Idea Transcript
Inventory Management (Session 3,4)
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Outline Elements of Inventory Management Inventory Control Systems Economic Order Quantity Models The Basic EOQ Model The EOQ Model with Non-Instantaneous Receipt The EOQ Model with Shortages Quantity Discounts Reorder Point Determining Safety Stocks Using Service Levels Order Quantity for a Periodic Inventory System 2
Elements of Inventory Management Role of Inventory
Inventory is a stock of items kept on hand used to meet customer demand.
A level of inventory is maintained that will meet anticipated demand.
If demand not known with certainty, safety (buffer) stocks are kept on hand.
Additional stocks are sometimes built up to meet seasonal or cyclical demand.
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Elements of Inventory Management Role of Inventory
Large amounts of inventory sometimes purchased to take advantage of discounts.
In-process inventories maintained to provide independence between operations.
Raw materials inventory kept to avoid delays in case of supplier problems.
Stock of finished parts kept to meet customer demand in event of work stoppage.
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Elements of Inventory Management Demand
Inventory exists to meet the demand of customers.
Customers can be external (purchasers of products) or internal (workers using material).
Management needs accurate forecast of demand.
Items that are used internally to produce a final product are referred to as dependent demand items.
Items that are final products demanded by an external customer are independent demand items.
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Elements of Inventory Management Inventory Costs
Carrying costs - Costs of holding items in storage. Vary with level of inventory and sometimes with length of time held. Include facility operating costs, record keeping, interest, etc. Assigned on a per unit basis per time period, or as percentage of average inventory value (usually estimated as 10% to 40%).
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Elements of Inventory Management Inventory Costs
Ordering costs - costs of replenishing stock of inventory. Expressed as dollar amount per order, independent of order size. Vary with the number of orders made. Include purchase orders, shipping, handling, inspection, etc.
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Elements of Inventory Management Inventory Costs
Shortage, or stockout costs - Costs associated with insufficient inventory. Result in permanent loss of sales and profits for items not on hand. Sometimes penalties involved; if customer is internal, work delays could result.
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Inventory Control Systems An inventory control system controls the level of inventory by determining how much (replenishment level) and when to order. Two basic types of systems -continuous (fixed-order quantity) and periodic (fixed-time). In a continuous system, an order is placed for the same constant amount when inventory decreases to a specified level. In a periodic system, an order is placed for a variable amount after a specified period of time.
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Inventory Control Systems Continuous Inventory System A continual record of inventory level is maintained. Whenever inventory decreases to a predetermined level, the reorder point, an order is placed for a fixed amount to replenish the stock. The fixed amount is termed the economic order quantity, whose magnitude is set at a level that minimizes the total inventory carrying, ordering, and shortage costs. Because of continual monitoring, management is always aware of status of inventory level and critical parts, but system is relatively expensive to maintain. 10
Inventory Control Systems Periodic Inventory System Inventory on hand is counted at specific time intervals and an order placed that brings inventory up to a specified level. Inventory not monitored between counts and system is therefore less costly to track and keep account of. Results in less direct control by management and thus generally higher levels of inventory to guard against stockouts. System requires a new order quantity each time an order is placed. Used in smaller retail stores, drugstores, grocery stores and offices. 11
Economic Order Quantity Models Economic order quantity, or economic lot size, is the quantity ordered when inventory decreases to the reorder point. Amount is determined using the economic order quantity (EOQ) model. Purpose of the EOQ model is to determine the optimal order size that will minimize total inventory costs. Three model versions to be discussed: Basic EOQ model EOQ model without instantaneous receipt EOQ model with shortages 12
Economic Order Quantity Models Basic EOQ Model A formula for determining the optimal order size that minimizes the sum of carrying costs and ordering costs. Simplifying assumptions and restrictions: Demand is known with certainty and is relatively constant over time. No shortages are allowed. Lead time for the receipt of orders is constant. The order quantity is received all at once and instantaneously. 13
Economic Order Quantity Models Basic EOQ Model
Figure The Inventory Order Cycle
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Basic EOQ Model Carrying Cost Carrying cost usually expressed on a per unit basis of time, traditionally one year. Annual carrying cost equals carrying cost per unit per year times average inventory level: Carrying cost per unit per year = Cc Average inventory = Q/2 Annual carrying cost = CcQ/2.
