Problem $1($Decision Tables$)$: Choosing an Insurance Policy

Tim Smith, the owner of Premier Mazda dealership in Kansas City, is trying to decide whether to

purchase an insurance policy to cover damage from hail storms to his inventory of more than $300$ cars.

Tim is considering the following insurance alternatives:

Plan Annual Payment

$($in $ thousands$)$ Terms

A $$0$ Self$-$insure. Tim pays for all damages out of pocket.

B $$25$ Insurance covers the amount of damage in excess of $$30,000.$

C $$33$ Insurance covers $80\%$ of the amount of damage in excess of $$10,000.$

D $$47$ Insurance covers $100\%$ of damages.

For example: Suppose total damages in a year end up being $$40$K$.$ With option A$,$ Tim has to pay the

entire $$40$K out of pocket. With option B$,$ Tim pays $$25$K for the insurance plus the first $$30$K of the

damages. That is a total cost of $$55$K$.$ With option C$,$ Tim pays $$33$K for the insurance, plus the first $$10$K

of damages, plus $20\%$ of $$30$K additional damages. That would be a total cost of $$49$K$.$ With option D$,$

Tim only pays $$47$K for the insurance.

Suppose that the amount of damages from hail storms in a typical year can range between $$0$ to

$$100,000.$ Tim likes to consider this cost in increments of $$20,000($that is$,$ assume the damage will be

either $$0,$ $$20,$ $$40,$ $$60,$ $$80,$ or $$100$ thousand, i$.$e$.,$ six possibilities$).$

a$)$ Construct a payoff $($cost$)$ table for this decision problem.

b$)$ Construct an opportunity loss $($regret$)$ table for this decision problem.

c$)$ Determine which insurance plan Tim should pick if:

$1.$ He is fully optimistic,

$2.$ He is fully pessimistic,

$3.$ He is only $40\%$ optimistic $($criterion of realism with $=0\mathrm{.}4),$

$4.$ He thinks the six outcomes are equally likely,

$5.$ He likes to avoid the maximum regret possible $($minimax$-$regret criterion$).$

Prescriptive Analytics $($ADMN $875)$

Hint: In this problem we are dealing with costs. So$,$ be careful of what $$optimistic$$ and $$pessimistic$$

mean, and that we are trying to pick the lowest$-$cost plan eventually. Looking at the self$-$practice

problem posted along with Decision Table lecture, about picking Car Lease options, might help.

d$)$ Looking back at historical annual damages, Tim believes that the amount of damages can be

described by the following probability distribution:

Damages

$($$ thousands$)020406080100$

Probability $0\mathrm{.}30\mathrm{.}250\mathrm{.}20\mathrm{.}10\mathrm{.}10\mathrm{.}05$

Calculate the expected value $($cost$)$ for each alternative and identify the best decision based on EMV.