Sensitivity Analysis............................................................2Definition:..............................................................................2Introduction:..........................................................................2Impacts of Sensitivity Analysis:...............................................2Uses of Sensitivity Analysis:....................................................3Effect of parameter and characteristics of the model on the optimality:.............................................................................5Range of Optimality:...............................................................5Example:................................................................................5Example:................................................................................7

Cowhide................................................................................................7Production Time...................................................................................7Production Limit...................................................................................8Non-negativity of Decision Variables....................................................8The Mathematical Model......................................................................8

Decision Analysis...............................................................9Definition:..............................................................................9Introduction:..........................................................................9Methodology:.......................................................................10Decision-making under uncertainty (or risk) Seven Steps:.......10Different Criterion:................................................................11Approaches for Decision under Risk:......................................11Approaches for Decision under uncertainty:...........................12

Optimistic criterion.............................................................................12Minimax (Regret) criterion..................................................................12Laplace criterion.................................................................................12Savage Minimax Regret criterion.......................................................13

Decision Tree with Examples:................................................13Decision tree example 1995 UG exam:..............................................13

Solution...............................................................................15

s

Sensitivity Analysis

Definition:

“Investigation into how projected performance varies along with changes in the

key assumptions on which the projections are based.”

Introduction:

Sensitivity analysis is used to determine how “sensitive” a model is to changes in the

value of the parameters of the model and to changes in the structure of the model.

Sensitivity analysis helps to build confidence in the model by studying the

uncertainties that are often associated with parameters in models. Many parameters in

system dynamics models represent quantities that are very difficult, or even

impossible to measure to a great deal of accuracy in the real world. Also, some

parameter values change in the real world. Therefore, when building a system

dynamics model, the modeler is usually at least somewhat uncertain about the

parameter values he chooses and must use estimates.

Impacts of Sensitivity Analysis:

Sensitivity analysis can be useful to computer modelers for a range of purposes

including,

Support decision making or the development of recommendations for decision makers

(e.g. testing the robustness of a result);

Enhancing communication from modelers to decision makers (e.g. by making

recommendations more credible, understandable, compelling or persuasive);

Increased understanding or quantification of the system (e.g. understanding

relationships between input and output variables); and

Model development (e.g. searching for errors in the model).

Uses of Sensitivity Analysis:

There is a very wide range of uses to which sensitivity analysis is put. 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 development.

1.Decision Making or Development of Recommendations for Decision

Makers

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1.

1Testing the robustness of an optimal solution.

1.

2

Identifying critical values, thresholds or break-even values where the

optimal strategy changes.

1.

3Identifying sensitive or important variables.

1.

4Investigating sub-optimal solutions.

1.

5Developing flexible recommendations, which depend on circumstances.

1.

6Comparing the values of simple and complex decision strategies.

1.

7Assessing the "riskiness" of a strategy or scenario.

   

2. Communication

2.

1

Making recommendations more credible, understandable, compelling or

persuasive.

2.

2Allowing decision makers to select assumptions.

2.

3Conveying lack of commitment to any single strategy.

   

3. Increased Understanding or Quantification of the System

3.

1Estimating relationships between input and output variables.

3.

2Understanding relationships between input and output variables.

3.

3Developing hypotheses for testing

   

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4. Model Development

4.

1Testing the model for validity or accuracy.

4.

2Searching for errors in the model.

4.

3Simplifying the model.

4.

4Calibrating the model.

4.

5Coping with poor or missing data.

4.

6Prioritizing acquisition of information.

 

In all models, parameters are more-or-less uncertain. The modeler 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)

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 ignored 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 (insensitive 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. Perhaps the base-case

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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.

Effect of parameter and characteristics of the model on

the optimality:

In many conditions, the parameter and characteristics of a linear programming model

may changeover a period of time. Also, the analyst may be interested to know the

effect of changing the parameters and characteristics of the model on the optimality.

This kind of sensitivity analysis can be carried out in the following ways:

Making changes in the right-hand side constants of the constraints

Making changes in the objective function coefficients

Adding a new constraint

Adding a new variable

Range of Optimality:

The range of optimality for each coefficient provides the range of values over which

the current solution will remain optimal. Managers should focus on those objective

coefficients that have a narrow range of optimality and coefficients near the endpoints

of the range.

Graphically, the limits of a range of optimality are found by changing the slope of the

objective function line within the limits of the slopes of the binding constraint lines.

Binding constraints are the constraints that intersect to form the optimal point.

We will focus on using the computer output to analyze the range of optimality for

objective function coefficients.

Example:

Before identifying the range of optimality.

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Now as discuss that for in graphical solution we can find the range of optimality by

changing the slope of the objective function such that it nearly lies within the

constraints line.

By changing coefficients in the objective function you are altering its slope. You can

verify this by simply changing a coefficient and graphing the new objective function.

