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Hyong-Mo Jeon
Reliability Models for Facility Location with Risk Pooling
ISE 2004 Summer IP Seminar
Jul 27 2004
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• Reliability Fixed-Charge Location Problem• Risk Pooling Effect• Location Model with Risk Pooling• Reliability Models for Facility Location with Risk
Pooling– Motivation– Approximation for Expected Failure Inventory Cost– Models– Solution Method– Computational Result
• The problems that we should solve
Contents
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Reliability Fixed-Charge Location Problem (Daskin, Snyder)
0
1
3
2
4
5
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Models
• Notation– fj = fixed cost to construct a facility at location j J
– hi = demand per period for customer i I
– dij = per-unit cost to ship from facility j J to customer i I
– m = |J|– q = probability that a facility will fail (0 q 1)
– Xj = 1 : if a facility is opened at location j
0 : otherwise
– Yijr = 1 : if demand node i is assigned to facility j as a level r
0 : otherwise
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Models
• Objective Function
1 =
2 =
– The Objective Function is 1 + (1 - ) 2
Ii Jj
ijijiJj
jj YdhXf 0
Ii Jj
m
rijr
riji Yqqdh
1
0
)1(
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Models
• The Formulation is Minimize 1 + (1 - ) 2
Subject to
}0,1{
}0,1{
1
1
1
0
ijr
j
m
rijr
jijr
Jjijr
Y
X
Y
XY
Y
Jj
JjIi ,
1,,0,, mrJjIi
1,,0, mrIi
1,,0,, mrJjIi
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Solution Method- Lagrangian Relaxation
• Relax the assignment constraint.Minimize
Subject to
Ii
m
ririjr
Jj Ii Jj
m
rijrjj YXfz
1
0
1
0
)(
}0,1{
}0,1{
11
0
ijr
j
m
rijr
jijr
Y
X
Y
XY
Jj
JjIi ,
1,,0,, mrJjIi
1,,0,, mrJjIi
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Solution Method- Lagrangian Relaxation
• Solve the relaxed problem– The benefit
– If j < 0, then set Xj = 1, that is, open facility j.
– Set Yijr = 1, if • facility j is open • < 0• r minimizes for s = 0, … , m-1.
}{min,0min1,...,0
ijrmr
Iijj f
ijr
ijs
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Solution Method
• Lower and Upper Bound– The Optimal objective value for the relaxed problem provides
a lower bound– Upper Bound : Assign customers to the open facilities level
by level in increasing order of distance and calculate the objective value.
• Branch and Bound – Branch on Xj variables with greatest assigned demand.
– Depth-first manner
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Risk Pooling Effects (Eppen, 1979)
N
iiK
1
N
ii
K1
22
1
N
iiK
DTC
CTC
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Location Model with Risk Pooling(Shen, Daskin, Coullard)
Minimize
Subject to
Jj Ii
ijij
Jjjj YdXf
^
Jj Ii
ijij YK ^
}0,1{
}0,1{
1
ij
j
jij
Jjij
Y
X
XY
Y
Jj
JjIi ,
JjIi ,
Ii
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Solution Method- Lagrangian Relaxation
• Relax the assignment constraint.
Minimize
Subject to
How could they solve this non-linear integer programming problem?
Ii
iJj Ii Ii
ijijijiijjj YKYdXf ^^
)(
}0,1{
}0,1{
ij
j
jij
Y
X
XY
Jj
JjIi ,
JjIi ,
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Solution Method- Sub-Problem Solving Procedure
• The Sub-Problem for each jSP(j)
Subject to
• Solving Procedure– Step 1 : Partition the Set I+={i: bi 0}, I0={i: bi < 0 and ci=0}
and I-={i: bi < 0 and ci > 0}
– Step 2 : Sort the element of I- so that b1/c1b2/c2…bn/cn
– Step 3 : Compute the partial sums
– Step 4 : Select m that minimize Sm
Ii Ii
iiiij ZcZbV min~
}0,1{iZ Ii
m
Iiiii
m
Iiiii
Iiii
Iiiim ZcZbZcZbS
,1,100
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Reliability Models for Facility Location with Risk Pooling - Motivation
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Objective Function
• Fixed Cost and Expected Failure Transportation Cost
• Expected Failure Inventory Cost
– Above Expected Failure Inventory Cost is incorrect. Why? Because f(E[x]) E[f(x)].
– It is too difficult to formulate the exact expected failure inventory cost. Approximation
Jj
m
rijri
Ii
rj YqqK
1
0
^
)1(
Jj
jjXf
Jj
m
rijrij
Ii
r Ydqq1
0
^
)1(
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Approximation for Expected Failure Inventory Cost
• The First Approximation [APP1]
• The Second Approximation[APP2]
– We believe : Exact Value APP2 APP1
Jj
m
rijri
Ii
rj YqqK
1
0
^
)1(
Jj
m
r Iiijri
rj YqqK
1
0
^
)1(
By Simulation
Proved (By Jensen’s
Inequality)
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Approximation for Expected Failure Inventory Cost (49 locations, q = 0.05)
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400 1600
APP1
APP2
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Model-Formulation
Minimize
Subject to
Jj
m
r Iiijrij
r
Jjjj YdqqXf
1
0
^
)1(
}0,1{
}0,1{
1
1
1
0
ijr
j
m
rijr
jijr
Jjijr
Y
X
Y
XY
Y
Jj
1,,0,, mrJjIi
1,,0, mrIi
JjIi ,
1,,0,, mrJjIi
Ii
ijrij YK ^
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Solution Method- Sub-Problem
• The Sub-Problem for each jSP(j) :
Subject to
– We Could not use the Shen’s Method because of the additional constraint.
– How can we solve this sub-problem?
1
0
~
minm
r Ii Iiiriririrj ZcZbV
}0,1{
11
0
ir
m
rir
Z
Z Ii
1,,0, mrIi
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• Approach 1 – Relax one more constraint
SP(j) :
Subject to
• Approach 2– The Sub-problem is same to a LMRP without fixed cost– Solve the each sub-problem as a LMRP– We have no idea whether this assignment problem is
NP-hard or not.
Iiiu
Solution Method- Sub-Problem – Two Approaches
1
0
~
minm
r Ii Iiiriririrj ZcZbV
Ii
}0,1{irZ 1,,0, mrIi
1
0
m
rirZ 1
iu
Ii
1
0
m
rirZ 1
^
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Computational Result
Number of
Locations25 49
Used Method Approach 1 Approach 2 Approach 1 Approach 2
Optimality Gap(%)
3.62 0.099 2.85 0.096
Iteration 1324 845 5877 575
Time 67 sec 1.2hr 2.17hr 8.6hr
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The Problems That We Should Solve
• Prove Exact Value App2
• Improve algorithm run times
• Different q for each facility.
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Questions?
Thank you