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Problem 170 Easy Difficulty

A store will order $q$ gallons of a liquid product to meet demand during a particular time period. This product can be dispensed to customers in any amount desired, so demand during the period. is a continuous random variable $X$ with cdf $F(x) .$ There is a fixed cost $c_{0}$ for ordering the product plus a cost of $c_{1}$ per gallon purchased. The per-gallon sale price of the product is $d .$ Liquid left unsold at the end of the time period has a salvage value of $e$ per gallon. Finally, if demand exceeds $q,$ there will be a shortage cost for loss of goodwill and future business; this cost is $f$ per gallon of unfulfilled demand. Show that the value of $q$ that maximizes expected profit, denoted by $q^{},$ satisfies
$$P(\text { satisfying demand })=F\left(q^{}\right)=\frac{d-c_{1}+f}{d-e+f}$$
Then determine the value of $F\left(q^{*}\right)$ if $d=\$ 35, c_{0}=\$ 25, c_{1}=\$ 15, e=\$ 5,$ and $f=\$ 25.5$ [Hint: Let $x$ denote a particular value of $X .$ Develop an expression for profit when $x \leq q$ and another expression for profit when $x>q .$ Now write an integral expression for expected profit (as a function of $q )$ and differentiate. $]$

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Video Transcript

Okay, So in this question, were given that a store will order que gallons of a liquid product to meet demand during a particular time period secure, which is in guns. So the management period is a continuous random variable x with C d f f of X assisted demand function. Yeah. Okay. And we know that if the value of Q the maximized expect profit is denoted by Q star, your story inches equal. Thio s thank you start, which is equal to demand C one plus f over demand she plus earth. Okay, we also know that the cost is equal to see, not plus C one X. The salvage value is e very gallant. No, um, on the good will cost. Yeah. Good. Little cost is s per gallon. Okay, so now, assuming that D is 35 she is 25 year old, 15 years, five on efforts 25. Just go ahead. Like everything to this function. Yeah, we're gonna have it is 35 minus 15 plus 25 over 35 minutes. Five plus 25 which is just gonna equal 9 11, which is a fun answer

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