1. A Hybrid Approach for the Inventory Routing Problem with Stochastic Demands José Cáceres Cruz Angel A. Juan {jcaceresc, ajuanp}@uoc.edu Department of Computer Science IN3 - Open University of Catalonia, Barcelona, SPAIN http://dpcs.uoc.edu | http://ajuanp.wordpress.com Barcelona, Spain June 14, 2012 Tolga Bektas t.bektas@soton.ac.uk Management School University of Southampton, UK Scott Grasman segeie@rit.edu Industrial & Systems Engineering Department Rochester Institute of Technology, USA

2. 0. Agenda 1.Introduction: IRPSD. 2.Observations. 3.Solving CVRP: SR-GCWS-CS. 4.Our Approach for the IRPSD. 5.Practical Example. 6.Advantages of Our Approach. 7.Computational Results. 8.Conclusions and Future Work 1 2012 IN3-HAROSA

3. 1. Introduction: The IRPSD  The Inventory Routing Problem (IRP):  Inventory Management  Vehicle Routing  For Retailers Centers: liquid air products, oil, gas, chemicals, etc. Campbell et al (2002).  Increase use of ICTs for transfer of information on real time. 2 2012 IN3-HAROSA

4. 1. Introduction: Costs of Logistic Activities * Differentiation for Performance: Excellence in Logistics (2004), ELA/AT Kearney 3 2012 IN3-HAROSA

5. 1. Introduction: The IRPSD  The IRP is a well-known NP-hard problem:  A set of customers’ demands must be supplied by a fleet of vehicles over a given planning horizon.  Resources are available from a depot.  Moving a vehicle from one node i to another j has associated costs c(i, j)  Customers consume the product at a rate, and can maintain an inventory of the product up to a level.  Several constraints must be considered: maximum load capacity per vehicle, service times, etc.  The IRP with Stochastic Demands (IRPSD) includes random demands (statistical distributions) considering stock levels. Depot (resources) Customers (demand) edge in a route InventoryRouting 4 2012 IN3-HAROSA

6. 2. Observations: Decision Making  how much inventory and when to ship to a retailer:  Retailer Centers itself.  Vendors (Vendor Managed Inventory - VMI) Kleywegt et al (2004). Depot (resources) Customers (demand) edge in a route InventoryRouting 5 2012 IN3-HAROSA

7. 2. Observations: Stock-Outs  Important observations:  The random behavior of Retailer Centers’ demands could cause an expected feasible solution to become an infeasible one if its final demand is not enough for satisfying their clients  route failure / stock-out.  Corrective actions must be introduced to deal with stock-outs. These actions (e.g. a vehicle reload at the depot) will increase the total inventory costs.  Goal: to construct solutions with a given probability of suffering stock-outs.  How?: to construct routes in which the associated expected demand (refill policy) will be somewhat adapted to the current level of stock.  But… be careful: a trade-off exists between inventory costs formula and routing costs minimization Stock-Outs  corrective/preventive policies  increase in inventory costs Stock-Outs  corrective/preventive policies  increase in inventory costs E[Total Costs] = Total Routing Costs + E[Inventory Costs] 6 2012 IN3-HAROSA

8. 2. Observations: Refill Policies Stock  Optimize over a Single-period or over a Multi-Period.  Consider different refill policies:  P1: Full-Refill  P2: ¾-Refill  P3: Half (1/2)-Refill  P4: ¼-Refill  P5: No-Refill  But, which is the best policy for each node separately?  Challenge: Creation of Test Instances with real nature! Jarugumilli, S.; Grasman, S.E.; Ramakrishnan, S. 2006. “A Simulation Framework for Real-Time Man- agement and Control of Inventory Routing Decisions”. In proceedings of 2006 Winter Simulation Conference, pp. 1485 – 1492. 7 2012 IN3-HAROSA

9. 2. Observations  Some initial ideas:  To explore how CVRP metaheuristics can be used to solve the VRPSD, i.e.: to transform the issue of solving a given VRPSD instance into an issue of solving several CVRP instances  To use DPCS to accelerate computations and provide many ‘good’ alternative solutions, so that the decision-maker can select the one that best fits his/her utility function  Yes, but…:  While efficient and flexible approaches have been already developed for the CVRP and VRPSD (Novoa and Storer 2009), it is not the case for IRPSD.  In real-life scenarios is not possible to model all costs, constraints and desirable solution properties in advance (Kant et al 2008) Juan, A.; Faulin, J.; Grasman, S.; Riera, D.; Marull, J.; Mendez, C. (2011): “Using Safety Stocks and Simulation to Solve the Vehicle Routing Problem with Stochastic Demands”. Transportation Research Part C, Vol. 19, pp. 751-765 Juan, A.; Faulin, J.; Jorba, J.; Caceres, J.; Marques, J. (2011): “Using Parallel & Distributed Computing for Solving Real-time Vehicle Routing Problems with Stochastic Demands”. Annals of Operations Research, pp. 1-22. 8 2012 IN3-HAROSA

