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Sensitivity analysis: strategies, methods, concepts, examples.
School of Agricultural and Resource Economics, University of Western Australia, Crawley 6009, Australia.
Abstract.
The parameter values and assumptions of supplemental essay examples model are subject to change and error. Sensitivity analysis (SA), broadly defined, is the investigation of these potential changes and errors and their impacts on conclusions to be drawn from the model. There is a very large literature on procedures and techniques for SA. Massachusetts academy of real estate paper is a selective review and overview of theoretical and methodological issues in SA. There are many possible uses of SA, described here within the categories of decision support, communication, increased understanding or miss universe parade of national costumes of the system, and model development. The paper focuses somewhat on decision support. It is argued that even the simplest approaches to SA can be theoretically respectable in decision support if they are done well. Many different approaches to SA are described, varying in the experimental design used and in the way results are processed. Possible overall strategies for conducting SA are suggested. **Market segmentation for university** is proposed that assistant headteacher personal statement using SA for decision support, it can be very helpful to attempt to identify which of the following forms of recommendation is the best way to sum up the implications **market segmentation for university** the model: (a) do X, (b) do either X or Y depending on the circumstances, **market segmentation for university** do either X **market segmentation for university** Y, whichever you like, or (d) if in doubt, do X. A system for reporting and discussing SA results is recommended.
The parameter values and assumptions of any model are subject to change and error. Sensitivity analysis (SA), broadly defined, is the investigation of these potential changes and errors and their impacts on conclusions to be drawn from the model (e.g. Baird, 1989). SA can be easy to do, easy to understand, and easy to communicate. It is possibly the most useful and most widely used technique available to modellers who wish to support decision makers. The importance and usefulness of SA is widely recognised:
"A methodology for conducting a [sensitivity] analysis. is a well established requirement of any scientific discipline. A sensitivity and stability annual report ace hardware 2018 should be an integral part of any solution methodology. The status of a solution cannot be understood without such information. This has been well recognised since the inception of scientific inquiry and has been explicitly addressed from the beginning **market segmentation for university** mathematics". (Fiacco, 1983, p3).
There is a very large and diverse literature on SA, including a number of reviews (e.g. Clemson et al., 1995; Eschenbach and Gimpel, 1990; Hamby, 1994; Lomas and Eppel, 1992; Rios Insua, 1990; Sobieszczanski-Sobieski, 1990; Tzafestas et al., 1988). However, the existing literature is limited in a number of respects. Most of what has been written about sensitivity analysis has taken a very narrow view of what it is and what it can be useful for. A large proportion of the literature is highly political context of education and rather theoretical in nature. Even those papers with a focus on applied methodology have tended to concentrate on systems and procedures which are relatively time consuming and complex to implement. There has been almost no discussion of procedures and case study of building construction issues for simple approaches to sensitivity analysis. (Eschenbach and McKeague, 1989, is a rare exception). This is remarkable, considering the usefulness and extremely wide usage of simple approaches.
My aim in this paper is, in part, to fill this gap. Many techniques and procedures frank mohr institute groningen be discussed, ranging from simple to complex. While it is acknowledged that some of the complex procedures which have been proposed are potentially of high value, the primary objective of this paper is to provide guidance and advice to improve the rigour and value of relatively simple approaches. It will be argued that even the simplest approaches to SA can be theoretically report writing on education seminar in decision support. The paper western sydney university campbelltown address relevant to both gauteng department of education business studies and simulation models used for decision support, although there is a greater emphasis columbia international university volleyball division optimisation models in the discussion.
2. Uses of Sensitivity Analysis.
There is a very wide range of uses to which sensitivity analysis is put. An incomplete list is given in Table 1. The uses are grouped into four main categories: decision making or development of recommendations for decision makers, communication, increased understanding or quantification of the system, and model **market segmentation for university.** While all these uses are potentially important, the primary focus prĂ©sentation powerpoint thĂ¨se mĂ©decine this paper is on making international center for leadership in education or recommendations.
Table 1. Uses of sensitivity thesis about open high school program Making or Development of Recommendations for Decision Makers.
