Transcript Document
Inventory Optimization under
Correlated Uncertainty
Abhilasha Aswal
G N S Prasanna,
International Institute of Information Technology –
Bangalore
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
1
Outline
Motivation
Optimizing with correlated demands
Generalized EOQ
Related work
Some Extensions:
Generalized base stock
Geman Tank
Relational Algebra
Conclusions
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
2
The EOQ model
The EOQ model (Classical – Harris 1913)
Q*
2CD
h
f*
Dh
2C
C: fixed ordering cost per order
h: per unit holding cost
D: demand rate
Q*: optimal order quantity
f*: optimal order frequency
Q*
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
3
Inventory optimization for multiple
products
EOQ(K)?
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
4
Motivation
Inventory optimization example
Car type I
Car type II
Car type III
Automobile
store
Tyre type I
Supplies
Tyre type II
Petrol
Drivers
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
5
Motivation
Ordering and holding costs
Product
Ordering Cost in Rs.
(per order)
Holding Cost in Rs.
(per unit)
Car Type I
1000
50
Car Type II
1000
80
Car Type III
1000
10
Tyre Type I
250
0.5
Tyre Type II
500 (intl shipment)
0.5
Petrol
600
1
Drivers
750
300
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
6
1 product versus 7 products
Exactly Known Demands, no uncertainty
EOQ solution and Constrained Optimization solution match exactly:
EOQ Solution
Product
Demand per
month
Car Type I
40
Car Type II
25
Car Type III
50
Tyre Type I
250
Tyre Type II
125
Petrol
300
0.5
600
Drivers
5
1
5
Constrained Optimization Solution
Order
Frequency
Order
Quantity
Cost
Order
Frequency
Order
Quantity
Cost
1
40
2000
1
40
2000
1
25
2000
1
25
2000
0.5
100
1000
0.5
500
250
0.25
500
250
600
0.5
600
600
1500
1
5
1500
UNREALISTIC!!!
0.5
100
1000
We cannot
know
the future
0.5
500
250
demands
0.25 exactly.
500
250
Total
7600
7600
But…
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
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1 product versus 7 products
Bounded Uncorrelated Uncertainty
Assuming the range of variation of the demands is known, we can get
bounds on the performance by optimizing for both the min value and the
max value of the demands.
EOQ solution and Constrained Optimization solution are almost the same.
EOQ solution
Product
Constrained Optimization
Order Frequency
Order Quantity
Order Frequency
Order Quantity
Min
Max
Min
Max
Min
Max
Min
Max
Car Type I
0.5
1
20
40
0.5
1
20
40
Car Type II
0
1
0
25
0
1
0
25
Car Type III
0.5
1
100
200
0.5
1
100
200
Tyre Type I
0.25
0.5
248.99
500
0.25
0.5
248
500
Tyre Type II
0.25
0.5
500
1000
0.25
0.5
500
1000
Petrol
0.25
0.5
300
600
0.25
0.5
300
600
Drivers
0.45
1
2.24
5
0.5
1
2
5
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
8
1 product versus 7 products
Beyond EOQ: Correlated Uncertainty in Demand
Considering the substitutive effects between a class of products (cars,
tyres etc.)
200 ≤ dem_tyre_1 + dem_tyre_2 ≤ 700
65 ≤ dem_car_1 + dem_car_2 + dem_car_3 ≤ 250
Considering the complementary effects between products that track each
EOQ cannot incorporate such
other
forms of uncertainty.
5 ≤ (dem_car_1 + dem_car_2 + dem_car_3) – dem_petrol ≤ 20
5 ≤ dem_car_2 – dem_drivers ≤ 20
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
9
1 product versus 7 products
Beyond EOQ: Correlated Uncertainty in Demand
Min-Max solution for different scenarios:
EOQ
Order
Frequency
Products
Order
Quantity
With Substitutive
Constraints
With Complementary
Constraints
With both Substitutive and
Complementary constraints
Order
Frequency
Order
Quantity
Order
Frequency
Order
Quantity
Order
Frequency
Order
Quantity
1
40 I
Car Type
0.75
25
0.5
38
0.5
40
1
25II
Car Type
0.5
13
0.5
22
1
10
0.5
100
Car Type
III
0.75
125
0.75
121
0.5
180
0.5
500I
Tyre Type
500II
Tyre Type
0.25
362
0.75
250
0.75
200
0.75
500
0.75
373
0.5
400
600
Petrol
5
Drivers
0.5
400
0.5
208
0.5
222.5
0.5
5
0.5
2
0.5
3
0.25
0.5
1
7600
Cost (Rs.)
