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Chapter 2:InventoryManagementand Risk Pooling
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CASE: Steel Works
Background of case and intentOverview of businessWhat does data tell you about Specialty?How much inventory might you expect?What opportunities are there for Custom?Wrap up
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Background & Intent
Abstraction from summer consulting job Intent is to examine a realistic, but
simplified inventory context and perform a diagnosis of problem – poor service and too much inventory
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Custom Products
Rapid growth, 1/3 of total sales ($133 MM)One customer per productVery high marginsHigh service level3 plants, co-located with R&D centerEach product produced at a single plant
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Specialty Products
Rapid growth, 2/3 of total sales ($267 MM) 6 product families 3 plants, each producing 2 product families 130 customers, 120 products Few big customers Highly volatile demand High service level
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Consultant Recommendation
Drop low volume products Improve forecastsConsolidate warehouses
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What Does Data Tell You?
cv
DB R10 15.5 13.2 0.85
DB R12 1008 256 0.25
DB R15 2464 494 0.20
DF R10 97 92.5 0.95
DF R12 18.5 11.4 0.62
DF R15 55 80 1.46
DF R23 35.5 45.9 1.29
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What Does Data Tell You?
Durabend R12:One customer accounts for 97% of demand
7 products:High volume (2) is not very volatileLow volume (5) is very volatile
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How Much Inventory Should You Expect?
Assume base stock model with periodic review
Review period = r = ?Lead time = L = ?
2
E I r z r L
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Cycle stock
Saf. stock
E[I] Act. Inv.
DB R10 15.5 13.2 8 26 34 72
DB R12 1008 256 504 510 1014 740
DB R15 2464 494 1232 990 2222 1875
DF R10 97 92.5 49 185 234 604
DF R12 18.5 11.4 9 23 32 55
DF R15 55 80 28 160 188 388
DF R23 35.5 45.9 18 92 110 190
1848 1986 3834 3824
Assumes r = 1; L=0.25; and z = 1.8
Cycle stock = r /2 Safety stock = z r+L
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What Are the Opportunities at Custom?
Combine production and inventory for common items, e. g. DF R23
Produce monthly: reduce setups by half and pool safety stocks
Produce twice a month: same number of setups but cut cycle stock and review period in half
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Wrap Up
Realistic diagnostic exercise In real life: not as clean, more data and
more considerationsYet simple models and principles can
provide valuable guidance
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2.1 IntroductionWhy Is Inventory Important?
Distribution and inventory (logistics) costs are quite substantial
Total U.S. Manufacturing Inventories ($m): 1992-01-31: $m 808,773 1996-08-31: $m 1,000,774 2006-05-31: $m 1,324,108
Inventory-Sales Ratio (U.S. Manufacturers): 1992-01-01: 1.56 2006-05-01: 1.25
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GM’s production and distribution network 20,000 supplier plants 133 parts plants 31 assembly plants 11,000 dealers
Freight transportation costs: $4.1 billion (60% for material shipments)
GM inventory valued at $7.4 billion (70%WIP; Rest Finished Vehicles)
Decision tool to reduce: combined corporate cost of inventory and transportation.
26% annual cost reduction by adjusting: Shipment sizes (inventory policy) Routes (transportation strategy)
Why Is Inventory Important?
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Why Is Inventory Required?
Uncertainty in customer demandShorter product lifecyclesMore competing products
Uncertainty in suppliesQuality/Quantity/Costs/Delivery Times
Delivery lead times Incentives for larger shipments
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Holding the right amount at the right time is difficult!
Dell Computer’s was sharply off in its forecast of demand, resulting in inventory write-downs 1993 stock plunge
Liz Claiborne’s higher-than-anticipated excess inventories 1993 unexpected earnings decline,
IBM’s ineffective inventory management 1994 shortages in the ThinkPad line
Cisco’s declining sales 2001 $ 2.25B excess inventory charge
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Inventory Management-Demand Forecasts
Uncertain demand makes demand forecast critical for inventory related decisions:What to order?When to order?How much is the optimal order quantity?
