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Introduction Mathematical Formulation Methodology Results Conclusions

A Column Generation Algorithm to Solve aSynchronized Log-Truck Scheduling Problem

Odysseus 2012

Greg Rix12 Louis-Martin Rousseau12 Gilles Pesant13

1Interuniversity Research Centre on Enterprise Networks, Logistics andTransportation (CIRRELT)

2Department of Mathematics and Industrial Engineering, École Polytechnique deMontréal

3Department of Computer Engineering, École Polytechnique de Montréal

May 22, 2012

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 1/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 2/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 3/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

MotivationTactical tool for planning transportation in the Canadianforestry industry which

takes into account the annual harvesting schedule,and produces an annual transportation plan divided into 262-week periods.

Multiple ObjectivesMinimize fleet size (heterogenous).

but balance fleet over all periodsMinimize total cost:

Storage costs (at mills and forests).Transportation costs:

Maximizing backhaul opportunities.Minimizing waiting times.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 4/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

MotivationTactical tool for planning transportation in the Canadianforestry industry which

takes into account the annual harvesting schedule,and produces an annual transportation plan divided into 262-week periods.

Multiple ObjectivesMinimize fleet size (heterogenous).

but balance fleet over all periodsMinimize total cost:

Storage costs (at mills and forests).Transportation costs:

Maximizing backhaul opportunities.Minimizing waiting times.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 4/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

Backhaul OpportunitiesWhen harvest plans are created, transportation costs areestimated (and minimized) based on delivering wood viaout-and-back routes.However, the use of backhaul has been seen to decreasetransportation costs by up to 30%.

Andersson et al. 2008.

Perhaps it is possible to plan wood flows in order togenerate backhaul opportunities...

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 5/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Backhaul

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 6/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Backhaul

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 6/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Backhaul

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 6/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Backhaul

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 6/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

Tactical LevelWhen to transport the wood which has been harvested ?From which forest should each mill be supplied ?How much wood should be stored where, when ?

Operational LevelWhat are the exact routes which should be operated ?Which truck should be used (self-loading or not) ?At which time should trucks be loaded or unloaded ?

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 7/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

Tactical LevelWhen to transport the wood which has been harvested ?From which forest should each mill be supplied ?How much wood should be stored where, when ?

Operational LevelWhat are the exact routes which should be operated ?Which truck should be used (self-loading or not) ?At which time should trucks be loaded or unloaded ?

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 7/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

Related Routing Problems (operational level)Pickup and Delivery

with K commodities (assortments),full truck loads and split deliveries and sync. constraints,define on a small network,but there high volumes means many trips.

InputsHarvest plan and mill demands:

Multiple forest sites (supply points).Multiple mills (demand points).Multiple log assortments: species, quality, length, diameter(commodities).

Truck types (capacity, self-loading or not):

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 8/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Introduction

Related Routing Problems (operational level)Pickup and Delivery

with K commodities (assortments),full truck loads and split deliveries and sync. constraints,define on a small network,but there high volumes means many trips.

InputsHarvest plan and mill demands:

Multiple forest sites (supply points).Multiple mills (demand points).Multiple log assortments: species, quality, length, diameter(commodities).

Truck types (capacity, self-loading or not):

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 8/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Relevant LiteratureThe pickup and delivery problem, which is NP-hard(Savelsbergh and Sol 1995), reduces to our problem.Column generation has previously been applied to thelog-truck scheduling problem:

Palmgren et al. 2004.Weintraub et al. 2009.

The synchronized log-truck scheduling problem has beenmodeled with integer programming and constraint basedlocal search:

El Hachemi 2009.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 9/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 10/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Variables

VariablesWood allocation from forest to mill for each period:

Sometimes geographically fixed.

Inter-period inventory at each node (forests and mills).Construction of log-truck routes to make deliveries:

Heterogeneous fleet.

Assignment of log-loaders to forests each period (millshave permanent loaders).

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 11/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Constraints

Global ConstraintsSatisfy mill demands.Not exceed forest supply.Wood delivered on time after harvest.

Modeled either with a constraintsor storage costs at forests sites.

Balanced number of routes each day.Limited number of loaders in forest each period.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 12/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Constraints

Routing Constraints

Trucks must begin and end their shift at the same mill:Mills act as depots.

Respect mill operating hours.Each loader can only serve one truck at a time (otherwisewaiting costs accrued).

