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Sample Problems for Second Exam With Solutions 1. A producer of antifreeze is faced with the problem of how much antifreeze to produce for the upcoming winter season. He has on hand 50,000 gallons that were not sold last year. His demand in millions of gallons is a random variable having the following probability density function: For every gallon stocked in excess of demand, it costs 50 cents for carrying this quantity in inventory until the next year. The production cost is $1.50 per gallon and the sale price is $2.50 per gallon. The cost of a shortage may be taken as the lost profit per gallon. Find the optimal amount of antifreeze to be produced this season. I 0 = 50,000 h = 0.50 v = 1.50 p = 2.50 C u = p v = 1 C o = h = 0.5 C u /( C u + C o ) = 1/1.5 = 2/3 F ( y ) = 0.67 y = l + (2/3)( u l ) = 1,666,667 gallons Q = 1,666,667 – 50,000 = 1,616,667 gallons Note: If we view this as an infinite horizon problem with no discounting, then the above interpretation of a C o value of just the holding cost is fine, because we didn’t “unnecessarily spend v on an unsold unit” (because we can sell it next year). On the other hand, if we forget about the future periods, then the overage cost might be interpreted as C o = v + h , and we would get C u /( C u + C o ) = 1/3. I wouldn’t penalize students who interpreted it this way.
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2. For a single-period problem, when demand is uniformly distributed between l and u , and overages result in a per unit holding cost h , and underages result in a per
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This note was uploaded on 02/15/2011 for the course EIN 6336 taught by Professor Staff during the Spring '08 term at University of Florida.

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