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EOQ Model 2/26/2016
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UTD-Branded Hover Boards!
Business Idea… UTD-Branded Hover Boards!
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Demand Information 𝑑=4 𝑢𝑛𝑖𝑡𝑠 𝑑𝑎𝑦
5 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑤𝑒𝑒𝑘 ×50 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑤𝑒𝑒𝑘𝑠 𝑦𝑒𝑎𝑟 =250 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑦𝑒𝑎𝑟 𝐷=4 𝑢𝑛𝑖𝑡𝑠 𝑑𝑎𝑦 ×250 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑦𝑒𝑎𝑟 =1,000 𝑢𝑛𝑖𝑡𝑠 𝑦𝑒𝑎𝑟
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Demand Information 𝑑=4 𝑢𝑛𝑖𝑡𝑠 𝑑𝑎𝑦 250 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑦𝑒𝑎𝑟
Assume that demand arrives at a perfectly CONSTANT rate. 250 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑦𝑒𝑎𝑟 Assume that we MUST satisfy ALL 1,000 units of demand. 𝐷=1,000 𝑢𝑛𝑖𝑡𝑠 𝑦𝑒𝑎𝑟
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1 Order/Year @ 1,000 Units/Order
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2 Orders/Year @ 500 Units/Order
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4 Orders/Year @ 250 Units/Order
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50 Orders/Year @ 20 Units/Order
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where Q is order quantity
Orders/Year 𝑂𝑟𝑑𝑒𝑟𝑠 𝑌𝑒𝑎𝑟 = 𝐷 𝑄 where Q is order quantity 𝑄=1,000: 1,000 𝑈𝑛𝑖𝑡𝑠/𝑌𝑒𝑎𝑟 1,000 𝑈𝑛𝑖𝑡𝑠/𝑂𝑟𝑑𝑒𝑟 =1 𝑂𝑟𝑑𝑒𝑟 𝑌𝑒𝑎𝑟 𝑄=20: 1,000 𝑈𝑛𝑖𝑡𝑠/𝑌𝑒𝑎𝑟 20 𝑈𝑛𝑖𝑡𝑠/𝑂𝑟𝑑𝑒𝑟 =50 𝑂𝑟𝑑𝑒𝑟𝑠 𝑌𝑒𝑎𝑟
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What is the best order size?
Depends on: Fixed ordering cost Inventory holding costs
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Fixed Shipping Cost (S)
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Inventory Holding Costs (H)
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Inventory Holding Costs (H)
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Comparative Statics: Shipping Cost
> 𝑄 $ ⋛ 𝑄 𝑓𝑟𝑒𝑒
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Comparative Statics: Shipping Cost
𝑆↓ ⟹ 𝑄↓ 𝑆↑ ⟹ 𝑄↑
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Comparative Statics: Holding Cost
𝑄 𝑁𝑌𝐶 ⋛ 𝑄 𝑁𝑜𝑤ℎ𝑒𝑟𝑒 <
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Comparative Statics: Holding Cost
𝐻↑ ⟹ 𝑄↓ 𝐻↓ ⟹ 𝑄↑
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𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑄 2 =500 𝑈𝑛𝑖𝑡𝑠
Q = 1,000 Units/Order 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑄 2 =500 𝑈𝑛𝑖𝑡𝑠
