IEGR 204 – INTRODUCTION TO IE AND COMPUTERS
MiniProject #1 (Slides Due no later than Monday,
5/11/15 via Bb by 11:59pm)
Using Microsoft PowerPoint (PP), complete a presentation about the
sub-area of interest in Industrial Engineering (IE) which you have
started your research on with the earlier assignments (Research
Assignments #1 and #2, and Lab #2). You must also find a 2nd
refereed research article on the same topic area, and a 3rd source of
your choice (webpage, magazine article, textbook, podcast, etc.).
Thus, a total of three sources are mandatory for this project (and
anything short of this will result in a deduction on your project
grade).
The PP slides must meet the following criteria:
8 slides minimum ( including the mandatory 1st slide for the
title page and a last slide entitled “References” for listing your
full APA-style bibliographic citations); both of these were
started during your Lab #2 assignment
The remaining slides in between must meet the following
specifications:
o Your 2nd slide must be entitled “Outline”
o The 3rd slide entitled “Background”, 4th slide entitled
“Research Problem”, 5th and 6th slides should be entitled
“Math Equations Used”, 7th slide entitled “Conclusions”,
and 8th entitled “References” as stated earlier
o
o
o
o
Keep in mind that on the “Math Equations Used”
slide, you must explain 2 or more equations used
within the articles in a brief statement or phrase
If there are no equations in the 1st article, then use
whatever is math related from the article…a
segment of a computer program described, a
mathematical algorithm shown, etc.
Be sure that the 2nd article contains mathematical
equations if the 1st article does not
If neither of the refereed articles contain
mathematical equations, your project will receive a
deduction
Remember, you must use your literature critique
assignment (from your 1st approved article) and a 2nd
refereed article for this project, as well as 3rd source
The presentation title should reflect the area of IE
which your project focuses on and not the title of
any of the articles (e.g., Robotics in IE, Energy
Systems in IE, Human Factors in IE, etc.).
Also, be sure to include your name, class, and due
date of project on the title slide
Remember, you can ‘pull’ some information from your
literature critique assignment into the “Background” or
“Research Problem” slides of the project
You can also use a non-refereed article for your 3rd source
to gather information for your “Background” slide
You must have a minimum of 3 sources/references as
stated earlier, you can gather more if they are related.
However, do not list any sources that are not used in this
project. In other words, having 5 sources will not gain
o
o
o
o
o
o
you any extra points on the project, but could only
enhance your project if utilized properly.
Thus, a minimum of 3 sources (2 refereed research
journal articles, and 1 additional source) must be
cited in the project on the “References” slide
You must include two slides with some IE mathematical
formulations and explain briefly how/why the author(s)
were using the math
You can include a figure(s) in your slides from your IE
research sources as long as you properly cite the
reference source (i.e., you must give the author(s) credit
for the information which you are utilizing!) – remember
to use the APA-format in-text citation with Author’s last
name and date in parenthesis at the end of the caption!
You must make good use of animation on slides and
between slides during transitions (if you need help on
this, ask/see the instructor!)
Again, you must use the APA journal style for your
bibliographic citations on the “References” slide as shown
on my Research webpages (i.e., the same format used on
your Literature Critique assignment – if done correctly)
If the citation(s) is/are incorrect, you will receive
deductions on the project
DO NOT use paragraphs of information on the slides and
please use a serif type font (such as Times New Roman,
Times Roman, etc.) and no size less than 14 pt font on
each slide
Use last week’s lab (from 4/29/15) which your submitted as part of
your PowerPoint presentation slides for this project (i.e., don’t
waste the work you have already done). While working on the
project (or if you complete the work before the due date), if you
have any additional questions, be sure to ask the instructor and not
just a classmate. Do not ask me questions such as (1) “Does this
look right?” or (2) “Is this OK?”…please only ask legitimate questions
which deal with uncertainty about a specific issue/item.
This is a major part of your grade, and if your friend gives you the
wrong information you will not be able to blame them! DO NOT
ASSUME anything if you are unsure about something.
Lastly, submit your PowerPoint presentation slides on-time along
with the 2 additional sources via Bb to receive full credit. If the Bb
submission time stamp is beyond 11:59pm on the due date, you will
receive a 20% deduction per day in points for lateness as on other
assignments. Take this very seriously! If no file is ever submitted
via Bb, you will NOT receive a project grade nor make it up at a
later time.
The timely submission will be part of the overall MiniProject grade.
The remaining portions of the grade will be made up from meeting
the specifications listed above. A project grading table for the PP
slides will be shown at a future date in class or posted via Bb. This
table will detail the full project requirements and allows for
everyone to know ‘up front’ what is necessary to achieve the
maximum number of points.
Do your best!
Be sure to submit your MiniProject Slides by the assigned due date and time.
It may be submitted ahead of time as well!
FINAL NOTE: Again, any assignment that has an Bb submission time stamp beyond the
assigned due time will be reduced (i.e., the grade assigned will have a deduction for
lateness of 20% off per 24 hour period). NO EXCEPTIONS!
If you have any questions, do one of the following: (1) please ask them in class, (2) ask via
email (Richard.Pitts@morgan.edu), or (3) come by during office hours to see me.
Created on May 2, 2015 by Dr. Richard Pitts, Jr.
Last Updated on May 3, 2015 by Dr. Richard Pitts, Jr.
INDUSTRIAL ENGINEERING
1
Industrial Engineering
Name:
Institution:
INDUSTRIAL ENGINEERING
2
This article discusses the challenges associated with travel time uncertainty and the
algorithmic solution employed to reduce route mechanism. Since on-time deliveries are very
critical for production purposes and responding to logistics are very important for customers it is
very vital to compute and ensure on- time the probability of vehicles in client locations are
known. Vehicles routing problem is an old concept that aims at designing a combination of
minimum- cost routes for logistics vehicles. The chosen routes start at the depot, cover all
customers within the geographical location and route back to the depot. Demand for each route
should coincide with vehicle capacity limit and duration should not exceed expected limits.
Three elements are very important in most VRP studies. These are demand, travel times
and customers. However, these elements can be stochastic due to the complexity of real logistic
application field. But the elements cannot be ignored if a feasible and optimal delivery solution
is to be achieved. After the literature review from past researchers a new proposal is presented.
The new VRP tries to solve both service time and travel time uncertainties associated with
vehicle routing. The new proposal is successful minimizing carrier costs and ensure on- time
arrivals for the different customers in different locations. This was achieved using an iterated
tabu search heuristic algorithm.
According to my opinion. The most intriguing part of this article is the instance when the
proposed model was to be tested on other six different problem instances. A larger increase in
INDUSTRIAL ENGINEERING
vehicle numbers, it looks to improve customer service, reduce time, reduce costs and ensure
location of the vehicles can be ensured at each specific time.
3
INDUSTRIAL ENGINEERING
4
References
Zhang, J., Lam, W., & Chen, B. (2013). A Stochastic Vehicle Routing Problem with Travel Time
Uncertainty: Trade-Off Between Cost and Customer Service. Netw Spat Econ, 13(4),
471-496. doi:10.1007/s11067-013-9190-x
Factory Farming
“For modern animal agriculture, the less the consumer knows about what’s happening
before the meat hits the plate, the better. Should we be reluctant to let people know what really
goes on, because we’re not really proud of it and concerned that it might turn them to
vegetarianism?” With an unknown author, it is almost certain this was said by a factory farm
employee. The worst part about this is that the speaker is not concerned with the fact that
millions of animals are being abused daily, but that if people know what happens to the food
before they buy it, they might not anymore. The quote above is not only approved by the
producers, but accepted by consumers. With animal cruelty it has turned into a don’t ask, don’t
tell, situation. People don’t want to hear about the abuse the animal go through that provided
them with dinner. The way these cows, pigs, chickens, and other farm raised animals are forced
to live their short lives is not humane. Everyone has turned a blind eye to factory farming.
Factory farming began in the 1920s right after scientists discovered a new use for
vitamins A and D. When these vitamins are added to the food, animals no longer need exercise
and sunlight to grow. This allowed large numbers of animals to be raised indoors year-round.
This eventually created problems. It allowed disease to spread quick and easy. In return to this
problem, antibiotics were made. The animals are pumped with medicine to prevent illness. This
solution can be even more dangerous to humans. Salmonella poising is becoming more difficult
to control because they are antibiotic resistant. Antibiotics are not the only chemicals given to the
animals; they are also filled with hormones, fertilizers, herbicides, and pesticides that collect in
the tissue of their bodies. They raise the animals in an assembly line way because it is considered
productive and cheat.
Unfortunately, this trend of mass production has resulted in incredible pain and suffering
for the animals. In the food industry, animals are not considered living creatures; they are looked
at as a money producing item. They are confined to small cages with metal bars, ammonia-filled
air and artificial lighting or no lighting at all. They are given horrible mutilations such as beak
searing, tail docking, ear cutting and castration. This is no way live.
A standard slaughterhouse kills 250 cattle every hour. This contributes to the 35 million
beef cattle that are slaughtered each year in the United States. It would be a different story if the
animals had a good life and were cared for properly, but this is not the case. These cows live in
inhumane conditions where they are not protected from freezing rain or the blazing sun. Most are
not fed or given water on a daily bases and some even die from starvation and dehydration.
Many develop diseases from the harsh living environment and are never taken to the vet or cared
for properly. Their living quarters are so tiny and hardly ever cleaned.
