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Order Quantities when Demand is Approximately Level Chapter 5 Inventory Management Dr. Ron Tibben-Lembke Inventory Costs Costs associated with inventory: Cost of the products Cost of ordering Cost of hanging onto it Cost of having too much / disposal Cost of not having enough (shortage) Shrinkage Costs How much is stolen? 2% for discount, dept. stores, hardware, convenience, sporting goods 3% for toys & hobbies 1.5% for all else Where does the missing stuff go? Employees: 44.5% Shoplifters: 32.7% Administrative / paperwork error: 17.5% Vendor fraud: 5.1% Inventory Holding Costs Category % of Value Housing (building) cost 6% Material handling 3% Labor cost 3% Opportunity/investment Pilferage/scrap/obsolescence Total Holding Cost 11% 3% 26% ABC Analysis Divides on-hand inventory into 3 classes Basis is usually annual $ volume A class, B class, C class $ volume = Annual demand x Unit cost Policies based on ABC analysis Develop class A suppliers more Give tighter physical control of A items Forecast A items more carefully Classifying Items as ABC % Annual $ Usage 100 80 60 A 40 B 20 C 0 0 50 100 % of Inventory Items 150 ABC Classification Solution Stock # Vol. 206 105 019 144 207 26,000 200 2,000 20,000 7,000 Total Cost $ Vol. $ 36 $936,000 600 120,000 55 110,000 4 80,000 10 70,000 1,316,000 % ABC ABC Classification Solution Stock # Vol. 206 105 019 144 207 26,000 200 2,000 20,000 7,000 Total Cost $ Vol. $ 36 $936,000 600 120,000 55 110,000 4 80,000 10 70,000 % 71.1 9.1 8.4 6.1 5.3 1,316,000 100.0 ABC A A B B C Economic Order Quantity Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs: A - setup cost per order v - unit cost r - holding cost per unit time EOQ Inventory Level Q* Optimal Order Quantity Decrease Due to Constant Demand Time EOQ Inventory Level Q* Optimal Order Quantity Instantaneous Receipt of Optimal Order Quantity Time EOQ Inventory Level Q* Reorder Point (ROP) Time Lead Time EOQ Inventory Level Q* Average Inventory Q/2 Reorder Point (ROP) Time Lead Time Total Costs Average Inventory = Q/2 Annual Holding costs = rv * Q/2 # Orders per year = D / Q Annual Ordering Costs = A * D/Q Annual Total Costs = Holding + Ordering Q D TC (Q) vr * A * 2 Q How Much to Order? Annual Cost Holding Cost = H * Q/2 Order Quantity How Much to Order? Annual Cost Ordering Cost = A * D/Q Holding Cost = H * Q/2 Order Quantity How Much to Order? Annual Cost Total Cost = Holding + Ordering Order Quantity How Much to Order? Total Cost = Holding + Ordering Annual Cost Optimal Q Order Quantity Optimal Quantity Total Costs = Q D vr * A * 2 Q Optimal Quantity Total Costs = Q D vr * A * 2 Q Take derivative with respect to Q = vr D A* 2 2 Q Optimal Quantity Total Costs = Q D vr * A * 2 Q Take derivative with respect to Q = vr D A* 2 0 2 Q Set equal to zero Optimal Quantity Total Costs = Q D vr * A * 2 Q Take derivative with respect to Q = vr D A* 2 0 2 Q Solve for Q: vr DA 2 2 Q Set equal to zero Optimal Quantity Total Costs = Q D vr * A * 2 Q Take derivative with respect to Q = vr D A* 2 0 2 Q Solve for Q: vr DA 2 2 Q 2 AS Q vr 2 Set equal to zero Optimal Quantity Total Costs = Q D vr * A * 2 Q Take derivative with respect to Q = vr D A* 2 0 2 Q Set equal to zero Solve for Q: vr DA 2 2 Q 2 AS Q vr 2 2 AS Q vr Sensitivity Suppose we do not order optimal EOQ, but order Q instead, and Q is p percent larger Q = (1+p) * EOQ Percentage Cost Penalty given by: p2 PCP 50 1 p EOQ = 100, Q = 150, so p = 0.5 50*(0.25/1.5) = 8.33 a 8.33% cost increase Figure 5.3 Sensitivity Percentage Cost Penalty using Q different from the EOQ 30 25 20 PCP 15 10 5 0 -0.6 -0.4 -0.2 0 -5 p 0.2 0.4 0.6 A Question: If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy? Benefits of EOQ Profit function is very shallow Even if conditions don’t hold perfectly, profits are close to optimal Estimated parameters will not throw you off very far Tabular Aid 5.1 For A = $3.20 and r = 0.24% Calculate Dv =total $ usage (or sales) Find where Dv fits in the table Use that number of months of supply D = 200, v = $16, Dv=$3,200 From table, buy 1 month’s worth Q = D/12 = 200/12 = 16.7 = 17 How do you get a table? Decide which T values you want to consider: 1 month, etc. Use same v and r values for whole table For each neighboring set of T’s, put them into 288 A Dv T1T2 r How do you get a table? For example, A = $3.20, r = 0.24 To find the breakpoint between 0.25 and 0.5 Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24) = 921.6 / 0.03 = 30,720 So if Dv is less than this, use 0.25, more than that, use 0.5 Find 0.5 and 0.75 breakpoint: Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240 Why care about a table? Some simple calculations to get set up No thinking to figure out lot sizes Every product with the same ordering cost and holding cost rate can use it Real benefit - simplified ordering Every product ordered every 1 or 2 weeks, or every 1, 2, 3, 4, 6, 12 months Order multiple products on same schedule: Get volume discounts from suppliers Save on shipping costs Savings outweigh small increase from non-EOQ orders Uncoordinated Orders Time Simultaneous Orders Time Same T = number months supply allows firm to order at same time, saving freight and ordering expenses Adjusted some T’s, changed order times Offset Orders Same T = number months supply allows firm to control maximum inventory level by coordinating replenishments With different T, no consistency Quantity Discounts How does this all change if price changes depending on order size? Explicitly consider price: 2 AD Q vr Discount Example D = 10,000 A = $20 Price v = 5.00 4.50 3.90 Quantity Q < 500 501-999 Q >= 1000 r = 20% EOQ 633 666 716 Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 500 1,000 Order Size Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 500 1,000 Order Size Discount Example Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70 Order 10,000/666 * 20 = $300.00 Mat’l 10,000*4.50 = $45,000.00 45,599.70 Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2=$390.00 Order 10,000/1,000 * 20 = $200.00 Mat’l 10,000*3.90 = $39,000.00 39,590.00 Discount Model 1.Compute EOQ for each price 2.Is EOQ ‘realizeable’? (is Q in range?) If EOQ is too large, use lowest possible value. If too small, ignore. 3.Compute total cost for this quantity 4.Select quantity/price with lowest total cost. Adding Lead Time Use same order size Order before inventory depleted R = DL where: 2DA Q vr D = annual demand rate L = lead time in years