Transcript Diapositive 1 - Marie

A tabu search heuristic to solve the split
delivery Vehicle Routing Problem with
Production and Demand Calendars
(VRPPDC)
Marie-Claude Bolduc
Gilbert Laporte, Jacques Renaud, Fayez Boctor
CORS/Optimization Days – May 12, 2008
[email protected]
Presentation outline
• Problem definition (VRPPDC);
• Literature;
• Solutions;
– Mathematical formulation;
– Tabu search heuristic.
• Computational results;
• Conclusion.
Problem definition
Split delivery Vehicle Routing Problem with Production
and Demand Calendars (VRPPDC)
• Network:
• Objective: Served all customers
– P products, T periods
– Private vehicle
– 1 distribution center (DC)
• Start and end at the DC;
– n customers with demand
• Capacity Q, length L, 1 route.
calendars (expressing due dates)
– Availability of the products
– Private limited fleet of m
(according to the production
homogeneous vehicles
calendar)
– Common carrier
– Respect of the due date
(according to the demand
– Production calendar (a priori set
calendar)
by the factory)
• Question: How to determine the distribution calendar (the routes of the
private fleet, as well as the customers served by a common carrier) in
order to minimize the overall transportation and inventory costs over a
given time horizon?
Literature
• Most case: production calendars + 3rd-party logistic
for transportation
• Objective: determine both distribution and
production calendars (adapted to feed the former)
– Fumero & Vercellis (1999)
• Mathematical VRPPDC model with private fleet only (no
common carrier)
• “On time" deliveries only
– Boudia, Louly & Prins (2006)
• Single product problem (no common carrier)
• GRASP with a reactive mechanism
Mathematical formulation
n
Minimize
n
m
T
  c x
i  0 j  0 k 1 t 1
j i
ij ijkt
subject to
n m
spt  sp ,t 1  gpt   qpikt
i 1 k  0
m
t
 q
k  0 h 1
n m
t
pikh
  d pih
h 1
 x0 jkt  m
j 1 k 1
n
x
j 0
j h
n
n
hjkt
  xihkt
i 0
i h
n
 t x
i 0 j 0
j i
ij ijkt
L
P
T
 eit    hp spt
p 1 t 1
P
q
p 1
n
n
pikt
 Q x jikt
j 0
j i
n
 x
i 0 j 0
j i
P
where eit  f  cl li l   cq qit q 
ijkt
  n  1 ykt
n
 b q
p 1 i 1
p pikt
 Qykt
uikt  u jkt   n  1 xijkt  n  2
i  j 
xijkt 0,1  i  j  , ykt 0,1 , uikt  0 , spt  0
Tabu search heuristic:
Step-by-step
1. Initial solution phase (two-phase method)
2. Tabu phase (with tabu iterations)
– Adjust the aspiration criterions sikt
– Evaluate all the neighbors (i,k’,t’) with respect of
the due dates
– Adjust the diversitification criterion ri
– If the solution of the best neighbor is feasible and
improve s*, save it and update the aspiration
criterions sikt.
– Update a (penalty quantity factor), b penalty
distance factor) and the tabu list
3. Improvement phase
Tabu search heuristic:
Initial feasible solution
• Period decomposition method based on VRPPC (Vehicle
Routing Problem with Private fleet and Common carrier)
• Two-phase method using SRI (Bolduc, Renaud & Boctor, 2006) and
RIP (Bolduc et al., 2006)
For each period…
Phase 1:
– List the customer-product (= a unique customer) with a demand due
in-time (current period, demand calendars). ( split)
– Common carrier cost.
– Solve using SRI or RIP heuristic.
Phase 2:
– List the customers (single unified demand) with a demand due in
current period.
– Common carrier cost.
– Solve using SRI or RIP heuristic.
Solution: Best of phase 1 and 2.
Tabu search heuristic:
Tabu search phase
• Penalized cost function:
n
n
m
f (s)  c(s)  h(s)  a q(s)  b d (s)
P
T
T
where c(s)    cij xijkt  eit , h(s)   hp spt ,
i  0 j  0 k 1 t 1
j i
p 1 t 1

q(s)     bpqpikt
k 1 t 1  p 1 i 1
m
T
P
n
and  x   max 0, x .


 n n


 Q  , d (s)     t ij xijkt  L  ,


k 1 t 1 i  0 j  0

 j  i


m
T
• Attribute set: (customer-truck-period) B(s)   i , k ,t 
• Neighbor:  i , k ',t ' where k '  k or t '  t
• Movable quantity:
– Complete switch (moved everything in respect to inventory
+ due dates)
– Partial switch (complete switch | space used in truck) ( split)
Tabu search heuristic:
Neighbor reduction strategy
• Random strategy
– Setting a priori proportion.
– Randomly selected some neighbors for evaluation.
• Distance strategy
– Setting a priori the longest distance between
customer and route.
– Customer is evaluated only if he is within the fixed
distance.
Tabu search heuristic:
Improvement phase
• Advance the delivery of
– complete blocks of product demand;
– partial blocks of product demand;
• Switch an internal customer (i.e., one served by a
private vehicle) with an external customer (i.e., one
served by a common carrier).
– Osman (1993) 1-1-exchange only between private fleet and
common carrier
Results: SRITabu
Solution procedure A
Solution procedure B
5,0%
5,0%
4,5%
4,5%
4,0%
4,0%
3,5%
3,5%
2.62%
2,719 sec.
3,0%
2,5%
2.50%
5,715 sec.
2,0%
1.12%
2,879 sec.
1,5%
deviation
deviation
a, b,  and : updated considering
iterations with infeasible or without
a  1 , b  1 ,    7.5log10 n  and   1.5
improvement
no strategy
distance 25%
random 50%
2.59%
2,874 sec.
3,0%
2,5%
2.42%
2,800 sec.
2,0%
1.77%
5,165 sec.
1,5%
1,0%
1,0%
0
20 000 40 000 60 000 80 000 100 000
iterations
0
20 000 40 000 60 000 80 000 100 000
iterations
Conclusion
• Our tabu search (procedure A, random 50% and
100 000 iterations): deviation of 1.12% from the
best-known solution in about 48 min.
• Good results, efficient neighbor reduction
• Solution to a rich variant of VRP
Thanks! Questions?
References
• Bolduc M.-C., Renaud J. & Boctor F.F. "A heuristic for the routing and
carrier selection problem." European Journal of Operational Research,
183, 2007, 926–932.
• Bolduc M.-C., Renaud J., Boctor F.F. & Laporte G. "A perturbation
metaheuristic for the vehicle routing problem with private fleet and
common carriers." Journal of the Operational Research Society, 2006, to
appear.
• Boudia M., Louly M.A.O. & Prins C. "A reactive GRASP and path relinking
for a combined production–distribution problem." Computers &
Operations Research, 34, 2007, 3402–3419.
• Cordeau J.-F., Gendreau M. & Laporte G. "A tabu search heuristic for
periodic and multi-depot vehicle routing problem." Networks, 30, 1997,
105–119.
• Fumero F. & Vercellis C. "Synchronized development of production
inventory, and distribution schedules." Transportation Science, 33, 1999,
330–340.
• Osman I.H. "Metastrategy simulated annealing and tabu search algorithms
for the vehicle routing problem." Annals of Operations Research, 41, 1993,
421–451.