CHAPTER 4 INVENTORY MANAGEMENT

LEARNING OBJECTIVES Define inventory and functions of inventory Conduct an ABC analysis. Explain and apply the EOQ and POQ model to solve typical problems. Compute a ROP and safety stock

Inventory A stock or store of goods A stock of item kept to meet demand Inventory Management How much When Classified Accuracy

Objective of Inventory Control To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds To keep enough inventory to meet customer demand and also be cost effective. Level of customer service Costs of ordering and carrying inventory

Functions of Inventory To meet anticipated demand To decouple operations To protect against stock-outs To take advantage of order cycles To help hedge against price increases To take advantage of quantity discounts

Types of Inventory

Types of Inventory Raw material Purchased but not processed Work-in-process Undergone some change but not completed A function of cycle time for a product Maintenance/repair/operating (MRO) Necessary to keep machinery and processes productive

Types of Inventory Finished-goods inventories (manufacturing firms) or merchandise (retail stores) Goods in transit Completed product awaiting shipment

The Material Flow Cycle Cycle time 95% 5% Input Wait for Wait to Move Wait in queue Setup Run Output inspection be moved time for operator time time

Effective Inventory Management A system to keep track of inventory A reliable forecast of demand Knowledge of lead times Reasonable estimates of Holding costs Ordering costs Shortage costs A classification system

Inventory Counting Systems Periodic System Physical count of items made at periodic intervals Perpetual Inventory System keeps track of removals from inventory continuously, thus monitoring current levels of each item

Inventory Counting Systems Two-Bin System Two containers of inventory; reorder when the first is empty Universal Bar Code printed on a label that has information about the item to which it is attached 214800 232087768

Key Inventory Terms Lead time: time interval between ordering and receiving the order Item cost: Cost per item plus any other direct costs associated with getting the item to the plant Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year Ordering costs: costs of ordering and receiving inventory Shortage costs: costs when demand exceeds supply

Independent Versus Dependent Demand Independent Demand A Dependent Demand B(4) C(2) D(2) E(1) D(3) F(2) Independent demand is uncertain. Dependent demand is certain.

ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A - very important B - moderately important C - least important High A Annual $ value of items B C Low Low High Percentage of Items

ABC Analysis A Items 80 – 70 – 60 – Percent of annual dollar usage 80 – 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – | | | | | | | | | | 10 20 30 40 50 60 70 80 90 100 Percent of inventory items A Items B Items C Items

ABC Analysis Example

Inventory Models for Independent Demand Need to determine when and how much to order Basic economic order quantity Production order quantity Quantity discount model

Basic EOQ Model Demand is known, constant, and independent Lead time is known and constant Receipt of inventory is instantaneous and complete Quantity discounts are not possible Only variable costs are setup and holding Stockouts can be completely avoided

Cycle-Inventory Levels Receive order On-hand inventory (units) Time Inventory depletion (demand rate) Q Average cycle inventory Q — 2 1 cycle

Total Annual Cycle-Inventory Costs Objective is to minimize total costs Annual cost Order quantity Total cost = (H) + (S) D Q 2 Curve for total cost of holding and setup Setup (or order) cost curve Minimum total cost Optimal order quantity (Q*) Holding cost curve Holding cost = (H) Q 2 Ordering cost = (S) D Q Table 11.5

Minimum Total Cost Q 2 H D S = The total cost curve reaches its minimum where the carrying and ordering costs are equal. Q 2 H D S =

Example Bird feeder sales are 18 units per week, and the supplier charges $60 per unit. The cost of placing an order (S) with the supplier is $45. Annual holding cost (H) is 25% of a feeder’s value, based on operations 52 weeks per year. Management chose a 390-unit lot size (Q) so that new orders could be placed less frequently. What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size?

