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Transcript of Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr....
![Page 1: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/1.jpg)
Operations Research II Course ,, September 2013 1
Part 3: Inventory Models
Operations Research II
Dr. Aref Rashad
![Page 2: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/2.jpg)
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
Topics
Operations Research II Course ,, September 2013 2
![Page 3: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/3.jpg)
Chapter 16 - Inventory Management 3
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.
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
Elements of Inventory ManagementRole of Inventory
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Chapter 16 - Inventory Management 4
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.
Elements of Inventory ManagementDemand
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Chapter 16 - Inventory Management 5
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%).
Elements of Inventory ManagementInventory Costs (1 of 2)
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Chapter 16 - Inventory Management 6
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.
Elements of Inventory ManagementInventory Costs (2 of 2)
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.
![Page 7: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/7.jpg)
Chapter 16 - Inventory Management 7
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.
Inventory Control Systems
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Chapter 16 - Inventory Management 8
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.
Inventory Control SystemsContinuous Inventory Systems
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Chapter 16 - Inventory Management 9
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.
Inventory Control SystemsPeriodic Inventory Systems
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Chapter 16 - Inventory Management 10
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
Economic Order Quantity Models
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Chapter 16 - Inventory Management 11
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.
Economic Order Quantity ModelsBasic EOQ Model (1 of 2)
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Chapter 16 - Inventory Management 12
The Inventory Order Cycle
Economic Order Quantity ModelsBasic EOQ Model (2 of 2)
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Chapter 16 - Inventory Management 13
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.
Basic EOQ ModelCarrying Cost (1 of 2)
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Chapter 16 - Inventory Management 14
Average Inventory
Basic EOQ ModelCarrying Cost (2 of 2)
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Chapter 16 - Inventory Management 15
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.
Basic EOQ ModelOrdering Cost
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Chapter 16 - Inventory Management 16
Figure 16.5The EOQ Cost Model
Basic EOQ ModelTotal Inventory Cost
2QCc
QDCoTC Total annual inventory cost is sum of ordering and
carrying cost:
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Chapter 16 - Inventory Management 17
EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost:
The EOQ model is robust because Q is a square root and errors in the estimation of D, Cc and Co are dampened.
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Basic EOQ ModelEOQ and Minimum Total Cost
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Chapter 16 - Inventory Management 18
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Carpet Discount Store
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.
Basic EOQ ModelExample (1 of 2)
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Chapter 16 - Inventory Management 19
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Basic EOQ ModelExample (2 of 2)
![Page 20: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/20.jpg)
Chapter 16 - Inventory Management 20
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For any time period unit of analysis, EOQ is the same.
Shag Carpet example on monthly basis:
Basic EOQ ModelEOQ Analysis Over Time (1 of 2)
![Page 21: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/21.jpg)
Chapter 16 - Inventory Management 21
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Basic EOQ ModelEOQ Analysis Over Time (2 of 2)
![Page 22: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/22.jpg)
Chapter 16 - Inventory Management 22
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.
EOQ ModelNon-Instantaneous Receipt Description (1 of 2)
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Chapter 16 - Inventory Management 23
The EOQ Model with Non-Instantaneous Order Receipt
EOQ ModelNon-Instantaneous Receipt Description (2 of 2)
![Page 24: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/24.jpg)
Chapter 16 - Inventory Management 24
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Non-Instantaneous Receipt ModelModel Formulation (1 of 2)
![Page 25: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/25.jpg)
Chapter 16 - Inventory Management 25
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curve cost total of point lowest at 12
Non-Instantaneous Receipt ModelModel Formulation (2 of 2)
![Page 26: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/26.jpg)
Chapter 16 - Inventory Management 26
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Non-Instantaneous Receipt ModelExample (1 of 2)
Super Shag carpet manufacturing facility:
![Page 27: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/27.jpg)
Chapter 16 - Inventory Management 27
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![Page 28: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/28.jpg)
Chapter 16 - Inventory Management 28
The EOQ Model with Shortages
EOQ Model with ShortagesDescription
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
![Page 29: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/29.jpg)
Chapter 16 - Inventory Management 29
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![Page 30: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/30.jpg)
Chapter 16 - Inventory Management 30
Cost Model with Shortages
EOQ Model with ShortagesModel Formulation (2 of 2)
![Page 31: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/31.jpg)
Chapter 16 - Inventory Management 31
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EOQ Model with ShortagesModel Formulation (1 of 3)
I-75 Carpet Discount Store allows shortages; shortage cost Cs, is $2/yard per year.
