1 Inventory Models

2 Overview of Inventory Issues Proper control of inventory is crucial to the success of an enterprise. Typical inventory problems include: – Basic inventory – Planned shortage – Quantity discount – Periodic review – Production lot size – Single period Inventory models are often used to develop an optimal inventory policy, consisting of: – An order quantity, denoted Q. – A reorder point, denoted R.

3 Inventory analyses can be thought of as cost- control techniques. Categories of costs in inventory models: – Holding (carrying costs) – Order/ Setup costs – Customer satisfaction costs – Procurement/Manufacturing costs Type of Costs in Inventory Models

4 Holding Costs (Carrying costs): These costs depend on the order size – Cost of capital – Storage space rental cost – Costs of utilities – Labor – Insurance – Security – Theft and breakage – Deterioration or Obsolescence C h = Annual holding cost per unit in inventory H = Annual holding cost rate C = Unit cost of an item C h = Annual holding cost per unit in inventory H = Annual holding cost rate C = Unit cost of an item C h = H * C Type of Costs in Inventory Models

5 Order/Setup Costs These costs are independent of the order size. – Order costs are incurred when purchasing a good from a supplier. They include costs such as Telephone Order checking Labor Transportation – Setup costs are incurred when producing goods for sale to others. They can include costs of Cleaning machines Calibrating equipment Training staff Type of Costs in Inventory Models C o = Order cost or setup cost

6 Customer Satisfaction Costs – Measure the degree to which a customer is satisfied. – Unsatisfied customers may: Switch to the competition (lost sales). Wait until an order is supplied. – When customers are willing to wait there are two types of costs incurred: Type of Costs in Inventory Models C b  = Fixed administrative costs of an out of stock item ($/stockout unit). C s = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year). C b  = Fixed administrative costs of an out of stock item ($/stockout unit). C s = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year).

7 Procurement/Manufacturing Cost – Represents the unit purchase cost (including transportation) in case of a purchase. – Unit production cost in case of in- house manufacturing. Type of Costs in Inventory Models C = Unit purchase or manufacturing cost.

8 Demand is a key component affecting an inventory policy. Projected demand patterns determine how an inventory problem is modeled. Typical demand patterns are: – Constant over time (deterministic inventory models) – Changing but known over time (dynamic models) – Variable (randomly) over time (probabilistic models) Demand in Inventory Models D = Demand rate (usually per year)

9 Inventory can be classified in various ways: Used typically by accountants at manufacturing firms. Enables management to track the production process. Items are classified by their relative importance in terms of the firm’s capital needs. Management of items with short shelf life and long shelf life is very different Inventory Classifications

10 Two types of review systems are used: – Continuous review systems. The system is continuously monitored. A new order is placed when the inventory reaches a critical point. – Periodic review systems. The inventory position is investigated on a regular basis. An order is placed only at these times. Review Systems

11 – Continuous review systems. The system is continuously monitored. A new order is placed when the inventory reaches a critical point. – EOQ Review Systems continuous review system

12 ALLEN APPLIANCE COMPANY (AAC) AAC wholesales small appliances. AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units. Management wishes to determine an optimal ordering policy for the Citron brand juicer

13 Data – C o = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) – C h = $1.40 [HC = (14%)($10)] – C = $10. – H = 14% (10% ann. interest rate) + (4% miscellaneous) – D = demand information of the last 10 weeks was collected: ALLEN APPLIANCE COMPANY (AAC)

14 Data – The constant demand rate seems to be a good assumption. – Annual demand = (120/week)(52weeks) = 6240 juicers. ALLEN APPLIANCE COMPANY (AAC)

15 Current ordering policy calls for Q = 600 juicers. TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $ EOQ = sqrt((2*6240*12)/1,4) = 327,065 = 327,065 = 327 AAC – Solution: EOQ and Total Variable Cost Savings of 16% TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $ Savings of 16%

16 TC(327) = ($10) + (13)($1.40) = $62, Under the current ordering policy AAC holds 13 units safety stock (how come? observe): AAC is open 5 day a week. – The average daily demand = 120/week/5 = 24 juicers. – Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers. – Reorder point without Safety stock = LD = 192. – Current policy: R = 205. – Safety stock = 205 – 192 = 13. For safety stock of 13 juicers the total cost is TV(327) + Procurement + Safety stock cost holding cost AAC – Solution: Reorder Point and Total Cost

17 Changing the order size – Suppose juicers must be ordered in increments of 100 (order 300 or 400) – AAC will order Q = 300 juicers in each order. – There will be a total variable cost increase of $1.71. – This is less than 0.5% increase in variable costs. Changes in input parameters – Suppose there is a 20% increase in demand. D=7500 juicers. – The new optimal order quantity is Q * = 359. – The new variable total cost = TV(359) = $502 – If AAC still orders Q = 327, its total variable costs becomes TV(327) = (327  2)($1.40) + (7500  327)($12) = $ Only 0.4% increase AAC – Solution: Sensitivity of the EOQ Results

18 For an order size of 327 juicers we have: – T = (327  6240) = year. = (52)(5) = 14 days. – This is useful information because: Shelf life may be a problem. Coordinating orders with other items might be desirable. AAC – Solution: Cycle Time working days per week

19 AAC – Excel Spreadsheet =SQRT(2*$B$10*$B $14/$B$13) =1/E11 Copy to cell H12 =E10/B10 Copy to cell H11 =$B$10*$B$11+E14+$B$13*B16 Copy to Cell H15 =(E10/2)*$B$13+($B$10/E10)*$B$14 Copy to cell H14 =$B$15*$B$10+$B$16- INT(($B$15*$B$10+$B$16)/E10)*E10 Copy to cell H13

20 Service Levels and Safety Stocks Service Levels and Safety Stocks

21 Determining Safety Stock Levels Businesses incorporate safety stock requirements when determining reorder points. A possible approach to determining safety stock levels is by specifying desired service level.

