Post-optimal analysis of LPP
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Post-Optimal AnalysisLinear optimization problem
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Contents
Introduction to Post-Optimal Analysis
Changes Affection Feasibility
Changes in the right-hand side (b)
Addition of a new constraint
Changes Affecting Optimality
Changes in the Objective Coefficients (c_j)
Addition of new activity (x_j)
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Post-Optimal Analysis
The sensitivity of the optimum solution
By determining the ranges for the different LP
parameters that keeps the optimum basic variable
unchanged
Deal with making changes in the parameters of the model
and finding the new optimum solution
These changes require periodic re-calculation of the
optimum solution and
The new computations are rooted in the use duality
and the primal-dual relationships
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Post-Optimal Analysis
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Example : Assembling 3 types of toys
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Example : Assembling 3 types of toys
Optimum tableau for the primal is
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Changes Affection Feasibility
Two possibilities which can affect feasibility
are
Changes in the right-hand side (b)
A new constraint is added
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The dual problem uses exactly the same parameters as the primal problem, but in different location.
Primal and Dual Problems
Primal Problem Dual Problem
Max
s.t.
Min
s.t.
n
j
jj xcZ1
,
m
i
ii ybW1
,
n
j
ijij bxa1
,
m
i
jiij cya1
,
for for.,,2,1 mi .,,2,1 nj
for.,,2,1 mi
for .,,2,1 nj ,0jx ,0iy
RHS means
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Changes in the right-hand side (b)
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Changes in the right-hand side (b)
New RHS of the problem
The current basic variable remain feasible at the new values and
The optimum revenue is $1880
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Changes in the right-hand side (b)
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Changes in the right-hand side (b)
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Addition of a new constraint
The new constraint for the operation 4 is
3x1 +3x2 + x3 ≤ 500
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Addition of a new constraint
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Addition of a new constraint
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Addition of a new constraint The tableau shows the x7 = 500 which is not consistent with the values
of x2 and x3 in the rest of the table….
Reason: the basic variable x2 and x3 have not been substituted out in
the new constraint…
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Addition of a new constraint
Application of the dual simplex method will produce the new optimum solution
x1 = 0, x2 = 90, x3 = 230, and z = $1370 (verify!)
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Changes Affecting Optimality
The changes in the Objective
Coefficients
The addition of a new
economic activity (variable)
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Changes in the Objective Coefficients
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Changes in the Objective Coefficients
Affect only the optimality of the solution and require recomputing the
z-row coefficients (reduced costs):-
1) Compute the dual values using the Method 2
2) Substitute the new dual values in Formula 2, to determine the new
reduced costs (z-row coefficients).
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Recapitulate :
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Recapitulate :
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Changes in the Objective Coefficients
Maximize z = 2x1 + 3x2 + 4x3 is the new objective
function then,
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Changes in the Objective Coefficients
Maximize
z=
2x1 +
3x2 +
4x3
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Changes in the Objective Coefficients
Maximize z = 6x1 + 3x2 + 4x3 is the new objective function then,
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Changes in the Objective Coefficients
The optimum solution :- x1 = 102.5 x2 = 215 and z = $12270.50 (verify)
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Addition of new activity
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Addition of new activity A new activity signifies adding a new variable to the model
Intuitively, the new activity desirable only if it is profitable
Formula 2 will help in checking this
To compute the reduced cost of the new variable
Let x7 represents the new product in the TOYCO and
Revenue per new toy is $4
Operation 1 1 minute
Operation 2 1 minute
Operation 3 2 minutes The new column in the
coefficient matrix of the
constraints
4x_7 will be the extra term in the
primal objective function
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Addition of new activity
Given that (y_1, y_2, y_3) = (1,2,0) are the optimal dual variable,
Reduce cost of x7 is
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Addition of new activityNot Optimal but
feasible solution
The new optimum is obtained by letting x7 enter the basis and x6 must leave the basis…..
The new optimum is x1 = 0, x2 = 0, x3 = 125 and x7 = 210 with the revenue z = $1465
(verify)
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Thanks for your attention