Inventory Management and Risk
Pooling
Designing & Managing the Supply Chain
Chapter 3
Byung-Hyun Ha
bhha@pusan.ac.kr
Outline
 Introduction to Inventory Management
 Effect of Demand Uncertainty




(s,S) Policy
Supply Contracts
Continuous/Periodic Review Policy
Risk Pooling
 Centralized vs. Decentralized Systems
 Managing Inventory in Supply Chain
 Practical Issues in Inventory Management
 Forecasting
Case: JAM USA, Service Level Crisis
 Background
 Subsidiary of JAM Electronics (Korean manufacturer)
• Established in 1978
• Five Far Eastern manufacturing facilities, each in different countries
• 2,500 different products, a central warehouse in Korea for FGs
 A central warehouse in Chicago with items transported by ship
 Customers: distributors & original equipment manufacturers
(OEMs)
 Problems
 Significant increase in competition
 Huge pressure to improve service levels and reduce costs
 Al Jones, inventory manage, points out:
• Only 70% percent of all orders are delivered on time
• Inventory, primarily that of low-demand products, keeps pile up
Case: JAM USA, Service Level Crisis
 Reasons for the low service level:
 Difficulty forecasting customer demand
 Long lead time in the supply chain
• About 6-7 weeks
 Large number of SKUs handled by JAM USA
 Low priority given the U.S. subsidiary by headquarters in Seoul
Monthly demand for item xxx-1534
Inventory
 Where do we hold inventory?
 Suppliers and manufacturers / warehouses and distribution
centers / retailers
 Types of Inventory
 Raw materials / WIP (work in process) / finished goods
 Reasons of holding inventory
 Unexpected changes in customer demand
• The short life cycle of an increasing number of products.
• The presence of many competing products in the marketplace.
 Uncertainty in the quantity and quality of the supply, supplier
costs and delivery times.
 Delivery lead time and capacity limitations even with no
uncertainty
 Economies of scale (transportation cost)
Single Warehouse Inventory Example
 Contents
 Economy lot size model
 News vendor model
• Effect of demand uncertainty
• Supply contract
 Multiple order opportunities
• Continuous review policy
• Periodic review policy
Single Warehouse Inventory Example
 Key factors affecting inventory policy





Customer demand characteristics
Replenishment lead time
Number of products
Length of planning horizon
Cost structure
• Order cost
• Fixed, variable
• Holding cost
• Taxes, insurance, maintenance, handling, obsolescence, and
opportunity costs
 Service level requirements
Economy Lot Size Model
 Assumptions






Constant demand rate of D items per day
Fixed order quantities at Q items per order
Fixed setup cost K when places an order
Inventory holding cost h per unit per day
Zero lead time
Zero initial inventory & infinite planning horizon
Inventory
Order
Size
Avg. Inven
Time
Economy Lot Size Model
 Inventory level
Q  TD
Q
T
D
Inventory
Order
Size
(Q)
Avg. Inven
Time
Cycle time (T)
 Total inventory cost in a cycle of length T
hTQ
K
2
 Average total cost per unit of time
KD hQ

Q
2
Economy Lot Size Model
 Trade-off between order cost and holding cost
160
140
Total Cost
120
Cost
100
Holding Cost
80
60
Order Cost
40
20
0
0
500
1000
Order Quantity
1500
Economy Lot Size Model
 Optimal order quantity
 Economic order quantity (EOQ)
2KD
Q 
h
*
 Important insights
 Tradeoff between setup costs and holding costs when
determining order quantity. In fact, we order so that these costs
are equal per unit time
 Total cost is not particularly sensitive to the optimal order
quantity
Order Quantity
50%
80%
90%
100%
110%
120%
150%
200%
Cost Increase
25.0%
2.5%
0.5%
-
0.4%
1.6%
8.0%
25.0%
Effect of Demand Uncertainty
 Most companies treat the world as if it were
predictable
 Production and inventory planning are based on forecasts of
demand made far in advance of the selling season
 Companies are aware of demand uncertainty when they create a
forecast, but they design their planning process as if the forecast
truly represents reality
 Recent technological advances have increased the level of
demand uncertainty
• Short product life cycles
• Increasing product variety
Effect of Demand Uncertainty
 Three principles of all forecasting techniques:
 Forecasting is always wrong
 The longer the forecast horizon the worst is the forecast
 Aggregate forecasts are more accurate
 News vendor model
 Incorporating demand uncertainty and forecast demand into
analysis
 Characterizing impact of demand uncertainty on inventory policy
Case: Swimsuit Production
 Fashion items have short life cycles, high variety of
competitors
 Swimsuit production
 New designs are completed
 One production opportunity
 Based on past sales, knowledge of the industry, and economic
conditions, the marketing department has a probabilistic
forecast
 The forecast averages about 13,000, but there is a chance that
demand will be greater or less than this
Case: Swimsuit Production
 Information
Production cost per unit (C) = $80
Selling price per unit (S) = $125
Salvage value per unit (V) = $20
Fixed production cost (F) = $100,000
Production quantity (Q)
Demand Scenarios
Sales
00
0
18
00
0
16
00
0
14
00
0
12
00
0
10
00
30%
25%
20%
15%
10%
5%
0%
80
Probability





Case: Swimsuit Production
 Possible scenarios
 Make 10,000 swimsuits, demand ends up being 12,000
• Profit = 125(10,000) - 80(10,000) - 100,000 = $350,000
 Make 10,000 swimsuits, demand ends up being 8,000
• Profit = 125(8,000) - 80(10,000) - 100,000 + 20(2,000) = $140,000
 …
Swimsuit Production Solution
 Problem
 Find optimal order quantity Q* that maximizes expected profit
 Notation
 di – ith possible demand where di < di+1
• (d1, d2, d3, d4, d5, d6) = (8K, 10K, 12K, 14K, 16K, 18K)
 pi – probability that ith possible demand occurs
• (p1, p2, p3, p4, p5, p6) = (0.11, 0.11, 0.28, 0.22, 0.18, 0.10)
 ES(Q) – expected sales when producing Q
 EI(Q) – expected left over inventory when producing Q
 EP(Q) – expected profit when producing Q
 Expected profit
 EP(Q) = SES(Q) – CQ – F + VEI(Q)
Swimsuit Production Solution
 Expected sales
6
ES(Q)   pi min{di , Q}
 Case 1: Q  d1
i 1
ES(Q)  Q
 Case 2: dk < Q  dk+1, k=1,…,5
k
6
i 1
i  k 1
ES (Q)   pi di  Q  pi
 Case 3: d6  Q
6
ES(Q)   pi di
i 1
 Expected left over inventory (why?)
 EI(Q) = Q – ES(Q)
Swimsuit Production Solution
 Quantity that maximizes average profit
Expected Profit
$400,000
Profit
$300,000
$200,000
$100,000
$0
8000
12000
16000
Order Quantity
20000
Swimsuit Production Solution
 Question
 Average demand is 13,000
 Will optimal quantity be less than, equal to, or greater than
average demand?
 Note that fixed production cost is sunk cost
• We have to pay in all cases
Swimsuit Production Solution
 Look at marginal cost vs. marginal profit
 If extra swimsuit sold, profit is 125-80 = 45
 If not sold, cost is 80-20 = 60
 In case of Scenario Two (make 10,000, demand 8,000)
• Profit = 125(8,000) - 80(10,000) - 100,000 + 20(2,000)
= 125(8,000) - 80(8,000 + 2,000) - 100,000 + 20(2,000)
= 45(8,000) - 60(2,000) - 100,000 = $140,000
 That is,
• EP(Q) = SES(Q) – CQ – F + VEI(Q)
= SES(Q) – C{ES(Q) + EI(Q)} – F + VEI(Q)
= (S – C)ES(Q) – (C – V)EI(Q) – F
 So we will make less than average
Swimsuit Production Solution
 Tradeoff between ordering enough to meet demand
and ordering too much
 Several quantities have the same average profit
 Average profit does not tell the whole story
 Question: 9000 and 16000 units lead to about the
same average profit, so which do we prefer?
Expected Profit
$400,000
Profit
$300,000
$200,000
$100,000
$0
8000
12000
16000
Order Quantity
20000
Swimsuit Production Solution
 Risk and reward
Probability
100%
80%
60%
Q=9000
40%
Q=16000
20%
-3
00
00
0
-1
00
00
0
10
00
00
30
00
00
50
00
00
0%
Revenue
Consult Ch13 of Winston, “Decision making under uncertainty”
Case: Swimsuit Production
 Key insights
 The optimal order quantity is not necessarily equal to average
forecast demand
 The optimal quantity depends on the relationship between
marginal profit and marginal cost
 As order quantity increases, average profit first increases and
then decreases
 As production quantity increases, risk increases (the probability
of large gains and of large losses increases)
Case: Swimsuit Production
 Initial inventory
 Suppose that one of the swimsuit designs is a model produced
last year
 Some inventory is left from last year
 Assume the same demand pattern as before
 If only old inventory is sold, no setup cost
 Question: If there are 5,000 units remaining, what
should Swimsuit production do?
Case: Swimsuit Production
 Analysis for initial inventory and profit
 Solid line: average profit excluding fixed cost
 Dotted line: same as expected profit including fixed cost
 Nothing produced
 225,000 (from the figure) + 80(5,000) = 625,000
 Producing
 371,000 (from the figure) + 80(5,000) = 771,000
 If initial inventory was 10,000?
500000
300000
200000
100000
Production Quantity
16000
15000
14000
13000
12000
11000
10000
9000
8000
7000
6000
0
5000
Profit
400000
Case: Swimsuit Production
 Initial inventory and profit
500000
300000
200000
100000
Production Quantity
16000
15000
14000
13000
12000
11000
10000
9000
8000
7000
6000
0
5000
Profit
400000
Case: Swimsuit Production
 (s, S) policies
 For some starting inventory levels, it is better to not start
production
 If we start, we always produce to the same level
 Thus, we use an (s, S) policy
• If the inventory level is below s, we produce up to S
 s is the reorder point, and S is the order-up-to level
 The difference between the two levels is driven by the fixed costs
associated with ordering, transportation, or manufacturing
Supply Contracts
 Assumptions for Swimsuit production
 In-house manufacturing
 Usually, manufactures and retailers
 Supply contracts





