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 decision-making 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 post-optimality 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. For a deeper understanding, check out Understanding Linear Programming Problems in Decision Making.
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. To learn more about how opportunity costs affect resource allocation, see Understanding Scarcity and Opportunity Cost in Economics.
- 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 Right-hand Side of Constraints: This examines how alterations in constraints, such as resource availability, impact the optimal solutions. You can explore more about these constraints with Understanding Linear Programming Problems Using Graphical Method and Excel Solver.
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 decision-making:
- 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. For a comprehensive approach to graphical solutions, consider exploring Mastering Linear Programming: A Step-by-Step Guide to Graphical Solutions.
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 decision-making in business management.
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dear student in the previous lecture i have discussed about the graphical solutions using a graphical calculator
and also i have discussed about how to use solver in this lecture we can discuss
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
post optimality analysis so sensitivity analysis is the study of
how the changes in the coefficient of an optimization model affect the optimal solution there are two way it will
affect how will a change in a coefficient of objective function affect the optimal
solution other one is how will a change in right hand side of a constraint affect the
optimal solution in this lecture we will discuss about the first case
that is if there is if there is any change in coefficient of objective function
how our solution optimal solution get affected
this was the problem what we have used right from the beginning maximize
ten years plus nine d there are four constraint so this problem
will ah i will explain how to do sensitivity analysis what is the center to analysis we are talking about if any
changes in the coefficient of this objective function for example if the 10 may be changed
the tennis is the profit contribution obviously the profit contribution will not be same
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
to that is a deluxe bags is equal to 252. so
based on the profit contribution fig figures of ten dollar per standard bag and nine
dollar per deluxe bag suppose we later learned that
a price reduction causes the profit contribution of the standard back fall from
10 dollar to say 8.5 dollar so sensitivity analysis
can be used to determine whether the production schedule calling for 540 standard bags
and 252 deluxe bags is still best if there is a drop in price
our solution that is s equal to 540 and d equal to 252 is the still best or where are this drop
in profit contribution will affect our final result or not that is the purpose
of sensitivity analysis if it is solving a modified linear programming problem
by changing the coefficient of objective function as the new objective function will not
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
the whole process is not required so the management would especially want to consider
how the optimal production quantity quantities should be revised if the profit contribution per
deluxe bag were to drop so that is if the price of the product is dropping
how that will affect our our production quantities there is another situation where
the sensitivity analysis can be done suppose look at the right hand side of the constraint 630 in the cutting and
dying department the total available hours is 630 that is what i have
brought here the another aspect of sensitivity analysis concern changes in the right hand side value of
the constraint in the problem the optimal solution
used all available time in the cutting and dying department and finishing department
you might have recollect from our previous previous lectures so this 630 hours
this constraint is our binding constraint what is the meaning of binding constraint we have utilized all
630 hours this is happening for to the finishing department also that's how i
brought in different colors so what would happen to the optimal solution
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 final solution or that will maximize or that will
affect our objective function whether it is a maximization or minimization type this is another way of doing sensitive
analysis just i am recollecting there are two way to do the sensitivity
analysis one is by changing the coefficient of the objective function
the second one is by changing the value on the right hand side of our constraints
so sensitivity analysis can help determine how much each additional hour
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
and sometime what will happen when you keep on adding the resources instead of it is
ah positively affecting our objective function sometime it will decrease so by
adding more resources will not help for our objective function so that is the case of this diminishing returns but in
this class we discuss about the first case that is if any changes in the
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
is ten dollar per unit for the standard bag and nine dollar per unit for the deluxe
bag it seems obvious that an increase in the profit
contribution for one of the bags might lead management to increase the production of that bag
and a decrease in the profit contribution for one of the bags might lead
management to decrease the production of that bag so the profit contribution is increases
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
would have to change before the management would want to
change the production quantities so what is the meaning of this we know that already we got the result
that is s equal to five hundred and forty standard bags and deluxe bag is two
fifty two if there is any small change in bag price
right sometime the small change in bag price will not affect your production quantities
but if the bag price is significantly changing so what it is required sometime you have
to change this production quantities also so what we are going to do in this one
how much allowable change is permitted how much how much profit contribution how much how much change in profit
contribution is permitted so that our solution remain changed so
i am going to introduce the term called range of optimality the current optimal solution to this problem call for
producing 540 standard golf bags and 252 deluxe golf bags the range of optimality for
each objective function coefficient provides the range of values over which the
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
profit contribution so this much change is permitted
so this change will not affect your final solution that is called range of optimality
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
range range of optimality and the coefficient near the end
end point of the range obviously when then when the the range is very narrow
the management should provide more attention for that because immediately your solution need to be
changed because if there is slight variation in the product
the product price that may affect your scheduling so
with this coefficient a small change can necessitate modifying the optimal solution
now i will explain this range of optimality graphically we know that this region is our feasible
region which we have discussed in our previous lectures this point
is our where we got the optimal solution so now we will go back to the problem
the graphical solution of the problem with the slope of objective function line between
slope of line a and b the extreme point three is optimal so
this point is optimal point for the given problem see there are i am writing what is line
a and line b the cutting and dying department is called that line is i am calling it line a
the finishing department corresponding constraint i am calling it as line b ok
so the range of optimality so we have the objective function objective function is
this n ah which is mentioned in the orange color this is 10 s plus 90.
