1

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

4

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

5

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(

6

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