Depending on the cows use, it may be treated different ways. A baby cow being
used for veal is only kept alive 4 months. The calves live in small crates. This allows very little
movement which prevents muscle growth so their flesh will remain tender. They are also made
to eat a low iron diet to keep their flesh pale and appealing to the consumer. Veal calves spend
each day alone with no companionship and hardly ever see light.
Dairy cows are bred today for high milk production. Many cows are raised in
complete confinement, where they suffer emotionally from being alone and not living naturally.
Dairy cows produce milk for about 10 months after giving birth so they are impregnated
continuously to keep up the milk flow. Female calves are kept to reproduce. When cows become
unable to produce the expected amount of milk they are sent to slaughter house so money can be
made from their flesh. The cows are kept in a holding facility where they are given food and
water. They have their waste removed mechanically and are allowed out only twice a day to be
milked by machines. Because of the unnatural amounts of milk they produce, there udders
become swollen and the milking process is extremely painful.
When the cows can no longer stand, let alone walk, they are considered “downed”. This
means from neglect, these animals are forced to lie in their own waste with food and water.
These animals are not done suffering yet. After lying for days without care they are expected to
be slaughtered. When this time comes workers drag, beat, and push the sick creature with
equipment to get it to the slaughter house. They’ll do whatever it takes.
The federal Humane Slaughter Act says an animal must be rendered unconscious before
being killed. This is usually done by using a machine that sends a blow to the cows head. This
machine however, is not 100% effective. Many times conscience animals are hung by their legs,
kicking and moaning. Eventually they are stabbed in the throat and bleed to death. This is shown
in detail in an April 2001 Washington Post article, which describes typical slaughter house
conditions: “The cattle were supposed to be dead before they got to Moreno. But too often they
weren't. They blink. They make noises, [he said softly]. The head moves, the eyes are wide and
looking around. [Still Moreno would cut]. On bad days, [he says], dozens of animals reached his
station clearly alive and conscious. Some would survive as far as the tail cutter, the belly ripper,
the hide puller. They die, [said Moreno], piece by piece...”
90% of pigs raised for food are confined at some point in their lives. Pigs are highly
social, affectionate and intelligent creatures, and suffer both physically and emotionally when
they are confined in narrow cages where they cannot even turn around. Many pigs become crazy
with boredom and develop nervous ticks while others fight and turn to cannibalism. Pigs are born
and raised inside buildings that have automated water, feed and waste removal. They are never
taken outside until they are shipped to the slaughter house. Dust, dirt and poorly ventilated trucks
result in the death of some pigs on the way.
Pigs used for reproduction are forced to lie on their side most of their lives to allow
piglets to nurse. Piglets are taken from their mother just 10 days after being born. They are kept
in tiny areas with many other pigs. Since they were not allowed fulfill their natural sucking
instincts, they bit other pigs tails off. The workers solution for this was to chop the pigs’ tails off
using no anesthesia. This results in the tail being extremely sensitive and when the piglets bite or
suck on them, it causes excruciating pain.
Chickens raised for their flesh are called broiler chickens. The broiler chicken industry
produces 6 billion chickens a year to be killed. This industry is ruled by only 60 companies
which have created an oligopoly. Broiler chickens are selectively bred and genetically altered to
produce bigger thighs and breasts, the parts in demand. When bred in this fashion the birds
become so heavy that their bones and organs can no longer do their jobs. This makes it difficult
for them to stand and causes some to have heart attacks and organ failure. The birds are grown so
fast they reach a weight of 3 1/2 pounds in seven weeks. These chickens are raised in
overcrowded broiler houses instead of cages to prevent bruising of flesh which would make their
meat undesirable. Their beaks and toes are cut off and they live in complete darkness to prevent
fighting among the birds. This process known as "debeaking" is inhumane. After the beaks are
clipped they chickens live in pain for weeks. Some cannot even eat.
There are 250 million hens in U.S. egg factories that supply 95% of the eggs in the
United States. In these facilities the birds are held in battery cages that are very small with
slanted wire floors which cause discomfort and foot deformation. Usually eight birds are
crammed in 14 square inch cages. Since the birds have no room, they become very aggressive
and attack the other birds in their cage. The chicks are sorted at birth and newborn males are
separated and suffocated in trash bags. The layer hens are subjected to constant light to
encourage egg production. At the end of their laying cycle they are either slaughtered or given no
food and water to push them into another laying cycle. Many birds become have no minerals
because of this excessive egg production and either die from fatigue or can no longer produce
eggs and are sent to the slaughterhouse.
The general image of farm animals is a pleasant one. The children's rhyme “old
McDonald” for example does not mention that the animals are killed for food. But somehow we
are left with the idea that except for being killed at the end, farm animals are well cared for.
Farm animals in our minds enjoy beautiful surroundings and lead natural live. We picture cows
grazing on lush green grass, chickens scratching on the ground, and pigs rooting through the
mud. But how accurate is this picture? Sadly, image and reality do not always agree. Most farm
animals do not live in the barnyard or on a farm at all. Instead they stay in crowded cages and are
confined inside buildings. They and are forced to go through mutations and given chemicals and
hormones. These animals never live a normal life. Do we want to be part of a system which
produces such pain, suffering, and death, day in order to produce inexpensive meat? Meat
consumption is detrimental even to our own health.
Bibliography
"Factory Farming." Sustainable Table. 7 Dec. 2008
.
"Factory Farming." Farm Sanctuary. 2008. 7 Dec. 2008
.
"Factory Farming." HFA [Factory Farming]. The Humane Farming Association. 7 Dec. 2008
.
Netw Spat Econ (2013) 13:471–496
DOI 10.1007/s11067-013-9190-x
A Stochastic Vehicle Routing Problem with Travel
Time Uncertainty: Trade-Off Between Cost
and Customer Service
Junlong Zhang & William H. K. Lam & Bi Yu Chen
Published online: 30 May 2013
# Springer Science+Business Media New York 2013
Abstract On-time shipment delivery is critical for just-in-time production and quick
response logistics. Due to uncertainties in travel and service times, on-time arrival
probability of vehicles at customer locations can not be ensured. Therefore, on-time
shipment delivery is a challenging job for carriers in congested road networks. In this
paper, such on-time shipment delivery problems are formulated as a stochastic vehicle
routing problem with soft time windows under travel and service time uncertainties. A
new stochastic programming model is proposed to minimize carrier’s total cost, while
guaranteeing a minimum on-time arrival probability at each customer location. The aim
of this model is to find a good trade-off between carrier’s total cost and customer service
level. To solve the proposed model, an iterated tabu search heuristic algorithm was
developed, incorporating a route reduction mechanism. A discrete approximation method is proposed for generating arrival time distributions of vehicles in the presence of
time windows. Several numerical examples were conducted to demonstrate the applicability of the proposed model and solution algorithm.
Keywords Vehicle routing . Time window . Customer service . Stochastic programming
. Tabu search . Discrete approximation
1 Introduction
The Vehicle Routing Problem (VRP), introduced by Dantzig and Ramser (1959),
involves the design of a set of minimum-cost routes for the vehicles of a logistics
J. Zhang (*) : W. H. K. Lam : B. Y. Chen
Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
e-mail: junlong.zhang@connect.polyu.hk
B. Y. Chen
State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing,
Wuhan University, Wuhan 430079, China
472
J. Zhang et al.
company. The routes must start at the depot, serve a group of geographically scattered
customers, and finally return to the depot. Each customer can only be visited once by
one single vehicle. The total demand of each route cannot exceed the capacity of the
vehicle and the duration of each route cannot exceed a given limit. VRP has broad
applications in distribution and logistics management fields. In the last 50 years,
strides have been made in the development of efficient and effective solution algorithms using both exact and heuristic approaches (Laporte 2009). More than one
hundred software companies are now selling commercial vehicle routing software
and thousands of logistics companies are using VRP software (Drexl 2012).
In the literature, a number of VRP variants have been intensively studied (Toth and
Vigo 2002; Golden et al. 2008; Leung et al. 2011; Norouzi et al. 2012; Yu et al. 2011;
Escuín et al. 2012; Li et al. 2012). Vehicle routing problems with time windows
(VRPTW) form a large proportion of the matters studied. One example is the request for
a vehicle to start service within a given time interval (i.e. time window). The time
window constraints can be modeled as either hard or soft. In the hard time window case,
customers refuse the service of late arrival vehicles (Solomon 1987; Savelsbergh 1992;
Cordeau et al. 2002; Nagata et al. 2009; Yu and Yang 2011). In the soft time window
case, customers accept the vehicle service regardless of arrival time, but nonetheless
penalties for earliness or tardiness are incurred (Koskosidis et al. 1992; Balakrishnan
1993; Taillard et al. 1997; Chiang and Russell 2004; Liberatore et al. 2011). In both the
hard and soft cases, early arrival vehicles must wait until the customer’s requested
service time arrives.
