Decision making is an integral part of operations management. It may be useful to a decision maker to have some indication of how sensitive an alternative choice might be to the changes in one or more of those values. Unfortunately, it is not possible to explore all the possible combinations of all the variables in a typical problem. In spite of this, there are some elements that a decision maker can use to assess the sensitivity of assumption probabilities. One of the tools useful for the analysis in some decision making problems is sensitivity analysis. It provides a range of feasibility over which the choice of alternative remains the same. Successful decision making consists of several steps, the first and most important being carefully defining the problem. Given that linear problems can be extensive and complex, they are solved by using sophisticated computer methods. This paper will present software solutions available for personal computers (Lindo, POM). For a manager taking the decision, however, a solution model is only part of the answer. Sensitivity analysis offers a better understanding of the problem, different effects of limitations and “what if“ questions. The insights obtained are frequently much more valuable that a specific numerical answer. One of the advantages of linear programming lies in the fact that it provides rich information on sensitivity analysis as a direct part of the solution.feasibility range, linear programming, Lindo, POM, optimum solution, optimum range, sensitivity analysis.

Martina Bris

English

Research Papers in Economics

287
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION 
MAKING TOOL
Martina Briš, B.Sc.(Econ)
Faculty of Economics in Osijek
SUMMARY
Decision making is an integral part of operations management. It may be useful to a 
decision maker to have some indication of how sensitive an alternative choice might be to 
the changes in one or more of those values. Unfortunately, it is not possible to explore all 
the possible combinations of all the variables in a typical problem. In spite of this, there 
are some elements that a decision maker can use to assess the sensitivity of assumption 
probabilities. One of the tools useful for the analysis in some decision making problems is 
sensitivity analysis. It provides a range of feasibility over which the choice of alternative 
remains the same.
Successful decision making consists of several steps, the ﬁ  rst and most important 
being carefully deﬁ  ning the problem. Given that linear problems can be extensive and 
complex, they are solved by using sophisticated computer methods. This paper will present 
software solutions available for personal computers (Lindo, POM). For a manager taking 
the decision, however, a solution model is only part of the answer. Sensitivity analysis 
offers a better understanding of the problem, different effects of limitations and “what 
if“ questions. The insights obtained are frequently much more valuable that a speciﬁ  c 
numerical answer. One of the advantages of linear programming lies in the fact that it 
provides rich information on sensitivity analysis as a direct part of the solution. 
Key words: feasibility range, linear programming, Lindo, POM, optimum solution, 
optimum range, sensitivity analysis.
1. ON SENSITIVITY ANALYSIS
Sensitivity analysis is a process of varying input parameters of a model within allowed 
area and observing the resulting changes in the model solution. It explores how changes 
in the model output can be qualitatively and quantitatively attributed to different change 
sources.
The  purpose  of  sensitivity  analysis  is  to  indicate  the  sensitivity  of  simulation  to 
uncertainties in the values of input data in the model.
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL288
Sensitivity analysis attempts to establish how the model depends on the attributed 
values, on its structure and on the assumptions it is based on. It is an important method 
for testing the quality of a given model, and it is also used for checking the reliability of 
the analysis.
Sensitivity analysis was originally devised to deal only with uncertainties in input 
variables and model parameters. Over time the analysis was expanded, insecurity was 
included also in the model structure, its assumptions and speciﬁ  cations. Generally speaking, 
sensitivity analysis is used to increase the reliability of a given model and its predictions 
by providing understanding of how model variables react to the input changes, whether 
they are data used for adjustments in the model structure, or factors, i.e. independent 
model variables. 
Sensitivity analysis can be used to determine the following:
￿  Similarity of the model with the observed process,
￿  Quality of the model deﬁ  nition,
￿  Factors contributing the most to output variables,
￿  Areas in the input factor space for which there is maximum variation,
￿  Optimum area in the space of factors, which are used in subsequent exploration 
of adjustments,
￿  Interaction between factors.1
Using sensitivity analysis is popular in ﬁ  nancial applications, risk analysis, neural 
networks, and in all the areas where models are developed. Sensitivity analysis can also 
be used in model based policy assessment studies.
Sensitivity analysis can be useful in different cases: when identifying key assumptions, 
or when comparing alternative model structures, when initiating future data collections, 
establishing important criteria, optimizing resource allocation, simplifying a model, etc.
2. LINEAR PROGRAMMING
Linear programming methods are the most important instrument in operations research, 
and one of the disciplines in mathematical optimizing.
