IEEM 517
Inventory Control –
(Q, r) Model
LEARNING OBJECTIVES
1.
Understand modeling assumptions, formulation, and optimal solution of the
(Q,r) inventory model
2.
Learn how to model leadtime uncertainty (one type of supply uncertainty)
3.
Compare inventory models learned in this course
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CONTENTS
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• (Q, r) Inventory Model
• Extension to Stochastic Leadtime
• Comparison of Inventory Models
• Summary
EXAMPLE (1)
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A retail outlet sells a product that is ordered/replenished from a supplier. Due to the
transportation and order-processing time, it takes 30 days from placing an order to
receiving the order.
The cost parameters for the retailer are the following: The setup cost for each
replenishment is $10.00. The unit purchase cost is $0.50. The inventory holding
cost is $0.20 per unit per year. The backorder cost $2.00 per unit per year.
Demand from consumers arrives according to a Poisson process with a rate of 2
units per day.
1) What are the inventory policy and policy parameters the retailer should choose?
2) For backorders, instead of charging the backorder cost mentioned above, the
retailer can use another cost metric, the stockout cost. For the stockout cost,
whenever a demand unit is backordered, regardless of its backordered time, a
stockout cost of $0.10 is charged. How will the inventory policy be changed?
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COMMONLY USED TERMS
Leadtime
Time from placing an order to receiving it
Replenishment orders
Orders being placed but having not arrived
On-hand inventory
Physical inventory in stock
Backorder level
The quantity of demand units being backordered
Stockout
The event that a demand unit is backordered
Net inventory level
On-hand inventory minus backorder level
Inventory position
Net inventory level plus replenishment orders
Fill rate
Fraction of demands that are filled from stock immediately
MODELING ASSUMPTIONS
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1. Constant leadtime required for delivery
2. Demand is stochastic
3. A production/replenishment incurs a fixed setup cost
4. Consider the case of a single product
5. Backorders are allowed
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CONCEPTUAL FRAMEWORK OF (Q,r) POLICY
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(Q,r) Policy
Whenever the inventory position drops to the reorder point r, a replenishment
order for quantity Q is placed immediately
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MODEL PARAMETERS & DECISION VARIABLES (1)
L
Replenishment leadtime (in days)
A
Setup cost per production/order (in dollars)
c
Unit production/purchase cost (in dollars per unit)
h
Annual unit holding cost (in dollars per unit per year)
b
Annual unit backorder cost (in dollars per unit per year)
k
Cost per stockout (in dollars)
D
Expected demand per year (in units)
X
Demand during the replenishment leadtime (in units), a random variable
θ
= E[X] = DL/365 = expected demand during replenishment leadtime (in units)
σ
Standard deviation of demand during replenishment leadtime (in units)
p(x) = P(X = x) = probability that demand during replenishment leadtime equals x
G(x) = P(X ≤ x) = Σi=0x p(i) = probability that demand during replenishment
leadtime is less than or equals x
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MODEL PARAMETERS & DECISION VARIABLES (2)
Q
Replenishment quantity (in units)
r
Reorder point (in units)
Decision
Variables
F(Q,r)
Order frequency as a function of Q and r
S(Q,r)
Fill-rate (fraction of demands filled from stock immediately)
as a function of Q and r
B(Q,r)
Average number outstanding backorders as a function of Q and r
I(Q,r)
Average on-hand inventory level (in units) as a function of Q and r
COMMENTS AND ISSUES (1)
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We treat time as being continuous
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Inventory position only takes the values of r+1, r+2, …, r+Q
•
In the steady state, inventory position is equally likely to be the values from
r+1 to r+Q
•
We treat the demand quantity as being discrete in modeling
(i.e., we denote by a probability mass function p(x))
But we can treat it as being continuous in analysis
(i.e., we can use a probability density function g(x) if necessary)
•
The objective is still to minimize total setup, inventory holding,
backorder/stockout, and production/order costs
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COMMENTS AND ISSUES (2)
•
We consider two types of cost for each demand unit not filled immediately
1) Backorder cost: for each demand unfilled and for its backordered time
2) Stockout cost: for each demand unfilled (regardless of its backordered
time)
•
We have two decision variables in this inventory model, the replenishment
quantity Q and the reorder point r:
1) The replenishment quantity Q determines the frequency of
replenishment orders and thus trades between the setup cost and the
inventory cost
2) The reorder point r determines the likelihood of a stockout and thus
trades between inventory holding cost and backorder/stockout cost
RELEVANT COSTS (1)
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Production/order cost
Expected production/order cost per year = cD
Setup cost
Expected order frequency = F(Q,r) = D/Q
Expected setup cost per year = A(D/Q)
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RELEVANT COSTS (2)
r+Q
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Explanation for computing backorder/stockout cost
Possible
inventory
position
0 Steady state
r+2
1 Want to analyze the
backorder level /
stockout situation
at this time t
r+1
t-L
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2 Leadtime L
Demand X
during leadtime L
Inventory position
at time t - L
Net inventory level
at time t - L
t
time
Net inventory level
at this time t
Replenishment orders
arriving during leadtime L
On-hand inventory
– Backorder level
RELEVANT COSTS (3)
Backorder cost
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: for each demand unfilled and for its backordered time
Expected backorder cost per year = bB(Q,r), where
B(Q,r) = Average number outstanding backorders
= E[ Backorder level at time t ]
1
Backorder level at time t | Inventory position at t – L is y ]
Q
1
1
r +Q
= [B(r + 1) + … + B(r + Q)] ≈ B(r) , where
= ∑ y =r +1B(y)
Q Q
=
∑
r +Q
y =r +1 E[
B(y) = E[ Backorder level at time t | Inventory position at t – L is y ]
