Degeneracy
Degeneracy refers to the concept of getting a degenerate basic feasible solution in a LPP. The degeneracy in a LPP may occur
At the starting stage, when at least one basic variable is zero in the initial basic feasible solution.
At any following iteration when more than one basic variable is suitable to leave the basic and therefore one or more variables becoming zero in the subsequent iteration and the problem is said to be degenerate. There is no guarantee that the value of the objective function will get better, as the new solutions may stay degenerate. Consequently, it is possible to repeat the identical sequence of simplex iterations continuously without improving the solutions. This concept is called as cycling or circling.
Rules to avoid cycling
Divide every element in the tied rows with the positive coefficients of the key column in that particular row.
Compare the resultant ratios, column by column, first of all in the identity and then in the body, from left to right.
The row which firstly consists of the smallest algebraic ratio now contains the leaving variable.
Example 1
Max Z = 3x1 + 9x2
Subject to
x1 + 4x2 ≤ 8
x1 + 2x2 ≤ 4
& x1 ≥ 0, x2 ≥ 0
Answer
Standard LPP
Max Z = 3x1 + 9x2 + 0s1 + 0s2
x1 + 4x2 + s1 = 8
x1 + 2x2 + s2 = 4
x1 , x2 , s1, s2 ≥ 0
Cj→
3
9
0
Basic Variables
CB
XB
X1
X2
S1
S2
XB / XK
S1 / X2
s1
8
1
4
1/4
s2
2
0/2→
Z = 0
-3
↑
-9
←Δj
-1
x2
1/2
Z =18
3/2
9/2
As all Δj ≥ 0, optimal basic feasible solution is achieved. Thus the solution is Max Z = 18, x1 = 0, x2 = 2
Note - As there is a tie in minimum ratio (degeneracy), we determine minimum of s1 /xk for these rows for which the tie exists.
Example 2
Max Z = 2x1 + x2
4x1 + 3x2 ≤ 12
4x1 + x2 ≤ 8
4x1 - x2 ≤ 8
Max Z = 2x1 + x2 + 0s1 + 0s2 + 0s3
4x1 + 3x2 + s1 = 12
4x1 + x2 + s2 = 8
4x1 - x2 + s3 = 8
x1 , x2 , s1, s2, s3 ≥ 0
Basic Varibles
S3
S1 / X1
S2 / X1
12
12/4=3
8/4=2
4/0=0
s3
0/4=0→
-2
4/4=1
0→
x1
-1/4
-
Z = 4
-3/2
-1/2
1/8
16
3/4
0 4
1 2
2 3/2
-1/8
3/8
Z = 5
As all Δj ≥ 0, optimal basic feasible solution is achieved. Hence the solution is Max Z = 5, x1 = 3/2, x2 = 2
tutorsglobe.com b. pyrophosphoric acid assignment help-homework help by online compounds of phosphorus tutors
tutorsglobe.com haemocytometer assignment help-homework help by online medical lab methods tutors
tutorsglobe.com plant viral disease assignment help-homework help by online viruses tutors
tutorsglobe.com entomophily assignment help-homework help by online agents of pollination tutors
tutorsglobe.com treasury management assignment help-homework help by online working capital management tutors
www.tutorsglobe.com offers chemical reactivity homework help, chemical reactivity assignment help, online tutoring assistance, organic chemistry solutions by online qualified tutor's help.
tutorsglobe.com ionisation potential assignment help-homework help by online chemical periodicity tutors
The statement of financial position and the income statement should not be viewed in any way like substitutes for one another. Rather they should be observed like performing different roles.
Historical Viewpoint tutorial all along with the key concepts of Histories of Cells Discoveries, Bridge between Life and Non-life, Protoplasmic Constituents and The Neuron Theory
tutorsglobe.com classification of herpes virus assignment help-homework help by online herpes viruses tutors
tutorsglobe.com leaf-spines assignment help-homework help by online leaf modification tutors
Theory and lecture notes of Theory of Expected Utility all along with the key concepts of theory of expected utility, Petersburg Paradox, Nicholas Bernoulli. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Theory of Expected Utility.
tutorsglobe.com difference between chemical and nuclear reactions assignment help-homework help by online nuclear reaction tutors
input devices — “how to tell it what to do” for instance, mouse and keyboard are the standard way to interact with the computer. other devices that are joysticks and game pads employed mainly for games.
Breeding methods tutorial all along with the key concepts of Mode of reproduction, Self fertilizing crops or autogamous crops, Mass selection, Selection of cross-pollinated crops, Mass selection, Recurrent selection, Breeding of Asexually Propagated Crops
1943807
Questions Asked
3689
Tutors
1443713
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!