so as long as the slope of the objective function is between the line a and b
right as long as the slope of our objective function is between line and b
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
cutting and dying constraint line and the slope of line b which coincides with finishing
constraint line the extreme point 3 with s equal to 540
d equal to 252 will remain optimal so changing an objective function
coefficient for s r d will cause the slope of objective function line to change
if there is any change in the coefficient of s and d that
will affect the slope of that objective function in figure we see that such changes cause
the objective function line to rotate around the extreme point three so if any changes in the coefficient of
objective function that will cause rotating the objective function
however as long as the objective function line stays within the shaded region the
extreme point three will remain optimal what is the shaded region this region
this region rotating the objective function counter clockwise
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
objective function line rotates counter clockwise because the slope is increased
enough to coincide with the line a so suppose this line
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
any further counter clockwise rotation of the objective function line will cause
extreme point 3 to be non optimal so beyond this beyond this red line if you rotate this
orange line beyond that so what will what will become then the solution point three will not be the
optimal solution hence the slope of line a right the slope of line a provides an
upper limit for the slope of the objective function line now we will see the other case
rotating the objective function line clockwise ok now the objective objective function
line is going to be rotated clockwise causes the slope to become more negative and slope decreases
when the objective function line rotates clockwise then slope will decrease
enough to coincide with line b what is line b which is in the red color
line b we obtain alternate optimal solution between three
and two so what will happen in this region your objective objective
function line will sit there so when it is coinciding so all these points between two and three will become
your optimal solution you will get multiple optimal solution
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
as long as slope of line b is less than or equal to
slope of line objective function line is less than or equal to slope of line a
what it says that if the slope of your objective function line
is between slope of line b and a your optimal solution will remain same
that will not disturb your optimal solution in given figure which is given on the right hand side we see that
the equation for line a line a is the this one line a that is the cutting and
dying constraint line is like this
what is that line because it is equal to written seven upon ten years plus one d equal to
six hundred and thirty so i want to know the slope of this equation
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
to bring in this form y equal to m x plus b so that i can find out what is the slope
so when you simplify d ok you will get minus seven upon ten s
plus six hundred and thirty so the minus seven upon ten says the slope of your line a
the 630 is the intercept of line a on d d axis
ok similarly we will find the slope of line b
in figure three in figure we see that the equation for line b is this one one s plus
two upon three d equal to seven hundred and eight so two upon
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
line b now i am going to write our objective function in the generic form
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
so when i simplify c d equal to minus c s e s plus p whole divided by
d so if i further simplify i get d equal to minus c s by c d s
plus p by c d we hold the profit contribution for the deluxe bag
fixed at at initial value c d equal to 9 we know that it is 10 s
plus 9 d so i am going to fix
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
thirteen point five similarly if i take the right hand side inequality
minus cs by nine less than or equal to minus seven upon ten so i am getting the cs value six point
three so what we got i have got the coefficient value for the standard bag
between 6.3 to 13.5 so what this number says that
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
in the original problem the standard back had a profit contribution of 10 dollar
so the resulting optimal solution was s equal to 540 d equal to 252
so the range of optimality for c s tells for the given problem
with other coefficients unchanged what is other coefficient the c d
c cd is unchanged thats what we have substituted cd equal to 9 the profit contribution for the standard
bag can be anywhere between 6.3 dollar and 13.5 dollar
and the production quantities of 540 standard bags and 252 deluxe bags will remain optimal
that is the interpretation of in the what i the slope which i have written in the previous slide
even though the production quantities will not change the total profit contribution
that is the value of objective function will change the value of objective function will
change due to change in profit contribution of standard backs so what is that our objective function z
equal to 10 s plus 90. so the value of this
coefficient of standard biax even though it is goes up to 13.5 and 6.3
our solution which we got it s equal to 540 d equal to 252 will remain same that is
the interpretation of this range of optimality so range of optimality for standard bags
similarly we can find the range of optimality for the deluxe bag also that is what we have done it here
the same equations so here c s by c d we know that c s by
c d minus so the c s value i have substituted equal to 10 because our
objective function is 10 s plus 9 d
so i am going to find out what is the allowable range for the coefficient of deluxe bags
by assuming that the coefficient of standard bag is 10. so the 10 i have substituted
and then i have readjusted on the left hand side so i got 6.67 is lower limit
