06Consolidation TL and LTL Shipment Column Generation Approach

7/30/2019 06Consolidation TL and LTL Shipment Column Generation Approach

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Consolidation

John H. Vande Vate

Spring, 2007


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Motivations

Speed

LTL shipments are consolidated, routed to

intermediate terminals, sorted,

TL shipments can be faster

Cost

Remember concave cost structure

Typically TL is less expensive per unit


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Context

Manufacturer/Distributor shipping to regular

customers

Default Option: LTL shipment to each customer

Consolidation: TL several order to LTL terminal near customers

LTL from terminal to customers

Typically not dynamic:

Where is the customer?

How large is the order?


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Interrelated Questions

Where do we consolidate (what

terminals)?

Which customers (orders) do we servethrough each terminal?


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Assumptions

Single Plant or origin for suppliesWe are not allocating customers to

production plants. Thats already been done

We know our customersNot always the case

Can use geographic regions in place ofactual customer locations

We have adjusted last years orders toreflect next years projections


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A Model

Identify a candidate set of consolidation points (terminals) More choices allows exploring more options

More choices slows computation

On the order of 30 say

Key Decisions Open: Do we use a candidate consol pt or not?

One for each candidate consol pt

Assign: Does a consol pt serve a customer or not?

One for each candidate consol pt and (reasonably close) customer

ServeDirect: Do we serve the customer directly via LTL or not?

One for each

Trucks: Annual (say) number of TL shipments to a candidate consol

pt.

One for each candidate consol pt.


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Objective: Transportation Cost

LTL shipments direct to customers Easy to rate these, weve been shipping this way

Recommend using rating engine to rate them anyways

Compute discount rate: DR = (Rated Cost -Actual Cost)/Rated Cost

Cost to serve * Serve Direct

Truck load shipments to consolidation points Might use $/mile and get distances from PC Miler or CzarLite

Might distinguish by region of country

Cost per truck * Trucks to Consol Pt.

LTL shipments from consol pts to customers These are painful to get

Use rating engine to rate historical shipments apply discount rate DR

Cost to Serve from Consol pt * Assign to Consol pt


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Elaborations Consider inventory costs

Handling charges at consol pts

Amortized capital charge or rent for consol pts

Time to customer


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Transport Requirements

TL shipments cost depends on capacity

How many trucks

Homogeneous commodity Either weight or cube or floor space drives

capacity Translate each customers annual demand into a

demand for this unit of capacity, e.g., weight

Heterogeneous commodities

Treat like homogeneous commodity based onbasket of products or

Translate each customers annual demand intoweight and cube (or floor space)


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Constraints

Every Customer is Served

For each customer:

ServeDirect + Sum over consol pts Assign = 1

Trucks required to each consol pt

For each consol pt (and type of capacity,

e.g., weight, cube, floor space)

Trucks*Load Factor Sum over customersAssign*Requirement/Capacity

Demand

varies.

Trucks

wont be full

e.g.,Weight of

customers

orders

e.g.,Weight

limit of Truck


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Frequency

Time matters

Minimum level of service to consol pt E.g., once per week or thrice per week

Amounts to a fixed (operating cost) foropening a consol pt.

ServiceLevelConstraint:

For each consol ptTrucks Minimum Service level*Open

e.g.,156 = 52*3


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Logic

Cant assign a customer to a consol pt

unless it is open

For each customer and consol pt (withinreason)

Assign Open


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Peculiarities

Typical of integer optimization

Does strange things to ensure we get the most

out of the fixed operating cost associated with

opening a consol pt.See assignments bypassing consol pts

Adding a nearby customer may force us to use

another truck, but adding a smaller one fartheraway may not


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Odd Assignments

Reasonable to use recommended consol pts?

Reasonable to use recommended

assignments?

