The truck scheduling problem at cross-docking terminals L. - - PowerPoint PPT Presentation

The truck scheduling problem at cross-docking terminals

• L. Berghman, C. Briand, R. Leus and P. Lopez;

PMS 2012

ORSTAT, KU Leuven; CNRS; LAAS

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Outline

• What is cross-docking?
• Exclusive versus mixed mode
• Problem statement and notations
• Time-indexed formulation
• Branch-and-bound
• Computational results
• Conclusions and future research

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

What is cross-docking?

• Items are immediately sorted out, reorganized, based
• n customer demands, and loaded into outbound

trucks

• The storage capacity and the length of the stay of a

product in the warehouse are limited

• Appropriate coordination of inbound and outbound

trucks is needed

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

The truck scheduling problem

• It decides on the succession of truck processing at

the dock doors

• Trucks are allocated to the different docks so as to

minimize the storage usage during the product transfer

• The internal organization of the warehouse is not

explicitly taken into consideration

• We do not model the resources that may be needed

to load or unload the trucks

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Exclusive mode

Each dock serves exclusively either as an outbound dock

• r as an inbound dock throughout the schedules execution

http://people.hofstra.edu/geotrans/eng/ch5en/conc5en/crossdocking.html http://www.lean.org/Common/LexiconTerm.aspx?termid=195

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Mixed mode

Each dock can be used both for loading and unloading

Shakeri M., Low M.Y.H. and Li Z. 2008. A generic model for crossdock truck scheduling and truck-to-door assignment problems. Proc. of the 6th IEEE int.

• conf. on industrial informatics. pp 857-864.

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Problem statement and notations

• A set of incoming trucks i ∈ I need to be unloaded
• A set of outgoing trucks o ∈ O need to be loaded
• The processing time of truck j ∈ I ∪ O equals pj
• Every truck has its release time rj (planned arrival

time) and its deadline ˜ dj (latest allowed departure time)

• There are precedence relations (i, o) ∈ P ⊂ I × O:

wio represents the number of pallets transshipped from i to o

• sj is the starting time of the handling of truck j
• There are n docks that can be used in mixed-mode

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Conceptual problem statement

min

• (i,o)∈P

wio(so − si) subject to sj ≥ rj ∀j ∈ I ∪ O sj + pj ≤ ˜ dj ∀j ∈ I ∪ O so − si ≥ 0 ∀(i, o) ∈ P |At| ≤ n ∀t ∈ T At = {j ∈ I ∪ O|sj ≤ t < sj + pj} the set containing all tasks being executed at time t T the set containing all time instants considered

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Time-indexed (linear) formulation

For all inbound trucks i ∈ I and all time periods τ ∈ Ti, xiτ =      1 if the unloading of inbound truck i is started during time period τ,

• therwise,

with Ti = [ri + 1, ˜ di − pi + 1], the relevant time window for inbound truck i. For all outbound trucks o ∈ O and all time periods t ∈ To, yoτ =      1 if the loading of outbound truck o is started during time period τ,

• therwise,

with To = [ro + 1, ˜ do − po + 1].

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Time-indexed formulation

min

• (i,o)∈P
• τ∈T

wioτ (yoτ − xiτ) subject to

• τ∈Ti

xiτ = 1 ∀i ∈ I (1)

• τ∈To

yoτ = 1 ∀o ∈ O (2)

• τ∈T

τ (xiτ − yoτ) ≤ 0 ∀(i, o) ∈ P (3)

• i∈I

τ

• u=τ−pi+1

xiu +

• ∈O

τ

• u=τ−po+1

you ≤ n ∀τ ∈ T (4) xiτ, yoτ ∈ {0, 1}

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Two different precedence constraints

