Big-M Two Phase Methods

7172019 Big-M Two Phase Methods

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In the standard simplex table

bull The coefficients of the basic variables in the z-rowshould be zero

bull Th r iv i i n in h n r in m rix

1

must be identity coefficients

bull In case it is not we have to use elementary row

operations to keep it in the standard simplex formbefore we apply the simplex method



Contents

bull Artificial Variables Techniques

bull Big M Method

bull Two Phase Method

2



If in a starting simplex tableau we donrsquot have an

identity sub matrix (ie an obvious starting BFS) then

we introduce artificial variables to have a starting BFS

This is known as artificial variable technique

There are two methods to find the starting BFS and

so ve t e prob em ndash

(1) Big M Method

(2) Two-Phase Method

3



Big M Method

4



Suppose that ith constraint equation doesnt have aslack variable ie

ith constraint in the original LPP is either ge or = type

Then an artificial variable Ri is added to form the ithunit vector column

Since the artificial variables are not part of theoriginal LPP they are assigned a very high penalty inthe objective function and thus forcing them to equal

zero in the optimum solution

5



Artificial variable objective coefficient

= - M (Sum of artificial variables) in a maximization

problem

= sum o t e art c a var a es n a m n m zat onproblem

where M is a very large positive number

Mathematically M rarrinfinrarrinfinrarrinfinrarrinfin

6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6

x1ge0 x2ge0 r t ng z as equat on an a ngslack and surplus variables to the

constraints

z ndash 4x1 ndash 3x2 = 0

2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7



Addition of artificial variables to obtain a starting BFS

Artificial variables

R1 and R2 added tothe constraints

2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6

These artificial variables are penalized in the objectivefunction by assigning some very large value (say M) to

them in the objective function

8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints

Where M is a very large penalty assigned onartificial variables R1 and R2

9



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Tabular form

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Since the z column does not change in the iterations it may be

deleted

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The initial BFS as obtained from the table is

R1 = 10 R2 = 6 and z = 0 Is this tablecorrect

11



But if R1 = 10 R2 = 6 then the value ofobjective function should be z = 16 M sincewe have defined the objective function as

z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r

This is because of the fact that the elementsof the basic variable have a non zero

coefficient in the z-row

The starting table now becomes

12



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Apply the usual simplex method for solvingminimization problem

13



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Selecting pivot row and pivot column for a minimization problem

Minratio

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102

61

---

Select the most positive element of the z row

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Minratio

Second iteration

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512

112

2172

983089983087983090

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Final iteration

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Solution x1 = 4 x2 = 2 z = 22

16



Consider the following LPP

Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17



Introducing surplus and artificial variables s2 R1 and R2 the

LPP is modified as follows

Maximize

Subject to

1 2 3 1 2

2 3 5 z x x x M R M R= + minus minus minus

1 2 3 1 7 x x x R+ + + =

Now we solve the above LPP by the Simplex method

1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M

Basic z x1 x2 x3 s2 R1 R2 Sol

z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457



The optimum (Maximum) value of

z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20



If in any iteration there is a tie for leaving variablebetween an artificial variable and other variable

(decision surplus or slack) we must prefer the

artificial variable to leave the basis

into the basis

If in the final optimal tableau an artificial variable ispresent in the basis at a non-zero level this means our

original problem has no feasible solution

21



Tutorial - 3

22



(1) Solve the following LPPs using Simplex method

(i) Max

subject to

1 21 2 1 5 z x x= minus minus

1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2

0 u n r e s t r i c e d

x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le

minus minus ge minus

+ le

ge

(iv) Minsubject to

24

1 2 36 2 6 z x x x= minus minus minus

1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge



(2) Solve the following LPPs using Big-M method

(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26



Why use two phase method when Big M - methodis already there

One major disadvantage of Big M - method is that itis computationally inefficient

Mathematically it is said that M should be a verylarge number but sometimes very large values of Mmay lead to a wrong result

