Biobjective Inventory Routing Problem

Transcription

1 Biobjective Inventory Routing Problem problem solution and decision support Martin Josef Geiger 1 Marc Sevaux 1,2 m.j.geiger@hsu-hh.de marc.sevaux@univ-ubs.fr 1 Helmut Schmidt University Logistic Management Dept. Hamburg, Germany 2 University of South-Brittany Lab-STICC, CNRS Lorient, France INOC Hamburg, Germany June /27

2 The Inventory Routing Problem () One depot A set of customers 2/27

3 The Inventory Routing Problem () One depot A set of customers A time horizon 2/27

4 The Inventory Routing Problem () One depot A set of customers A time horizon A known demand for each customer and each period 2/27

5 The Inventory Routing Problem () One depot A set of customers A time horizon A known demand for each customer and each period Minimize Inventory cost 2/27

6 The Inventory Routing Problem () One depot A set of customers A time horizon A known demand for each customer and each period Minimize Inventory cost Minimize Routing cost 2/27

7 The Inventory Routing Problem () One depot A set of customers A time horizon A known demand for each customer and each period Minimize Inventory cost Minimize Routing cost is a real bi-objective optimization problem 2/27

8 Important decisions For solving the, we must make the following decisions: Decisions 1 When deliver customers? 2 How much deliver? 3 With which routes? All these decisions are linked together: increase delivery quantities change routes or frequency change frequency adapt delivery quantities... 3/27

9 Important references First papers [Bell et al., 1983] Improving the distribution of industrial gases with and on-line computerized routing and scheduling optimizer Interfaces 13(6):4 23, [Federgruen and Zipkin, 1984] A combined vehicle routing and inventory allocation problem Operations Research 32(5): , 1984 Recent surveys [Bertazzi et al., 2008] Inventory routing in The Vehicle Routing Problem: Latest Advances and New Challenges 49 72, [Cordeau et al., 2011] Short-haul routing in Handbooks in Operations Research and Management Science: Transportation. To appear. 4/27

10 variants There exists a large number of variants planning horizon (finite/infinite) inventory costs and capacities production/demand rates (of single/multiple product) specific restrictions/regulations deterministic/stochastic demand/production initial inventory fleet (homogeneous/heterogeneous) And different objectives usually a combination of inventory level and routing cost 5/27

11 Practical What does mean Practical? Main goal: helping companies where is important Propose a simple output Use simple rules (that can be understood) Use simple implementation that can be reproduced Give different alternatives We want to help the decision maker! 6/27

12 A major assumption in our work To follow a common strategy in companies dealing with We have separated the decisions Determine quantities for each period Compute best routing for each period 7/27

13 The choice of policies Standard policies: DD Day-to-day delivery policy If not enough in stock, deliver the missing demand OU Order-up-to level policy When you ship, ship the maximum (customer capacity) ML Maximum level policy (bad name!) Any quantity less than the maximum level (but which quantity?) 8/27

14 The choice of policies Standard policies: DD Day-to-day delivery policy If not enough in stock, deliver the missing demand OU Order-up-to level policy When you ship, ship the maximum (customer capacity) ML Maximum level policy (bad name!) Any quantity less than the maximum level (but which quantity?) Assumptions Serve only if current inventory is not enough If served, only full number of period demands 8/27

15 The frequency policy encoding Solutions are modeled as a frequency f of the deliveries for each customer f = 1 DD policy f = 2 serve for the next two consecutive periods... f = k serve for the next k consecutive periods... f = + OU policy The policies are simple to understand (even for companies) and easy to encode for us (vector of integer) 9/27

16 Evaluation of solutions Each solution is measured with the two criteria Inventory cost Sum of all inventory levels at customers at the end of each period. This can be computed in O(np) Routing cost Sum of all distances run by the trucks at every period. Solve a VRP for each period. This is a NP-hard problem!!! 10/27

17 Evaluation of solutions Each solution is measured with the two criteria Inventory cost Sum of all inventory levels at customers at the end of each period. This can be computed in O(np) Routing cost Sum of all distances run by the trucks at every period. Solve a VRP for each period. This is a NP-hard problem!!! Routing engines Clarke & Wright + savings heuristic Record-to-record travel algorithm 10/27

18 The Inventory Routing Solver 11/27

19 The Inventory Routing Solver (alternatives) 11/27

20 The Inventory Routing Solver (Cost details) 11/27

21 The Inventory Routing Solver (Inventory) 11/27

22 The Inventory Routing Solver (Periods) 11/27

23 The Inventory Routing Solver 11/27

24 A new set of instances A new set of instances has been generated inspired from Christofides, Mingozzi and Toth VRP instances Parameters Possible values # customers, n {50,75,100,120,150,200} Horizon, p {240} Truck capacity, K {140, 160, 200} Demand {Constant, Increasing, Sinus} This form a new set of 42 instances available at 12/27