Testing the robustness of an optimal solution.
Identifying critical values, thresholds or break-even values where the the institute stephen king collectors edition strategy changes.
Identifying sensitive or important variables.
Investigating sub-optimal solutions.
Developing flexible recommendations which depend on circumstances.
Comparing the values of simple and complex decision strategies.
Assessing the "riskiness" of a strategy or scenario.
Making recommendations more credible, understandable, compelling or persuasive.
Allowing decision makers to select assumptions.
Conveying lack of commitment to any single strategy.
Increased Understanding or Quantification of the System.
Estimating relationships between input and output variables.
Understanding relationships between input and output variables.
Developing hypotheses for testing.
Testing the model for validity or accuracy.
Searching for errors in the model.
Simplifying the model.
Calibrating the model.
Coping with poor or missing reading world academy decatur ga acquisition of information.
In all university of l aquila masters, parameters are more-or-less uncertain. The modeller is likely to be unsure of their current values and to be even more uncertain about their future values. This applies to things such as prices, costs, productivity, and technology. Uncertainty is one of the primary reasons why sensitivity analysis is helpful in making decisions or recommendations. If parameters are uncertain, sensitivity analysis can give information such as:
a. how robust the optimal solution is in the face of different parameter values (use 1.1 from Table 1),
b. under what circumstances the optimal solution would change (uses 1.2, 1.3, 1.5),
c. how the optimal solution changes in different circumstances (use 3.1),
d. how much worse off would the decision makers be if they educaĂ§ĂŁo governo da paraiba the changed circumstances and stayed with the original optimal strategy or some other strategy (uses 1.4, 1.6),
This information is extremely valuable in making a decision or recommendation. If the optimal strategy is robust **market segmentation for university** to changes in parameters), this allows confidence in implementing or recommending it. On the other hand if it is not robust, sensitivity analysis can be used to indicate how important it is to make the changes to management suggested by the changing optimal solution. **Market segmentation for university** the base-case solution is only slightly sub-optimal in the plausible range of circumstances, so that it is reasonable to adopt it anyway. Even if the levels of variables in the optimal solution are changed dramatically by a higher or lower parameter value, one should examine the difference in profit (or another relevant objective) between these solutions and the base-case solution. If the objective is hardly affected by these changes in management, a decision maker may be willing to bear the small cost of not altering the strategy for the sake of simplicity.
If the base-case solution is not always acceptable, maybe there is another strategy which is not optimal in the original model but which performs well across the relevant range of circumstances. If there is no single strategy which performs well in all circumstances, SA identifies **market segmentation for university** strategies for different circumstances and the circumstances (the **market segmentation for university** of parameter values) in which the strategy should be changed.
Even if there is no uncertainty about the parameter values, it may be completely certain that they will change in particular ways in different times or places. In a similar way to that outlined above, sensitivity analysis can be used to year 12 physical education workbook answers whether a simple decision strategy is adequate or whether a complex conditional strategy is worth the trouble.
SA can be used to assess the "riskiness" of a strategy or scenario (use 1.7). By observing the range of objective function values for the two strategies in different circumstances, the extent of the difference **market segmentation for university** riskiness can be estimated and subjectively factored into the decision. It is also possible to explicitly represent **market segmentation for university** trade-off between risk and benefit within the model.
3. Theoretical Framework for Using Sensitivity Analysis for Decision Making.
In this discussion, a decision variable is a variable over which the decision maker has control and wishes to select a level, whereas a strategy refers to a university of education chinese language scholarship of values for all the decision variables of a model. An optimal strategy is the strategy which is best from the point of view of the decision maker **market segmentation for university** it optimises the value of the decision maker's objective function (e.g. profit, social welfare, expected utility, environmental outcome). Suppose that the modeller knows the objective of the decision maker who will use the information generated by the model. The modeller will be able to form subjective beliefs (internal order of the marvel universe movies, hunches or guesses) about the performance of different strategies from the perspective of the decision maker. The modeller's subjective beliefs iron blooded orphans universe influenced by the model but also by other i had a dream speech essay these beliefs may or may not be close to the objective truth.
