1. Inventory Models in SC Environment By Debadyuti Das

2. Three levels of Inventory Decisions  Supply Chain Decisions (Strategic) - What are the potential alternatives to inventory? - How should the product be designed?  Deployment Decisions (Strategic) - What items should be carried as inventory? - In what form should they be maintained? - How much of each should be held and where?  Replenishment Decisions (Tactical/Operational) - How often should inventory status be determined? - When should a replenishment decision be made? - How large should the replenishment be?

3. Classification of Inventory  Financial / Accounting Categories - Raw Materials - Work in process (WIP) - Components, Semi-Finished Goods - Finished Goods  Functional Classification - Cycle Stock - Safety Stock - Pipeline Inventory - Decoupling Stock - Anticipation Inventory

4. Total Relevant Costs (TRC) TC = Purchase + Order + Holding + Shortage  What makes a cost relevant?  Four standard cost components - Purchase (unit value) cost - Ordering (Set up) cost - Holding (Carrying) cost - Shortage cost

5. What factors influence inventory replenishment models?  Demand  Lead time  Dependence of items  Review time  Number of echelons  Discounts  Shortages  Perishability  Planning horizon  Number of items

6. Basic EOQ Model: Assumptions  Demand - Constant - Known - Continuous  Lead time - Instantaneous  Dependence of items - Independent  Review time - Continuous  Number of echelons - One

7. Basic EOQ Model: Assumptions  Discounts - None  Shortages - None  Perishability - None  Planning horizon - Infinite  Number of items - One

8. The Inventory Cycle Profile of Inventory Level Over Time Quantity on hand Q Receive order Place order Receive order Place order Receive order Lead time Reorder point Usage rate Time

9. Basic Terminologies D: Annual demand S: Setup or Order Cost C: Cost per unit h: Holding cost per year as a fraction of product cost H: Holding cost per unit per year Q: Lot Size T: Reorder interval Material cost is constant and therefore is not considered in this model

10. Basic Terminologies Number of orders per year = D/Q Annual material cost = CD Annual order cost = (D/Q)S Annual holding cost = (Q/2)H = (Q/2) hC Total annual cost = (TC) = CD + (D/Q)S + (Q/2) hC

11. Cost Minimization Goal Order Quantity (Q) The Total-Cost Curve is U-Shaped Ordering Costs QOQO Annual Cost ( optimal order quantity)

12. Deriving the EOQ Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

13. Minimum Total Cost The total cost curve reaches its minimum where the carrying and ordering costs are equal.

14. EOQ: Important Observations*  Tradeoff between set-up costs and holding costs when determining order quantity. In fact, we order so that these costs are equal per unit time  Total Cost is not particularly sensitive to the optimal order quantity

15. Example (1) Demand, D = 12,000 computers per year d = 1000 computers/month Unit cost, C = $500 Holding cost fraction, h = 0.2 Fixed cost, S = $4,000/order Q* = Sqrt[(2)(12000)(4000)/(0.2)(500)] = 980 computers Cycle inventory = Q/2 = 490 Flow time = Q/2d = 980/(2)(1000) = 0.49 month Reorder interval, T = Q/d = 0.98 month

16. Example 1(continued) Annual ordering and holding cost = = (12000/980)(4000) + (980/2)(0.2)(500) = $97,980 Suppose lot size is reduced to Q=200, which would reduce flow time: Annual ordering and holding cost = = (12000/200)(4000) + (200/2)(0.2)(500) = $250,000 To make it economically feasible to reduce lot size, the fixed cost associated with each lot would have to be reduced

17. Example 2 If desired lot size = Q* = 200 units, what would S have to be? D = 12000 units C = $500 h = 0.2 Use EOQ equation and solve for S: S = [hC(Q*) 2 ]/2D = [(0.2)(500)(200) 2 ]/(2)(12000) = $166.67 To reduce optimal lot size by a factor of k, the fixed order cost must be reduced by a factor of k 2

18. Key Points from EOQ Model  In deciding the optimal lot size, the tradeoff is between setup (order) cost and holding cost.  If demand increases by a factor of k, it is optimal to increase batch size by a factor of Sqrt k. No of orders placed per year should increase by a factor of Sqrt k. Flow time attributed to Cycle inventory should decrease by a factor of Sqrt k.  If lot size is to be reduced, one has to reduce fixed order cost. To reduce lot size by a factor of 2, order cost has to be reduced by a factor of 4.

19. Single period Inventory Model  Model for ordering of perishables and other items with limited useful lives  One time purchasing decision (Example: newspapers, vegetables, fresh fruits etc.)  Shortage cost:(or the underestimated cost) generally the unrealized profits per unit (Revenue/unit minus cost/unit).  Excess cost: (or the overestimated cost) difference between purchase cost and salvage value of items left over at the end of a period.