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𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑄 2 =250 𝑈𝑛𝑖𝑡𝑠
Q = 500 Units/Order 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑄 2 =250 𝑈𝑛𝑖𝑡𝑠
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Symbol Recap 𝑄:𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑜𝑟𝑑𝑒𝑟
𝑄 2 :𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑙𝑒𝑣𝑒𝑙 𝑆:𝐹𝑖𝑥𝑒𝑑 𝑠ℎ𝑖𝑝𝑝𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝐷:𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑:𝐷𝑎𝑖𝑙𝑦 𝑑𝑒𝑚𝑎𝑛𝑑 𝐷 𝑄 :𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟𝑠/𝑦𝑒𝑎𝑟
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Unaffected by order size Q
Total Cost 𝑇𝑜𝑡𝑎𝑙 𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑜𝑠𝑡 = 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝐶𝑜𝑠𝑡𝑠 + 𝑆ℎ𝑖𝑝𝑝𝑖𝑛𝑔 𝐶𝑜𝑠𝑡𝑠 + 𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐺𝑜𝑜𝑑𝑠 𝑆𝑜𝑙𝑑 Unaffected by order size Q 1,000 𝑢𝑛𝑖𝑡𝑠×𝑢𝑛𝑖𝑡 𝑐𝑜𝑠𝑡=𝐹𝐼𝑋𝐸𝐷 𝑁𝑈𝑀𝐵𝐸𝑅
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Total Cost 𝑇𝑜𝑡𝑎𝑙 𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑜𝑠𝑡 =
𝐴𝑣𝑔𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 × 𝐴𝑛𝑛𝑢𝑎𝑙 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝐶𝑜𝑠𝑡 + 𝑁𝑢𝑚𝑏𝑒𝑟 𝑂𝑟𝑑𝑒𝑟𝑠 𝑝𝑒𝑟 𝑌𝑒𝑎𝑟 × 𝐹𝑖𝑥𝑒𝑑 𝑆ℎ𝑖𝑝𝑝𝑖𝑛𝑔 𝐶𝑜𝑠𝑡
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Total Cost 𝑇𝐶= 𝑄 2 ×𝐻 + 𝐷 𝑄 ×𝑆 Objective: Minimize TC by adjusting Q
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Remember Calculus?
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Remember Calculus?
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Solution to this equation:
Minimum Total Cost 𝜕 𝑇𝐶 𝜕 𝑄 = 𝜕 𝜕𝑄 𝑄 2 ×𝐻 + 𝜕 𝜕𝑄 𝐷 𝑄 ×𝑆 =0 Solution to this equation: 𝑄 ∗ = 𝑄 𝑂𝑝𝑡 = 𝑄 𝑂 = 2×𝐷×𝑆 𝐻
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Economic Order Quantity
𝑄 ∗ = 𝑄 𝑂𝑝𝑡 = 𝑄 𝑂 = 2×𝐷×𝑆 𝐻 𝑆↓ ⟹ 𝑄↓ 𝑆↑ ⟹ 𝑄↑ 𝐻↑ ⟹ 𝑄↓ 𝐻↓ ⟹ 𝑄↑
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Economic Order Quantity
𝑄 ∗ = 𝑄 𝑂𝑝𝑡 = 𝑄 𝑂 = 2×𝐷×𝑆 𝐻 # 𝑂𝑟𝑑𝑒𝑟𝑠 𝑌𝑒𝑎𝑟 = 𝐷 𝑄 ∗ 𝐷𝑎𝑦𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑜𝑟𝑑𝑒𝑟𝑠= 𝑄 ∗ 𝑑
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Economic Order Quantity
𝑄 ∗ = 𝑄 𝑂𝑝𝑡 = 𝑄 𝑂 = 2×𝐷×𝑆 𝐻 𝑇 𝐶 𝑚𝑖𝑛 = 𝑄 ∗ 2 ×𝐻 + 𝐷 𝑄 ∗ ×𝑆
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When to Reorder? 𝑅𝑂𝑃=𝑑×𝐿𝑇
Objective: Sell last unit just as order arrives 𝑑=4 𝑢𝑛𝑖𝑡𝑠 𝑑𝑎𝑦 𝑅𝑂𝑃=𝑑×𝐿𝑇 𝑅𝑂𝑃:𝑅𝑒𝑜𝑟𝑑𝑒𝑟 𝑃𝑜𝑖𝑛𝑡 𝐿𝑇:𝐿𝑒𝑎𝑑 𝑇𝑖𝑚𝑒 𝑖𝑛 # 𝐷𝑎𝑦𝑠 𝐿𝑇=2 𝑑𝑎𝑦𝑠 𝑅𝑂𝑃=4 𝑢𝑛𝑖𝑡𝑠 𝑑𝑎𝑦 ×2 𝑑𝑎𝑦𝑠 =8 𝑢𝑛𝑖𝑡𝑠
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Monday Watch for practice problem set!
