Inventory Theory. Inventory Theory

Transcription

1 Inventory Theory Contents 1 Definitions 2 Economic Order Quantity Model 3 EOQ Example 4 Economic Order Quantity Model with hortages 5 EOQ with hortages Example 6 ingle Period tochastic Inventories 7 Newsboy Example 8 Main Distributions 9 Airline Overbooking Definitions Inventory Theory Demand: The number of units/items needed/used/sold at time t Unit cost: Cost per unit/item. (Materials/labour/etc for production of one piece; price/shipping/etc for purchasing.) et-up cost: Also: Order cost. The initial cost of preparing for each batch produced/order placed. (etting up machinery, accounting, equipment needed for production/handling, etc.; cost of placing an order.) Holding cost: Cost for storing items before they are used/sold. (torage, cost of capital tied up in inventory) hortage cost: Cost of back-orders, cost of lost sales (estimate), etc. Total Cost = Batch Cost+ Order Cost cycle cycle year Holding Cost + cycle cycle year hortage Cost + cycle cycle year

2 Inventory Theory Q b d =(Q b) D(t t i ) Q o b tock Level in Inventory Constant Demand (linear) implified Inventory Model The imple Economic Order Quantity (EOQ) Model Assume Let 1. Constant linear demand 2. No shortages are allowed 3. No delay on orders (lead time); i.e., orders arrive immediately D = Total demand; Q = # units per order; Q/2 = Average inventory; The total cost is c = Cost per unit K = Fixed cost per order h = Holding cost per item Cost = (batch cost) + (order cost) + (holding cost) = c D + K D Q Minimizing w.r.t. Q gives Q = + h Q 2 2KD h ; the EOQ formula.

3 implified Inventory Model EOQ Total Cost EOQ Holding Cost Order Cost EOQ on the Total Cost Curve EOQ Example Example An accounting firm buys boxes of 1 DVDs for storing files. The supplier charges $5 per box of high-quality, high-capacity DVDs. The firm uses 1, DVDs per year. The cost of ordering is $2 (local processing, then shipping and handling charges). The holding cost is $1 per box. 1. What is the optimal ordering quantity? 2. How many orders are placed per year? 3. What is the maximum inventory level?

4 implified Inventory Model with hortages The EOQ Model with hortages Assume 1. Constant linear demand 2. No delay on orders (lead time); i.e., orders arrive immediately Let D = Total demand (unit/yr) c = Cost per unit Q = # units per order K = Fixed cost per order M/2 = Average inventory h = Holding cost (unit/yr) (Q M)/2 = Average shortage s = hortage cost (unit/yr) The total cost is Cost = (batch cost) + (order cost) + (holding cost) + (shortage cost) D M2 (Q M)2 K + h + s Q 2Q 2Q q q q q 2KD h+s 2KD s Minimizing wrt [Q, M] gives Q = and M = s h h h+s = D c + EOQ with hortages EOC urface

5 EOQ with hortages Example Example Midtown Optometry Clinic sells 1, frames per year. A local supplier charges $15 per frame; ordering incurs a $5 charge (shipping, handling, and processing). hortage costs are estimated at $15 per frame (lost sales estimate). Holding cost is $4.5 per frame. 1. What is the optimal ordering quantity? 2. How many orders are placed per year? 3. What is the maximum inventory level? 4. What is the maximum shortage? Basic tructure ingle Period tochastic Inventories Items are offered for sale for a single period ( perishable demand ). c: Purchase cost per unit p: ale price per unit a: alvage value per unit at the end of the period d: hortage cost per unit : Number of units ordered x: Number of units sold, random variable with pdf f (x) & CDF F(x) { +a( x) if x Profit = px c + d(x ) if x s E[Profit] = p x f dx + p f dx c + a ( x) f dx d (x ) f dx s