In most of VRP studies, the three elements including demand, customers and travel
times are assumed to be deterministic. In reality, however, all these elements can be
highly stochastic due to the complexity of the real logistics applications. For example,
travel times in urban road networks are highly stochastic due to roadway capacity
variations and traffic demand fluctuations (Lam et al. 2008; Chen et al. 2011, 2012;
Li et al. 2012; Wei et al. 2012). Ignoring the stochastic nature of these elements may lead
to sub-optimal even infeasible delivery solutions. In view of this, researchers have
investigated the stochastic vehicle routing problem (SVRP) by considering stochastic
demands and/or customers (Bertsimas and van Ryzin 1991; Gendreau et al. 1995;
Laporte et al. 2002; Lei et al. 2011) and uncertain travel times (Laporte et al. 1992;
Lambert et al. 1993; Kenyon and Morton 2003; Zhang et al. 2012).
Some researchers have also investigated SVRP with time windows under travel
time uncertainties. Ando and Taniguchi (2006) studied SVRP with soft time window
constraints. A model was proposed to minimize carrier’s total cost, which is comprised of the fixed vehicle employment cost, operating cost and penalty cost. The
operating cost was assumed to be proportional to the total mean travel time, while the
penalty cost was formulated as the expected earliness and tardiness of vehicles at
customer locations. Travel time distributions were estimated from probe vehicle data.
Russell and Urban (2008) developed a multiple-objective model for SVRP with
soft time window constraints. Priorities among different objectives were assumed in
order of the number of required vehicles, total distance traveled and time-window
penalties incurred. The model minimizes a weighted average of these objectives. A
tabu search heuristics was developed to solve the model. To reduce the number of
required vehicles, the fixed vehicle employment cost was multiplied by a large
weighting parameter in the objective function. The earliness and tardiness penalties
Vehicle Routing Problem with Travel Time Uncertainty
473
due to time window violation were deduced based on the assumption of Erlang travel
time distributions.
Li et al. (2010) investigated SVRP in both the hard and soft time window cases.
Uncertain service times were also considered. In the soft time window case, a two-stage
stochastic programming with recourse model was built. The objective of the model is to
design a set of routes in the first stage and to minimize the expected costs in the second
stage when random travel and service times are realized. A tabu search heuristics was
also developed to solve the model. Travel and service times were assumed to be
normally distributed and stochastic simulation was used for probability check and
computing expected values. In the hard time window case, a chance-constrained model
was proposed to ensure that the probability of vehicles arriving at customers within the
time windows is at least a pre-specified value. Expected earliness and tardiness of
vehicles were not included in carrier’s total cost in the hard time window case.
On the basis of the previous works, this paper aims to investigate SVRP with soft
time window constraints under travel and service time uncertainties. In the rest of the
paper, the SVRP with soft time window constraints is referred to as SVRPSTW. The
previous works is extended in the following two aspects.
(1) A new stochastic programming model is proposed in this paper not only to minimize
carrier’s total cost, but also to guarantee a minimum on-time arrival probability at
each customer location. The previous SVRPSTW models mainly focused on
reducing carrier’s total cost. This optimization approach is essentially formulated
from the perspective of the carrier in order to provide shipment delivery service at
minimum total cost. By this approach, the probability of late shipment delivery for
some customers may be quite high. In practice, inventories are limited due to high
holding costs and therefore production or sales processes may be disrupted by
frequently delayed shipments. It is common for customers to require a certain
probability of on-time shipment delivery, though late service is at times permitted.
In this paper, the on-time arrival probability at each customer location is explicitly
formulated in the proposed model by the introduction of a customer service level
constraint. The probability of on-time shipment delivery to each customer can be
then ensured. The advantages of the proposed model are listed below.
&
&
&
The proposed model is a generalization of the conventional recourse models
for SVRPSTW in the literature (Russell and Urban 2008; Li et al. 2010).
When no customer service level constraint is imposed, solutions of the
proposed model are the same as the previous SVRPSTW models.
The proposed model provides an easy way of exploring trade-offs between
carrier’s total cost and customer service level, simply by adjusting the
customer service level constraints.
The proposed model can easily be adapted to multi-class customers with
various service level preferences by imposing appropriate customer service
level constraints.
(2) An iterated tabu search heuristic algorithm is developed to solve the proposed
model. A route reduction mechanism is designed and incorporated in the
developed heuristics. When performing a neighborhood search, trial move costs
that remain unchanged in one iteration are kept in the memory for use in the next
474
J. Zhang et al.
iteration. On the basis of these cost records, every route that can be decomposed
and inserted into other routes is identified in an SVRPSTW solution. In this way,
the number of routes can be reduced. Additionally, inspired by Miller-Hooks
and Mahmassani (1998) and Chen et al. (unpublished), an approximation
method called α-discrete is proposed in this paper for generating arrival time
distributions of vehicles in the presence of time windows. Using this approximation method, SVRPSTW solutions can be evaluated without suffering restrictions on the assumption of travel and service time distributions.
The remainder of this paper is organized as follows. The proposed model is
formulated in the following section. The iterative tabu search heuristics
designed for solving the proposed model is presented in Section 3. The
approximation method for estimating arrival time distributions is described in
Section 4. Computational results are shown in Section 5. Finally, concluding
remarks are given in Section 6.
2 Model Formulation
Let G=(V0,A) be a complete digraph, where V0 ={0,…,n} is the vertex set and A={(i,j):
i,j∈V0,i≠j} is the arc set. Vertex 0 represents the depot where m0 identical vehicles with
capacity Q are available. The customer set is denoted as V ¼ V0 =f0g ¼ f1; …; ng. Each
customer i∊V has a nonnegative demand qi, a service time Si and a time window [ei,li]. It
is expected that service at customer i begins within [ei,li]. If the vehicle arrives at customer
i’s location before ei, it has to wait until ei; if it arrives at customer i’s location after li, a
penalty proportional to the lateness must be paid. A time window [e0,l0] is also
associated with the depot, where e0 represents the earliest possible departure
time from the depot and l0 represents the latest possible arrival time at the
depot. A travel time Tij is associated with each arc (i,j)∊A. Both Tij and Si are
random variables with distributions assumed to be known and independent of
everything else. Additional assumptions are: Q≥qi,i∊V (i.e. each vehicle can
serve at least one customer) and m0 is big enough (i.e. there are sufficient
vehicles at the depot). Additional notation is listed as follows:
M
f
m
m*
K
xijk
Rk
Ar j k
T ss
r jk
d0k
a sufficiently large number
fixed cost of employing one vehicle
number of required vehicles in a feasible solution, m≤m0
number of required vehicles in the optimal solution, m*≤m
the set of required vehicles in a feasible solution, K={1,2,…,m}
a binary variable associated with each arc (i,j)∊A. It is equal to 1 if and only if
arc (i,j) is traversed by vehicle
k and 0 otherwise, k∊K
route k defined as Rk ¼ r0 ¼ 0; r1 ; …; r j ; r jþ1 ; …; rnk ; rnk þ1 ¼ 0 , where nk
is the number of customers assigned to vehicle k, rj ∊V0,0≤j≤nk +1,k∊K
arrival time of vehicle k at vertex rj’s location, rj ∊Rk,1≤j≤nk +1,k∊K
service start time of vehicle k at customer rj, rj ∊Rk,1≤j≤nk,k∊K
departure time of vehicle k from the depot, k∊K
Vehicle Routing Problem with Travel Time Uncertainty
Dr j k
W r jk
Pr j k
Bk
h
Zk
l1i
l2i
l2,0
l3
αi
α0
β
475
departure time of vehicle k from customer rj’s location, rj ∊Rk,1≤j≤nk, k∊K
earliness (waiting time) of vehicle k at customer rj’s location, rj ∊Rk,1≤j≤nk, k∊K
tardiness of vehicle k at vertex rj’s location, rj ∊Rk,1≤j≤nk +1,k∊K
duration of route k, k∊K
upper bound of the duration of each route
excess route duration for vehicle k, k∊K
penalty coefficient for earliness at customer i’s location, i∊V
penalty coefficient for tardiness at customer i’s location, i∊V
penalty coefficient for tardiness when a vehicle returns to the depot
penalty coefficient for excess route duration
required probability of on-time shipment delivery (i.e. required service level)
by customer i, i∊V
required on-time arrival probability when a vehicle returns to the depot
required probability that duration of each route is smaller than h.
The stochastic programming model for SVRPSTW proposed in this paper is given
below.
Min
nk
k þ1
X X
X X
nX
Mf ⋅m þ
E T ij xijk þ
l1r j E W r j k þ
l2r j E Pr j k þ l3 EðZ k Þ
ði; jÞ∈A k∈K
Subject to
XX
xijk ¼ 1;
ð1Þ
j¼1
j¼1
k∈K
!
∀i∈V
ð2Þ
j∈V0 k∈K
X
x0jk ¼ 1;
∀k∈K
ð3Þ
xi0k ¼ 1;
∀k∈K
ð4Þ
j∈V
X
i∈V
X
xijk −
i∈V0
X
xjik ¼ 0;
∀ j∈V; k∈K
ð5Þ
i∈V0
X X
qi
xijk ≤ Q;
i∈V
∀k∈K
ð6Þ
∀r j ∈Rk ; 1≤ j≤ nk þ 1; k∈K
ð7Þ
j∈V0
P Ar j k ≤ l r j ≥αr j ;
PfBk ≤ hg≥ β; ∀k∈K
ð8Þ
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J. Zhang et al.
xijk ¼ f0; 1g;
∀i; j∈V0 ; k∈K
ð9Þ
The objective function Eq. (1) consists of three parts: 1. fixed vehicle employment
cost, 2. total mean travel time as the operating cost, and 3. weighted expected earliness,
tardiness and excess route duration as penalty cost. W r j k , Pr j k and Zk are given in Eqs.
(10), (11) and (13) respectively. Illustrations of the penalty coefficients l1i and l2i,i∊V
can be found in Fig. 1. The proposed model has a hierarchical optimization objective:
the primary objective is to minimize the number of required vehicles to satisfy constraints (2) to (9); the secondary objective is to minimize the operating and penalty costs
given the minimized number of vehicles m. This hierarchical optimization objective
implies that one SVRPSTW solution with fewer routes but higher operating and penalty
costs is better than another with more routes but lower operating and penalty costs.
W r j k ¼ max er j −Ar j k ; 0 ; r j ∈Rk ; 1 ≤ j ≤ nk ; k∈K
ð10Þ
Pr j k ¼ max Ar j k −l r j ; 0 ; r j ∈Rk ; 1 ≤ j≤ nk þ 1; k∈K
ð11Þ
Bk ¼ T r 0 r 1 þ
nk
X
W r j k þ S r j þ T r j r jþ1 ;
r j ∈Rk ; k∈K
ð12Þ
j¼1
Z k ¼ maxfBk −h; 0g;
ð13Þ
k∈K
Equation (2) indicates that each customer must be visited exactly once by one vehicle.
Equations (3) and (4) ensure that each vehicle starts and ends its route at the depot.
Equation (5) ensures that each vehicle departs from a customer location after it visits the
customer. Equation (6) is the capacity constraint. Equation (7) is the customer service level
constraint and ensures that the probability of on-time shipment delivery (i.e. service level)
to each customer is at least a predefined value αi (i.e. required service level). Similarly,
when a vehicle returns to the depot, the on-time arrival probability at the depot must be
larger than a threshold α0. The required service level of each customer can be adjusted
according to practical requests. Equation (8) ensures that each route is completed within h
with at least probability β. If αi in Eq. (7) and β in Eq. (8) are set to zero, the proposed
model then reduces to the conventional recourse models for SVRPSTW (Russell and
Urban 2008; Li et al. 2010). Equation (9) defines the domain of the decision variables.
Figure 1 shows the relationship between the time window specified by customer i and
possible arrival times of vehicle k. If the vehicle arrives earlier (later) than ei (li), an
earliness (tardiness) penalty cost will then be incurred with unit earliness (tardiness)
On-time arrival probability
Penalty
cost
Early arrival probability
Late arrival probability
PDF of arrival
time Aik
PDF2
PDF1
2i
1
1i
1
ei
1
2
li
Arrival time
Fig. 1 Time window [ei, li] and arrival time Aik of vehicle k at customer i′s location
Vehicle Routing Problem with Travel Time Uncertainty
477
penalized by l1i (l2i). The probability density functions (PDF) of three possible vehicle
arrival times are also shown in Fig. 1. In deterministic cases, arrival times μ1 and μ2 may
be accepted because both lie within the time window. While under travel time uncertainties, arrival times with PDF1 and PDF2 might not be accepted since PDF1 can
possibly lead to a long waiting time and PDF2 may result in a large late arrival
probability. In this sense, it is reasonable and necessary to incur an earliness penalty
cost and to impose a constraint on the vehicle’s minimum on-time arrival probability.
In the proposed model, objective function Eq. (1) represents carrier’s total cost and
constraint Eq. (7) guarantees customers’ required service level. In order to highlight the
primary optimization objective of the proposed model, fixed vehicle employment cost is
multiplied by a sufficiently large number M in Eq. (1). By adjusting αi,i∊V in Eq. (7),
various trade-offs between carrier’s total cost and customer service level can be obtained.
An alternative way to trade off between carrier’s total cost and customer service level is
to adjust the penalty coefficient l2i,i∊V in Eq. (1), without imposing any customer service
level constraint as Eq. (7). If M in Eq. (1) is a moderate number, l2i can be then increased to
a level such that tardiness penalty cost is comparable to the fixed vehicle employment cost.
In this way the trade-off between cost and service can also be found. However, the increase
of l2i may be irrational as fixed cost of employing one vehicle is usually much larger than
the unit penalty cost (e.g. 1000 versus 0.5 in Russell and Urban 2008). Irrational increase of
l2i may lead to unrealistic cost coefficients and even irrational routing solutions
(Koskosidis et al. 1992). Compared with this method, the proposed model in this paper
provides a more straightforward way of achieving the goal of exploring trade-offs between
carrier’s total cost and customer service level. The probability of on-time arrival to each
customer is explicitly formulated in the proposed model and guaranteed by customer
service level constraints. This has a clear implication in practical applications.
3 Solution Algorithm
In this section, a heuristic algorithm based on an iterated tabu search (ITS) by
Cordeau and Maischberger (2012) is developed for solving the proposed model. In
ITS, an iterated local search (Lourenço et al. 2002) is used as the general framework
and a tabu search is adopted as the local search improvement method. The major
strengths of ITS are simplicity, flexibility and efficiency. The neighborhood structure
of the tabu search heuristics in ITS is simple and a single type of solution perturbation
is adopted. ITS is also flexible as it can solve many VRP variants without changing
the methodology and parameter settings. Finally, ITS is reasonably fast and effective.
Cordeau and Maischberger (2012) reported solving classical VRP and seven VRP
variants by using ITS, and showed the competitiveness of ITS with other heuristics
for each particular problem variant.
The heuristic algorithm developed in this section is a modified version of ITS,
denoted as SVRP-ITS. Minor extensions of ITS are made in SVRP-ITS: (a) a direct
route reduction mechanism is incorporated in SVRP-ITS, aiming to reduce the
number of required vehicles in a SVRPSTW solution; (b) when performing a
neighborhood search, trial move costs that remain unchanged in one iteration are
kept in the memory for use in the next iteration. These cost records can be used to
avoid possible repetitive computing in both neighborhood search and route reduction
478
J. Zhang et al.
processes. In the following sections, the main framework of SVRP-ITS is shown first,
followed by brief descriptions of each component.
3.1 Main Framework of SVRP-ITS
The main framework of SVRP-ITS is summarized in Algorithm SVRP-ITS. Firstly a
feasible initial solution s0 is constructed (Section 3.2). In each iteration of SVRP-ITS,
tabu search (Section 3.3) is then used to improve the current solution s′, resulting in an
improved solution es. The best feasible solution s* is updated accordingly. Before the start
of the next iteration, s′ is renewed employing a perturbation mechanism (Section 3.5). If
es satisfies the acceptance criterion (Section 3.4), es is then perturbed, otherwise s* is
perturbed. Let η be the maximum number of tabu search iterations allowed to be
performed during the entire search process of SVRP-ITS. ζ represents the total number
of tabu search iterations performed so far. SVRP-ITS stops if ζ is larger than η. Note that a
tabu search iteration denotes an iteration within a tabu search process (see Step 2 in
Procedure Tabu Search); c(s) is set to the objective function Eq. (1).
Algorithm SVRP-ITS
Step 1. Initialization.
s0 ← Initial Solution Construction()
s* ← s0 , s' ← s0
Step 2. Iterated local search.
while ζ is smaller than η do
Step 2.1 Local search improvement.
s ← Tabu Search( s' )
Step 2.2 Update the best feasible solution s* found so far.
if s is feasible and c(s )
>
t erj
>
>X
Ar k
Ar k
Ar k
<
ε er j −bt j ; if bt erjj ≤ er j < bt erjj þ1 ; 1≤ t erj ≤ L 1
E W r j k ¼ t¼1
>
>
L
>
X
>
Ar k
Ar k
>
>
ε er j −bt j ; if er j ≥ bL j
:
t¼1
ð26Þ
Vehicle Routing Problem with Travel Time Uncertainty
485
8
L
A
X
>
Ar k
r k
>
>
ε bt j −l r j ; if l r j < b1 j
>
>
>
> t¼1
< X
L
A
Ar k
Ar k
E Pr j k ¼
r jk
ε
b
−l
; if bt lrjj ≤ l r j < bt lrjjþ1 ; 1 ≤ t lrj ≤ L−1
>
t
r
j
>
>
>
t¼t lrj þ1
>
>
>
Ar k
:
0; if l r j ≥ bL j
ð27Þ
8
L
A
X
>
Ar k
>
r k
>
>
ε bt j −d 0k −h ; if d 0k þ h < b1 j ; j ¼ nk þ 1
>
>
>
t¼1
<
L
A
X
EðZ k Þ ¼
Ar k
Ar j k
r k
>
ε bt j −d 0k −h ; if bt h j ≤ d 0k þ h < bt h þ1
; 1 ≤t h ≤L−1; j ¼ nk þ 1
>
>
>
t¼t h þ1
>
>
>
Ar k
:
0; if d 0k þ h≥ bL j ; j ¼ nk þ 1
ð28Þ
The probabilities in Eqs. (7) and (8) can also be estimated, as shown in Eqs. (29)
and (30) respectively.