10.  Our CVRP approach is based on the Clarke and Wright’s savings (CWS) algorithm (Clarke & Wright 1964).  CWS algorithm: 1.For each pair of nodes i and j, calculate the savings, s(i, j), associated to the edge connecting them, where: s(i, j) = c(0, i) + c(0, j) – c(i, j) 2.Construct a list of edges, sorting the edges according to their associated savings 3.Construct an initial feasible solution by routing a vehicle to each client node 4.Select the first edge in the savings list and, if no constraint is violated, merge the routes that it connects 5.Repeat step 4 until the savings list is empty Start savings(i, j) Savings list Initial solution Select first edge & Merge List empty? End 3. Solving CVRP: SR-GCWS-CS (1/3)  This parallel version of the CWS heuristic usually provides ‘acceptable solutions’ (average gap between 5% and 10%), especially for small and medium-size problems Juan, A.; Faulin, J.; Jorba, J.; Riera, D.; Masip; D.; Barrios, B. (2011): “On the Use of Monte Carlo Simulation, Cache and Splitting Techniques to Improve the Clarke and Wright Savings Heuristics”. Journal of the Operational Research Society, Vol. 62, pp. 1085-1097. 9 2012 IN3-HAROSA

11.  CWS  the first edge (the one with the most savings) is the one selected.  SR-GCWS introduces randomness in this process by using a quasi-geometric statistical distribution  edges with more savings will be more likely to be selected at each step, but all edges in the list are potentially eligible.  Notice: Each time SR-GCWS is run, a random feasible solution is obtained. By construction, chances are that this solution outperforms the CWS one  hundreds of ‘good’ solutions can be obtained after some seconds/minutes. Good results with 0.10 < α < 0.20 Good results with 0.10 < α < 0.20 3. Solving CVRP: SR-GCWS-CS (2/3) 10 2012 IN3-HAROSA

12. 1.Adding ‘memory’ to our algorithm with a hash table:  A hash table is used to save, for each generated route, the best-known sequence of nodes (this will be used to improve new solutions)  ‘Fast’ method that provides small improvements on the average Improvement #1: Hash Table Improvement #2: Splitting 2. Splitting (divide-and-conquer) method:  Given a global solution, the instance is sub-divided in smaller instances and then the algorithm is applied on each of these smaller instances  ‘Slow’ method that can provide significant improvements 3. Solving CVRP: SR-GCWS-CS (3/3) 11 2012 IN3-HAROSA

13. 4. Our Approach for the IRPSD (1/2) 1.Given a IRPSD instance  n customers, random demands Di 2.Estimate the Expected Inventory Cost for each customer-policy case by using MCS. Notice that: a)Random demands are generated and whenever a stock-out occurs, a corrective policy (  costs) is applied 3.Solve the CVRP with the full-refill policy by using the SR-GCWS-CS algorithm. Notice that: a)The solution for CVRP is expensive for having stocks up to maximum. 4.Compute marginal routing savings for each customer-policy case by using the SR- GCWS-CS algorithm and estimate: a)S = CVRP(Full Refill) – CVRP(not node i) b)Inventory Cost for Policy p - Saving 12 2012 IN3-HAROSA

14. 5.Sort total expected costs of policies for each node in a decreasing rank. Notice that: a)Depending on the current level, one policy would be appropriated. 6.Solve the CVRP with the Top policy of each node by using the SR-GCWS-CS algorithm. Notice that: a)This is the recommended refill rate for each client in the current level estimated. 7.Apply a biased randomization to each customer policy rank. 8.Solve the CVRP with the Top policy of each node by using the SR-GCWS-CS algorithm: 9.Repeat from Step 7 with a new order of policies for each customer, i.e.: explore different policy-based scenarios 4. Our Approach for the IRPSD (2/2) 13 2012 IN3-HAROSA

15. 4. Practical Example 12345 1P1P2P3P4P5 2P4P5P3P2P1 3P3P4P5P1P2 4P5P4P3P2P1 12345 1 2 3 4 Policy rank for customers Current Level for each customer Top Policy for each customer Biased Randomly Selection of Policies for each customer Customers P1 and P5 are extreme policies! 14 2012 IN3-HAROSA

16.  The idea of solving the Routing Problems with Stochastic Demands through solving a related CVRP is not new (Stewart and Golden 1983, Laporte et al. 1989). However, our approach differs from others in:  It contemplates the analysis of different policy scenarios  It uses a more practical perspective based on the combination of MCS, heuristics and realistic contexts.  It does not require strong assumptions on the variables that model customers’ demands  Potential benefits:  Wider scope: it is valid for any statistical distribution with a known mean.  Efficiency: it reduces the complexity of a IRPSD to a more tractable CVRP.  Flexibility: it offers different policy-based scenarios to the decision-maker. 5. Advantages of Our Approach 15 2012 IN3-HAROSA

