Question

Describe the purpose of sensitivity analysis in the context of Decision Theory in multistage decision making and how strategy tables and tornado charts can be used for this purpose

Answer 1

What is Sensitivity Analysis?

Sensitivity Analysis is a device utilized in money related displaying to dissect how the various estimations of a lot of free factors influence a particular ward variable under certain particular conditions. When all is said in done, sensitivity analysis is utilized in a wide scope of fields, extending from science and topography to financial aspects and designing.

It is particularly helpful in the examination and analysis of a "Discovery Procedure" where the yield is a murky capacity of a few sources of info. A hazy capacity or procedure is one which, for reasons unknown, can't be contemplated and investigated. For instance, atmosphere models in topography are typically mind-boggling. Thus, the specific connection between the sources of info and yields are not surely known.

Approaches to Sensitivity Analysis

In principle, sensitivity analysis 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.

4.1 What to vary

One might choose to vary any or all of the following:

a. the contribution of an activity to the objective,

b. the objective (e.g. minimise the 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 an activity), or

f. 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 certain situation will occur or it will not. Examples include:

·  What if the government legislates to ban a particular technology for environmental reasons?

·  In the shortest route problem, what if a new freeway were built between two major centres?

·  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, the 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 of the change.

4.2 What to observe

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 sub-optimal strategies (e.g. strategies which are optimal for other scenarios, or particular strategies suggested by the decision-maker),

c. the difference in objective function values between two strategies (e.g. between the optimal strategy and a particular strategy suggested by the decision-maker),

d. the values of decision variables,

e. in an optimisation model, the values of shadow costs, constraint slacks or shadow prices, or

f. the rankings of decision variables, shadow costs, etc.

Multiple-stage decisions refer to decision tasks that consist of a series of interdependent stages leading towards a final resolution. The decision-maker must decide at each stage what action to take next to optimize performance (usually utility). One can think of myriad examples of this sort: working towards a degree, troubleshooting, medical treatment, scheduling, budgeting, etc. Decision trees are a useful means for representing and analyzing multiple-stage decision tasks (Figure 1), where decision nodes [X] indicate decision-maker choices, event nodes (Y) represent elements beyond the control of the decision-maker, and terminal nodes • represent possible final consequences (cf. Gass 1985, Chapter 23). In this example, the pursuit of a graduate student towards a PhD is represented as a multiple-stage decision tree. The first decision Theory and Decision 51: 217–246, 2001. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 218 JOSEPH J. JOHNSON AND JEROME R. BUSEMEYER Figure 1.   

Example of a real-life situation represented as a decision tree and solved using the dynamic programming method. node concerns whether or not to apply to graduate school, which leads to the event node of being accepted. If accepted, a second decision is required concerning which degree to pursue, leading to probabilistic event nodes dictating the decision-makers chances of success for each. While optimal navigation of this rather small decision tree may not seem so overwhelming, one can imagine the difficulty in comprehending the different scenarios involved with larger trees, such as a foreign policy decision task.

Based on elements of utility theory for single-stage gambles, backward induction (also known as dynamic programming) is an accepted method of selecting the optimal path of decision tree navigation (see, Bertsekas 1976; DeGroot 1970; and Raiffa 1968). The method of backward induction is applied to the graduate student example at the bottom of Figure 1. First, the decision-maker assigns subjective utility values to all terminal nodes, reflecting his/her satisfaction with the final alternatives. Next, the decision-maker specifies the probabilities at the event nodes, to the best degree possible.

For example, by using enrollment and matriculation rates one could assign meaningful values to the event nodes in Figure 1. Using backwards induction, one can then compute the optimal path for any given decision tree. As in SEU theories, the expected utility for event nodes (2) and (3) are determined by weighting the utility of each outcome (A, B for (2))by its the probability of occurrence (0.3, MULTIPLE-STAGE DECISION-MAKING 219 0.7), resulting in EU(2) = 11.70 and EU(3) = 13.80. Then, the tree is effectively ‘pruned’ at the preceding decision node, removing the option with the lower expected utility, (2). Thus, whenever a decision-maker reaches node [2], s/he should always choose to proceed to event node (3) – effectively assigning the utility for (3) to [2]. This reasoning is continued down the tree, computing a probabilistic utility for (1) based on the probabilities of the terminal node {E} and the newly defined [2]. Finally, if the value EU(1) exceeds EU{F}, the student should apply to graduate school.