Each linear programming method consists of the following:
a)  LINEAR OBJECTIVE FUNCTION – The objective of optimizing comes to the 
foreground here. It can be maximizing proﬁ  ts, but also minimizing costs. The 
objective is chosen depending on the problem structure. The objective function is 
a set that should meet the requirements of the decision maker.
1  Sensitivity  analysis  –  Wikipedia,  The  free  encyclopedia,  http://en.wikipedia.org/wiki/sensitivity_analysis 
(20.01.2007.)
Martina Briš289
b)  LINEAR RESTRICTIONS – They come from the environment of the decision 
maker  and  impose  certain  restrictions  /  conditions  in  terms  of  achieving  the 
objective. The real limitation is the product of multiplication of time required 
to make one unit with the quantity. This product of multiplication must not be 
bigger than available capacity. The restrictions are always speciﬁ  ed in the form of 
equations or inequations.
c)  CONDITIONS FOR VARIABLE NON NEGATIVITY – for example, negative 
quantities of products cannot be produced.
THE GENERAL MATHEMATICAL FORMULA FOR LINEAR PROGRAMMIN
IS AS FOLLOWS: 2
  max Z =  n nx c x c x c + + + ... 2 2 1 1
      1 1 1 2 1 2 1 1 1 ... ... b x a x a x a x a n n j j ≤ + + + + +
   2 2 2 2 2 2 1 2 1 ... ... b x a x a x a x a n n j j ≤ + + + + +
     M           M                 M                  M         M
             m n m n j m j m m b x a x a x a x a ≤ + + + + + ... ... 2 2 1 1
     0 , 0 , 0 2 1 ≥ ≥ ≥ n x x x
where:
m – is the number of restrictions,
 n – is the volume of particular activities,
xj – is the unknown activity j (structural variable),
aij – are technical coefﬁ  cients which indicate how many units of a resource    
       (restriction) are being spent for producing a unit xj,
bi – is the size of the restricting factor,
            cj – is the proﬁ  t per unit j.
2 Barković, D.: Operacijska istraživanja, Ekonomski fakultet u Osijeku, Osijek, 2002, p. 58.
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL290
2.1. Computer software LINDO and POM for Windows
In order to explain sensitivity analysis, we shall use here the following problem in 
linear programming:
  max Z =  2 1 500 300 x x +
                  170 2 2 1 ≤ + x x
        + 1 x   150 2 ≤ x
               180 3 2 ≤ x
                 0 2 , 1 ≥ x
To achieve the optimum solution we have used the widely available software for linear 
programming LINDO and POM for Windows.
The name LINDO is an acronym for Linear INteractive Discrete Optimization. The 
term “discrete“ means skipping from one to another basic solution instead of circling 
within the permissible area seeking the optimum solution (if there is one).3 
This software has an emphasis on operations research. It is a specialized programme 
which  covers  the  topics  such  as  linear/non linear  programming  and  combinatorial 
optimizations. It is not really intended for solving managerial problems; rather, it is more 
suitable as a learning tool for the methods of ﬁ  nding solutions in operations research, as 
well as for learning how to formulate an optimization problem in mathematical terms. 
This software could not be easily incorporated into the curriculum of the Operations 
Management course. 
In contrast, software intended for studying operations management covers the same, if 
not a larger scope of material, and is generally well suited for solving the problems from 
one or several course books dealing with operations management. 
The software with an emphasis on operations management is called POM for Windows. 
The name POM is the acronym for Production and Operations Management.
This programme offers elementary solutions for a wide range of operating problems. 
It contains several modules connected to operations research. The additional modules are 
assignation, linear programming, simulation, transport, etc. 
Within the POM software, the user ﬁ  rst has to select the desired module, or the window 
for entering the data will not open. After that, parameters are selected that will be used in 
the formulation, and ﬁ  nally, the data are entered.
3 Barković, D.: Operacijska istraživanja u investicijskom odlučivanju, Ekonomski fakultet u Osijeku, Osijek, 2004., str. 
23.
Martina Briš291
3. AN EXAMPLE OF SENSITIVITY ANALYSIS
Both  software  programmes  presented  here  use  a  simplex  table.  The  majority  of 
information required for sensitivity analysis is contained in the ﬁ  nal simplex table. This 
and the data from the initial problem is sufﬁ  cient to perform an analysis.
Figures 1 and 2 indicate the solutions provided by those two software programmes.
Figure 1. SOLUTION WITH A COPIED PROBLEM LP FROM THE INITIAL WINDOW (LINDO  
                PROGRAMME)
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL292
Figure 2. TWO WAYS OF SHOWING THE SOLUTION (POM for Windows)
After running the programme, the computer printout of sensitivity analysis for this 
particular problem shows us that the optimum solutions are: 
x1 = 130, and   x2 = 20
at which the maximum proﬁ  ts of  $49,000.00 are achieved.
OPTIMUM SOLUTION is that combination of products which can result in the highest 
proﬁ  t, i.e. the lowest cost (depending on the optimization objective).
As stated above, sensitivity analysis is a means for assessing the scope and effectiveness 
of potential changes on parameters of LP model.
a) COEFFICIENT CHANGES IN THE LINEAR OBJECTIVE FUNCTION
OPTIMUM RANGE is the range of values over which optimum values of decision 
Martina Briš293
making variables will not be changed. (upper part of the table Ranging)
We differentiate LOWER BOUND and UPPER BOUND which indicates that the 
optimum solution will not change if the proﬁ  t coefﬁ  cient is moving between the lower 
and upper limit. 
 