Approach includes a set of techniquesINVENTORY POLICY!!
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Supply Chain Factors in Inventory Policy
Estimation of customer demand Replenishment lead time The number of different products being considered The length of the planning horizon Costs
Order cost: Product cost Transportation cost
Inventory holding cost, or inventory carrying cost: State taxes, property taxes, and insurance on inventories Maintenance costs Obsolescence cost Opportunity costs
Service level requirements
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2.2 Single Stage Inventory Control
Single supply chain stage Variety of techniques
Economic Lot Size Model Demand Uncertainty Single Period Models Initial Inventory Multiple Order Opportunities Continuous Review Policy Variable Lead Times Periodic Review Policy Service Level Optimization
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2.2.1. Economic Lot Size Model
FIGURE 2-3: Inventory level as a function of time
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Assumptions
D items per day: Constant demand rate Q items per order: Order quantities are fixed, i.e., each
time the warehouse places an order, it is for Q items. K, fixed setup cost, incurred every time the warehouse
places an order. h, inventory carrying cost accrued per unit held in
inventory per day that the unit is held (also known as, holding cost)
Lead time = 0 (the time that elapses between the placement of an order and its receipt)
Initial inventory = 0 Planning horizon is long (infinite).
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Deriving EOQ
Total cost at every cycle:
Average inventory holding cost in a cycle: Q/2
Cycle time T =Q/D Average total cost per unit time:
2
hTQK
2
hQ
Q
KD
h
KDQ
2*
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EOQ: Costs
FIGURE 2-4: Economic lot size model: total cost per unit time
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Sensitivity Analysis
b .5 .8 .9 1 1.1 1.2 1.5 2
Increase in cost
25% 2.5% 0.5% 0 .4% 1.6% 8.9% 25%
Total inventory cost relatively insensitive to order quantities
Actual order quantity: Q Q is a multiple b of the optimal order quantity Q*. For a given b, the quantity ordered is Q = bQ*
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2.2.2. Demand Uncertainty
The forecast is always wrong It is difficult to match supply and demand
The longer the forecast horizon, the worse the forecast It is even more difficult if one needs to predict
customer demand for a long period of time Aggregate forecasts are more accurate.
More difficult to predict customer demand for individual SKUs
Much easier to predict demand across all SKUs within one product family
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2.2.3. Single Period Models
Short lifecycle productsOne ordering opportunity onlyOrder quantity to be decided before
demand occurs
Order Quantity > Demand => Dispose excess inventory
Order Quantity < Demand => Lose sales/profits
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Single Period Models Using historical data
identify a variety of demand scenarios determine probability each of these scenarios will occur
Given a specific inventory policy determine the profit associated with a particular scenario given a specific order quantity
weight each scenario’s profit by the likelihood that it will occur determine the average, or expected, profit for a particular ordering
quantity.
Order the quantity that maximizes the average profit.
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Single Period Model Example
FIGURE 2-5: Probabilistic forecast
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Additional Information
Fixed production cost: $100,000 Variable production cost per unit: $80.During the summer season, selling price:
$125 per unit.Salvage value: Any swimsuit not sold
during the summer season is sold to a discount store for $20.
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Two Scenarios
Manufacturer produces 10,000 units while demand ends at 12,000 swimsuits Profit = 125(10,000) - 80(10,000) - 100,000 = $350,000
Manufacturer produces 10,000 units while demand ends at 8,000 swimsuits Profit= 125(8,000) + 20(2,000) - 80(10,000) - 100,000= $140,000
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Probability of Profitability Scenarios with Production = 10,000 Units
Probability of demand being 8000 units = 11%Probability of profit of $140,000 = 11%
Probability of demand being 12000 units = 27%Probability of profit of $140,000 = 27%
Total profit = Weighted average of profit scenarios
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Order Quantity that Maximizes Expected Profit
FIGURE 2-6: Average profit as a function of production quantity
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Relationship Between Optimal Quantity and Average Demand
Compare marginal profit of selling an additional unit and marginal cost of not selling an additional unit
Marginal profit/unit = Selling Price - Variable Ordering (or, Production) Cost
Marginal cost/unit =Variable Ordering (or, Production) Cost - Salvage Value
If Marginal Profit > Marginal Cost => Optimal Quantity > Average Demand
If Marginal Profit < Marginal Cost => Optimal Quantity < Average Demand
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For the Swimsuit Example
Average demand = 13,000 units. Optimal production quantity = 12,000 units.
