Integrated Vehicle Routing -Tank Sizing
Problem with Safety Stock Optimization
Fengqi You
Elisabet Capon
Ignacio E. Grossmann
Jose M. Pinto
EWO Meeting, Mar. 2009
Vehicle Routing – Tank Sizing Problem
• Tank Sizing:
Routing ?
Š New customers need new
tanks to be sized
Š All or some of existing
customers subject to tank
upgrades or downgrades
Š Selection among several
available tank sizes
• Vehicle Routing
Truck selection
Š Selection among several
truck sizes
Š Determine the routing and
timing decisions
Tank size for the
New Customer ?
Potential size change in existing tanks
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Tank Sizing Issues
• Major Features
Š Key tradeoff: capital (tank sizing) vs. distribution (routing) cost
Š Capture the effects of customer synergies and truck availability
• Challenges
Š Routing decisions and tank sizing decisions should be integrated
ƒ A small change of tank size at a particular customer may influence
distribution (routing) of all other customers
ƒ Many possible routes for each fixed tank sizing decision
Š Requires fast computational strategies for good solutions
ƒ Direct MILP model leads to large problem (millions of binary variables)
ƒ Difficult to obtain “good” solution in reasonable times
Š Account for the fluctuation of customer demand
3
Clustering - Integrated Approach
Customer Clustering
Pros: accurate
Cons: large scale MILP,
long CPU time
Select the first
clustering solution
Detailed Integrated Model
Total Cost
(obtain routing and tank sizing
decisions simultaneously)
Next clustering
solution
0
1
2
3
4
5
6
Termination?
Key Tradeoff: Routing Cost vs. Tank Cost
4
Route Selection – Tank Sizing Approach
Route Generation
Visiting a customer with
more than one route
Need all possible routes
Approximated inventory
Input all the
possible routes
Route Selection –Tank Sizing Model
(obtain routes and tank sizing decisions)
Fix tank sizes
Fix Routes
Smaller models,
faster computation
Detailed Routing Model (Reduced)
(obtain vehicle routing decisions)
Pros: fast for routing Cons: hard to estimate “global” gap
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Inventory for Multiple Replenishments
Tank size = max. Inv. ≈ Min. Inv. + max{working Inv.}
Inventory
Max. Inv.
working inv. 2
working inv. 1
working inv. 3
Constant
demand
rate
Min. Inv.
Rep.1
Rep.2
Rep.3
Time
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Continuous Approximation Approach
Customer Clustering
Smaller models,
faster computation
Select the first
clustering solution
Cont. Approximation Model
(obtain tank sizing decisions)
Fix tank sizes
Next clustering
solution
integer
cut
Detailed Routing Model
(obtain vehicle routing decisions)
Termination?
Pros: fast computation Cons: hard to estimate “global” gap
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Improved CAM Approach
Cont. Approximation Model
in Full Space (all customers)
(obtain tank sizing decisions)
Smaller models,
faster computation
Fix tank sizes
Customer Clustering
Select the first
clustering solution
Detailed Routing Model
(obtain vehicle routing decisions)
Next clustering soln.
Termination?
Note: more accurate, still fast, need to improve routing model
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“Cyclic” Inventory-Routing in CAM
• Key Assumption: each customer is replenished in a “cyclic”
way with fixed interval T
Inventory
• Required tank size ≥ max. inv. = min. inv. + working inventory
Max. Inv.
working inventory
Constant
demand
rate
Min. Inv.