Pricing and volume discounts
Minimum and maximum purchase quantities
Delivery lead times
Product or material quality
Product return policies
Supply Contracts
 Condition
Fixed Production Cost =$100,000
Variable Production Cost=$35
Wholesale Price =$80
Selling Price=$125
Salvage Value=$20
Manufacturer
Manufacturer DC
Retail DC
Stores
Demand Scenario and Retailer Profit
30%
25%
20%
15%
10%
5%
0%
80
00
10
00
0
12
00
0
14
00
0
16
00
0
18
00
0
Probability
Demand Scenarios
Sales
Expected Profit
500000
400000
300000
200000
100000
0
6000
8000
10000
12000
14000
Order Quantity
16000
18000
20000
Demand Scenario and Retailer Profit
 Sequential supply chain




Retailer optimal order quantity is 12,000 units
Retailer expected profit is $470,700
Manufacturer profit is $440,000
Supply Chain Profit is $910,700
 Is there anything that the distributor and
manufacturer can do to increase the profit of both?
 Global optimization?
Buy-Back Contracts
 Buy back=$55
$513,800
500,000
400,000
300,000
200,000
100,000
0
retailer
600,000
500,000
$471,900
400,000
300,000
200,000
100,000
0
60
00
70
00
80
00
90
00
10
00
0
11
00
0
12
00
0
13
00
0
14
00
0
15
00
0
16
00
0
17
00
0
18
00
0
Order Quantity
Manufacturer Profit
60
00
70
00
80
00
90
00
10
00
0
11
00
0
12
00
0
13
00
0
14
00
0
15
00
0
16
00
0
17
00
0
18
00
0
Retailer Profit
600,000
Production Quantity
manufacturer
Buy-Back Contracts
 Sequential supply chain




Retailer optimal order quantity is 12,000 units
Retailer expected profit is $470,700
Manufacturer profit is $440,000
Supply chain profit is $910,700
 With buy-back contracts