27



Phase - I

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Phase - II

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29



Example

Min z = 4x1 + 6x2 + 5x3

Subject to2x1 + 4x2 + 3x3 ge 32x + 2x + 4x ge 28

x1 x2 x3 ge 0

30



s1 s2 ndash surplus variableR1 R2 ndash artificial variables

Min z = 4x1 + 6x2 + 5x3

Subject to

2x1 + 4x2 + 3x3 ndash s1 + R1 = 32x1 + 2x2 + 4x3 ndash s2 + R2 = 28x1 x2 x3 ge 0

Artificial variables are added to get an identity sub matrix

31



Phase I (construct an auxiliary LPP)

construct a new objective function in which thecoefficients of artificial variables are 1 and thecoefficients of all other variables are zero

Min r = R1 + R2New objective function


Constraints

32



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form a new r row (similar to the manner in which new z- row is

formed in Big M method)

ThusNew r-row = r-row + R1 row + R2 row

33



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Standard simplex table

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Applying the usual simplex method for solving aminimization problem

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Minratio

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At this point both the artificial variables have completed theirpurpose and hence their columns can be removed from thetable

37



Phase II

We will use the BFS obtained in Phase-I as a starting

BFS of the given problem and use simplex method tofind the optimal solution

38



Form a simplex table

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Is this standard simplex table

39



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z ndashrow = z row + 6(x2 row) + 5(x3 row)

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Since all the conditions for optimality (for a minimization problem) arebeing satisfied Phase two ends at this point

40



Solve the following LPP using Two-Phase method

Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +

1 2 3 1 12 20 x x x s Rminus + minus + =

=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0

Basic r x1 x2 x3 s1 R1 R2 Sol

3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44



This is optimal tableau Thus Phase I has ended and wenow start Phase II with the starting BFS as the optimal

solution of Phase I

Phase II

subject to the same constraints as given in the original

problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5

Thus the optimal solution is x1 = 25 x2 = 0 x3 = 0 and the

optimal z = Min z = 5046



In Phase I we always minimize the sum of the

artificial variables

If in the optimal table of phase I an artificial variablesrema ns at non-zero eve n t e as s t en t e pro em

has no feasible solution

47



Tutorial - 4

48



Solve the following LPPs using Two-Phase Method

(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Contents

bull Artificial Variables Techniques

bull Big M Method

bull Two Phase Method

2









(1) Big M Method


3



Big M Method

4








5





problem




6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



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Tabular form

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10




deleted

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983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

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11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge









(1) Big M Method


3



Big M Method

4








5





problem




6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Big M Method

4








5





problem




6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge








5





problem




6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge





problem




6



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Example 1

Min z = 4x1 + 3x2

Subject to

2x1 + x2 ge 10x1 + x2 ge 6

-3x + 2x le 6


constraints


2x1 +x2 ndash S1 = 10x1 + x2 - S2 = 6

-3x1 + 2x2 + S3 = 6

7






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge






2x1 + x2 ndash S1 +R1 = 10

x1 + x2 ndash S2 +R2 = 6- 3x1 + 2x2 + S3 = 6



8



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



z = 4x1 + 3x2 + MR1 + MR2

2x1 + x2 ndash s1 +R1 = 10

x1 + x2 ndash s2 +R2 = 6- 3x1 + 2x2 + s3 = 6

Objective function

constraints


9



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983162 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983089 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122 983088 983090 983089 983085983089 983088 983089 983088 983088 983089983088

Tabular form

983122983090

983088 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983088 983085983091 983090 983088 983088 983088 983088 983089 983094

10




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




deleted

983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983085983091 983088 983088 983085 983117 983085 983117 983088 983088

983122983089

983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090 983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094



11




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




z = 4x1 + 3x2 + MR1 + MR2

Wh hi in n i n r




12



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122983089 983090 983089 983085983089 983088 983089 983088 983088 983089983088

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091 983085983091 983090 983088 983088 983088 983088 983089 983094


13



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983085983092 983083 983091983117 983085983091983083983090983117 983085983117 983085983117 983088 983088 983088 983089983094983117

983122 983090 983089 983085983089 983088 983089 983088 983088 983089983088


Minratio

983122983090

983089 983089 983088 983085983089 983088 983089 983088 983094

983123983091

983085983091 983090 983088 983088 983088 983088 983089 983094

102

61

---


14



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155

983145983139

983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983085983090983083983117

983090

983085983092983085983117

983090

983085983117 983092 991251 983091983117

983090

983088 983088 983090983088 983083983117

Minratio

Second iteration

983160983089

983089 983089 983090 983085983089 983090 983088 983089 983088 983088 983093

983122983090

983088 983089983087983090 983089983087983090 983085983089 983088 983089 983088 983089

