1. 1 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Chapter 15 Inventory Control

2. 2 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Definition and Purpose of Inventory Inventory Costs Independent vs. Dependent Demand Single-Period Inventory Model Multi-Period Inventory Models: Basic Fixed- Order Quantity Models Multi-Period Inventory Models: Basic Fixed- Time Period Model Miscellaneous Systems and Issues OBJECTIVES

3. 3 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Inventory System Defined Inventory is the stock of any item or resource used in an organization. They are carried in anticipation of use and can include: —raw materials —finished products —component parts —supplies, and —work-in-process An inventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be

4. 4 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Problems of Inventory Management When to order the products —Issue of timing How much to order —Issue of quantity These problems often addressed with the objective of inventory cost minimization —Total annual cost —And meet prescribed service levels

5. 5 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Purposes of Inventory 1. To maintain independence of operations 2. To meet variation in product demand 3. To allow flexibility in production scheduling 4. To provide a safeguard for variation in raw material delivery time 5. To take advantage of economic purchase-order size

6. 6 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Wrong Reasons for Carrying Inventory Poor quality Inadequate maintenance of machines Poor production schedules Unreliable suppliers Poor worker’s attitudes Just-in-case

7. 7 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Inventory Costs Holding (or carrying) costs: H —Costs for physical storage space —Handling —Taxes and insurance, etc. —Breakage, spoilage, deterioration, obsolescence —Cost of capital Setup (or production change) costs: S —Costs for arranging specific equipment setups, etc.

8. 8 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Inventory Costs: Continued  Ordering costs: S —Costs (clerical) of someone placing an order (phone, typing, etc.) —Supplies and order processing, etc.  Shortage costs: —Costs of canceling an order —Rain checks (backorders) —Loss of customer goodwill, etc.  Outdate costs (for perishable products)  Purchase cost: C

9. 9 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved E(1 ) Independent vs. Dependent Demand Independent Demand (Demand for the final end-product or demand not related to other items) Dependent Demand (Derived demand items for component parts, subassemblies, raw materials, etc) Finished product Component parts

10. 10 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Classification of Inventory Systems: Single Period Models Single-Period Inventory Model —One time purchasing decision: Example:  Vendor selling T-shirts at a football game  Newspaper sales (Newsboy Model) —Seeks to balance the costs of inventory overstock and under stock

11. 11 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Classifying Inventory Models: Multi-Period Models Fixed-Order Quantity Models — Event triggered (Example: running out of stock) — Inventory is monitored all the time — Order Q when inventory reaches R, the reorder point — Also known as FOSS, (Q,R) or EOQ system Fixed-Time Period Models — Time triggered (Example: Monthly sales call by sales representative) — Inventory is monitored periodically — Also known as FOIS, (S,S), or EOI system

12. 12 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Single-Period Inventory Model This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

13. 13 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Single Period 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 at 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; P ≤ $10 / ($10 + $5) =.667 Z.667 =.432 (use NORMSDIST(.667) or Appendix E) therefore we need 2,400 +.432(350) = 2,551 shirts

14. 14 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Multi-Period Models: Fixed-Order Quantity Model Model Assumptions (Part 1) Demand for the product is constant and uniform throughout the period Lead time (time from ordering to receipt) is constant Price per unit of product is constant

15. 15 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Multi-Period Models: Fixed-Order Quantity Model Model Assumptions (Part 2) Inventory holding cost is based on average inventory Ordering or setup costs are constant All demands for the product will be satisfied (No back orders are allowed)

16. 16 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Basic Fixed-Order Quantity Model and Reorder Point Behavior R = Reorder point Q = Economic order quantity L = Lead time L L QQQ R Time Number of units on hand 1. You receive an order quantity Q. 2. Your start using them up over time. 3. When you reach down to a level of inventory of R, you place your next Q sized order. 4. The cycle then repeats.

