Hyong-Mo Jeon

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

2. 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

3. Reliability Fixed-Charge Location Problem (Daskin, Snyder) 1 0 4 5 2 3

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

6. Models The Formulation is Minimize 1 + (1 - )  2 Subject to

7. Solution Method- Lagrangian Relaxation Relax the assignment constraint. Minimize Subject to

8. 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 minimizesfor s = 0, … , m-1.

9. 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

10. Risk Pooling Effects(Eppen, 1979)

11. Location Model with Risk Pooling(Shen, Daskin, Coullard) Minimize Subject to

12. Solution Method- Lagrangian Relaxation Relax the assignment constraint. Minimize Subject to How could they solve this non-linear integer programming problem?

13. Solution Method- Sub-Problem Solving Procedure The Sub-Problem for each j SP(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/c1b2/c2…bn/cn Step 3 : Compute the partial sums Step 4 : Select m that minimize Sm

14. Reliability Models for Facility Location with Risk Pooling - Motivation

15. 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

16. Approximation for Expected Failure Inventory Cost The First Approximation [APP1] The Second Approximation[APP2] We believe : Exact Value  APP2  APP1 Proved (By Jensen’s Inequality) By Simulation

17. Approximation for Expected Failure Inventory Cost (49 locations, q = 0.05)

18. Model-Formulation Minimize Subject to

19. Solution Method- Sub-Problem The Sub-Problem for each j SP(j) : Subject to We Could not use the Shen’s Method because of the additional constraint. How can we solve this sub-problem?

20. 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. Solution Method- Sub-Problem – Two Approaches " Î = - i I , r 0 , , m 1 K

21. Computational Result

22. The Problems That We Should Solve Prove Exact Value  App2 Improve algorithm run times Different q for each facility.

23. Questions? Thank you