4590.438
Abhilasha Aswal & G N S Prasanna
4593.688
IIIT-B
4654.188
INFORMS 2010
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1 product versus 7 products
Beyond EOQ: Correlated Uncertainty in Demand
Comparison of different uncertainty sets
Scenario sets
Absolute Minimum Cost
Absolute Maximum Cost
Bounds only
3349.5
9187.5
Bounds and Substitutive
constraints
3412.5
9100
Bounds and Complementary
constraints
4469.5
8972.5
Bounds, Substitutive and
Complementary constraints
4482.5
8910
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
11
Optimizing with Correlated Demands
Mathematical Programming
Formalism
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
12
Optimal Inventory policy using “ILP”
Minimizedecision Max uncertain
Subject to:
ytp htp Invtp1
T 1
N T-1
p
P
p
It C
yt
t 0
p 1 t 0
ytp S tp Invtp1
I
I
p
t
p
t
Fixed costs and
breakpoints: nonconvexities that
preclude strong-duality
from being achieved.
No breakpoints or fixed
costs: min-max
optimization QP
M S tp
1 M S tp
Invtp1 Invtp S tp Dtp 0
(CP) D E
S tp 0
Dtp 0
Abhilasha Aswal & G N S Prasanna
Min-max optimization,
not an LP.
Duality??
Heuristics have to be
used in general.
IIIT-B
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Optimal Inventory policy by Sampling
A simple statistical sampling heuristic
Begin
for i = 1 to maxIteration
{
parameterSample = getParameterSample(constraint Set)
bestPolicy = getBestPolicy(parameterSample)
findCostBounds(bestPolicy)
}
chooseBestSolution()
End
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
14
Optimizing with Correlated Demands:
Analytical Formulation: Generalized
EOQ(K)
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
15
Classical EOQ model
Per order fixed cost = f(Q)
holding cost per unit time = h(Q)
C Q h Q f Q D / Q
Q* 2 fD / h;C * Q 2 fDh
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
16
EOQ(K) with multiple products,
uncertain demands
Additive SKU costs
Case with 2 commodities, generalized to n commodities
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
17
EOQ(K) with multiple products,
uncertain demands
Holding cost linear, ordering cost fixed
Q1* 2 f1 D1 / h1 ;C1* D1 2 f1 D1h1
Q2* 2 f 2 D2 / h2 ;C2* D2 2 f 2 D2 h2
C * D1 , D2 C1* D1 C2* D2 2 f1 D1h1 2 f 2 D2 h2
Cmax max D1 , D2 CP 2 f1 D1h1 2 f 2 D2 h2
Cmin min D1 , D2 CP 2 f1 D1h1 2 f 2 D2 h2
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
18
Analytical solution: Substitutive
constraints
Holding cost linear, ordering cost fixed
Under a substitutive constraint D1 + D2 <= D
C * ( D1 , D2 ) C1* ( D1 ) C 2* ( D2 ) 2 f1 D1h1 2 f 2 D2 h2
D1 D2 D
f1h1 D
f 2 h2 D
2 D f1h1 f 2 h2
C max C *
,
f1h1 f 2 h2 f1h1 f 2 h2
C min min C * 0, D , C * D,0 2 D min f1h1 , f 2 h2
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
19
Analytical solution: Substitutive
constraints - Example
2 products, demands D1 & D2
Costs:
h1 = 2/unit
h2 = 3/unit
f1 = 5/order
f2 = 5/order
D1 + D2 = D = 100
Maximum cost
C max 2 D f1h1 f 2 h2
2 100 10 15 70.71
Minimum cost
2 100 15 44.72
C min min 2 D f1h1 , 2 D f 2 h2
min 2 100 10,
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
20
Analytical solution: Complementary
constraints
Holding cost linear, ordering cost fixed
Under a complementary constraint D1 – D2 <= D, with D1 and D2
limited to Dmax
Cmax C * Dmax D , Dmax
Cmin min C D,0, C 0, D
Abhilasha Aswal & G N S Prasanna
*
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*
INFORMS 2010
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Analytical solution: Complementary
constraints - Example
2 products, demands D1 & D2
Costs:
h1 = 2/unit
h2 = 3/unit
f1 = 5/order
f2 = 5/order
Demand constraints:
D1 - D2 = K = 20
D1 <= Dmax = 50
D2 <= Dmax = 50
Maximum cost
C max 2( Dmax K ) f1 h1 2 Dmax f 2 h2
2 30 10 2 50 15 45.83
Minimum cost
2 20 15 20
C min min 2 K f1 h1 , 2 K f 2 h2
min 2 20 10,
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
22
Both substitutive & complementary
constraints
Holding cost linear, ordering cost fixed
Under both substitutive and complementary constraints
C * D1 , D2 C1* D1 C2* D2 2 f1D1h1 2 f 2 D2 h2
Dmin D1 D2 Dmax
CP :
D1 D2
Cmax max D1 , D2 CP 2 f1 D1h1 2 f 2 D2 h2
Cmin min D1 , D2 CP 2 f1 D1h1 2 f 2 D2 h2
Convex optimization techniques are required for this
optimization.
