Simulated Annealing Algorithm for Customer-Centric Location Routing Problem

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1 Simulated Annealing Algorithm for Customer-Centric Location Routing Problem May 22, 2018 Eugene Sohn Advisor: Mohammad Moshref-Javadi, PhD 1

2 Agenda Why this research? What is this research? Methodology Mathematical Model & Computational Results Heuristic Algorithm & Computational Results Conclusion 2

3 Why This Research? E-Commerce Market Growing Rapidly Sales grew from $360b to $409b from 2016 to 2017 (14% growth) Expected sales in 2021 to be $603b (YoY 8.5% growth) Change in Consumer s Expectation on Deliveries and Shipping A quarter of consumers would pay a premium for same-day delivery 3

4 Why This Research? (cont.) Growing Market + Change in Consumer Expectation = Opportunities for Disruption and Market Share Gain By providing faster delivery for better customer shopping experience 4

5 What is This Research? Capacitated Latency Location Routing Problem: Location Routing Problem Determines location of depots, allocation of vehicles, and routes of the vehicles concurrently Facility Location Problem (NP-Hard) Vehicle Routing Problem (NP-Hard) Capacitated Latency Minimize Customer Waiting Time vs Cost Capacity Constraints on Vehicles and Warehouses 5

6 Capacitated Latency Location Routing Problem 3 7 D D1 4 D2 opened depot D3 Example of a location routing problem unopened depot customer # 6

7 Methodology Two Ways to Solve CLLRP Mathematical Model Heuristic Algorithm Mathematical Model Always provides the best solution Computationally inefficient Heuristic Algorithm Provides good enough solution Computes very quickly 7

8 Mathematical Model Assumptions Number and locations of candidate depots are known Number of depots to open and vehicles to use are pre-determined Capacities of depots and vehicles are pre-determined All the demands are satisfied Travel time between customer i and j are symmetric 8

9 Mathematical Model (cont.) Indices i,j,u k g Represent customers, totally NN cc customers Represents vehicle Represents candidate depots, totally NN ff Sets K Set of vehicles, KK G Set of candidate depots, GG = NN ff V Set of customers, VVV = NN cc V Set of all customers and candidate depots VV = NN = NN cc + NN ff 9

10 Mathematical Model (cont.) 10 Parameters NN vv Wg qq jj QQ kk cc iiii NN gg M Number of vehicles Capacity of depot g Demand quantity at customer j Capacity of vehicle k Travel time between nodes i and j Number of facilities to open Large positive constant Variables kk tt ii Arrival time of vehicle k at customer i kk xx iiii 1 if vehicle k traverses arc (i,j) from customer i to customer j ; otherwise, 0 ff gggg 1 if customer i is supplied from depot g; otherwise, 0 zz gg 1, if facility g is open; otherwise 0

11 Mathematical Model (cont.) Minimize: kk KK,ii VVV tt ii kk (1) s.t. ff gggg qq ii WW gg gg GG (2) ii VVV xx kk iiii = xx kk jjjj ii VV, kk jj VV KK jj VV (3) 11

12 Mathematical Model (cont.) kk KK,jj VV,ii jj gg GG xx iiii kk = 1 ii VVV ff gggg = 1 jj VVV xx kk iiii qq jj QQ kk kk KK ii VV,jj VV xx kk gggg = 1 kk KK gg GG,ii VV (4) (5) (6) (7) xx kk gggg + uu VV uu VV\ ii GG xx kk uuuu 1 + ff gggg ii VV, kk KK, gg (8) 12

13 Mathematical Model (cont.) tt kk ii + cc iiii 1 xx kk iiii MM tt kk jj, ii VV, jj VV, ii jj, kk KK, gg GG (9) ff gggg MMzz gg gg GG (10) ii VVV ff gggg zz gg gg GG (11) ii VVV zz gg = NN gg gg GG tt kk ii 0, ii VV, kk KK zz gg 0,1 gg GG xx kk iiii 0,1 ii jj (12) (13) 13

14 Computational Results Mathematical Model 800 CUSTOMER WAITING TIME FOR DIFFERENT NUMBER OF DEPOTS AND VEHICLES D4V4 D3V4 D3V3 D2V4 D2V3 D2V2 700 TOTAL WAITING TIME (SECONDS) # OF CUSTOMERS 14

15 Computational Results Mathematical Model 6000 COMPUTING TIME FOR DIFFERENT NUMBER OF DEPOTS AND VEHICLES D4V4 D3V4 D3V3 D2V3 D2V4 D2V2 COMPUTING TIME (SECONDS) # OF CUSTOMERS 15

16 Heuristic Algorithm Simulated Annealing Inspired by annealing process Accepts worse solutions initially in order to leave the local optimum 16

17 Simulated Annealing Parameters Initial Temperature, TT 00 Final Temperature, TT ff Cooling Rate, α Boltzmann Constant, K ee ΔΔ/(KKTT) = number between 0 and 1. 17

18 Simulated Annealing (cont.) Initial Solution Nearest Neighborhood algorithm with probabilistic centrality Operators Local operators: local insertion, local swap, flip Non-local operators: non-local insertion, non-local swap 18

19 Simulated Annealing (cont.) Local Operator Example Flip Vehicle 1 Depot 1 C1 C2 C3 C4 C5 C6 C7 C8 - Vehicle 1 Depot 1 C1 C2 C3 C7 C6 C5 C4 C8-19

20 Simulated Annealing (cont.) Non-Local Operator Example Non-local Swap Vehicle 1 Depot 1 C1 C2 C3 C4 - Vehicle 2 Depot 2 C5 C6 C7 C8 - Vehicle 1 Depot 1 C1 C2 C3 C7 - Vehicle 2 Depot 2 C5 C6 C4 C8-20

21 Simulated Annealing (cont.) Three Different Types of Simulated Annealing Adaptive Simulated Annealing (SA1) Uses all local operators within each prime operator and give preference to better performing prime operators Sequential Simulated Annealing (SA2) Uses all five operators in a row Iterative Simulated Annealing (SA3) Uses all local operators within each non-local operator 21

22 Computational Results Simulated Annealing Prins et al. Benchmark Test Problem Vehicle Depot Open LLRP SA1 Average instance Customers Capacity Capacity Depots Vehicles Depots Best Best Average Time b / b / / , b / , , , b , BIS , bBIS , / / , b / , , ,

23 Computational Results Iterative Simulated Annealing vs Mathematical Model Problem Vehicle Depot Open Math Math Time SA3 SA3 Time Instance Customers Capacity Capacity Depots Vehicles Depots Best (seconds) Best Average (seconds)

24 Conclusion Minimize total customer waiting time instead of cost Heuristic algorithm is necessary to solve large-sized problems All three simulated annealing algorithms perform competitively with the algorithms in the literature and the mathematical model 24

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