![Page 32: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/32.jpg)
Chapter 16 - Inventory Management 32
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![Page 33: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/33.jpg)
Chapter 16 - Inventory Management 33
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![Page 34: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/34.jpg)
Chapter 16 - Inventory Management 34
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:
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
PDQCcQDCoTC
2
Quantity Discounts
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Chapter 16 - Inventory Management 35
Quantity Discounts with Constant Carrying CostsAnalysis Approach
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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Chapter 16 - Inventory Management 36
University bookstore: For following discount schedule offered by Comptek, should bookstore buy at the discount terms or order the basic EOQ order size?
Determine optimal order size and total cost:
Quantity Price 1- 49 50 – 89 90 +
$1,400 1,100 900
Quantity Discounts with Constant Carrying CostsExample (1 of 2)
572190
20050022Cc
2CoDQopt
200 D unit per $190 Cc $2,500 Co
.))(,(
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Chapter 16 - Inventory Management 37
Compute total cost at eligible discount price ($1,100):
Compare with total cost of with order size of $90 and price of $900:
Because $194,105 < $233,784, maximum discount price should be taken and 90 units ordered.
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Quantity Discounts with Constant Carrying CostsExample (2 of 2)
![Page 38: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/38.jpg)
Chapter 16 - Inventory Management 38
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
Quantity Discounts with Carrying CostsPercentage of Price Example (1 of 3)
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Chapter 16 - Inventory Management 39
Compute optimum order size for purchase price without discount and Cc = $210:
Compute new order size:
Quantity Price Carrying Cost
0 - 49$1,4001,400.)15 = ($21050 - 89 1,1001,100.)15 = (165
90+ 900 900.)15 = (135
Quantity Discounts with Carrying CostsPercentage of Price Example (2 of 3)
69210
200500222 ))(,(CcCoDQopt
877165
20050022 .))(,( Qopt
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Chapter 16 - Inventory Management 40
Compute minimum total cost:
Compare with cost, discount price of $900, order quantity of 90:
Optimal order size computed as follows:
Since this order size is less than 90 units , it is not feasible,thus optimal order size is 90 units.
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Quantity Discounts with Carrying CostsPercentage of Price Example (3 of 3)
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Chapter 16 - Inventory Management 41
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
Reorder Point (1 of 4)
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Chapter 16 - Inventory Management 42
Reorder Point and Lead Time
Reorder Point (2 of 4)
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Chapter 16 - Inventory Management 43
Inventory Model with Uncertain Demand
Reorder Point (3 of 4)
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.
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Chapter 16 - Inventory Management 44
Inventory model with safety stock
Reorder Point (4 of 4)
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Chapter 16 - Inventory Management 45
Determining Safety Stocks Using Service Levels
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.
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Chapter 16 - Inventory Management 46
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LdZLdR
Reorder Point with Variable Demand (1 of 2)
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Chapter 16 - Inventory Management 47
Reorder Point for a Service Level
Reorder Point with Variable Demand (2 of 2)
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Chapter 16 - Inventory Management 48
26.1. : formula point reorder in term second is stockSafety
yd13261263001056511030
)appendix A 1,- A(Table 1.65 Z level, service 95% For
day per yd5 ddays 10 L
day per yd30
..))()(.()(
LdZLdR
d
I-75 Carpet Discount Store Super Shag carpet.