8
Ar j k
>
> 0; if l r j < b1
<
Ar k
Ar k
ð29Þ
P Ar j k ≤ l r j ¼ t lrj ε; if bt lrjj ≤ l r j < bt lrjjþ1 ; 1 ≤ t lrj ≤ L−1
>
>
Ar j k
:
1; if l r j ≥bL
8
Ar j k
>
>
< 0; if d 0k þ h < b1 ; j ¼ nk þ 1
A
A jk
PfBk ≤ hg ¼ t h ε; if bt hr j k ≤ d 0k þ h < bt rþ1
; 1 ≤ t h ≤L−1; j ¼ nk þ 1 ð30Þ
h
>
>
Ar j k
:
1; if d 0k þ h ≥ bL ; j ¼ nk þ 1
Expressions for estimating violations of route duration and service level constraints (see Section 3.3.2) in solution s are shown in Eqs. (31) and (32) respectively.
d ðsÞ ¼
X
n
h . i
o
max vAr j k β ε ⋅ε −d 0k −h; 0 ;
r j ∈Rk ; j ¼ nk þ 1
ð31Þ
k∈K
wðsÞ ¼
k þ1
X nX
n
o
h . i
max vAr j k αr j ε ⋅ε −l r j ; 0 ;
r j ∈Rk
ð32Þ
k∈K j¼1
5 Computational Results
5.1 Accuracy of the α-Discrete Approximation Method
In this section, the accuracy of the proposed α-discrete approximation method was
tested by comparison with Chang’s method (Chang et al. 2009). Stochastic simulation
(Li et al. 2010) provided the ground true value for the comparison. The test example
486
J. Zhang et al.
is a simple partial route shown in Fig. 2, where D denotes the depot and Ci represents
the customer, i=1,…,5. In Fig. 2, numbers within square brackets are customers’ soft
time windows, whereas numbers within round brackets represent the mean and
variance of link travel times. Departure time from D was set to zero.
For the proposed α-discrete approximation method, L was set to 100. For stochastic simulation, a total of 106 iterations were performed. Since arrival times were
assumed to be normally distributed
method (Chang et al. 2009),
in Chang’s
expressions for computing E W r j k , E Pr j k and customer service levels under the
assumption of normal arrival time distribution are given below,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ð33Þ
E W r j k ¼ Var Ar j k ½ϕðz1 Þ þ z1 Φðz1 Þ; r j ∈Rk ; 1≤ j≤ nk
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
E Pr j k ¼ Var Ar j k fϕðz2 Þ−z2 ½1−Φðz2 Þg; r j ∈Rk ; 1 ≤ j≤ nk þ 1
P Ar j k ≤ l r j ¼ Φðz2 Þ;
r j ∈Rk ; 1 ≤ j≤ nk þ 1
ð34Þ
ð35Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
where z1 ¼ er j −E Ar j k = Var Ar j k ; z2 ¼ l r j −E Ar j k = Var Ar j k ; ϕ(∙) is
the probability density function of standard normal distribution; Φ(∙) is the cumulative distribution function of standard normal distribution.
The test was conducted in two scenarios. Link travel times in Fig. 2 were assumed
to follow normal distribution in Scenario 1 and lognormal distribution in Scenario 2.
Given mean and variance of link travel times, parameters for the lognormal distribution of link travel times can be computed using Eqs. (36) and (37). In both scenarios,
service times at customers were assumed to follow the same normal distribution
N(10,52). The test results are shown in Tables 1 to 4.
2 .qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi
Var T r j r jþ1 þ E T r j r jþ1
μ T r j r jþ1 ¼ log E T r j r jþ1
;
σ T r j r jþ1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
.
2
¼ log Var T r j r jþ1 E T r j r jþ1 þ 1 ;
r j ∈Rk ; 0≤ j ≤ nk ð36Þ
r j ∈Rk ; 0≤ j≤ nk
ð37Þ
In Scenario 1, it can be seen from Tables 1 to 3 that Chang’s method performed well at
the first customer C1, but not at the rest of the customers. This result is expected, since the
arrival time at C1 is exactly normally distributed in Scenario 1, but arrival times at
customers C2 to C5 are not, due to the impact of time windows. In Scenario 2, the
estimation error is even larger than that in Scenario 1 by Chang’s method. For example,
the relative error between the estimated expected tardiness at C3 by Chang’s method and
[50,80]
[100,140]
[190,220]
[250,275]
[280,300]
C1
C2
C3
C4
C5
D
(60, 202 )
Fig. 2 A simple partial route
(50, 202 )
(40,152 )
(40, 202 )
(30,102 )
Vehicle Routing Problem with Travel Time Uncertainty
487
Table 1 Estimated service level at each customer by different methods
Type of
travel time
distribution
Approximation Estimated service level (%) at each customer (relative error %)
method
C2
C3
C4
C5
C1
Normal
Stochastic
(Scenario 1) simulation
84.07
74.31
Chang’s
method
84.13(0.07)
73.60(−0.96) 94.79(1.47)
α-discrete
83.50(−0.68) 73.50(−1.10) 92.50(−0.99) 88.50(−0.67) 62.50(−0.84)
Lognormal
Stochastic
(Scenario 2) simulation
85.26
77.04
93.42
92.24
89.10
63.03
89.85(0.84)
56.92(−9.69)
87.69
65.73
89.85(2.47)
56.92(−13.40)
Chang’s
method
84.13(−1.32) 73.60(−4.47) 94.79(2.77)
α-discrete
84.50(−0.89) 76.50(−0.70) 91.50(−0.81) 87.50(−0.21) 65.50(−0.35)
A bolded number represents the smallest one in relevant rows of a particular column
that by stochastic simulation is −37.09 % in Scenario 1, while it is −63.52 % in Scenario 2
(Table 3). The reason is clear that travel and arrival times are not normally distributed in
Scenario 2. This, however, is a basic assumption of Chang’s method.
For the proposed α-discrete approximation method, it was found that larger errors
were produced when estimating expected tardiness. The largest relative error between
the results of α-discrete and stochastic simulation is −1.44 % in Tables 1 and 2, while it
is −9.89 % in Table 3. However, the estimation error by α-discrete is still much smaller
compared with that of Chang’s method, as shown in Tables 1 to 3. In addition, the
accuracy of α-discrete can be further improved by increasing the value of L (e.g. 200).
Table 4 shows that the difference between the estimation error by Chang’s method
and that by α-discrete is small in terms of the total cost of the partial route (−0.52 %
Table 2 Estimated expected earliness at each customer by different methods
Type of travel
time distribution
Approximation Estimated expected earliness at each customer (relative error %)
method
C2
C3
C4
C5
C1
Normal
(Scenario 1)
Stochastic
simulation
3.95
2.14
Chang’s
method
3.96(0.11)
2.36(10.57) 18.92(−3.62)
α-discrete
3.94(−0.42) 2.11(−1.22) 19.61(−0.12)
11.81(−0.09) 0.75(−1.19)
Stochastic
simulation
3.17
12.00
Chang’s
method
3.96(24.62) 2.36(46.30) 18.92(−11.70) 11.90(−0.83) 1.25(125.57)
α-discrete
3.16(−0.44) 1.61(−0.03) 21.46(0.14)
Lognormal
(Scenario 2)
1.61
19.63
21.43
11.82
0.76
11.90(0.67)
1.25(64.98)
0.56
11.98(−0.15) 0.55(−1.44)
A bolded number represents the smallest one in relevant rows of a particular column
488
J. Zhang et al.
Table 3 Estimated expected tardiness at each customer by different methods
Type of travel
time
distribution
Approximation Estimated expected tardiness at each customer (relative error %)
method
C2
C3
C4
C5
C1
Normal
(Scenario 1)
Stochastic
simulation
1.67
4.28
0.94
Chang’s
method
1.67(0)
4.08(−4.68)
0.59(−37.09) 1.13(−31.92) 5.21(−7.52)
α-discrete
1.65(−1.51)
4.22(−1.37)
0.88(−6.07)
1.58(−4.59)
5.55(−1.61)
Stochastic
simulation
2.29
4.91
1.61
2.83
6.82
Chang’s
method
1.67(−27.31) 4.08(−16.95) 0.59(−63.52) 1.13(−60.20) 5.21(−23.53)
α-discrete
2.25(−2.04)
Lognormal
(Scenario 2)
4.78(−2.56)
1.65
1.45(−9.89)
5.64
2.59(−8.48)
6.53(−4.30)
A bolded number represents the smallest one in relevant rows of a particular column
versus −0.14 % in Scenario 1 and −2.23 % versus −0.42 % in Scenario 2). This is
because penalty cost takes up only a small part of the total cost in this test example
and mean travel times are the same for both methods. Even though this difference can
be ignored, however, the difference in estimating service levels at each customer by
these two methods is sometimes large (Table 1) and thus can not be neglected.
In conclusion, the accuracy of the proposed α-discrete approximation method is
higher than that of Chang’s method, especially when travel times are not normally
distributed. Thus α-discrete is selected as the approximation method for solution
evaluation and used in the remainder of this paper.
Table 4 Estimated cost of the partial route by different methods
Type of travel
time
distribution
Normal
(Scenario 1)
Lognormal
(Scenario 2)
Approximation Estimated
method
No. of
vehicles
used
cost of the partial route (relative error %)
Mean
travel
time
Penalty cost
Sum of mean travel
time and penalty cost
Earliness
penalty
Tardiness
penalty
Stochastic
simulation
1
220
38.30
14.18
Chang’s
method
1
220
38.39(0.23)
12.67(−10.65) 271.06(−0.52)
α-discrete
1
220
38.22(−0.21) 13.88(−2.12)
272.10(−0.14)
Stochastic
simulation
1
220
38.77
277.23
Chang’s
method
1
220
38.39(−0.98) 12.67(−31.37) 271.06(−2.23)
α-discrete
1
220
38.76(0)
18.46
17.60(−4.66)
A bolded number represents the smallest one in relevant rows of a particular column
272.48
276.36(−0.42)
Vehicle Routing Problem with Travel Time Uncertainty
489
5.2 Performance of the Proposed Model for SVRPSTW
In this section, several of Solomon’s benchmark problems (Solomon 1987) were
adapted as test instances to demonstrate the performance of the proposed model. In
the remainder of this section, the test dataset and experiment settings are introduced
first, followed by the computational results of the proposed model on this dataset.
