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

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

What is cross-docking? • Items are immediately sorted out, reorganized, based on customer demands, and loaded into outbound Cross-docking trucks Modes Notations • The storage capacity and the length of the stay of a Formulation product in the warehouse are limited Branch-and- • Appropriate coordination of inbound and outbound bound Results trucks is needed Conclusions

The truck scheduling problem • It decides on the succession of truck processing at Cross-docking Modes the dock doors Notations • Trucks are allocated to the different docks so as to Formulation minimize the storage usage during the product Branch-and- bound transfer Results • The internal organization of the warehouse is not Conclusions explicitly taken into consideration • We do not model the resources that may be needed to load or unload the trucks

Exclusive mode Each dock serves exclusively either as an outbound dock or as an inbound dock throughout the schedules execution Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions http://people.hofstra.edu/geotrans/eng/ch5en/conc5en/crossdocking.html http://www.lean.org/Common/LexiconTerm.aspx?termid=195

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. Cross-docking conf. on industrial informatics . pp 857-864. Modes Notations Formulation Branch-and- bound Results Conclusions

Problem statement and notations • A set of incoming trucks i ∈ I need to be unloaded Cross-docking • A set of outgoing trucks o ∈ O need to be loaded Modes • The processing time of truck j ∈ I ∪ O equals p j Notations • Every truck has its release time r j (planned arrival Formulation time) and its deadline ˜ Branch-and- d j (latest allowed departure bound time) Results Conclusions • There are precedence relations ( i , o ) ∈ P ⊂ I × O : w io represents the number of pallets transshipped from i to o • s j is the starting time of the handling of truck j • There are n docks that can be used in mixed-mode

Conceptual problem statement � min w io ( s o − s i ) Cross-docking ( i , o ) ∈ P Modes subject to Notations Formulation s j ≥ r j ∀ j ∈ I ∪ O Branch-and- bound s j + p j ≤ ˜ ∀ j ∈ I ∪ O d j Results Conclusions s o − s i ≥ 0 ∀ ( i , o ) ∈ P | A t | ≤ n ∀ t ∈ T A t = { j ∈ I ∪ O | s j ≤ t < s j + p j } the set containing all tasks being executed at time t T the set containing all time instants considered

Time-indexed (linear) formulation For all inbound trucks i ∈ I and all time periods τ ∈ T i ,  1 if the unloading of inbound truck i is started Cross-docking   Modes x i τ = during time period τ, Notations  0 otherwise ,  Formulation with T i = [ r i + 1 , ˜ Branch-and- d i − p i + 1], the relevant time window for bound inbound truck i . Results For all outbound trucks o ∈ O and all time periods t ∈ T o , Conclusions  1 if the loading of outbound truck o is started   y o τ = during time period τ,  0 otherwise ,  with T o = [ r o + 1 , ˜ d o − p o + 1].

Time-indexed formulation � � min w io τ ( y o τ − x i τ ) τ ∈T ( i , o ) ∈ P Cross-docking subject to Modes � x i τ = 1 ∀ i ∈ I (1) Notations τ ∈T i Formulation � y o τ = 1 ∀ o ∈ O (2) Branch-and- bound τ ∈T o Results � τ ( x i τ − y o τ ) ≤ 0 ∀ ( i , o ) ∈ P (3) Conclusions τ ∈T τ τ � � � � x iu + y ou ≤ n ∀ τ ∈ T (4) i ∈ I u = τ − p i +1 o ∈ O u = τ − p o +1 x i τ , y o τ ∈ { 0 , 1 }

Two different precedence constraints Cross-docking � τ ( x i τ − y o τ ) ≤ 0 ∀ ( i , o ) ∈ P Modes τ ∈T Notations τ τ Formulation � � x iu − y ou ≤ 0 ∀ ( i , o ) ∈ P ; ∀ τ ∈ T Branch-and- bound u =1 u =1 Results Conclusions • 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

Branch-and-bound I • At each node, an uncapacitated cross-docking problem is considered Cross-docking Modes � min w io ( s o − s i ) Notations Formulation ( i , o ) ∈ P Branch-and- subject to bound Results s o − s i − δ i , o ≥ 0 ∀ ( i , o ) ∈ P Conclusions δ i , o ∈ [ δ i , o , δ i , 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.

