Rule-based Price Discovery Methods in Transportation

Procurement Auctions

Jiongjiong SongAmelia Regan

Institute of Transportation StudiesUniversity of California, Irvine

INFORMS Revenue Management Conference 2004

Outline

• Introduction to Procurement Auctions• The Business Rule based Bid Analysis

Problem– Shippers’ business considerations – An integer programming model

• Our solution methodologies– Construction heuristics and Lagrangian heuristics– Experimental results

• Conclusion and extensions

Procurement Auctions

• Combinatorial auction– An allocation mechanism for multiple items– Multiple items put out for bid simultaneously– Bidders can submit complicated bids for any

combinations of items

• Unit auction– Packages are pre-defined and are mutually exclusive

• Applications in freight transportation– Freight transportation exhibits economies of scope– Shippers gain more benefits to bundle lanes– Carriers dislike this combinatorial auction idea

Procurement Auctions

• Combinatorial auction– Complicated optimization problems for both

shippers and carriers– Shippers lose control over bundles, carriers have

more freedom

• Unit auction– Shippers gain control– Carriers have much simpler pricing problem to

solve

• Shippers still have a difficult optimization problem to solve

Business Considerations

• If price is the sole reason for assigning bids – the unit auction problem is simple to solve

• However, shippers have additional considerations

• Caplice and Sheffi (2003) identify the primary considerations for the trucking industry case

Business Considerations

• Minimum/maximum number of winning carriers (core carriers)

• Favor of Incumbents

• Backup concerns

• Minimum/maximum coverage

• Threshold volumes

• Complete regional coverage

Business Considerations

• Performance factors – these are necessary to ensure that high priced carriers don’t “Lose the auction but win the freight”

Our Model

• We include the following: – maximum / minimum number of winning

carriers– maximum / minimum coverage– incumbent preference– performance factors (penalty cost)

Our Model

• We assume that:– backup considerations– regional coverage

• Can be taken care of in pre-processing and pre-screening steps

The General Model

,

min

. . 1 (1)

(2)

(0,1) (3)

Where:

is a bid package in set

is a bidding carrier in set

kj kjj J k K

kjk K

kj

kj

c x

s t x j J

x

j J

k K

c

is the cost for carrier to serve package

1 if carrier k wins package j =

0 otherwise

are any business or logical constraints

kj

k j

x

Our Model

min max

min max

min

. .

1, (4)

, (5)

, (6)

, (0,1)

kj kj k kk j k

kjk

kk

k kk kj k

j

k kj

c x p y

s t

x j J

K y K

T y x T y k K

y x

(7)

Our Model

min max

mi

Where:

is the penalty cost for carrier to be included in the winning bids

1 if carrier k wins one or more package

0 otherwise

, are the minimum and maximum number of winning carriers

k

k

p k

y

K K

T

n max , are the minimum and maximum number of packages that can be

assigned to carrier

k kT

k

Our Model

• Our objective function problem minimizes total procurement costs including the bid prices and the penalty costs to manage multiple carrier accounts

# of Carriers

Cost

Relationship between procurement costs and number of winners

Our Model

• The penalty cost can also be used to capture the shipper’s favoring of specific carriers at the system level– incumbents have a zero penalty cost and non-

incumbents have a positive penalty cost

• This could be extended to specific packages• Though we model the maximum and minimum

volume constraints at the system level, these could be applied at the regional or facility level

Our Model

• Even with the simplification of some business constraints to the network level this problem can easily be shown to be NP-Complete

• Solving problems of reasonable size (thousands of lanes, hundreds of carriers) using exact methods is not feasible– CPLEX failed to solve such as a case in two

days with a moderately fast computer

Our Solution Approach

• Simple construction techniques based on the relationship between our problem and the capacitated facility location problem– MDROP and MADD for Modified DROP and

ADD

• Lagrangian Relaxation– Constraint (4) is relaxed (a lane is only

assigned to a single carrier)– Network flow based algorithms to solve the

relaxed problem

Test Data

• Input data for each problem includes:– Each carrier’s bid prices for each lane– penalty cost for each carrier– minimum and maximum number of lanes if this carriers

is a winner– minimum and maximum number of winners– a carrier’s bid price is randomly distributed between 10

and 100– the penalty cost is randomly distributed between 0 and

3% of total bid prices

Results

• Small Problems

Case Index 1 2 3 4

# of carriers 20 20 20 30

# of lanes 200 300 400 300

Lower / Upper 99.8% 99.9% 99.3% 99.6%

Upper / CPLEX 1.0 1.0 1.0 1.0

MADD / CPLEX 1.01 1.0 1.001 1.007

MDROP / CPLEX 1.0 1.0 1.001 1.0

Results

• Small Problems

Case Index 5 6 7 8 9

# of carriers 30 40 40 40 50

# of lanes 400 300 400 500 400

Lower / Upper 96.9% 97.4% 97.9% 97.5% 97.9%

Upper / CPLEX 1.0 1.001 1.001 1.0 1.0

MADD / CPLEX 1.003 1.009 1.004 1.002 1.003

MDROP / CPLEX 1.0 1.003 1.001 1.001 1.001

Solution Times (minutes)

• Small Problems

Case Index 5 6 7 8 9

CPLEX 66.3 66.2 137.5 231.0 192.5

Lagrangian 0.7 0.6 0.8 0.7 0.7

MADD 0.04 0.05 0.06 0.06 0.07

MDROP 0.03 0.03 0.04 0.04 0.05

Results

• Larger Problems

Case Index 11 12 13 14

# of carriers 100 100 200 200

# of lanes 2000 4000 4000 6000

Lower/Upper 99.2% 96.9% 97.9% 99.0%

MADD/Upper 1.057 1.051 1.063 1.063

MDROP/Upper 1.056 1.050 1.058 1.062

Results

• Larger Problems

Case Index 15 16 17 18 19

# of carriers 300 300 400 400 500

# of lanes 6000 8000 8000 10000 10000

Lower/Upper 99.6% 99.3% 99.0% 99.1% 99.0%

MADD/Upper 1.070 1.067 1.068 1.090 1.080

MDROP/Upper 1.065 1.066 1.067 1.076 1.071

Solution Times (minutes)

• Larger Problems

Case Index 11 12 13 14

Lagrangian 6 14 31 48

MADD 0.4 0.4 0.6 1

MDROP 0.5 1.1 3.9 6.6

Case Index 15 16 17 18 19

Lagrangian 76 101 136 181 225

MADD 1.1 1.4 2.1 4 7.6

MDROP 13.9 20 34 46 69

Conclusion

• We show that unit auctions with side constraints can be solved in reasonable time and with a high degree of confidence

• The Lagrangian Relaxation solution method could be used to make final decisions while the heuristics (or improved versions of these) could be used to conduct sensitivity analysis

Extensions

• Shippers may have additional or more complicated business rules

• As optimization tools improve, requirements will increase

• Eventually, pure combinatorial auctions (for large shippers and large carriers) may be feasible and preferable – we are working to solve bidding and winner determination problems for those auctions

Thank You