doing my Powerpoint with eight slide

User Generated

Snvfnynyx

Engineering

Description

IEGR 204- INTRODUCTION TO IE AND.pdf In this file I have all instruction I need someone to follow these steps one by one. Do exactly same the file.

 engineering subject for summary.pdf this is my article. 

 Industrial Engineering summary 1.docx and here my summary. 

Here you have all the instruction you need. Please  do it like that. 


Unformatted Attachment Preview

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 476 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 Copyright of Networks & Spatial Economics is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email 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
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer


Anonymous
Really useful study material!

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags