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 plant viruses assignment help-homework help by online classification of virus tutors
Other Types of Bonding tutorial all along with the key concepts of Metallic bonding, Explaining the Physical Properties of Metals, Hydrogen bonding, Hydrogen Bonds in Biology, Other intermolecular forces, London dispersion forces, Dipole-dipole interactions
tutorsglobe.com dependence of dissociation constant assignment help-homework help by online chemical equilibrium tutors
Want first-class Human Neuropsychology Assignment Help at your doorstep? Approach us and get benefitted with A++ grades.
Two broad approaches to equity maintenance have raced for acceptance - Maintaining the owners’ investment, Maintaining business operations
tutorsglobe.com poverty and environment assignment help-homework help by online environmental science tutors
tutorsglobe.com cell as a physiological unit assignment help-homework help by online plant physiology tutors
Lagrange and Hamilton Formulation of Mechanics tutorial all along with the key concepts of D'Alembert's Principle, Generalized Equations of Motion, Generalized Conservative Forces, Euler-Lagrange Equations, Lagrange's Mechanics, Conservative Systems, Generalized Coordinates, Generalized Momenta, 2-D motion with time-varying constraint
A resource will only be considered as an asset and involved on the statement of financial position if it can be calculated in monetary terms, along with a appropriate degree of reliability.
Hire Environmental Management Assignment Help tutors and obtain non-plagiarized papers from industry best experts at viable prices!
tutorsglobe.com the visual pathway assignment help-homework help by online receptor organs tutors
Air-borne nematode tutorial all along with the key concepts of Enterobius vermicularis, human pin-worm, Morphology of Enterobius vermicularis, Life cycle of Enterobius vermicularis
www.tutorsglobe.com offers Advantages and Disadvantages of Data Dictionary homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
tutorsglobe.com method of work assignment help-homework help by online mendels experiments tutors
Theory and lecture notes of Nonlinear Systems-Newtons Method all along with the key concepts of ector Notation, Linear Approximation for Vector Functions, Newton’s Method. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Nonlinear Systems-Newtons Method.
1957517
Questions Asked
3689
Tutors
1461810
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!