Marginal profit = $45 Marginal cost = $60.
Thus, Marginal Cost > Marginal Profit
=> optimal production quantity < average demand.
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Risk-Reward Tradeoffs
Optimal production quantity maximizes average profit is about 12,000
Producing 9,000 units or producing 16,000 units will lead to about the same average profit of $294,000.
If we had to choose between producing 9,000 units and 16,000 units, which one should we choose?
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Risk-Reward Tradeoffs
FIGURE 2-7: A frequency histogram of profit
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Risk-Reward Tradeoffs Production Quantity = 9000 units
Profit is: either $200,000 with probability of about 11 % or $305,000 with probability of about 89 %
Production quantity = 16,000 units. Distribution of profit is not symmetrical. Losses of $220,000 about 11% of the time Profits of at least $410,000 about 50% of the time
With the same average profit, increasing the production quantity: Increases the possible risk Increases the possible reward
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ObservationsThe optimal order quantity is not necessarily
equal to forecast, or average, demand. As the order quantity increases, average
profit typically increases until the production quantity reaches a certain value, after which the average profit starts decreasing.
Risk/Reward trade-off: As we increase the production quantity, both risk and reward increases.
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2.2.4. What If the Manufacturer Has an Initial Inventory?
Trade-off between:Using on-hand inventory to meet demand and
avoid paying fixed production cost: need sufficient inventory stock
Paying the fixed cost of production and not have as much inventory
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Initial Inventory Solution
FIGURE 2-8: Profit and the impact of initial inventory
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Manufacturer Initial Inventory = 5,000
If nothing is produced, average profit =
225,000 (from the figure) + 5,000 x 80 = 625,000 If the manufacturer decides to produce
Production should increase inventory from 5,000 units to 12,000 units.
Average profit =
371,000 (from the figure) + 5,000 • 80 = 771,000
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No need to produce anything average profit > profit achieved if we produce to
increase inventory to 12,000 units If we produce, the most we can make on
average is a profit of $375,000. Same average profit with initial inventory of 8,500
units and not producing anything. If initial inventory < 8,500 units => produce to raise
the inventory level to 12,000 units. If initial inventory is at least 8,500 units, we should not
produce anything (s, S) policy or (min, max) policy
Manufacturer Initial Inventory = 10,000
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2.2.5. Multiple Order Opportunities
REASONS To balance annual inventory holding costs and annual fixed order
costs. To satisfy demand occurring during lead time. To protect against uncertainty in demand.
TWO POLICIES Continuous review policy
inventory is reviewed continuously an order is placed when the inventory reaches a particular level or reorder point. inventory can be continuously reviewed (computerized inventory systems are
used)
Periodic review policy inventory is reviewed at regular intervals appropriate quantity is ordered after each review. it is impossible or inconvenient to frequently review inventory and place orders if
necessary.
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2.2.6. Continuous Review Policy Daily demand is random and follows a normal distribution. Every time the distributor places an order from the
manufacturer, the distributor pays a fixed cost, K, plus an amount proportional to the quantity ordered.
Inventory holding cost is charged per item per unit time. Inventory level is continuously reviewed, and if an order is
placed, the order arrives after the appropriate lead time. If a customer order arrives when there is no inventory on
hand to fill the order (i.e., when the distributor is stocked out), the order is lost.
The distributor specifies a required service level.
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AVG = Average daily demand faced by the distributor
STD = Standard deviation of daily demand faced by the distributor
L = Replenishment lead time from the supplier to the
distributor in days h = Cost of holding one unit of the product for
one day at the distributor α = service level. This implies that the probability
of stocking out is 1 - α
Continuous Review Policy
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(Q,R) policy – whenever inventory level falls to a reorder level R, place an order for Q units
What is the value of R?
