ANT/OR. Risk-constrained Cash-in-Transit Vehicle Routing Problem. Luca Talarico, Kenneth Sörensen, Johan Springael. 19 June 2012

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1 University of Antwerp Operations Research Group ANT/OR Risk-constrained Cash-in-Transit Vehicle Routing Problem Luca Talarico, Kenneth Sörensen, Johan Springael VEROLOG 212, Bologna, Italy 19 June 212

2 Summary Introduction Risk Constraint Problem Formulation Solution Strategies Strategies Description Metaheuristic Components Experimental Results Conclusions Future works 2/39

3 Introduction Context = Cash and valuables in transportation sector Risk = Crime is a real challenge In the U.K. alone, there is an estimated 5 billion transported each year ( 1.4 billion per day); In 28, there were 1, documented attacks against CIT couriers in the UK; With the downturn in the global economy, CIT crime is on the rise; Money stolen in CIT attacks is a major source of funding for organized crime. 3/39

4 Risk Constraint RISK Probability that the = Exposure of the vehicle unwanted event occurs outside of the depot x x Consequences of the = Loss of the money unwanted event or valuables transported 4/39

5 Risk Constraint RISK Probability that the = Exposure of the vehicle unwanted event occurs outside of the depot x x Consequences of the = Loss of the money unwanted event or valuables transported Risk threshold 4/39

6 Example Risk Constraint 5 2 C B 4 D Risk Threshold R=15 km $ Current Values 1 A 1 5 E 1 CC=1 $ TT=1 km RE= 1 = km $ F CC=cash carried TT=travel time RE=risk exposure 5/39

7 Example Risk Constraint C B 4 D Risk Threshold R=15 km $ Current Values 1 A 1 5 E 1 CC=3 $ TT=18 km RE=1 8 = 8 km $ F CC=cash carried TT=travel time RE=risk exposure 6/39

8 Example Risk Constraint C B 4 D Risk Threshold R=15 km $ Current Values 1 5 CC=8 $ A 1 E 1 TT=28 km RE= = 38 km $ F CC=cash carried TT=travel time RE=risk exposure 7/39

9 Example Risk Constraint C B 13 4 D Risk Threshold R=15 km $ Current Values 1 5 CC=8 $ A 1 E 1 TT=41 km RE= = 142 km $ F CC=cash carried TT=travel time RE=risk exposure 8/39

10 Example Risk Constraint C B 13 4 D Risk Threshold R=15 km $ Current Values 1 A E 1 CC=4 $ TT=12 km RE= 12 = km $ F CC=cash carried TT=travel time RE=risk exposure 9/39

11 Example Risk Constraint C B A 1 CC=cash carried TT=travel time RE=risk exposure 4 D 7 5 E 1 F Risk Threshold R=15 km $ Current Values CC=9 $ TT=19 km RE=4 7 = 28 km $ 1/39

12 Example Risk Constraint C B A 1 13 CC=cash carried TT=travel time RE=risk exposure 4 D 7 5 E 1 F Risk Threshold R=15 km $ Current Values CC=9 $ TT=32 km RE= = 145 km $ 11/39

Example Risk Constraint 5 2 1 C B 8 13 1 12 A 1 12 15 CC=cash carried TT=travel time RE=risk

13 Example Risk Constraint C B A CC=cash carried TT=travel time RE=risk exposure 4 D 7 5 E 1 F Risk Threshold R=15 km $ Current Values CC=1 $ TT=15 km RE= 15 = km $ 12/39

Example Risk Constraint 5 2 1 C B 8 13 1 12 A 1 12 15 15 CC=cash carried TT=travel time RE=risk

14 Example Risk Constraint C B A CC=cash carried TT=travel time RE=risk exposure 4 D 7 5 E 1 F Risk Threshold R=15 km $ Current Values CC=1 $ TT=3 km RE=1 15 = 15 km $ 13/39

15 RCTVRP Problem Risk-constrained Cash-in-Transit Vehicle Routing Problem Determine a set of routes, where a route is a tour that begins at the depot, traverses a subset of the customers in a specified sequence and returns to the depot. Each customer must be assigned to exactly one of the routes. The total risk for each route must not exceed the risk threshold. The routes should be chosen to minimize total travel cost. [Talarico, Sörensen and Springael in 212] 14/39