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Example:

Wilson Problem: Wilson Manufacturing produces both baseballs and softballs, which

it wholesales to vendors around the country. Its facilities permit the manufacture of a

maximum of 500 dozen baseballs and a maximum of 500 dozen softballs each day.

The cowhide covers for each ball are cut from the same processed cowhide sheets.

Each dozen baseballs require five square feet of cowhide (including waste), whereas,

one dozen softballs require six square feet of cowhide (including waste). Wilson has

3600 square feet of cowhide sheets available each day. Production of baseballs and

softballs includes making the inside core, cutting and sewing the cover, and

packaging. It takes about one minute to manufacture a dozen baseballs and two

minutes to manufacture dozen softballs. A total of 960 minutes is available for

production daily.

Formulate a set of linear constraints that characterize the production process at Wilson

Manufacturing.

Decision Variables

X1= number of dozen baseballs produced daily

X2= number of dozen softballs produced daily

Constraints

In addition to non-negativity constraints (i.e., the implied constraints) for the decision

variables, there are three functional constraints.

The use of cowhide.

The daily limit for production time.

The maximum production limit of total units.

CowhideThe total amount of cowhide used daily cannot exceed the amount of cowhide

available daily

5X1 + 6X2 £ 3600

Production TimeThe amount of production minutes used daily cannot exceed the total number of

production minute’s available daily

X1 + 2X2 £ 960

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Production LimitThe total number of dozen units produced daily cannot exceed the marketing limits

X1 £ 500

X2 £ 500

Non-negativity of Decision VariablesNegative Production of baseballs and softballs is impossible. Thus,

X1, X2 3 0

The Mathematical Model

Max 7X1+ 10X2 (Objective Function)

Subject to:

5X1 + 6X2 £ 3600 (Cowhide)

X1 + 2X2 £ 960 (Production time)

X1 £ 500 (Production limit of baseballs)

X2 £ 500 (Production limit of softballs)

X1, X2 3 0 (Non-negativity)

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Decision Analysis

Definition:

Decision analysis is a logical and systematic way to address a wide variety of

problems involving decision-making in an uncertain environment. We introduce the

method of decision analysis and the analytical model of constructing and solving a

decision tree with the following prototypical decision problem.

Introduction:

The term decision analysis was coined in 1964 by Ronald A. Howard, who since then,

as a professor at Stanford University, has been instrumental in developing much of the

practice and professional application of DA.

Graphical representation of decision analysis problems commonly use influence

diagrams and decision trees. Both of these tools represent the alternatives available to

the decision maker, the uncertainty they face, and evaluation measures representing

how well they achieve their objectives in the final outcome. Uncertainties are

represented through probabilities and probability distributions. The decision maker's

attitude to risk is represented by utility functions and their attitude to trade-offs

between conflicting objectives can be made using multi-attribute value functions or

multi-attribute utility functions (if there is risk involved). In some cases, utility

functions can be replaced by the probability of achieving uncertain aspiration levels.

Decision analysis advocates choosing that decision whose consequences have the

maximum expected utility (or which maximize the probability of achieving the

uncertain aspiration level). Such decision analytic methods are used in a wide variety

of fields, including business  healthcare research and management, energy

exploration, litigation and dispute resolution, etc. An applied branch of decision

theory.

Decision analysis offers individuals and organizations a methodology for making

decisions; it also offers techniques for modeling decision problems mathematically

and finding optimal decisions numerically. Decision models have the capacity for

accepting and quantifying human subjective inputs: judgments of experts and

preferences of decision-makers. Implementation of models can take the form of

simple paper-and-pencil procedures or sophisticated computer programs known as

decision aids or decision systems.

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Methodology:

The methodology is rooted in postulates of rationality—a set of properties which

preferences of rational individuals must satisfy. One such property is transitivity: if an

individual prefers action ato action b and action b to action c, he or she should

prefer a to c. From the rationality postulates, principles of decision-making are

derived mathematically. The principles prescribe how decisions ought to be made, if

one wishes to be rational. In that sense, decision analysis is normative.

The methodology is broad and must always be adapted to the problem at hand. An

illustrative adaptation to a class of problems known as decision-making under

uncertainty (or risk) is outlined in the illustration and consists of seven steps:

Decision-making under uncertainty (or risk) Seven

Steps:

The problem is structured by identifying feasible actions, one of which must be

decided upon; possible events, one of which occurs thereafter; and outcomes, each of

which results from a combination of decision and event. Problem structuring can be

facilitated by displays such as decision trees and decision matrices.

At the time of decision-making, the event that will actually occur cannot be predicted

perfectly. The degree of certainty about the occurrence of an event, given all

information at hand, is quantified in terms of the probability of the event.

Preferences are personal: the same outcome may elicit different degrees of desirability

from different individuals. The subjective value that a decision-maker attaches to an

outcome is quantified and termed the utility of outcome.

