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![Page 1: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/1.jpg)
Sensitivity analysis
LI Xiao-lei
![Page 2: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/2.jpg)
A graphical introduction to sensitivity analysis
Sensitivity analysis is concerned with how changes in an LP’s parameters affect the LP’s optimal solution.
![Page 3: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/3.jpg)
A graphical introduction to sensitivity analysis
The Giapetto problemmax z=3x1+2x2
s.t. 2x1 + x2≤100 (finishing constraint)
x1 + x2≤80 (carpentry constraint)
x1 ≤40 (demand constraint)
x1 , x2 ≥0Where
x1=number of soldiers produced per week
x2=number of trains produced per week
![Page 4: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/4.jpg)
A graphical introduction to sensitivity analysis
0 20 40 60 80 1000
20
40
60
80
100
x
y
3 x+2 y-180 = 0
z=60 z=100
z=180
The optimal solution is z=180, x1=20, x2=60
![Page 5: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/5.jpg)
Graphical analysis of the effect of a change in an objective function coefficient
Let c1 be the contribution to profit by each soldier. For what values of c1 does the current basis remain optimal?
At present, c1=3 and the profit line has the form 3x1+2x2=constant,
or x2=-3x1/2 + constant/2.
![Page 6: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/6.jpg)
Figure 1
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
x1
x 2
3 x1+2 x
2-120 = 0
finishing constraint, slope=-2
carpentry constraint, slope =-1
demand constraint
profit line z=120, slope=-3/2
B
C
D
A
E
![Page 7: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/7.jpg)
Graphical analysis of the effect of a change in an objective function coefficient
If a change in c1 cause the profit lines to be flatter than the carpentry constraint, the optimal solution will change from point B to a new optimal solution (point A).The slope of each profit line is –c1/2
The profit line will be flatter than the carpentry constraint if –c1/2>-1, or c1<2, and the new optimal solution will be (0,80), point A.
![Page 8: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/8.jpg)
Graphical analysis of the effect of a change in an objective function coefficient
If the profit line s are steeper than the finishing constraint, the optimal solution will change from point B to point C.The slope of the finishing constraint is -
2.
If –c1/2<-2, or c1>4, and the new optimal solution will be (40,20).
![Page 9: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/9.jpg)
Graphical analysis of the effect of a change in an objective function coefficient
In summary, if all other parameters remain unchanged, the current basis remains optimal for 2≤c1≤4. and Giapetto should still manufacture 20 soldiers and 60 trains.
But Giapetto’s profit will change. For instance, if c1=4, Giapetto’s profit will now be 4(20)+2(60)=$200 instead of $180.
![Page 10: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/10.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Whether a change in the right-hand side of a constraint will make the current basis no longer optimal?
Let b1 the number of available finishing hours. Currently, b1=100.
![Page 11: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/11.jpg)
Figure 2
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
x1
x 22 x
1+x
2-120 = 0
finishing constraint, slope=-2
carpentry constraint, slope =-1
demand constraint
profit line z=120, slope=-3/2
finishing constraint b1=80finishing constraint b1=120 A
B
C
D
![Page 12: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/12.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
A change in b1 shifts the finishing constraint parallel to its current position.
The current optimal solution is where the carpentry and finishing constraints are binding.
Then as long as the point where the finishing and carpentry constraints are binding remains feasible, the optimal solution will still occur where these constraints intersect.
![Page 13: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/13.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
From figure 2, for 80≤b1≤120, the current basis remains optimal.
With the changing of b1, the values of the decision variables and the objective function value change.
For example, if 80≤b1≤100, the optimal solution will change from point B to some point on the line segment AB. Similarly, if 100≤b1≤120, the optimal solution will change from point B to some point on the line segment BD.
![Page 14: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/14.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
To determine how a change in the right-hand side of a constraint changes the values of the decision variables.Let b1=number of available finishing hours.
If we change b1 to 100+Δ, the current basis remains optimal for -20≤Δ≤20.
Note: as b1 changes, the optimal solution to the LP is still the point where the finishing-hour and carpentry-hour constraints are binding.
![Page 15: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/15.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Thus, we can find the new values of the decision variables by solving 2x1+x2=100+Δ
and x1+x2=80
This yields x1=20+Δ and x2=60-Δ
Thus, a increase in the number of available finishing hours results in an increase in the number of soldiers produced and a decrease in the number of trains produced.
