SariyerRobotics is a company that

specializes in developing custom industrial robots. A manufacturer

approaches SariyerRobotics $($SR$)$ for a custom$-$made warehouse material handling robot that will be contracted through a bidding process

where the best proposal among the bidders would be selected and receive a contract of $$100,000.$ To prepare a proposal for the bid has

an estimated cost of $$20,000.$ If they decide to prepare and submit a

proposal $($bid$),$ they think that their chances of winning the contract

is $\%50.$

If they win the contract, then they are committed to making and delivering the actual product within the specified deadline $($they cannot

cancel once they win the bid$).$ At this point, it is not clear to SR how

the final robot will be developed and some alternatives are being considered. A first alternative is to modify an existing model with some

more mechanical additions. They are certain that this approach would

work and the development cost will be estimated at $$50,000.$ A second

alternative is to completely redesign the controller while keeping the mechanical system similar to the existing model. This is estimated to

cost $$20,000$ but they think that this design is risky and has only a $40\%$

probability of success. The third alternative is somewhere in between

the first two the controller is updated with minor modifications. The

estimated cost is then $$30,000$ and the chance of success is estimated

as $\%75.$ SR can work on only one approach at a time and has time

to try at most two approaches. If they try either the second or third

method and the attempt fails, they would be forced to use the first

alternative to guarantee a successful model. Assume that SariyerRobotics have logarithmic utility

U$($x$)=$ ln$($x $+$ w$)$ where w is their starting wealth $($i$.$e$.$ cashflow

position$).$ Find the Find the optimal strategy of the company as

w ranges from $$80,000$ to $$1,000,000$