Suppose that a decision maker faced with four decision alternatives and four states of nature develops the following profit payoff table.

States of Nature

Decision Alternative s$1$ s$2$ s$3$

d$1-41021$

d$2-71428$

d$33815$

d$48910$

a$.$ Please draw a decision tree to illustrate the decision problem $($must have square node, circle nodes and branches$).$

b$.$ If the decision maker knows nothing about the probabilities of the four states of nature, what is the recommended decision using the optimistic approach? $($details are required as the example in lecture notes$)?$

D$1$: Max payoff $=14$

D$2$: Max payoff $=14$

D$3$: Max payoff $=28$

D$4$: Max payoff $=15$

c$.$ What is the recommended decision using the conservative approaches $($details are required as the example in lecture notes$)?$

D$1$: Min payoff $=10$

D$2$: Min payoff $=7$

D$3$: Min payoff $=15$

D$4$: Min payoff $=8$

d$.$ What is the recommended decision using minimax regret approaches $($details are required as the example in lecture notes$)?$

D$1$: Min Payoff $=$

e$.$ Suppose the decision maker obtained the probability assessments P$($s$1)=0\mathrm{.}10,$ P$($s$2)=0\mathrm{.}30$ and P$($s$3)=0\mathrm{.}60.$ Use the expected value approach to determine the recommended alternative.