A hybrid heuristic for an inventory routing problem

1
A hybrid heuristic for an inventory routing
problem

• C.Archetti, L.Bertazzi, M.G. Speranza
• University of Brescia, Italy
• A.Hertz
• Ecole Polytechnique and GERAD, Montréal, Canada

DOMinant 2009, Molde, September 20-23
2
The literature

• Surveys
• Federgruen, Simchi-Levi (1995), in Handbooks in
  
• Operations Research and Management Science
• Campbell et al (1998), in Fleet Management and
  Logistics
• Cordeau et al (2007) in Handbooks in Operations
  Research and
• Management Science Transportation
• Bertazzi, Savelsbergh, Speranza (2008) in The
  vehicle routing problem..., Golden,
  Raghavan, Wasil (eds)
• Pioneering papers
• Bell et al (1983), Interfaces
• Federgruen, Zipkin (1984), Operations Research
• Golden, Assad, Dahl (1984), Large Scale
  Systems
• Blumenfeld et al (1985), Transportation Research
  B
• Dror, Ball, Golden (1985), Annals of Operations
  Research

3
The literature

• Deterministic product usage - no inventory
  holding costs in the objective function
• Jaillet et al (2002), Transp. Sci.
• Campbell, Savelsbergh (2004), Transp. Sci.
• Gaur, Fisher (2004), Operations Research
• Savelsbergh, Song (2006), Computers and
  Operations Research
• Deterministic product usage - inventory holding
  costs in the objective function
• Anily, Federgruen (1990), Management Sci.
• Speranza, Ukovich (1994), Operations Research
• Chan, Simchi-Levi (1998), Management Sci.
• Bertazzi, Paletta, Speranza (2002), Transp.
  Sci.
• Archetti et al (2007), Transp. Sci.

4
The literature

• Deterministic product usage - inventory holding
  costs in the objective function production
  decision
• Bertazzi, Paletta, Speranza (2005), Journal of
  Heuristics
• Archetti, Bertazzi, Paletta, Speranza ,
  forthcoming
• Boudia, Louly, Prins (2007), Computers and
  Operations Research
• Boudia, Prins (2007), EJOR
• Boudia, Louly, Prins (2008), Production
  Planning and Control

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The problem
Data
n customers H time units 1 vehicle
1
Availability at t
Demand of s at t
0
2
Capacity of s
3
initial inventory travelling costs
inventory costs vehicle capacity
How much to deliver to s at time t to minimize
routing costs inventory costs
No stock-out No lost sales
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Replenishment policies
1
0
2
3

• Order-Up-to Level (OU)
• Maximum Level (ML)

Constraints on the quantities to deliver
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Order-Up-to level policy (OU)
Inventory at customer s
Maximum level
Us
Initial level
Time
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The Maximum Level policy (ML)
Every time a customer is visited, the shipping
quantity is such that at most the maximum level
is reached
Us
9
Basic decision variables
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Problem formulation

• Inventory definition at the supplier

• Stock-out constraints at the supplier

11

• Inventory definition at the customers

• Stock-out constraints at the customers

• Capacity constraints

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• Order-up-to level constraints

The quantity shipped to s at time t is
Maximum level constraints
13

• Routing constraints

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Known algorithms
Local search Very fast Error?

• A heuristic
• (for the OU policy)
• Bertazzi, Paletta, Speranza (2002), Transp.
  Science

Instances up to H3, n50 H6, n30

• A branch-and-cut algorithm
• (for the OU and for the ML policies)
• Archetti, Bertazzi, Laporte, Speranza (2007),
  Transp. Science