      x1   250   500
      x2  300   600
The optimum solution remains, x1 = 130, and x2 = 20, however, the proﬁ  t is no longer 
$49,000.00. If the proﬁ  t per unit x1 is increased from 300 to 350, the total proﬁ  t is then:
  350 × 130 + 500 × 20 = $55,500.00
since we receive  $50.00 of additional proﬁ  t per product unit x1.
Let us assume that we have to report to our supervisor on the quantities of particular 
products our company should produce in order to maximize proﬁ  ts. Our superviser could 
pose the following problem for us: our engineers have found the way of increasing the 
proﬁ  t for the product x2 from $500.00 to $550.00, and we should think about this and 
propose a new product mix. We do not really need time for considering this and can 
answer immediately. The optimum product mix remains x1 = 130, and x2 = 20 because 
the proﬁ  t of $550.00 is within optimality limits (which is observable from the LOWER 
– UPPER BOUND). The proﬁ  t will be changed as we are now earning additional 50 
dollars per product x2.
The proﬁ  t will increase by 50 × 20 = $1,000.00. 
We cannot use sensitivity analysis to evaluate the changes in several coefﬁ  cients 
simultaneously.
b) CHANGES IN RESTRICTION VALUES
FEASIBILITY RANGE is the range of values for RHS (right hand side) restrictions 
over which dual prices remain the same. 
Dual values or shadow prices are deﬁ  ned as the amount at which the values of the 
objective function will be changed with the change of one unit in the value of restriction. 
Dual values are marginal values for restrictions in our problem. A marginal value is the 
amount we are prepared to pay for an additional resource unit. This is not the original 
sales price for a resource but the price we need to pay in order to get more of a resource. 
(lower part of the table Ranging)
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL294
Dual values indicate that we will not be willing to engage an additional unit of restriction 
3, since we are left with 120 units already when producing the optimum product mix.
  0 × 130 + 3 × 20 ≤ 180
               60 ≤ 180
These are UNTIED RESTRICTIONS which have either slack or surplus4. An additional 
unit of such a restriction would result only in the increase of slack or surplus, and not in 
the increase of proﬁ  ts (or decrease of costs).
Unlike untied restrictions, TIED RESTRICTIONS force us to go beyond the optimum 
point. They have a certain amount for dual values (different from 0), and slack/surplus 
is 0. An additional unit of this restriction can enable us to increase the total proﬁ  t (or 
decrease the total cost) by ﬁ  nding a new optimum solution.
Dual values 200 and 100 indicate that we will be willing to pay $200.00 for each 
additional unit of restriction 1, and $100.00 for each additional unit of restriction 2. We 
should not be prepared to pay more than $200.00 for the restriction 1, because if we get 
one unit of restriction 1 more, the proﬁ  t will rise by only $200.00. Marginal values help in 
determining resource prices and making decisions on the purchase of new resources.
The purchase of additional units will change the optimum solution.
Dual values are not viable for unlimited supply of restrictions. From LOWER – UPPER 
BOUND we can see that the dual value for restriction 1 is valid in the range from 150 to 
210.
We cannot use the feasibility range to evaluate the changes on more than one restriction 
simultaneously.
Let us assume that our boss says that he can acquire 30 additional units of restriction 
1 for $75.00 per unit. We would immediately advise him to buy all the 30 units since the 
quantity to date, i.e. 170 + 30 = 200, which is within lower and upper limit. Each additional 
unit within feasibility range adds $200.00 of proﬁ  t. The total increase in proﬁ  ts will be 
 
