Lec 11

# Lec 11 - IEOR 151 – Notes Reasons To Keep Inventory 1...

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Unformatted text preview: IEOR 151 – October 11, 2010 Notes Reasons To Keep Inventory 1.) Hedging – future price 2.) Economies of Scale – cheaper 3.) Demand is random 4.) Supply can be random too Costs 1.) Inventory Holding Cost/Perishability 2.) Ordering Cost 3.) Penalty Cost: Loss of Sales Loss of Goodwill Beer Seller Problem D: demand is random R: revenue/unit (R > C) C: ordering cost/unit V: salvage value/unit Decision Variable: Y Order Quantity Maximize: Expected Profit Revenue: R*min(D,Y) Cost: C*Y Salvage Value: V*(Y‐D)+ What does (Y­D)+ mean? X+ = X if X > 0 0 Otherwise What is the objective of the problem? Objective: max E[R*min(D,Y) – (C*Y) + V*(Y‐D)+] Rule 1: min(D,Y) = D – (D‐Y)+ Rule 2: Y = D – (D‐Y)+ + (Y‐D)+ Taking these 2 rules into account, the new objective function is… Objective: P(Y) = max (R‐C)*E[D] – E[(R‐C)*(D‐Y)+ + (C‐V)(Y‐D)+] IEOR 151 – October 11, 2010 Notes Below is what each term in the above objective function represents. (R‐C)*E[D] is the profit if there is no randomness Cu = (R‐C) underage cost (D‐Y)+ is the number of underage units Co = (C‐V) overage cost (Y‐D)+ is the number of overage units From here on out, (R­C) will be referred to as Cu and (C­V) will be referred to as Co. So what we really want to do is… min G(Y) = E[Cu(D‐Y)+ + Co(Y‐D)+] ‐ In order to do so, take derivative! ‐ How do we take the derivative of a term like (D‐Y)+? Notice that for the term (D­Y)+, there are two cases: if D <= Y an increase of Y by 1 unit has no impact if D > Y an increase of Y by 1 unit will decrease the objective value by 1 So… G’(Y) = Cu[Prob(D<=Y)*0 + Prob(D>Y)*(‐1)] + Co[Prob(D<=Y)*1 + Prob(D>Y)*0] = 0 Simplified… G’(Y) = ‐Cu*Prob(D>Y) + Co*Prob(D<=Y) Now notice… Prob(D>Y) = 1 – Prob(D<=Y) Therefore… G’(Y) = ‐Cu*[1 – Prob(D<Y)] + Co*Prob(D<Y) = 0 Use the following to solve… Prob(D<Y) = (Cu)/(Cu + Co) = (R‐C)/(R‐V) ...
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## This document was uploaded on 02/01/2012.

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