Time
Replenishment
Interval T
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Deterministic Case: Example 1
• Problem Size:
5,250.87 L/Month
Š A cluster with 2 customers, N15 is new
Š N16 has an existing tank of 13,000 L
N15
1,100 km
25 km
Plant
Š 6 available tank size, 4 types of trucks
N16
1,100 km
3,500.58 L/Month
• Integrated Model
Š CPU time: ~ 8 min. (0% gap)
Š Total cost: $18,112, Tank size: 13,000 L for N15, no change for N16
• Routing Selection – Tank Sizing (RSTS)
Š CPU time: 43 sec. for RSTS, 53 sec. for routing problem (0% gap)
Š Total cost: $19,085, Tank size: 6,000 L for N15, no change for N16
• Continuous Approximation Model (CAM)
Š CPU time: < 1 sec. for CAM, 74 sec. for routing problem (0% gap)
Š Total cost: $18,112, Tank size: 13,000 L for N15, no change for N16
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Deterministic Case: Example 2
12,835.47 L/Month
• Problem Size:
N14
1,123.61 km
Š A cluster with 4 customers, N14 is new
Š All customers have a tank of 10,000 L
Š 6 available tank size, 4 types of trucks
1,253.00 km
600 km
950 km
N18
Plant
6417.74 L/M
1,100 km
N15
5250.88 L/M
290 km
993.28 km
• Integrated Model
N21
1,137.59 km
23337.22 L/Month
Š CPU time: ~15 hours (4.39% gap, out of memory)
Š Total cost: $30,620; Tank size: 16,000 L for N14, no change for others
• Routing Selection – Tank Sizing (RSTS)
Š CPU time: ~10 hours. for RSTS (out of memory)
• Continuous Approximation Model (CAM)
Š CPU time: < 1 sec. for CAM, 2360 sec. for routing problem (0% gap)
Š Total cost: $30,299; Tank size: 16,000 L for N14, no change for others
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Inventory Policy
Inventory Level
Order Quantity
(Q)
Order
placed
Replenishment
Reorder
Point
(r)
Time
Lead Time
• (r, Q) Inventory Policy
Š When inventory level falls to r, order a quantity of Q
Š Reorder Point (r) = Demand over Lead Time
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Stochastic Base-Stock Inventory Model
Inventory
Maximum Inv. = Working Inv. + Safety Stock
Lead time
Max. Inv.
Inventory
on hand
Reorder
Point (r)
Time
Safety Stock
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Safety Stock Level
(Service Level)
Lead time = T
Safety Stock
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“Cyclic” Inventory-Routing in CAM
• Key Assumption: each customer is replenished in a “cyclic” way
with fixed interval T
Inventory
• Required tank size ≥ max. inv. = Safety Stock + demand rate× T
Max. Inv.
working inventory
Constant
demand
rate
Safety Stock
Time
Replenishment
Interval T
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Routing & Replenishment in CAM
• T = R / (ave. speed)
Š T - replenishment interval
Š R - minimum distance to replenish all
the customer in a cluster once
Š Average travelling speed is known
plant
customer
• If only one trip for each replenishment
Š R = TSP distance of the cluster & plant
• If allowing multiple trips for replenishment
Š R=?
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CAM for Capacitated Routing Problems*
• Bounds for minimum routing distance R
Š n – # of customers in the cluster
Š q – capacity, max. # of customers that can be visited in one trip or volume
in terms of # of customers with unit demand
Š r – average distance from customers to the plant
Š TSP – traveling salesman distance to visit all customers once
• Examples
Š Cluster 1: q=1, TSP=0, r = 67
Š Cluster 2: q=1, same as Cluster 1,
67
1,100
25
1,100
Š Cluster 2: q=2, TSP=50, r = 1,100
Cluster 1
Cluster 2
* M Haimovich, AHG Rinnooy Kan, “Bounds and heuristics for CRP”, Math. of Oper. Res., 1985, 10(4), 527-541
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Improved CAM Approach
Cont. Approximation Model
+ safety stock (all customers)
(obtain tank sizing decisions)
Determine safety
stock level in the
strategic level
Fix tank sizes
Safety stock
Customer Clustering
Select the first
clustering solution
Detailed Routing Model
(obtain vehicle routing decisions)
Next clustering soln.
Termination?
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Stochastic Case: Example 3
• Problem Size:
5,250.87 L/Month
1,100 km
Š A cluster with 2 customers, N15 is new
Š 6 available tank size, 4 types of trucks
25 km
Plant
Š Demands follow normal dist. (σ=μ)
N15
1,100 km
N16
3,500.58 L/Month
• Fix safety stock to 15% of tank size
Š Total cost: $16,792; Tank size: 13,000 L for N15, no change for others
Š Safety stocks: N15: 1,950 L, N16: 1,950 L
Š Service level: N15: ~100%, N16: ~100%,
• Safety stock optimization (97.75% service level)
Š Total cost: $16,646; Tank size: 13,000 L for N15, no change for others
Š Safety stocks: N15: 1,189L, N16: 793L
• What is the implications of different total costs and tank sizes?
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Conclusion / Future Work
• Conclusion
Š Two approaches to reduce the computational efforts for the
integrated vehicle routing – tank sizing problem
Š Integrate safety stock optimization in the continuous
approximation approach to account for demand fluctuation
• Future Work
Š Refine the models and validate the assumptions
Š Consider uncertainties such as adding or losing customers
Š Quantify the economic benefits of considering uncertainties
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