$1.[-/12\mathrm{.}5$ Points$]$

The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multirow$\{2\}\{*\}\{\backslash$begin$\{$tabular$\}\{$c$\}$

Decision $\backslash \backslash$

Alternative

$\backslash$end$\{$tabular$\}\}$ & $\backslash$multicolumn$\{3\}\{|$c$|\}\{$ States of Nature $\}\backslash \backslash$

$\backslash$cline $\{2-4\}$ & $\backslash (\backslash$boldsymbol$\{$s$\}\_\{\backslash$mathbf$\{1\}\}\backslash )$ & $\backslash (\backslash$boldsymbol$\{$s$\}\_\{\backslash$mathbf$\{2\}\}\backslash )$ & $\backslash (\backslash$boldsymbol$\{$s$\}\_\{\backslash$mathbf$\{3\}\}\backslash )\backslash \backslash$

$\backslash$hline $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$mathbf$\{1\}\}\backslash )$ & $260$ & $110$ & $35\backslash \backslash$

$\backslash$hline $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$mathbf$\{2\}\}\backslash )$ & $110$ & $110$ & $85\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($a$)$ Construct a decision tree for this problem.

Decision Tree

Description

$($i$)\backslash$begin$\{$tabular$\}\{|$c$|$l$|$l$|$l$|\}$

$\backslash$hline $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$mathbf$\{1\}\}\backslash )$ & $260$ & $110$ & $35\backslash \backslash$

$\backslash$hline $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$mathbf$\{2\}\}\backslash )$ & $110$ & $110$ & $85\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($a$)$ Construct a decision tree for this problem.

$\backslash (\backslash$square $\backslash )$

The recommended decision using the optimistic approach is $.$ The recommended decision using the conservative approach is

$\backslash (\backslash$square $\backslash ).$ The recommended decision using the minimax regret approach is

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