Chapter 2 – Inventory Management
There is a trade-off: Expensive inventory VS Happy customers
Trade-offs are caused by:
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Order costs
Holding costs
Out of stocks costs
Production rates
Quantity discount
Demand uncertainty
Safety stocks
Service levels
Lead times
ABC-Analysis: it’s another approach to distinguish different types of inventories
The most fundamental model is the Economic Order Quantity (EOQ)
Economic Order Quantity: it assumes that we buy goods from the supplier, and we want to
optimize the optimal order quantity
The Reorder Point is an important threshold when a new order is placed
Newsvendor model is widely used and well suited for products with seasonality
Locations and Functions of Inventory
What types of inventories can we distinguish?
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We can distinguish them according to their locations:
o Raw materials inventory: Goods received from supplier
o Work-in-progress inventory (WIP)
o Finished goods inventory: Goods ready to be dispatched to the customer
The rule of thumb is that the raw material inventory represents 30%, the WIP represents 40%
and the finished goods inventory accounts for 30% of the total inventory.
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We can distinguish them according to their functions:
o Buffer and safety: to cope with randomness in demand and supply
o Cycles: Multiple products from one operation => keep supply while other
products are produced
o Decoupling: Match the rate of processing at different points in the production
processes (doing inventories to not depend on the production units ahead (if
they are not producing at the same pace))
o Quantity game: To enable bulk and other discount
o Pipeline/ movement: Compensate for transportation and distribution times
ABC Analysis
Pareto’s law: with 20% of the effort, you can generate 80% of the value
A part: 20% of the items inventory (expenses)  usually makes 80% of the inventory value
B parts: 30% of the items inventory
C part: 50% of the items inventory
A and B parts must be managed carefully because they account for a lot
Economic Order Quantity (EOQ) – Understanding the Problem
How many items should I order each time?
The EOQ is based on the fact that the annual demand is well known (given)
Now, we must define how often do we need to order and how much per order?
It’s the trade-off of the ordering costs VS the holding costs
We should assume that demand is constant
Economic Order Quantity (EOQ) – Scenario Analysis
For total holding cost, we need to divide the order quantity by 2 in order to get the average
inventory.
Economic Order Quantity – Calculation of the Optimal Order Quantity
D = Annual demand Q = Optimal order quantity
Co = Ordering Costs Ch = Holding costs
Total ordering cost: D/Q * Co
Total holding costs: Q/2 * Ch
The goal is minimizing the total costs
TC = D/Q * Co + Q/2 * Ch
To minimize  TC’ = 0
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- D/Q2 * Co + ½ * Ch = 0  Q = ((2*D*Co)/ Ch)1/2
Economic Order Quantity – Reorder Point (ROP)
Reorder Point: it’s the inventory level which triggers the next order
Lead time: is the time between you’ve ordered something and the delivery to your inventory
ROP = Lead time * Demand per business days
Demand per business day = Annual demand / Business days per year
Economic Production Quantity (EPQ) – Formula
D: Demand forecast
Cs: Production Setup
d: Demand per business days
Ch: Holding cost per units and year
P: Production rate
Q: Optimal Production quantity
Production surplus: P – d  gives us how fast do we build up an inventory
We need to produce until we’ve reached Q
Our Inventory doesn’t go to Q but until Imax
Imax
= t * (P-d)
t = Q/P
= Q (1 – d/P)
To determine the Production Quantity, we have to take the Order Quantity formula and
modify it a bit:
Q = ((2*D*Co)/ Ch)1/2
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Q = ((2*D*Cs)/ CH*(1-d/P))1/2
Holding costs need to be adjusted compared to the Order Quantity Formula:
Ch = Alternative production quantity/2 * Holding costs * (1 – d/P)
Economic Production Quantity – Scenario Analysis
When Production Rate = Demand per business day  Just-in-time inventory
Where EOQ focuses on ordering goods from suppliers, EPQ focuses on producing its own
supplies.
