Inventory Control Part 1 Subject to Known Demand By Ming Dong

1. Inventory Control Part 1 Subject to Known Demand By Ming Dong Department of Industrial Engineering & Management Shanghai Jiao Tong University

2. Contents • Types of Inventories • Motivation for Holding Inventories; • Characteristics of Inventory Systems; • Relevant Costs; • The EOQ Model; • EOQ Model with Finite Production Rate

3. Introduction • Definition: Inventory is the stock of any item or resource used in an organization. • An inventory system is the set of policies and controls that monitors levels of inventory or determines what levels should be maintained. • Generally, inventory is being acquired or produced to meet the need of customers; • Dependant demand system - the demand of components and subassemblies (lower levels depend on higher level) -MRP; • The fundamental problem of inventory management : • When to place order for replenishing the stock ? • How much to order?

4. Introduction • Inventory:plays a key role in the logistical behavior of virtually all manufacturing systems. • The classical inventory results:are central to more modern techniques of manufacturing management, such as MRP, JIT, and TBC. • The complexity of the resulting model depends on the assumptions about the various parameters of the system -the major distinction is between models for known demand and random demand.

5. Introduction • The current investment in inventories in USA is enormous; • It amounted up to $1.37 trillion in the last quarter of 1999; • It accounts for 20-25% of the total annual GNP (general net product); • There exists enormous potential for improving the efficiency of economy by scientifically controlling inventories; Breakdown of total investment in inventories

6. Types of Inventories • A natural classification is based on the value added from manufacturing operations • Raw materials: Resources required in the production or processing activity of the firm. • Components: Includes parts and subassemblies. • Work-in-process (WIP): the inventory either waiting in the system for processing or being processed. • The level of WIP is taken as a measure of the efficiency of a production scheduling system. • JIT aims at reducing WIP to zero. • Finished good: also known as end items or the final products.

7. Why Hold Inventories (1) • For economies of scale • It may be economical to produce a relatively large number of items in each production run and store them for future use. • Coping with uncertainties • Uncertainty in demand • Uncertainty in lead time • Uncertainty in supply • For speculation • Purchase large quantities at current low prices and store them for future use. • Cope with labor strike

8. Why Hold Inventories (2) • Transportation • Pipeline inventories is the inventory moving from point to point, e.g., materials moving from suppliers to a plant, from one operation to the next in a plant. • Smoothing • Producing and storing inventory in anticipation of peak demand helps to alleviate the disruptions caused by changing production rates and workforce level. • Logistics • To cope with constraints in purchasing, production, or distribution of items, this may cause a system maintain inventory.

9. Characteristics of Inventory Systems • Demand (patterns and characteristics) • Constant versus variable • Known versus random • Lead Time • Ordered from the outside • Produced internally • Review • Continuous: e.g., supermarket • Periodic: e.g., warehouse • Excess demand • demand that cannot be filled immediately from stock • backordered or lost. • Changing inventory • Become obsolete: obsolescence

10. Relevant Costs - Holding Cost • Holding cost (carrying or inventory cost) • The sum of costs that are proportional to the amount of inventory physically on-hand at any point in time • Some items of holding costs • Cost of providing the physical space to store the items • Taxes and insurance • Breakage, spoilage, deterioration, and obsolescence • Opportunity cost of alternative investment • Inventory cost fluctuates with time • inventory as a function of time

11. Inventory as a Function of Time Relevant Costs - Holding Cost

12. Relevant Costs - Order Cost • It depend on the amount of inventory that is ordered or produced. • Two components • The fixed cost K: independent of size of order • The variable cost c: incurred on per-unit basis

13. Order Cost Function Relevant Costs - Order Cost

14. Relevant Costs - Penalty Cost • Also know as shortage cost or stock-out cost • The cost of not having sufficient stock on-hand to satisfy a demand when it occurs. • Two interprets • Backorder case: include delay costs may be involved • Lost-sale case: include “loss-of-goodwill” cost, a measure of customer satisfaction • Two approaches • Penalty cost, p, is charged per-unit basis. • Each time a demand occurs that cannot be satisfied immediately, a cost p is incurred independent of how long it takes to eventually fill the demand. • Charge the penalty cost on a per-unit-time basis.

