Inventory ManagementTime
Impact of lead time and uncertainty in demand
Lead time has NO impact if the demand is deterministic and at a constant rate.
Uncertainty in the demand creates the need for safety stock
Lead time under uncertain demand requires even a larger safety stock!
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Time
Choosing an Appropriate Service-Level Policy
Service level (Cycle-service level): The desired probability of not running out of stock in any one ordering cycle, which begins at the time an order is placed and ends when it arrives.
Protection interval: The period over which safety stock must protect the user from running out (in this case, it will be the leadtime period).
Reorder point (R) = DL + Safety stock (SS)
Safety stock (SS) = zsL
z = The number of standard deviations needed for a given cycle-service level.
sL=Standard deviation of the demand during lead time
DL
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Average demand (D) during lead time
Cycle-service level = 85%
Probability of stockout
(1.0 – 0.85 = 0.15)
Finding Safety Stock and R
Records show that the demand for dishwasher detergent during the lead time is normally distributed, with an average of 250 boxes and L = 22. What safety stock should be carried for a 99 percent cycle-service level? What is R?
Safety stock (SS) = zsL
= 250 + 51
= 301 boxes
2.33 is the number of standard deviations, z, to the right of average demand during the lead time that places 99% of the area under the curve to the left of that point.
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In Class Example
Suppose that the demand for an item during the lead time period is normally distributed with and an average of 85 and a standard deviation of 40.
Find the safety stock and reorder point for a service level of 95%
How much reduction is safety stock will result if the desired service level is reduced to 85%
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st = 15
st = 15
Continuous Review Systems
Selecting the reorder point with variable demand and constant lead time
Reorder point = Average demand during lead time
+ Safety stock
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Standard deviation of demand during lead time
Safety stock and reorder point
Safety stock = zσdLT
where
z = number of standard deviations needed to achieve the cycle-service level
σdLT = stand deviation of demand during lead time
σdLT = σd2L = σd L
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Continuous Review Systems
General Cost Equation
+ Annual ordering cost
C = (H) + (S) + (H) (Safety stock)
Q
2
D
Q
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Safety stock = zsL = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
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 percent cycle-service level. What is the total cost of the Q system? (t = 1 week; d = 18 units per week; L = 2 weeks)
C = $562.50 + $561.60 + $135 = $1259.10
Demand distribution for lead time must be developed:
sL = st L = 5 2 = 7.1
75
2
936
75
Class Example:
The following info is available for the purchase of kitty litter:
Demand: 100 bags/week with a standard deviation of 10 bags/week (assume 50 weeks/year)
Price: $10/bag
Desired service level: 99%
Lead time: 4 weeks
What is the Order Quantity and the Reorder Point that assures this service level while minimizing inventory costs. What is the minimum inventory costs?
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Often the case that both are variable
The equations are more complicated
Safety stock = zσdLT
+ Safety stock
Solved Problem
Grey Wolf Lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all room service items, including a special pine-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year.
a. What is the economic order quantity for the bar of soap?
b. What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service level?
c. What is the total annual cost for the bar of soap, assuming a Q system will be used?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Solved Problem
SOLUTION
a. We have D = (275)(365) = 100,375 bars of soap; S = $10; and H = $0.30. The EOQ for the bar of soap is
2DS
H
EOQ = =
2(100,375)($10)
$0.30
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Solved Problem
Consult the body of the Normal Distribution appendix for 0.9900. The closest value is 0.9901, which corresponds to a z value of 2.33. We calculate the safety stock and reorder point as follows:
Safety stock = zσdLT =
(275)(5) + 660 = 2,035 bars
b. We have d = 275 bars/day, σd = 30 bars, L = 5 days, and σLT = 1 day.
