The planning and production factor

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ISE 2004 Summer IP Seminar
Reliability Models for Facility Location with
Risk Pooling
Hyong-Mo Jeon
Jul 27 2004
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Contents

• 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

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Reliability Fixed-Charge Location Problem
(Daskin, Snyder)
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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

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Models

• The Formulation is
• Minimize ??1 (1 - ?) ? 2
• Subject to

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Solution Method- Lagrangian Relaxation

• Relax the assignment constraint.
• Minimize
• Subject to

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Solution Method- Lagrangian Relaxation

• Solve the relaxed problem
• The benefit
• If ?j lt 0, then set Xj 1, that is, open
  facility j.
• Set Yijr 1, if
• facility j is open
• lt 0
• r minimizes for s 0, , m-1.

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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)
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Location Model with Risk Pooling(Shen, Daskin,
Coullard)

• Minimize
• Subject to

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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?

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Solution Method- Sub-Problem Solving Procedure

• The Sub-Problem for each j
• SP(j)
• Subject to
• Solving Procedure
• Step 1 Partition the Set Ii bi ? 0, I0i
  bi lt 0 and ci0
• and I-i bi lt 0 and ci gt 0
• Step 2 Sort the element of I- so that
  b1/c1?b2/c2??bn/cn
• Step 3 Compute the partial sums
• Step 4 Select m that minimize Sm

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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(Ex) ? Ef(x).
• It is too difficult to formulate the exact
  expected failure inventory cost. ? Approximation

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Approximation for Expected Failure Inventory Cost

• The First Approximation APP1
• The Second ApproximationAPP2
• We believe Exact Value ? APP2 ? APP1

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Approximation for Expected Failure Inventory Cost
(49 locations, q 0.05)
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Model-Formulation

• Minimize
• Subject to

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Solution Method- Sub-Problem

• The Sub-Problem for each j
• SP(j)
• Subject to
• We Could not use the Shens Method because of the
  additional constraint.
• How can we solve this sub-problem?

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Solution Method- Sub-Problem Two Approaches

• 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.

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Computational Result
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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