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A Hybrid Approach for the Inventory Routing Problem with Stochastic Demands
José Cáceres CruzAngel 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, SpainJune 14, 2012
Tolga Bektas
Management School
University of Southampton, UK
Scott Grasman
Industrial & Systems Engineering Department
Rochester Institute of Technology, USA
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
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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.
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1. Introduction: Costs of Logistic Activities
7%
26%
11%
13%
43%
12%
31%
Packaging
Warehousing
Inventory
Administration
Urban Transportation
Inter-UrbanTransportation
* Differentiation for Performance: Excellence in Logistics (2004), ELA/AT Kearney3
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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
Inve
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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
Inve
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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]
E[Total Costs] = Total Routing Costs + E[Inventory Costs]
if 0 (surplus)
cost of a roundtrip from to depot if <0 (shortage) i i
ii
surplus surplusIC
i surplus
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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.
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.
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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.; 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.
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.
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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.
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
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11)( kkXP
sk ,...,2,1
s
k
k
sk
k
1
1
1
1 111
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 with0.10 < α < 0.20
Good results with0.10 < α < 0.20
3. Solving CVRP: SR-GCWS-CS (2/3)
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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 #1: Hash Table
1. Select routes on the SE area(area below the diagonal)
2. Consider the new CVRP subproblem
3. Solve the subproblem and re-construct the solution
Improvement #2: SplittingImprovement #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)
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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
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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)
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4. Practical Example
1 2 3 4 5
1 P1 P2 P3 P4 P5
2 P4 P5 P3 P2 P1
3 P3 P4 P5 P1 P2
4 P5 P4 P3 P2 P1
1 2 3 4 5
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!
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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
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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.
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).
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6. Design of Experiments (2/2): MCS
Stock
P2
P3
P4
P5
Possible Current Levels
E[di]
E[di] ~ LogNormal(Demandi)
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 (2di) and a Current Level (all related to the Expected Demand):
3 9 15
1 5 7
4 8
0 if is odd and multiple of 3 (e.g. , , , ...)
0.5 if is odd and not multiple of 3 (e.g. , , , ...)
if is even and multiple of 4 (e.g. , ,i
ii
i L L L
d i L L LL
d i L L
12
2 6 10
, ...)
1.5 if is even and not multiple of 4 (e.g. , , , ...)i
L
d i L L L
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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
InstanceRouting
CostInventory
CostTotalCost
RoutingCost
InventoryCost
TotalCost
RoutingCost
InventoryCost
TotalCost
(1)-(3) (2)-(3)
A-n32-k5 981.38 4.72 986.1 600.08 24.12 624.21 547.49 50.66 598.15 64.86% 4.36%
A-n33-k5 805.34 4.62 809.96 421.17 41.7 462.87 421.17 41.7 462.87 74.99% 0.00%
A-n33-k6 841.43 5.72 847.15 504.88 27.2 532.08 503.92 23.49 527.41 60.62% 0.88%
A-n63-k9 1,941.07 9.32 1,950.40 952.33 72.16 1,024.49 952.33 72.16 1,024.49 90.38% 0.00%
A-n65-k9 1,372.32 9.83 1,382.16 672.13 62.72 734.86 672.13 62.72 734.86 88.09% 0.00%
A-n80-k10 2,153.31 10.65 2,163.96 1,019.82 148.77 1,168.59 1,019.82 148.77 1,168.59 85.18% 0.00%
B-n31-k5 807.06 4.27 811.33 495.18 12.66 507.84 453.4 34.5 487.89 66.29% 4.09%
B-n35-k5 1,179.77 5.49 1,185.27 713.53 18.87 732.4 532.69 48.76 581.45 103.85% 25.96%
B-n39-k5 652.67 6.05 658.72 357.73 18.35 376.08 357.73 18.35 376.08 75.15% 0.00%
B-n41-k6 931.99 5.87 937.86 536.08 24.35 560.43 510.57 35.99 546.56 71.59% 2.54%
B-n68-k9 1,541.41 9.59 1,551.00 735.64 67.35 803 735.64 67.35 803 93.15% 0.00%
B-n78-k10 1,423.15 9.99 1,433.14 764.51 56.54 821.05 764.51 56.54 821.05 74.55% 0.00%
Averages 1,170.95 7.34 1,178.29 622.39 44.46 666.85 608.28 49.55 657.83 79.12% 1.37%
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7. Computational Results (2/3)
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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
InstanceRouting
CostInventory
CostTotalCost
RoutingCost
InventoryCost
TotalCost
RoutingCost
InventoryCost
TotalCost
(1)-(3) (2)-(3)
A-n32-k5 981.38 4.72 986.1 600.08 24.12 624.21 547.49 50.66 598.15 64.86% 4.36%
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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
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A Hybrid Approach for the Inventory Routing Problem with Stochastic Demands
José Cáceres CruzAngel 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, SpainJune 14, 2012
Tolga Bektas
Management School
University of Southampton, UK
Scott Grasman
Industrial & Systems Engineering Department
Rochester Institute of Technology, USA
Thanks for your attention!Thanks for your attention!