![Page 16: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/16.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Let b2=the number of available carpentry hours If we change b2 to 80+Δ, the current ba
sis remains optimal for -20≤Δ≤20.
Note: as b2 changes, the optimal solution to the LP is still the point where the finishing-hour and carpentry-hour constraints are binding.
![Page 17: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/17.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Thus, if b2=80+ Δ, the optimal solution to the LP is the solution to 2x1+x2=100
and x1+x2=80+ Δ
This yields x1=20-Δ and x2=60+2Δ.
It shows that an increase in the amount of available carpentry hours decrease the number of soldiers produced and increases the number of trains produced.
![Page 18: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/18.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Let b3=the demand for soldiers.If b3 is changed to 40+Δ, the current basis remains
optimal for Δ≥-20. for Δ in this range, the optimal solution to the LP will still occur where the finishing and carpentry constrains are binding.
Thus, the optimal solution to the LP is the solution to
2x1+x2=100
and x1+x2=80
This yields x1=20 and x2=60
![Page 19: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/19.jpg)
Graphical analysis of the effect of a change in a right-hand side on the LP’s optimal solution
Note:
In a constraint with positive slack( or positive excess) in an LP’s optimal solution, if we change the right-hand side of the constraint to a value in the range where the current basis remains optimal, the optimal solution to the LP is unchanged.
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Shadow prices
We define the shadow price for the ith constraint of an LP to be the amount by which the optimal z-value is improved if the right-hand side of the ith constraint is increased by 1.
This definition applies only if the change in the right-hand side of constraint i leaves the current basis optimal.
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Shadow prices
For example, for the Δchanging in the finishing hours, the optimal solution is x1=20+Δ and x2=60-Δ. Then the optimal z-value will equal 3x1+2x2=3(20 +Δ) +2(60-Δ) =180+ Δ. Thus, as long as the current basis remains optimal, a unit increase in the number of available finishing hours will increase the optimal z-value by $1.so the shadow price of the first constraint is $1.
![Page 22: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/22.jpg)
Shadow prices
For the second constraint, if Δchanging in the finishing hours, the optimal solution is x1=20-Δ and x2=60+2Δ. Then the optimal z-value will equal 3x1+2x2=3(20 -Δ) +2(60+2Δ) =180+ Δ. Thus, as long as the current basis remains optimal, a unit increase in the number of available finishing hours will increase the optimal z-value by $1.so the shadow price of the second constraint is $1.
![Page 23: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/23.jpg)
Shadow prices
For the third constraint, the optimal values of the decision variables remain unchanged, as long as the current basis remains optimal. Then the optimal z-value will also remain unchanged, which shows that the shadow price of the third constraint is $0.
It turns out that whenever the slack variable or excess variable for a constraint is positive in an LP’s optimal solution, the constraint will have a zero shadow price.
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Shadow prices
Each unit by which constraint i’s right-hand side is increased will increase the optimal z-value( for a max problem) by the shadow price.
Suppose we increase the rhs of the ith constraint of an LP by Δbi , thus, the new optimal z-value is given by(new optimal z-value)=(old optimal z-value) +(constraint i’s shadow price)Δbi
For a minimization problem,(new optimal z-value)=(old optimal z-value) -(constraint i’s shadow price)Δbi
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Importance of sensitivity analysis
If a parameter changes, sensitivity analysis often makes it unnecessary to solve the problem again.
A knowledge of sensitivity analysis often enables the analyst to determine from the original solution how changes in an LP’s parameters change the optimal solution.
![Page 26: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/26.jpg)
Some important formulas
Assume a max LP problem with m constraints and n variables. Although some of these variables may be slack, excess, or artificial ones, we choose to label them x1,x2,…,xn. max z=c1x1+c2x2+…+cnxn
s.t. a11x1+a12x2+…+a1nxn=b1
a21x1+a22x2+…+a2nxn=b2 (1) : : : :
am1x1+am2x2+…+amnxn=bm
xi≥0 (i=1,2,…,n)
![Page 27: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/27.jpg)
Some important formulas
For the Dakota problem,
max z=60x1+30x2+20x3+0s1+0s2+0s3
s.t. 8x1+ 6x2+ x3+ s1 =48(lumber)
4x1+ 2x2+1.5x3+ s2 =20 (finishing) (1’) 2x1+1.5x2+0.5x3+ s3=8 (carpentry)
x1,x2,x3,s1,s2,s3≥0
![Page 28: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/28.jpg)
Some important formulas
Let BVi be the basic variable for row i of the optimal tableau. Also define BV={BV1, BV2 , … , BVm} to be the set of basic variables in the optimal tableau, and define the m × 1 vector xBv=[xBv1, xBv2 , …, xBvm]T
We also defineNBV=the set of nonbasic variables in the opti
mal tableau
xNBv=(n-m) ×1 vector listing the nonbasic variables.
