Part I Introduction
8.5 Case Studies
SVRPs differ from their deterministic ones in several aspects. Solution method- ologies are more intricate and combine the characteristics of stochastic and integer programs. SVRPs are often computationally intractable; therefore, only relatively small instances can be solved to optimality, and good heuristics are hard to design and assess [79]. SVRPs can be cast within the framework of stochastic programming.
SVRPs are usually modeled using mixed- or pure-integer stochastic programs or as Markov decision processes. All known exact algorithms belong to the first category.
Tillman in 1969 [80] addressed the CVRPSD for the first time. He considered a multidepot variant of the CVRP with Poisson-distributed demands. The model con- sidered a cost trade-off between exceeding the vehicle capacity and finishing the route with excess capacity [81].
The published papers covering all kinds of SVRP are categorized in Table 8.6 by specifying the parameters that have been assumed stochastic.
8.4.7 Fuzzy VRP
There is widespread evidence that the exact values of the mean demands, travel times, numbers and locations of customers, and so on that follow probability distri- butions are very difficult to obtain. In some new systems, it is also hard to describe the parameters of the problem as random variables because of insufficient data to analyze the distribution. Using methods from fuzzy sets theory makes it possible to successfully model problems that contain an element of uncertainty, subjectivity, ambiguity, and vagueness.
Fuzzy logic was used by Teodorovic and Pavkovic [130] in VRP when the demands were uncertain. The model was based on the heuristic sweeping algo- rithm, rules of fuzzy arithmetic, and fuzzy logic.
Cheng and Gen [131] introduced the concept of fuzzy die-time in the vehicle- routing and scheduling context. They represented the fuzzy time window in two types: the tolerable interval of service time and the desirable time for service.
Usual approaches consider the tolerable interval of service time without minding customers’ desired time. Their fuzzy approach can handle both kinds of customers’
preferences simultaneously. Table 8.7 categorizes the papers on VRPF.
Table 8.6 VRPSD Literature
Author(s) Year Type Author(s) Year Type
Tillman [80] 1969 MDVRPSD Golden and Stewart
[82]
1978 VRPSD Golden and Yee [83] 1979 VRPSD Yee and Golden
[84]
1980 VRPSD Stewart and Golden
[85]
1980 VRPSD Stewart [86] 1981 VRPSD
Bodin et al. [87] 1983 VRPSD Stewart and Golden [88]
1983 VRPSD Jezequel [89] 1985 VRPSD, VRPSC,
VRPSDC
Dror and Trudeau [90]
1986 VRPSD Jaillet [91] 1987 VRPSD, VRPSC,
VRPSDC
Bertsimas [92] 1988 VRPSD, VRPSC Jaillet and Odoni [93] 1988 VRPSD, VRPSC,
VRPSDC
Laporte et al. [94] 1989 VRPSD Waters [95] 1989 VRPSD VRPSC Dror et al. [96] 1989 VRPSD Laporte and
Louveaux [97]
1990 VRPSD Bertsimas et al.
[98]
1990 VRPSD Bastian and Rinnooy
Kan [99]
1992 VRPSD Benton and Rosset
[100]
1992 VRPSCD Dror [101] 1992 VRPSD, VRPSC Laporte et al. [102] 1992 VRPSD Trudeau and Dror
[103]
1992 VRPSCD Bertsimas [104] 1992 VRPSD,
VRPSCD Dror et al. [105] 1993 VRPSD Bouzdiene-Ayari
et al. [106]
1993 SDVRPSD Laporte and
Louveaux [107]
1993 VRPSD Gendreau et al. [79] 1995 VRPSD, VRPSCD Gendreau et al. [58] 1996 VRPSD, VRPSCD Secomandi [108] 1998 VRPSD Hjorring and Holt
[109]
1999 VRPSD Yang et al. [110] 2000 SVRP
Secomandi [111] 2001 VRPSD Laporte et al. [112] 2002 CVRPSD Markovic et al. [113] 2005 VRPSD Chepuri et al. [114] 2005 VRPSD Dessouky et al. [115] 2006 VRPSD Novoa et al. [116] 2006 VRPSD Ak and Erera [117] 2007 VRPSD Tan et al. [118] 2007 VRPSD Haugland et al. [119] 2007 VRPSD Sungur et al. [120] 2008 CVRPSD Jula et al. [121] 2008 SVRPT Shen et al. [122] 2009 VRPSDT Christiansen and
Lysgaard [123]
2009 VRPSD Secomandi and
Margot [124]
2009 VRPSD Novoa and Storer
[125]
2009 VRPSD Li et al. [126] 2010 VRPSTST
Smith et al. [127] 2010 VRPSD Rei et al. [128] 2010 VRPSD Mendoza et al. [129] 2010 VRPSD
MDVRPSD, multidepot VRPSD; VRPSC, VRP with stochastic customers; VRPSDC, VRP with stochastic demand and customer; VRPSTST, VRP stochastic travel time and service time.