25 Initial solutions We have implemented several initial solutions 13/27

26 Initial solutions We have implemented several initial solutions Identical frequency: f is the same for each customer Routing Inventory 13/27

27 13/27 Initial solutions We have implemented several initial solutions Identical frequency: f is the same for each customer Totally random: f is chosen at random for each customer Routing Inventory

28 13/27 Initial solutions We have implemented several initial solutions Identical frequency: f is the same for each customer Totally random: f is chosen at random for each customer Controlled random: f is chosen at random between two bounds Routing Inventory

29 Neighborhood Encoding is a vector of integers Many moves are possibles but think of the neighborhood size Initial solution s /27

30 Neighborhood Encoding is a vector of integers Many moves are possibles but think of the neighborhood size Initial solution s Move 1: Change frequency by ±1 (here +1) s /27

31 Neighborhood Encoding is a vector of integers Many moves are possibles but think of the neighborhood size Initial solution s Move 1: Change frequency by ±1 (here +1) s Move 2: Change frequency by ±k (here 3) s /27

32 General algorithm Algorithm 1: MOEA- Initialization: create an initial population Identical frequency + controlled random Cleanup: remove dominated solutions repeat Improve all solutions with a local search steepest descent algorithm with Move 1 Rebuild archive with non-dominated solutions only until no more improvements Other possible options NSGA-II or SPEA Path Relinking with elite population 15/27

33 Problem: the gap effect Output for an instance with 50 customers Inventory-Routing alternatives Totally random frq. Controlled random frq. Identical frq. Total routing cost Total inventory cost 16/27

34 Problem: the gap effect 16/27

35 Problem: the gap effect (cont d) Visualizing the Pareto front, we noticed important gaps how to avoid them? how to fill them? 17/27

36 Problem: the gap effect (cont d) Visualizing the Pareto front, we noticed important gaps how to avoid them? how to fill them? 17/27

37 Problem: the gap effect (cont d) Visualizing the Pareto front, we noticed important gaps how to avoid them? how to fill them? Potential solutions: the more controlled random strategy the controlled local search strategy 17/27

38 Problem: the swing effect 18/27

39 Problem: the swing effect 18/27

40 Problem: the swing effect 18/27

41 Problem: the swing effect 18/27

42 Problem: the swing effect 18/27

43 Problem: the swing effect (cont d) In the -solver, we noticed repeating identical patterns Customers are groups together because they run out of goods at the same time No cleaver grouping strategy Customers may be served together even if they are far away from each others Potential solutions: the double frequency encoding the binary encoding 19/27

44 Too many solutions are generated Output for an instance with 50 customers (2485 alternatives) Inventory-Routing alternatives Initial Pop. Final Pop. Total routing cost Total inventory cost 20/27

45 Too many solutions are generated 20/27

46 How to handle this number of solutions This approach generates too many solutions thousands of solutions, how to handle them? Implemented solution: the reference set strategy 21/27

47 How to handle this number of solutions This approach generates too many solutions thousands of solutions, how to handle them? Implemented solution: the reference set strategy (here with 7 reference points) 21/27

48 How to handle this number of solutions This approach generates too many solutions thousands of solutions, how to handle them? Implemented solution: the reference set strategy (here with 7 reference points) 21/27

49 Evolution of the search Inventory-Routing alternatives Total routing cost Zooming section Total inventory cost 22/27

50 Evolution of the search Search direction on the zoomed section Direction Path of the search Total routing cost Total inventory cost 22/27

51 Changing the number of reference points Even if we change this number, the method is still robust Inventory-Routing alternatives RP=3 RP=5 RP=11 Total routing cost Total inventory cost 23/27

52 Forthcoming improvements Already in the box The double frequency encoding solves the swing effect Path Relinking might solves the gap problem Other improvements should be considered improve the routing software find new operators (dedicated or combined) but be careful with the time for exploration 24/27

53 Improved routing Output for an instance with 50 customers Inventory-Routing alternatives Improved controlled frq. Improved identical frq. Former Pareto front New Pareto front Total routing cost Total inventory cost 25/27

54 Improved routing ost /27

55 Improved routing 25/27

56 Give the hand to the Decision Maker Add interaction in the select a solution and manually adapt it 26/27

57 Give the hand to the Decision Maker Add interaction in the select a solution and manually adapt it Integrate our model in a more general framework for the On-line Inventory Routing Problem Company have forecasts (predictive demands) At period k, estimation of demand for next 5, 20, 60 days A solution can be computer for a short horizon Implemented but revised at every period 26/27

58 Visit our web site... or the project page Contact: 27/27