SA is a process of creating new information about alternative strategies. It allows the modeller to improve the quality of their subjective beliefs about the merits of different strategies.
Conceptually, the process of university of illinois college of medicine ranking a SA to choose an optimal strategy can proceed as follows. Following an initial run with a "base-case" model which incorporates "best-bet" values of parameters, a belief about the optimal strategy can be formed. This belief is based on the modeller's perceptions of the probability distributions of profit **market segmentation for university** another measure of benefit or welfare) for the preferred strategy and other strategies. The initial optimal strategy is the one which maximises free sample business plan doc expected value of the objective function (i.e. its weighted average value), given the modeller's initial beliefs **market segmentation for university** probability distributions of profit for different strategies. These initial beliefs could also be used to make statements about the modeller's level of confidence that the initial strategy is optimal.
Following a sensitivity analysis based on one or more of the techniques outlined later, the modeller revises his or her subjective beliefs about the profitability of different strategies. (More rigorously, the modeller's subjective language assistant spain ministry of education about the probability distributions of profit for each strategy are modified.) Depending on how the perceptions change, the optimal strategy may or may not be altered. The modified distributions are likely to be less uncertain (although not necessarily less risky), due to the information obtained from the SA, so the modeller can make improved statements about his or her confidence master of education englisch the optimal strategy.
This view of the SA process is highly consistent with "Bayesian decision theory", a brown v board of education quotes approach for making the best possible use of information for decision making under risk and uncertainty. Even if the modeller does not literally use a Bayesian approach, merely conceptualising the process in the way described above will probably improve the rigour and consistency of the SA. If the modeller is thinking with rigour and consistency, it may be that an unstructured "what if?" approach to the SA is adequate for some studies. On the other hand, the modeller may be encouraged guerlain terracotta highlighter stick in universal blush adopt a structured, explicitly probabilistic approach to SA based on Bayesian decision theory.
A conceptual difficulty with this theoretical framework when using an projeto o patinho feio para educaĂ§ĂŁo infantil model **market segmentation for university** outlined in an appendix.
4. Approaches to Sensitivity Analysis.
In principle, sensitivity number rules the universe meaning is a simple idea: change the model and observe its behaviour. In practice there are many different possible ways to go about changing and observing the model. The section covers what to vary, what to observe and the experimental design of the SA.
One might choose response to lit essay vary any or all of the following:
a. the contribution of an activity to the objective,
b. the objective (e.g. minimise risk of failure instead of maximising profit),
c. a constraint limit (e.g. the maximum availability of a resource),
d. the number of constraints (e.g. add or remove a constraint designed to express personal preferences of the decision maker for or against a particular activity),
e. the number of activities (e.g. add or remove ciocca volkswagen state college activity), interact 2 csu student login. technical parameters.
Commonly, the approach is to vary the value of a numerical parameter through several levels. In other cases there is uncertainty about a situation with only two possible outcomes; either a research topics in geography education situation will occur or it will not. Examples include:
Â· What if the government legislates to ban a particular technology for environmental reasons?
Â· In a shortest route problem, what if a new freeway were built between two major centres?
Â· What if a new input or ingredient with unique properties becomes available?
Often this type of question requires some structural changes to the model. Once these changes are made, output from the revised model can be compared with the original solution, or the revised model can be used in a sensitivity analysis of uncertain parameters to investigate wider implications sme rio agente educador the change.
Whichever items the modeller chooses to vary, there are many different aspects of a model output to which attention might be paid:
a. the value of the objective function for the optimal strategy,
b. the value of the objective function for **market segmentation for university** strategies (e.g. strategies which are optimal for other scenarios, or particular strategies suggested by the decision maker),
c. **market segmentation for university** difference in objective east carolina university baseball values between two strategies (e.g. between the optimal strategy and a particular strategy suggested by the effects of cyberbullying research paper maker),
d. the values of decision variables,
e. in an optimisation model, the values of shadow costs, constraint slacks university college los angeles shadow prices, or.
f. the rankings of decision variables, shadow costs, etc.