17. 6. Design of Experiments (1/2)  Knowing that in the IRPSD literature there are no commonly used benchmarks, many authors construct their own instances, generated using different statistical distributions.  The previous situation implies that many results are dependent on the statistical assumptions.  Now, we have designed a methodology based on introducing randomness in demands of the well-known CVRP instances. Different designs of demands in means and variances.  We have modeled random demands, Di, considering E[Di] = di (original demands) and Var[Di] = wE[Di] with w=0.25 (moderate variance). 16 2012 IN3-HAROSA

18. 6. Design of Experiments (2/2): MCS Stock P2 P3 P4 P5 Possible Current Levels E[d i ] E[di] ~ LogNormal(Demand i )  Use classical CVRP benchmarks, and estimate the Expected Inventory Cost by a given “real” formula which:  Relates the inventory costs’ magnitudes with routing costs’ ones,  Considers a Max Level for each node (2d i ) and a Current Level (all related to the Expected Demand): 17 2012 IN3-HAROSA

19.  In a IRPSD, a scenario with less stock-outs provides lower total expected costs. 7. Computational Results (1/3) FULL-REFILL POLICY (1)TOP POLICY (2) BIASED RANDOMIZED POLICY (3) GAPS Instance Routing Cost Inventory Cost Total Cost Routing Cost Inventory Cost Total Cost Routing Cost Inventory Cost Total Cost (1)-(3)(2)-(3) A-n32-k5 981.384.72986.1600.0824.12624.21547.4950.66598.15 64.86%4.36% A-n33-k5 805.344.62809.96421.1741.7462.87421.1741.7462.87 74.99%0.00% A-n33-k6 841.435.72847.15504.8827.2532.08503.9223.49527.41 60.62%0.88% A-n63-k9 1,941.079.321,950.40952.3372.161,024.49952.3372.161,024.49 90.38%0.00% A-n65-k9 1,372.329.831,382.16672.1362.72734.86672.1362.72734.86 88.09%0.00% A-n80-k10 2,153.3110.652,163.961,019.82148.771,168.591,019.82148.771,168.59 85.18%0.00% B-n31-k5 807.064.27811.33495.1812.66507.84453.434.5487.89 66.29%4.09% B-n35-k5 1,179.775.491,185.27713.5318.87732.4532.6948.76581.45 103.85%25.96% B-n39-k5 652.676.05658.72357.7318.35376.08357.7318.35376.08 75.15%0.00% B-n41-k6 931.995.87937.86536.0824.35560.43510.5735.99546.56 71.59%2.54% B-n68-k9 1,541.419.591,551.00735.6467.35803735.6467.35803 93.15%0.00% B-n78-k10 1,423.159.991,433.14764.5156.54821.05764.5156.54821.05 74.55%0.00% Averages 1,170.957.341,178.29622.3944.46666.85608.2849.55657.83 79.12%1.37% 18

20. 7. Computational Results (2/3) 19 2012 IN3-HAROSA

21. 7. Computational Results (3/3)  Solutions obtained for A-n32-k5 using the ‘top’ refill and ‘biased-randomized’ refill policies. FULL-REFILL POLICY (1) TOP POLICY (2) BIASED RANDOMIZED POLICY (3) GAPS Instance Routing Cost Inventory Cost Total Cost Routing Cost Inventory Cost Total Cost Routing Cost Inventory Cost Total Cost (1)-(3)(2)-(3) A-n32-k5 981.384.72986.1600.0824.12624.21547.4950.66598.15 64.86%4.36% 20 2012 IN3-HAROSA

22.  We have presented a hybrid approach for solving the IRPSD with stock-outs. This approach combines MCS and the SR-GCWS-CS algorithm.  A set of benchmarks for the IRPSD were developed and a realistic expression to model inventory costs was also proposed.  Our approach provides the decision-maker with a set of alternative solutions with different properties (number of served customers, inventory and routing costs, refill policies, etc.)  It offers flexibility since it does not assume any particular behavior of the customers’ stochastic demands. Therefore, the statistical distributions which describe demands can be generic.  We are currently developing a deepest version of this study for a Journal publication (more variances, λ formula parameter, more policies, etc.). 8. Conclusions 21 2012 IN3-HAROSA

23. A Hybrid Approach for the Inventory Routing Problem with Stochastic Demands José Cáceres Cruz Angel A. Juan {jcaceresc, ajuanp}@uoc.edu Department of Computer Science IN3 - Open University of Catalonia, Barcelona, SPAIN http://dpcs.uoc.edu | http://ajuanp.wordpress.com Barcelona, Spain June 14, 2012 Tolga Bektas t.bektas@soton.ac.uk Management School University of Southampton, UK Scott Grasman segeie@rit.edu Industrial & Systems Engineering Department Rochester Institute of Technology, USA Thanks for your attention!