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

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)

4 Models Notation –f j = fixed cost to construct a facility at location j  J –h i = demand per period for customer i  I –d ij = 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) –X j = 1 : if a facility is opened at location j 0 : otherwise –Y ijr = 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 X j = 1, that is, open facility j. –Set Y ijr = 1, if facility j is open < 0 r minimizes for 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 X j 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: b i  0}, I 0 ={i: b i < 0 and c i =0} and I - ={i: b i 0} –Step 2 : Sort the element of I - so that b 1 /c 1  b 2 /c 2  …  b n /c n –Step 3 : Compute the partial sums –Step 4 : Select m that minimize S m

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 By Simulation Proved (By Jensen’s Inequality)

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 1,,0,  mrIi 

21 Computational Result Number of Locations 2549 Used MethodApproach 1Approach 2Approach 1Approach 2 Optimality Gap(%) Iteration Time67 sec1.2hr2.17hr8.6hr

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

23 Questions? Thank you