15. EOQ History • Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once” • Product types: A, B and C • A-B-C-A-B-C: 6 times of setup • A-A-B-B-C-C: 3 times of setup • A factory producing various products and switching between products causes a costly setup (wages, material and overhead). Therefore, a trade-off between setup cost and production lot size should be determined. • Early application of mathematical modeling to Scientific Management

16. EOQ Modeling Assumptions 1.Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously. 2.Delivery is immediate – there is no time lag between production and availability to satisfy demand. 3.Demand is deterministic – there is no uncertainty about the quantity or timing of demand. 4.Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5.A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant. 6.Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.

17. Notation  demand rate (units per year) c proportional order cost at c per unit ordered (dollars per unit) K fixed or setup cost to place an order (dollars) h holding cost (dollars per year); if the holding cost consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the order or lot size

18. Q Inventory Q/ 2Q/ 3Q/ 4Q/ Time Inventory vs Time in EOQ Model slope = - Order cycle: T=Q/

19. Costs • Holding Cost: • Setup Costs: K per lot, so • The average annual cost:

20. MedEquip Example • Small manufacturer of medical diagnostic equipment. • Purchases standard steel “racks” into which components are mounted. • Metal working shop can produce (and sell) racks more cheaply if they are produced in batches due to wasted time setting up shop. • MedEquip doesn't want to hold too much capital in inventory. • Question: how many racks should MedEquip order at once?

21. MedEquip Example Costs •  = 1000 racks per year • c = $250 • K = $500 (estimated from supplier’s pricing) • h = i*c + floor space cost = (0.1)($250) + $10 = $35 per unit per year

22. Costs in EOQ Model

23. Economic Order Quantity Solution (by taking derivative and setting equal to zero): Since Q”>0，G(Q) is convex function of Q EOQ Square Root Formula MedEquip Solution

24. Another Example • Example 2 • Pencils are sold at a fairly steady rate of 60 per week; • Pencils cost 2 cents each and sell for 15 cents each; • Cost $12 to initiate an order, and holding costs are based on annual interest rate of 25%. • Determine the optimal number of pencils for the book store to purchase each time and the time between placement of orders • Solutions • Annual demand rate =6052=3,120; • The holding cost is the product of the variable cost of the pencil and the annual interest-h=0.02 0.25=0.05

25. Reorder Point Calculation for Example 2 The EOQ Model-Considering Lead Time • Since there exists lead time (4 moths for Example 2), order should be placed some time ahead of the end of a cycle; • Reorder point R-determines when to place order in term of inventory on hand, rather than time.

26. Reorder Point Calculation for Lead Times Exceeding One Cycle The EOQ Model-Considering Lead Time • Determine the reorder point when the lead time exceeds a cycle. Computing R for placing order 2.31 cycles ahead is the same as that 0.31 cycle ahead. • Example: • Q=25; • =500/yr; • =6 wks; • T=25/500=2.6 wks; • /T=2.31---2.31 cycles are included in LT. • Action: place every order 2.31 cycles in advance.

27. EOQ Modeling Assumptions 1.Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously. 2.Delivery is immediate – there is no time lag between production and availability to satisfy demand. 3.Demand is deterministic – there is no uncertainty about the quantity or timing of demand. 4.Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5.A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant. 6.Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products. relax via EOQ Model for Finite Production Rate

28. The EOQ Model for Finite Production Rate • The EOQ model with finite production rate is a variation of the basic EOQ model • Inventory is replenished gradually as the order is produced (which requires the production rate to be greater than the demand rate) • Notice that the peak inventory is lower than Q since we are using items as we produce them

29. Notation – EOQ Model for Finite Production Rate  demand rate (units per year) P production rate (units per year), where P> c unit production cost, not counting setup or inventory costs (dollars per unit) K fixed or setup cost (dollars) h holding cost (dollars per year); if the holding cost is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the production lot size decision variable

30. Inventory vs Time 1. Production run of Q takes Q/P time units slope = - (P-)(Q/P) - P- (P-)(Q/P)/2 Inventory Time slope = P- 2. When the inventory reaches 0, production begins until Q products are produced (it takes Q/P time units). During the Q/P time units, the inventory level will increases to (P-)(Q/P) Time Inventory increase rate

31. Solution to EOQ Model with Finite Production Rate • Annual Cost Function: • Solution (by taking first derivative and setting equal to zero): holding setup • tends to EOQ as P • otherwise larger than EOQ because replenishment takes longer EOQ model

32. Example: Non-Slip Tile Co. • Non-Slip Tile Company (NST) has been using production runs of 100,000 tiles, 10 times per year to meet the demand of 1,000,000 tiles annually. The set-up cost is $5000 per run and holding cost is estimated at 10% of the manufacturing cost of $1 per tile. The production capacity of the machine is 500,000 tiles per month. The factory is open 365 days per year.