5.2.1 Test Dataset and Experiment Settings
The well-known Solomon’s benchmark problems (Solomon 1987) have been chosen
as test datasets by many previous studies on VRPTW (Taillard et al. 1997; Cordeau et
al. 2001; Chiang and Russell 2004; Nagata et al. 2009). Several factors were
considered when these benchmark problems were generated, such as geographical
locations of customers and tightness of time windows. In this study, seven of the
benchmark problems were chosen and adapted for the demonstration of the proposed
model. Their major characteristics are listed in Table 5.
Three types of problems according to the geographical locations of customers are
shown in Table 5. For each problem type, two or three instances were chosen with
different time window width sizes. For each problem instance, there are a total of 20
customers and the sum of customer demands is listed in Table 5. Vehicle capacity is
200 units in each problem instance. Travel times were assumed to follow lognormal
distribution and service times were assumed to be normally distributed. The COV of
the travel times was randomly generated from [0.2, 0.6]. Given the mean and variance
of travel times, parameters for the lognormal distribution of travel times can be
computed using Eqs. (36) and (37). This dataset is chosen to demonstrate the
applicability of the proposed model on problem instances with different characteristics. Additional data (e.g. coordinates and time windows) of the dataset can be found
on http://web.cba.neu.edu/~msolomon/problems.htm.
The proposed solution algorithm SVRP-ITS was implemented in Matlab 7.5.0 and
run on a PC with a four-core Inter Core i7 3.40 GHz CPU (only one core was used)
and 4 GB RAM. The maximum number of tabu search iterations η was set to 2000. In
local search improvement phases, tabu search stops if the best feasible solution has
not been improved for 200 consecutive iterations. For the proposed α-discrete
approximation method, L was set to 100.
Each problem instance was tested in four scenarios, with parameter settings of the
proposed model in these scenarios shown in Table 6. In the last scenario, customers
were divided into two groups: the first 6 customers being the first group and the rest
14 the second group. It was assumed that customers in the first group require higher
service levels than those in the second group.
5.2.2 Computational Results of the Proposed Model on a Typical Problem Instance
In this section, one of the problem instances RC101.20 in Table 5 was selected
as a typical example to demonstrate the performance of the proposed model.
RC101.20 was considered typical because of the mixed (randomly distributed
75~177
30
Random
Clustered
Clustered
Rand./Clus.b
Rand./Clus.
Rand./Clus.
R109.20
C101.20
C106.20
RC101.20
RC106.20
RC107.20
430
430
430
360
360
265
265
Sum of customer demands
b
a
Geographical locations of customers in this problem type are randomly distributed and/or clustered
With vehicle velocity set to 1, the value of mean travel time is equal to the value of the distance traveled;
A bolded number represents the smallest one in relevant rows of a particular column
41~155
60
37~89
37~83
30
Random
R105.20
Time window width
Geographical locations of customers
Problem
Table 5 Characteristics of the test dataset
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Dist.
Dist.
Dist.
Dist.
Dist.
0.2~0.6
0.2~0.6
0.2~0.6
0.2~0.6
0.2~0.6
0.2~0.6
0.2~0.6
Dist.
Dist.a
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Distribution
s.d.
Distribution
Mean
Service time
Travel time
12
12
12
90
90
12
12
Mean
0.4
0.4
0.4
0.4
0.4
0.4
0.4
s.d.
490
J. Zhang et al.
Vehicle Routing Problem with Travel Time Uncertainty
491
Table 6 Parameter settings of the proposed model for each problem instance in different scenarios
Scenario
Parameter settings of the proposed model
l1i,i∊V
l2i,i∊V
αi,i∊V
l1,0
l2,0
l3
α0
β
f
1
0.5
2
0
0
1
0
0
0
1000
2
0.5
2
0.5
0
1
0
0
0
1000
3
0.5
2
0.8
0
1
0
0
0
1000
4
0.5
4a, 2b
0.8a, 0.5b
0
1
0
0
0
1000
A bolded number represents the smallest one in relevant rows of a particular column
a
Parameter setting for customers in the first group
b
Parameter setting for customers in the second group
and/or clustered) geographical locations of customers in that problem and
because the time window width (30 min) is common in just-in-time production
(Chang et al. 2009). Computational results of the proposed model on RC101.20
are shown in Table 7.
Table 7 shows that the number of vehicles used m in Scenario 1 is the
smallest among all the four scenarios. Because in achieving the primary objective of the proposed model (i.e. minimizing m while satisfying all the constraints, Section 2), constraints are the weakest in Scenario 1 with no customer
service level constraints imposed (Table 6). In such a case, capacity constraint
Eq. (6) determines the number of required vehicles in the solution. As the sum
of customer demands is 430 units in RC101.20 (Table 5) and the vehicle
capacity is 200, at least 3 vehicles are required in the solution, as shown in
Table 7. With limited number of vehicles used in Scenario 1, customer service
levels cannot be ensured. The tardiness penalty cost is the largest and the mean
customer service level is the lowest in this scenario.
Table 7 Computational results of the proposed model on RC101.20
Scenario
Computational results of the proposed model
No. of
vehicles
used
Mean
service
a
Tardiness level
penalty
Sum of mean travel
time and penalty cost
Total
costb
29.72
176.53
69.30%
544.58
3544.58
51.05
22.62
93.92 %
502.99
4502.99
455.75
85.21
6.49
98.15 %
547.45
5547.45
433.58
47.26
40.93
91.27 %
521.77
4521.77
Mean
travel
time
Penalty cost
RC101.20_1 3
338.33
RC101.20_2 4
429.31
RC101.20_3 5
RC101.20_4 4
Earliness
penalty
A bolded number represents the smallest one in relevant rows of a particular column
a
Mean value of the service levels at the 20 customers in the problem;
b
Total cost equals to the sum of fixed vehicle employment cost, mean travel time and penalty cost
492
J. Zhang et al.
In Scenario 2, a minimum service level of 50 % was specified for each customer in
RC101.20. Consequently, the number of required vehicles increases in this scenario
(Table 7). With one more vehicle used, the mean customer service level is greatly
improved compared with that in Scenario 1. This result implies that carriers may
consider adding one more vehicle to their fleet to improve their service quality. In
Scenario 3, required customer service levels were further increased to 80 %, resulting
in a high mean customer service level (Table 7). Whereas the number of required
vehicles in this scenario is the largest among all the four scenarios.
In Scenario 4, only thirty percent of the customers in RC101.20 (i.e. customers in
the first group) were guaranteed a minimum service level of 80 %, while for others
(i.e. customers in the second group) the value was 50 %. This differentiation among
customers according to their diverse service level preferences leads to a reduction in
the number of required vehicles in Scenario 4, compared with that in Scenario 3
(Table 7).
It can be also found from Table 7 that mean travel time increases when more
vehicles are used. For example, mean travel time increases by 26.89 % in Scenario 2
compared with that in Scenario 1. Earliness penalty cost also increases in line with the
increase in the number of vehicles used. The reason is that, in such a case, on average,
fewer customers will be assigned to one vehicle. Additionally, the duration of each
route will not necessarily become shorter, due to the existence of time windows.
Therefore more time might be spent in waiting at customer locations. Finally, it is
shown in Table 7 that the total cost is the highest in Scenario 3, since the number of
vehicles used is the largest in that scenario.
5.2.3 Computational Results of the Proposed Model on Other Problem Instances
To demonstrate the applicability of the proposed model on different types of
problems, the proposed model was solved on the other six problem instances
in Table 5 (except RC101.20) in this section. The computational results are
displayed in Table 8.
A large increase in the number of vehicles used in the solution (from 2 to 5) is
found from Scenario 1 to Scenario 2 for problem R105.20 in Table 8. The trade-off is
that the mean customer service level is greatly improved (from 30.65 % in Scenario 1
to 90.40 % in Scenario 2). For Scenarios 2 to 4 of problem R105.20, the number of
vehicles used is the same, although the required customer service levels differ in these
scenarios (Table 5). In these cases, the trade-off is between the mean travel time and
the tardiness penalty cost, with the number of required vehicles kept the same. Similar
trends can be found for problem R109.20, except that the increase in the number of
vehicles used from Scenario 1 to Scenario 2 is smaller for R109.20 than that for
R105.20. The reason is that the time windows specified by customers were tighter in
Problem R105.20 than those in R109.20 (Table 5).
For problems C101.20 and C106.20, the total cost is the smallest in Scenario 2
among all four scenarios as shown in Table 8. This is because the cost paid for
additional vehicles is compensated by the reduction in the tardiness penalty cost. In
Scenario 4 of these two problem instances, a reduction in the number of required
vehicles is seen again because of the differentiation among customers according to
their diverse service level preferences.
