Understanding Sensitivity Analysis in Linear Programming
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Introduction
In the world of optimization, sensitivity analysis plays a crucial role in understanding how changes in certain coefficients of an optimization model can affect the optimal solution. This analysis is particularly relevant for decisionmaking processes in management where profit contributions, resource allocations, and product pricing can vary. In this article, we will delve into the concept of sensitivity analysis, particularly focusing on how variations in the coefficient of the objective function can influence the optimal solution, alongside a discussion of range of optimality.
What is Sensitivity Analysis?
Sensitivity analysis, also known as postoptimality analysis, is the study of how the changes in the coefficients of an optimization problem affect the solution's feasibility and optimality. Understanding sensitivity analysis is essential, especially in linear programming, as it helps predict how changes in parameters can influence results.
Importance of Sensitivity Analysis
 Evaluation of Optimal Solution Stability: It informs managers whether their current optimal production schedules remain valid under new conditions.
 Resource Allocation: Helps in determining the best allocation of resources when profit contributions change.
 Risk Assessment: Aids in identifying risks associated with changes in input values and helps develop strategies to mitigate those risks.
Components of Sensitivity Analysis
Sensitivity analysis considers two primary aspects:
 Change in Coefficient of Objective Function: This involves understanding how variations in profit contributions per product affect overall production quantities and profitability.
 Changes in Righthand Side of Constraints: This examines how alterations in constraints, such as resource availability, impact the optimal solutions.
Change in Coefficient of Objective Function
Case Study: Profit Contribution Dynamics
Consider a linear programming model aiming to maximize profit derived from two types of products: standard bags and deluxe bags with initial profit contributions of $10 and $9, respectively. The problem emphasizes how a decrease in the profit contribution for standard bags from $10 to $8.5 affects the current optimal solution of producing 540 standard bags and 252 deluxe bags.
Analyzing the Profit Contributions

Understanding Profit Contribution Changes:
 When the profit contribution increases, the production of that item is likely to also increase.
 Conversely, if the profit contribution decreases, production may need to be reduced.

Sensitivity of Optimal Solutions:
 If the new optimal profit contribution remains within certain thresholds, the production schedule remains optimal. For example, calculating the allowable decrease in profit contribution can help determine if adjustments are necessary in production schedules.
 By deriving the range of optimality, we understand the flexibility in profit contribution allowed without altering the optimal output.
Range of Optimality
The range of optimality is defined as the allowable variation in the coefficients of the objective function that ensures the current optimal solution remains unchanged. Understanding this range helps in effective decisionmaking:
 Upper Limit: Maximum allowable increase in profit contribution that does not necessitate changing the production strategy.
 Lower Limit: Minimum allowable reduction in profit contribution that still retains the optimal solution.
Calculating the Range of Optimality
Assuming a $10 profit contribution:
 The upper limit might be $13.5, while the lower might be $6.3. Hence, the optimal solution remains valid as long as the profit contribution stays within this range.
 For the deluxe bags, the corresponding values must also be evaluated using similar methods to ascertain their impact.
Effect of Simultaneous Changes
While sensitivity analysis allows us to evaluate the impact of one variable changing at a time, simultaneous changes can complicate matters:
 Individual Changes: If only one coefficient is altered, there remains a clear interpretation of the range of optimality.
 Combined Changes: In scenarios where both the contribution of standard bags and deluxe bags change simultaneously, the overall effect needs to be recalculated, as it may lead to a different optimal solution.
Case Study: Combined Contribution Changes
Suppose the profit contribution for standard bags rises to $13 while deluxe bags drop to $8. Evaluating the new objective function
 The relationship derived from the coefficients needs assessment to see if the new ratios fall within established limits determined from prior analyses. If they exceed those limits, the optimal solution would drastically change, requiring a full resolution of the model.
