Exercise $2$

a$)32\mathrm{.}56\%$

b$)25\mathrm{.}14\%$

c$)Q=1300$ units

d$)ES=846\mathrm{.}7$ units

e$)353\mathrm{.}3$ units

f$)28\mathrm{.}1\%$

g$)$ $$33716$

h$)$ $$24000$

i$)15\mathrm{.}9\%$

j$)$ about $1150$ unitsExercise $2$ FIGURE $14\mathrm{.}6$ Density Function FIGURE $14\mathrm{.}7$ Distribution Function

Flextrola, Inc., an electronics systems integrator, is planning to design a key component for their next$-$

generation product with Solectrics. Flextrola will integrate the component with some software and then

sell it to consumers. Given the short life cycles of such products and the long lead times quoted by

Solectrics, Flextrola only has one opportunity to place an order with Solectrics prior to the beginning of

its selling season. Flextrola's demand during the season is normally distributed with a mean of $1,000$ and

a standard deviation of $600.$

Solectrics' production cost for the component is $$52$ per unit and it plans to sell the component for $$72$

per unit to Flextrola. Flextrola incurs essentially no cost associated with the software integration and

handling of each unit. Flextrola sells these units to consumers for $$121$ each. Flextrola can sell unsold

inventory at the end of the season in a secondary electronics market for $$50$ each. The existing contract

specifies that once Flextrola places the order, no changes are allowed to it$.$ Also, Solectrics does not

accept any returns of unsold inventory, so Flextrola must dispose of excess inventory in the secondary

market.

a$)$ What is the probability that Flextrola's demand will be within $25$ percent of its forecast?

b$)$ What is the probability that Flextrola's demand will be more than $40$ percent greater than its

forecast?

c$)$ Under this contract, how many units should Flextrola order to maximize its expected profit?

For parts d through i$,$ assume Flextrola orders $1,200$ units.

d$)$ What are Flextrola's expected sales?

e$)$ How many units of inventory can Flextrola expect to sell in the secondary electronics market?

f$)$ What is Flextrola's expected gross margin percentage, which is $($Revenue $-$ Cost$)/$Revenue$?$

g$)$ What is Flextrola's expected profit?

h$)$ What is Solectrics' expected profit?

i$)$ What is the probability that Flextrola has lost sales of $400$ units or more?

j$)$ A sharp manager at Flextrola noticed the demand forecast and became wary of assuming that

demand is normally distributed. She plotted a histogram of demands from previous seasons for

similar products and concluded that demand is better represented by the log normal

distribution. Figure $14\mathrm{.}6$ plots the density function for both the log normal and the normal

distribution, each with mean of $1,000$ and standard deviation of $600.$ Figure $14\mathrm{.}7$ plots the

distribution function for both the log normal and the normal. Using the more accurate forecast

$($i$.$e$.,$ the log normal distribution$),$ approximately how many units should Flextrola order to

maximize its expected profit? $($Answer the question by visually inspecting the graph$)$