D. Anthony CheversSBCO 6240 - Production and Operations Management
Lecture 7 – Inventory Planning & Control |
Lecture #5 – Inventory Management
Definitions & Purpose Inventory Costs & Trade-off Inventory Models
• Independent DemandBasic economic order quantity (EOQ) modelQuantity Discount Model
• Safety Stock & Customer Service Level• Fixed-Period (P) System
Exercises
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Lecture 7 – Inventory Planning & Control |
Inventory Planning & Control
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Supply
The operation
Operations resources
Delivery of products and services when
required
Need for products and services at a
particular time
Demand
The market
Customer requirements
Lecture 7 – Inventory Planning & Control |
Definitions
Inventory - A physical resource that a firm holds in stock with the intent of selling it or transforming it into a more valuable state.
Inventory System - A set of policies and controls that monitors levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be.
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Lecture 7 – Inventory Planning & Control |
Reasons for Inventories
Improve customer service Economies of purchasing Economies of production Transportation savings Hedge against future Unplanned shocks
(labour strikes, natural disasters, surges in demand, etc.)
To maintain independence of supply chain
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Lecture 7 – Inventory Planning & Control |
Role of Inventory in Services
Need to inventory goods that facilitate service
Helps to “deliver” service on time Helps to accommodate
seasonal demand Helps to hedge against
increases in cost
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Lecture 7 – Inventory Planning & Control |
Reasons Against Inventory
Non-value added costs Opportunity cost Complacency Inventory deteriorates,
becomes obsolete, lost, stolen, etc.
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Lecture 7 – Inventory Planning & Control |
Gaining Competitive Advantage
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•Value added activitiesValue added activities•Cost adding activitiesCost adding activities
VALUE ADDED
COST ADDED
Lecture 7 – Inventory Planning & Control |
Inventory Costs
Procurement costs Carrying costs Out-of-stock costs
• Interest or Opportunity Costs• Storage and Handling Costs• Insurance and Shrinkage Costs• Ordering and Setup Costs• Transportation Costs
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Lecture 7 – Inventory Planning & Control |
Identifying Critical Inventory Items with ABC
Analysis Thousands of items are held in inventory Only a small percentage of them deserves
management’s closest attention and tight control
ABC analysis is the process of dividing items into three classes, according to their dollar usage, so that managers can focus on items that have the highest dollar value e.g. – Class A items typically represent only about 20% of the
items but account for 80% of the dollar usage
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Lecture 7 – Inventory Planning & Control |
Class C
Class A
Class B
ABC Classification
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Per
cen
tag
e o
f d
oll
ar v
alu
e100 —
90 —
80 —
70 —
60 —
50 —
40 —
30 —
20 —
10 —
0 —10 20 30 40 50 60 70 80 90 100
Percentage of items
Lecture 7 – Inventory Planning & Control |
Classifying Inventory Items
ABC Classification (Pareto Principle)
A ItemsA Items: very tight control, complete and accurate records, frequent review
B ItemsB Items: less tightly controlled, good records, regular review
C ItemsC Items: simplest controls possible, minimal records, large inventories, periodic review and reorder
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Lecture 7 – Inventory Planning & Control |
Types of Inventory
The stocks held in any organization take a variety of forms but generally can be classified under four main headings:
Raw Material & IngredientsRaw Material & Ingredients – material purchased, but not processed.
Work-in-ProcessWork-in-Process – partially completed inventory.MROMRO – Inventories devoted to maintenance/ repair/
operating supplies. Exist because the need and timing for maintenance and repair of some equipment are unknown.