983123983091 983088 983095983087983090 983085983091983087983090 983088 983088 983088 983089 983090983089

512

112

2172

983089983087983090

15



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983123983089

983123983090

983122983089

983122983090

983123983091

983155983151983148983157983156983145983151983150

983162 983088 983088 983085983089 983085983090 983089 991251 983117 983080983090983085983117983081983087983090 983088 983090983090

983160983089

983089 983088 983085983089 983088 983089 983088 983088 983092

Final iteration

983160983090

983088 983089 983089 983085983089 983088 983089 983088 983090

983123983091

983088 983088 983085983089983091983087983090 983088 983088 983088 983089 983089983092

Solution x1 = 4 x2 = 2 z = 22

16




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

17





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge





Maximize

Subject to

1 2 3 1 2


1 2 3 1 7 x x x R+ + + =


1 2 3 2 2

1 2 3 2 1 2

2 5 1 0 0

x x x s R

x x x s R R

minus + minus + =

ge

18



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



R2 0 2 -5 1 -1 0 1 10

-2-3M -3+4M 5-2M M 0 0 -17M


z 1 -2 -3 5 0 M M 0

R1 0 1 1 1 0 1 0 7

z 1 0 -8 - 6 - -1 - 0 1 + 10 -

x1 0 1 -52 12 -12 0 12 5

7M2 M2 M2 3M2 2MR1 0 0 72 12 12 1 -12 2

x2 0 0 1 17 17 27 -17 47

z 1 0 0 507 17 167 + -17 1027M +M

x1 0 1 0 67 -17 57 17 457




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




z = 1027

and it occurs at

x1 = 457 x2 = 47 x3 = 0

20






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge






into the basis



21



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Tutorial - 3

22




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




(i) Max

subject to


1 2

1 2

1 2

4 3 1 22 5 1 0

0

x x x x

x x

+ le

+ le

ge

(ii) Max

subject to

23

1 22 3 z x x= minus

1 2

1 2

1 2

1 2

2

2 2

2


x x

x x

x x

x x

minus + le

minus le

minus minus le

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



(iii) Max

subject to

1 2 33 2 5 z x x x= + +

1 2 3

1 3

1 2

1 2 3

2 4 3 0

3 2 4 6 0

4 4 2 0

0

x x x

x x

x x

x x x

+ + le


+ le

ge

(iv) Minsubject to

24


1 2 3

1 2 3

1 2 3

1 2 3

2 3 1 4

4 4 1 0 4 6

2 2 4 3 7

2 1 3

x x x

x x x

x x x

x x x

minus + le

minus + + le

+ minus le

ge ge ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




(i) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 23 3

4

x x

x x

x

+ ge

+ le

le

(ii) Min

subject to

1 2 0 x x ge

25

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 5

2 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



TWO PHASE METHOD

26






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge






27



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Phase - I

983127983141 983139983151983150983155983156983154983157983139983156 983137983150 983137983157983160983145983148983145983137983154983161 983116983120983120 983080983137983148983159983137983161983155 983137 983149983145983150983145983149983145983162983137983156983145983151983150

983152983154983151983138983148983141983149983081983086

983124983144983141 983151983138 983141983139983156983145983158983141 983157983150983139983156983145983151983150 983145983155 983155983157983149 983151 983137983154983156983145 983145983139983145983137983148 983158983137983154983145983137983138983148983141983155

983159983145983156983144 983156983144983141 983151983154983145983143983145983150983137983148 983139983151983150983155983156983154983137983145983150983156983155

983124983144983141 983151983152983156983145983149983137983148 983155983151983148983157983156983145983151983150 983151983142 983156983144983141 983137983157983160983145983148983145983137983154983161 983152983154983151983138983148983141983149 983159983145983148983148 983143983145983158983141

983137 983155983156983137983154983156983145983150983143 983106983110983123 983142983151983154 983156983144983141 983151983154983145983143983145983150983137983148 983152983154983151983138983148983141983149983086

28



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Phase - II

983127983141 983157983155983141 983156983144983141 983138983137983155983145983139 983142983141983137983155983145983138983148983141 983155983151983148983157983156983145983151983150 983137983158983137983145983148983137983138983148983141 983142983154983151983149 983156983144983141