Vehicle Routing Problem with Travel Time Uncertainty
493
Table 8 Computational results of the proposed model on the other 6 problem instances
Scenario
Computational results of the proposed model
Sum of mean travel
time and penalty cost
Total
costb
Earliness
penalty
Mean
service
a
Tardiness level
penalty
350.78
12.13
1893.61
30.65%
2256.52
4256.52
455.91
84.78
32.62
90.40 %
573.31
5573.31
5
524.18
76.29
24.80
94.47 %
625.27
5625.27
R105.20_4
5
487.19
79.78
39.44
91.43 %
606.41
5606.41
R109.20_1
2
318.0
10.06
1259.10
48.35%
1587.16
3587.16
R109.20_2
4
392.16
42.01
20.05
95.40 %
454.21
4454.21
R109.20_3
4
428.62
27.46
13.28
96.65 %
469.36
4469.36
R109.20_4
4
394.15
57.63
24.94
93.93 %
476.71
4476.71
C101.20_1
2
160.82
29.45
1612.38
47.0%
1802.64
3802.64
C101.20_2
3
250.18
217.52
82.81
90.85 %
550.51
3550.51
C101.20_3
4
283.04
252.36
49.40
93.80 %
584.80
4584.80
C101.20_4
3
275.17
205.02
108.69
90.55 %
588.88
3588.88
C106.20_1
2
160.82
26.34
1505.31
50.95%
1692.46
3692.46
C106.20_2
3
248.76
194.60
78.02
91.70 %
521.37
3521.37
C106.20_3
4
281.62
230.93
45.17
94.55 %
557.72
4557.72
C106.20_4
3
250.18
201.11
81.56
92.15 %
532.85
3532.85
3417.09
No. of
vehicles
used
Mean
travel
time
Penalty cost
R105.20_1
2
R105.20_2
5
R105.20_3
RC106.20_1 3
329.76
22.33
65.0
91.92%
417.09
RC106.20_2 3
329.76
22.33
65.0
91.92%
417.09
3417.09
RC106.20_3 4
387.13
32.47
9.19
98.25 %
428.79
4428.79
RC106.20_4 3
329.76
22.33
66.42
91.92%
418.51
3418.51
RC107.20_1 3
289.67
25.12
36.46
93.75 %
351.25
3351.25
RC107.20_2 3
289.67
25.12
36.46
93.75 %
351.25
3351.25
RC107.20_3 3
353.48
25.22
23.90
96.37 %
402.60
3402.60
RC107.20_4 3
307.59
25.22
45.88
93.22%
378.69
3378.69
A bolded number represents the smallest one in relevant rows of a particular column
a
Mean value of the service levels at the 20 customers in the problem;
b
Total cost equals to the sum of fixed vehicle employment cost, mean travel time and penalty cost
For problems RC106.20 and RC107.20, it can be seen from Table 8 that solutions
in some of the scenarios are the same. For example, the solution of problem
RC106.20 in Scenario 1 is the same as that in Scenario 2. The reason is that for
problem RC106.20 at least 3 vehicles were required in Scenario 1 with no customer
service level imposed (for the same reason as stated in Section 5.2.2 for problem
RC101.20). With this number of vehicles used, customer service levels in Scenario 1
494
J. Zhang et al.
already reached the required level in Scenario 2. Therefore the problem solution for
Scenario 2 is the same as that in Scenario 1.
6 Conclusions
A stochastic vehicle routing problem with soft time windows considering travel and
service time uncertainties is presented in this paper. A new stochastic programming
model has been proposed to minimize carrier’s total cost while ensuring a minimum
on-time arrival probability at each customer location. To solve the proposed model,
an iterated tabu search heuristic algorithm was developed. A route reduction mechanism was incorporated in this heuristics to reduce the number of required vehicles. A
discrete approximation method has been proposed to estimate the arrival time distributions of vehicles in the presence of time windows.
Computational results showed that the accuracy of the proposed α-discrete approximation method is higher than that of Chang’s method (Chang et al. 2009), especially
when travel times are not normally distributed. Although travel times were assumed to
be normal or lognormal in the test example, it is noted that α-discrete can also be applied
for other travel time distributions. In addition, several numerical examples were carried
out to demonstrate the performance of the proposed SVRPSTW model. Computational
results confirmed that the proposed model can be applied on different types of problems.
Various trade-offs between carrier’s total cost and customer service level can be explored
using the proposed model. Results indicated that the proposed model can easily be
adapted to multi-class customers with various service level preferences by imposing
appropriate customer service level constraints.
The computational efficiencies of the proposed α-discrete approximation method,
Chang’s method (Chang et al. 2009) and stochastic simulation (Li et al. 2010) will be
investigated in future work. More effort is also required to improve the efficiency of
the proposed solution algorithm for solving larger problem instances. In addition, it is
noted that VRP has been studied in combination with other problems. For example,
location-routing problems (Bozkaya et al. 2010; Toyoglu et al. 2012; Silva and Gao
2012) combine the problems of facility location and vehicle routing. It is interesting
to investigate the effects of travel time uncertainties on these problems.
Acknowledgments The work described in this paper was jointly supported by a research grant
from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project
No. PolyU 5196/10E) and a research grant from the National Science Foundation of China
(41201466). The first author is sponsored by a Postgraduate Stipend of the Hong Kong Polytechnic
University. Thanks are due to two anonymous reviewers for their valuable comments that improved
both the content and presentation of the paper.
References
Ando N, Taniguchi E (2006) Travel time reliability in vehicle routing and scheduling with time windows.
Netw Spat Econ 6:293–311
Balakrishnan N (1993) Simple heuristics for the vehicle routing problem with soft time windows. J Oper
Res Soc 44:279–287
Vehicle Routing Problem with Travel Time Uncertainty
495
Bertsimas DJ, van Ryzin G (1991) A stochastic and dynamic vehicle routing problem in the Euclidean
plane. Oper Res 39:601–615
Bozkaya B, Yanik S, Balcisoy S (2010) A GIS-based optimization framework for competitive multi-facility
location-routing problem. Netw Spat Econ 10:297–320
Brandão J (2004) A tabu search algorithm for the open vehicle routing problem. Eur J Oper Res 157:552–
564
Chang T-S, Wan Y-W, Ooi WT (2009) A stochastic dynamic traveling salesman problem with hard time
windows. Eur J Oper Res 198:748–759
Chen BY, Lam WHK, Sumalee A, Shao H (2011) An efficient solution algorithm for solving multi-class
reliability-based traffic assignment problem. Math Comput Model 54:1428–1439
Chen BY, Lam WHK, Sumalee A, Li QQ, Shao H, Fang Z (2012) Finding reliable shortest paths in road
networks under uncertainty. Netw Spat Econ. doi:10.1007/s11067-012-9175-1
Chiang W-C, Russell RA (2004) A metaheuristic for the vehicle-routeing problem with soft time windows.
J Oper Res Soc 55:1298–1310
Cordeau J-F, Maischberger M (2012) A parallel iterated tabu search heuristic for vehicle routing problems.
Comput Oper Res 39:2033–2050
Cordeau J-F, Laporte G, Mercier A (2001) A unified tabu search heuristic for vehicle routing problems with
time windows. J Oper Res Soc 52:928–936
Cordeau J-F, Desaulniers G, Desrosiers J, Solomon MM, Soumis F (2002) VRP with time windows. In:
Toth P, Vigo D (eds) The vehicle routing problem. SIAM, Philadelphia, pp 157–193
Dantzig GB, Ramser JH (1959) The truck dispatching problem. Manage Sci 6:80–91
Drexl M (2012) Synchronization in vehicle routing—A survey of VRPs with multiple synchronization
constraints. Transp Sci 46:297–316
Escuín D, Millán C, Larrodé E (2012) Modelization of time-dependent urban freight problems by using a
multiple number of distribution centers. Netw Spat Econ 12:321–336
Fosgerau M, Karlström A (2010) The value of reliability. Transp Res B Methodol 44:38–49
Gendreau M, Laporte G, Séguin R (1995) An exact algorithm for the vehicle routing problem with
stochastic demands and customers. Transp Sci 29:143–155
Golden B, Raghavan S, Wasil E (2008) The vehicle routing problem: Latest advances and new challenges.
Springer, New York
Kenyon AS, Morton DP (2003) Stochastic vehicle routing with random travel times. Transp Sci 37:69–82
Koskosidis YA, Powell WB, Solomon MM (1992) An optimization-based heuristic for vehicle routing and
scheduling with soft time window constraints. Transp Sci 26:69–85
Lam WHK, Shao H, Sumalee A (2008) Modeling impacts of adverse weather conditions on a road network
with uncertainties in demand and supply. Transp Res B Methodol 42:890–910
Lambert V, Laporte G, Louveaux F (1993) Designing collection routes through bank branches. Comput
Oper Res 20:783–791
Laporte G (2009) Fifty years of vehicle routing. Transp Sci 43:408–416
Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transp
Sci 26:161–170
Laporte G, Louveaux F, van Hamme L (2002) An integer L-shaped algorithm for the capacitated vehicle
routing problem with stochastic demands. Oper Res 50:415–423
Lei H, Laporte G, Guo B (2011) The capacitated vehicle routing problem with stochastic demands and time
windows. Comput Oper Res 38:1775–1783
Leung SCH, Zhou X, Zhang D, Zheng J (2011) Extended guided tabu search and a new packing algorithm
for the two-dimensional loading vehicle routing problem. Comput Oper Res 38:205–215
Li X, Tian P, Leung SCH (2010) Vehicle routing problems with time windows and stochastic travel and
service times: models and algorithm. Int J Prod Econ 125:137–145
Li X, Leung SCH, Tian P (2012) A multistart adaptive memory-based tabu search algorithm for the
heterogeneous fixed fleet open vehicle routing problem. Expert Syst Appl 39:365–374
Li Z-C, Huang H-J, Lam WHK (2012) Modelling heterogeneous drivers’ responses to route guidance and
parking information systems in stochastic and time-dependent networks. Transportmetrica 8:105–129
Liberatore F, Righini G, Salani M (2011) A column generation algorithm for the vehicle routing problem
with soft time windows. 4OR-Q J. Oper Res 9:49–82
Lourenço HR, Martin OC, Stützle T (2002) Iterated local search. In: Glover F, Kochenberger G (ed)
Handbook of metaheuristics. International series in operations research & management Science, vol.