![Page 29: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/29.jpg)
Some important formulas
The optimal tableau for the LP 1’ z + 5x2 +10s2+10s3=280
- 2x2 + s1 +2 s2 - 8s3=24
- 2x2+ x3+ 2s2 - 4s3 =8
x1+1.25x2 -0.5s2+1.5s3=2
For this optimal tableau, BV1=s1, BV2=x3, and BV3=x1, then xBV=[s1,x3,x1]’
We may choose NBV={x2,s2,s3}, then xNBV=[x
2,s2,s3]’
![Page 30: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/30.jpg)
Some important formulas
DEFINATION cBV is the 1×m row vector [cBV1 cBV2 … cB
Vm].
Thus, the elements of cBV are the objective function coefficients for the optimal tableau’s basic variables. For the Dakota problem, BV={s1,x3,x1}, then form (1’) we find that cBV=[0 20 60].
![Page 31: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/31.jpg)
Some important formulas
DEFINATION cNBV is the 1×(n-m) row vector whose el
ements are the coefficients of the nonbasic variables( in the order of NBV).
If we choose to list the nonbasic variables for the Dakota problem in the order NBV={x2,s2,s3}, then form (1’) we find that cNBV=[30 0 0].
![Page 32: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/32.jpg)
Some important formulas
DEFINATION
The m×m matrix B is the matrix whose jth column is the column for BVj in (1).
For the Dakota problem, the first column of B is the s1 column in (1’); the second, the x3 column; and the third, the x1 column. Thus,
25.00
45.10
811
B
![Page 33: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/33.jpg)
Some important formulas
DEFINATION aj is the column (in the constraints) for
the variable xj in (1).
For example, in the Dakota problem,
5.1
2
6
2a
![Page 34: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/34.jpg)
Some important formulas
DEFINATION
N is the m×(n-m) matrix whose columns are the columns for the nonbasic variables (in the NBV order) in (1).
For the Dakota problem, we write NBV={x2,s2,s3}, then
105.1
012
006
N
![Page 35: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/35.jpg)
Some important formulas
DEFINATION
The m×1 column vector b is the right-hand side of the constraints in (1).
For the Dakota problem,
8
20
48
b
![Page 36: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/36.jpg)
Matrix notation
(1) may be written as
(3)
0,
..
NBVBV
NBVBV
NBVNBVBVBV
xx
bNxBxts
xcxcz
0,
..
NBVBV
NBVBV
NBVNBVBVBV
xx
bNxBxts
xcxcz
![Page 37: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/37.jpg)
Matrix notation
The Dakota problem can be written as
0
0
0
,
0
0
0
8
20
48
101.5
012
006
25.00
45.10
811
..
003060200 max
3
2
2
1
3
1
3
2
2
1
3
1
3
2
2
1
3
1
s
s
x
x
x
s
s
s
x
x
x
s
ts
s
s
x
x
x
s
z
0
0
0
,
0
0
0
8
20
48
101.5
012
006
25.00
45.10
811
..
003060200 max
3
2
2
1
3
1
3
2
2
1
3
1
3
2
2
1
3
1
s
s
x
x
x
s
s
s
x
x
x
s
ts
s
s
x
x
x
s
z
![Page 38: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/38.jpg)
Matrix notation
Multiplying the constraints in (3) through by B-1, we obtain
or (4)
In (4),BVi occurs with a coefficient of 1 in the ith constraint and a zero coefficient in each other constraint. Thus, BV is the set of the basic variables for (4), and (4) yields the constraints for the optimal tableau.
bBNxBBxB NBVBV111
bBNxBx NBVBV11
![Page 39: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/39.jpg)
Matrix notation
For the Dakota problem,
Then,
Of course, these are the constraints for the optimal tableau.