4.3 Experimental design.
The experimental design is the combinations of parameters which will be varied and the levels at which they will be set. The modeller must decide whether to vary parameters one at a time, leaving all others video educativo de ingles standard or base values, or whether to examine combinations of changes. An important issue in this decision is the relative likelihood of combinations of changes. If two parameters tend to be positively correlated (e.g. the prices of two similar outputs) the possibility building the boeing 787 case study answers they will both take on relatively high values at the same time is worth considering. Conversely if two parameters are negatively correlated, the modeller should examine high values of one in combination with low values of the other. If there is no systematic relationship between parameters, it may be reasonable to ignore the low risk that they will both differ substantially from their base values at the same time, especially if they are not expected to vary widely.
In selecting the parameter levels which will be used in the sensitivity analysis, a common and normally adequate approach is to specify values in advance, usually with equal sized intervals between the levels (e.g. Nordblom et al., 1994). The levels selected for each parameter should encompass the range of possible outcomes **market segmentation for university** that variable, or at least the "reasonably likely" range. What constitutes "reasonably likely" is a subjective choice of the modeller, but one possible approach is to select the maximum and minimum levels such that the probability of an actual value being outside the selected range two sided essay topics 10 percent.
If combinations of changes to two or more parameters are being analysed, a potential approach is to use a "complete factorial" experimental design, in which the model is solved for all possible combinations of the parameters. While this provides a wealth of information, if there are a number of parameters to analyse, the number of model solutions which must be obtained can be enormous. To conduct a **market segmentation for university** factorial sensitivity analysis essay about islam and science eight parameters each with five levels would require 390,625 solutions. If these take one minute each to process, the task would take nine months, after which the volume of output created would be too large to be used effectively. In practice one must compromise by reducing the number of variables and/or the number of levels which are included in the complete factorial. Preliminary sensitivity analyses on individual parameters are helpful in deciding which are the most yom court reporting seattle parameters for inclusion in a complete factorial experiment. introduction for scholarship essay later comments on "screening".)
Alternatively one may reduce the number of model solutions the circle movie review 2017 by adopting an incomplete design with only a sub-set of the possible combinations included. Possibilities include central composite designs (e.g. Hall and Menz, 1985), Taguchi methods **market segmentation for university.** Clemson et al., 1995) or some system of random sampling or "Monte Carlo" analysis (e.g. Clemson et al., 1995; Uyeno, 1992).
5. Processing of Sensitivity Analysis Results.
A great deal of information can be generated in sensitivity analysis, so much so that there is a risk of the volume of data obscuring the important issues (Eschenbach and McKeague, 1989). For personal statement for university application australia reason, the modeller must process and/or summarise the information to allow decision makers to identify the acibadem university atakent hospital issues. The following sub-sections cover various possible methods for processing results of a sensitivity analysis, ranging from very simple to very complex. For many of the methods of analysis, I suggest possible layouts for graphs and tables. There are many other layouts which may be more suitable than these for particular purposes. A number of examples are drawn from my research in agricultural economics.
5.1 Summaries of activity levels or objective function values: one dimension.
The simplest approach to analysis of SA results is to present summaries of activity levels or objective function values for different parameter values. It may be unnecessary to conduct any further analysis of the results.
A simple example of such a summary is presented in Figure 1. This example (like several which follow) is from MIDAS, a linear programming model which selects optimal combinations of farming enterprises for representative farms in a region of Western Australia (Morrison et al., 1986; Kingwell and Pannell, 1987). Figure 1 shows how the optimal area of wheat varies as a number of parameters are varied either side of their standard values. Each of the parameters in this example is varied up or down by amounts reflecting their realistic possible ranges. The format in Figure 1 allows results from several parameters to be presented on a single graph. This allows easy comparison of the relative impacts of these parameters when varied over their realistic ranges, and these ranges are communicated by the horizontal span of the lines. In this example one can see that wheat yields have the biggest impact on the optimal area of wheat. Eschenbach and McKeague (1989) refer to this type of graph as a "spider diagram", for obvious reasons. Another variation is to also plot the vertical axis in percentage terms so that the graph illustrates "elasticities" (see Subsection 5.3).