Continuous Review Policy
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Continuous Review Policy
Average demand during lead time: L x AVG
Safety stock:
Reorder Level, R:
Order Quantity, Q:
LSTDz
LSTDzAVGL
h
AVGKQ
2
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Service Level & Safety Factor, z
Service Level
90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 99.9%
z 1.29 1.34 1.41 1.48 1.56 1.65 1.75 1.88 2.05 2.33 3.08
z is chosen from statistical tables to ensure that the probability of stockouts during lead time is exactly 1 - α
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Inventory Level Over Time
LSTDz Inventory level before receiving an order =
Inventory level after receiving an order =
Average Inventory =
LSTDzQ
LSTDzQ 2
FIGURE 2-9: Inventory level as a function of time in a (Q,R) policy
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Continuous Review Policy Example
A distributor of TV sets that orders from a manufacturer and sells to retailers
Fixed ordering cost = $4,500Cost of a TV set to the distributor = $250Annual inventory holding cost = 18% of
product costReplenishment lead time = 2 weeksExpected service level = 97%
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Month Sept Oct Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug
Sales 200 152 100 221 287 176 151 198 246 309 98 156
Continuous Review Policy Example
Average monthly demand = 191.17 Standard deviation of monthly demand = 66.53
Average weekly demand = Average Monthly Demand/4.3Standard deviation of weekly demand = Monthly standard deviation/√4.3
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Parameter Average weekly demand
Standard deviation of weekly demand
Average demand during lead time
Safety stock
Reorder point
Value 44.58 32.08 89.16 86.20 176
87.052
25018.0
Weekly holding cost =
Optimal order quantity = 67987.
58.44500,42
Q
Average inventory level = 679/2 + 86.20 = 426
Continuous Review Policy Example
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Average lead time, AVGL Standard deviation, STDL. Reorder Level, R:
222 STDLAVGSTDAVGLzAVGLAVGR
2.2.7. Variable Lead Times
222 STDLAVGSTDAVGLz Amount of safety stock=
h
AVGKQ
2Order Quantity =
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Inventory level is reviewed periodically at regular intervals
An appropriate quantity is ordered after each review Two Cases:
Short Intervals (e.g. Daily) Define two inventory levels s and S During each inventory review, if the inventory position falls below s,
order enough to raise the inventory position to S. (s, S) policy
Longer Intervals (e.g. Weekly or Monthly) May make sense to always order after an inventory level review. Determine a target inventory level, the base-stock level During each review period, the inventory position is reviewed Order enough to raise the inventory position to the base-stock level. Base-stock level policy
2.2.8. Periodic Review Policy
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(s,S) policy
Calculate the Q and R values as if this were a continuous review model
Set s equal to RSet S equal to R+Q.
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Base-Stock Level Policy Determine a target inventory level, the base-
stock level Each review period, review the inventory
position is reviewed and order enough to raise the inventory position to the base-stock level
Assume:r = length of the review periodL = lead time AVG = average daily demand STD = standard deviation of this daily demand.
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Average demand during an interval of r + L days=
Safety Stock= LrSTDz
AVGLr )(
Base-Stock Level Policy
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Base-Stock Level Policy
FIGURE 2-10: Inventory level as a function of time in a periodic review policy
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Assume: distributor places an order for TVs every 3 weeks Lead time is 2 weeks Base-stock level needs to cover 5 weeks
Average demand = 44.58 x 5 = 222.9 Safety stock = Base-stock level = 223 + 136 = 359 Average inventory level =
Distributor keeps 5 (= 203.17/44.58) weeks of supply.
Base-Stock Level Policy Example
58.329.1
17.203508.329.1258.443
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Optimal inventory policy assumes a specific service level target.