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Basic EOQ Model Carrying Cost
Figure: Average Inventory
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Basic EOQ Model Ordering Cost Total annual ordering cost equals cost per order (Co) times number of orders per year. Number of orders per year, with known and constant demand, D, is D/Q, where Q is the order size: Annual ordering cost = CoD/Q Only variable is Q, Co and D are constant parameters. Relative magnitude of the ordering cost is dependent on order size.
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Basic EOQ Model Total Inventory Cost Total annual inventory cost is sum of ordering and carrying cost:
TC = C D + C Q c2 oQ
TC = Total Inventory Cost Co = Ordering Cost Cc = Inventory Carrying Cost Q = Order Quantity D = Demand
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Basic EOQ Model Total Inventory Cost
Figure: The EOQ Cost Model
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Basic EOQ Model EOQ and Min. Total Cost EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost: Q C D = o + C opt TC c 2 min Qopt 2C D o Q = opt C c
The EOQ model is robust because Q is a square root and errors in the estimation of D, Cc and Co are dampened. 20
Example 1 I-75 Carpet Discount Store, Super Shag carpet sales. Given following data, determine number of orders to be made annually and time between orders given store is open every day except Sunday, Thanksgiving Day, and Christmas Day.
Model parameters : C = $0.75, C = $150, D = 10,000yd c o Optimal order size : 2C D o = 2(150)(10,000) = 2,000 yd Q = opt C (0.75) c
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Total annual inventory cost : Q = C D + C opt = (150)10,000 + (0.75) (2,000) = $1,500 TC c 2 oQ 2,000 2 min opt Number of orders per year : D = 10,000 = 5 2,000 Q opt Order cycle time = 311 days = 311 = 62.2 store days 5 D/Q opt 22
Basic EOQ Model EOQ Analysis over time For any time period unit of analysis, EOQ is the same. Shag Carpet example on monthly basis: Model parameters : C = $0.0625 per yd per month c C = $150 per order o D = 833.3 yd per month Optimal order size : 2C D o = 2(150)(833.3) = 2,000 yd Q = opt C (0.0625) c
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Basic EOQ Model EOQ Analysis over time Total monthly inventory cost : Q = C D + C opt = (150) (833.3) + (0.0625) (2,000) TC c 2 oQ 2,000 2 min opt = $125 per month Total annual inventory cost = ($125)(12) = $1,500
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EOQ Model Non-Instantaneous Receipt Description In the non-instantaneous receipt model the assumption that orders are received all at once is relaxed. (Also known as gradual usage or production lot size model.) The order quantity is received gradually over time and inventory is drawn on at the same time it is being replenished.
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EOQ Model Non-Instantaneous Receipt Description
Figure: The EOQ Model with Non-Instantaneous Order Receipt
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EOQ Model Non-Instantaneous Receipt Description p = daily rate at which the order is received over time d = daily rate at which inventory is demanded
Model ⎛ ⎞ d ⎜ Maximum inventory level = Q⎜1 − ⎟⎟ Formulation p⎠ ⎝ ⎛
⎞
Average inventory level = Q ⎜⎜1 − d ⎟⎟ 2 ⎝ p⎠ ⎛ ⎞ Q d ⎜ Total carrying cost = C ⎜1− ⎟⎟ c 2 ⎝ p⎠ ⎛ ⎞ Total annual inventory cost = C D + C Q ⎜⎜1 − d ⎟⎟ o Q c 2 ⎝ p⎠
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EOQ Model Non-Instantaneous Receipt Description
⎛ ⎞ C Q ⎜⎜1 − d ⎟⎟ = C D at lowest point of total cost curve c 2 ⎝ p⎠ o Q
2C D o Optimal order size : Q = opt C (1 − d / p) c
Equation shown above is the optimal order size under non-Instantaneous receipt 28
Example Super Shag carpet manufacturing facility: C = $150 o C = $0.75 per unit c D = 10,000 yd per year = 10,000/311 = 32.2 yd per day p = 150 yd per day 2C D o Optimal order size : Q = = 2(150)(10,000) opt ⎛ ⎞ ⎛ ⎞ 32 . 2 d ⎜ ⎟ ⎜ ⎟⎟ 0 . 75 1 − C ⎜1 − ⎟ ⎜ c⎝ p ⎠ 150 ⎠ ⎝ = 2,256.8 yd 29
Example ⎛
⎞
Total minimum annual inventory cost = C D + C Q ⎜⎜1 − d ⎟⎟ c 2 ⎝ p⎠ oQ ⎛ ⎞ = (150 ) (10,000 ) + (.075 ) (2,256 .8) ⎜⎜1 − 32 .2 ⎟⎟ = $1,329 150 ⎠ 2 (2,256 .8) ⎝ Production run length = Q = 2,256 .8 = 15 .05 days p 150 Number of orders per year (productio n runs) = D Q = 10,000 = 4.43 runs 2,256 .8 ⎛
⎞
⎛
⎞
Maximum inventory level = Q ⎜⎜1 − d ⎟⎟ = 2,256 .8⎜⎜1 − 32 .2 ⎟⎟ = 1,772 yd p⎠ 150 ⎠ ⎝ ⎝ 30
EOQ Model With Shortages
In the EOQ model with shortages, the assumption that shortages cannot exist is relaxed. Assumed that unmet demand can be backordered with all demand eventually satisfied.
