can

solve problems of realistic size.

For this purpose, there exists a method called

the Simplex Algorithm developed by George.B.

Dantzig.


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A few basic concepts must be learnt

before the algorithm is introduced.Basic Solution:

Consider the following set of equations :


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There are four variables and two equations. Hencethis cannot be solved as simultaneous equations.

However, it is possible to find a set of solutions

called basic solutions which are obtained as follows

In the given system of equations, if we set the 'extra'

variables = 0, we can solve for the remaining

variables.

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For instance, if we set

We get

This is called a basic solution.

We can see that there are as many as

solutions.Of these, the solutions that also satisfy the

non-negativity condition are called basic

feasible solutions

.

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In general, if we have a system of m

equations with n variables (where n > m),

we can obtain basic solutions by setting

(n - m) variables = 0 and solving for the

remaining variables.

There will be basic solutions

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Canonical form :

If there exists a system of equations such that

each equation has one variable with coefficient

of 1 in that equation and a coefficient of 0 in all

the other equations, such a system is said to be

in Canonical form.

The given system is seen to be in Canonicalform since satisfy this condition.

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Other Definitions :

The variables that are set = 0 are callednon basic variables while the remaining

variables are called basic variables.

Now let us consider the same example:

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Remove the inequalities and rewrite the constraints

as equations by introducing 'slack variables' are

shown below

This is a system of two equations with four variables.

Choose the variables in canonical form, i.e., asthe basic variables for the initial solution.

Rewrite the equations, expressing the basic variables interms of non-basic variables.

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Both the non basic variables have positive

coefficients in the objective function and

hence have the potential to increase the

value of Z .

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Let us now consider makingX1 a basic variable.

To do this, one of the existing basic variables must

become non basic since there can be only two basic

variables at any stage.

To find out which variable is to be replaced, we find

the maximum possible value forX1 in equations (1)

and (2).

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Of these, only the lower value will satisfy both

the constraints. Hence

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When we examine for the maximum possible

value ofX2 in (3) and (4), we see thatX3

displacesX4 as basic variable.

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Here, the coefficients of both the non-

basic variables in the objective functionare negative.

Hence, there is no further scope forincreasing the value of Z. So the final

solution is

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T.T. Narendran

Department of Management StudiesIndian Institute of Technology Madras

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Simplex Algorithm : Tabular form

The algorithm explained above can be

implemented in a tabular form for greater

working convenience.

This form is also easier for coding and

automation purposes. The key features of the

tabular form are as follows:

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Each variable is represented as column.

The corresponding coefficients appear in

the boxes of the table.

CB = Coefficient of Basic Variable in the

objective function

It can be seen that the tabular form is just

another way of representing the same set of

equations used above

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The solution at this stage reads as

X3 = 60, X4 = 48, Z = 0

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In the row Cj Zj , the positive coefficients for

the non basic variables X1

and X2

indicate that

the value of Z will increase if one of these

variables becomes a basic variable.

We choose X1to become the basic variable.This requires one of the existing basic

variables to become non basic.

In other words, X1is the entering variable. We

have to find the departing variable.

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Now, evaluate the ratio bi / aij for all aij >0 in

this column.

The ratios are

60 / 4 = 15 and 48 / 2 = 24

The minimum ratio determines the departing

variable.

Hence, in this case X3 is the departing variable.

The coefficient 4 corresponding to this

operation is called thepivot.

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TTN

DoMS, IIT Madras, 17-Apr-13

The next table is obtained by performing

the following operations.In the 'basis' column X1 replaces X3.

Divide the pivotal row

(4 2 1 0 | 60) by the pivot (4).

We get (1 0 | 15)

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To get the X4 row, do the following :

Multiply the new row just obtained by the

coefficient in the pivotal column (2). Weget (2 1 0 | 30)

Subtract corresponding elements from the

old X4 row. That is

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This constitutes the new X4 row in the

table. The Cj Zj row is evaluated as

follows :

Zj = CB aij

For example,Z2 = 8 () + 0 (3) = 4 and

C2-Z2 = 6-4 = 2

In this manner, all the values ofCj Zj are

computed

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At the end of this iteration, the solution

reads as

X1 = 15, X4 = 18, Z = 120,

Since the X2 column in the Cj Zjrow shows

a positive value, we now bring in X2 as the

entering variable.

Proceeding as before, we find18 / 6 = 3 is less than 15 / () = 30

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Therefore, 3 is the pivot and X4 is the

departing variable.

The next table is obtained using the same

steps described earlier

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Now the solution reads

X1 = 12, X2 = 6, Z = 132

The coefficients of the non basic variables in

the Cj Z

jrow are all non positive.

Therefore, there is no further scope for

increasing the value of Z.

Hence the algorithm stops at this stage. The

given solution is the optimal solution

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The steps involved in the simplex algorithm

are given below :

Rewrite inequalities as equations using slack

variables.

Set up L.P in tabular form.

Determine the entering variable as one with a

positive Cj Zjcoefficient ;

Generally, the variable with the maximum Cj

Zj ischosen as the entering variable though this is not a

strict criterion.

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Determine the departing variable as the row

in which the aij coefficient yields the

minimum positive bi / aij (aij >0).

The aij corresponding to the departing

variable is called the pivot.

In order to obtain the next table, perform the

following steps.

Divide the pivotal row throughout by thepivot.

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Multiply this new row by coefficients in the

pivotal column and subtract the product from

the corresponding old row;

Repeat this process till the table is complete.

Check if there is any Cj

Zj > 0. If so, repeatsteps (3) - (7).

If not, read off the optimal solution from the

last table.

02 lp_graphical method simplex algorithm

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