The main conceptual difference between EOQ and EPQ is that the inventory no longer
restocks infinitely fast, instead, inventory is received at a constant rate for a certain period of
time.
With EPQ:
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The smaller the production quantity, the higher the annual setup costs
The higher the production quantity, the higher the holding cost
If the production rate P equals the demand rate d, this is referred to as a just-in-time
inventory
If the demand rate d is greater than the production rate p, it doesn’t make sense to
apply the EPQ model
Low production rate lead to a slow build-up of inventory, which means that inventory holding
costs are not as high. This leads to higher production quantities per production batch.
Economic Production Quantity and Discounts
Total cost = Ordering costs + Holding costs + Unit costs (purchase cost)
Ordering costs: Number of orders * Costs per order
Holding costs: Alternative order quantity/2 * Discount unit cost * Holding cost
Unit costs: Annual demand * Discount unit cost
When they ask a comparison between the EOQ Model and cost savings, we need to take the
lowest value of the exercise and the EOQ value
 EOQ value – Lowest cost of the exercise = Difference
Safety Stock (SS) – The problem
What if demand is unknown? How can we deal with uncertainty?
Safety stock is a reserve in the inventory which support the company with changes in
demand.
Stockout: what’s not possible to deliver to customers because not enough inventory
Reorder Point (ROP) is the available stock during the lead time
We now want to find a new ROP including the safety stock, to avoid stockout
The ROP must be larger than the actual demand during lead time (ADDLT)
Now, we want to calculate the expected Demand during lead time with:
= SUMPRODUCT (Units; Probability)
Safety Stock – Solution
We’ll determine the optimal safety stock
You can have negative safety stock (they are going to reduce the Reorder Point)
IF DDLT > ROP then stockout cost  (DDLT – ROP) * Stockout cost per unit * Number of
order per year
IF DDLT < ROP then inventory cost  (ROP – DDLT) * 5
When a Number of order is given, it doesn’t represent the one we need to use
 always check with: Annual demand / Optimal Order Quantity
=SUMPRODUCT(B2:B6 ; C2:C6)
Annual demand / Optimum Order Quantity
=SUMPRODUCT (B2:F2 ; $B$9:$F$9)
=G4 -G6
= IF ($A2 >= B$1 ; ($A2 – B$1) * holding costs ; (B$1-$A2) * stockout cost * number of orders per years )
Discrete Newsvendor model
Very specific  different from Economic Order Model
This model can be applied to fashion products and perishable goods
The characteristics of this problem:
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Single period problem  ordering decision is made every period
Demand is uncertain
Order is placed before demand materializes
It’s possible to order only once per period
There is cost for ordering too much and cost for ordering too few items
IF OQ < Demand  OQ*Margin
OQ: Order Quantity Margin: Sale price – Unit cost
IF OQ > Demand  Demand * Sales Price – OQ * Unit cost+ Overstock * Salvage value
 Overstock: OQ – Sales
NEED TO CHECK THE PROFIT INCREASE IN 2.4C
=SUMPRODUCT (B2:G2; $B$13:$G$13)
= IF ($A2 > B$1 ; ($A2 – B$1) * salvage price + B$1 * selling price ; $A2 * selling price) – setup cost - $A2 * production costs
Uncertainty and Continuous Demand
Newsvendor Model – Normal Distribution
1. We need to understand demand
We don’t have discrete probabilities anymore, but we use a distribution function:
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Probability density function
Cumulative distribution function
Overage cost (co) = overstock (storage) costs
Underage cost (cu) = stockout cost (when demand is not fulfilled)
Critical ratio (CR): cu / (cu + co)
2. We must determine costs and critical ratio
When:
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0 < CR < 0,1 
co > > cu
 Order very carefully to avoid overage cost
 applied to expensive products
CR = 0,5
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co = cu
 Neutral
With a critical ratio over 0.5, the overstock cost has lower weight than the out-of-stock