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EOQ Model With Shortages
Figure: The EOQ Model with Shortages
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EOQ Model With Shortages 2 S Total shortage costs = C s 2Q
2 Q S ( ) − Total carrying costs = C c 2Q
Total ordering cost = C D oQ 2 2 Q S ( ) − S Total inventory cost = C +C +C D s 2Q c 2Q oQ 2Co D ⎛⎜ Cs + Cc ⎞⎟ Optimal order quantity = Q = opt Cc ⎜ Cs ⎟ ⎝
Shortage level = S
opt
⎛ C ⎜ c =Q ⎜ opt ⎜ C + C s ⎝ c
⎠
⎞ ⎟ ⎟ ⎟ ⎠
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EOQ Model With Shortages
Figure: Cost Model with Shortages
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Example I-75 Carpet Discount Store allows shortages; shortage cost Cs, is $2/yard per year. C = $150 o C = $0.75 per yd c C = $2 per yd s D = 10,000 yd Optimal order quantity : 2C D ⎛⎜ C + C ⎞⎟ s c = 2(150)(10,000) ⎛⎜ 2 + 0.75 ⎞⎟ = 2,345.2 yd o Q = ⎜ ⎟⎟ opt 0.75 2 ⎟⎠ C ⎜⎜ C ⎝ s ⎠ c ⎝ 35
Example Shortage level: S opt
⎞ ⎛ C ⎜ c ⎟ = 2,345.2⎛⎜ 0.75 ⎞⎟ = 639.6 yd =Q ⎜ ⎜ 2 + 0.75 ⎟ opt ⎜ C + C ⎟⎟ ⎝ ⎠ s⎠ ⎝ c
Total inventory cost : 2 2 Q S ( ) − S TC = C +C +C D s 2Q c 2Q oQ 2 (0.75)(1,705.6)2 (150)(10,000) ( 2 )( 639 . 6 ) = + + 2,345.2 2(2,345.2) 2(2,345.2) = $174.44 + 465.16 + 639.60 = $1,279.20 36
Example Number of orders = D = 10,000 = 4.26 orders per year Q 2,345.2 Maximum inventory level = Q − S = 2,345.2 − 639.6 =1,705.6 yd Time between orders = t =
days per year = 311 = 73.0 days number of orders 4.26
Time during which inventory is on hand = t1 = Q − S = 2,345.2 -639.6 = 0.171 or 53.2 days D 10,000 Time during which there is a shortage = t 2 = S = 639.6 = 0.064 year or 19.9 days D 10,000 37
Quantity Discounts Price discounts are often offered if a predetermined number of units is ordered or when ordering materials in high volume. Basic EOQ model used with purchase price added:
TC = C D + C Q + PD oQ c 2 where: P = per unit price of the item D = annual demand Quantity discounts are evaluated under two different scenarios: With constant carrying costs With carrying costs as a percentage of purchase price 38
Quantity Discounts with Constant Cc
Optimal order size is the same regardless of the discount price. The total cost with the optimal order size must be compared with any lower total cost with a discount price to determine which is the lesser.
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Example University bookstore: For following discount schedule offered by Comptek, should bookstore buy at the discount terms or order the basic EOQ order size?
Quantity 1 – 49 50-89 90 +
Price $ 1400 1100 900
Determine optimal order size and total cost: C = $2,500 o
C = $190 per unit c
2C D o = 2(2,500)(200) = 72.5 Q = opt C 190 c
D = 200
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Example Compute total cost at eligible discount price ($1,100): Q C D TC = o + C opt + PD c 2 min Qopt = (2,500 )( 200 ) + (190 ) (72 .5) + (1,100 )( 200 ) = $233,784 2 (72 .5)
Compare with total cost of with order size of 90 and price of $900: C D TC = o + C Q + PD c2 Q = (2,500)(200) + (190)(90) + (900)(200) = $194,105 2 (90) Because $194,105 < $233,784, maximum discount price should be taken and 90 units ordered.