and the upper limit is fourteen point two nine so what this number
implies that as long as the coefficient of deluxe bags this one
goes up to six point six seven and fourteen point two nine our solution that is our
s equal to 540 d equal to 252 remain same now how to see this range of optimality in our
solver output so i have brought the output of the solver in the variable cell you see that
the final value that is our answer this is objective function coefficient 10 and 9
so here the allowable increase is 3.5 that is 13.5 allowed decrease
three point seven similarly for the deluxe bag the
currently it is nine it can be increased ah nine plus 5.2
and decreased by 9 minus 2.3 so this is the place where you can get
the range of optimality so the range of optimality is providing
kind of your flexibility in the price changes even though the price changes we need
not bother about that because that will not affect our our optimal solution now i will show this
output in our excel sheet now this problem which we have solved in the
previous class we got the answer is 540 and 252 when i go to
this answer report you have to go to sensitivity report look at this location this location that
is final value is our answer there is a objective function there is
allowable increase and allowable decrease so this says the range of optimality
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
vertical see this green color saves our given given our
objective function sometime if the objective function is vertical there will be either no upper limit or
no lower limit for the slope as it appears in the form of expression
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
nine d say this is the new objective function in this case
the the extreme point two so now this point two provides the optimal solution
so rotating this objective function line counter clockwise counter clockwise
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
coincides with the line b o this is our line b we showed previously
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
clockwise result in the slope becoming more and more negative approaching a value of
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
holding c d constant and the original value that is c d equal to nine
so we have minus c s by nine less than or equal to minus three by two
so when you simplify this we are getting c s is thirteen point five in reviewing this figure we note that
the extreme point two remains optimal for all values of c s about thirteen point five
thus we obtain the following range of optimality for cs at extreme point so
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
simultaneous changes what is the meaning of simultaneous changes
coefficient of standard bags and coefficient of deluxe bags so simply
compute the slope of the objective function minus c s by c d for the new coefficient
values if this ratio is greater than or equal to the lower
limit on the slope of the objective function r
less than or equal to the upper limit then the changes made will not cause a change in optimal solution
if there is a simultaneous change you have to find the ratio of minus cs by cd
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
now both the things are happening together recall that changes for optimality for c
s and c d both computed in one at a time manner the way we have arrived c s the range of
optimality because we have kept the coefficient of one
for example c s is kept constant we found the answer for the d similarly then d kept constant so we got
the answer for yes but now there is a simultaneous changes
so c s is 6.3 to 13.5 now c d is 6.67 to 14.29
so for these ranges of optimality we can conclude that
changing either cs to 13 dollar rcd to 8 but not both
any one changes can take place would not cause change in optimal solution s equal to 540
d equal to 252 even though there are two changes are taking place
but we are considering only one change at a time
if there is any one change taking place at a time both the
changes are within the limit that will not affect our our optimal solution
but we cannot conclude from the ranges of optimality that changing both coefficients
simultaneously would not result in change in the objective function
so what will happen if there is simultaneous changes so that will affect our
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
the number line 0 minus so as long as this ratio is within this limit
that will not disturb our optimal solution if the c s is changed to 13
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
will get minus thirteen upon eight so minus one point six two five so this minus one
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
optimal solution the solution which you are already are having that will be disturbed
because it is going beyond the ranges yeah minus one point six two five because minus
one point sixty five is less than the lower limit of minus 3 by 2 the current solution
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
not both if both the coefficients are changing at a time we have to resolve the problem
range of optimality simultaneous changes looking at the ranges of optimality we conclude that
changing either cs to 13 or cd to 18 but not both would not cause change in the optimal
solution but in a recomputing the slope of the objective function with simultaneous
changes for both c s and c d we
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
we got will not be valid
now we will summarize what we have learnt in this class we have learnt
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
how that will affect our optimal solution ok student the next class
we will learn how a change in right side value of constraint affect the optimal solution
so far we have seen if any changes in the coefficient of objective function how data will affect
our optimal solution but in the next lecture if the right hand side of the constraint
is changing how that will affect our optimal solution
thank you [Music] [Applause]
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hey [Music] you
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