15
History of the hybrid heuristic design
Exact approach allowed us to compute errors
generated by the local search Design of a
tabu search Design of a hybrid heuristic (tabu
search MILP models)
Often large errors, rarely optimal Sometimes la
rge errors, sometimes optimal Excellent results
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HAIR (Hybrid Algorithm for Inventory Routing)
Initialize generates initial solution A Tabu
search is run Whenever a new best solution is
found Improvements is run Every JumpIter
iterations without improvements Jump is run
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OU policy - Initialize
Each customer is served as late as possible
Initial solution may be infeasible (violation of
vehicle capacity or stock-out at the supplier)
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OU policy Tabu search
Search space feasible solutions
infeasible solutions (violation of vehicle
capacity or stock-out at the
supplier) Solution value total cost two
penalty terms Moves for each customer Removal
of a day Move of a day Insertion of a day Swap
with another customer After the moves Reduce
infeasibility Reduce costs
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OU policy Improvements MILP 1

• Route assignment
• Goal to find an optimal assignment of routes to
  days optimizing the quantities delivered at the
  same time. Removal of customers is allowed.

Optimal solution of a MILP model
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OU policy Improvements MILP 1

• The route assignment model

Binary variables assignment of route r to time
t removal of customer s from route r Continuous
variables quantity to customer s at time
t inventory level of customer s at time
t inventory level of the supplier at time t
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OU policy Improvements MILP 1

• The route assignment model

Min inventory costs saving for
removals s.t. Stock-out constraints OU policy
defining constraints Vehicle capacity
constraints Each route can be assigned to one day
at most Technical constraints on possibility to
serve or remove a customer
of binary variables (nH)( of routes)nH
NP-hard
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OU policy Improvements MILP 1
Day 6
Day 5
Day 1
Day 2
Day 3
Day 4
Node removed
Incumbent solution
Unused
The optimal route assignment
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OU policy Improvements MILP 2

• Customer assignment
• Objective to improve the incumbent solution by
  merging a pair of consecutive routes. Removal of
  customers from routes, insertion of customers
  into routes and quantities delivered are
  optimized.

Optimal solution of a MILP model
For each merging and possible assignment day of
the merged route a MILP is solved
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OU policy Improvements MILP 2

• The customer assignment model

Binary variables removal of customer s from time
t insertion of customer s into time t Continuous
variables quantity to customer s at time
t inventory level of customer s at time
t inventory level of the supplier at time t
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OU policy Improvements MILP 2

• The customer assignment model

Min inventory costs insertion costs saving
for removals s.t. Stock-out constraints OU
policy defining constraints Vehicle capacity
constraints Each route can be assigned to one day
at most Technical constraints on possibility to
insert or remove a customer
of binary variables nH
NP-hard
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OU policy - Jump
After a certain number of iterations without
improvements a jump is made. Jump move
customers from days where they are typically
visited to days where they are typically not
visited. (In our experiments a jump is made
only once)
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The hybrid heuristic for the ML policy
The ML policy is more flexible than the OU policy
The entire hybrid heuristic has been adapted
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Tested instances

• 160 benchmark instances from Archetti et al
  (2007), TS
• Known optimal solution
• H 3, n 5,10, ., 50
• H 6, n 5,10, ., 30
• Inventory costs low, high

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Summary of results
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Summary of results
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Summary of results
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Large instances

• HAIR has been slightly changed
• the improvement procedure is called only if at
  least 20 iterations were performed since its last
  application
• the swap move is not considered.

Optimal solution unknown

• H 6
• n 50, 100, 200
• Inventory costs low, high
• 10 instances for each size for a total of 60
  instances

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Summary of results large instances
Running time for OU 1 hour Running time for ML
30 min Errors taken with respect to the best
solution found Running time BPS always less
than 3 min
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Summary of results large instances
Running time for OU 1 hour Running time for ML
30 min Errors taken with respect to the best
solution found Running time BPS always less
than 3 min
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Hybrid vs tabu OU policy
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Hybrid vs tabu ML policy
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Conclusions

• Tabu search combined with MILP models very
  successful
• Use of the power of CPLEX
• Ad hoc designed MILP models used to explore in
  depth parts of the solution space
• It is crucial to find the appropriate MILP models
  (models that are needed, models that explore
  promising parts of the solution space, trade-off
  between size of search and complexity)