Sensitivity Analysis Using a Tornado Chart

One of the easiest ways to increase the effectiveness of your optimization is to remove decision variables that require a lot of effort to evaluate and analyze, but that does not affect the objective very much. If you are unsure how much each of your decision variables affects the objective, you can use the Tornado Chart tool in Crystal Ball (see the Oracle Crystal Ball User's Guide for more information on the Tornado Chart).

The Tornado Chart tool shows how sensitive the objective is to each decision variable as they change over their allowed ranges. The chart shows all the decision variables in order of their impact on the objective.

Figure 119, Crystal Ball Tornado Chart shows a Crystal Ball tornado chart. When you view a tornado chart, the most important variables are at the top. This arrangement makes it easier to see the relative importance of all the decision variables. The variables listed at the bottom are the least important in that they affect the objective the least. If their effect is significantly smaller than those at the top, you can probably eliminate them as variables and just let them assume constant value.

Figure 119. Crystal Ball Tornado Chart

Strategy Tables

Frequently the decision structure of the problem is complex, especially when the decision being considered relates to a large-scale initiative involving many sub-decisions. The Strategy Table helps both decision-makers and the decision team to simply and clearly define the alternatives.

The example below illustrates how this works for the case of a decision involving a space mission to Europa, a moon of Jupiter. As background, Europa is a moon shrouded with ice that is believed to harbour a liquid water ocean beneath the ice covering. Some scientists suspect this ocean could contain living microorganisms, perhaps similar to the “extremophiles” that inhabit the thermal vents in the deep oceans of Earth. The heat source permitting liquid water to exist, and perhaps thermal vents, arises from the action of tidal forces generated by Jupiter that continually stress Europa’s core.

Mission planners must consider several important decisions, including significantly different design options and how aggressive to make the schedule. The Strategy Table is shown below:

The decisions are the column headers of the table—target launch date, scientific scope, probe vehicle, and degree of reliability to build in. In each column are the options associated with each decision. Different options have different cost implications, timing implications, and different implications for the success of the mission.

The team then defines the alternatives to be analyzed. In this case, the alternatives are called “strategies,” sensibly-connected combinations of decision options. A strategy alternative is a pathway through the table, illustrated below:

For example, the “Maximum Discovery” (green) alternative is a strategy that seeks to maximize the likelihood of scientific success by including a science package (A++) that contemplates several different possibilities for the circumstances in which microbial life may be found, and design for the probe vehicle that will have flexibility and controllability of operation once it has penetrated the ice sheet. Designing and fabricating such a package will require longer development time, but this longer time allows more thorough testing so redundancies can be reduced while maintaining substantial reliability.

The “Assured Arrival” alternative builds in aggressive redundancies to maximize the likelihood that the probe will arrive on target and function properly. Development and fabrication can be accomplished more quickly than with Maximum Discovery if a less sophisticated science package (A) is chosen. The probe vehicle design can still be of the more flexible kind, but the less sophisticated science package it carries reduces the likelihood of finding life.

The “Cheap and Fast” alternative addresses the beliefs of some decision-makers that more sophisticated science packages and probe design are overkill for the science mission; that the most likely microbial forms will be readily apparent if they exist; that the mission can be accomplished much more cheaply than some propose; and that there is high scientific value to proving or excluding simpler life forms as quickly as feasible for benefit of the scientific community. It also offers the potential for a quick and badly-needed political “win” for the organization. So this alternative uses a simpler scientific package (B) and a simple probe (Design #1) that has limited mobility and controllability once it reaches liquid water.