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

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

Sarah RootJune 8, 2005

Joint work with advisor Amy M. Cohn

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

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

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

HF

NO

HF

NO

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

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

Planning Process

Load matching problem

cijs ≤ cij

d ≤ 2cijs

ji

i j

i j

cijs

cijd

cijs cij

s

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

Planning Process Load routing problem

Tractors can stop at intermediate nodes to reconfigure the trailers it pulls Time feasibility—travel time and allowances

Relationship between load matching and load routing Which loads should be matched together? How should

these loads be routed? Strongly interconnected—both decided simultaneously

1 2

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Load 1 Load 1Load 3

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Load 1 Load 1Load 3

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Load 1 Load 1Load 3

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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

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

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

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)


Define the following parameters: cij

s =the cost of a single load flowing on arc (i,j) cij

d=the cost of a double load flowing on arc (i,j) bjk =supply or demand of commodity k at node j

bjk> 0 if node j has a supply of commodity k

bjk < 0 if node j has a demand for commodity k A=the set of all arcs (i,j) V=the set of all nodes j F=the set of all facilities f K=the set of all commodities k Vf =the set of nodes corresponding to facility f


min cijs sij+ cij

d 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)(i,j)єєAA (i,j)(i,j)єєAA

i:(i:(j,i)j,i)єєAA i:(i,j)i:(i,j)єєAA

kkєєKK

jjєєVVff kkєєKK i:(j,i)i:(j,i)єєAA i:(i,j)i:(i,j)єєAA


Large number of constraints (|V||K|+|A|+|F|) 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

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

Alternative Cluster-based Approach

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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 c

l = 1 if cluster c contains load l; 0 otherwise c

f = 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


min cc xc

s.t. cl xc = 1 l in L

cf xc = 0 f in F

xc in Z+

cc

cc

cc

Alternative Cluster-based Approach For a given cluster, it is easy to identify

cost and whether or not it is time feasible |L|+|F| 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

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 clusters—time 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

Alternative Cluster-based Approach Indifference

Minimally dependent—cannot 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’’| > 0

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Implementation Details

A BABA BABA BABABA BABABABABA B∅A A∅A B∅∅A A∅∅∅∅A BAB∅A AAA∅AA∅A BAC∅ AAAAAA AAA∅A AAA∅AA∅ AAAAAAAAAAACB CACBCAACAACB CACBCB CACBCACBCA BABAC BACA BABACA BABACABAC BACBACA BABAC CAC∅A AAAAAAAAA AAA∅AA∅

A BABA BAB

A BAB ABA BAB ABAB AB

A B∅A B∅

A B∅ ∅A B∅ ∅∅ ∅

A BAB ∅A BAB ∅AB ∅ A BAC ∅ CAC BCA BAC ∅A BAC ∅AC ∅ CAC BCAC BC

A AC B CAC BCA ACA AC B CAC BCB CAC BCAC BC

A BAB AC CAC ∅A BAB ACAB AC CAC ∅AC ∅

A BAB AC CACA BAB ACA BAB ACAB AC CAC CAC

Use of “cluster templates” to create initial clusters

Leverage the idea of dominance


B CAC BDA AC DBDB CAC BDAC BDA AC DBD

B CAC BD DBD ∅A AC B CAC BD DBD ∅B CAC BDAC BD DBD ∅BD ∅A AC

A BAC ∅ CAC BD DBDA BAC ∅A BAC ∅AC ∅ CAC BDAC BD DBD

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A BAC ∅ CACA BAC ∅A BAC ∅AC ∅ CAC CAC


In the data we’ve 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

Implementation Details Consider an example

406 407

425

424 426

Building 93

406 407

425

424 426

406 407

425

424 426

406 407

425

424 426

Building 93

410 412

411 413

Building 129

410 412

411 413

410 412

411 413

410 412

411 413

Building 129

Load 1880

Orig: 425Dest: 411ED: 2192LA: 2476

Load 1882

Orig: 425Dest: 412ED: 2192LA: 2800

Travel time

253

410 412

411 413

Building 129

410 412

411 413

410 412

411 413

410 412

411 413

Building 129

406 407

425

424 426

Building 93

406 407

425

424 426

406 407

425

424 426

406 407

425

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Building 93Building 93 Building 129


We need to fit these loads into a cluster template Using original nodes

Using building numbers

425 4111880 1882 4121882

93 1291880 1882


Three legs in the cluster if we consider moves between node numbers

Single leg in the cluster if we consider moves between building numbers

410 412

411 413

Building 129

410 412

411 413

410 412

411 413

410 412

411 413

Building 129

406 407

425

424 426

Building 93

406 407

425

424 426

406 407

425

424 426

406 407

425

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Building 93


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

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 objective—$482,849 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


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

combinations—does not converge to a provably optimal solution in 2 hours

UPS solution without load routing—$609,854 UPS solution with load routing—$585,128 Michigan solution—$584,881


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 we’ve been using Looser time windows in the US network allow for

more potential to match loads and realize cost savings

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

clusters—column generation Further integrate steps in the planning

process Assigning volume to trailers

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

Questions?

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

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