Summary
In this lecture, we explored sensitivity analysis's vital role in optimization, focusing particularly on how changes to the coefficient of an objective function might reflect on the optimal solutions. By defining the range of optimality, we determined how minor tweaks to production contributions can lead to varying effects on output decisions. Moreover, while analyzing simultaneous changes in coefficients, we underscored the necessity of recalibrating our models to ascertain the latest optimal state. Overall, sensitivity analysis remains an indispensable tool for effective operational decisionmaking in business management.
dear student in the previous lecture i have discussed about the graphical solutions using a graphical calculator
sensitivity analysis the agenda for this lecture is sensitivity analysis in that if there is any change in in the
coefficient of objective function how that will affect our result so the another name for sensitivity analysis is
how the changes in the coefficient of an optimization model affect the optimal solution there are two way it will
will ah i will explain how to do sensitivity analysis what is the center to analysis we are talking about if any
sometime the profit contribution will be more sometime the profit contribution will be less
but what will be effect of this profit contribution on our final solution i have brought the same problem here
the optimal solution for the given problem which is on the right hand side s equal to 540 standard bags and d equal
our solution that is s equal to 540 and d equal to 252 is the still best or where are this drop
be necessary so obviously you may ask this question if the profit contribution is decreasing
we can simply change that coefficient value in the excel then we can get the final answer but the recall redoing of
how the optimal production quantity quantities should be revised if the profit contribution per
the sensitivity analysis can be done suppose look at the right hand side of the constraint 630 in the cutting and
brought here the another aspect of sensitivity analysis concern changes in the right hand side value of
this constraint is our binding constraint what is the meaning of binding constraint we have utilized all
and the total profit contribution if that company could obtain additional quantities of either of the resources
so what is the meaning here suppose if the company is trying to bring some additional resources
they say plus one something by bringing this additional resources we need to check whether that will
affect our objective function whether it is a maximization or minimization type this is another way of doing sensitive
of production time is worth and how many hours can be added before diminishing return certain
so what is the meaning of this is if i add one extra resources whether the adding this extra resources
is worth or not how much it is worth how much it is worth means how that will affect my affect my objective function
adding more resources will not help for our objective function so that is the case of this diminishing returns but in
coefficient of the objective function how that will affect our final result let us consider how the changes in the
objective function coefficient might affect the optimal solution the current contribution to the profit
contribution for one of the bags might lead management to increase the production of that bag
obviously will company go for manufacturing more bags if it is decreases they will try to reduce the
number of quantities manufactured it is not as obvious however how much the profit contribution
change the production quantities so what is the meaning of this we know that already we got the result
how much allowable change is permitted how much how much profit contribution how much how much change in profit
i am going to introduce the term called range of optimality the current optimal solution to this problem call for
current solution remain optimal i think this picture will explain you clear idea you see this is our coefficient of our
objective function so there is a lower limit for profit contribution that is upper limit for
even though there is a variation in the coefficient of your objective function that variation will not disturb your
solutions that allowable variation the range of variation is nothing but your range of optimality
the managerial attention should be focused on those objective function coefficient that have narrow
the management should provide more attention for that because immediately your solution need to be
now i will explain this range of optimality graphically we know that this region is our feasible
a and line b the cutting and dying department is called that line is i am calling it line a
that will not affect our result so that will provide your range of optimality as long as the slope of objective
function line is between slope of line a see that which if it is slope of line a is this one which coincides with the
the objective function line to rotate around the extreme point three so if any changes in the coefficient of
suppose the objective function is in orange color suppose if you rotate this objective function counterclockwise
the slope to become less negative ok then the slope increases ok so when you rotate this way when the
coincide with line a we obtain alternative optimal solutions between extreme points three and four
what will happen if this orange line is coinciding with this red line so what will happen in this region
ok see all the points which are in this region will be your optimal solution so we will get alternate optimal solution
orange line beyond that so what will what will become then the solution point three will not be the
line is going to be rotated clockwise causes the slope to become more negative and slope decreases
function line will sit there so when it is coinciding so all these points between two and three will become
any further clockwise rotation of the objective function line will cause extreme point 3 to be non optimal so if
you rotate beyond this so 3 will not be your optimal solution hence the slope of line b provides a
lower limit for the slope of the objective function line so now we got the upper limit and lower limit
