## Case Study A hypothetical decision tree analysis

The market for a particular crop is increasing and hence a number of farmers who have not normally produced the crop are now turning to it as an alternative source of income. Farms that grow the crop have suffered losses due to a high incidence of an insect pest which can be controlled by insecticides. One of the farmers adopting the new crop needs to obtain use of spraying equipment suitable for applying insecticides against the insect pest. The farmer believes that the equipment will be used often, but since there is some doubt as to the likely incidence of an outbreak on his particular farm he does not know whether to hire or purchase the equipment. The farmer can choose to consult an independent advisor but this will cost £850. What decision should the farmer take?

To carry out a decision tree analysis a great deal of information is required. Firstly, a set of conditional probabilities must be determined based on the demand for the spraying equipment by other farmers (who had or had not taken the advice of the independent consultant) (Table 9.9). Secondly, a set of prior probabilities is required that identifies the actual demand for the sprayer by farmers who did not pay for independent advice (Fig. 9.15). These prior and conditional probabilities are then used to determine a set of posterior probabilities which are used in the sequential decision tree analysis.

At the chance node A the overall probabilities associated with the advice

Table 9.9. The conditional probabilities of sprayer demand when the advice from the consultant was either favourable or unfavourable.

Actual situation

High demand (H)

Moderate demand (M)

Low demand (L)

Favourable (F) Unfavourable (U)

being either favourable or unfavourable are computed. These probabilities are computed using the prior probabilities (0.5, 0.3, 0.2) and the conditional probabilities (Table 9.9), i.e. where the advice was favourable:

PF = P(FIH).P(H) + P(FIM).P(M) + P(FIL).P(L) PF = 0.4 + 0.18 + 0.06 Pp. = 0.64

The probability of 0.36 (0.1 + 0.12 + 0.14) was similarly obtained for the situation in which the advise was unfavourable. The next step in the analysis is to consider the probabilities associated with the chance nodes - the conditional probabilities. These are computed using Baye's theorem, the general form being:

where:

Ai = set of n mutually exclusive and exhaustive events B = a known end effect, or outcome of an experiment P(Ai) = the prior probability for event i

P(BIAi) = the conditional probability of end effect B given the occurrence of Ai.

The posterior probability is thus a revision of the prior probability using new or additional information. At each chance node the posterior probability P(AiIB) is the probability of the event 'high, medium or low demand' given the outcome of either favourable or unfavourable advice (new information). At each chance node (B-E) the probability of high, moderate or low demand given that the advice was favourable would be obtained by:

The probabilities for all other conditions (FIM), (FIU), (UIH), (UIM) and (UIL) were calculated similarly replacing PF with PU as appropriate. Using these conditional probabilities the expected monetary value (MEV) at each chance node can be determined by:

EMVb = (0.625) (29,150) + (0.281) (13,650) + (0.094) (1250) = 13,843 + 3245 + 296 = 22,170

EMVC = (0.625) (22,150) + (0.281) (11,550) + (0.094)(3150) = 13,843 + 3245 + 296 = 17,384

etc. for each chance node.

From the expected monetary values shown at chance nodes B and C the largest EMV £22,170 is selected for the decision node 2 and £13,104 for decision node 3. Using the EMVs at decision nodes 2 and 3 the EMV at chance node A can be computed:

EMVa = (0.64) (22,170) + (0.36) (13,104) A = 14,188 + 4717 = 18,905

From the results of this decision tree analysis it can be concluded that the best strategy for the farmer would be to not seek any advice from the independent consultant but to purchase the spray equipment.

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