17. 17 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Cost Minimization Goal Ordering Costs Holding Costs Order Quantity (Q) COSTCOST Annual Cost of Items (DC) Total Cost Q OPT By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Q opt inventory order point that minimizes total costs

18. 18 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Basic Fixed-Order Quantity (EOQ) Model Formula Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost ++ TC=Total annual cost D =Demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H =Annual holding and storage cost per unit of inventory TC=Total annual cost D =Demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H =Annual holding and storage cost per unit of inventory

19. 19 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Deriving the EOQ Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Q opt We also need a reorder point to tell us when to place an order

20. 20 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved EOQ Example (1) Problem Data Annual Demand = 1,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = $2.50 Lead time = 7 days Cost per unit = $15 Given the information below, what are the EOQ and reorder point?

21. 21 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved EOQ Example (1) Solution In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

22. 22 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved EOQ Example (2) Problem Data Annual Demand = 10,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = 10% of cost per unit Lead time = 10 days Cost per unit = $15 Determine the economic order quantity and the reorder point given the following…

23. 23 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved EOQ Example (2) Solution Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

24. 24 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Fixed-Order Quantity Model Model With Safety Stock Safety stock: —An amount added to R due to uncertainty of demand during the lead time —Could be constant or variable (based on known service level)   Where, = number of standard deviations for a given service probability = Standard deviation of usage during the lead time

25. 25 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Fixed-Time Period Model with Safety Stock Formula q = Average demand + Safety stock – Inventory currently on hand

26. 26 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Multi-Period Models: Fixed-Time Period Model: Determining the Value of s T+L The standard deviation of a sequence of random events equals the square root of the sum of the variances

27. 27 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Example of the Fixed-Time Period Model Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units. Given the information below, how many units should be ordered?

28. 28 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Example of the Fixed-Time Period Model: Solution (Part 1) The value for “z” is found by using the Excel NORMSINV function, or as we will do here, using Appendix D. By adding 0.5 to all the values in Appendix D and finding the value in the table that comes closest to the service probability, the “z” value can be read by adding the column heading label to the row label. So, by adding 0.5 to the value from Appendix D of 0.4599, we have a probability of 0.9599, which is given by a z = 1.75

29. 29 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Example of the Fixed-Time Period Model: Solution (Part 2) So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

30. 30 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Variants of EOQ: Price-Break or Discount Models Seller offers you incentive to buy more Larger lot sizes result in cheaper/unit cost Two possibilities: —Holding cost is constant or variable: Compute Q using the appropriate H If Q corresponds to lowest price break, it is optimal Else, find TC for Q and those for lower price breaks The Q corresponding to lowest TC is optimal

31. 31 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Price-Break (Discount) Model Formula Based on the same assumptions as the EOQ model, the price- break model has a similar Q opt formula: i = percentage of unit cost attributed to carrying inventory C = cost per unit Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

32. 32 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Discount Example: Fixed Holding Cost Demand for tires is 12,000 units/year. Normal cost for tires is $30/unit. Orders between 100-499 will cost $27.75/unit, 500-999 will cost $25.5/unit, and 1000 units of more will cost $24/unit. Order cost is $150/order and holding cost is $5/unit/year. What policy should the company adopt?

33. 33 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Discount Example: Variable Holding Cost Suppose in the discount tire example that holding cost is charged at 18% of the cost of the tires. What policy should the company adopt? —The Q could be infeasible —The Q could be semi-feasible —The Q could be feasible

34. 34 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Discount Example: VHC Continued Flagstaff Products offers the following discount schedule for its 4’x8’ sheets of plywood. Order QuantityUnit Cost 9 Sheets or less$18.00 10 to 50 Sheets$17.50 More than 50 Sheets$17.25 Home Sweet Home Company (HSH) orders plywood from Flagstaff Products. HSH has an ordering cost of $45.00. The carrying cost is 20% of cost, and the annual demand is 100 sheets. What do recommend as the ordering policy for HSH?