5.15.00
420
8211B
2
8
24
5.15.01.25
422-
822-
3
2
2
1
3
1
s
s
x
x
x
s
![Page 40: Sensitivity analysis LI Xiao-lei. A graphical introduction to sensitivity analysis Sensitivity analysis is concerned with how changes in an LP’s parameters.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f305503460f94c4a5a6/html5/thumbnails/40.jpg)
Matrix notation
From (4), we see that the column of a nonbasic variable xj in the constraints of the optimal tableau is given by
B-1(column for xj in (1) )=B-1aj
The following two equations summarize the preceding discussion:
Column for xj in optimal tableau’s constraints=B-1aj (5)Right-hand side of optimal tableau’s constraints=B-1b (6)
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Matrix notation How to express row 0 of the optimal tableau in terms
of BV?We multiply the constraints through by the vector cBVB-1:
(7)
And rewrite the original objective function as,
(8)
Adding (7) to (8), we can eliminate the optimal tableau’s basic variables,
(9)
bBcNxBcxc BVNBVBVBVBV11
0 NBVNBVBVBV xcxcz
bBcxcNBcz BVNBVNBVBV11 )(
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Matrix notation
From (9), the coefficient of xj in row 0 is
cBVB-1(column of N for xj)-
(coefficient for xj in cNBV)=cBVB-1aj-cj
the right hand side of row 0 is cBVB-1b
To summarize, we let be the coefficient of xj in the optimal tableau’s row 0. then we have shown that
(10)
and
right-hand side of optimal tableau’s row 0=cBVB-1b (11)
jjBVj caBcc 1
jc
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Matrix notation
To illustrate, we determine row 0 of the Dakota problem’s optimal tableau.
Then cBVB-1=[0 10 10], from (10) we find that the coefficients of the nonbasic variables in row 0 are
5.15.00
420
821
]60200[ 1BandcBV
530
5.1
2
6
10100221
2
caBcc BV
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Matrix notation
and
Of course, the optimal tableau’s basic variables (x1,x3,and s1) will have zero coefficients in row 0.
100
0
1
0
0 12
BcrowoptimalinsoftCoefficien BV
100
1
0
0
0 13
BcrowoptimalinsoftCoefficien BV
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Matrix notation
From (11), the right-hand side of row 0 is
Putting it all together, we see that row 0 is
z+5x2+10s2+10s3=280
280
8
20
48
101001
bBcBV
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Matrix notation
Simplifying formula (10) for slack, excess, and artificial variablesIf xj is the slack variable si, the coefficient of
si in the objective function is 0, and the column for si in the original tableau has 1 in row I and 0 in all other rows. Then (10) yields
Coefficient of si in optimal row 0
= ith element of cBVB-1-0
=ith element of cBVB-1 (10’)
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Matrix notation
Similarly, if xj is the excess variable ei, the coefficient of ei in the objective function is 0 and the column for ei in the original tableau has -1 in row i and 0 in all other rows. Then (10) reduces to
Coefficient of ei in optimal row 0
=-( ith element of cBVB-1)-0
=-(ith element of cBVB-1) (10’’)
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Matrix notation
Finally, if xj is an artificial variable ai, the objective function coefficient of ai( for a max problem) is –M and the original column for ai has 1 in row i and 0 in all other rows. Then (10) reduces to
Coefficient of ai in optimal row 0
=( ith element of cBVB-1)-(-M)
=(ith element of cBVB-1)+(M) (10’’’)
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Summary of formulas for computing the optimal tableau from the initial LP
Column for xj in optimal tableau’s constraints=B-1aj (5)
Right-hand side of optimal tableau’s constraints=B-1b (6)
Coefficient of si in optimal row 0=ith element of cBVB-1 (10’)
Coefficient of ei in optimal row 0=-(ith element of cBVB-1) (10’’)
Coefficient of ai in optimal row 0=-(ith element of cBVB-1)+(M)
(10’’’)
right-hand side of optimal tableau’s row 0=cBVB-1b (11)
We must first find B-1 because it is necessary in order to compute all parts of the optimal tableau.
jjBVj caBcc 1
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Example 1
Solution After adding slack variables s1 and s2,
we obtain:
max z=x1+4x2
s.t. x1+2x2+s1 =6
2x1+ x2 +s2=8
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Example 1
For the following LP, the optimal basis is BV={x2,s2}, computer the optimal tableau.