Costing out a Lot Sizing Policy Museum of Natural History Gift Shop: What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size? D = (18 /week)(52 weeks) = 936 units H = 0.25 ($60/unit) = $15 C = (H) + (S) = (15) + (45) Q 2 D 936 390 C = $2925 + $108 = $3033

Lot Sizing at the Museum of Natural History Gift Shop D = 936 units; H = $15; S = $45; Q = 390 units; C = $3033 Q = 468 units; C = ? Would a lot size of 468 be better? C = (H) + (S) = (15) + (45) Q 2 D 936 468 C = $3510 + $90 = $3600 Q = 468 is a more expensive option. The best lot size (EOQ) is the lowest point on the total annual cost curve!

Lot Sizing at the Museum of Natural History Gift Shop 3000 — 2000 — 1000 — 0 — | | | | | | | | 50 100 150 200 250 300 350 400 Lot Size (Q) Annual cost (dollars) Current cost Q Total cost Holding cost Lowest cost Best Q (EOQ) Ordering cost

Computing the EOQ Bird Feeders: C = $1,124.10 EOQ = 2DS H D = annual demand S = ordering or setup costs per lot H = holding costs per unit D = 936 units H = $15 S = $45 EOQ = 2(936)45 15 = 74.94 or 75 units C = (H) + (S) Q 2 D C = (15) + (45) 75 2 936 C = $1,124.10

Computing EOQ using the Excel Solver

Expected number of orders Demand Order quantity D Q* Expected time between orders Number of working days per year N EOQ D TBOEOQ =

Understanding the Effect of Changes A Change in the Demand Rate (D): When demand rises, the lot size also rises, but more slowly than actual demand. A Change in the Setup Costs (S): Increasing S increases the EOQ and, consequently, the average cycle inventory. A Change in the Holding Costs (H): EOQ declines when H increases.

EOQ: Robust Model ? Errors in Estimating D, H, and S: Total cost is fairly insensitive to errors, even when the estimates are wrong by a large margin. The reasons are that errors tend to cancel each other out and that the square root reduces the effect of the error.

When to Reorder with EOQ Ordering Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level - Probability that demand will not exceed supply during lead time.

Determinants of the Reorder Point The rate of demand The lead time Demand and/or lead time variability Stockout risk (safety stock)

Safety Stock Quantity Maximum probable demand Expected demand LT Time Expected demand during lead time Maximum probable demand ROP Quantity Safety stock Safety stock reduces risk of stockout during lead time

Number of working days in a year Lead time for a new order in days Reorder Point Inventory level (units) Time (days) Q* d = D Number of working days in a year ROP = Lead time for a new order in days Demand per day ROP (units) Slope = units/day = d = d x L Lead time = L

Production Order Quantity Model Used when inventory builds up over a period of time after an order is placed Used when units are produced and sold simultaneously Inventory level Time Part of inventory cycle during which production (and usage) is taking place Demand part of cycle with no production Maximum inventory t

Production Order Quantity Model

Production Order Quantity Model Q = Number of pieces per order p = Daily production rate H = Holding cost per unit per year d = Daily demand/usage rate t = Length of the production run in days = – Maximum inventory level Total produced during the production run Total used during the production run = pt – dt

Production Order Quantity Model = – Maximum inventory level Total produced during the production run Total used during the production run = pt – dt However, Q = total produced = pt ; thus t = Q/p Maximum inventory level = p – d = Q 1 – Q p d Holding cost = (H) = 1 – H d p Q 2 Maximum inventory level

Production Order Quantity Model Setup cost = (D/Q)S Holding cost = HQ[1 - (d/p)] 1 2 (D/Q)S = HQ[1 - (d/p)] 1 2 Q2 = 2DS H[1 - (d/p)] Q* = 2DS H[1 - (d/p)] p EPQ.xls

Quantity Discount Models Reduced prices are often available when larger quantities are purchased Trade-off is between reduced product cost and increased holding cost Total cost = Setup cost + Holding cost + Product cost TC = S + H + PD D Q 2