Finished GoodsFinished Goods – Completed inventory (occurs because customer demands for a given period is unknown)
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Lecture 7 – Inventory Planning & Control |
Basic EOQ Model Assumptions
Demand is known, constant & independent Lead time is known & constant Receipt of inventory is instantaneous &
complete Quantity discounts are not possible The only variable costs are the cost of setting
up or placing an order & the cost of holding inventory over time
Stock-outs can be completely avoided if orders are placed at the right time
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Lecture 7 – Inventory Planning & Control | 16
Rel
evan
t T
otal
Cos
ts (
Dol
lars
)
2,000
4,000
6,000
8,000
10,000
5,434
600 1,200 1,800 2,400988EOQ
Annual relevant
carrying costs
Annual relevant total costs
Annual relevant ordering costs
Order Quantity (Units)
Economic-Order-Quantity Decision Model Example
20 - 15
Lecture 7 – Inventory Planning & Control |
Inventory Models for Independent Demand
Basic Economic Order Quantity (EOQ) Model • EOQ = √(2xDxS)/H Or √(2xDxOC)/H
Production Order Quantity Model• Q*p = √(2xDxS)/{H[1 – (d/p)]}
Quantity Discount Model• Q* = √(2xDxS)/(IxP)
Where D = Annual demand; S = Set-up or ordering cost for each order; H = Holding or carrying cost per unit per year; Q* = Optimum number of pieces per order (EOQ); Q = Number of pieces per order; p = Daily production rate; d = Daily demand rate or usage rate; t = Length of the production run in days; IP is holding cost expressed as a percentage of unit price; OC = Order cost
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Lecture 7 – Inventory Planning & Control |
Diagram –Perpetual Inventory
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Qu
anti
ty
227 Maximum Level
EOQ/ROQ
ROL/ROP33
Minimum Level
30
0 Time
Lecture 7 – Inventory Planning & Control |
Perpetual Equations
Where D is annual demand, OC is order cost, H is holding cost, CC is carrying cost, UP is unit price, U is usage, LT is lead time, TIC is total inventory cost, EOQ is economic order quantity, ROQ is reorder quantity, ROL is reorder level and Q is EOQ.
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EOQ √(2 x D x OC)/H
ROQ equivalent to EOQ
Mini LT x U
ROL (LT x U) x 1.10 [for 10% buffer stock
Maxi EOQ + (ROL – Mini)
# Orders D/EOQ
Interval Annual work days/# of orders
TIC [(Q/2)(H)] + [(D/Q) x OC]
Lecture 7 – Inventory Planning & Control |
Exercise – Local TV Distributor
A local TV distributor has found from experience that demand for a certain TV model is fairly constant at a rate of 24 sets per week. Holding cost is $28 per unit per year and ordering cost is estimated to be $50 per order. If lead time is 1 week.
a) How many sets should be ordered to minimize inventory cost?b) What is the reorder level using a 4% safety buffc) What is the total inventory cost?d) How many orders will be processed per year?e) What is the time interval between orders?
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Lecture 7 – Inventory Planning & Control |
Solution – Inventory –TV Distribitor
[Demand is equivalent to Usage]Given:
Usage (U) = 24/week
LT = 1 week
H = $28
OC= $50
Buffer = 4%
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(a) EOQ = √ (2 x 1,248 x 50)/28 = √124,800/28 = √ 4,457.14 = 66.76 TV sets
(b) ROL = (LT X U) 1.04 = (1 x 24) 1.04 = 24 x 1.04 = 24.96 TV sets
(c ) TIC = [Q/2(H)] + [D/Q(OC)] = [66.76/2(28)] + [1,248/66.76(50)] =
934.64 + 934.50 = $1,869.14
(d) # of Orders = D/EOQ = 1,248/66.76 = 18.69 times per year
(e) Interval = Annual work days / # of orders
Assume 365 days/year Assume 260 days/year365/18.69 = 19.53 days 260/18.69 = 13.91 days
Calculations
D = (24x52) = 1,248
Assumptions
Lecture 7 – Inventory Planning & Control |
Quantity Discount Model To increase sales, many companies offer quantity
discounts to their customers
Quantity discount is a reduced price for items purchased in large quantities. For e.g.
Table1Table1
Might be tempted to order 2,000 units to take advantage of lower product cost
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Discount # Discount Qty % Price
1 0 – 999 -- $5.00
2 1,000-1,999 4 $4.80
3 2,000 & over 5 $4.75
Lecture 7 – Inventory Planning & Control |
Trade-Off
As discount quantity goes up, the product cost goes down.