For following data, determine reorder point and safety stock for service level of 95%.
Reorder Point with Variable Demand Example
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Chapter 16 - Inventory Management 49
stocksafety
timelead during demand ofdeviation standard
timelead ofdeviation standard timelead average
demanddaily constant
:where
LZdL
dL
Ld
LZdLdR
Reorder Point with Variable Lead Time
For constant demand and variable lead time:
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Chapter 16 - Inventory Management 50
yd 5.4485.148300)3)(30)(65.1()10)(30(
level service 95% afor 1.65
days 3
days 10
dayper yd 30
LZdLdR
Z
L
L
d
Reorder Point with Variable Lead Time Example
Carpet Discount Store:
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Chapter 16 - Inventory Management 51
stocksafety 22
)(2
)(Z
timelead during demand ofdeviation standard 22
)(2
)(
timelead average demanddaily average
:where
22)(
2)(
dLLd
dLLd
Ld
dLLdZLdR
When both demand and lead time are variable:
Reorder PointVariable Demand and Lead Time
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Chapter 16 - Inventory Management 52
yds 450.8 150.8 300
(30)(3)(3)(30) (5)(5)(10)(1.65) (30)(10)
22)(
2)(
level service 95%for 1.65 days 3days 10
dayper yd 5 dayper yd 30
dLLdZLdR
ZL
Ld
d
Carpet Discount Store:
Reorder PointVariable Demand and Lead Time Example
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Chapter 16 - Inventory Management 53
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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Chapter 16 - Inventory Management 54
For normally distributed variable daily demand:
stockin inventory
stocksafety
demand ofdeviation standard timelead
ordersbetween timefixed therate demand average
:where
)(
I
LbtdZd
Lbtd
ILbtdZLbtdQ
Order Quantity for Variable Demand
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Chapter 16 - Inventory Management 55
bottles 398 85 60)(1.65)(1.2 5) (6)(60
)(
level service 95%for 1.65 bottles 8 days 5
days 60
bottles 1.2 dayper bottles 6
ILbtdZLbtdQ
ZILbtd
d
Corner Drug Store with periodic inventory system.
Order size to maintain 95% service level:
Order Quantity for Variable Demand Example
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Chapter 16 - Inventory Management 56
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.
Example Problem SolutionElectronic Village Store (1 of 3)
computers personal 779170
200145022
450$170
computers personal 1,200
.),)((
$
CcCoDQ
CoCcD
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Chapter 16 - Inventory Management 57
Step 2 (part b): Compute the EOQ with Shortages.
$13,549.91
7792001450
2779170
2 cost Total
.,.
QDCoQCc
Example Problem SolutionElectronic Village Store (2 of 3)
computers personal 390
600170600
170120045022
600
.
))((
$
CsCcCs
CcCoDQ
Cs
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Chapter 16 - Inventory Management 58
Example Problem SolutionElectronic Village Store (3 of 3)
$11,960.98
3902001450
39022919390170
39022919600
22
22 cost Total
computers personal 19.9600170
170390
.,
).()..(
).().)((
)(
.
QCoD
QSQCc
QCsS
CsCcCcQS
![Page 59: Operations Research II Course,, September 20131 Part 3: Inventory Models Operations Research II Dr. Aref Rashad.](https://reader036.fdocuments.in/reader036/viewer/2022062315/5697bfc21a28abf838ca496d/html5/thumbnails/59.jpg)
Chapter 16 - Inventory Management 59
level) service 98% a(for 05.2
dayper monitors 0.4
days 15
dayper monitors 1.6
Z
d
L
d
Example Problem SolutionComputer Products Store (1 of 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.
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Chapter 16 - Inventory Management 60
Example Problem SolutionComputer Products Store (2 of 2)
Step 2: Solve for R.
monitors 18.2718.324
15)04)(.05.2()15)(6.1(
LdZLdR