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EPQ Model 2/29/2015
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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.
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Quiz Solutions ― D d S H QO LT SS Days Worked per Year 240 days/year
Symbol Value Days Worked per Year ― 240 days/year Annual Demand D 2,400 units/year Daily Demand d 10 units/day Setup Cost per Order S $100.00 /order Annual Holding Cost per Unit H $3.00 /unit per year Optimal Order Size QO units/order Supplier Lead Time LT 3 days Safety Stock SS 20 units
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Quiz Solutions 𝑄 𝑂 = 2×𝐷×𝑆 𝐻 𝑄 𝑂 = 2×2,400×$100 $3 = 160,000 𝑄 𝑂 =400
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But what if you had to PRODUCE that inventory over time?
In EOQ model, shipments from supplier allows inventory to increase instantly. But what if you had to PRODUCE that inventory over time?
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Limited Production Capacity
Inventory Time Production Phase Consumption Phase
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Limited Production Capacity
Inventory Time Production Consumption Production Consumption
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Limited Production Capacity
𝐻: Annual Holding Cost per Unit 𝑆: Setup Cost to Start a Production Run 𝑝: Daily Production Rate in Units 𝑢: Daily Usage Rate in Units (same as Daily Demand 𝑑) Inventory Time Average Inventory Production Consumption Production Consumption
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Limited Production Capacity
Inventory Time Some Customers Will Arrive Here and Find Zero Inventory
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Limited Production Capacity
Inventory Time
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Limited Production Capacity
Inventory Time
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Limited Production Capacity
Inventory Time Production & Consumption Consumption Only Production & Consumption Consumption Only
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Limited Production Capacity
Inventory Time Total Cumulative Production Production & Consumption Consumption Only Production & Consumption Consumption Only Total Consumption During P&C Phase Inventory Level
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UNlimited Production Capacity
Inventory Time Production & Consumption Consumption Only Production & Consumption Consumption Only
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Economic ORDER Quantity Model
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Economic PRODUCTION Quantity
0≪𝑢≪𝑝≪∞ Inventory Time Production & Consumption Consumption Only Production & Consumption Consumption Only 𝐻: Annual Holding Cost per Unit 𝑆: Setup Cost to Start a Production Run 𝑝: Daily Production Rate in Units 𝑢: Daily Usage Rate in Units (same as Daily Demand 𝑑)
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Economic Production Quantity
𝑄 𝑂 = 2×𝐷×𝑆 𝐻 × 𝑝 𝑝−𝑢 𝐼 𝑚𝑎𝑥 = 𝑄 𝑂 𝑝 × 𝑝−𝑢 𝐼 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝐼 𝑚𝑎𝑥 2 Inventory Time Production & Consumption Consumption Only Production & Consumption Consumption Only 𝑄 𝑂 𝐼 𝑚𝑎𝑥 𝐼 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝐻: Annual Holding Cost per Unit 𝑆: Setup Cost to Start a Production Run 𝑝: Daily Production Rate in Units 𝑢: Daily Usage Rate in Units (same as Daily Demand 𝑑)
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Economic Production Quantity
𝑇𝐶= 𝐼 𝑚𝑎𝑥 2 ×𝐻 + 𝐷 𝑄 ×𝑆 Runs per Year= 𝐷 𝑄 𝑂 Inventory Time Production & Consumption Consumption Only Production & Consumption Consumption Only 𝑄 𝑂 𝐼 𝑚𝑎𝑥 𝐼 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝐻: Annual Holding Cost per Unit 𝑆: Setup Cost to Start a Production Run 𝑝: Daily Production Rate in Units 𝑢: Daily Usage Rate in Units (same as Daily Demand 𝑑)
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