33. Example: Non-Slip Tile Co. (Cont.) • This is a “EOQ Model with Finite Production Rate” problem with  = 1,000,000 P = 500,000*12 = 6,000,000 h = 0.1 K = 5,000

34. Example: Non-Slip Tile Co. (Cont.) • Find the Optimal Production Lot Size • How many runs should they expect per year? • How much will they save annually using EOQ Model with Finite Production Rate?

35. Example: Non-Slip Tile Co. (Cont.) • Optimal Production Lot Size • Number of Production Runs Per Year • The number of runs per year = /Q* = 2.89 times per year

36. Example: Non-Slip Tile Co. (Cont.) • Annual Savings: Holding cost Setup cost TC = 0.04167Q + 5,000,000,000/ Current TC = 0.04167(1,000,000) + 5,000,000,000/(1,000,000) = $54,167 Optimal TC = 0.04167(346,410) + 5,000,000,000/(346,410) = $28,868 Difference = $54,167 - $28,868 = $25,299

37. Sensitivity of EOQ Model to Quantity • Optimal Unit Cost: • Optimal Annual Cost: Multiply G* by  and simplify,

38. Sensitivity of EOQ Model to Quantity (cont.) • Annual Cost from Using Q': • Ratio:

39. Sensitivity of EOQ Model to Quantity (cont.) Example: If Q' = 2Q*, then the ratio of the actual cost to optimal cost is (1/2)[2 + (1/2)] = 1.25 If Q' = Q*/2, then the ratio of the actual cost to optimal cost is (1/2)[(1/2)+2] = 1.25 A 100% error in lot size results in a 25% error in cost.

40. EOQ Takeaways • Batching causes inventory (i.e., larger lot sizes translate into more stock). • Under specific modeling assumptions the lot size that optimally balances holding and setup costs is given by the square root formula: • Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive).

41. Inventory Control Part 2 Inventory Control Subject to Unknown Demand By Ming Dong Department of Industrial Engineering & Management Shanghai Jiao Tong University

42. The Wagner-WhitinModel Change is not made without inconvenience, even from worse to better. – Robert Hooker

43. EOQ Assumptions • 1. Instantaneous production. • 2. Immediate delivery. • 3. Deterministic demand. • 4. Constant demand. • 5. Known fixed setup costs. • 6. Single product or separable products. WW model relaxes this one

44. Dynamic Lot Sizing Notation t a period (e.g., day, week, month); we will consider t = 1, … ,T, where T represents the planning horizon. Dt demand in period t (in units) ct unit production cost (in dollars per unit), not counting setup or inventory costs in period t At fixed or setup cost (in dollars) to place an order in period t ht holding cost (in dollars) to carry a unit of inventory from period t to period t +1 Qt the unknown size of the order or lot size in period t decisionvariables

45. Wagner-Whitin Example Data Lot-for-Lot Solution Since production cost c is constant, it can be ignored.

46. Wagner-Whitin Example (cont.) Data Fixed Order Quantity Solution

47. Wagner-Whitin Property A key observation If we produce items in t (incur a setup cost) for use to satisfy demand in t+1, then it cannot possibly be economical to produce in t+1 (incur another setup cost) . Either it is cheaper to produce all of period t+1’s demand in period t, or all of it in t+1; it is never cheaper to produce some in each. Under an optimal lot-sizing policy • either the inventory carried to period t+1 from a previous period will be zero (there is a production in t+1) • or the production quantity in period t+1 will be zero (there is no production in t+1) Does fixed order quantity solution violate this property? Why?

48. Basic Idea of Wagner-Whitin Algorithm By WW Property, either Qt=0 or Qt=D1+…+Dk for some k. If jk*= last period of production in a k period problem, then we will produce exactly Dk+…DT in period jk*. Why? We can then consider periods 1, … , jk*-1 as if they are an independent jk*-1 period problem.

49. Wagner-Whitin Example • Step 1: Obviously, just satisfy D1 (note we are neglecting production cost, since it is fixed). • Step 2: Two choices, either j2* = 1 or j2* = 2.

50. Wagner-Whitin Example (cont.) • Step3: Three choices, j3* = 1, 2, 3.