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Quantity Discounts with Cc % of Price University Bookstore example, but a different optimal order size for each price discount. Optimal order size and total cost determined using basic EOQ model with no quantity discount. This cost then compared with various discount quantity order sizes to determine minimum cost order. This must be compared with EOQ-determined order size for specific discount price. Data: Co = $2,500 D = 200 computers per year
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Quantity Discounts with Cc % of Price Quantity
Price
Carrying Cost
0 - 49 50 - 89 90 +
$1,400 1,100 900
1,400(.15) = $210 1,100(.15) = 165 900(.15) = 135
Compute optimum order size for purchase price without discount and Cc = $210: 2C D o = 2(2,500)(200) = 69 Q = opt C 210 c
Compute new order size: Q = 2(2,500)(200) = 77.8 opt 165 43
Quantity Discounts with Cc % of Price Compute minimum total cost: C D TC = o + C Q + PD = (2,500)(200) + 165 (77.8) + (1,100)(200) c2 77.8 2 Q = $232,845
Compare with cost, discount price of $900, order quantity of 90: TC = (2,500)(200) + (135)(90) + (900)(200) = $191,630 90 2 Optimal order size computed as follows: Q = 2(2,500)(200) = 86.1 opt 135 Since this order size is less than 90 units , it is not feasible,thus optimal order size is 90 units. 44
Reorder Point The reorder point is the inventory level at which a new order is placed. Order must be made while there is enough stock in place to cover demand during lead time. Formulation: R = dL where d = demand rate per time period L = lead time For Carpet Discount store problem: R = dL = (10,000/311)(10) = 321.54 45
Reorder Point
Figure: Reorder Point and Lead Time
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Reorder Point Inventory level might be depleted at slower or faster rate during lead time. When demand is uncertain, safety stock is added as a hedge against stockout.
Figure: Inventory Model with Uncertain Demand
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Reorder Point (4 Reorder of 4)
Point
Figure: Inventory model with safety stock
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Determining Safety Stock by Service Level Service level is probability that amount of inventory on hand is sufficient to meet demand during lead time (probability stockout will not occur). The higher the probability inventory will be on hand, the more likely customer demand will be met. Service level of 90% means there is a .90 probability that demand will be met during lead time and .10 probability of a stockout. 49
Reorder Point with Variable Demand R = d L + Zσ d L where: R = reorder point d = average daily demand L = lead time
σ d = the standard deviation of daily demand Z = number of standard deviations corresponding to service level probability Zσ d L = safety stock 50
Reorder Point with Variable Demand
Figure: Reorder Point for a Service Level
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Example I-75 Carpet Discount Store Super Shag carpet. For following data, determine reorder point and safety stock for service level of 95%. d = 30 yd per day L = 10 days σ = 5 yd per day d For 95% service level, Z = 1.65 (Table A -1, appendix A) R = d L + Zσ = 326.1 yd
d
L = 30(10) + (1.65)(5)( 10 ) = 300 + 26.1
Safety stock is second term in reorder point formula : 26.1.
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Reorder Point with Variable Lead Time For constant demand and variable lead time:
R = d L + Zdσ L where: d = constant daily demand L = average lead time σ L = standard deviation of lead time dσ L = standard deviation of demand during lead time Zdσ L = safety stock
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Example Carpet Discount Store:
d = 30 yd per day L = 10 days
σ L = 3 days Z = 1.65 for a 95% service level R = d L + Zdσ L = (30)(10) + (1.65)(30)(3) = 300 +148.5 = 448.5 yd
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Reorder Point with Variable Demand and Lead Time When both demand and lead time are variable:
2
2 2 R = d L + Z (σ d ) L + (σ L ) d where: d = average daily demand L = average lead time 2
2 2 (σ d ) L + (σ L) d = standard deviation of demand during lead time 2
2 2 Z (σ d ) L + (σ L ) d = safety stock 55
Example Carpet Discount Store:
d = 30 yd per day σ d = 5 yd per day
L = 10 days σ L = 3 days Z = 1.65 for 95% service level 2
2 2 R = d L + Z (σ d ) L + (σ L) d = (30)(10) + (1.65) (5)(5)(10) + (3)(3)(30)(30) = 300 + 150.8 = 450.8 yds 56
Order Quantity for a Periodic Inventory System A periodic, or fixed-time period inventory system is one in which time between orders is constant and the order size varies. Vendors make periodic visits, and stock of inventory is counted. An order is placed, if necessary, to bring inventory level back up to some desired level. Inventory not monitored between visits. At times, inventory can be exhausted prior to the visit, resulting in a stockout. Larger safety stocks are generally required for the periodic inventory system.