cost. Therefore, the order quantity is larger than the average value
1 > CR > 0,9 
co < < cu
 Order aggressively to avoid underage cost
 avoid underage cost like penalty because not delivered in time
3. Determine Optimal Order Quantity
Standard normal distribution (textbook way): average/ mean = 0 and a standard deviation = 1
(Normal distribution: mean = x and standard deviation = y  x and y can be chosen)
μ = mean
z = F-1 (CR)
σ = standard deviation (68/95)
Q= optimal order quantity
Q = μ +/- σ * z (aggressive ordering  adding / carefully ordering  subtracting)
We use z as a probability and take the cumulative distribution function of the standard normal
distribution and derive a corresponding z-value
We take the normal standard distribution inv. =NORM.S.INV(CR)) to derive z
Optimal Order Quantity = SUM (Mean; standard deviation * z)
4. Determine costs (probability density function)
C = (cu + co) * f01 (z) * σ
f01 (z): Norm.S.Dist (z; FALSE)
False because we use the probability density function and not the cumulative
distribution anymore
5. Determine Profit
P = (p - c) * μ – C
p: sales price
c: unit cost
Newsvendor Model - Service levels
Demand is normal distributed with a mean of 100 and a standard deviation of 20
Service level is one of the most important metrics in business today
SLA = Service Level Agreement
SLA define service level KPI’s
Service availability (service level) is never assured 100%
When service level = 95%, we are looking for the order quantity (stock/inventory) that
assures us that we can deliver in 95% of the cases
Optimal order quantity: = NORM.INV (probability; mean or expected demand; standard
deviation)
The closer we get to 100% of service availability, the faster the order quantity increases
If we really go for 100%  OQ = Infinite
The cost are so high to satisfy 100% of customers that often we have to traded off with costs
Inventory Management in Practice
Existing Inventory Management Approaches
Fundamental management policies based on Reorder Point (ROP)
Applied when demand is not stable:
The R-Q approach: Demand can vary (not always reordering at the same period) but the
ordered items quantity to refill our inventory is always the same (q doesn’t change) .
The R-T approach (R-Q approach with a target level): Each time we have to reorder, we
would fill up the inventory to the target level  the quantity of items ordered vary
Periodic inventory management approaches:
Applied when there is a constant period between orders
The T-Q approach: applied when demand is stable, the quantity of items ordered is always
the same
The T-T approach: applied when there is a target level, demand is constant but the ordered
quantity is variable (in order to always reach the target level)
Periodic Order Management with Target Inventory Level –
Implementation
Inventory Position = Inventory at the beginning of the period + Open Orders
Order quantity = Target inventory – Inventory Position
Inventory at the end of the period = Inventory at the beginning + Orders received – Demand
during the period
Periodic Order Management with Target Inventory Level – Determining
the Target Level
Critical ratio: Underage costs / (overage + underage costs)
 allow us to get the z-value: Norm.S.Inv of critical ratio
Z- value can also be obtained only with the service level  Norm.S.INV (service level)
The items we ordered are only available to us after Lead time + 1  week when we place
the order + LT
Standard deviation within LT+1
= SQRT((Lead time + 1) * Standard deviation ^2 )
Average demand within LT+1:
= Average demand * (LT + 1)
Target inventory level:
=ROUND ((Average demand within LT + 1) + (Standard deviation within LT+1) * Z-Value; 0)
Pipeline inventory: Sum of the orders within LT
Inventory Position = Physical inventory + Pipeline Inventory
Order Quantity = Target Inventory – Inventory Position
Physical Inventory in the end = Physical Inventory at the beginning + Orders received –
Demand fulfilled
Safety stock = Target inventory level – Average demand within Lead time (LT)
With increasing lead time, the safety stock remains quite low compared to the pipeline
inventory (which is good because it means not much holding costs)