6 Algebra E[Profit] = p ingle Period tochastic Inventories s x f dx + p = p µ c + a f dx c + a ( x) f dx (d + p) = p µ c + ae e (d + p)e ( x) f dx d (x ) f dx (x ) f dx where E e is the expected surplus and E is the expected shortage. Calculus d d E[Profit] = c + a f dx + (d + p) f dx d E[Profit] = = c + af() + (d + p)(1 F()) = d F( ) = p + d c p + d a ingle Period tochastic Inventory Example Example (Classic Newsboy Problem ) Determine how many newspapers to purchase for the day. c = $.1: Purchase cost per newspaper p = $.25: ale price per newspaper a = $.2: alvage value per newspaper at the end of the day d = $.15: hortage cost per newspaper (from p c) =? : Number of newspapers ordered x: Newspapers sold, normal random variable N [µ = 25 & σ = 5] olution: 1. N [,1] ( ) = p+d c p+d a = z = from (N [,1] (.84).789) 3. = µ + z σ = E[Profit] = Maple

7 Main Distributions Continuous Normal Distribution PDF CDF N [µ,σ] (x) = 1 1 (x µ) 2 2πσ 2 e 2 σ 2 = σ 1 N x µ [,1]( σ [ ( )] ˆN [µ,σ] (x) = erf x µ σ 2 Discrete Poisson Distribution PDF P λ (k) = λ k e λ CDF ˆP λ (k) = k! Γ( k + 1,λ) k! A Poisson distribution is appropriate if: Events k =,1,2,... are discrete Events occur independently. The rate events occur at is constant Events cannot occur simultaneously Probability of an event in a small interval length(interval). ) Maple ingle Period Inventory: Airline Overbooking Overbooking as an Inventory Problem Consider the seats for a flight as units of inventory. Ticket: Nonrefundable reservation for a unit (seat on the flight) No-how: Failure to arrive for the flight Overbooking: elling more reservations than seats Purchase price: Cost of ticket Denied-boarding cost: compensation and loss of goodwill Model Parameters r: Revenue per seat L: Number of seats on the flight n: Number of tickets sold p: Probability reservation arrives s: hortage cost for unsatisfied demand Random Variables D(n): Demand given n seats sold P[D(n)=d] = ( n) d p d (1 p) n d U(n): Unsatisfied { demand D(n) L = D(n) L D(n) > L E[U(n)] = n d=l+1 (d L)P[D(n) = d]

8 ingle Period Inventory: Airline Overbooking Marginal Analysis The binomial distribution is discrete; use marginal analysis to replace the derivative. (A continuous normal distribution N [np, np(1 p)] approximates a binomial for large n.) Profit P(n) = r n su(n). Then E(P(n)) = r n se(u(n)). o that E(P(n)) = E(P(n + 1)) E(P(n)) = r s E(U(n)) = r s pp[d(n) L]. When n < L, then E(U(n)) = ; if n > L, then E(U(n)) increases as n grows. o search for the value n such that E(P(n 1)) > and E(P(n )) or P[D(n 1) L] < r s p and r s p P[D(n ) L] witching to continuous: N [np, np(1 p)] gives n approximately from N n [np, np(1 p)] (n)dn = r s p Approximating a Binomial with a Normal Courtesy of the Central Limit Theorem Let Y = X 1 + X X n be independent, identically-distributed random binomial variables P n (d), then Z = Y np np(1 p) d N [,1] And P n (d) N [np, np(1 p)] General Rule of Thumb The sample size n is sufficiently large when np 5 and n(1 p) 5 Maple

9 Airline Overbooking Example Example (Transcontinental Airlines [H&L, pg 86]) Transcontinental Airlines has a daily flight from FO to ORD with 15 seats. The average fare of $3 is nonrefundable; no-shows forfeit $3. Typically, 2% are no-shows. Customers bumped from this flight are put on the next available flight to ORD or on another airline at an average cost of $2, and are given a voucher worth $4 (costing the company $3) for use on a future flight; an additional $5 is assessed as an intangible cost for loss of goodwill. The total cost of bumping a customer is $1,. Each flight is currently overbooked by 15 seats. Is this optimal? Parameters: p =.8, r = $3, s = $1,, L = 15. olution: Maple