MODELING AND ANALYSIS OF AN AUCTION-BASED LOGISTICS MARKET

Barış TanSemra Ağralı and Fikri Karaesmen

Department of Industrial EngineeringKoç University, Istanbul, Turkey

May 23rd, 2005

FIFTH INTERNATIONAL CONFERENCE ON"Analysis of Manufacturing Systems -

Production Management"May 20-25, 2005 - Zakynthos Island, Greece

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Motivation

The Logistics Center established by Eskisehir Chamber of Industry in the Organized Industrial Zone in 2003.

The goal is to satisfy the logistics needs of the producers located in the Industrial Zone at the lowest cost by using an auction mechanism

A hub for logistics firms and truck owners Attracts truck owners to the center with all the

necessary facilities A reduction of 20-30% in transportation costs is

achieved through the market mechanism

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Eskişehir Chamber of Commerce Logistics Center

www.esolojistik.com

ProductionConsumption Consumption

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Realized Transportation Prices

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Comparison with Market Price

0

200

400

600

800

1000

0 200 400 600 800 1000 1200

Distance (km)

Tra

ns

po

rta

tio

n P

ric

e

Realized Auction Price Market Price

0%

10%

20%

30%

40%

50%

60%

0 200 400 600 800 1000Distance (km)

Red

uct

ion

wrt

Mar

ket

Pri

ce

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Modeling and Analysis Issues

Final price is determined by an auction The number of bidders and their costs affect the price Different costs of transportation for different trucks for

the same order and the same destination Random arrival of orders and trucks Possible abandonment of orders and trucks Limited capacity of the market

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Operation of the Logistics Market - 1

Istanbul 200 YTL

220 YTL

180 YTL

Logistics Center

Payment: 180 YTL (First Price)

200 YTL (Second Price)

Industrial Zone

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Operation of the Logistics Market - 2

Adana

250 YTLLogistics Center

Payment: 250 YTL

Industrial Zone

İzmir

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Research Questions

What is the gain that will be obtained by using an auction in logistics for the shippers and for the logistic firms?

What are the effects of various system parameters on the gains?

What will be the effect of the auction type used? What will be the effect of the auction process?

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Approach Analyze the second-price auction in a static setting with a given

number of bidders and obtain the expected auction price. Develop an analytical model with some simplifying assumptions

and obtain closed-form expressions for the performance measures.

Develop a state-space model and determine the performance measures from the steady-state probabilities of the continuous-time Markov chain.

Develop a simulation model to validate the analytical model and also to handle other extensions

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Literature Survey

Economics– Vickrey, 1961 – Myerson, 1981– McAfee and McMillan, 1987– Klemperer, 1999– Bapna et al., 2002– Holt, 1980; Riley and Samuelson, 1981 – Milgrom and Weber, 1982– Graham and Marshall, 1967; McAfee and

McMillan, 1987 – Wilson, 1967; Weverbergh, 1979; Fibich

et al., 2004; Griesmer et al., 1967; Maskin and Riley, 2000; Fibich and Gavious, 2003;Campo et al., 2003

Empirical Analysis Literature– Hendricks and Paarsch, 1995– Paarsch, 1989– Hendricks and Porter, 1988– Paarsch, 1989; Laffont, Ossard, and

Vuong, 1995; and Elyakime et al., 1997– Laffont, Ossard, and Vuong, 1995– Elyakime et al., 1997

Operations Research/Operations Management

– Goldsteins, 1952– Stark and Rothkopf, 1979– Lucking-Reiley, 2000– Wagner and Schwab, 2003– Kameshwaran and Narahari,

2001– Ledyard et al., 2002– Song and Regan, 2003– Chen et al., in progress– Vakrat and Seidmann, 2000 – Emiliani and Stec, 2002– Talluri and Ragatz, 2004– Qi, 2002

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Model Assumptions

Logistics Center

Industrial Zone

Type L

Type B

l

la

b

lb

Second Price AuctionMarket Price PM

Cost: cdf: Fl(v), E[v].

