Tricia Barry, the store manager of Kmart in Keysborough, is considering signing a new contract with one of the two courier companies in the area such as ‘Pack & Send’ and ‘Send it Easy’ for the daily deliveries of products. Courier companies may reject the contract, but Tricia Barry may also negotiate to sign the contract for 6 months or 2 years. Tricia Barry must decide to sign the contract with Pack & Send or Send it Easy, but she knows that if she signs with Send it Easy, her profit will not be as high in comparison to Pack & Send. If any contract is rejected, it will incur a cost to the Kmart store in Keysborough. Tricia Barry would like to maximise the store profit. The profit for the different contract lengths are summarised in the following payoff table in thousands of Australian dollars).

Profits/losses for various contract lengths
DECISION ALTERNATIVE (di)= Pack & Send (d1), Send it easy (d2)
Reject (S1) = -3 , -2
6 Months (S2) = 5 ,3
2 Years (S3) = 15 , 8

(i) Apply the Opportunity Loss Approach (with the minimax regret criterion) to recommend a decision to Tricia Barry.

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(ii) Tricia Barry has done an initial assessment on what she thinks two courier companies might do and came up with the following probabilities:
Pack & Send:
P(Reject) = 0.3
P(6 Months) = 0.4
Send it Easy:
P(Reject) = 0.2,
P(6 Months) = 0.5
Draw a decision tree and recommend a decision strategy to Tricia Barry on whom she should sign the contract with. Clearly show your calculations and provide a justification for your decision strategy.

(iii) For a consulting fee of $3,000, Whereto Consultancy Australia (WCA) will review the plans for the Pack & Send contract and indicate the overall chances of a favourable outcome to Tricia Barry. Denote favourable review by F, and unfavourable review by U. Tricia Barry believes that the following conditional probabilities are realistic appraisals of WCA’s evaluation accuracy. No review is conducted on Send it Easy contract because there was insufficient information for WCA to do so.
P(F|Rejection) = 0.2,
P(F16 Months) = 0.6
P(F|2 Years) = ?
P(U|Rejection) = 0.8,
P(U16 Months) = ?
P(U12 Years) = 0.3
Apply Bayes’ Theorem to compute the posterior probabilities for Pack & Send for both the Favourable and Unfavourable review cases

(iv) Draw the decision tree, determine the recommended decision strategy(ies) and the expected value for the case where Tricia Barry has already decided to conduct the review on Pack & Send. Make the assumption that after receiving the review from WCA, Tricia Barry only has two options of whom she can sign the contract with: namely, Pack & Send or the Send it Easy. Draw the decision tree with clear labels, showall your calculations and provide justification(s) for your decision strategy(ies) to Tricia Barry.

(v) Construct a risk profile for the optimal decision strategy from part (iv).
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