Plant in

FL!?*


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Translation to Implementation

Suggests the value of dynamic

assignments that change from week to

week Reasonable to drop integrality of Assign

One Project: Implement and evaluate the

impact of dynamic assignments


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Next Step: Multi-Stop Routes

Can we improve performance by sharingthe fixed operating cost across severalconsol pts

Advantage: Allows smaller consol pts Disadvantage: Lower efficiency in TLshipmentsDo you really want to run trucks half empty

half way across the country?Stop charges: e.g., $50 per stop


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Typical Multi-Stop Rt

Clustered

destinations


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Objective

Transportation Costs

TL to Consol Pts

Multi-stop TL to Consol Pts LTL to Consol

Pts

LTL Direct to Customer

Multi-Stop TL costs include

Mileage charge

Stop charges


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Constraints Every Customer is Served

For each customer:


Trucks required to each consol pt For each consol pt (and type of capacity, e.g., weight, cube, floor space)

Sum over routes that stop at the consol pt Route Volume + Trucks*Load Factor

Sum over customers Assign*Requirement/Capacity Service Level Constraint For each consol pt.

Trucks +

Sum over routes that stop at the consol pt MultiStopTrucks

Minimum Service level*Open

Logic: For each customer and consol pt (within reason)

Assign Open

Multi-Stop Trucks For each multi-stop route

MultiStopTrucks*Load Factor Sum over stops on the route Route Volume


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Problems

How do we know all the (interesting) routes?

How many are there?

If we have ~ 50 consol pts and limit routes to say4 stops, we get 5.5 million potential routes!


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Column Generation

Turns out this is a bit more complicated

Illustrate the basic concept first

Apply to our Multi-Stop problem


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Column Generation

Illustrate with a pure Multi-Commodity

Flows problem

Multi-Commodity Network FlowsNetwork flows with several products

(commodities)

Joint capacity constraints Total volume of all commodities moving on a link

2 commodities


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Example MCNF Problem

Costs

From\To 1 2 3 4 5

1 0 9 4 8 60

2 0 0 9 70 3

3 0 6 0 4 1

4 7 7 2 0 7

5 3 6 5 7 0

Product 1 10 0 0 0 -10

Product 2 0 20 0 -20 0

Capacities

From\To 1 2 3 4 5

1 0 1 2 1 10

2 1 0 1 20 1

3 1 1 0 2 1

4 2 2 1 0 1

5 1 1 1 2 0

2 commodities

Prod 1 from 1 to 5

Prod 1 from 2 to 4

2 to 4 has

lots of

capacity

But it is

expensive


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Understand Problem?

Capacity Constraints:

Capacity on 1-3 is 1

Either 1 unit of Product 1 or 1 unit of Product 2, not

both

Can send 0.5 units of Product 1 & 0.5 units of Product

2.

How to solve this if there are no capacityconstraints?


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A Flows on Paths Model

Variables: For Product 1: Each path from node 1 to node 5

For Product 2: Each path from node 2 to node 4

1

23

45


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Constraints

Product 1 Demand:Total Flow of Product 1 on paths from 1

to 5 is 10 Product 2 Demand:

Total Flow of Product 2 on paths from 2to 4 is 20

And?


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Capacity Constraints

One for each edge in the network (in

this case 20)

Example: Capacity on 2-3 is 1:

Total Flow of Product 1 on paths that use

edge from 2 to 3 +

Total Flow of Product 2 on paths that use

edge from 2 to 3 1


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Column Generation Approach

Start with a small initial set of paths E.g., just the single-edge path from 1 to 5 for

Product 1 and from 2 to 4 for Product 2

Solve the Flows on Paths Model with thesepaths

Use the Dual Prices or Shadow Prices fromthis solution to determine if any new paths

will improve the solution. If there are no better paths, youre done.Otherwise add the paths to the formulationand repeat.

The art &

science


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The Dual Prices

One for each constraint Tell us the change in the objective per unit

increase in the right-hand-side of theconstraint (its a rate, i.e., $/unit)

Examples: Product 1 Demand Constraint: The dual pricetells us how much more it would cost if weinsisted on sending 10 + units of Product 1from 1 to 5

Capacity Constraint on Edge 2-3: The dualprices tells us how much more (less) it wouldcost if we increase the capacity on this edge by

What does intuition suggest about the signs?


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Try It


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Reduced Cost

Sending flow on a new path has two impacts:

We have to pay to send the flow

We reduce flows on the current paths

Computing the cost of sending the flow iseasy: Cost of the path * Units sent

Cost of the path is?

Computing the cost of the corresponding

changes in the flows on the current paths

turns out to be easy too.


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Use the Shadow Prices

Sending flow of Product 1 on the path from 1 to 3 andthen 3 to 5 has 4 effects: It incurs the cost to send flow on this path

It reduces the requirements for sending flow of Product 1from node 1 to node 5 on the current paths: Whats the value

of this? It reduces the capacity on the edge 1-3 available to thecurrent paths: Whats the value of this?