20. Single period Inventory Model  Seeks to balance the costs of inventory overstock and under stock.  The goal is to identify the order quantity that will minimize the long run excess and shortage costs.  Service level is the key to determining the optimal stock level in this model.

21. Single-Period Inventory Model

22. Single Period Inventory Model Example  Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell just before the game, but lose $5 on every shirt not sold. How many shirts should we make for the game? C u = $10 and C o = $5; SL = $10 / ($10 + $5) =.667 Z.667 =.432 therefore we need 2,400 +.432(350) = 2,551 shirts

23. Quantity Discount Model  Why quantity discounts? Coordination in the supply chain Price discrimination to maximize supplier profits  Lot size based  Volume based  How should buyer react?

24. All unit Quantity Discount Model  Pricing schedule has specified quantity break points q 0, q 1, …, q r, where q 0 = 0  If an order is placed that is at least as large as q i but smaller than q i+1, then each unit has an average unit cost of C i  The unit cost generally decreases as the quantity increases, i.e., C 0 >C 1 >…>C r  The objective for the company (a retailer in our example) is to decide on a lot size that will minimize the sum of material, order, and holding costs

25. Total Cost with Constant Carrying Costs OC EOQ Quantity Total Cost TC a TC c TC b Decreasing Price CC a,b,c

26. Total Cost with Carrying Costs expressed as % of unit price OC EOQ Quantity Total Cost TC a TC c TC b Decreasing Price CC c CC a CC b

27. Case 1: All unit Quantity Discount Procedure  Compute the common minimum point  Identify the feasible minimum point  If the feasible minimum point is on the lowest price range, that is the optimal order quantity.  If the feasible minimum point is on any other range, compute the total cost for the minimum point and for the price breaks of all lower unit costs.  Compare the total costs, the quantity that yields the lowest total cost is the optimal order quantity.

28. Case 2: All unit Quantity Discount Procedure Step 1: Calculate the EOQ for the lowest price. If it is feasible (i.e., this order quantity is in the range for that price), then stop. This is the optimal lot size. Calculate TC for this lot size. Step 2: If the EOQ is not feasible, calculate the TC for this price and the smallest quantity for that price. Step 3: Calculate the EOQ for the next lowest price. If it is feasible, stop and calculate the TC for that quantity and price. Step 4: Compare the TC for Steps 2 and 3. Choose the quantity corresponding to the lowest TC. Step 5: If the EOQ in Step 3 is not feasible, repeat Steps 2, 3, and 4 until a feasible EOQ is found.

29. All-Unit Quantity Discounts: Example Cost/Unit $3 $2.96 $2.92 Order Quantity 5,000 10,000 Order Quantity 5,000 10,000 Total Material Cost

30. All-Unit Quantity Discount: Example Order quantityUnit Price 0-5000$3.00 5001-10000$2.96 Over 10000$2.92 q0 = 0, q1 = 5000, q2 = 10000 C0 = $3.00, C1 = $2.96, C2 = $2.92 D = 120000 units/year, S = $100/lot, h = 0.2

31. All-Unit Quantity Discount: Example Step 1: Calculate Q2* = Sqrt[(2DS)/hC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410 Not feasible (6410 < 10001) Calculate TC2 using C2 = $2.92 and q2 = 10001 TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+(12 0000)(2.92) = $354,520

32. All-Unit Quantity Discount: Example Step 2: Calculate Q1* = Sqrt[(2DS)/hC1] =Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367 Feasible (5000<6367<10000)  Stop TC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96) + (120000)(2.96) = $358,969 TC2 < TC1  The optimal order quantity Q* is q2 = 10001

33. Coordination for Commodity Products  D = 120,000 bottles/year  S R = $100, h R = 0.2, C R = $3  S S = $250, h S = 0.2, C S = $2 Retailer’s optimal lot size = 6,324 bottles Retailer cost = $3,795; Supplier cost = $6,009 Supply chain cost = $9,804

34. Coordination for Commodity Products  What can the supplier do to decrease supply chain costs? Coordinated lot size: 9,165; Retailer cost = $4,059; Supplier cost = $5,106; Supply chain cost = $9,165  Effective pricing schemes All-unit quantity discount  $3 for lots below 9,165  $2.9978 for lots of 9,165 or more Pass some fixed cost to retailer (enough that he raises order size from 6,324 to 9,165)

35. Quantity Discounts When Firm Has Market Power  No inventory related costs  Demand curve 360,000 - 60,000p What are the optimal prices and profits in the following situations? The two stages coordinate the pricing decision  Price = $4, Profit = $240,000, Demand = 120,000 The two stages make the pricing decision independently  Price = $5, Profit = $180,000, Demand = 60,000

36. Lessons from Discounting Schemes  Lot size based discounts increase lot size and cycle inventory in the supply chain  Lot size based discounts are justified to achieve coordination for commodity products  Volume based discounts with some fixed cost passed on to retailer are more effective in general Volume based discounts are better over rolling horizon