Cost: cdf: Fb(v), E[c].

o

oa

One truck load(no split)

Maximum L,B trucksO orders

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Empirical Analysis: Order Arrivals

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Estimating the Cost Distribution from the Bid Distribution

150 200 250 300 350 400 450 500 550 6000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Bid b(v)

%

Izmir

1

1

1 ( )( )

1 ( )

v n

vn

F x dxb v v

F v

0 100 200 300 400 500 600 7000

0.002

0.004

0.006

0.008

0.01

0.012

Izmir

Cost v

%

First price

Second price b(v) = v

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Analysis of an AuctionGiven that there are l carriers (bidders) at the logistics center: In a single-unit second-price auction, or the Vickrey auction,

the carrier that submits the lowest bid wins and the winning bidder is paid at the second lowest bid.

In a second-price auction, the optimal strategy is bidding the actual cost

The expected auction price is the expected value of the second lowest cost in a group of l bidders:

When there is one bidder, it is paid at the market price without an auction

1

(2)( ) 1 ( ) ( ) 1 ( )v v

l l

l L L L

v v

p l E v v F x lF x dx F x dx

pl(1)=PM.

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Analysis of an Auction

The winning carrier bids its cost which is the minimum of the costs of l bidders.

Then the expected profit is the difference between the expected auction price and the expected cost.

1( ) 1 ( ) ( )

vl

l

v

q l L F x F x dx

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Effect of the Number of Bidders

Analytical ModelEmprical Results

Uniform cost distribution

2( )

1

v vv

l

pl(l)

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6

Number of Bidders

Ave

rag

e au

ctio

n p

rice

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State Space Model

The state of the system: S(t) = [No(t), Nl(t), Nb(t)]

– No(t): number of orders at time t

– Nl(t): number of Type L carriers at time t

– Nb(t): number of Type B carriers at time t

The process {S(t), t≥0} is a Continuous-time Markov Chain. The steady-state probabilities:

The state space model gives the probabilities of having No(t)=o orders and Nl(t)=l, Nb(t)=b carriers in the steady state.

lim Pr[ ( ) ]it

S t i

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Steady State Analysis

Combining with the steady-state distribution of the number of carriers, all the performance measures can be determined:

Pav: the expected average auction price

Qav: expected profit of the carriers

Tav the expected average number of carriers waiting at the center,

Oav the expected average number of waiting orders,

Mo the probability of rejecting an order,

Ml and Mb probability of rejecting carrier because of the capacity constraint for Type L and Type B carriers

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Steady-State Analysis: Special Case

Only Type L carriers; no abandonment of orders and carriers; and no capacity constraint for arriving orders.

The state of the system: the number of outstanding orders at time t: S(t) = No(t)- Nl(t) + L,

Identical to an M/M/1 queue

λb=0; λoa =0; λla = 0; O → ∞.

(1 ) ii o

l

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Average Auction Price and Profit1 1

1

0 1 01 1

0 1

( ) (1 ) ( )

(1 )

L Li L

o i l l i M o l l Mi i L i

av L L Lo l

o i l ii i L

p L i P p L i PP

where pl (k) and ql (k) are determined before

1 1

1

0 1 0 11 1

0 1

( ) (1) (1 ) ( ) (1 ) )

(1 )

L Li L i

o i l l i l o l l M li i L i i L

av L L Lo l

o i l ii i L

q L i q q L i P E vQ

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Average Number of Carriers and Orders

11 1

0 0

(1 )( ) (1 ) ( ) (1 ) ( 1)

1

LL Li L L

av ii i

T L i L i L L

1

( ) (1 ) ( )1

Li

av ii L i L

O i L i L

Rejection Probability

0 1lM

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Steady-State Analysis: General Case (L+1)(B+1)+O states in the state space. The steady state probabilities satisfy the following set of

transition equations

( , ,0) ( 1, ,0) ( , 1,0) ( , 1,0) ( 1, ,0)( ( 1) ) ( 1)l b o la ba i j l i j b i j o ba i j la i ji j j i

i=1,…,L-1, j=1,…,B-1

(0, ,0) (0, 1,0) (0, 1,0) (1, ,0)( ( 1) )l b o ba j b j o ba j la jj j j=1,…,B-1

( , ,0) ( 1, ,0) ( , 1,0) ( , 1,0)( ( 1) )l b o la ba L j l L j b L j o ba L jL j j j=1,…,B-1

( ,0,0) ( 1,0,0) ( ,1,0) ( 1,0,0)( ) ( ( 1) )l b o la i l i o ba i o la ii i i=1,…,L-1

( , ,0) ( 1, ,0) ( , 1,0) ( 1, ,0)( 1)l o la ba i B l i B b i B la i Bi B i i=1,…,L-1