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 13/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Route Variables

Route VariablesEach variable yjpd is the number of times route j istraversed on day d of period p.A route starts at a mill and iterates between forests andmills, returning to the same mill.The number of these variables is exponential, andenumerating them all would be impractical.We dynamically generate improving routes through columngeneration.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 14/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Full ModelX

p2P

X

d2Dp

X

j2J

cjyjpd +X

m2M

X

p2P

X

d2Dp

X

t2T

ctTmpdt

+X

f2F

X

l2L

X

p2P

cflpzflp +X

m2M

X

l2L

X

p2P

cmlpzmlp (1)

subject to:

zml0 = iml, 8 m 2M, l 2 L, (2)

zfl0 = ifl, 8 f 2 F, l 2 L, (3)

zmlp +X

f2F

X

t2T

xfmlpt � dmlp = zml,p+1, 8 m 2M, l 2 L, p 2 P , (4)

zflp + hflp �X

m2M

X

t2T

xfmlpt = zfl,p+1, 8 f 2 F, l 2 L, p 2 P , (5)

min{wflp,p�1}X

w=1

hfl,p�w � zflp, 8 f 2 F, p |P | + 1, (6)

X

j2Jt

ktlafmlj

X

d2Dp

yjpd � xfmlpt, 8 f 2 F, m 2M, l 2 L, p 2 P, t 2 T ,

(7)X

j2Jmt

yjpd = Tmpdt, 8 m 2M, p 2 P, d 2 Dp, t 2 T (8)

Tmpdt nTmpt, 8 m 2M, p 2 P, d 2 Dp, t 2 T (9)

X

m2M

X

t2T

Tmpdt �1� ✏X

p2P

|Dp|X

m2M

X

p2P

X

d2Dp

X

t2T

Tmpdt, (10)

8 p 2 P, d 2 Dp,X

m2M

X

t2T

Tmpdt 1 + ✏X

p2P

|Dp|X

m2M

X

p2P

X

d2Dp

X

t2T

Tmpdt, (11)

8 p 2 P, d 2 Dp,X

f2F

Lfp nLp , 8 p 2 P , (12)

8

X

j2Jt

Ujmiyjpd nLmp, 8 m 2M, p 2 P, d 2 Dp, i 2 I, (13)

X

t2T

vt

X

j2J

Ljfiyjpd Lfp, 8 f 2 F, p 2 P, d 2 Dp, i 2 I, (14)

X

t2T

vt

X

d2Dp

X

m2M

X

l2L

xfmlpt ⌦Lfp, 8 f 2 F, p 2 P , (15)

X

m2M

X

j2Jmt

X

i2I

Ujmiyjpd X

m2M

X

j2Jmt

X

i2I

Ujmiyjp,d+1, (16)

8 p 2 P, d |Dp|� 1,

Lfp 2 {0, 1}, 8 f 2 F, p 2 P . (17)

yjpd, Tmpdt 2 Z+, 8 m 2M, j 2 Jm, p 2 P, d 2 Dp, t 2 T . (18)

zmlp, zflp, xfmlp 2 R+, 8 f 2 F, m 2M, l 2 L, p 2 P . (19)

We denote this problem by (P ). The objective function (1) minimizes total

costs associated with driving and storage. Constraints (2) and (3) set the initial

inventories at the mills and forests, respectively. Constraints (4) and (5) link the

storage variables of successive periods at the mills and forests, respectively. The

non-negativity of all variables ensure that forest supply and mill demands are

respected. Constraints (6) ensure that wood is not left at the forest longer than

allowed. Constraints (7) force the quantity delivered to respect the capacities of

all trucks making that trip. Constraints (8) fix the number of routes originating

from each mill in each period, and constraints (9) bound this by the associated

availability. Constraints (10) and (11) ensure a balanced schedule in terms of

the number of truck routes traversed every day of the horizon. Constraints (12)

limit the total number of loaders assigned to forests in a period. Constraints

(13) and (14) assign each loader to only one truck at any time. Constraints

(15) are redundant constraints that force a loader to be assigned to a forest in

any period in which a truck requires one, with ⌦ a su�ciently large constant.

Constraints (16) break the symmetry between the days that define a period.

Constraints (17) force the loader-to-forest assignment variables to be binary.

Finally, constraints (18) and (19) enforce the non-negativity of the other vari-

ables, as well as discretize those that count log-truck routes. We denote by Z+

and R+ the sets of non-negative integers and reals, respectively.

We note that, in cases where the truck fleet is homogenous, we will express

volumes in truckloads as is more common in LTSP literature. The only necessary

9

initial inventory

Storage F & M

Freshness

Truck capacity and availability

Route balancing

F loaders used

Loaders-Truck sync.