σdLT = Lσd2 + d2σLT2 =
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Solved Problem
c. The total annual cost for the Q system is
C = (H) + (S) + (H)(Safety stock)
Q
2
D
Q
2,587
2
100,375
2,587
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Periodic Review System (P)
Four of the original EOQ assumptions maintained
No constraints are placed on lot size
Holding and ordering costs
Lead times are certain
Order is placed to bring the inventory position up to the target inventory level, T, when the predetermined time, P, has elapsed
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Periodic Review System (P)
P
P
T
L
L
L
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How Much to Order in a P System
EXAMPLE
A distribution center has a backorder (BO) for five 36-inch color TV sets. No inventory is currently on hand (OH), and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled (SR)?
SOLUTION
IP = OH + SR – BO
That is, 405 sets must be ordered to bring the inventory position up to T sets.
= 0 + 0 – 5 = –5 sets
T – IP =
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Periodic Review System
Selecting the period of time between reviews (P)
The order-up-to level (T) when demand is variable and lead time is constant will be equal to the average demand during the protection period (P+L) + Safety Stock
T = d(P + L) + safety stock for protection interval
Safety stock = zσP + L , where σP + L =
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Continuous Review Model Example
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
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 percent cycle-service level. What is the total cost of the Q system? (t = 1 week; d = 18 units per week; L = 2 weeks)
C = $562.50 + $561.60 + $135 = $1259.10
Demand distribution for lead time must be developed:
sL = st L = 5 2 = 7.1
75
2
936
75
Calculating P and T
What is the equivalent P system to the bird feeder example? Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system calls for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Calculating P and T
SOLUTION
We first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week:
D =
EOQ
D
(52) = 4.2 or 4 weeks
With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Calculating P and T
We now find the standard deviation of demand over the protection interval (P + L) = 6:
Before calculating T, we also need a z value. For a 90 percent cycle-service level z = 1.28. The safety stock becomes
Safety stock = zσP + L =
= (18 units/week)(6 weeks) + 16 units = 124 units
T = Average demand during the protection interval + Safety stock
= d(P + L) + safety stock
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Periodic Review System
Use simulation when both demand and lead time are variable
Total costs for the P system are the sum of the same three cost elements as in the Q system
Order quantity and safety stock are calculated differently
C = (18 units/week)*(4 weeks)/2*(15) + 936/(18*4)*(45) + (15)*1.28*(12.25)
C = 36*15 + 13*45 + 15*16 = 540+585+240 = $1,365
C = (H) + (S) + HzσP + L
dP
2
D
dP
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In Class Example
Demand = 10 units/wk (assume 52 weeks per year) = 520
EOQ = 62 units (with reorder point system)
Lead time (L) = 3 weeks
Standard deviation in weekly demand = 8 units
Cycle-service level of 70% (z = 0.525 )
Choose the Reorder interval P such as this system is approximates the EOQ model.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Comparative Advantages
Convenient
Primary advantages of Q systems
Review frequency may be individualized
Fixed lot sizes can result in quantity discounts
Lower safety stocks
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Single Period Model
Assume that you want to have a certain level of confidence that you won’t run out of stock and that the demand follows a normal distribution, then the inventory level you should carry will be equal to:
Q = D + zs
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Example
So if the demand for newspapers on Monday’s is normally distributed with a mean of 90 and standard deviation of 10, and the newsboy wants to be 80% certain that he/she will not run out of papers, then the number of papers he/she should order will be equal to:
Q = D + zs
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Cost per unit of overestimating demand
Cost per unit of underestimating demand
Then:
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Example continued
If we assume that the newspaper boy pays 20 cents per paper and he sells it for 50 cents. How many newspapers should he order if the demand is normally distributed with a mean of 90 and standard deviation of 10?
Cost of underestimating (Lost sales)= .5 - .2 = .3
Cost of overestimation (stock piling) = .2
Probability of stock outs <= .3/(.2+.3) <= .6 <= 60%
Z = .253
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In Class Example
Assume you are helping a Christmas tree retailer determine how many trees to order for this year’s season. Assuming that you know from past experience that the average demand for Christmas trees in his area is 500 but that the demand over the past 25 years has varied depending on the economy and the offers on plastic trees. The standard deviation of the demand is 100 trees. If this person can buy each tree at an average cost of $5 and sell them at $50, then how many trees would you recommend he orders?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
L
P
d