57. Kluwer Academic Publishers, Norwell, MA, pp 321–353
Miller-Hooks E, Mahmassani HS (1998) Optimal routing of hazardous materials in stochastic, time-varying
transportation networks. Transp Res Rec 1645:143–151
496
J. Zhang et al.
Nagata Y, Bräysy O, Dullaert W (2009) A penalty-based edge assembly memetic algorithm for the vehicle
routing problem with time windows. Comput Oper Res 37:724–737
Norouzi N, Tavakkoli-Moghaddam R, Ghazanfari M, Alinaghian M, Salamatbakhsh A (2012) A new
multi-objective competitive open vehicle routing problem solved by particle swarm optimization. Netw
Spat Econ 12:609–633
Pisinger D, Ropke S (2007) A general heuristic for vehicle routing problems. Comput Oper Res 34:2403–2435
Russell RA, Urban TL (2008) Vehicle routing with soft time windows and Erlang travel times. J Oper Res
Soc 59:1220–1228
Savelsbergh MWP (1992) The vehicle routing problem with time windows: minimizing route duration.
INFORMS J Comput 4:146–154
Silva F, Gao L (2012) A joint replenishment inventory-location model. Netw Spat Econ. doi:10.1007/
s11067-012-9174-2
Solomon MM (1987) Algorithms for the vehicle routing and scheduling problems with time window
constraints. Oper Res 35:254–265
Taillard E, Badeau P, Gendreau M, Guertin F, Potvin J-Y (1997) A tabu search heuristic for the vehicle
routing problem with soft time windows. Transp Sci 31:170–186
Thompson RG, Taniguchi E, Yamada T (2011) Estimating the benefits of considering travel time variability
in urban distribution. Transp Res Rec 2238:86–96
Toth P, Vigo D (2002) The vehicle routing problem. SIAM, Philadelphia
Toyoglu H, Karasan OE, Kara BY (2012) A new formulation approach for location-routing problems. Netw
Spat Econ 12:635–659
van Lint JWC, van Zuylen HJ, Tu H (2008) Travel time unreliability on freeways: why measures based on
variance tell only half the story. Transp Res A Policy Pract 42:258–277
Wei C, Asakura Y, Iryo T (2012) A probability model and sampling algorithm for the inter-day stochastic
traffic assignment problem. J Adv Transp 46: 222–235
Yu B, Yang ZZ (2011) An ant colony optimization model: the period vehicle routing problem with time
windows. Transp Res E Logist Transp Rev 47:166–181
Yu B, Yang ZZ, Xie JX (2011) A parallel improved ant colony optimization for multi-depot vehicle routing
problem. J Oper Res Soc 62:183–188
Zhang T, Chaovalitwongse WA, Zhang Y (2012) Scatter search for the stochastic travel-time vehicle
routing problem with simultaneous pick-ups and deliveries. Comput Oper Res 39:2277–2290
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articles for individual use.
INDUSTRIAL ENGINEERING
1
Industrial Engineering
Name:
Institution:
INDUSTRIAL ENGINEERING
2
This article discusses the challenges associated with travel time uncertainty and the
algorithmic solution employed to reduce route mechanism. Since on-time deliveries are very
critical for production purposes and responding to logistics are very important for customers it is
very vital to compute and ensure on- time the probability of vehicles in client locations are
known. Vehicles routing problem is an old concept that aims at designing a combination of
minimum- cost routes for logistics vehicles. The chosen routes start at the depot, cover all
customers within the geographical location and route back to the depot. Demand for each route
should coincide with vehicle capacity limit and duration should not exceed expected limits.
Three elements are very important in most VRP studies. These are demand, travel times
and customers. However, these elements can be stochastic due to the complexity of real logistic
application field. But the elements cannot be ignored if a feasible and optimal delivery solution
is to be achieved. After the literature review from past researchers a new proposal is presented.
The new VRP tries to solve both service time and travel time uncertainties associated with
vehicle routing. The new proposal is successful minimizing carrier costs and ensure on- time
arrivals for the different customers in different locations. This was achieved using an iterated
tabu search heuristic algorithm.
According to my opinion. The most intriguing part of this article is the instance when the
proposed model was to be tested on other six different problem instances. A larger increase in
INDUSTRIAL ENGINEERING
vehicle numbers, it looks to improve customer service, reduce time, reduce costs and ensure
location of the vehicles can be ensured at each specific time.
3
INDUSTRIAL ENGINEERING
4
References
Zhang, J., Lam, W., & Chen, B. (2013). A Stochastic Vehicle Routing Problem with Travel Time
Uncertainty: Trade-Off Between Cost and Customer Service. Netw Spat Econ, 13(4),
471-496. doi:10.1007/s11067-013-9190-x
INDUSTRIAL ENGINEERING
1
This article discusses the challenges associated with travel time uncertainty and the
algorithmic solution employed to reduce route mechanism. Since on-time deliveries are very
critical for production purposes and responding to logistics are very important for customers it is
very vital to compute and ensure on- time the probability of vehicles in client locations are
known. Vehicles routing problem is an old concept that aims at designing a combination of
minimum- cost routes for logistics vehicles. The chosen routes start at the depot, cover all
customers within the geographical location and route back to the depot. Demand for each route
should coincide with vehicle capacity limit and duration should not exceed expected limits.
Three elements are very important in most VRP studies. These are demand, travel times
and customers. However, these elements can be stochastic due to the complexity of real logistic
application field. But the elements cannot be ignored if a feasible and optimal delivery solution
is to be achieved. After the literature review from past researchers a new proposal is presented.
The new VRP tries to solve both service time and travel time uncertainties associated with
vehicle routing. The new proposal is successful minimizing carrier costs and ensure on- time
arrivals for the different customers in different locations. This was achieved using an iterated
tabu search heuristic algorithm.
According to my opinion. The most intriguing part of this article is the instance when the
proposed model was to be tested on other six different problem instances. A larger increase in
Alkharsan
vehicle numbers, it looks to improve customer service, reduce time, reduce costs and ensure
location of the vehicles can be ensured at each specific time.
Alkharsan
References
Zhang, J., Lam, W., & Chen, B. (2013). A Stochastic Vehicle Routing Problem with Travel Time
Uncertainty: Trade-Off Between Cost and Customer Service. Netw Spat Econ, 13(4),
471-496. doi:10.1007/s11067-013-9190-x
A STOCHASTIC VEHICLE ROUTING
P R O B L E M W I T H T R AV E L T I M E
U N C E R T A I N T Y: T R A D E - O F F B E T W E E N C O S T
A N D C U S T O M E R S E RV I C E
Faisal Alkharsan
IEGR 204.001
REFERENCES
❖ Zhang, J., Lam, W., & Chen, B. (2013). A Stochastic Vehicle
Routing Problem with Travel Time Uncertainty: Trade-Off Between
Cost and Customer Service. Netw Spat Econ, 13(4), 471-496.
doi:10.1007/s11067-013-9190-x
EQUATIONS
𝐴
𝐴
σ𝐿𝜏=1 ℰ 𝑏𝜏 𝑟𝑗𝑘 − 𝑙𝑟𝑗 , if 𝑙𝑟𝑗 < 𝑏1 𝑟𝑗𝑘
❖ 𝐸 𝑃𝑟𝑗 𝑘
𝐴𝑟𝑗𝑘
𝐴𝑟𝑗𝑘
𝐿
𝐴𝑟 𝑗𝑘
σ
ℰ
𝑏
−
𝑙
,
if
𝑏
𝜏=𝜏
𝑟𝑗
𝜏
𝜏𝑙𝑟𝑗 ≤ 𝑙𝑟𝑗 < b
=
𝑙𝑟𝑗+1
𝜏𝑙𝑟𝑗+1, 1≤ 𝜏 𝑙𝑟𝑗 ≤ L−1
0, if 𝑙𝑟𝑗 ≥ 𝑏𝐴𝑟𝑗𝑘
𝐿
𝑛𝑘 +1
❖ w 𝑠 = σ𝑘∈𝐾 σ𝑗=1
max {𝑣 𝐴 𝑟𝑗𝑘 ሺ ቂ𝛼𝑟𝑗 Τ𝜀 ሿ ∙ 𝜀 ሻ − 𝑙𝑟𝑗, 0}, 𝑟𝑗∈ 𝑅𝑘
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