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An Integrated Approach to Load Matching, Routing, and Equipment Balancing Problem

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Title: An Integrated Approach to Load Matching, Routing, and Equipment Balancing Problem


1
An Integrated Approach to Load Matching, Routing,
and Equipment Balancing Problem
  • Sarah Root
  • June 8, 2005
  • Joint work with advisor Amy M. Cohn

2
Overview
  • Problem description
  • Load matching, routing and equipment balancing
    problems
  • Literature review
  • Modeling approaches
  • Traditional multi-commodity flow model
  • Alternative cluster-based modeling approach
  • Implementation details
  • Initial computational results
  • Future research goals and directions

3
Planning Process
  • Load planning (Package routing)
  • Determine routing or path for each package
  • Service commitments and sort capacities must not
    be violated
  • Load matching (FeederOpt)
  • Match loads together to leverage cost
    efficiencies
  • Assign volume to a trailer type
  • Equipment balancing
  • Delivering loads from origin to destination
    causes some areas of the network to accumulate
    trailers and others to run out
  • Redistribute trailers so that no such imbalances
    occur
  • Driver scheduling (FSOS)
  • Take output of load matching and equipment
    balancing problems and assign drivers to each
    tractor movement
  • Load planning (Package routing)
  • Determine routing or path for each package
  • Service commitments and sort capacities must not
    be violated
  • Load matching (FeederOpt)
  • Match loads together to leverage cost
    efficiencies
  • Assign volume to a trailer type
  • Equipment balancing
  • Delivering loads from origin to destination
    causes some areas of the network to accumulate
    trailers and others to run out
  • Redistribute trailers so that no such imbalances
    occur
  • Driver scheduling (FSOS)
  • Take output of load matching and equipment
    balancing problems and assign drivers to each
    tractor movement

4
Current Solution Approach
  • Ensuring a feasible solution to the problem is
    difficult!
  • Minimizing cost is even harder!!!
  • Assume all data is deterministic
  • Break problem into sequential problems
  • Solve each problem individually
  • Renders problem tractable
  • Fails to capture interaction between different
    levels of the planning process, negatively
    impacting solution quality

5
Research Goals
  • Improve solution quality by integrating different
    steps of the planning process
  • Develop novel modeling approaches and algorithms
    to exploit underlying problem structure
  • We have initially integrated the load matching,
    load routing, and equipment balancing stages of
    the planning process (LMREBP)
  • Allows loads and empties to be routed together to
    realize cost savings
  • Improve solution quality by integrating different
    steps of the planning process
  • Develop novel modeling approaches and algorithms
    to exploit underlying problem structure
  • We have initially integrated the load matching,
    load routing, and equipment balancing stages of
    the planning process (LMREBP)
  • Allows loads and empties to be routed together to
    realize cost savings

6
Planning Process
  • Load matching problem

cijs
  • Assume all volume is assigned to 28 trailers
  • Non-linear cost structure single trailer
    combination vs. double trailer combination
  • Each load must be delivered to its destination
    within its time window

cijs
cijs
cijd
cijs cijd 2cijs
7
Planning Process
  • Load routing problem
  • Tractors can stop at intermediate nodes to
    reconfigure the trailers it pulls
  • Time feasibilitytravel time and allowances
  • Relationship between load matching and load
    routing
  • Which loads should be matched together? How
    should these loads be routed?
  • Strongly interconnectedboth decided
    simultaneously

8
Planning Process
  • Equipment balancing problem
  • Some nodes in the network have more inbound loads
    than outbound loads these nodes accumulate
    trailers
  • Some nodes in network the have more outbound
    loads than inbound loads these nodes run out of
    trailers
  • How should empty trailers be redistributed such
    that each node in the network is balanced?
  • Assume trailers do not have time windows

9
Literature Review
  • Specific LMREBP not considered in the literature
    to the best of our knowledge
  • Bodies of related literature
  • General multi-commodity flows
  • Multi-commodity flows with non-linear arc costs
  • Express package industry
  • Time windows
  • Empty balancing
  • See prelim proposal for a more detailed
    literature review

10
Traditional Approach to Modeling LMREBP
  • LMREBP is at its core a multi-commodity flow
    (MCF) problem
  • More difficult than a traditional MCF problem
  • Time windows
  • Non-linear cost structure
  • Network size
  • 2,500 nodes 24,000 arcs 15,000 commodities
    routed daily in the United States network
  • Traditional MCF formulation can be modified to
    capture the LMREBP

11
Traditional MCF Formulation
  • Let a node j represent a location and time
  • Define the following variables
  • xijk number of commodity k flowing on arc (i,j)
  • yij number of empty trailers flowing on arc
    (i,j)
  • sij number of single loads flowing on arc (i,j)
  • dij number of double loads flowing on arc (i,j)

12
Traditional MCF Formulation
  • Define the following parameters
  • cijs the cost of a single load flowing on arc
    (i,j)
  • cijdthe cost of a double load flowing on arc
    (i,j)
  • bjk supply or demand of commodity k at node j
  • bjkgt 0 if node j has a supply of commodity k
  • bjk lt 0 if node j has a demand for commodity k
  • Athe set of all arcs (i,j)
  • Vthe set of all nodes j
  • Fthe set of all facilities f
  • Kthe set of all commodities k
  • Vf the set of nodes corresponding to facility f

13
Traditional MCF Formulation
  • min ? cijs sij ? cijd dij
  • s.t. ? xjik - ? xijk bjk ? j in V, k in K
  • sij 2dij ? xijk yij ? (i,j) in A
  • ? (? bjk ? yji - ? yji ) 0 ? f in F
  • xijk, yij, sij, dij in Z

(i,j)?A
(i,j)?A
i(j,i)?A
i(i,j)?A
k?K
j?Vf
k?K
i(j,i)?A
i(i,j)?A
14
Traditional MCF Formulation
  • Large number of constraints (VKAF)
  • Huge number of nodes and large number of
    commodities!
  • VERY fractional LP relaxation
  • Incentive to send ½ double trailers instead of
    single trailers because of cost structure
  • Problem does not converge to a feasible solution
    even after relaxing time requirements
  • Motivates the need for an alternative modeling
    approach

15
Alternative Cluster-based Approach
  • Instead of considering the movement of trailers
    along each arc, consider groups of trailers which
    move together
  • A cluster is a set of loads, a set of empties,
    the routes they take, and the tractor
    configurations that pull them
  • Every load in the cluster moves completely from
    origin to destination
  • Only define clusters which are feasible

16
Alternative Cluster-based Approach
17
Alternative Cluster-based Approach
18
Alternative Cluster-based Approach
19
Alternative Cluster-based Approach
  • Let a node represent a facility (i.e., physical
    location)
  • Define the following variables
  • xc the number of cluster c used
  • Define the following parameters
  • cc cost of cluster c
  • ?cl 1 if cluster c contains load l 0 otherwise
  • ?cf the impact of cluster c on trailer balance
    at facility f
  • L the set of all loads l
  • F the set of all facilities f

20
Alternative Cluster-based Approach
  • min ? cc xc
  • s.t. ? ?cl xc 1 ? l in L
  • ? ?cf xc 0 ? f in F
  • xc in Z

c
c
c
21
Alternative Cluster-based Approach
  • For a given cluster, it is easy to identify cost
    and whether or not it is time feasible
  • LF constraints
  • Much stronger LP relaxation
  • Moves as part of a ½ double combination
    prohibited where both loads do not exist
  • Converges quickly to an integer solution
  • Flexibility in further expanding the problem
    scope
  • Non-facility meets, triple trailers, allowance
    times

