OPERATIONS RESEARCHChapter 03 - The Simplex Method Principles
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The Simplex MethodPrinciples
Definition: A variable is said to a basic variable in a given equation if it appears with a unit coefficient in that equation and zero coefficients in
all other equations.Other variables are called nonbasic variables.
Remark: Recall the reduced row-echelon form of the augmented matrix of a system of linear equations.
Definition: A pivot operation is a sequence of elementary operations that reduces a given system to an equivalent system in which a specified variable has a unit coefficient in one equation and zero elsewhere. (this
is a basic variable).Example :
yields the system
After a sequence of elementary row operations. This is called the canonical form of the system .
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54321
54321
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2 3 2 6423
5432
5431
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3 3
The Simplex MethodPrinciples
Definition: A variable is said to be a basic variable in a given system of linear equations if it appears with a unit coefficient in one equation and zero coefficients in other equations. Other variables are called nonbasic variables.
Definition: A pivot operation is a sequence of elementary row operations that reduces a given system to an equivalent system in which a specified variable has a unit coefficient in one equation and zero elsewhere. (Basic variable).
Remark: The number of basic variables is determined by the number of equations in the system. (no. of basic variables is less than or equal to the no. of equations).
Definition: The solution obtained from a canonical system by setting the nonbasic variables to zero and solving for the basic variables is called a basic solution.
A basic feasible solution is a basic solution in which the values of the basic variables are all nonnegarive.
In the previous example, the basic feasible solution is
1x
.2 xand 6 21 x
4 4
The Simplex MethodPrinciples
The simplex method is an iterative process for solving LPP’s expressed in standard form . In addition to that, the constraint equations are expressed in a canonical system.
Steps:1. Start with an initial basic feasible solution in canonical form.2. Improve the initial solution (if possible) by finding another bfs with a
better objective function value. The SM implicitly eliminates from consideration all those bfs’s whose objective function values are worse than the present (current) solution.
3. Continue until a particular bfs cannot be improved further. It becomes an optimal solution, and the method terminates.
Definition: A bfs that differs from the present bfs by exactly one basic variable is called an adjacent bfs.
Definition: The relative profit of a variable is the change in the value of the objective function that results from increasing the value of this variable by 1.
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Example 1
MaximizeS.T :
Iteration # 1Step 1: The system is in canonical form with respect to . Take
Notice that the current value of z= -1. (basic: , nonbasic: )Step 2: Compute relative profits of the nonbasic variables, as follows:
1)
Z=5-7+4=2, relative profit=2-(-1)=3. 2)
Z=2-6+3=-1, relative profit=-1-(-1)=0.3)
54321 325 xxxxxZ
7x 43x8 22
5321
4321
xxxxxx
7,8 takeand ,0 54321 xxxxx.,, 321 xxx
11 x
54 , xx
54 , xx
4,773
854
51
41
xxxxxx
12 x 3,67482
5452
42
xxxxxx
13 x 6,67
8254
53
43
xxxxxx
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Example 1
z= 3-6+6=3, relative profit=3-(-1)=4. is the new basic variable (highest rel.profit). Highest increase in is the
minimum {4,7} = 4. Why?. Now,
Iteration # 2: Step 1: Rewrite the system in canonical form with respect to
3x 3x
153)4(33,4,0,
3,07
82
53421
5453
43
zxxxxxSo
xxxxxx
:get to, and 53 xx
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25
4 21
21
5421
4321
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{4,7 :}what number we put instead of x3 in the
two equations to get x4 and x5 = 0
7 7
Example 1
Step 2: Compute rel profits of the nonbasic variables.1)
2) 1.15-16profit rel.
1621
2215
21,
27
3x x25
4 x 21
1
53
51
31
1
Z
xxx
x
0.-415-11profit rel.11092
0,3 3x 3x
4 x
1
5352
32
2
Z
xxx
x
8
Example 1
3)
is the new basic variable. = min{8,6/5} = 6/5.
0.-215-13profit .
13271
221
27,
27
321
4 21
1
53
54
43
4
rel
Z
xxxx
xx
x
1x 1x
0 x:nonbasic ,,:581
5516
517
534
54231
3
xxxxbasic
Z
x
9
Example 1Iteration # 3Step 1: Rewrite the system in canonical form with respect to
1)
2)
:get to, and 31 xx
517
51
53x
52
56 x
52-
51-
56
5432
5421
xxx
xxx
035
1558111 .,1192
3,0
517
52
56
56
1
31
32
21
2
profitrelZ
xxxx
xx
x
0359
581
572 .,
5721
5427
514,
57
517
53
56
51
1
31
43
41
4
profitrelZ
xxxx
xx
x
10
Example 1
3)
Conclusion: All relative profits in this iteration are negative. Therefore, there is no new entering variables. The results of the previous iteration give the optimal solution. i.e
DONE
052
581
579 .,
5791
5544
518,
54
517
51
56
52
1
31
53
51
5
profitrelZ
xxxx
xx
x
581,
517,
56
31 Zxx
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Example 2
Maximize
I. Write the LPP in standard form, to get:Maximize
21 23 xxZ
0, x2 x 1x- 82x 62:.
21
2
21
21
21
x
xxxxTS
21 23 xxZ
0,...,, x2x x 1 x x- 8 x 2x 6 2:.
621
62
521
421
321
xx
xx
xxxTS
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Example 2
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15
16
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Summary
Summary of the simplex method:1. Start with an initial basic feasible solution (bfs) in canonical form.2. Check if the current solution is optimal or not as follows:
(i) If the relative profits of the nonbasic variables are all zero or negative, then this is the optimal solution. STOP. (ii) Else, choose the nonbasic variable with highest relative profit as an entering variable. The leaving variable is determined by the constraint that gives the minimum value to the entering variable. (The minimum ratio rule).
3. Rewrite the system in canonical form with respect to the new basic variables.
4. GO TO STEP 2.