(0,0, ) (0,0, 1) (0,0, 1)( ( 1) )l b o oa z l b oa z o zz z z=1,…,O-1

(0,0,0) (0,1,0) (1,0,0) (0,0,1)( ) ( ) ( )l b o o ba o la l b oa

(0, ,0) (0, 1,0) (1, ,0)l o ba B b B la Bj

( ,0,0) ( 1,0,0) ( ,1,0)( )b o la L l L o ba Li

( , ,0) ( 1, ,0) ( , 1,0)o la ba L B l L B b L BL B

(0,0, ) (0,0, 1)l b o oa O o Oz

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State-Transition Diagram

0,0,5

λL+ λB+5λOAλO

0,0,4

λO λL+ λB+4λOA

0,0,0

0,0,3

λO λL+ λB+3*λOA

0,0,2

λL+ λB+2*λOAλO

0,0,1

λO λL+ λB+λOA

λL

1,0,0

λL

2,0,0

λL

5,0,0

λL

λO+ λBA

5,1,0

λB

λO+ 2λBA

5,2,0

λB

λO+ 3λBA

5,3,0

λB

λO+4 λBA

5,4,0

λB

λO+5 λBA

5,5,0λB

3,0,0

λL

4,0,0

λO+ λBA

0,1,0

λB

λO+ 2λBA

0,2,0

λB

λO+ 3λBA

0,3,0

λB

λO+4 λBA

0,4,0

λB

λO+5 λBA

0,5,0

λB

λO+ λLA

1,0,0

λLλO+ 2λLA

2,0,0

λLλO+ 3λLA

λO+ 5λLA λL

3,0,0

λLλO+ 4λLA

4,0,0

λL

1,1,0

λLλO+ 2λLA

2,1,0

λLλO+ 3λLA

λO+ 5λLA λL

3,1,0

λLλO+ 4λLA

4,1,0

λL

1,2,0

λLλO+ 2λLA

2,2,0

λLλO+ 3λLA

λO+ 5λLA λL

3,2,0

λLλO+ 4λLA

4,2,0

λL

1,3,0

λLλO+ 2λLA

2,3,0

λLλO+ 3λLA

λO+ 5λLA λL

3,3,0

λLλO+ 4λLA

4,3,0

λO+ λLA λL

1,4,0

λLλO+ 2λLA

2,4,0

λLλO+ 3λLA

λO+ 5λLA λL

3,4,0

λLλO+ 4λLA

4,4,0

L=5, B=5, O=5

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Performance Measures

(0,0, ) (1,0,0) (0,1,0) ( ,1,0) (1) ( ,0,0) ( , ,0)1 1 2 0 2

(0,0, ) ( , ,0) (0,0,0)1 0 0

( ) ( ) [ ( )] ( ) ( )

( )

O L L L B

l b z o M o i i l i j bz i i i j

av O L B

l b z o i jz i j

P E c i p i p j

P

(0,0, ) (1,0,0) (0,0, ) (0,1,0) ( ,1,0) (1) ( ,0,0 ) ( , ,0)1 1 1 2 0 2

(0,0, ) ( , ,0) (0,0

(1) (1) (1) [ ( ) [ ]] ( ) ( )

( )

O O L L L B

l z o l b z o b b o i i l i j bz z i i i j

av B

l b z o i jj

q q q E v i E c q i q i

Q

0,0)1 0

O L

z i

( , ,0)0 0

( )L B

av i ji j

T i j

, (0,0, )1

O

av zz

O z

, ( , ,0)0

B

l L jj

M

, ( , ,0)0

L

b i Bi

M

, (0,0, )o OM

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Numerical ResultsPav Qav Simulation Simulation