max z=x1+4x2
s.t. x1+2x2≤6
2x1+ x2≤8
x1,x2≥0
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Example 1
First we compute B-1,
12
1
02
1
and 11
02 1BB
12
1
02
1
and 11
02 1BB
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Example 1
Use (5) and (6) to determine the optimal tableau’s constraints.The column for x1 in the optimal tableau is
The other nonbasic variable is s1,
23
21
2
1
121
021
11aB
212
1
0
1
121
021
tableauoptimalin for Column 11
1 aBs
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Example 1
From (6)
Since BV is listed as {x2,s2},
5
3
8
6
121
021
tableauoptimal of side hand-Right 1bB
0212
1
021
04c 1BV
B
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Example 1
Then (10) yields
From (10’)
112
102
tableauoptimal of 0 rowin oft coefficien
111
1
caBc
x
BV
2 ofelement first
tableauoptimalin oft coefficien1
1
Bc
s
BV
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Example 1
(11) yields
Of course, the basic variables x2 and s2 will have zero coefficients in row 0.
Thus, the optimal tableau’s row 0 is
z+x1+2s1=12
126
602
0 row s tableau'optimal theof side hand-right the
1
bBcBV
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Example 1
The complete optimal tableau is
z +x1 +2s1 =12
½x1+x2+ ½ s1 =3
3/2x1 - ½ s1 +s2=5
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Sensitivity analysis
The study of how an LP’s optimal solution depends on its parameters is called sensitivity analysis.
We let BV be the set of basic variables in the optimal tableau. Given a change in an LP, we want to determine whether BV remains optimal.
Important observation:A simplex tableau( for a max problem) for a set of
BV is optimal if and only if each constraint has a nonnegative right-hand side and each variable has a nonnegative coefficient in row 0.
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Sensitivity analysis
Suppose we have solved an LP and have found that BV is an optimal basis. We can use the following procedure to determine if any change in the LP will cause BV to be no longer optimal.Step 1 Using the preceding formulas, determine
how changes in the LP’s parameters change the right-hand side and row 0 of the optimal tableau.
Step 2 If each variable in row 0 has a nonnegative coefficient and each constraint has a nonnegative right-hand side, BV is still optimal. Otherwise , BV is no longer optimal.
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Sensitivity analysis
Two reasons why a change in an LP’s parameters causes BV to be no longer optimal. A variable in row 0 may have a negative coefficient.
In this case , a better bfs can be obtained by pivoting in a nonbasic variable with a negative coefficient in row 0. we say that the BV is now a suboptimal basis.
A constraint may now have a negative right-hand side. In this case, at least one member of BV will now be negative and BV will no longer yield a bfs. We say that BV is now an infeasible basis.
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Illustrate in example
The Dakota furniture example x1=number of desks manufactured
x2=number of tables manufactured
x3=number of chairs manufactured
The objective function was
max z=60x1+30x2+20x3
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Illustrate in example
The initial tableau was z -60x1 -30x2 - 20x3 =0
8x1+ 6x2+ x3 +s1 =48 (lumber constraint)
4x1+ 2x2+1.5x3 +s2 =20 (finishing constraint)
2x1+1.5x2+0.5x3 +s3=8 (carpentry constraint)
The optimal tableau was z +5x2 +10s2 +10s3 =280
-2x2 + s1+ 2s2 -8s3=24
-2x2+x3 +2s2 -4s3=8
x1+1.25x2 -0.5s2 +1.5s3=2
BV={s1,x3,x1} and NBV={x2,s2,s3}. The optimal bfs is z=280,s1=24,x3=8,x1=2,x2=0,s2=0,s3=0
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Illustrate in example
Six types of changes in an LP’s parameters: Change 1 changing the objective function coeffici
ent of a nonbasic variable Change 2 changing the objective function coeffici
ent of a basic variable Change 3 changing the right-hand side of a const
raint Change 4 changing the column of a nonbasic var
iable Change 5 adding a new variable or activity Change 6 adding a new constraint
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Illustrate in example
Changing the objective function coefficient of a nonbasic variable
The only nonbasic decision variable is x2. the objective function coefficient of x2 is c2=30.
How would a change in c2 affect the optimal solution?
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Illustrate in example
Suppose we change c2 from 30 to 30+Δ. From (6), since the B-1 and b are unchanged, the ri
ght-hand side of BV’s tableau (B-1b) has not changed, so BV is still feasible.
From (10), BV will remain optimal if c2≥0, and BV will be suboptimal if c2<0.
From preceding, we know that cBVB-1=[0 10 10].