of your objective function line so that the point 3 will remain optimal so the extreme point 3 will be optimal solution
that will not disturb your optimal solution in given figure which is given on the right hand side we see that
so i am going to write rearrange we know that very very gen y equal to m x plus b where m is the slope
here y is nothing but your d number of deluxe bags so i am going to write this this equation
three d i am simplifying when i brought on the right hand side minus s plus seven zero eight
so when you simplify this d equal to minus three by two s plus one thousand sixty two now this is your slope of your
so p equal to c s s plus c d d c s means coefficient of standard back c d means coefficient of deluxe back
this value of c d equal to nine we know that minus three by two less than or equal to minus c s by c d
so instead of c d i have substituted the value nine less than or equal to minus seven by ten
so when i simplify suppose i take only on the left hand side when i simplify this minus three by two
less than or equal to minus c s by nine so when minus get cancelled so the value of c s what i am getting is
minus cs by nine less than or equal to minus seven upon ten so i am getting the cs value six point
as long as the coefficient of standard bag is within this range that will not disturb our final answer
there is a final solution so that will not affect our optimality so how we are interpreting
c cd is unchanged thats what we have substituted cd equal to 9 the profit contribution for the standard
that is the interpretation of in the what i the slope which i have written in the previous slide
change due to change in profit contribution of standard backs so what is that our objective function z
similarly we can find the range of optimality for the deluxe bag also that is what we have done it here
not bother about that because that will not affect our our optimal solution now i will show this
this answer report you have to go to sensitivity report look at this location this location that
now we can see the a special case in range of optimality in cases where the rotation of the
objective function line about an optimal extreme point causes the objective function line is become
objective function sometime if the objective function is vertical there will be either no upper limit or
so we know that minus three by two less than or equal to minus c s by c d less than or equal to minus seven by ten
to show this special situation can occur suppose the objective function for the problem is eighteen c s plus
this way rotating in counter clockwise around the extreme point 2 provides an upper limit for the slope
when the objective function line coincides with the line b so rotating the objective function line
counter clockwise around extreme point 2 provides an upper limit for the slope when the objective function line
the slope of line b is minus three upon two so the upper limit for the slope of the objective function line must be
minus three by two there is upper limit there is no problem however rotating the objective function
in clockwise suppose if you rotate this objective function this objective function clockwise
minus infinity as the objective function line become vertical in this case the slope of the objective
function has no lower limit using the upper limit of minus 3 upon 2 we can write minus c s by c d less than or
equal to minus 3 by 2 that is the slope of our objective function following the previous procedure of
so when you simplify this we are getting c s is thirteen point five in reviewing this figure we note that
the val the lower limit for this coefficient of standard bag is 13.5 and there is no upper limit this is the
special case of optimality when the objective function is vertical now there may be another situation
less than or equal to the upper limit then the changes made will not cause a change in optimal solution
if that ratio is within the limit so our final result will not change so consider changes in both of the
objective function coefficients for the problem which we are discussing suppose the profit contribution
for the standard bag is increased to 13 and the profit contribution per deluxe bag is simultaneously reduced to eight
for example c s is kept constant we found the answer for the d similarly then d kept constant so we got
range of optimality simultaneous changes we showed that the extreme point 3 remains optimal as long as
ok minus 3 upon 2 less than or equal to minus c s by c d minus 7 by 7 upon 10. so i brought in the
and simultaneously c d is changed to 8 the new objective function slope will will be given by minus c s by c d so we
point six two by uh minus 1.625 will go on this side so what will happen if both the things
are changes are taking place simultaneously so you will not get the optimal suit that is you will get
s equal to 540 and d equal to 252 will now will no longer be optimal so in this case by resolving the problem with
the c s equal to thirteen and c d equal to eight we will find the extreme point two is the new optimal solution now this
is two is the new optimal solution so what we are learning from here if there is a simultaneous changes then we have
to resolve the problem so sensitivity analysis what we have discussed so far is any one value changes at a time
range of optimality simultaneous changes looking at the ranges of optimality we conclude that
saw that the optimal solution did change so what we are learning if there is a simultaneous changes
our optimal solution will change so the result emphasize this fact that the range of optimality by itself
can only be used to draw a conclusion about changes made to one objective function coefficient at a
time if both the coefficients are changing simultaneously the range of optimality the answer which
how will a change in the coefficient of objective function affect the optimal solution then we have learnt a new term
called range of optimality then we have seen if there is a simultaneous changes of coefficient of objective function
so far we have seen if any changes in the coefficient of objective function how data will affect