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Summary of results
Summary of results
Summary of results – large instances
Summary of results – large instances
Hybrid vs tabu – OU policy
Hybrid vs tabu – OU policy
Hybrid vs tabu – ML policy
Hybrid vs tabu – ML policy
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A hybrid heuristic for an inventory routing problem

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1A hybrid heuristic for an inventory 18total cost + two penalty terms Moves for
routing problem. C.Archetti, L.Bertazzi, each customer: Removal of a day Move of a
M.G. Speranza University of Brescia, Italy day Insertion of a day Swap with another
A.Hertz Ecole Polytechnique and GERAD, customer After the moves: Reduce
Montr?al, Canada. DOMinant 2009, Molde, infeasibility Reduce costs.
September 20-23. 19OU policy – Improvements – MILP 1.
2The literature. Surveys Federgruen, Route assignment Goal: to find an optimal
Simchi-Levi (1995), in ‘Handbooks in assignment of routes to days optimizing
Operations Research and Management the quantities delivered at the same time.
Science’ Campbell et al (1998), in ‘Fleet Removal of customers is allowed. Optimal
Management and Logistics’ Cordeau et al solution of a MILP model.
(2007) in ‘Handbooks in Operations 20OU policy – Improvements – MILP 1. The
Research and Management Science: route assignment model. Binary variables:
Transportation’ Bertazzi, Savelsbergh, assignment of route r to time t removal of
Speranza (2008) in ‘The vehicle routing customer s from route r Continuous
problem...’, Golden, Raghavan, Wasil (eds) variables: quantity to customer s at time
Pioneering papers Bell et al (1983), t inventory level of customer s at time t
Interfaces Federgruen, Zipkin (1984), inventory level of the supplier at time t.
Operations Research Golden, Assad, Dahl 21OU policy – Improvements – MILP 1. The
(1984), Large Scale Systems Blumenfeld et route assignment model. NP-hard. Min
al (1985), Transportation Research B Dror, inventory costs – saving for removals s.t.
Ball, Golden (1985), Annals of Operations Stock-out constraints OU policy defining
Research …… constraints Vehicle capacity constraints
3The literature. Deterministic product Each route can be assigned to one day at
usage - no inventory holding costs in the most Technical constraints on possibility
objective function Jaillet et al (2002), to serve or remove a customer. # of binary
Transp. Sci. Campbell, Savelsbergh (2004), variables: (n+H)*(# of routes)+n*H.
Transp. Sci. Gaur, Fisher (2004), 22OU policy – Improvements – MILP 1. Day
Operations Research Savelsbergh, Song 5. Day 6. Day 1. Day 2. Day 3. Day 4.
(2006), Computers and Operations Research Incumbent solution. The optimal route
…… Deterministic product usage - inventory assignment. Node removed. Unused.
holding costs in the objective function 23OU policy – Improvements – MILP 2.
Anily, Federgruen (1990), Management Sci. Customer assignment Objective: to improve
Speranza, Ukovich (1994), Operations the incumbent solution by merging a pair
Research Chan, Simchi-Levi (1998), of consecutive routes. Removal of
Management Sci. Bertazzi, Paletta, customers from routes, insertion of
Speranza (2002), Transp. Sci. Archetti et customers into routes and quantities
al (2007), Transp. Sci. …… delivered are optimized. Optimal solution
4The literature. Deterministic product of a MILP model. For each merging and
usage - inventory holding costs in the possible assignment day of the merged
objective function – production decision route a MILP is solved.
Bertazzi, Paletta, Speranza (2005), 24OU policy – Improvements – MILP 2. The
Journal of Heuristics Archetti, Bertazzi, customer assignment model. Binary
Paletta, Speranza , forthcoming Boudia, variables: removal of customer s from time
Louly, Prins (2007), Computers and t insertion of customer s into time t
Operations Research Boudia, Prins (2007), Continuous variables: quantity to customer
EJOR Boudia, Louly, Prins (2008), s at time t inventory level of customer s
Production Planning and Control. at time t inventory level of the supplier
5The problem. Data. Availability at t. at time t.