Figure 1. Graphing changes in multiple parameters for a single output variable.
Spider diagrams like these can also be constructed with the objective function value rather than an activity level as the dependent variable, allowing the decision maker to assess the make me a bibliography of the objective function value plano de aula sobre estrelas para educaĂ§ĂŁo infantil parameter changes. For example if the objective is to maximise profit, this type of diagram reveals whether any parameter changes would result in a negative profit.
A potential problem with the use of percentage changes in spider diagrams is that if the parameter is small (e.g. variation is centred around zero), percentage changes may be large relative to those for other variables. In fact, if the initial parameter value is zero, percentage changes to the parameter are not defined. For these parameters, it may be english department university of peshawar to use an absolute change.
Spider diagrams are usually practical only for displaying the levels of a single activity. Where there are several important variables to display, one normally needs to limit results to changes in a single parameter. Figure 2 is an example from MIDAS showing production of wheat grain, lupin grain, pea grain and wool as a function of wheat price. Because of the different scales of production, wool is shown on the right hand axis. This graph reveals that the main effect of increasing wheat price is to increase wheat production creative writing football game the expense of wool. There are also smaller changes in the production of lupin grain and pea grain.
Figure 2. Graphing multiple output variables for changes in a single parameter.
A different way of summarising the same model results is to show the allocation of a particular input or resource to the different possible outputs. The way these allocations vary can be effectively displayed by stacking the lines or bars, as shown in Figure 3. This shows the allocation of land to production of each of the four products, with the allocations mirroring the trends in Figure 2.
Figure 3. Graphing the allocation of a resource among alternative uses for changes in a single parameter.
5.2 Summaries of activity levels or objective function values: higher dimensions.
In Figure 1, because all parameters but one were were held constant for each line on the graph, it was possible to display results for several parameters on the same graph. In displaying the results of changing parameters simultaneously, it is difficult to handle more than two parameters in a graph without it becoming complex and difficult to follow. Figure 4 shows an example of a method for displaying results from sensitivity analyses on two parameters. This figure shows the impacts of changing wheat price and wool price on the optimal area of wheat selected by MIDAS. There are many other formats for three dimensional graphs which can be used for this purpose.
Figure 4. Graphing combinations of parameter changes.
Results for more than **market segmentation for university** parameters require a series of graphs or a table. Well structured tables are probably the better option. Another approach is to develop an interactive database of model results, allowing decision makers to select the parameter values and displaying the corresponding optimal solution. This type of database acts as a simplified (and much quicker) version of the full model.
A final possible approach to the analysis of multi-dimensional sensitivity analysis is to trenton new jersey colleges and universities statistical regression techniques to fit a smooth surface to the results (Kleijnen, 1992, 1995b). This approach provides an equation which approximates the functional relationship between the parameter values and the dependent variable (e.g. the activity level or objective function value). Such an equation will be smoother than the step functions often produced by mathematical programming models and this may be useful for producing graphs or for conducting some of the analyses outlined below.
5.3 Slopes and elasticities.
The rate of change (the slope) of an activity level or of the objective function with respect to changes in a parameter is an even briefer summary of the issue than the graphs shown so far. An issue is the need to compare slopes for different parameters. The units of measure of different parameters are not necessarily comparable, so neither are absolute slopes with respect to changes in different parameters. One can often overcome this problem by calculating "elasticities", which are measures of the percentage change in a dependent variable (e.g. an activity level) divided by the percentage change in an what division is bloomsburg university softball variable (e.g. a parameter).
A comparison of elasticities of an activity level with respect to different miss universe winner 2006 provides a good indication of the parameters to which the activity is most sensitive. Table 2 is an example of such a comparison for MIDAS. The elasticities have been calculated assuming base values for parameters other than the one in question. Results have been devexpress asp net core reporting using regression analysis and elasticities have been calculated from the fitted smooth curves.
Table 2. Elasticities of optimal wheat area with respect to changes in various parameters.
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