Consolidation

11

Consolidation

John H. Vande Vate

Spring, 2007

22

Agenda

• Combining LTL into TL shipments– Motivation– Models– Issues

• Multi-Stop TL shipments– Column Generation Approach

33

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

44

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?

55

Interrelated Questions

• Where do we consolidate (what terminals)?

• Which customers (orders) do we serve through each terminal?

66

Assumptions

• Single Plant or origin for supplies– We are not allocating customers to

production plants. That’s already been done

• We know our customers– Not always the case– Can use geographic regions in place of

actual customer locations

• We have adjusted last year’s orders to reflect next year’s projections

77

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.

88

Objective: Transportation Cost• LTL shipments direct to customers

– Easy to rate these, we’ve 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

99

Elaborations• Consider inventory costs

• Handling charges at consol pts

• Amortized capital charge or rent for consol pts

• Time to customer

• …

1010

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 on

basket of products or– Translate each customers annual demand into

weight and cube (or floor space)

1111

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 customers Assign*Requirement/Capacity

Demand varies. Trucks

won’t be full

e.g.,Weight of customer’s

orders

e.g.,Weight limit of Truck

1212

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) for

opening a consol pt.

• ServiceLevelConstraint:– For each consol ptTrucks Minimum Service level*Open

e.g.,156 = 52*3

1313

Logic

• Can’t assign a customer to a consol pt unless it is open– For each customer and consol pt (within

reason)

Assign ≤ Open

1414

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 farther away may not

1515

Odd Assignments

• Reasonable to use recommended consol pts?• Reasonable to use recommended assignments?

Plant in FL!?*

1616

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

1717

Next Step: Multi-Stop Routes• Can we improve performance by sharing

the fixed operating cost across several consol pts

• Advantage: Allows smaller consol pts• Disadvantage: Lower “efficiency” in TL

shipments– Do you really want to run trucks half empty

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

1818

Typical Multi-Stop Rt

Clustered destinations

1919

Model: Key Decisions• ServeDirect: Do we serve customer via direct LTL

shipments• Open: Do we open a candidate consol pt.

– One for each candidate consol pt.

• Assign: Do we assign customer to consol pt. – One for each customer and (reasonable) consol pt.

• Trucks: Annual trucks running to consol pt– One for each candidate consol pot

• RouteTrucks: Annual trucks running on multi-stop route– One for each candidate multi-stop route

• RouteVolume: Annual volume at each consol pt that is picked up by each multi-stop route– One for each candidate route and stop on the route

2020

Assumptions

• Volume to a consol pt can be split among direct trucks and (potentially several) multi-stop routes

• The operating fixed cost imposed by the frequency requirement can be shared among these, i.e., there’s a lower bound on the number of times we “stop” at the consol pt each year.

2121

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

2222


– 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)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 routeMultiStopTrucks*Load Factor Sum over stops on the route Route Volume

2323

Problems

• How do we know all the (interesting) routes?• How many are there?• If we have ~ 50 consol pts and limit routes to say

4 stops, we get 5.5 million potential routes!

2424

Good News!

• We can find good routes as we solve the problem

• Use technique called Column Generation

• Big Idea: – Use Shadow Price information from current

solution to identify attractive routes– When no new routes are attractive, we’ve found

all the interesting ones (well sort of)

2525

Column Generation

• Turns out this is a bit more complicated

• Illustrate the basic concept first

• Apply to our Multi-Stop problem

2626

Column Generation

• Illustrate with a “pure” Multi-Commodity Flows problem

• Multi-Commodity Network Flows– Network flows with several products

(commodities)– Joint capacity constraints

• Total volume of all commodities moving on a link

2727

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

2828

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 capacity constraints?

2929

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

3030

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 2

to 4 is 20

• And?

3131

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

3232

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 these

paths• Use the Dual Prices or Shadow Prices from

this solution to determine if any new paths will improve the solution.

• If there are no better paths, you’re done. Otherwise add the paths to the formulation and repeat.

The art & science

3333

The Dual Prices• One for each constraint• Tell us the change in the objective per unit

increase in the right-hand-side of the constraint (it’s a rate, i.e., $/unit)

• Examples: – Product 1 Demand Constraint: The dual price

tells us how much more it would cost if we insisted on sending 10 + units of Product 1 from 1 to 5

– Capacity Constraint on Edge 2-3: The dual prices tells us how much more (less) it would cost if we increase the capacity on this edge by

– What does intuition suggest about the signs?