• τ∈T

τ (xiτ − yoτ) ≤ 0 ∀(i, o) ∈ P

τ

• u=1

xiu −

τ

• u=1

you ≤ 0 ∀(i, o) ∈ P; ∀τ ∈ T

• Aggregated versus disaggregated constraint
• Disaggregated is theoretically stronger
• The additional CPU time needed to solve the larger

linear program does not always counterbalance the significant improvement of the bound

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Branch-and-bound I

• At each node, an uncapacitated cross-docking

problem is considered

min

• (i,o)∈P

wio(so − si) subject to so − si − δi,o ≥ 0 ∀(i, o) ∈ P δi,o ∈ [δi,o, δi,o] ∀(i, o) ∈ P

• The dual of this problem is a max-cost flow problem

that can be solved efficiently, which gives a lower bound.

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Branch-and-bound II

• Initially, [δi,o, δi,o] = [0, ˜

do − ri − po]

• At each node, the relaxed solution is analyzed in
• rder to find a time τ ∗ for which the gate capacity n

is exceeded (if the solution respects the capacity, it is a local optimum);

• Then a pair (i∗, o∗) ∈ P such that either i∗ or o∗ is

in progress at time τ ∗ and wi∗,o∗ is minimal.

• Two child nodes are considered:
• Child 1: [δi,o, δi,o] ←
• ⌈(δi,o − δi,o)/2)⌉, δi,o
• Child 2: [δi,o, δi,o] ←
• δi,o, ⌊(δi,o − δi,o)/2)⌋

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Generation of instances

• n = 10 and |I| = 30
• |O| = α × |I| with α = {0.8, 1, 1.2}
• pi ∈ [β, 30] with β = {10, 20, 30}
• wio ∈ [ 0.8×pi

γ

, 1.2×pi

γ

] with γ ∈ [1, pi]

• ri ∈ [1, δ× pi

n

] with δ = {0.3, 0.6, 0.9}

• ˜

do ∈ [1.5 × do, 8 × do] with do = max(i,o)∈P{ri + po}

• ro ∈ [max(i,o)∈P{ri}, do − po]
• ˜

di ∈ [1.5 × (ri + pi), min(i,o)∈P{do}]

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Preliminary computational results I

• Solving with Cplex (Tcpu ≤ 5 minutes)

|O| β δ # opt # feas # infeas

• bjective value

24 10 0.3 1 5 4 18181 24 10 0.6 9 1 25842 24 10 0.9 1 6 3 40231 24 20 0.3 5 5 25209 24 20 0.6 9 1 38763 24 20 0.9 1 5 4 41819 30 10 0.3 6 4 17919 30 10 0.6 7 3 36644 30 10 0.9 7 3 33085 30 20 0.3 5 5 26200 30 20 0.6 1 8 1 42643 30 20 0.9 8 2 48581 30 30 0.3 1 6 3 39727 30 30 0.6 8 2 71194 30 30 0.9 9 1 71040 36 10 0.3 10 23012 36 10 0.6 7 3 28343 36 10 0.9 5 5 48968 36 20 0.3 8 2 25043 36 20 0.6 1 7 2 43823 36 20 0.9 8 2 49757

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Preliminary computational results II

• Experimenting our branch-and-bound procedure
• First results are encouraging but further

improvements are needed ...

• Solving the problem relaxation is very fast
• But the procedure fails in finding feasible solutions

for almost all the instances

• Nodes are cut only when they become unfeasible
• Improvement ideas:
• Modify the branching strategy to favor deadline

satisfactions

• Use a greedy algorithm for trying to find a feasible

solution at each node (intensification)

• Any other (good) idea is welcomed!

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Conclusions and future research

Conclusions

• Truck scheduling problem at cross-docking terminals
• Time-indexed (integer programming) formulation
• Branch-and-bound

Future research

• Branch-and-bound improvements
• Comparison between the mixed mode strategy and

the exclusive one

• Analysis of the special case pi = p

Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions

Future research: staging

Shakeri M. Truck scheduling problem in logistics of crossdocking. Technical Report NTU-SCE-1101. Nanyang Technological University.