Topic: Post-Optimality Analysis
By: -
1.
2.
Malik Faizan
Ishtiaq Ahmad
1.
2.
Post-Optimality Analysis constitutes a very major and
important part of most operations research studies,
particularly in typical linear programming applications.
Suppose an optimum solution to the given LPP is obtained,
it is necessary to study the changes in the current optimum
solution due to the changes in the parameters of the LPP.
The changes in the parameters of the LPP are of two
types:
Discrete
Continuous
This is focuses on the effect of the discrete changes in
the parameters on the optimum solution, which is called
post-optimality Analysis. This analysis also sometimes
referred to as “what-if analysis” because it studies what
would happen to the optimum solution if different
assumptions are made about future conditions. On the
other hand, this chapter concentrates on the continuous
changes in parameters, which are called parametric
programming.
 For
example we take the example of poultry
industry where an LP model is commonly
used to determine the optimal feed mix per
broiler. The weekly consumption per broiler
varies from 0.26 lb (120 grams) for a oneweek-old bird to 2.1 lb (950 grams) for an
eight-week-old bird. Additionally, the cost of
the ingredients in the mix may change
periodically. These changes require periodic
recalculation of the optimum solution. PostOptimal Analysis determine the new solution
in an efficiently way.
 Feasibility
Analysis, is an analysis of the
viability of an idea. It describes a preliminary
study undertaken to determine and
document a project’s viability. The results of
this analysis are used in making the decision
whether to proceed with the project or not.
 The
feasibility of the current optimum solution my
be affected only if
 (1) the right hand side of the constraints is
changed.
 (2) a new constraint is added to the model.
In above both cases, infeasibility occurs when at
least one element of the right-hand side of the
optimal tableaus become negative. That is one or
more of the current basic variaibel become
negative.
 This
change requires recomputing the right-hand
side of the tableau using the following formula: -
 Recall
that the right-hand side of the tableaus gives
the values of the basic variables
Addition of New Constraints
The addition of a new constraint to an existing
model can lead to one of the two cases:_
1. The new constraint is redundant, meaning that it
is satisfied by the current optimum solution, and
hence can be dropped from the model altogether.
2. The current solution violates the new constraint,
in which case the dual simplex method is used to
restore feasibility.

Changes Affecting Optimality

1.
2.
This considers two particular situations that
could affect the optimality of the current
solution:
Changes in the original objective
coefficients.
Addition of the new economic activity
(variable) to the model.
Changes in the Objective Function
Coefficients:

1.
2.
The Changes affect only the optimality of
the solution. Such changes thus require
recomputing the z-row coefficients
(reduced cost) according the following
procedure:Compute the dual value.
Use the new dual values.
 In
this question, we examine how changes in
the coefficients of the objective function
change the optimal solution. If one of the
coefficients of the objective function
increases or decreases, the feasible region
remains the same. Since the feasible region
is defined by the constraints and not the
objective function, there are no changes to
the corner points. A change in one of the
coefficients changes the slope of the isoline,
the line that corresponds to setting the
objective function equal to a constant.
 The
objective function takes on the optimal
value P 3450 at the corner point (15, 20).
 The isoprofit line 70x +120y = 3450 just
touches the feasible region at that point
verifying this assertion.
 As
the coefficient on x is decreased, the
isoprofit line for the optimal solution gets
less steep. At a value of 40, the isoprofit line
passes through (0, 25) and (15, 20). At this
point, the isoprofit line forms the border of
the feasible region there so all points on the
line between (0, 25) and (15, 20) yield the
same maximum profit. If the coefficient is
decreased to a value less than 40, (15, 20) is
no longer part of the optimal solution.

As the coefficient on x is increased, the isoprofit line for
the optimal solution gets steeper. At a value of 120, the
isoprofit line passes through (0, 25) and (25, 10). The
isoprofit line forms the border of the feasible region there,
so all points on the line between (0, 25) and (25, 10) yield
the same maximum profit. If the coefficient is increased to
a value greater than 120, (15, 20) is no longer part of the
optimal solution.


As long as the coefficient on x in the objective function falls
between 40 to 120, the optimal solution includes x, y
15,20 . If the coefficient is equal to 40 or 120, any ordered
pair on a line connecting 15, 20 to one of the adjacent
corner points is optimal. However, if the coefficient on x falls
outside of 40 to 120 the optimal solution moves to a new
corner point.
Recall that the coefficient on x in the objective function is
the profit for each frame bag. If the profit per frame bag
were to increase by up to $50 (to $120) or decrease by up to
$30 (to $40), the optimal solution would not change. This
solution, 15 frame bags and 20 panniers, is fairly insensitive
to changes in the coefficient on x. Even though the location
of the optimal solution does not change over this range, the
profit does not change. Notice that we are only varying one
coefficient at a time. Varying both coefficients
simultaneously is beyond the scope of this presentation
Thank You