Facility Location-Routing-Scheduling Problem: LRS

Transcription

1 Facility Location-Routing-Scheduling Problem: LRS Zeliha Akca Lehigh University Rosemary Berger Lehigh University ➀ Definition of LRS ➁ Branch and Price Algorithm ➂ Mater and Pricing Problem ➃ Solution of Pricing Problem ➄ What We Have Done ➅ What We Will Do June 15,

2 LRS Location Routing and Scheduling Problem: 3 dependent problems: ➀ locate facilities ➁ construct routes for vehicles ➂ assign routes to vehicles capacitated facilities capacitated vehicles time restriction for the vehicles RS 2

3 Location-Routing and Scheduling Problem: In literature, heuristic solution for LRS problem (no IP formulation) Exact solutions for RS and LR We choose, Branch and Price Algorithm: IP formulation includes many constraints (s.t. sub tour elimination constraints) Can be written in set partitioning problem easily Easy to think routes in terms of columns With set partitioning formulation, many possible columns Other methods to solve: Lagrangian Relaxation Branch and bound and cut Heuristic design? RS 3

4 Problems in Literature Facility Location -too many Vehicle Routing -too many Routing Scheduling Location Routing Location Routing and Scheduling routing: Location one-to-one relation btw routes and vehicles. Location scheduling: not necessarily one-to-one relation assignment of one vehicle to many paths. RS 4

5 Objective: Minimize total cost. IP FORMULATION Constraints: TotalCost Fixed cost of Facility and Vehicle Operating cost of Vehicles ➀ Each demand node should be served once ➁ # of a vehicle entering a node must be equal to # of the vehicle leaves this node ➂ Capacity restriction for facility ➃ Capacity restriction for vehicles ➄ Flow balance equations (to satisfy demand and eliminate the subtours) ➅ Time restriction to the routes FORMULATION 5

6 Set Partitioning Model: ALTERNATE FORMULATION Pairing Concept: Set of routes assigned to a vehicle and can be served within the given time limit. PAIR Customer facility LTERNATE FORMULATION

7 #! $%=!$:set of customers; &!! Set Partitioning Model: ALTERNATE FORMULATION Variables for set partitioning based on pairing concept: 1 if pairing "otherwise is chosen for facility,, :set of feasible pairs of facility j 1 if facility j is open, "otherwise :set of facilities; LTERNATE FORMULATION 7

8 5 of " A 7 # /5 # 7 4 / =1 if node? # = )8 ) ;: 5 <( + >= + )* ) ' )* )* )3! (' #,+-.' / Set Partitioning Model: (1) s.t. $(2),!,!,!(3) (4) (5) 'is in pairing. LTERNATE FORMULATION 8

9 BRANCH AND PRICE ALGORITHM Restricted Master Prblem (RMP) Initial Columns Solve LP LP LOWER BD IP UPPER BD. Dual Variables info Until no new column new columns Pricing Problem Find columns with ( ) reduced cost ROOT NODE If LP is not integral Branching Modified RPM Cut is added Solve LP modified Pricing Problem RANCH AND PRICE ALGORITHM 9

10 Restricted Master Problem: Initial pairs are formed pair pair 3 Y13 Y12 2 pair Y14 4 pair Y11 fac pair Y21 fac.2 pair Y22 Each pair represent a column in set partitioning formulation Restricted-since includes set of columns, not all columns RANCH AND PRICE ALGORITHM 10

11 B Pricing problem: Create pair : a column for If 3rd const changed to: )3 4 5 = / / F B D E C ' #!and $() We have:,, dual variables Reduced Cost for HG (7) / )8 )8 )8 D 5 C 5 5 I ;: 5 <( Operating cost of the vehicle ( Jtravel time) + Fixed Cost of a vehicle Independent pricing problem for each facility RANCH AND PRICE ALGORITHM 11

12 K ELEMENTARY SHOR THEST PATH WITH RESOURCE CONSTRAINT Pricing Problem = ESPRC If: Set up a network, including all customers and a source and sink nodes Arc costs: LMON QSRT LM\[ e ^ f em (8) d P UVW ] M^ ba Z cy _ `R XY Z find minimum cost path to the sink in our case allow visits more than once to sink If Total cost of path + Vehicle fixed Cost gih, add the column to restricted master problem stop when the shorthest path does not give negative cost column LEMENTARY SHORTHEST PATH WITH RESOURCE CONSTRAINT 12

13 j 7 ELEMENTARY SHOR THEST PATH WITH RESOURCE CONSTRAINT What is an elementary path? Each node can be visited at most once. Why elementary instead of walks? Trade of between more difficult pricing problem and more depth in branch and bound tree In our case: # of visits to sink In each visit to sink, current truck load is set to zero Adapt the Labelling Algorithm for ESPRC by Feillet, Dejax, Gendreau, Geuguen. LEMENTARY SHORTHEST PATH WITH RESOURCE CONSTRAINT 13

14 ESPRC Problem: too many feasible paths Keep resource consumptions, visited nodes, and cost Keep unreachable nodes for each label A node may be unreachable from other if not enough resource or is already visited. Eliminate dominated labels with respect to resource consumption and unreachable nodes. SPRC 14

15 CURRENTLY What we have done Design Master Problem and Pricing Problem Adapted ESPRC algorithm to solve Pricing Problem Do the column generation Solve the root node URRENTLY 15

16 MINTO MINTO:Mixed INTeger Optimizer MINTO uses LP solver and do branch and bound algorithm MINTO can do many applications such as preprocessing, constraint generation, primal heurisitcs MINTO allows user to write own algorithm (for column generation, constraint generation, heuristics,..) specific to the problem Prof. Linderoth supports MINTO in our University WHAT ELSE? COIN-BCP:(Common Optimization INterface) and SYMPHONY Open source allows parallellization in branch and bound tree supported by Prof. Ralphs HAT ELSE? 1

17 NEXT What we will do More implementation Test problems Determine the right number of columns to be generated in each time Different LP algorithms, to find better reduced costs See how well the root node solution Create column pool Branching strategies parallelization Alternate solution: 2-sub problems approach IP formulation Focus on the pricing problem EXT 17

18 o n v p q rw u o n k Q p q t e q x e q m u m 2-SUB PROBLEM APPROACH Master problem includes 3 set of variables: Location variables 1 if pairing is chosen for facility, Qsr, p r el N hotherwise 1 if path wis chosen for facility,, p r el N hotherwise 2 nested sub problems: generating paths,and combining these paths as pairs. SP1: Generating paths: vehicle routing problem or elementary shortest path with 2 resources SP2: Combining paths: knapsack problem -SUB PROBLEM APPROACH 18

19 2-SUB PROBLEM ALGORITHM Solve LP RMP Get dual variables for each customer and facility Solve SP1: New Paths with ( ) cost Form pairs heuristically add to MP With new paths and old paths Solve Master Problem Solve SP2: generate pairs with ( ) cost Get dual variables -SUB PROBLEM ALGORITHM 19

20 THANKS... Any Questions? HANKS... 20