The preceding steps conform to the principle of decomposition: probabilities of events

and utilities of outcomes must be measured independently of one another. They are

next combined in a criterion for evaluating decisions. The utility of a decision is

defined as the expected utility of the outcome. The optimal, or the most preferred,

decision is one with the maximum utility.

The probability encodes the current state of information about a possible event. Often,

additional information can be acquired in the hope of reducing the uncertainty. The

monetary value of such information is computed before purchase and compared with

the cost of information. Thus, one can determine whether or not acquiring information

is economically rational.

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The source of information may be a real-world experiment, a laboratory test, a

mathematical model, or the knowledge of an expert. The in formativeness of the

source is described in terms of a probabilistic relation between information and event.

This relation, known as the likelihood function, makes it possible to revise the prior

probability of the event and to obtain a posterior probability of the event, conditional

on additional information. The revision is carried out via Bayes' rule.

Given the additional information, prior probabilities can be replaced by posterior

probabilities, and the analysis can be repeated from step 4 onward. Steps 4–6 may be

cycled many times, until the cost of additional information exceeds its value, at which

moment the optimal decision is implemented.

Decision analysis provides a rich set of concepts and techniques to aid the decision

maker in dealing with complex decision problems under uncertainty.

Decision-making can be broadly classified into three broad categories.

Decision making under certainty

Decision Making under Risk

Decision making under uncertainty

Different Criterion:

Approaches for Decision under Risk:

Most of the decisions are made on the basis of some criterion. When there is certainty

or the outcome is sure, decision-making is simpler. When the outcome is not sure,

then different criteria are used. They are for decision making under risk.

Expected Value Criterion

Combined expected value and variance criterion

Known aspiration level criterion

Most likely occurrence criterion

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Approaches for Decision under uncertainty:

Optimistic criterionAn optimistic decision maker would use the optimistic approach. The decision with

the largest possible payoff is chosen. If the payoff table were in terms of costs, the

decision with the lowest cost would be chosen.

Minimax (Regret) criterionThe minimax regret approach requires the construction of a regret table or an

opportunity loss table. This is done by calculating for each state of nature the

difference between each payoff and the largest payoff for that state of nature. Then,

using this regret table, the maximum regret for each possible decision is listed. The

decision chosen is the one corresponding to the minimum of the maximum regrets.

Laplace criterionEqually likely, also called Laplace, criterion finds decision alternative with highest

average payoff.

First calculate average payoff for every alternative.

Then pick alternative with maximum average payoff.

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Savage Minimax Regret criterionThe Minimax Regret criterion focuses on avoiding regrets that may result from

making a non-optimal decision. Although regret is a subjective emotional state, the

assumption is made that it is quantifiable in direct (linear) relation to the rewards of

the payoff matrix.

Regret is defined as the opportunity loss to the decision maker if action alternative Ai

is chosen and state of nature Sj happens to occur. Opportunity loss is the payoff

difference between the best possible outcome under Sj and the actual outcome

resulting from choosing Ai. Formally:

OLij = (row j maximum payoff) - Rij for positive-flow payoffs (profits, income)

OLij = Rij - (row j minimum payoff) for negative-flow payoffs (costs, losses)

where Rij is the reward value (payoff) for column i and row j of the payoff matrix R.

Note that opportunity losses are defined as nonnegative numbers. The best possible

OL is zero (no regret) and the higher the OL value, the greater the regret.

Decision Tree with Examples:

Decision tree example 1995 UG exam:Your company is considering whether it should tender for two contracts (MS1 and

MS2) on offer from a government department for the supply of certain components.

The company has three options:

Tender for MS1 only; or

Tender for MS2 only; or

Tender for both MS1 and MS2.

If tenders are to be submitted the company will incur additional costs. These costs

will have to be entirely recouped from the contract price. The risk, of course, is that if

a tender is unsuccessful the company will have made a loss.

The cost of tendering for contract MS1 only is £50,000. The component supply cost if

the tender is successful would be £18,000.

The cost of tendering for contract MS2 only is £14,000. The component supply cost if

the tender is successful would be £12,000.

The cost of tendering for both contracts MS1 and contract MS2 is £55,000. The

component supply cost if the tender is successful would be £24,000.

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For each contract, possible tender prices have been determined. In addition, subjective

assessments have been made of the probability of getting the contract with a particular

tender price as shown below. Note here that the company can only submit one tender

and cannot, for example, submit two tenders (at different prices) for the same

contract.

In the event that the company tenders for both MS1 and MS2 it will either win both

contracts (at the price shown above) or no contract at all.

What do you suggest the company should do and why?

What are the downside and the upside of your suggested course of action?

A consultant has approached your company with an offer that in return for £20,000 in

cash she will ensure that if you tender £60,000 for contract MS2 only your tender is

guaranteed to be successful. Should you accept her offer or not and why?

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Solution

The decision tree for the problem is shown below.

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