16 RCTVRP Problem Risk-constrained Cash-in-Transit Vehicle Routing Problem Determine a set of routes, where a route is a tour that begins at the depot, traverses a subset of the customers in a specified sequence and returns to the depot. Each customer must be assigned to exactly one of the routes. The total risk for each route must not exceed the risk threshold. The routes should be chosen to minimize total travel cost. [Talarico, Sörensen and Springael in 212] No single tour as a consequence of the risk threshold! 14/39

17 Problem Formulation A necessary and sufficient condition on the RCTVRP feasibility is: T max i N {p i t i } The condition guarantees that at least the problem has a feasible solution containing n routes. p3 3 p4 p2 4 p5 p pn pi n i 15/39

18 Time (s.) Exact Approach Small instances of the RCTVRP have been solved to optimality using CPLEX The time needed to solve small istances of the problem increases exponentially; We were unable to solve problems with more than 1 nodes in a reasonable time (Instances with 16 nodes solved in 15 min) Nodes Time Expon. (Time) We need efficient Metaheuristics! 16/39

19 Solving Strategies Seven Metaheuristics have been developped and coded in JAVA combining the following components: 4 Constructive Heuristics: Modified Clarke & Wright with Grasp (Stochastic); Nearest Neigborhood with Grasp (Stochastic); TSP Nearest Neigborhood with Grasp plus Splitting (Stochastic); TSP Lin Kernighan plus Splitting (Deterministic). 17/39

20 Solving Strategies Seven Metaheuristics have been developped and coded in JAVA combining the following components: 4 Constructive Heuristics: Modified Clarke & Wright with Grasp (Stochastic); Nearest Neigborhood with Grasp (Stochastic); TSP Nearest Neigborhood with Grasp plus Splitting (Stochastic); TSP Lin Kernighan plus Splitting (Deterministic). 1 Local Search Block (6 LS Operators); 17/39

21 Solving Strategies Seven Metaheuristics have been developped and coded in JAVA combining the following components: 4 Constructive Heuristics: Modified Clarke & Wright with Grasp (Stochastic); Nearest Neigborhood with Grasp (Stochastic); TSP Nearest Neigborhood with Grasp plus Splitting (Stochastic); TSP Lin Kernighan plus Splitting (Deterministic). 1 Local Search Block (6 LS Operators); 2 Structures: Multi-Start; Perturbation. 17/39

22 Metaheuristics Overview MSMC&WG MC&WGP start Initialize Meta- Heuristic start Initialize Meta- Heuristic no Find an IS using MC&WG Find a CS using MC&WG Improve the CS using LSO Perturb the CS Improve the CS using LSO no Exit yes Is the # of restart reached? no Is CS better than BS? Exit yes Is the # of restart reached? no Is CS better than BS? yes yes Update the BS Update the BS 18/39

23 Metaheuristics Overview MSNNG NNGP start Initialize Meta- Heuristic start Initialize Meta- Heuristic no Find an IS using NNG Find a CS using NNG Improve the CS using LSO Perturb the CS Improve the CS using LSO no Exit yes Is the # of restart reached? no Is CS better than BS? Exit yes Is the # of restart reached? no Is CS better than BS? yes yes Update the BS Update the BS 19/39

24 Metaheuristics Overview MSTSPNNGS TSPNNGSP start Initialize Meta- Heuristic start Initialize Meta- Heuristic no Solve the beneath TSP using NNG Solve the beneath TSP using NNG Find a CS splitting the Big Tour Find a CS splitting the Big Tour Improve the CS using LSO Perturb the CS Improve the CS using LSO no Exit yes Is the # of restart reached? no Is CS better than BS? Exit yes Is the # of restart reached? no Is CS better than BS? yes yes Update the BS Update the BS 2/39

25 Metaheuristics Overview TSPLKSP start Initialize Meta- Heuristic Solve the beneath TSP using LK Find a CS splitting the Big Tour Perturb the CS Improve the CS using LSO no Exit yes Is the # of restart reached? no Is CS better than BS? yes Update the BS 21/39

26 Modified Clarke & Wright Heuristic i j Route R1 i R1 R2 j Route R2 R2 R1 i j 22/39

27 Local Search Block Six different Local Search Operators have been used: 2 Intra-Route LSO; 4 Inter-Route LSO. 23/39

28 Intra-Route Local Search Operators Two-opt Or-opt p A A p C C p A A p C C p A A p D D p A A p D D p C p B p C p B D p D B p B D p D B p B p F F C B p E E p F F C B p E E 24/39