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Order Quantity for Variable Demand For normally distributed variable daily demand:
Q = d (tb + L) + Zσ d tb + L − I where: d = average demand rate tb = the fixed time between orders L = lead time σ d = standard deviation of demand Zσ d tb + L = safety stock I = inventory in stock
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Order Quantity for Variable Demand Corner Drug Store with periodic inventory system. Order size to maintain 95% service level:
d = 6 bottles per day σ d = 1.2 bottles tb = 60 days L = 5 days I = 8 bottles Z = 1.65 for 95% service level Q = d (tb + L) + Zσ d tb + L − I = (6)(60 + 5) + (1.65)(1.2) 60 + 5 −8 = 398 bottles 59
Example 1 (Electronic Village Store) Electronic Village store stocks and sells a particular brand of PCs. It costs the store $450 each time it places an order with the manufacturer for the PCs. The annual cost of carrying the PCs in inventory is $ 170. the store manager estimates the annual demand for the PCs will be 1200 units. a) Determine the optimal order quantity and the total min. inventory cost. b) Assume that shortages are allowed and shortage cost is $600 per unit per year. Compute the optimal order quantity and the total minimum inventory cost. 60
Example 1 For data below determine: Optimal order quantity and total minimum inventory cost. Assume shortage cost of $600 per unit per year, compute optimal order quantity and minimum inventory cost. Step 1 (part a): Determine the Optimal Order Quantity.
D = 1,200 personal computers C = $170 c C = $450 o 2C D o = 2(450)(1,200) = 79.7 personal computers Q= C 170 c
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Example 1 ⎛ ⎞ ⎛ ⎞ Total cost = C Q + C D = 170⎜⎜ 79.7 ⎟⎟ = 450⎜⎜1,200 ⎟⎟ c2 oQ ⎝ 2 ⎠ ⎝ 79.7 ⎠
= $13,549.91 Step 2 (part b): Compute the EOQ with Shortages. Cs = $600 2C D ⎛⎜ C + C ⎞⎟ o s c = 2(450)(1200) ⎛⎜ 600 + 170 ⎞⎟ Q= ⎜ ⎟⎟ C ⎜⎜ C 170 600 ⎟⎠ ⎝ c ⎝ s ⎠ = 90.3 personal computers 62
Example 1 S
⎛ C ⎞ ⎛ 170 ⎞⎟ = 19.9 personal computers ⎜ ⎟ c ⎜ = Q⎜ = 90 . 3 ⎜170 + 600 ⎟ ⎜ C + C ⎟⎟ ⎝ ⎠ s⎠ ⎝ c
2 C D C S2 − ( ) Q S + o Total cost = s + C c 2Q 2Q Q 2 2 ⎛ ⎞ − ( 600 )( 19 . 9 ) ( 90 . 3 19 . 9 ) = +170 + 450⎜⎜1,200 ⎟⎟ 2(90.3) 2(90.3) ⎝ 90.3 ⎠ = $11,960.98
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Example 2 (Computer Product Store) A computer products store stocks color graphics monitors, and the daily demand is normally distributed with a mean of 1.6 monitors and a standard deviation of 0.4 monitors. The lead time to receive an order from the manufacturer is 15 days. Determine the reorder point that will achieve a 98% service level.
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Example 2 Sells monitors with daily demand normally distributed with a mean of 1.6 monitors and standard deviation of 0.4 monitors. Lead time for delivery from supplier is 15 days. Determine the reorder point to achieve a 98% service level. Step 1: Identify parameters. d = 1.6 monitors per day L =15 days
σ d = 0.4 monitors per day Z = 2.05 (for a 98% service level) 65
Example 2 Step 2: Solve for R. R = d L + Zσ d L = (1.6)(15) + (2.05)(.04) 15 = 24 + 3.18 = 27.18 monitors
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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
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% of $ Value 30 0
% of Use
30
A B
C
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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 67
Inventory Accuracy and Cycle Counting Defined
Inventory accuracy refers to how well the inventory records agree with physical count
Cycle Counting is a physical inventorytaking technique in which inventory is counted on a frequent basis rather than once or twice a year 68
End of Session
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