22
Alternative Cluster-based Approach
  • Though the number of variables is very large,
    there are a number of ways to address this
    obstacle
  • Feasibility
  • Huge number of ways in which loads and empty
    trailers can be combined
  • Many of these ways are infeasible
  • Particularly true in complex clusterstime to
    reconfigure tractors and drive circuitous mileage
    can lead to violation of loads time windows
  • Dominance
  • Many ways to combine a given set of loads
  • Each potential combination corresponds to the
    same column in the constraint matrix
  • We only need to consider the cluster with the
    cheapest cost, as it would be chosen in an
    optimal solution

23
Alternative Cluster-based Approach
  • Indifference
  • Minimally dependentcannot be broken into smaller
    pieces
  • For all sets of trailers T? T and T? T such
    that T?? T ?? and T ? T T, there is
    at least one leg containing both a trailer from
    T and T if T,T gt 0

24
Implementation Details
  • Use of cluster templates to create initial
    clusters
  • Leverage the idea of dominance

25
Implementation Details
26
Implementation Details
  • In the data weve been given, nodes represent
    sorts
  • Multiple sorts can occur in the same building
    throughout a day these will correspond to
    multiple nodes
  • We balance each building, not each node!
  • This node definition can limit matching
    opportunities when we use cluster templates

27
Implementation Details
  • Consider an example

Building 93
Building 129
28
Implementation Details
  • We need to fit these loads into a cluster
    template
  • Using original nodes
  • Using building numbers

29
Implementation Details
  • Three legs in the cluster if we consider moves
    between node numbers
  • Single leg in the cluster if we consider moves
    between building numbers

30
Implementation Details
  • Benefits in initially considering moves between
    building numbers
  • More opportunities to match loads together in
    clusters
  • 10,000 clusters using original node numbers
  • 50,000 clusters using building numbers
  • Equipment balancing is done at the building
    level, not the node level

31
Computational Results
  • Moderately sized data set
  • 2,000 loads 600 nodes 250 buildings
  • Time windows extremely tight for the loads
  • Lower bound on the optimal objective482,849
  • Route each load directly from origin to
    destination at cost of half a double combination
  • Balance the empties in the system using a
    transportation problem where arc costs are for
    half double combinations
  • Add cost of routing loads and balancing empties

32
Computational Results
  • Traditional MCF model did not converge to an
    integer feasible solution
  • Alternative formulation converged to an integer
    solution within 15 seconds
  • Within 2 minutes, within 1 of the optimal
    solution using the subset of clusters
  • Still incentive to split empties into ½ double
    empty combinationsdoes not converge to a
    provably optimal solution in 2 hours
  • UPS solution without load routing609,854
  • UPS solution with load routing585,128
  • Michigan solution584,881

33
Computational Results
  • More than 75 of trailers move at least one leg
    as part of a double combination
  • Solution will improve as we add new cluster
    templates
  • Addition of a single cluster template has saved
    5,000 in this network
  • Ideas for cluster templates are appreciated!
  • Extremely tight time windows in the data set
    weve been using
  • Looser time windows in the US network allow for
    more potential to match loads and realize cost
    savings

34
Future Research Directions
  • Finish up this portion of the research
  • Include new cluster types when solving the
    problem
  • Leverage symmetry of the problem
  • Investigate sensitivity of solution to time
    windows
  • Move to larger networks (e.g., US network)
  • Use dual information to generate promising
    clusterscolumn generation
  • Further integrate steps in the planning process
  • Assigning volume to trailers

35
Summary
  • Integrated multiple steps in a complex planning
    process
  • Load matching, routing and equipment balancing
  • Developed a novel modeling approach that
    overcomes the traditional difficulties associated
    with traditional modeling approach
  • Framework can capture complexities associated
    with the real-world decisions to be made and
    allows us to extend the problem scope
  • Initial computational results are promising
  • Can quickly find a good solution
  • More matching opportunities in the US network

36
Questions?
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