A hybrid heuristic for an inventory routing problem

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, Montr?al, Canada

DOMinant 2009, Molde, September 20-23

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 ……

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. ……

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

The problem

Data

Availability at t

Demand of s at t

Capacity of s

+ initial inventory + travelling costs + inventory costs + vehicle capacity

n customers H time units 1 vehicle

How much to deliver to s at time t to minimize routing costs + inventory costs

No stock-out No lost sales

1

0

2

3

Replenishment policies

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

Constraints on the quantities to deliver

1

0

2

3

Order-Up-to level policy (OU)

Inventory at customer s

Maximum level

Us

Initial level

Time

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

Basic decision variables

Problem formulation

Inventory definition at the supplier

Stock-out constraints at the supplier

Inventory definition at the customers

Stock-out constraints at the customers

Capacity constraints

Order-up-to level constraints

The quantity shipped to s at time t is

Maximum level constraints

Routing constraints

Known algorithms

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

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

Local search Very fast Error?

Instances up to H=3, n=50 H=6, n=30

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 large errors, sometimes optimal Excellent results

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

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)

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

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

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

OU policy – Improvements – MILP 1

The route assignment model

NP-hard

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: (n+H)*(# of routes)+n*H

OU policy – Improvements – MILP 1

Day 5

Day 6

Day 1

Day 2

Day 3

Day 4

Incumbent solution

The optimal route assignment

Node removed

Unused

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

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

OU policy – Improvements – MILP 2

The customer assignment model

NP-hard

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: n*H

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)

The hybrid heuristic for the ML policy

The ML policy is more flexible than the OU policy

The entire hybrid heuristic has been adapted

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

Summary of results

Summary of results

Summary of results

Large instances

Optimal solution unknown

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

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

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

Hybrid vs tabu – OU policy

Hybrid vs tabu – ML policy

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)