But, holding cost increases because orders are larger Trade off between reduced product cost and increased
holding cost Equation for total annual inventory cost, when cost of
product is included:
• Total cost [TC] = (Ordering + Holding + Product) cost• TC = (D/Q)OC + (QH/2) + PDWhere Q = Quantity ordered; D = Annual demandOC = Ordering cost or Setup cost per order or per setupP = Price per unit; H = Holding cost per unit per year
[Note: Order cost = Setup cost]
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Lecture 7 – Inventory Planning & Control |
4 Stepsto determine quantity that will minimize total
annual inventory cost
1. For each discount, calculate a value for optimal order size Q*, using:1. Q* = √(2xDxS)/(IP) Or √(2xDxO)/(IxP) Where S is setup & O is
order cost
2. Express the holding cost (I) as a percentage of unit price (P) instead of as a constant cost per unit per year, H.
2. If the order quantity is low to qualify for the discount, adjust the order quantity upwards to the lowest quantity that will qualify for the discount1. [[Total cost (TC) = Order cost +Holding cost +Product cost]]
3. Use total cost equation and compute a total cost for every Q*, Equation: [TC = (D/Q)OC + (QH)/2 + PD]
4. Select the Q* that has the lowest total cost24
Lecture 7 – Inventory Planning & Control |
Example #1 – Quantity Discount
Work’s Discount stocks toy race cars. Recently, the store has a quantity discount schedule as shown below:
Furthermore, ordering cost is $49.00 per order, annual demand is 5,000 race cars, and inventory carrying charge, as a percentage of cost, I, is 20%.
What order quantity will minimize the total inventory cost?
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Discount Discount NumberNumber
Discount Discount QuantityQuantity
Discount Discount (%)(%)
DiscountDiscountPrice (P)Price (P)
1 0 – 999 No discount $5.00
2 1,000-1,999 4 $4.80
3 2,000 & over 5 $4.75
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Solution – [Q* = √(2xDxO)/(IxP)]
Step #2 - Adjust upwards those values of Q* that are below the allowable discount range. Since Q1* is between 0 - 999, it is not adjusted.
Q1* = 700
Q2* = 1,000 - adjusted
Q3* = 2,000 - adjusted
Step #1 - Q1* = √ 2(5,000)(49) = 700 cars
(0.2)(5.00)
Q2* = √ 2(5,000)(49) = 714 cars
(0.2)(4.80)
Q3* = √ 2(5,000)(49) = 718 cars
(0.2)(4.75)
Lecture 7 – Inventory Planning & Control |
Solution cont’d …
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Step #3 - Use total cost equation and compute a total cost for every order qty.
Equation: [TC = (D/Q)OC + (QH)/2 + PD] [H = IP]5000 49 0.2 Order C Carrying C Annual Dem Total Cost
Discount Number Unit Price Order Quantity
Annual Ordering
Cost
Annual Holding
Cost
Annual Product
Cost Total1 $5.00 700 $350 $350 $25,000 $25,700
2 $4.80 1,000 $245 $480 $24,000 $24,725
3 $4.75 2,000 $123 $950 $23,750 $24,823
Step #4 - Select that order quantity with the lowest total costOrder quantity of 1,000 toy cars will minimize the total cost
Annual Product Cost: ($5 x 5,000) = $25,000
Annual Ordering Cost: Q1* = (5,000/700)49 = $350
Annual Holding Cost: Q1* = (700 x 0.2 x $5)/2 = $350
Lecture 7 – Inventory Planning & Control |
Further Problem When it is difficult or impossible to determine the cost
of being out of stock, a manager may decide to follow a policy of keeping enough safety stock on hand to meet a prescribed customer service level
Define the service level as meeting 95% of the demand (or conversely, having stock-outs only 5% of the time)
Assuming that demand during lead time (the reorder period) follows a normal curve, only the mean and standard deviation are needed to define the inventory requirements for any given service level.
Sales data are usually adequate for computing the mean (µ) and standard deviation (σ).
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Lecture 7 – Inventory Planning & Control |
Equations & Z-Values
Safety stock = x - µ [known][known]
But Z = (x - µ)/σThen x - µ = Z σ; So Safety stock = Zσ
ROP = Expected demand during lead time + safety stock
Z value @ 95% = 1.645Z value @ 97% = 1.88Z value @ 99% = 2.33
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Lecture 7 – Inventory Planning & Control |
Exercise
Member Hospital stocks a “code blue” rescue kit that has a normally distributed demand during the reorder period. The mean (average) demand during the reorder period is 350 kits, and the standard deviation is 10 kits. The hospital administrator wants to follow a policy that results in stock-outs occurring only 5% of the time.