What is the appropriate level of service? May be determined by the downstream
customerRetailer may require the supplier, to maintain a
specific service levelSupplier will use that target to manage its own
inventoryFacility may have the flexibility to choose the
appropriate level of service
2.2.9. Service Level Optimization
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Service Level Optimization
FIGURE 2-11: Service level inventory versus inventory level as a function of lead time
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Trade-Offs
Everything else being equal:the higher the service level, the higher the
inventory level. for the same inventory level, the longer the
lead time to the facility, the lower the level of service provided by the facility.
the lower the inventory level, the higher the impact of a unit of inventory on service level and hence on expected profit
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Retail Strategy
Given a target service level across all products determine service level for each SKU so as to maximize expected profit.
Everything else being equal, service level will be higher for products with:high profit marginhigh volumelow variabilityshort lead time
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Profit Optimization and Service Level
FIGURE 2-12: Service level optimization by SKU
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Target inventory level = 95% across all products.
Service level > 99% for many products with high profit margin, high volume and low variability.
Service level < 95% for products with low profit margin, low volume and high variability.
Profit Optimization and Service Level
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2.3 Risk Pooling
Demand variability is reduced if one aggregates demand across locations.
More likely that high demand from one customer will be offset by low demand from another.
Reduction in variability allows a decrease in safety stock and therefore reduces average inventory.
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Demand Variation
Standard deviation measures how much demand tends to vary around the averageGives an absolute measure of the variability
Coefficient of variation is the ratio of standard deviation to average demandGives a relative measure of the variability,
relative to the average demand
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Acme Risk Pooling Case Electronic equipment manufacturer and distributor 2 warehouses for distribution in New York and New
Jersey (partitioning the northeast market into two regions)
Customers (that is, retailers) receiving items from warehouses (each retailer is assigned a warehouse)
Warehouses receive material from Chicago Current rule: 97 % service level Each warehouse operate to satisfy 97 % of demand
(3 % probability of stock-out)
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Replace the 2 warehouses with a single warehouse (located some suitable place) and try to implement the same service level 97 %
Delivery lead times may increase But may decrease total inventory investment
considerably.
New Idea
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Historical Data
PRODUCT A
Week 1 2 3 4 5 6 7 8
Massachusetts 33 45 37 38 55 30 18 58
New Jersey 46 35 41 40 26 48 18 55
Total 79 80 78 78 81 78 36 113
PRODUCT B
Week 1 2 3 4 5 6 7 8
Massachusetts 0 3 3 0 0 1 3 0
New Jersey 2 4 3 0 3 1 0 0
Total 2 6 3 0 3 2 3 0
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Summary of Historical DataStatistics Product Average Demand Standard
Deviation of Demand
Coefficient of Variation
Massachusetts A 39.3 13.2 0.34
Massachusetts B 1.125 1.36 1.21
New Jersey A 38.6 12.0 0.31
New Jersey B 1.25 1.58 1.26
Total A 77.9 20.71 0.27
Total B 2.375 1.9 0.81
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Inventory LevelsProduct Average
Demand During Lead Time
Safety Stock Reorder Point
Q
Massachusetts A 39.3 25.08 65 132
Massachusetts B 1.125 2.58 4 25
New Jersey A 38.6 22.8 62 31
New Jersey B 1.25 3 5 24
Total A 77.9 39.35 118 186
Total B 2.375 3.61 6 33
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Savings in Inventory
Average inventory for Product A: At NJ warehouse is about 88 units At MA warehouse is about 91 units In the centralized warehouse is about 132 units Average inventory reduced by about 36 percent
Average inventory for Product B: At NJ warehouse is about 15 units At MA warehouse is about 14 units In the centralized warehouse is about 20 units Average inventory reduced by about 43 percent
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The higher the coefficient of variation, the greater the benefit from risk pooling The higher the variability, the higher the safety stocks
kept by the warehouses. The variability of the demand aggregated by the single warehouse is lower
The benefits from risk pooling depend on the behavior of the demand from one market relative to demand from another risk pooling benefits are higher in situations where
demands observed at warehouses are negatively correlated
Reallocation of items from one market to another easily accomplished in centralized systems. Not possible to do in decentralized systems where they serve different markets
Critical Points
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2.4 Centralized vs. Decentralized Systems
Safety stock: lower with centralization Service level: higher service level for the same
inventory investment with centralization Overhead costs: higher in decentralized system Customer lead time: response times lower in the
decentralized system Transportation costs: not clear. Consider
outbound and inbound costs.