Retailer optimal order quantity is 14,000 units
Retailer expected profit is $513,800
Manufacturer expected profit is $471,900
Supply chain profit is $985,700
 Manufacture sharing some of risk!
Revenue-Sharing Contracts
 Wholesale price from $80 to $60, RS 15%
$504,325
500,000
400,000
Supply chain profit is
$985,700
300,000
200,000
100,000
0
60
00
70
00
80
00
90
00
10
00
0
11
00
0
12
00
0
13
00
0
14
00
0
15
00
0
16
00
0
Manufacturer Profit
17
00
0
18
00
0
700,000
Order Quantity
retailer
600,000
500,000
$481,375
400,000
300,000
200,000
100,000
0
60
00
70
00
80
00
90
00
10
00
0
11
00
0
12
00
0
13
00
0
14
00
0
15
00
0
16
00
0
17
00
0
18
00
0
Retailer Profit
600,000
Production Quantity
manufacturer
Other Types of Supply Contracts
 Quantity-flexibility contracts
 Supplier providing full refund for returned (unsold) items up to a
certain quantity
 Sales rebate contracts
 Direct incentive to retailer by supplier for any item sold above a
certain quantity
 …
 Consult Cachon 2002
Global Optimization
 What is the most profit both the supplier and the
buyer can hope to achieve?
 Assume an unbiased decision maker
 Transfer of money between the parties is ignored
 Allowing the parties to share the risk!
 Marginal profit=$90, marginal loss=$15
 Optimal production quantity=16,000
1,000,000
$1,014,500
800,000
600,000
400,000
200,000
0
60
00
70
00
80
00
90
00
10
00
0
11
00
0
12
00
0
13
00
0
14
00
0
15
00
0
16
00
0
17
00
0
18
00
0
 Decision-making power
 Allocating profit
Supply Chain Profit
 Drawbacks
1,200,000
Production Quantity
Global Optimization
 Revised buy-back contracts
 Wholesale price=$75, buy-back price=$65
 Global optimum
 Equilibrium point!
 No partner can improve his profit by deciding to deviate from the
optimal decision
 Consult Ch14 of Winston, “Game theory”
 Key Insights
 Effective supply contracts allow supply chain partners to replace
sequential optimization by global optimization
 Buy Back and Revenue Sharing contracts achieve this objective
through risk sharing
Supply Contracts: Case Study
 Example: Demand for a movie newly released video
cassette typically starts high and decreases rapidly
 Peak demand last about 10 weeks
 Blockbuster purchases a copy from a studio for $65
and rent for $3
 Hence, retailer must rent the tape at least 22 times before
earning profit
 Retailers cannot justify purchasing enough to cover
the peak demand
 In 1998, 20% of surveyed customers reported that they could not
rent the movie they wanted
Supply Contracts: Case Study
 Starting in 1998 Blockbuster entered a revenuesharing agreement with the major studios
 Studio charges $8 per copy
 Blockbuster pays 30-45% of its rental income
 Even if Blockbuster keeps only half of the rental
income, the breakeven point is 6 rental per copy
 The impact of revenue sharing on Blockbuster was
dramatic
 Rentals increased by 75% in test markets
 Market share increased from 25% to 31% (The 2nd largest
retailer, Hollywood Entertainment Corp has 5% market share)
Multiple Order Opportunities
 Situation
 Often, there are multiple reorder opportunities
 A central distribution facility which orders from a manufacturer
and delivers to retailers
• The distributor periodically places orders to replenish its inventory
 Reasons why DC holds inventory
 Satisfy demand during lead time
 Protect against demand uncertainty
 Balance fixed costs and holding costs
Continuous Review Inventory Model
 Assumptions





Normally distributed random demand
Fixed order cost plus a cost proportional to amount ordered
Inventory cost is charged per item per unit time
If an order arrives and there is no inventory, the order is lost
The distributor has a required service level
• expressed as the likelihood that the distributor will not stock out
during lead time.
 (s, S) Policy
 Whenever the inventory position drops below a certain level (s)
we order to raise the inventory position to level S
Reminder: The Normal Distribution
Standard Deviation = 5
Standard Deviation = 10
Average = 30
0
10
20
30
40
50
60
A View of (s, S) Policy
S
Inventory Level
Inventory Position
Lead
Time
s
0
Time
(s, S) Policy
 Notations






AVG = average daily demand
STD = standard deviation of daily demand
LT = replenishment lead time in days
h = holding cost of one unit for one day
K = fixed cost
SL = service level (for example, 95%)
• The probability of stocking out is 100% - SL (for example, 5%)
 Policy
 s = reorder point, S = order-up-to level
 Inventory Position
 Actual inventory + (items already ordered, but not yet delivered)
(s, S) Policy - Analysis
 The reorder point (s) has two components:
 To account for average demand during lead time:
LTAVG
 To account for deviations from average (we call this safety
stock)
zSTDLT
where z is chosen from statistical tables to ensure that the
probability of stock-outs during lead-time is 100% - SL.
 Since there is a fixed cost, we order more than up to
the reorder point:
Q=(2KAVG)/h
 The total order-up-to level is:
S=Q+s
(s, S) Policy - Example
 The distributor has historically observed weekly demand of:
AVG = 44.6
STD = 32.1
 Replenishment lead time is 2 weeks, and desired service level
SL = 97%
 Average demand during lead time is: 44.6  2 = 89.2
 Safety Stock is: 1.88  32.1  2 = 85.3
 Reorder point is thus 175, or about 3.9 weeks of supply at
warehouse and in the pipeline
 Weekly inventory holding cost: .87
 Therefore, Q=679
 Order-up-to level thus equals:
 Reorder Point + Q = 176+679 = 855
Periodic Review
 Periodic review model
 Suppose the distributor places orders every month
 What policy should the distributor use?
 What about the fixed cost?
 Base-Stock Policy
r
r
Inventory Level
Base-stock
Level
L
L
L
Inventory
Position
0
Time
Periodic Review
 Base-Stock Policy
 Each review echelon, inventory position is raised to the basestock level.
 The base-stock level includes two components:
• Average demand during r+L days (the time until the next order
arrives):
(r+L)*AVG
• Safety stock during that time:
z*STD* r+L
Risk Pooling
 Consider these two systems:
 For the same service level, which system will require more
inventory? Why?
 For the same total inventory level, which system will have better
service? Why?
 What are the factors that affect these answers?
Warehouse One
Market One
Warehouse Two
Market Two
Supplier
Market One
Supplier
Warehouse
Market Two
Risk Pooling Example
 Compare the two systems:





two products
maintain 97% service level
$60 order cost
$.27 weekly holding cost
$1.05 transportation cost per unit in decentralized system, $1.10
in centralized system
 1 week lead time
Week
1
2
3
4
5
6
7
8
Prod A,
Market 1
Prod A,
Market 2
Prod B,
Market 1
Product B,
Market 2
33
45
37
38
55
30
18
58
46
35
41
40
26
48
18
55
0
2
3
0
0
1
3
0
2
4
0
0
3
1
0
0
Risk Pooling Example
 Risk Pooling Performance
Warehouse Product AVG
STD CV
s
S
Market 1
A
39.3
13.2
.34
65
197
Avg.
%
Inven. Dec.
91
Market 2
A
38.6
12.0
.31
62
193
88
Market 1
Market 2
B
B
1.125 1.36
1.25 1.58
1.21 4
1.26 5
29
29
14
15
Cent.
Cent
A
B
77.9 20.7
2.375 1.9
.27
.81
118 304
6
39
132
20
36%
43%
Risk Pooling: Important Observations
 Centralizing inventory control reduces both safety
stock and average inventory level for the same
service level.
 This works best for
 High coefficient of variation, which increases required safety
stock.
 Negatively correlated demand. Why?
 What other kinds of risk pooling will we see?
Centralized vs. Decentralized Systems
 Trade-offs that need to be considered in comparing
centralized and decentralized systems





Safety stock
Service level
Overhead costs
Customer lead time
Transportation costs
Inventory in Supply Chain
 Centralized distribution systems
 How much inventory should
management keep at each location?
 A good strategy:
 The retailer raises inventory to level Sr
each period
 The supplier raises the sum of
inventory in the retailer and supplier
warehouses and in transit to Ss
 If there is not enough inventory in the
warehouse to meet all demands from
retailers, it is allocated so that the
service level at each of the retailers will
be equal.
Supplier
Warehouse
Retailers
Practical Issues
 Top seven effective strategies







Periodic inventory reviews
Tight management of usage rates, lead times, and safety stock
Reduced safety stock levels
Introduce or enhance cycle counting practice
ABC approach
Shift more inventory, or inventory ownership, to suppliers
Quantitative approaches
Forecasting
 Recall the three rules
 Nevertheless, forecast is critical
 General overview:




Judgment methods
Market research methods
Time series methods
Causal methods
Forecasting
 Judgment methods




Assemble the opinion of experts
Sales-force composite combines salespeople’s estimates
Panels of experts – internal, external, both
Delphi method
• Each member surveyed
• Opinions are compiled
• Each member is given the opportunity to change his opinion
Forecasting
 Market research methods
 Particularly valuable for developing forecasts of newly introduced
products
 Market testing
• Focus groups assembled
• Responses tested
• Extrapolations to rest of market made
 Market surveys
• Data gathered from potential customers
• Interviews, phone-surveys, written surveys, etc.
Forecasting
 Time series methods
 Past data is used to estimate future data
 Examples include
• Moving averages – average of some previous demand points.
• Exponential Smoothing – more recent points receive more weight
• Methods for data with trends:
• Regression analysis – fits line to data
• Holt’s method – combines exponential smoothing concepts with
the ability to follow a trend
• Methods for data with seasonality
• Seasonal decomposition methods (seasonal patterns removed)
• Winter’s method: advanced approach based on exponential
smoothing
• Complex methods (not clear that these work better)
Forecasting
 Causal methods
 Forecasts are generated based on data other than the data
being predicted
 Examples include:
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Inflation rates
GNP
Unemployment rates
Weather
Sales of other products
Forecasting
 Selecting appropriate forecasting technique
 What is purpose of forecast? How is it to be used?
 What are dynamics of system for which forecast will be made?
 How important is past in estimating future?
Summary
 Matching supply and demand
 Inventory management
 Reducing cost and providing required service level
 Taking into account holding and setup cost, lead time, forecast
demand, and information about demand variability
 Demand aggregation
 Globally optimal inventory policies
 Global optimization