It reduces the capacity on the edge 3-5 available to thecurrent paths: Whats the value of this?

Reduced Cost of Path 1-3-5:Cost of using edge 1-3 + Cost of using edge 3-5Minus Shadow Price for demand of Product 1

Minus Shadow Price for capacity on edge 1-3



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Is Path 1-3-5 Attractive?

Is Reduced Cost of Path 1-3-5 < 0?Cost of using edge 1-3 + Cost of using edge 3-5

Minus Shadow Price for demand of Product 1



< 0?

Reduced Cost of Path:

Sum over the edges ofCost of edge Shadow Price for capacity on edge

< Shadow Price for Demand

The net cost

(including the value

of the consumedcapacities) to send

a unit of flow

The net

value


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Finding Attractive Paths

Reduced Cost of Path:Sum over the edges of

Cost of edge Shadow Price for capacity on edge

< Shadow Price for Demand

If we fix the commodity, the right-hand-side is aconstant

Find a shortest path for this commodity using themodified costs for the edges

If the length of this path is less than the Shadow Price for Demand, we have a candidate

Greater than the Shadow Price for Demand, there is nocandidate path for this commodity


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Try It

Shadow Price for Demand for Product 1

is 60 (Explain)

No edge is at capacity so all shadow

prices for capacities are 0

Find a shortest path from 1 to 5, if it is

less than 60, it is better than sending

flows direct.


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Shortest Path

For Product 1

1-3-5 has cost 5 < 60 so its reduced cost is

-55. It is attractive, add it.

For Product 2

2-1-4 has cost 8 < 70 so its reduced cost is

-62. It is attractive, add it.


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Solve the Master

Uses the new paths to capacity

Objective value drops to 1883

Edge 3-5 at capacity.Shadow Price55

Modified cost for 3-5 is 1(-55) = 56

Edge 2-1 at capacityShadow Price62

Modified cost for 2-1 is 0(-62) = 62

Since these edges

are at capacity,

using them in a

new path wouldforce us to give

up some of the

gains


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Next Iteration

A Most attractive path for Product 1

1-2-5 with cost 12

A Most attractive path for Product 2

2-5-4 with cost 10

Master Problem objective drops to 1823

Shadow Price for capacity on 2-5 is -60

Competing

for capacity


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Etc.

After 4 iterations, the Objective value in theMaster Problem has fallen to 1721

The Shadow Prices for demand are still Product 1: 60

Product 2: 70

The lengths of the Shortest Paths usingmodified costs are

Product 1: 60 Product 2: 70

We have an optimal answer.


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Questions?

Everyone understand the basics of

column generation

Comment: Computationally this is only

different from basic LP in so far as we

used the Shortest Path Problem to find

an attractive path rather than simply

work through a list of variables, pricingthem out one by one.


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Back to Multi-Stop Routes

Lets apply Column Generation to solve ourMulti-Stop Consolidation Problem

Recall Shipping to customers from a single plant

Consolidating LTL shipments throughconsolidation points

Serving the consolidation points via TL and/orMulti-Stop TL

Modeled as though we knew all the Multi-Stop

Routes Use Column Generation to produce the

Routes


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Objective

Transportation Costs

TL to Consol Pts

Multi-stop TL to Consol Pts LTL to Consol

Pts

LTL Direct to Customer

Multi-Stop TL costs include

Mileage charge

Stop charges

T


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Constraints

Every Customer is Served For each customer:


Trucks required to each consol pt For each consol pt (and type of capacity, e.g., weight, cube, floor space)

Sum over routes that stop at the consol pt Route Volume + Trucks*Load Factor

- Sum over customers Assign*Requirement/Capacity 0

Service Level Constraint For each consol pt.

Trucks +


- Minimum Service level*Open 0

Logic:

For each customer and consol pt (within reason)Open - Assign 0


MultiStopTrucks*Load Factor - Sum over stops on the route Route Volume 0

Two aspects

of a route:

Trucks &Volume


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Two Issues

Issue #1: What columns do we generate?

MultiStop Trucks?

Route Volume?

Both?


