Inventory Theory. Inventory Theory

Transcription

1 Inventory Theory Contents 1 Definitions 2 Economic Order uantity Model 3 EO Example 4 Economic Order uantity Model with hortages 5 EO 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, euipment 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 b d =( b) D(t t i ) o b tock Level in Inventory Constant Demand (linear) implified Inventory Model The imple Economic Order uantity (EO) 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; = # units per order; /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 Minimizing w.r.t. gives = + h 2 2KD h ; the EO formula.

3 implified Inventory Model EO Total Cost EO Holding Cost Order Cost EO on the Total Cost Curve EO Example Example An accounting firm buys boxes of 1 DVDs for storing files. The supplier charges $5 per box of high-uality, 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 uantity? 2. How many orders are placed per year? 3. What is the maximum inventory level?

4 implified Inventory Model with hortages The EO 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 = # units per order K = Fixed cost per order M/2 = Average inventory h = Holding cost (unit/yr) ( M)/2 = Average shortage s = hortage cost (unit/yr) The total cost is Cost = (batch cost) + (order cost) + (holding cost) + (shortage cost) D M2 ( M)2 K + h + s 2 2 2KD h+s 2KD s Minimizing wrt [, M] gives = and M = s h h h+s = D c + EO with hortages EOC urface

5 EO 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 uantity? 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