35. 35 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Price-Break Example Problem Data (Part 1) A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units? Order Quantity(units)Price/unit($) 0 to 2,499 $1.20 2,500 to 3,999 1.00 4,000 or more 0.98

36. 36 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Price-Break Example Solution (Part 2) Annual Demand (D)= 10,000 units Cost to place an order (S)= $4 First, plug data into formula for each price-break value of “C” Carrying cost % of total cost (i)= 2% Cost per unit (C) = $1.20, $1.00, $0.98 Interval from 0 to 2499, the Q opt value is feasible Interval from 2500-3999, the Q opt value is not feasible Interval from 4000 & more, the Q opt value is not feasible Next, determine if the computed Q opt values are feasible or not

37. 37 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Price-Break Example Solution (Part 3) Since the feasible solution occurred in the first price-break, it means that all the other true Q opt values occur at the beginnings of each price-break interval. Why? 0 1826 2500 4000 Order Quantity Total annual costs So the candidates for the price-breaks are 1826, 2500, and 4000 units Because the total annual cost function is a “u” shaped function

38. 38 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Price-Break Example Solution (Part 4) Next, we plug the true Q opt values into the total cost annual cost function to determine the total cost under each price-break TC(0-2499) =(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999) = $10,041 TC(4000&more) = $9,949.20 TC(0-2499) =(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999) = $10,041 TC(4000&more) = $9,949.20 Finally, we select the least costly Q opt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

39. 39 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Variant of the EOQ: The EPQ Model Product produced and consumed simultaneously Plant within a plant situation EOQ definitions apply plus: —d = constant usage rate —p = production rate —EPQ = —TC =

40. 40 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved EPQ: Example A product, X, is assembled with another component, Xi, which is produced in another department at a rate of 100 units/day. The use rate for Xi at assembly department is 40/day. Find optimal production order size, reorder point, and total cost if there are 250 working days/year, S=$50, H=$.5/unit/year, P=$7/Xi, L=7 days, and D=10000. Solution: —EPQ = —R = dL = 40(7) = 280 units —TC =

41. 41 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Miscellaneous Systems: Optional Replenishment System Maximum Inventory Level, M M Actual Inventory Level, I q = M - I I Q = minimum acceptable order quantity If q > Q, order q, otherwise do not order any.

42. 42 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Miscellaneous Systems: Bin Systems Two-Bin System FullEmpty Order One Bin of Inventory One-Bin System Periodic Check Order Enough to Refill Bin

43. 43 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved ABC Classification System Items kept in inventory are not of equal importance in terms of: — dollars invested — profit potential — sales or usage volume — stock-out penalties 0 30 60 30 60 A B C % of $ Value % of Use So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 65% are the “C” items

44. 44 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved ABC Classification System Based on Pareto Study —20% of people 80% of wealth (80/20 rule) —Used in most areas of business The scheme —Class A items Fewest in number—10-20% Highest in dollar value—70-80% Exercise greatest control here

45. 45 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved ABC Classification System The Scheme —Class B items Moderate in number--30-35% Moderate in dollar value--15-20% Exercise average control here —Class C items Highest in number--about 50% Lowest in dollar value--5-10% Exercise the least control here Natural break or company policy scheme

46. 46 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved ABC Classification: Example Use ABC method to classify the following five products in a company’s inventory. ItemAnnual DemandCost/Unit 1 50$200 2 10$200 3100$800 4 50$100 5 15$200 Steps for the ABC Classification Find annual dollar usage (ADU) for each item Rank ADU in descending order of magnitude Find percentage of usage to total inventory cost Find running sum of percentage of dollar usage and classify

47. 47 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved Inventory Accuracy and Cycle Counting Defined Inventory accuracy refers to how well the inventory records agree with physical count Cycle Counting is a physical inventory- taking technique in which inventory is counted on a frequent basis rather than once or twice a year