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Issue #2

Trucks required to each consol pt For each consol pt (and type of capacity, e.g., weight, cube, floor

space)

Sum over routes that stop at the consol pt Route Volume +Trucks*Load Factor

- Sum over customers Assign*Requirement/Capacity 0 Service Level Constraint

For each consol pt.

Trucks +




MultiStopTrucks*Load Factor - Sum over stops on the route Route

Volume

0

We wont write this tillwe have generated the

route! But wont we

need the Shadow Priceon this to generate the

route?


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Case 1: Issue #2


space)



For each consol pt.

Trucks +




MultiStopTrucks*Load Factor - Sum over stops on the route Route

Volume

0

If its not

tight

dropping it

has no effect.

W t b th th R t V l


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Case 1: Relevant Constraints

Trucks required to each consol pt For each consol pt (and type of capacity, e.g., weight,

cube, floor space)

Sum over routes that stop at the consol pt Route Volume

+ Trucks*Load Factor


Service Level Constraint

For each consol pt.Trucks +

Sum over routes that stop at the consol pt

MultiStopTrucks


We want both the Route Volumes

& the MultiStopTrucks to price out

I R V l A i ?


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Is a RouteVolume Attractive?

What are the effects (direct and indirect)

of increasing a RouteVolume variable?

For clarity we should write that as

RouteVolume[route, consol]: the volume

for the consolidation point that is

delivered on this route.

Is there a direct cost for the

RouteVolume[route, consol] variable?

No. We pay for trucks

and LTL. We will

handle the cost of the

multi-stop route whenwe ensure Multi-Stop

Trucks prices out

Pricing Out


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Pricing Out

RouteVolume[route, consol]

So there is no direct cost

Just indirect costs, (like consuming

capacity on an edge or satisfying

demand in the multi-commodity flowproblem)

What Shadow Prices do we need to

look at?

R V l [ l]


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Which Shadow Prices? Trucks required to each consol pt

For each consol pt (and type of capacity, e.g., weight,

cube, floor space)







MultiStopTrucks


Just the one for the trucks required

at the consol point


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The Shadow Price

Trucks required to each consol pt

For each consol pt (and type of capacity,

e.g., weight, cube, floor space)

Sum over routes that stop at the consol ptRoute Volume + Trucks*Load Factor

- Sum over customers

Assign*Requirement/Capacity

0

What happens to cost if

we increase this?

R V l [ l]

0 i th Sh d P i f


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Reduced Cost

The Shadow Price for trucks required at

consol point is the cost of satisfying

another truck load of demand there

0 if the service constraint is the driver

Something positive otherwise

Whats the reduced cost of

RouteVolume[route, consol]?

When is RouteVolume[route, consol]

attractive?

0 minus the Shadow Price fortrucks required at consol point

As long as service isnt thedriver there!


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Is MultiStop Trucks Attractive?

What are the effects (direct and indirect)

of increasing a MultiStop Trucks

variable?

For clarity we should write that as

MultiStop Trucks[route]

Is there a direct cost for the

MultiStop Trucks[route] variable?

Yes. The cost of a

truck on that route

Pricing Out


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Pricing Out


So the direct cost is Route Cost

What indirect costs?

What Shadow Prices do we need to

look at?

M ltiSt T k [ t ]


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Which Shadow Prices?


cube, floor space)







MultiStopTrucks


The route provides service to each

consol point it visits!


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The Shadow Price


For each consol pt.

Trucks +


MultiStopTrucks


What happens to cost if

we increase this?

Reduced Cost of


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Reduced Cost of


Direct CostIndirect Costs < 0

Route CostSum of Shadow Prices for

Service on the route < 0

Route Cost < Sum of Shadow Prices for

Service on the route

The value of the services exceeds the

cost of the route!

I th R t Att ti ?


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Is the Route Attractive?

For each consol pt on the routeRouteVolume[route, consol]

prices out

0 < Shadow Price for trucks at consol pt (i.e.,service isnt the driver, the trucks are full)

Does


price out?

Route Cost < Sum over stops on the route ofFrequency Shadow Prices


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A Resolution

Two Cases:

Case 1: MultiStopTrucks*Load Factor - Sum

over stops on the route Route Volume> 0

Case 2: MultiStopTrucks*Load Factor - Sum

over stops on the route Route Volume= 0


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Case 2

MultiStopTrucks = (Sum over stops on

the route Route Volume)/ Load Factor

Eliminate MultiStopTrucks

Insist each Route Volume be attractive

(price out)Otherwise, we would short-

cut the route and not stop at that Consol

pt.