Demand of s at t. Capacity of s. + initial 25OU policy – Improvements – MILP 2. The
inventory + travelling costs + inventory customer assignment model. NP-hard. Min
costs + vehicle capacity. n customers H inventory costs + insertion costs – saving
time units 1 vehicle. How much to deliver for removals s.t. Stock-out constraints OU
to s at time t to minimize routing costs + policy defining constraints Vehicle
inventory costs. No stock-out No lost capacity constraints Each route can be
sales. 1. 0. 2. 3. assigned to one day at most Technical
6Replenishment policies. Order-Up-to constraints on possibility to insert or
Level (OU) Maximum Level (ML). Constraints remove a customer. # of binary variables:
on the quantities to deliver. 1. 0. 2. 3. n*H.
7Order-Up-to level policy (OU). 26OU policy - Jump. After a certain
Inventory at customer s. Maximum level. number of iterations without improvements
Us. Initial level. Time. a jump is made. Jump: move customers from
8The Maximum Level policy (ML). Every days where they are typically visited to
time a customer is visited, the shipping days where they are typically not visited.
quantity is such that at most the maximum (In our experiments a jump is made only
level is reached. Us. once).
9Basic decision variables. 27The hybrid heuristic for the ML
10Problem formulation. Inventory policy. The ML policy is more flexible
definition at the supplier. Stock-out than the OU policy. The entire hybrid
constraints at the supplier. heuristic has been adapted.
11Inventory definition at the customers. 28Tested instances. 160 benchmark
Stock-out constraints at the customers. instances from Archetti et al (2007), TS
Capacity constraints. Known optimal solution H = 3, n = 5,10,
12Order-up-to level constraints. The …., 50 H = 6, n = 5,10, …., 30 Inventory
quantity shipped to s at time t is. costs low, high.
Maximum level constraints. 29Summary of results.
13Routing constraints. 30Summary of results.
14Known algorithms. A branch-and-cut 31Summary of results.
algorithm (for the OU and for the ML 32Large instances. Optimal solution
policies) Archetti, Bertazzi, Laporte, unknown. H = 6 n = 50, 100, 200 Inventory
Speranza (2007), Transp. Science. A costs: low, high 10 instances for each
heuristic (for the OU policy) Bertazzi, size for a total of 60 instances. HAIR has
Paletta, Speranza (2002), Transp. Science. been slightly changed: the improvement
Local search Very fast Error? Instances up procedure is called only if at least 20
to H=3, n=50 H=6, n=30. iterations were performed since its last
15History of the hybrid heuristic application; the swap move is not
design. Exact approach allowed us to considered.
compute errors generated by the local 33Summary of results – large instances.
search Design of a tabu search Design of a Running time for OU: 1 hour Running time
hybrid heuristic (tabu search +MILP for ML: 30 min Errors taken with respect
models). Often large errors, rarely to the best solution found Running time
optimal Sometimes large errors, sometimes BPS: always less than 3 min.
optimal Excellent results. 34Summary of results – large instances.
16HAIR (Hybrid Algorithm for Inventory Running time for OU: 1 hour Running time
Routing). Initialize generates initial for ML: 30 min Errors taken with respect
solution A Tabu search is run Whenever a to the best solution found Running time
new best solution is found Improvements is BPS: always less than 3 min.
run Every JumpIter iterations without 35Hybrid vs tabu – OU policy.
improvements Jump is run. 36Hybrid vs tabu – ML policy.
17OU policy - Initialize. Each customer 37Conclusions. Tabu search combined with
is served as late as possible. Initial MILP models very successful Use of the
solution may be infeasible (violation of power of CPLEX Ad hoc designed MILP models
vehicle capacity or stock-out at the used to explore in depth parts of the
supplier). solution space It is crucial to find the
18OU policy – Tabu search. Search space: appropriate MILP models (models that are
feasible solutions infeasible solutions needed, models that explore promising
(violation of vehicle capacity or parts of the solution space, trade-off
stock-out at the supplier) Solution value: between size of search and complexity).
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A hybrid heuristic for an inventory routing problem

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