Total Cost With PD Cost Adding Purchasing cost doesn’t change EOQ EOQ/POQ TC with PD TC without PD PD Quantity Adding Purchasing cost doesn’t change EOQ

Quantity Discount Models A typical quantity discount schedule Discount Number Discount Quantity Discount (%) Discount Price (P) 1 0 to 999 no discount $5.00 2 1,000 to 1,999 4 $4.80 3 2,000 and over 5 $4.75 Table 12.2

Quantity Discount Models Steps in analyzing a quantity discount For each discount, calculate Q* If Q* for a discount doesn’t qualify, choose the smallest possible order size to get the discount Compute the total cost for each Q* or adjusted value from Step 2 Select the Q* that gives the lowest total cost

Quantity Discount Models Total cost $ Order quantity 1,000 2,000 Total cost curve for discount 2 Total cost curve for discount 1 Total cost curve for discount 3 Q* for discount 2 is below the allowable range at point a and must be adjusted upward to 1,000 units at point b a b 1st price break 2nd price break Figure 12.7

Quantity Discount Models 2DS IP Calculate Q* for every discount Q1* = = 700 cars/order 2(5,000)(49) (.2)(5.00) Q2* = = 714 cars/order 2(5,000)(49) (.2)(4.80) Q3* = = 718 cars/order 2(5,000)(49) (.2)(4.75)

Quantity Discount Models 2DS IP Calculate Q* for every discount Q1* = = 700 cars/order 2(5,000)(49) (.2)(5.00) Q2* = = 714 cars/order 2(5,000)(49) (.2)(4.80) 1,000 — adjusted Q3* = = 718 cars/order 2(5,000)(49) (.2)(4.75) 2,000 — adjusted

Quantity Discount Models Discount Number Unit Price Order Quantity Annual Product Cost Annual Ordering Cost Annual Holding Cost Total 1 $5.00 700 $25,000 $350 $25,700 2 $4.80 1,000 $24,000 $245 $480 $24,725 3 $4.75 2,000 $23.750 $122.50 $950 $24,822.50 Table 12.3 Choose the price and quantity that gives the lowest total cost Buy 1,000 units at $4.80 per unit

Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10 each and requires minimum quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering hats at $9 in lots of 4000. Who should he buy from? Since the EOQ of 516 is not feasible, calculate the total cost (C) for each price to make the decision 4000 hats at $9 each saves $19,320 annually. Space?

Probabilistic Models and Safety Stock Used when demand is not constant or certain Use safety stock to achieve a desired service level and avoid stockouts ROP = d x L + ss Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit x the number of orders per year

Safety Stock Example Number of Units Probability 30 .2 40 ROP  50 .3 ROP = 50 units Stockout cost = $40 per frame Orders per year = 6 Carrying cost = $5 per frame per year Number of Units Probability 30 .2 40 ROP  50 .3 60 70 .1 1.0

A safety stock of 20 frames gives the lowest total cost Safety Stock Example ROP = 50 units Stockout cost = $40 per frame Orders per year = 6 Carrying cost = $5 per frame per year Safety Stock Additional Holding Cost Stockout Cost Total Cost 20 (20)($5) = $100 $0 $100 10 (10)($5) = $ 50 (10)(.1)($40)(6) = $240 $290 $ 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960 A safety stock of 20 frames gives the lowest total cost ROP = 50 + 20 = 70 frames

Probabilistic Demand Inventory level Time Figure 12.8 Safety stock 16.5 units ROP  Place order Inventory level Time Minimum demand during lead time Maximum demand during lead time Mean demand during lead time ROP = 350 + safety stock of 16.5 = 366.5 Receive order Lead time Normal distribution probability of demand during lead time Expected demand during lead time (350 kits) Figure 12.8

Probabilistic Demand Probability of no stockout 95% of the time Mean demand 350 Risk of a stockout (5% of area of normal curve) ROP = ? kits Quantity Safety stock Number of standard deviations z

Probabilistic Demand Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined ROP = demand during lead time + ZsdLT where Z = number of standard deviations sdLT = standard deviation of demand during lead time

