Problem $4.$ Johnson Marketing is interested in producing and selling an innovative new food processor.$$ The decision they face is the typical $$make or buy$$ decision often faced by manufacturers.$$ On one hand, Johnson could produce the processor itself, subcontracting different sub$-$assemblies, such as the motor or the housing.$$ Cost estimates in this case are as follows:

$$

$$ Alternative:$$ Make Food Processor

$$

$$ Cost Per Unit $($$$)$ Chance $(\%)$

$$

$35\mathrm{.}0025$

$42\mathrm{.}5025$

$45\mathrm{.}0037$

$49\mathrm{.}0013$

$$

$$ The company also could have the entire machine made by a subcontractor.$$ The subcontractor, however, faces similar uncertainties regarding the costs and has provided Johnson$$ Marketing with the following schedule of costs and chances:

$$

$$ Alternative:$$ Buy Food Processor

$$

$$ Cost per Unit $($$$)$ Chance $(\%)$

$$

$37\mathrm{.}0010$

$43\mathrm{.}0040$

$46\mathrm{.}0030$

$50\mathrm{.}0020$

$$

a$)$ If Johnson Marketing wants to minimize its expected cost of production in this case, should it make or buy?$(10$ points$)$

$$

$$

$$

b$)$ Construct cumulative risk profiles to support your recommendation. $(10$ points$)$

$$

$$

$$

c$)$ Does one alternative deterministically dominate the other?$$ Why or why not? $(5$ points$)$

$$

$$

$$

d$)$ Does one alternative stochastically dominate the other?$$ Why or why not? $(5$ points$)$

$$

Compare the grower$$s expected values for the three alternatives he has, considering the various possible loss scenarios for the burner and the sprinklers. Which alternative would you suggest the grower take? Why? $(20$ Points$)$

Suppose Napoleon was using Bayes$$ theorem to revise this information.$$ To do so$,$ he would have to make some judgment about P$($Prussian and English join Forces $$ Napoleon Wins$)$ and P$($Prussian and English join Forces $$Napoleon Loses$).$ In particular, he would have had to make judgment about the ratio of these two probabilities.$$ What is that ratio?$(15$ Points$)$