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Inventory decisions are given by a single decision maker whose objective is to minimize the system-wide cost
The decision maker has access to inventory information at each of the retailers and at the warehouse
Echelons and echelon inventoryEchelon inventory at any stage or level of the system
equals the inventory on hand at the echelon, plus all downstream inventory (downstream means closer to the customer)
2.5 Managing Inventory in the Supply Chain
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Echelon Inventory
FIGURE 2-13: A serial supply chain
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Reorder Point with Echelon Inventory
Le = echelon lead time, lead time between the retailer and the
distributor plus the lead time between the distributor and its supplier, the wholesaler.
AVG = average demand at the retailer STD = standard deviation of demand at
the retailerReorder point ee LSTDzAVGLR
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4-Stage Supply Chain Example
Average weekly demand faced by the retailer is 45
Standard deviation of demand is 32 At each stage, management is attempting
to maintain a service level of 97% (z=1.88) Lead time between each of the stages,
and between the manufacturer and its suppliers is 1 week
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Costs and Order Quantities
K D H Q
retailer 250 45 1.2 137
distributor 200 45 .9 141
wholesaler 205 45 .8 152
manufacturer 500 45 .7 255
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Reorder Points at Each Stage
For the retailer, R=1*45+1.88*32*√1 = 105For the distributor, R=2*45+1.88*32*√2 =
175For the wholesaler, R=3*45+1.88*32*√3 =
239For the manufacturer, R=4*45+1.88*32*√4
= 300
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More than One Facility at Each Stage
Follow the same approach Echelon inventory at the warehouse is the
inventory at the warehouse, plus all of the inventory in transit to and in stock at each of the retailers.
Similarly, the echelon inventory position at the warehouse is the echelon inventory at the warehouse, plus those items ordered by the warehouse that have not yet arrived minus all items that are backordered.
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Warehouse Echelon Inventory
FIGURE 2-14: The warehouse echelon inventory
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2.6 Practical Issues Periodic inventory review. Tight management of usage rates, lead times, and
safety stock. Reduce safety stock levels. Introduce or enhance cycle counting practice. ABC approach. Shift more inventory or inventory ownership to
suppliers. Quantitative approaches. FOCUS: not reducing costs but reducing inventory levels. Significant effort in industry to increase inventory turnover
LevelInventoryAverage
SalesAnnualRatioTurnoverInventory
__
___
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Inventory Turnover Ratios for Different Manufacturers
Industry Upper quartile Median Lower quartile
Electronic components and accessories
8.1 4.9 3.3
Electronic computers 22.7 7.0 2.7
Household audio and video equipment
6.3 3.9 2.5
Paper Mills 11.7 8.0 5.5
Industrial chemicals 14.1 6.4 4.2
Bakery products 39.7 23.0 12.6
Books: Publishing and printing
7.2 2.8 1.5
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2.7 Forecasting
RULES OF FORECASTING The forecast is always wrong. The longer the forecast horizon, the
worse the forecast. Aggregate forecasts are more accurate.
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Utility of Forecasting
Part of the available tools for a managerDespite difficulties with forecasts, it can be
used for a variety of decisionsNumber of techniques allow prudent use
of forecasts as needed
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Techniques Judgment Methods
Sales-force composite Experts panel Delphi method
Market research/survey Time Series
Moving Averages Exponential Smoothing
Trends Regression Holt’s method
Seasonal patterns – Seasonal decomposition Trend + Seasonality – Winter’s Method Causal Methods
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The Most Appropriate Technique(s)
Purpose of the forecastHow will the forecast be used?Dynamics of system for which forecast will
be madeHow accurate is the past history in
predicting the future?
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SUMMARY
Matching supply with demand a major challenge Forecast demand is always wrong Longer the forecast horizon, less accurate the
forecast Aggregate demand more accurate than
disaggregated demand Need the most appropriate technique Need the most appropriate inventory policy