Probabilistic Example Average demand = m = 350 kits Standard deviation of demand during lead time = sdLT = 10 kits 5% stockout policy (service level = 95%) Using Appendix I, for an area under the curve of 95%, the Z = 1.65 Safety stock = ZsdLT = 1.65(10) = 16.5 kits Reorder point = expected demand during lead time + safety stock = 350 kits + 16.5 kits of safety stock = 366.5 or 367 kits

Other Probabilistic Models When data on demand during lead time is not available, there are other models available When demand is variable and lead time is constant When lead time is variable and demand is constant When both demand and lead time are variable

Other Probabilistic Models Demand is variable and lead time is constant ROP = (average daily demand x lead time in days) + ZsdLT where sd = standard deviation of demand per day sdLT = sd lead time Variance = daily variance x no. of days of lead time Standard D . (sum of daily variance during lead time)

Probabilistic Example Average daily demand (normally distributed) = 15 Standard deviation = 5 Lead time is constant at 2 days 90% service level desired Z for 90% = 1.28 From Appendix I ROP = (15 units x 2 days) + Zsdlt = 30 + 1.28(5)( 2) = 30 + 9.02 = 39.02 ≈ 39 Safety stock is about 9 iPods

Other Probabilistic Models Lead time is variable and demand is constant ROP = (daily demand x average lead time in days) + Z x (daily demand) x sLT where sLT = standard deviation of lead time in days

Probabilistic Example Z for 98% = 2.055 From Appendix I Daily demand (constant) = 10 Average lead time = 6 days Standard deviation of lead time = sLT = 3 98% service level desired ROP = (10 units x 6 days) + 2.055(10 units)(3) = 60 + 61.65 = 121.65 Reorder point is about 122 cameras

Other Probabilistic Models Both demand and lead time are variable ROP = (average daily demand x average lead time) + ZsdLT where sd = standard deviation of demand per day sLT = standard deviation of lead time in days sdLT = (average lead time x sd2) + (average daily demand)2 x sLT2

Probabilistic Example Average daily demand (normally distributed) = 150 Standard deviation = sd = 16 Average lead time 5 days (normally distributed) Standard deviation = sLT = 1 day 95% service level desired Z for 95% = 1.65 From Appendix I ROP = (150 packs x 5 days) + 1.65sdLT = (150 x 5) + 1.65 (5 days x 162) + (1502 x 12) = 750 + 1.65(154) = 1,004 packs

Total Q System Costs Total cost = Annual Holding Cost + Annual setup/ordering Cost + Annual safety stock holding cost

Fixed-Period (P) Systems Orders placed at the end of a fixed period Inventory counted only at end of period Order brings inventory up to target level Only relevant costs are ordering and holding Lead times are known and constant Items are independent from one another

Fixed-Period (P) Systems Target quantity (T) On-hand inventory Time Q1 Q2 Q3 Q4 P P Figure 12.9

Fixed-Period (P) Example 3 jackets are back ordered No jackets are in stock It is time to place an order Target value = 50 Order amount (Q) = Target (T) - On-hand inventory - Earlier orders not yet received (SR)+ Back orders Q = 50 - 0 - 0 + 3 = 53 jackets

Periodic Review Systems: Calculations for TI Targeted Inventory level: TI = d(p + LT) + SS d = average period demand p = order interval (days, wks) LT = lead time (days, wks) SS = zσd Replenishment Quantity (Q)=TI-OH

Periodic Review Systems Example The KVS Pharmacy stocks a popular brand of over-the-counter flu and cold medicine. The average demand for the medicine is 6 packages per day, with a standard deviation of 1.2 packages. A vendor for the pharmaceutical company checks KVS’s stock every 60 days. During one visit the store had 8 packages in stock. The lead time to receive an order is 5 days. Determine the order size for this order period that will enable KVS to maintain a 95% service level. Q = d(p + LT) + zσd - OH = 6(60 + 5) + 1.65(1.2) - 8 = 397.96