The first case study considers a central warehouse of a dairy company that hosts a heterogeneous fleet of vehicles and stores perishable foods. The foods have to be delivered to a set of customers through daily deliveries. The distance between each pair of customers and also the central warehouse and each customer’s location is known. After deliveries, each vehicle route ends at the central warehouse.
The second actual case study considers a distribution center of a concrete com- pany in which a heterogeneous fleet of concrete-mixer trucks load ready-to-pour concrete for delivery to a set of construction sites. Every construction site requires a specific type of concrete-mixer truck of different capacity that can carry different blends of concrete. Concrete-mixer trucks return to the distribution center after unloading the demands.
Tarantilis and Kiranoudis [140] formulated these problems as HVRP with differ- ent fixed and variable cost for each vehicle and solved them by a heuristic based on adaptive memory. Computational outcome on the first case results in substan- tially improving on the current practice of the company by using 24 vehicles instead of 27 and saving at least 28.23% of the total duration of the trips. The same results are attained for the next case: the number of concrete-mixer trucks reduces from 13 to 10 and a 49.66% saving in total duration of trips.
8.5.2 The Collection of Urban Recyclable Waste
The next case is about the collection of recyclable waste in Portugal’s central coastal region. There are two central depots in which the waste of 1642 distinct
Table 8.7 FVRP Literature
Author(s) Year Type Algorithm
Cheng and Gen [131] 1995 VRPFTW Genetic algorithm Teodorovic and
Pavkovic [130]
1996 VRPFD Heuristic sweeping algorithm, rules of fuzzy arithmetic and fuzzy logic
Werners and Drawe [132]
2003 VRPF Fuzzy modeling based on mixed-integer linear programming
Kuo et al. [133] 2004 VRPFT Ant colony optimization
Sheng et al. [134] 2005 FVRP Compares fuzzy measure method with other programming methods
He and Xu [135] 2005 VRPFD Genetic algorithm
Zheng and Liu [136] 2006 VRPFT Fuzzy simulation, genetic algorithm Lin [137] 2008 VRPFTW Genetic algorithm (multiobjective) Erbao and Mingyong
[138]
2009 VRPFD Stochastic simulation Erbao and Mingyong
[139]
2010 OVRPFD Stochastic simulation
VRPFTW, VRPF with fuzzy time window; VRPFD, VRP with fuzzy demand; VRPFT, VRP with fuzzy travel time;
OVRPFD, OVRP with fuzzy demand.
collection sites is unloaded by five vehicles. Three types of waste must be carried separately. Because 70% of the operational cost is dedicated to the transportation, creating the best collection routes minimizes the total distance of vehicles with the restrictions in the vehicle’s capacity and route duration that must be managed in one work shift.
The problem is modeled as a PVRP and develops routes for every day of each month. This model is repeated in each month with 20 workdays and two work shifts in each day. It is noted that because the vehicles are busy about half the time with the exit and return trips to depots, a single route is created in one shift.
The problem is solved in three phases using heuristic algorithms. For each zone and for each work shift of each day, the decision variables are the type of waste, the sites, and the routes in which the waste must be collected [141].
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