983085

983152983154983151983138983148983141983149983086

29



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Example

Min z = 4x1 + 6x2 + 5x3


x1 x2 x3 ge 0

30




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




Min z = 4x1 + 6x2 + 5x3

Subject to



31







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge







Constraints

32



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983122983089

983090 983092 983091 983085983089 983088 983089 983088 983091983090

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096




33



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085


983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096

34




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091 983094 983095 983085983089 983085983089 983088 983088 983094983088

983085

Minratio

983089

983122983090

983089 983090 983092 983088 983085983089 983088 983089 983090983096 284983092

35



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983122983089

983122983090

983155983151983148983157983156983145983151983150

983154 983091983087983092 983093983087983090 983088 983085983089 983095983087983092 983088 983085983095983087983092 983089983089

983122983089

983093983087983092 983093983087983090 983088 983085983089 983091983087983092 983089 983085983091983087983092 983089983089983093983087983090

983160983091

983089983087983092 983089983087983090 983089 983088 983085983089983087983092 983088 983089983087983092 983095

36



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089 983160983090 983160983091 983155983089 983155983090 983122983089 983122983090 983155983151983148983157983156983145983151983150

983154 983088 983088 983088 983088 983088 983085983089 983085983089 983088

983160983090

983089983087983090 983089 983088 983085983090983087983093 983091983087983089983088 983090983087983093 983085983091983087983089983088 983090983090983087983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983085983089983087983093 983090983087983093 983090983092983087983093


37



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Phase II



38




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085983092 983085983094 983085983093 983088 983088 983088

983160 983089 983090 983089 983088 983085983090 983093 983091 983089983088 983090983090 983093

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


39



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



983106983137983155983145983139 983160983089

983160983090

983160983091 983155983089

983155983090

983155983151983148983157983156983145983151983150

983162 983085 983089 983088 983088 983085983089983090983087983093 983085983090 983090983093983090983087983093


983090

983160983091

983088 983088 983089 983089983087983093 983085983090983087983093 983090983092983087983093


40




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




Minimize

Sub ect to

321 352 x x x z ++=

1 2 3

1 2 3

1 2 3

2 4 5 0

0

x x x

x x x

x x x

minus +

+ + =

ge

41



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Phase I

Minimize

Subject to

1 2r R R= +


=1 2 3 2

1 2 3 1 1 2 0 x x x s R R ge

42



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



r 1 0 0 0 0 -1 -1 0


3 2 2 -1 0 0 70

R1 0 1 -2 1 -1 1 0 20

R2 0 2 4 1 0 0 1 50

x1 0 1 -2 1 -1 1 0 20

r 1 0 8 -1 2 -3 0 10

R2 0 0 8 -1 2 -2 1 10

43



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



r 1 0 8 -1 2 -3 0 10


x1 0 1 -2 1 -1 1 0 20

R2 0 0 8 -1 2 -2 1 10

x1 0 1 0 34 -12 12 14 452

r 1 0 0 0 0 -1 -1 0

x2 0 0 1 -18 14 -14 18 54

44




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




solution of Phase I

Phase II


problem

321

45




z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



z 1 -2 -5 -3 0 0

1 2 3 1 1 2

x1 0 1 0 34 -12 452

x2 0 0 1 -18 14 54

- -

0 0 -178 14 2054

x1 0 1 2 12 0 25

s1 0 0 4 -12 1 5









47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge







47



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



Tutorial - 4

48




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge




(i) Maximize

Subject to

1 2 32 3 5 z x x x= + minus

1 2 3 7 x x x+ + =

1 2 3

1 2 3

2 5 10

0

x x x

x x x

minus + ge

ge

49



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



(ii) Maximize

Subject to

321 422 x x x z ++=

1 2 32 2 x x x+ + le

1 2 3

1 2 3 0

x x x

x x x

+ +

ge

50



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge



(iii) Max

subject to

1 23 z x x= minus +

1 2

1 2

1

2 2

3 34

0

x x

x x

x

x x

+ ge

+ le

le

ge

(iv) Min

subject to

51

1 2 39 2 z x x x= + +

1 2 3

1 2 3

1 2 3

4 2 52 3 4

0

x x x

x x x

x x x

+ + ge

+ + ge

ge

Documents

Big-M Two Phase Methods