λo

λB

λL

λoa

λBa

λLa Model

Average 95% C.I. Model

Average 95% C.I. 1 2 4 1.5 1 0.5 86.8158 86.8199 (86.7376 86.9022) 7.1988 7.1883 (7.1470 7.2297) 2 2 4 1.5 1 0.5 89.7265 89.6986 (89.6429 89.7542) 7.3126 7.2912 (7.2608 7.3216) 4 2 4 1.5 1 0.5 93.7508 93.7589 (93.7211 93.7967) 7.9909 8.0047 (7.9738 8.0356) 8 2 4 1.5 1 0.5 98.3035 98.3037 (98.2795 98.3279) 10.3863 10.3996 (10.3730 10.4261) 3 1 4 1.5 1 0.5 94.9411 94.9545 (94.9219 94.9871) 6.4886 6.4996 (6.4739 6.5254) 3 2 4 1.5 1 0.5 91.9208 91.9066 (91.8598 91.9534) 7.5271 7.5306 (7.5060 7.5552) 3 4 4 1.5 1 0.5 85.1962 85.2074 (85.1439 85.2708) 6.7448 6.7353 (6.7042 6.7663) 3 8 4 1.5 1 0.5 77.2961 77.2976 (77.2545 77.3408) 3.7696 3.7697 (3.7448 3.7946) 3 2 1 1.5 1 0.5 95.2993 95.3307 (95.2683 95.3932) 14.1584 14.1705 (14.1177 14.2234) 3 2 2 1.5 1 0.5 93.9171 93.9104 (93.8569 93.9639) 10.4235 10.4089 (10.3656 10.4523) 3 2 4 1.5 1 0.5 91.9208 91.9066 (91.8598 91.9534) 7.5271 7.5306 (7.5060 7.5552) 3 2 8 1.5 1 0.5 90.6867 90.6923 (90.6526 90.7319) 6.4542 6.4351 (6.4115 6.4587) 3 2 4 0.5 1 0.5 92.0826 92.0632 (92.0097 92.1167) 7.6100 7.5799 (7.5490 7.6108) 3 2 4 1 1 0.5 91.9879 91.9941 (91.9386 92.0495) 7.5615 7.5735 (7.5441 7.6030) 3 2 4 1.5 1 0.5 91.9208 91.9066 (91.8598 91.9534) 7.5271 7.5306 (7.5060 7.5552) 3 2 4 2 1 0.5 91.8695 91.8338 (91.7831 91.8845) 7.5001 7.4985 (7.4693 7.5276) 3 2 4 1.5 0.5 0.5 90.3576 90.4070 (90.3488 90.4652) 7.3725 7.3944 (7.3688 7.4200) 3 2 4 1.5 1 0.5 91.9208 91.9066 (91.8598 91.9534) 7.5271 7.5306 (7.5060 7.5552) 3 2 4 1.5 1.5 0.5 92.8599 92.8653 (92.8224 92.9083) 7.5069 7.5255 (7.4987 7.5523) 3 2 4 1.5 2 0.5 93.4957 93.4884 (93.4542 93.5225 7.4246 7.4177 (7.3904 7.4451) 3 2 4 1.5 1 0.5 91.9208 91.9066 (91.8598 91.9534) 7.5271 7.5306 (7.5060 7.5552) 3 2 4 1.5 1 1 92.9154 92.8955 (92.8421 92.9489) 8.4779 8.4954 (8.4635 8.5273) 3 2 4 1.5 1 1.5 93.6158 93.5884 (93.5438 93.6330) 9.2264 9.2498 (9.2182 9.2813)

Table 1. Comparison with Simulation for Different Arrival and Abandonment Rates

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Validation

77.17.27.37.47.57.67.77.87.9

88.1

1 2 3 4

λ o

Qav

Qav-Analytical

Qav-Simulation

Qav-CI-lb

Qav-CI-ub

8686.5

8787.5

8888.5

8989.5

9090.5

9191.5

9292.5

9393.5

94

1 2 3 4

λ o

Pav

Pav-Analytical

Pav-Simulation

Pav-CI-lb

Pav-CI-ub

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Effect of Arrival Rates

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Effect of Abandonment Rates

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Observations

Average auction price is less than the market price

As the truck arrival rate or the order abandonment rate increases the auction price decreases

As the order arrival rate or the truck abandonment rate increases, the auction price increases

When different types of carriers are accepted, the average auction price decreases

The average auction price decreasing in the capacity for carriers and increasing in the capacity for orders

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Conclusions

An analytical model that captures the auction mechanism with the dynamics of the system is developed.

The model allows the users to examine the effects of various system parameters on the performance measures

The analytical results answer various design questions

– Should a first price or second price auction be used?

– Should the total number of bidders be revealed during the auction?

– ...

Utilization of the logistics auction market allows

– producers to reduce their transportation costs

– transporters to utilize their capacity in more efficient way

– logistics companies to create value by being an intermediary between producers and transporters