22 The unit service level – The percentage of demands that are filled without incurring any delay. – Applied when the percentage of unsatisfied demand should be under control. Two Types of Service Level The cycle service level –The probability of not incurring a stockout during an inventory cycle. –Applied when the likelihood of a stockout, and not its magnitude, is important for the firm. Service levels can be viewed in two ways.

23 In many cases short run demand is variable even though long run demand is assumed constant. Therefore, stockout events during lead time may occur unexpectedly in each cycle. Stockouts occur only if demand during lead time is greater than the reorder point. The Cycle Service Level Approach

24 To determine the reorder point we need to know: – The lead time demand distribution. – The required service level. In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by The Cycle Service Level Approach R =  L + z   L 1 –  = service level

25  =192 P(D L > R) = P(Z > (R –  L )/  L ) = . Since P(Z > Z  ) = , we have Z  = (R –  L )/  L, which gives… The Cycle Service Level Approach P(D L >R) =  Service level = P(D L <R) = 1 –  R R =  L + z   L

26 Assume that lead time demand is normally distributed. Estimation of the normal distribution parameters: – Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week. – Estimation of the variance of the weekly demand = Sample variance = juicers 2. AAC - Cycle Service Level Approach

27 To find  L and  L the parameters  (per week) and  (per week)  must be adjusted since the lead time is longer than one week. – Lead time is 8 days =(8 / 5) weeks = 1.6 weeks. Estimates for the lead time mean demand and variance of demand  L  (1.6)(120) = 192;  2 L  (1.6)(83.33) = AAC - Cycle Service Level Approach

28 Let us use the current reorder point of 205 juicers. 205 =192 + z (11.55 ) z = 1.13 From the normal distribution table we have that a reorder point of 205 juicers results in an 87% cycle service level. AAC - Service Level for a given Reorder Point

29 Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is R = (11.55) = 219 juicers. Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is R = (11.55) = 219 juicers. AAC – Reorder Point for a given Service Level

30 AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers. What is the equivalent service level for this strategy? AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers. What is the equivalent service level for this strategy? AAC – Acceptable Number of Stockouts per Year

31 AAC – Acceptable Number of Stockouts per Year There will be an average of 6240  327 = lead times per year. The likelihood of stockouts = 1/19 = This translates into a service level of 94.76% There will be an average of 6240  327 = lead times per year. The likelihood of stockouts = 1/19 = This translates into a service level of 94.76%

When lead time demand follows a normal distribution service level can be calculated as follows: – Determine the value of z that satisfy the equation L(z) =  Q *   L – Solve for R using the equation R =  L + z  L 32 The Unit Service Level Approach

33 =NORMDIST(B8,B5,B6,TRUE) AAC – Cycle Service Level (Excel spreadsheet) =NORMINV(B7,B5,B 6)

34 Quantity Discounts are Common Practice in Business – By offering discounts buyers are encouraged to increase their order sizes, thus reducing the seller’s holding costs. – Quantity discounts reflect the savings inherent in large orders. – With quantity discounts sellers can reward their biggest customers without violating the Robinson - Patman Act. EOQ Models with Quantity Discounts

35 Quantity Discount Schedule – This is a list of per unit discounts and their corresponding purchase volumes. – Normally, the price per unit declines as the order quantity increases. – The order quantity at which the unit price changes is called a break point. – There are two main discount plans: All unit schedules - the price paid for all the units purchased is based on the total purchase. Incremental schedules - The price discount is based only on the additional units ordered beyond each break point. EOQ Models with Quantity Discounts

36 To determine the optimal order quantity, the total purchase cost must be included TC(Q) = (Q  2)C h + (D  Q)C o + DC i + C h SS C i represents the unit cost at the i th pricing level. All Units Discount Schedule

37 AAC - All Units Quantity Discounts AAC is offering all units quantity discounts to its customers. Data

38 Should AAC increase its regular order of 327 juicers, to take advantage of the discount? Should AAC increase its regular order of 327 juicers, to take advantage of the discount?

39 AAC – All units discount procedure – Step 1: Find the optimal order Q i * for each discount level “i”. Use the formula – Step 2: For each discount level “i” modify Q i * as follows If Q i * is lower than the smallest quantity that qualifies for the i th discount, increase Q i * to that level. If Q i * is greater than the largest quantity that qualifies for the i th discount, eliminate this level from further consideration. – Step 3: Substitute the modified Q * i value in the total cost formula TC(Q * i ). – Step 4: Select the Q i * that minimizes TC(Q i * )

40 Step 1: Find the optimal order quantity Q i * for each discount level “i” based on the EOQ formula AAC – All units discount procedure

41 – Step 2 : Modify Q i * Q1*Q1* Q1*Q1* 300 $10/unit Q2*Q2* Q2*Q2* $9.75/unit Q3*Q3* Q3*Q3* 336 $9.50 AAC – All Units Discount Procedure

42 – Step 2 : Modify Q i * Q1*Q1* Q1*Q1* 300 $10/unit 331 Q2*Q2* Q2*Q2* Q3*Q3* Q3*Q3* 336 $9.50 AAC – All Units Discount Procedure Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3*

43 – Step 3: Substitute Q I * in the total cost function – Step 4 AAC should order 5000 juicers AAC – All Units Discount Procedure

44 AAC – All Units Discount Excel Worksheet