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.
 Spring '09

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