29 Inter-Route Search Operators Two-opt A pa D pd B pb C pc A pa D pd B pb C pc Relocate A pa B pb C pc D pe E pe A pa B pb C pc D pd E pe 25/39

30 Inter-Route Local Search Operators Exchange A pa B pb C pc F pf E pe D pd A pa B pb C pc F pf E pe D pd Cross Exchange A pa B pb C pc D pd E pe F pf G pg H ph A pa B pb C pc D pd E pe F pf G pg H ph 26/39

31 Instances Description Since RCTVRP has not been studied before, no test instances are available in the literature. 55 Instance have been used to test the 7 Metaheuristics: 11 Basic Instances taken from VRP lib; 5 diffferent Risk Constraint Levels. Name Nodes E22-4g 22 E26-8m 26 E3-3g 3 E36-11h 36 E45-4f 45 E51-5e 51 E72-4f 72 E11-8e 11 E121-7c 121 E135-7f 135 E151-12b 151 Risk Level Value RL1 T = max i N {p i t i } RL2 1.5 RL1 RL3 2. RL1 RL4 2.5 RL1 RL5 3. RL1 27/39

32 Metaheuristic Parameters Two different kind of parameters have been analyzed: Instance characteristics; Metaheuristics parameters; Parameter Values Nr. of Levels Nodes 22, 26, 3, 36, 45, 51, 72, 11, 121, 135, Risk Level 1, 1.5, 2, 2.5, 3 5 Restart 1, 2, 3, 4, 5 5 TSP-Repetition 1, 2, 3, 4 4 Perturbation %, 5%, 1%, 15%, 2%, 25% 6 Or-Opt-Internal on, off 2 Two-opt-Internal on, off 2 Exchange on, off 2 Relocate on, off 2 Cross-Exchange on, off 2 Two-opt-External on, off 2 28/39

33 Time (s) Computational Results Computational Time Instances with RL Nodes MSNNG MSMC&WG MC&WGP NNGP MSTSPNNGS TSPNNGSP TSPLKSP 29/39

34 Computational Results 3/39

35 Computational Results 31/39

36 Computational Results 32/39

37 Computational Results 33/39

38 Conclusions The main contributions of this works are: Introduction and mathematical formulation of a new Risk index; Introduction of a new unexplored variant of the vehicle routing problem named RCTVRP; Mathematical formulation of the RCTVRP; Exact approach to solve small instances of the problem; 7 metaheuristic approaches to solve the RCTVRP in a reasonable time. 34/39

39 Future research Completing the statistical analysis of the results obtained Handling the risk as an additional objective to be minimized Finding some real applications of the problem involving CIT companies Several extensions of the problem can be proposed for future research taking in consideration some real-life constraints as: route length restrictions; time windows; precedence relations. 35/39

40 Thank you! Questions? 36/39

41 Mathematical Formulation 1/3 s.t. m n n min t ij xij r r= i= j= n xj r = n xi r r =,..., m (1) j= i= n xj = 1 (2) j= n xi r n x r+1 j r =,, m 1 (3) i= j= m n n n n 1 + x r ij n r= i= j= h=1 n n xhj r xjk r = h= k= x r+1 h r =,, m 1 (4) j =,, n r =,, m (5) 37/39

42 Mathematical Formulation 2/3 P r = r =,, m (6a) ( ) Pi r + p j 1 xij r HP Pj r P r i j= i =,, n j = 1,, n r =,, m (6b) n n xij r p i i =,, n (6c) i= R r j R r j ( ) P r t j 1 xj r HC ( ) Ri r + Pi r t ij 1 xij r HC n Ri r T xij r j= j = 1,, n r =,, m i =,, n j = 1,, n r =,, m i =,, n r =,, m (7a) (7b) (7c) 38/39

43 Mathematical Formulation 3/3 n ui r uj r + n xij r n xl r l=1 i =,, n j = 1,, n i j r =,, m (8a) u r = r =,, m (8b) n ui r (n 1) x r i = 1,, n ij (8c) r =,, m j= u r ij {, 1} i =,, n j = 1,, n r =,, m (8d) x r ij {, 1} i =,, n j = 1,, n r =,, m (9) 39/39

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