30 × ($200.00   $75.00) = $3,750.00
On the other hand, if our boss were to tell us he can acquire 20 additional units of 
restriction 3 for only $0.50, we would advise him not to buy any additional units of 
restriction 3 as 180 + 20 = 200, which is within feasibility range, and in that case the 
4 SLACK is an ADDITIONAL VARIABLE, it is an unused resource. SURPLUS is  the production surplus.
Martina Briš295
dual value is 0. We already have 120 units more than we need (slack), and if we buy 
additional 20 units, our surplus will rise to 140 units.
The restriction is 180, and we have:
       60     =>     SLACK
          220     =>     SURPLUS
4. CONCLUSION
Decision  makers  are  often  concerned  with  more  things  than  just  a  solution  to  a 
problem. He or she could beneﬁ  t from the results of sensitivity analysis in several ways. 
While preparing the problem for processing by means of linear programming, a decision 
maker  could  use  subjective  assumptions  for  particular  parameters  (e.g.  restriction 
coefﬁ  cients, objective function coefﬁ  cients and values of right hand side restrictions). 
It is understandable that a decision maker wishes to know how sensitive the optimum 
solution is to the changes in values of one or more parameters. If the solution is relatively 
insensitive to reasonable changes, the decision maker can have more conﬁ  dence in the 
application of his solution. In contrast, if the solution is sensitive to changes, he or she 
will want to determine much more precise assumptions on the basis of parameters in 
question. 
The problem related to sensitivity analysis includes speciﬁ  c changes in parameters. For 
example, there can be a change in price which can alter the objective function coefﬁ  cient. 
Furthermore, a change in the production process can require a change in the coefﬁ  cient 
of one of the restrictions. Sensitivity analysis can provide an adequate insight into the 
problem for the decision maker. The third possibility is the question whether the purchase 
of high quantities of a resource would be beneﬁ  cial. This question often arises when large 
quantities of resources are not used in the optimum solution; by adding more of other 
resources it is possible to achieve a disproportionate increase in the objective function 
value.
There are obviously numerous reasons why a decision maker may want to use sensitivity 
analysis.
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL296
5. REFERENCES
1. Barković, D.: Operacijska istraživanja, Ekonomski fakultet u Osijeku, Osijek, 
    2002.
2. Barković, D.: Operacijska istraživanja u investicijskom odlučivanju, Ekonomski 
    fakultet u Osijeku, Osijek, 2004.
3. Bonini, Ch. P., Hausman, W.H., Bierman, H.: Quantitative Analysis for 
    Management, Irwin/McGraw Hill, New York, 1997.
4. Stevenson, W. J.: Production/Operations Management, Irwin, New York, 1993.
INTERNET SITES:
  http://www.lindo.com  (17 January 2007)
  http://en.wikipedia.org/wiki/sensitivity_analysis   (20 January 2007)
  http://sensitivity analysis.jrc.ec.europa.eu/forum/docs/Overview.html  (20 Janu
            ary 2007)
  http://eu.wiley.com  (22 January 2007)
  http://www.llnl.gov/casc/nsde/sensitivity.html  (25 January 2007)
  http://www.hearne.com.au  (25 January 2007)
  http://www.indiana.edu/~statmath/math/lindo/overview.html   (30 January 2007)
Martina Briš

Operacijska istraživanja u investicijskom odlučivanju, Ekonomski fakultet u Osijeku,

Operacijska istraživanja, Ekonomski fakultet u Osijeku,

Quantitative Analysis for Management,

Sensitivity Analysis as a Managerial Decision Making Tool

http://www.efos.hr/repec/osi/journl/PDF/InterdisciplinaryManagementResearchIII/IMR3a9.pdf

Sensitivity Analysis as a Managerial Decision Making Tool

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