(a)What is the appropriate value of z?
(b)How much safety stock should the hospital maintain?
(c)What reorder point should be used.
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Lecture 7 – Inventory Planning & Control |
Solution
(a) (a) We use the properties of a standardized normal curve to get a Z value for an area under the normal curve of 0.95 (or 1-0.05). Using a normal table (Appendix 1), we find a Z vague of 1.65 standard deviation form the mean
(b) (b) Safety stock = x –u
Because z = x-u
σ
Then Safety stock = Zσ ---- (1)
Solving for safety stock, as in Equation (1), gives
Safety stock = (1.65(10) =16.5 kits
(c) (c) The reorder point is:
ROP = Expected demand during lead time + safety stock
350 kits + 16.5kits of safety stock =366.5 or 367 kits31
Lecture 7 – Inventory Planning & Control |
Reorder Point for a Service Level
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Risk of a Risk of a stockoutstockout
(5% of area of(5% of area ofnormal curve)normal curve)
ROP = ? kitsROP = ? kitsMeanMeanDemandDemand
350350
Probability of Probability of No stock 95% of the timeNo stock 95% of the time
QuantityQuantity
SafetySafetyStockStock
Number of Number of standard deviationsstandard deviations
00 zz
Lecture 7 – Inventory Planning & Control |
Assumptions The following equations:
• ROP = Expected demand during lead time + Zσ• Safety stock = Zσ
Assume that both an estimate of expected demand during lead times and its standard deviation are available
When data on lead time demand are not at hand, these equations cannot be applied
Need to determine if:• Demand is variable and lead time is constant OR• Only lead time is variable OR • Both demand and lead time are variable
For each, a different equation is needed to compute ROP
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Inventory Systems
Fixed-quantity (Q) system• An EOQ ordering system, with the same order
amount each time [models considered so far]
Perpetual inventory system• A system that keeps track of each withdrawal or
addition to inventory continuously, so records are always current [fixed-quantity model]
Fixed-period (P) system• A system in which inventory orders are made at
regular time intervals.
Lecture 7 – Inventory Planning & Control |
Q and P Systems
Q SystemsQ Systems Based on reorder point –
when inventory is depleted to ROP, order replenishment of quantity EOQ
Individuals review frequencies Possible quantity discounts Lower, less-expensive safety
stocks
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P SystemsP Systems Alternative ROP/Q-System
control is periodic review method Q-System – each stock item rendered at different times – complex, no economies of scale or common prod./transport runs P-system – inventory levels for multiple stock items reviewed at same time – can be reordered together higher carrying costs – not optimum, but more practical
Convenient to administer Orders may be combined IP may require review
Lecture 7 – Inventory Planning & Control |
Fixed-Period (P) System
Inventory is ordered at the end of a given period At that time, on-hand inventory is counted Only the amount necessary to bring total inventory up
to a pre-specified target level is ordered
Assumptions (similar to basic EOQ fixed-quantity system)
• Only relevant cost are ordering & holding costs• Lead times are known & constant• Items are independent of one another
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Lecture 7 – Inventory Planning & Control |
Fixed-Period (P) SystemVarious amounts (Q1, Q2, etc) are ordered at regular
time intervals (P) based on the qty. necessary to bring inventory up to target maxi. (T)
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On-
hand
Inv
ento
ry
Time
Target maximum (T)
Lecture 7 – Inventory Planning & Control |
Fixed-Period (P) System
Hard Rock Little London has a back order for three (3) leather jackets in its retail store. There are no jackets in stock, none are expected from earlier orders, and it is time to place an order. The target value is 50 jackets. How many jackets should be ordered?
