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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