= E(X – y)+
= ∑ ∞x = y (x − y)p(x)
= θ − ∑ yx -=10 [1 − G(x)]
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RELEVANT COSTS (4)
Stockout cost
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: for each demand unfilled (regardless of its backordered time)
Expected stockout cost per year = kD[1 – S(Q,r)]
Expected number of stockouts per year = D[1 – S(Q,r)], where
S(Q,r) = Fill-rate = fraction of demands filled from stock immediately
= P(a demand unit is filled from stock immediately)
= P(net inventory level at time t is ≥ 1)
1
t is ≥ 1 | Inventory position at t – L is y)
Q
1
y - 1 | Inventory position at t – L is y)
Q
1
r +Q
y =r +1P(demand in leadtime L is ≤ y - 1)
Q
1
r +Q
y =r +1G(y - 1)
Q 1
= 1 - [ B(r) – B(r + Q) ]
Q
1
≈ 1 - B(r)
Q
∑
=∑
= ∑
= ∑
=
r +Q
y =r +1 P(net inventory level at time
r +Q
y =r +1P(demand in leadtime L is ≤
RELEVANT COSTS (5)
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Inventory holding cost
Expected inventory holding cost = hI(Q,r)
Expected net inventory level
= Expected on-hand inventory level – Expected backorder level
= I(Q,r) - B(Q,r)
Å Assuming that demand comes as being deterministic
≈ Average inventory in order cycle of Q + (reorder point – demand during leadtime)
= (Q + 1)/2 + (r – θ)
Expected on-hand inventory level = I(Q,r) ≈ (Q + 1)/2 + (r – θ) + B(Q,r)
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RELEVANT COSTS (6)
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Explanation for computing inventory holding cost
BACKORDER COST APPROACH (1)
Total cost
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: setup cost + backorder cost + inventory holding cost
First order condition
Note: use the continuous analogy for the B(r) function
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BACKORDER COST APPROACH (2)
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Second order condition
Optimal solution (under approximation)
Q* =
2AD
h
G(r*) =
b
b+h
STOCKOUT COST APPROACH (1)
Total cost
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: setup cost + stockout cost + inventory holding cost
First order condition
Note: use the continuous analogy for the B(r) function
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STOCKOUT COST APPROACH (2)
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Second order condition
Solution (under approximation)
Q* =
G(r*) =
2AD
h
kD
kD + hQ *
EXAMPLE (2)
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Backorder cost approach
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EXAMPLE (3)
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Stockout cost approach
CONTENTS
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• (Q, r) Inventory Model
• Extension to Stochastic Leadtime
• Comparison of Inventory Models
• Summary
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CHANGES IN ASSUMPTION AND NOTATION
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Change in assumption:
The leadtime L is now stochastic, instead of being constant
Consequence of the change:
The demand X in leadtime now depends on the uncertainty on both the
demand side and the supply side, instead of depending only on the demand side
Changes in notation:
L
Replenishment leadtime (in days), a random variable
l
= E[L] = expected replenishment leadtime (in days)
σL
Standard deviation of replenishment leadtime (in days)
Dt
Demand on day t (in units), a random variable ⇔ daily demand
d
= E[Dt] = expected daily demand
σD Standard deviation of daily demand (in units)
ANALYSIS
X
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= Demand during the replenishment leadtime (in units)
= D1 + D2 + … + DL
θ
= Expected demand during replenishment leadtime (in units)
= E[L]E[Dt]
= ld
Var(X) = Variance of demand during replenishment leadtime
= E[L]Var(Dt) + (E[Dt])2Var(L)
= lσD2 + d2σL2
σ
= Standard deviation of demand during replenishment leadtime (in units)
= √ Var(X)
= √ lσD2 + d2σL2
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EXAMPLE (4)
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We continue with the previous example. We have assumed that the leadtime is
constant and equals 30 days. Now instead of being constant, the leadtime becomes
a random variable with mean 30 days and variance 102. All other data keep the
same. How the values of the (Q,r) policy change? Discuss both the backorder
approach and the stockout approach.
Note that:
Demand from consumers arrives according to a Poisson process with a rate of 2
units per day
⇔
⇔
EXAMPLE (5)
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Backorder cost approach: stochastic leadtime
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EXAMPLE (6)
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Stockout cost approach: stochastic leadtime
CONTENTS
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• (Q, r) Inventory Model
• Extension to Stochastic Leadtime
• Comparison of Inventory Models
• Summary
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COMPARISON OF INVENTORY MODELS
EOQ
DLS
Newsvendor
(Q,r)
Demand
Time
Setup cost
Backorders
Leadtime
Solution
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ABC ANALYSIS
ABC analysis is a way to determine what items in inventory are important
cumulative %
of contribution
cumulative % of ranked items
Type
% of items
% of contribution
Action on inventory
A
5-10%
50%
Very close scrutiny
B
50%
45%
Moderate scrutiny
C
40-45%
5%
minimal effort
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CONTENTS
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• (Q, r) Inventory Model
• Extension to Stochastic Leadtime
• Comparison of Inventory Models
• Summary
SUMMARY
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• (Q, r) inventory model is a continuous-review inventory model where the
tradeoff decision is made among setup, inventory holding,
backorder/stockout costs
• Uncertainty in leadtime can add some additional variability to the (Q,r)
inventory model
• Different inventory models are applied in different settings, and different
magnitudes of effort are used to manage items with different levels of
importance
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ANNOUNCEMENTS
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• For the (Q,r) model, read Section 2.4.3-2.5
• Lab 4 will be held on Tuesday, April 12
- Time:
5:00pm – 6:50pm
- Location:
IS Lab of IEEM Dept (Room 3207)
- Content:
Modeling (Q,r) inventory system by Vensim
- Preparation:
Review Chapter 1-3 in User’s Guide of Vensim
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