3434

Try It

ColumnGeneration.xls

3535

Finding Attractive Paths

• Use the Dual Prices from this solution to determine if any new paths will improve the solution.

• If the Reduced Cost of a path is negative, it is attractive, i.e., adding it (can) improve the solution.

• Reduced Cost of a Path?

3636

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 is easy: 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.

3737

Use the Shadow Prices

• Sending flow of Product 1 on the path from 1 to 3 and then 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 1

from node 1 to node 5 on the current paths: What’s the value of this?

– It reduces the capacity on the edge 1-3 available to the current paths: What’s the value of this?

– It reduces the capacity on the edge 3-5 available to the current paths: What’s 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 Minus Shadow Price for capacity on edge 3-5

3838

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 3-5Minus Shadow Price for demand of Product 1 Minus Shadow Price for capacity on edge 1-3 Minus Shadow Price for capacity on edge 3-5 < 0?

• Reduced Cost of Path:Sum over the edges of Cost of edge – Shadow Price for capacity on edge< Shadow Price for Demand

The net cost (including the value

of the consumed capacities) to send

a unit of flow

The net value

3939

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 a constant

• Find a shortest path for this commodity using the modified 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 no

candidate path for this commodity

4040

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.

ColumnGeneration.xls

4141

Shortest Path

• For Product 1– 1-3-5 has cost 5 < 60 so it’s reduced cost is

-55. It is attractive, add it.

• For Product 2– 2-1-4 has cost 8 < 70 so it’s reduced cost is

-62. It is attractive, add it.

4242

Repeat

• The Master Problem now has 4 paths– For Product 1:

• 1-5 with cost 60 and capacity 10

• 1-3-5 with cost 5 and capacity ?

1

– For Product 2:• 2-4 with cost 70 and capacity 20

• 2-1-4 with cost 8 and capacity ?1

4343

Solve the Master

• Uses the new paths to capacity

• Objective value drops to 1883

• Edge 3-5 at capacity. – Shadow Price – 55– Modified cost for 3-5 is 1 – (-55) = 56

• Edge 2-1 at capacity– Shadow Price – 62– Modified cost for 2-1 is 0 – (-62) = 62

Since these edges are at capacity, using them in a new path would force us to give up some of the

gains

4444

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

4545

Etc.• After 4 iterations, the Objective value in the

Master Problem has fallen to 1721• The Shadow Prices for demand are still

– Product 1: 60– Product 2: 70

• The lengths of the Shortest Paths using modified costs are – Product 1: 60– Product 2: 70

• We have an optimal answer.

4646

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, “pricing them out” one by one.

4747

Back to Multi-Stop Routes• Let’s apply Column Generation to solve our

Multi-Stop Consolidation Problem• Recall

– Shipping to customers from a single plant– Consolidating LTL shipments through

consolidation points– Serving the consolidation points via TL and/or

Multi-Stop TL – Modeled as though we knew all the Multi-Stop

Routes• Use Column Generation to produce the

Routes

4848

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

4949


– 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)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 + Sum over routes that stop at the consol pt MultiStopTrucks - Minimum Service level*Open 0

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

Open - Assign 0• Multi-Stop Trucks

– For each multi-stop routeMultiStopTrucks*Load Factor - Sum over stops on the route Route Volume 0

Two aspects of a route: Trucks & Volume

5050

Two Issues

• Issue #1: What columns do we generate?– MultiStop Trucks?– Route Volume?– Both?

• …

5151

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


• Multi-Stop Trucks– For each multi-stop routeMultiStopTrucks*Load Factor - Sum over stops on the route Route

Volume 0

We won’t write this till we have generated the route! But won’t we

need the Shadow Price on this to generate the

route?

5252

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

What’s the shadow price for this constraint in this case?

0!

5353

Case 1: Issue #2• Trucks required to each consol pt






Volume 0

If it’s not “tight”

dropping it has no effect.

5454

Case 1: Relevant Constraints• Trucks required to each consol pt




• Service Level Constraint – For each consol pt.