5)30(
5.1
2
6
101002c
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Illustrate in example
Thus, c2≥0 holds if 5-Δ≥0,or Δ≤5.This means that if the price of tables is de
creased or increased by $5 or less, BV remains optimal.
If BV remains optimal after a change in a nonbasic variable’s objective function coefficient, the values of the decision variables and the optimal z-value remain unchanged.
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Illustrate in example For example, if the price of tables increase to
$33(c2=33), the optimal solution to the Dakota problem remains unchanged.
On the other hand, if c2>35, BV will no longer be optimal, because c2<0.
For example, if c2=40,
540
5.1
2
6
101002
c
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Illustrate in example
Now the BV “final” tableau is, Basic
variables ratio z -5x2 +10s2+10s3=280 z=280
-2x2 +s1+ 2s2 -8s3=24 s1=24 none
-2x2+x3 +2s2 -4s3=8 x3=8 none
x1+1.25x2 -0.5s2+1.5s3=2 x1=2 1.6*
This is not an optimal tableau, and we can increase z by making x2 a basic variable in row 3.
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Illustrate in example
Changing the objective function coefficient of a basic variable
In the Dakota problem, x1 and x3 are basic variables. Since we are not changing B or b, (6) shows
that the right-hand side of each constraint will remain unchanged, and BV will remain feasible.
Since we are changing cBV. From (10), this may change more than one coefficient in row 0.
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Illustrate in example
The Dakota problem,Suppose that c1 is changed from 60 to 60+Δ.
5.1105.0100
5.15.00
420
821
60200
5.15.00
420
821
25.00
45.10
811
1
1
Bc
BB
BV
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Illustrate in example Noting that,
The coefficient of each nonbasic variable in the new row 0 is,
20,30,60,
5.0
5.1
1
,
5.1
2
6
,
2
4
8
321321
cccaaa
5.110 ofelement second0 rowin oft coefficien
5.010 ofelement second0 rowin oft coefficien
25.1530
5.1
2
6
5.1105.0100
13
12
221
2
Bcs
Bcs
caBcc
BV
BV
BV
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Illustrate in example
Thus, row 0 of the optimal tableau is now z+(5+1.25Δ)x2+(10-0.5Δ)s2+(10+1.5Δ)s3=?
BV will remain optimal if anf only if the following hold:
5+1.25Δ≥0 Δ≥-4
10-0.5Δ≥0 Δ≤20
10+1.5Δ≥0 Δ≥-20/3
the current basis will remain optimal if and only if -4≤Δ≤20.
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Illustrate in example
Thus, as long as 56=60-4≤c1≤60+20=80, the current basis remains optimal.
Suppose c1=70, z=70x1+30x2+20x3.
revenue should increase by 2(10)=$20, and new revenue=280+20=$300.
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Illustrate in example If c1<56 or c1>80, the current basis is no longer op
timal. Suppose c1=100(Δ=40), how can we determine th
e new optimal solution?
360
8
20
48
701000 row of side handright
705.1100 rowin t coefficien
105.0100 rowin t coefficien
00 rowin t coefficien
0,5525.15,0
1
3
2
1
321
bBc
s
s
s
ccc
BV
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Illustrate in example
The BV tableau is , Basic
variables ratio z +55x2 -10s2+70s3=360 z=360
-2x2 +s1+ 2s2 -8s3=24 s1=24 12
-2x2+x3 +2s2 -4s3=8 x3=8 4*
x1+1.25x2 -0.5s2+1.5s3=2 x1=2 none
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Illustrate in example
Changing the right-hand side of a constraint Effect on the current basis
Since b does not appear in (10), changing the right-hand side of a constraint will leave row 0 of the optimal tableau unchanged;
changing a right-hand side cannot cause the current basis to become suboptimal.
Will affect the right-hand side of the constraints in the optimal tableau.
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Illustrate in example
Note As long as the right-hand side of each
constraint in the optimal tableau remains nonnegative, the current basis remains feasible and optimal.
If at least one right-hand side in the optimal tableau becomes negative, the current basis is no longer feasible and therefore no longer optimal.
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Illustrate in example
Suppose we change the amount of finishing hours (b2) in the Dakota problem.