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
23
Both substitutive & complementary
constraints - Optimization
Objective function: concave
Minimization: HARD!
Envelope based bounding schemes
Heuristics to find upper bound.
Simulated annealing based
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
24
Both substitutive & complementary
constraints - Example
2 products, demands D1 & D2
Costs:
h1 = 2/unit
h2 = 3/unit
f1 = 5/order
f2 = 5/order
Demand constraints:
150 <= D1 + D2 <= 200
-20 <= D1 – D2 <= 20
Maximum cost: 99.88
Minimum cost
Enumerating all vertices (exact)
85.39
Error: 0.111247 %
Simulated annealing heuristic
85.48499
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
25
Both substitutive & complementary
constraints – Example (contd)
5 products, demands
D1, D2, D3, D4 & D5
Costs:
Demand constraints:
h1 = 2/unit
h2 = 3/unit
h3 = 4/unit
h4 = 5/unit
h5 = 6/unit
f1, f2, f3, f4, f5 = 5/order
Abhilasha Aswal & G N S Prasanna
IIIT-B
D1 + D2 + D3 + D4 + D5 <= 1000
D1 + D2 + D3 + D4 + D5 >= 500
2 D1 - D2 <= 400
2 D1 - D2 >= 100
5 D5 - 2 D4 <= 900
5 D5 - 2 D4 >= 150
D2 + D4 <= 400
D2 + D4 >= 250
D1 <= 350
D1 >= 100
D3 >= 150
D3 <= 300
D4 >= 75
D4 <= 200
INFORMS 2010
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Both substitutive & complementary
constraints – Example (contd)
Maximum cost: 436.6448
Minimum cost:
Enumerating all vertices (exact)
323.5942
Simulated annealing heuristic
324.4728
Error: 0.271505 %
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
27
Inventory constraints
Constrained Inventory Levels
If the inventory levels Qi and demands Di, are constrained as
Q1 , Q2 , D1 , D2 0
The vector constraint above can incorporate constraints like
Limits on total inventory capacity (Q1+Q2 <= Qtot)
Balanced inventories across SKUs (Q1-Q2) <= ∆
Inventories tracking demand (Q1-D1<=Dmax)
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
28
Inventory constraints
Constrained Inventory Levels
C1 Q1 , D1 h1 Q1 f1 Q1 D1 / Q1
C2 Q2 , D2 h2 Q2 f 2 Q2 D2 / Q2
C Q1 , Q2 , D1 , D2 C1 Q1 C2 Q2
[ D1 , D2 ] CP
Q1 , Q2 , D1 , D2 0
C * D1 , D2 min Q1 ,Q2 C Q1 , Q2 , D1 , D2
Cmax max[ D1 , D2 ]CP C * D1 , D2
Cmin min[ D1 , D2 ]CP C * D1 , D2
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
29
Related Work
McGill (1995)
Inderfurth (1995)
Dong & Lee (2003)
Stefanescu et. al.
(2004)
Abhilasha Aswal & G N S Prasanna
Bertsimas, Sim,
Thiele et. al.
IIIT-B
INFORMS 2010
30
Related work
Bertsimas, Sim, Thiele - “Budget of uncertainty”
Uncertainty: aij aij , aij aij
zij
Normalized deviation for a parameter:
aij aij
aij
n
Sum of all normalized deviations limited:
z
j 1
ij
i , i
N uncertain parameters polytope with 2N sides
In contrast, our polyhedral uncertainty sets:
More general
Much fewer sides
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
31
Extensions:
Generalized basestock
German Tank
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
32
Basestock with correlated inventory
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
33
The German Tank Problem
Classical German Tank
Biased estimators
Generalization
Maximum likelihood
Unbiased estimators
Minimum Variance
unbiased estimator
(UMVU)
Maximum Spacing
estimator
Bias-corrected maximum
likelihood estimator
Abhilasha Aswal & G N S Prasanna
IIIT-B
Given correlated data
samples, drawn from a
uniform distributionestimating the bounded
region formed by correlated
constraints enclosing the
samples.
Estimating the constraints
without bias and with
minimum variance.
INFORMS 2010
34
Information Theory and Relational
Algebra
Uncertainty can be identified with Information.
Information polyhedral volume
Relational algebra between alternative
constraint polyhedra
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
35
Conclusions
Generalized EOQ to Correlated Demands
Analytical Solutions
Computational Solutions
Enumerative versus Simulated Annealing
Extensions of formulations
Generalized Basestock
German Tank
Information Theory and Relational Algebra
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
36
Thank you
Abhilasha Aswal & G N S Prasanna
IIIT-B
INFORMS 2010
37