EOQ Model 2/26/2016 Demand Information π’πππ‘π π=4 πππ¦ π€ππππππ πππ¦π π€ππππππ π€ππππ π€ππππππ πππ¦π 5 × 50 = 250 π€πππ π¦πππ π¦πππ π’πππ‘π π€ππππππ πππ¦π π’πππ‘π π·=4 × 250 = 1,000 πππ¦ π¦πππ π¦πππ Demand Information π’πππ‘π π=4 πππ¦ Assume that demand arrives at a perfectly CONSTANT rate. π€ππππππ πππ¦π 250 π¦πππ Assume that we MUST satisfy ALL 1,000 units of demand. π’πππ‘π π· = 1,000 π¦πππ 1 Order/Year @ 1,000 Units/Order 2 Orders/Year @ 500 Units/Order 4 Orders/Year @ 250 Units/Order 50 Orders/Year @ 20 Units/Order Orders/Year ππππππ π· = ππππ π where Q is order quantity 1,000 ππππ‘π /ππππ πππππ π = 1,000: =1 1,000 ππππ‘π /πππππ ππππ 1,000 ππππ‘π /ππππ ππππππ π = 20: = 50 20 ππππ‘π /πππππ ππππ What is the best order size? Depends on: o Fixed ordering cost o Inventory holding costs Fixed Shipping Cost (S) Inventory Holding Costs (H) Inventory Holding Costs (H) Comparative Statics: Shipping Cost π$5 β > πππππ Comparative Statics: Shipping Cost πβ βΉ πβ πβ βΉ πβ Comparative Statics: Holding Cost ππππΆ β < ππππ€βπππ Comparative Statics: Holding Cost π»β βΉ πβ π»β βΉ πβ Q = 1,000 Units/Order π π΄π£πππππ πΌππ£πππ‘πππ¦ = = 500 ππππ‘π 2 Q = 500 Units/Order π π΄π£πππππ πΌππ£πππ‘πππ¦ = = 250 ππππ‘π 2 Symbol Recap π: ππ’ππππ ππ π’πππ‘π ππ πππβ πππππ π : π΄π£πππππ πππ£πππ‘πππ¦ πππ£ππ 2 π: πΉππ₯ππ π βππππππ πππ π‘ π·: π΄πππ’ππ ππππππ π: π·ππππ¦ ππππππ π· : ππ’ππππ ππ ππππππ /π¦πππ π Total Cost πππ‘ππ π΄πππ’ππ πΆππ π‘ = π»ππππππ πΆππ π‘π + πβππππππ πΆππ π‘π + πΆππ π‘π ππ πΊππππ ππππ 1,000 π’πππ‘π × π’πππ‘ πππ π‘ = πΉπΌππΈπ· ππππ΅πΈπ Unaffected by order size Q Total Cost πππ‘ππ π΄πππ’ππ πΆππ π‘ = π΄π£ππππππ πΌππ£πππ‘πππ¦ × π΄πππ’ππ π»ππππππ πΆππ π‘ + ππ’ππππ ππππππ πππ ππππ × πΉππ₯ππ πβππππππ πΆππ π‘ Total Cost π π· ππΆ = ×π» + ×π 2 π Objective: Minimize TC by adjusting Q Remember Calculus? Remember Calculus? Minimum Total Cost π ππΆ π π π π· = ×π» + ×π =0 ππ ππ 2 ππ π Solution to this equation: β π = ππππ‘ = ππ = 2×π·×π π» Economic Order Quantity β π = ππππ‘ = ππ = 2×π·×π π» πβ βΉ πβ π»β βΉ πβ πβ βΉ πβ π»β βΉ πβ Economic Order Quantity β π = ππππ‘ = ππ = # ππππππ π· = β ππππ π 2×π·×π π» πβ π·ππ¦π πππ‘π€πππ ππππππ = π Economic Order Quantity β π = ππππ‘ = ππ = ππΆπππ 2×π·×π π» πβ π· = ×π» + β×π 2 π When to Reorder? Objective: Sell last unit just as order arrives π’πππ‘π π=4 πππ¦ π ππ = π × πΏπ π ππ: π ππππππ πππππ‘ πΏπ: πΏπππ ππππ ππ # π·ππ¦π πΏπ = 2 πππ¦π π’πππ‘π π ππ = 4 × 2 πππ¦π πππ¦ = 8 π’πππ‘π Monday Watch for practice problem set! EPQ Model 2/29/2015 Quiz A skateboard retailer enjoys a constant demand of exactly 200 customers every month. The cost of ordering and receiving shipments is $100 per order. Their accounting department estimates that annual carrying costs are $3.00 per unit. The lead time for shipments of new products is 3 days. The store operates 240 days per year. Each order is received from the supplier in a single delivery. Company policy is to carry a safety stock of 20 units. There are no quantity discounts. Quiz Solutions Symbol Value Days Worked per Year β 240 days/year Annual Demand D 2,400 