Currently,b2=20, we change b2 to 20+Δ, from (6)
The current basis will remain optimal if and only if the following hold:
24+2Δ≥0 Δ≥-12
8+20Δ≥0 Δ≥-4
2-0.50Δ≥0 Δ≤4
5.02
28
224
8
20
48
5.15.00
420
821
8
20
481B
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Illustrate in example
For -4≤Δ≤4, 20-4≤b2≤20+4, or 16≤b2≤24, the current basis remains feasible and therefore optimal.
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Illustrate in example
Effect on decision variables and zEven if the current basis remains optimal, the values of
the decision variables and z change.
The new values of the basic variables are as follows
From (11),
1
12
28
8
22
48
5.15.00
420
8211
1
3
1
bB
x
x
s
300
8
22
48
10100) new( New 1
bBcvaluez BV
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Illustrate in example
When the current basis is no longer optimalSuppose we change b2 to 30. from (6), the right-
hand side of the constraints in the tableau for BV={s1,x3,x1} is
From (11), the right-hand side of row 0 is
3
28
44
8
30
48
5.15.00
420
8211bB
380
8
30
48
101001
bBcBV
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Illustrate in example
The tableau is, Basic
variables
z +5x2 +10s2+10s3=380 z=380
-2x2 +s1+ 2s2 -8s3=44 s1=44
-2x2+x3 +2s2 -4s3=28 x3=28
x1+1.25x2 -0.5s2+1.5s3=-3 x1=-3
Since x1=-3, BV is no longer feasible or optimal.
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Illustrate in example
Changing the column of a nonbasic variable x2 is a nonbasic variable in the optimal solution,
this means that Dakota should not manufacture any tables at present.
Suppose, the price of tables increased to $43 and, because of changes in production technology, a table required 5 board feet of lumber, 2 finishing hours and 2 carpentry hours.
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Illustrate in example
Here we are changing elements of the column for x2 in the original problem. The change leaves B and b unchanged. Thus, the right-hand side of the optimal tableau remains unchanged.
From (10),the only part of row 0 that is changed is c2; the current basis will remain optimal if and only if c2≥0 holds.
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Illustrate in example
Use (10) to compute the new coefficient of x2 in row 0.
Since c2<0, the current basis is no longer optimal. c2=-3 means that each table now increases revenues by $3.
0343
2
2
5
10100
2
2
5
and 43
10100
2
22
1
c
ac
BcBV
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Illustrate in example
From (5),
The tableau is now, Basic
variables
z -3x2 +10s2+10s3=280 z=280
-7x2 +s1+ 2s2 -8s3=24 s1=24
-4x2+x3 +2s2 -4s3=8 x3=8
x1+2x2 -0.5s2+1.5s3=2 x1=2*
2
4
7
2
2
5
5.15.00
420
821
21aB
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Illustrate in example
Adding a new activity Suppose that Dakota is considering
making footstools. A stool sells for $15 and requires 1 board foot of lumber, 1 finishing hour, and 1 carpentry hour. Should the company manufacture any stools?
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Illustrate in example
Our new initial tableau is z -60x1 -30x2 -20x3-15x4 =0
8x1 +6x2 +x3 +x4+s1 =48
4x1 +2x2+1.5x3 +x4 +s2 =20
2x1+1.5x2+0.5x3 +x4 +s3=8
We call the addition of the x4 column to the problem adding a new activity.
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Illustrate in example
From (6), we see that the right-hand sides of all constraints in the optimal tableau will remain unchanged.
From (10), we see that the coefficient of each of the old variables in row 0 will remain unchanged.
Compute c4, the current basis will remain optimal if c4≥0.
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Illustrate in example
Since
The current basis is still optimal.This means that each stool manufactured will
decrease revenues by $5. for this reason, we choose not to manufacture any stools.
515
1
1
1
10100
1
1
1
and 15
4
44
c
ac
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Summary of sensitivity analysis
Change in initial problem
Effect on optimal tableau
Current basis is still optimal if
Changing nonbasic objective function coefficient cj
Coefficient of xj in optimal row 0 is changed
Coefficient of cj in row 0 for current basis is still nonnegative
Changing basic objective function coefficient cj
Entire row 0 may change
Each variable still has a nonnegative coefficient in row 0
Changing right-hand side of a constraint
Right-hand side of constraints and row 0 are changed
Right-hand side of each constraint is still nonnegative
Changing the column of a nonbasic variable xj or adding a new variable xj
Changes the coefficient for xj in row 0 and xj’s constraint column in optimal tableau
The coefficient of xj in row 0 is still nonnegative