Total P System Costs Same three cost element as Q system Order quantity, Q will be the average consumption of inventory during the p periods between order; Q =dP

Q System Example Suppose that the average demand for bird feeders is 18 units per week with a standard deviation of 5 units. The lead time is constant at 2 weeks. Determine the safety stock and reorder point for a 90% cycle-service level. What is the total cost of the Q system? sdLT = sd LT = 5 2 = 7.1 Safety stock = zsdLT = 1.28(7.1) = 9.1 or 9 units Reorder point = dL + safety stock = 2(18) + 9 = 45 units C = ($15) + ($45) + 9($15) 75 2 936 C = $562.50 + $561.60 + $135 = $1259.10

P System Example d(P+LT) = sd P + LT = 5 6 = 12 units Bird feeder demand is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units, operating 52 weeks a year. Lead time (L) is 2 weeks and EOQ is 75 units with a safety stock of 9 units and a cycle-service level of 90%. Annual demand (D) is 936 units. What is the equivalent P system and total cost? P = (52) = (52) = 4.2 or 4 weeks EOQ D 75 936 Time between reviews = Standard deviation of demand over the protection period d(P+LT) = sd P + LT = 5 6 = 12 units T = Average demand during the protection interval + Safety stock = d (P + LT) + zsd(P + LT) = (18 units/week)(6 weeks) + 1.28(12 units) = 123 units

P System Example continued © 2007 Pearson Education P System Example continued d = 18 units L = 2 weeks Cycle/service level = 90% EOQ = 75 units D = (18 units/week)(52 weeks) = 936 units Safety Stock during P = 15 Holding Costs = $15/unit Ordering Costs = $45 The time between reviews (P) = 4 weeks Average demand during P + Safety stock = T = 123 units The total P-system cost for the bird feeders is: C = ($15) + ($45) + 15($15) 4(18) 2 936 C = $540 + $585 + $225 = $1350 The P system requires 15 units in safety stock, while the Q system only needs 9 units. If cost were the only criterion, the Q system would be the choice.

Fixed-Period (P) Systems Inventory is only counted at each review period May be scheduled at convenient times May require only periodic checks of inventory levels May result in stockouts between periods May require increased safety stock

Comparison of Q and P Systems Convenient to administer Orders for multiple items from the same supplier may be combined Inventory Position (IP) only required at review Systems in which inventory records are always current are called Perpetual Inventory Systems Q Systems Review frequencies can be tailored to each item Possible quantity discounts Lower, less-expensive safety stocks

Single Period Inventory Model The SPI model is designed for products that share the following characteristics: Sold at their regular price only during a single-time period Demand is highly variable but follows a known probability distribution Salvage value is less than its original cost so money is lost when these products are sold for their salvage value Objective is to balance the gross profit of the sale of a unit with the cost incurred when a unit is sold after its primary selling period

SPI Model Example: Tee shirts are purchase in multiples of 10 for a charity event for $8 each. When sold during the event the selling price is $20. After the event their salvage value is just $2. From past events the organizers know the probability of selling different quantities of tee shirts within a range from 80 to 120 Payoff Table Prob. Of Occurrence .20 .25 .30 .15 .10 Customer Demand 80 90 100 110 120 # of Shirts Ordered Profit 80 $960 $960 $960 $960 $960 $960 90 $900 $1080 $1080 $1080 $1080 $1040 Buy 100 $840 $1020 $1200 $1200 $1200 $1083 110 $780 $ 960 $1140 $1320 $1320 $1068 120 $720 $ 900 $1080 $1260 $1440 $1026 Sample calculations: Payoff (Buy 110)= sell 100($20-$8) –((110-100) x ($8-$2))= $1140 Expected Profit (Buy 100)= ($840 X .20)+($1020 x .25)+($1200 x .30) + ($1200 x .15)+($1200 x .10) = $1083