Trucks +

Sum over routes that stop at the consol pt MultiStopTrucks

- Minimum Service level*Open 0

We want both the Route Volumes& the MultiStopTrucks to price out

5555

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 when we ensure Multi-Stop

Trucks prices out

5656

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 flow problem)

• What Shadow Prices do we need to look at?

5757

RouteVolume[route, consol] Which Shadow Prices?

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

cube, floor space)




Trucks +



Just the one for the trucks required at the consol point

5858

The Shadow Price

• Trucks required to each consol pt– For each consol pt (and type of capacity,

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



What happens to cost if we increase this?

5959

RouteVolume[route, consol]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

• What’s the reduced cost of

RouteVolume[route, consol]?

• When is RouteVolume[route, consol] attractive?

0 minus the Shadow Price for trucks required at consol pointAs long as service isn’t the driver there!

6060

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

6161

Pricing Out MultiStop Trucks[route]

• So the direct cost is Route Cost

• What indirect costs?

• What Shadow Prices do we need to look at?

6262

MultiStop Trucks[route] Which Shadow Prices?

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

cube, floor space)




Trucks +



The route provides service to each consol point it visits!

6363

The Shadow Price


Trucks +



What happens to cost if we increase this?

6464

Reduced Cost of MultiStop Trucks[route]

• Direct Cost – Indirect Costs < 0

• Route Cost – Sum 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!

6565

Is the Route Attractive?

• For each consol pt on the route RouteVolume[route, consol] prices out • 0 < Shadow Price for trucks at consol pt (i.e.,

service isn’t the driver, the trucks are full)• Does MultiStop Trucks[route] price out?• Route Cost < Sum over stops on the route of

Frequency Shadow Prices

ConsolSubProblem.xls

6666

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

6767

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.

6868

Case 2• Trucks required to each consol pt






Volume 0

Sum over stops on the route

Route Volume /Load Factor

6969

Relevant Constraints• Trucks required to each consol pt





Trucks +

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 if one Route volume

is attractive…

7070

Route Volume Attractive

• We must consider– What we pay for the Route Volume (later)– It’s influence on the Trucks required at the

Consol Pt (Shadow Price of the Trucks required to carry the weight at the consol pt)

– It’s influence on the Frequency constraint for every consol pt on the route (Shadow Prices of these constraints/Load Factor)

This is where we replaced

Multi-Stop Trucks with

the sum

7171

What we pay for Route Volume

• In the objective, we also replaced Cost per Multi-Stop Truck * Multi-Stop

TrucksWithCost 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/Load Factor

7272


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

Cost per Multi-Stop Truck/Load Factor2. It’s influence on the Trucks required at the

Consol Pt? Shadow Price of the Trucks required to carry

the weight at the consol pt3. It’s influence on the Frequency constraint for

every consol pt on the route Sum over all the stops on the route of the Shadow Prices of the Frequency constraint/Load Factor

7373


Is Route 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 thing that changes from

consol pt to consol pt

7474

Is the Route Attractive?

Is

Route Cost

- Load Factor*Weight Price at Consol Pt

- Sum of Frequency Prices

< 0

For every Consol Pt on the route?

Get this from sensitivity infoGet these from sensitivity info

7575

How to generate routes?

• Have to decide which consol pts are on the 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?

7676

Constraints• Limit number of stops (practical)

– At least 2 (so it’s multi-stop)

– At most 4 (say)

Bounds on the total number of legs

• Find 1 Route – One leg out of the origin

• Can’t go from consol pt B to consol pt C unless some leg takes you to BNumber of legs out of B ≤ Number of legs into B

7777

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 don’t know what’s on the route!

• Define

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

• Disjunctive Constraint

7878

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)

7979

Try It


8080

New Problem

• Sub-Tours:

16

5

8181

Resolutions• Practical:

– Each subsequent stop must be farther from the plant.

• Subtour Elimination (Less Practical)– For each three consol pts, we can choose at most

two legs– For each two consol pts, we can choose at most

one 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)

8282

Try It


8383

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

8484

Issues

• Our procedure for generating multi-stop routes does not consider the integer decisions about what consol pts to use.

• Heuristic resolution: At the end, repeat the column generation procedure with the consol pt decisions fixed.

8585

Next Time

• Change in Emphasis:

• This time– Service level was fixed– Reduce transport cost

• Next time– Transport cost “fixed” – Load Driven– Increase service

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