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Case 2


space)



For each consol pt.

Trucks +




MultiStopTrucks*Load Factor - Sum over stops on the route RouteVolume 0

Sum over stops on the route

Route Volume/Load Factor

Route Volume is specific to the


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Relevant Constraints


cube, floor space)






Sum over routes that stop at the consol pt (Sum over

stops on the route Route Volume/Load Factor)


Route Volume is specific to the

Route AND the Consol Pt

But this is the sum over all the

stops on the route

So, to determine ifone Route volume

is attractive


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Route Volume Attractive

We must consider

What we pay for the Route Volume (later)

Its influence on the Trucks required at the

Consol Pt (Shadow Price of the Trucksrequired to carry the weight at the consol

pt)

Its influence on the Frequency constraint

for every consol pt on the route (Shadow

Prices of these constraints/Load Factor)

This is

where wereplaced

Multi-Stop

Trucks with

the sum

h f l


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What we pay for Route Volume

In the objective, we also replacedCost per Multi-Stop Truck * Multi-Stop

Trucks

WithCost per Multi-Stop Truck * (Sum over stops

on the route Route Volume/Load Factor)

So, each Route Volume bears the full

cost of the Multi-Stop Route/LoadFactor


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Three Factors:1. What we pay for the Route Volume?

Cost per Multi-Stop Truck/Load Factor

2. Its influence on the Trucks required at the

Consol Pt?Shadow Price of the Trucks required to carrythe weight at the consol pt

3. Its influence on the Frequency constraint for

every consol pt on the routeSum over all the stops on the route of theShadow Prices of the Frequencyconstraint/Load Factor



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IsRoute Cost/Load Factor

- Weight Price at Consol Pt

- Sum of Frequency Prices/Load Factor

< 0?

Is

Route Cost

- Load Factor*Weight Price at Consol Pt- Sum of Frequency Prices

< 0?

The only thingthat changes from

consol pt to

consol pt


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H t t t ?


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How to generate routes?

Have to decide which consol pts are onthe route

Decision Variables

Is Consol pt first on a multi-stop route?

Does consol pt A follow consol pt B on a

multi-stop route?


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Constraints

Limit number of stops (practical)

At least 2 (so its multi-stop)

At most 4 (say)

Bounds on the total number of legs Find 1 Route

One leg out of the origin

Cant go from consol pt B to consol pt C

unless some leg takes you to B

Number of legs out of B Number of legs into B


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Price Constraints

For each consol pt on the route

Route Cost < Load Factor * Weight Shadow

Price for consol pt + Sum of Frequency

Shadow Prices on route

But we dont know whats on the route!

Define

OnRoute = sum of legs into consol pt (0 or 1)

Disjunctive Constraint


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Price Constraints

For each consol pt on the route

Route Cost < Load Factor * Weight Shadow

Price for consol pt + Sum of Frequency

Shadow Prices on route + M*(1-OnRoute)

Define

OnRoute = sum of legs into consol pt (0 or 1)

Try It


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Try It


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New Problem

Sub-Tours:

16

5


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Resolutions

Practical: Each subsequent stop must be farther from the

plant.

Subtour Elimination (Less Practical)

For each three consol pts, we can choose at mosttwo legs

For each two consol pts, we can choose at mostone leg

Generally, for each N consol pts, we can choose at

most N-1 legs (but we limited routes to 4 legs) Dynamic Programming type algorithm or

iterative heuristic (software)

Try It


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Try It

S


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Summary

Solve the Master LP (relax integrality)without routes

Get Shadow Prices

Generate Routes (Case 1 & Case 2)

If there are attractive routes, add them

and solve the Master LP again

If there are no attractive routes, solve to

an Integer Optimum


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Issues

Our procedure for generating multi-stoproutes does not consider the integer

decisions about what consol pts to use.

Heuristic resolution: At the end, repeatthe column generation procedure with

the consol pt decisions fixed.


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Next Time

Change in Emphasis:

This time

Service level was fixed

Reduce transport cost

Next time

Transport cost fixed Load Driven

Increase service

Documents

06Consolidation TL and LTL Shipment Column Generation Approach