SolutionOrder amount (Q) = Target (T) – OnHand inventory – Earlier orders not yet received + Back ordersQ = 50 – 0 – 0 + 3 = 53 jackets
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Lecture 7 – Inventory Planning & Control |
Advantages: Fixed-Period No physical count of inventory items after an item is
withdrawn Count occurs only when the time for the next review
comes up Procedure is convenient to administer, especially if
employee has several duties Appropriate when vendors make routine visits to
customers to take fresh orders Appropriate when purchasers want to combine orders
to save ordering & transportation costs. E.g., a vending machine company may refill its machine every Tuesday
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(Heizer, 2004)
Lecture 7 – Inventory Planning & Control |
Disadvantages: Fixed-Period
No tally of inventory during review period can cause stock-out during period
A large order (pick list) can draw the inventory level down to zero
Hence, higher level of safety stock (as compared to a fixed-quantity system) needs to be maintained to provide protection against stock-out during both the time between reviews and lead time.
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(Heizer, 2004)
Lecture 7 – Inventory Planning & Control |
Summary
Inventory represents a major investment for many firmsThis investment is often larger than it should be because firms find it easier to have “Just-in-case” inventory rather than “Just-in-time” inventory.
A basic inventory management question is whether to order large quantities infrequently or to order small quantities frequently. The EOQ provides guidance for this choice by indicating the lot size that minimizes the sum of holding and ordering costs over some period of time, such as a year.
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Lecture 7 – Inventory Planning & Control |
Normal Distribution
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Lecture 7 – Inventory Planning & Control |
Equations
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Lecture 7 – Inventory Planning & Control |
Exercises – Lot Size
A museum of natural history opened a gift shop two years ago. Managing inventories has become a problem. Low inventory turnover is squeezing profit margins and causing cash-flow problems. One of the top-selling items in the container group at the museum’s gift shop is a bird feeder. Sales are 18 units per week, and the supplier charges $60 per unit. The cost of placing an order with the supplier is $45. Annual holding cost is 25% of a feeder’s value and the museum operates 52 weeks per year. Management chose a 390-unit lot size so that new orders could be placed less frequently. What is the annual cost of the current policy of using a 390-unit lot size?
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Lecture 7 – Inventory Planning & Control |
Solution – Lot Size D = (18 units/wk)(52 wks/yr) = 936 units H = 0.25($60/unit) = $15 Total annual cost for the current policy is:
• C = [Q/2(H) + D/Q(S)]• C = [390/2($15) + 936/390($45)]• C = $2,925 + $108 = $3,033
Total annual cost for alternative lot size is:• C = [468/2($15) + 936/469($45)]• C = $3,510 + $90 = $3,600
The lot size of 468 units, which is a half-year supply, would be a more expensive option than the current policy. The savings in order costs are more than offset by the increase in holding costs. Management should use the total annual cost equation to explore other lot-size alternatives.
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Lecture 7 – Inventory Planning & Control |
Exercise – Basic EOQ Model
The economic order quantity, EOQ, is 75 units when annual demand, D, is 936 units/year, setup cost, S, is $45, and holding cost, H, is $15/unit/year. Suppose that we mistakenly estimate inventory holding cost to be $30/unit/year.
• (a) What is the new order quantity, Q, if D = 936 units/year, S, = $5 and H = $30/unit/year?
• (b) What is the change in order quantity, expressed as a percentage of the economic order quantity (75 units)?
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Lecture 7 – Inventory Planning & Control |
Solution – Basic EOQ Model
(a) The new order quantity isEOQ = √(2DS)/H = √[2(936)($45)]/$30√2,808 = 52.99 or 53 units
(b) The change in percentage is[(53 – 75)/75] x (100) = -29.33%
The new order quantity (53) is about 29 percent smaller than the correct order quantity (75)
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Lecture 7 – Inventory Planning & Control |
Discussion Questions
What is the relationship between inventory and the nine competitive priorities (low-cost operations, top quality, consistent quality, delivery speed, on-time delivery, development speed, customization, variety and volume flexibility)?
Suppose that two competing manufacturers, Company H and Company L, are similar except that Company H has much higher investments in raw materials, work-in-process, and finished goods inventory than Company L. In which of the nine competitive priorities will Company H have an advantage?
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Lecture 7 – Inventory Planning & Control |
Case Discussion
Next Class – Case Study Analysis• Catherine’s Confectionaries (Notes #1)