units/year Daily Demand d Setup Cost per Order S Annual Holding Cost per Unit H $3.00 /unit per year Optimal Order Size QO β units/order Supplier Lead Time LT 3 days Safety Stock SS 20 units 10 units/day $100.00 /order Quiz Solutions ππ = ππ = 2×π·×π π» 2 × 2,400 × $100 = $3 ππ = 400 160,000 One Year 450 400 Units of Inventory 350 300 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Day Order EoD Inv 140 150 160 170 180 190 200 210 220 230 240 In EOQ model, shipments from supplier allows inventory to increase instantly. One Year 450 400 Units of Inventory 350 300 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 Day Order But what if you had to PRODUCE that inventory over time? 230 240 Inventory Limited Production Capacity Time Production Phase Consumption Phase Inventory Limited Production Capacity Time Production Consumption Production Consumption Inventory Limited Production Capacity π»: Annual Holding Cost per Unit π: Setup Cost to Start a Production Run π: Daily Production Rate in Units π’: Daily Usage Rate in Units (same as Daily Demand π) Average Inventory Time Production Consumption Production Consumption Inventory Limited Production Capacity Time Some Customers Will Arrive Here and Find Zero Inventory ο Inventory Limited Production Capacity Time Inventory Limited Production Capacity Time Inventory Limited Production Capacity Production & Consumption Consumption Only Production & Consumption Consumption Only Time Inventory Limited Production Capacity Total Cumulative Production Inventory Level Production & Consumption Consumption Only Production & Consumption Consumption Only Total Consumption During P&C Phase Time Inventory UNlimited Production Capacity Production & Consumption Consumption Only Production & Consumption Consumption Only Time Economic ORDER Quantity Model One Year 450 400 Units of Inventory 350 300 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Day Order EoD Inv 140 150 160 170 180 190 200 210 220 230 240 Economic PRODUCTION Quantity Inventory 0βͺπ’βͺπβͺβ Production & Consumption Consumption Only Production & Consumption π»: Annual Holding Cost per Unit π: Setup Cost to Start a Production Run π: Daily Production Rate in Units π’: Daily Usage Rate in Units (same as Daily Demand π) Consumption Only Time Economic Production Quantity Inventory ππ = 2×π·×π π × π» πβπ’ Production & Consumption πΌπππ₯ = Consumption Only ππ × πβπ’ π Production & Consumption πΌππ£πππππ = πΌπππ₯ 2 Consumption Only ππ πΌπππ₯ πΌππ£πππππ π»: Annual Holding Cost per Unit π: Setup Cost to Start a Production Run π: Daily Production Rate in Units π’: Daily Usage Rate in Units (same as Daily Demand π) Time Economic Production Quantity Inventory ππΆ = πΌπππ₯ π· ×π» + ×π 2 π Production & Consumption Consumption Only Runs per Year = Production & Consumption π· ππ Consumption Only ππ πΌπππ₯ πΌππ£πππππ π»: Annual Holding Cost per Unit π: Setup Cost to Start a Production Run π: Daily Production Rate in Units π’: Daily Usage Rate in Units (same as Daily Demand π) Time
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