Loader needed in F

Day symmetry breaking

Variable definition

Exponentiel # of y

Driving Cost Fixed Cost

Forest Storage Mill Storage

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 15/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 16/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Methodology

Column GenerationThe methodology used on this problem is columngeneration, a procedure for solving large linear programs.We discretized time in into 40 minutes slices (roughly(un)loading time).We first relax our problem to a linear program:

Partial routes allowed.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 17/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Column Generation

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 18/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Methodology

Initial Route PoolWe start with a subset of the variables:

No backhaul.No attempt to minimize waiting times.Loader constraints relaxed with a penalty in the objectivevalue.

We then generate improving variables as needed via asubproblem.

A variable can only improve the objective function if it has anegative reduced cost.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 19/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Column Generation

Optimality GapTo solve the original MIP to optimality would require abranch and price algorithm:

Barnhart et al. 1998

However solving the linear relaxation to optimality withcolumn generation yields a lower bound on the problem.This gives us an optimality gap on any integer solutionsfound in the final MIP.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 20/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Computing the Reduced Costs

Reduced Cost of a RouteHow do we determine the reduced cost of the routeassociated with variable yjpd?Equal to the cost of the route subtracted by a set of dualvalues associated with:

Starting a shift at the depot mill on day d .Each assortment-forest-mill delivery on the route.Loading capacity at each forest for each time slide.Unloading at each mill capacity for each time slice.

If a route has negative reduced cost, we add it to the LP.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 21/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Dynamically Generating Routes

Column Generation SubproblemDetermining negative reduced cost routes can be modeledas a set of Shortest Path Problem with ResourceConstraints (SPPRC).Much easier than the Elementary SPPRC common invehicle routing problems:

Large volumes allow cycles.

Solved using a dynamic programming (DP)-based labelingalgorithm (Ahuja et al. 1993) on a space-time networkrepresentation of the problem.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 22/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Space-Time Network Representation

AssortmentsWe don’t have to represent the assortments explicitly, as wecan pick the one with the smallest dual value at each node.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 23/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Space-Time Network Representation

AssortmentsWe don’t have to represent the assortments explicitly, as wecan pick the one with the smallest dual value at each node.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 23/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Column Generation Parameters

StabilizationTo generate interior dual values, we solve the linearprograms using a barrier methodology.

Column Pool ManagementRather than all negative reduced cost routes generated ateach iteration, only the best 500 are added to the LP.Columns are removed for the pool if they have not beenused in 30 consecutive LP solutions.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 24/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 25/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Case Studies

Data from industrial partner FPI6 case studies provided by FPInnovations.Data from companies in Quebec, New Brunswick and B.C.

Instance |F | |M| |L| VW1 6 5 3 29,745W2 6 5 3 16,065A1 43 7 5 722,531A2 8 1 1 372,670A3 8 1 2 462,272A4 3 1 3 743,600

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 26/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Resources

Hardware and SoftwareLP and MIP Solver: Gurobi 4.6.Time limit: 40 minutes.Machine: intel core i7, 2.67 GHz processor with 4.0 GB ofmemory.

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 27/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Results

ComparisonMethodology compared against two-phase decomposed IPapproach (El Hachemi, 2009):

Instance Gap Colgen ImprovementW1 0.87% 1.05%W2 3.53% −2.42%A1 0.11% 0.55%A2 0.28% 8.25%A3 0.12% 3.43%A4 0.17% −−−

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 28/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Results

ComparisonMethodology compared against two-phase decomposed IPapproach (El Hachemi, 2009):

Instance Gap Colgen ImprovementW1 0.87% 1.05%W2 3.53% −2.42%A1 0.11% 0.55%A2 0.28% 8.25%A3 0.12% 3.43%A4 0.17% −−−

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 28/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Outline

1 Introduction

2 Mathematical Formulation

3 MethodologyColumn GenerationComputing the Reduced CostDynamically Generating Routes

4 Results

5 Conclusions

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 29/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Conclusions

RemarksWe can generate near-optimal solutions to this multi-periodsynchronized LTSP in reasonable time.Preliminary integration with FPI’s FPSuite software.Ongoing negotiation for deployment in western Canada.

Future workSynchronizing tactical transportation decisions withharvest scheduling.Generalizing the routing constraints to allow forheterogeneous driver profiles in addition to truck fleet.(allow for planning of workforce).

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 30/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Conclusions

RemarksWe can generate near-optimal solutions to this multi-periodsynchronized LTSP in reasonable time.Preliminary integration with FPI’s FPSuite software.Ongoing negotiation for deployment in western Canada.

Future workSynchronizing tactical transportation decisions withharvest scheduling.Generalizing the routing constraints to allow forheterogeneous driver profiles in addition to truck fleet.(allow for planning of workforce).

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 30/ 31

Introduction Mathematical Formulation Methodology Results Conclusions

Questions?

Greg Rix, Louis-Martin Rousseau, Gilles Pesant 31/ 31