Branch-and-bound II • Initially, [ δ i , o , δ i , o ] = [0 , ˜ d o − r i − p o ] Cross-docking • At each node, the relaxed solution is analyzed in Modes order to find a time τ ∗ for which the gate capacity n Notations is exceeded (if the solution respects the capacity, it is Formulation a local optimum); Branch-and- bound • Then a pair ( i ∗ , o ∗ ) ∈ P such that either i ∗ or o ∗ is Results in progress at time τ ∗ and w i ∗ , o ∗ is minimal. Conclusions • Two child nodes are considered: � � • Child 1: [ δ i , o , δ i , o ] ← ⌈ ( δ i , o − δ i , o ) / 2) ⌉ , δ i , o � � • Child 2: [ δ i , o , δ i , o ] ← δ i , o , ⌊ ( δ i , o − δ i , o ) / 2) ⌋

Generation of instances • n = 10 and | I | = 30 Cross-docking Modes • | O | = α × | I | with α = { 0 . 8 , 1 , 1 . 2 } Notations • p i ∈ [ β, 30] with β = { 10 , 20 , 30 } Formulation • w io ∈ [ 0 . 8 × p i , 1 . 2 × p i ] with γ ∈ [1 , p i ] Branch-and- γ γ bound • r i ∈ [1 , δ × � p i Results ] with δ = { 0 . 3 , 0 . 6 , 0 . 9 } n Conclusions • ˜ d o ∈ [1 . 5 × d o , 8 × d o ] with d o = max ( i , o ) ∈ P { r i + p o } • r o ∈ [max ( i , o ) ∈ P { r i } , d o − p o ] • ˜ d i ∈ [1 . 5 × ( r i + p i ) , min ( i , o ) ∈ P { d o } ]

Preliminary computational results I • Solving with Cplex ( T cpu ≤ 5 minutes) | O | # opt # feas # infeas objective value β δ 24 10 0.3 1 5 4 18181 Cross-docking 24 10 0.6 0 9 1 25842 Modes 24 10 0.9 1 6 3 40231 Notations 24 20 0.3 0 5 5 25209 24 20 0.6 0 9 1 38763 Formulation 24 20 0.9 1 5 4 41819 Branch-and- 30 10 0.3 0 6 4 17919 bound 30 10 0.6 0 7 3 36644 Results 30 10 0.9 0 7 3 33085 Conclusions 30 20 0.3 0 5 5 26200 30 20 0.6 1 8 1 42643 30 20 0.9 0 8 2 48581 30 30 0.3 1 6 3 39727 30 30 0.6 0 8 2 71194 30 30 0.9 0 9 1 71040 36 10 0.3 0 10 0 23012 36 10 0.6 0 7 3 28343 36 10 0.9 0 5 5 48968 36 20 0.3 0 8 2 25043 36 20 0.6 1 7 2 43823 36 20 0.9 0 8 2 49757

Preliminary computational results II • Experimenting our branch-and-bound procedure • First results are encouraging but further Cross-docking improvements are needed ... Modes • Solving the problem relaxation is very fast Notations • But the procedure fails in finding feasible solutions Formulation Branch-and- for almost all the instances bound • Nodes are cut only when they become unfeasible Results • Improvement ideas: Conclusions • 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!

Conclusions and future research Conclusions Cross-docking • Truck scheduling problem at cross-docking terminals Modes Notations • Time-indexed (integer programming) formulation Formulation • Branch-and-bound Branch-and- bound Future research Results • Branch-and-bound improvements Conclusions • Comparison between the mixed mode strategy and the exclusive one • Analysis of the special case p i = p

Future research: staging Cross-docking Modes Notations Formulation Branch-and- bound Results Conclusions Shakeri M. Truck scheduling problem in logistics of crossdocking. Technical Report NTU-SCE-1101. Nanyang Technological University.