An Integrated Approach to Load Matching, Routing, and Equipment Balancing Problem Sarah Root June 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 HFNO

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 c ij s ≤ c ij d ≤ 2c ij s ji ijij c ij s c ij d c ij 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

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:  x ijk = number of commodity k flowing on arc (i,j)  y ij = number of empty trailers flowing on arc (i,j)  s ij = number of single loads flowing on arc (i,j)  d ij = number of double loads flowing on arc (i,j)

Traditional MCF Formulation Define the following parameters:  c ij s =the cost of a single load flowing on arc (i,j)  c ij d =the cost of a double load flowing on arc (i,j)  b jk =supply or demand of commodity k at node j b jk > 0 if node j has a supply of commodity k b jk < 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  V f =the set of nodes corresponding to facility f

Traditional MCF Formulation min  c ij s s ij  +  c ij d d ij s.t.  x jik -  x ijk = b jk  j in V, k in K s ij + 2d ij =  x ijk + y ij  (i,j) in A  (  b jk +  y ji -  y ji ) = 0  f in F x ijk, y ij, s ij, d ij in Z + (i,j)єA i:( j,i) єA i:(i,j)єA kєKkєKkєKkєK jєVfjєVfjєVfjєVf kєKkєKkєKkєK i:(j,i)єA i:(i,j)єA

Traditional MCF Formulation 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:  x c = the number of cluster c used Define the following parameters:  c c = 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

Alternative Cluster-based Approach min  c c x c s.t.   c l x c = 1  l in L   c f x c = 0  f in F x c in Z + c c c

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 = +

Implementation Details Use of “cluster templates” to create initial clusters Leverage the idea of dominance

Implementation Details

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 Load 1880 Orig: 425 Dest: 411 ED: 2192 LA: 2476 Load 1882 Orig: 425 Dest: 412 ED: 2192 LA: 2800 Travel time 253 Building 93 Building 129

Implementation Details We need to fit these loads into a cluster template  Using original nodes  Using building numbers

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

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

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

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 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

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 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?