
Research Article


A Mathematical Model for Vehicle Routing Problem in a Flexible Supply Network


Hamed Fazlollahtabar,
Iraj Mahdavi,
Nezam MahdaviAmiri
and
Amir Mohajeri



ABSTRACT

The development of robust supply chains requires careful attention to both the location of individual supplier facilities and the opportunities for effective transportation among them. Here, we propose a supply chain which considers multiple depots, multiple vehicles, multiple products and multiple customers, with multi time periods. The supplier receives the order and forwards it to depots of multiple products. The depots investigate the capacity level and accept/refuse supplying the order. Considering the location of the customers, the depots decide upon sending the suitable vehicles. Each vehicle has its specific traveling time and cost. We present a mathematical model for the allocation of orders to depots and vehicles minimizing the total cost. We also provide a Lingo encoding and use it to solve an illustrating example to show the effectiveness of the proposed model.





Received:
November 12, 2011; Accepted: February 13, 2012;
Published: February 29, 2012 

INTRODUCTION The rapid industrialization and economic growth of many countries around the world have spurred the development of various supply chains reaching around the world. This has provided opportunities for manufacturers to cut costs and be closer to emerging and highly grown markets but it has also created new risks. As supply chains become increasingly dependent on the efficient movement of materials among geographically dispersed facilities, there is more opportunity for disruption.
Supply chain coordination has gained considerable notice lately from both practitioners
and researchers. In some markets there is only one chain in which there is only
one retailer being called perfect competing market, to maximize the profit we
need to integrate supply chain vertically (Jeuland and Shugan,
1983; Cachon, 2003; Bernstein
and Federgruen, 2005).
A market with two competing supply chains was investigated in the seminal work
of McGuire and Staelin (1983). The researchers considered
two suppliers price competition (i.e., Bertrand) each selling through an independent
retailer. Coughlan (1985) applied this research to the
electronics industry and Moorthy (1988) further explained
why the decentralized chains could lead to higher profits for the manufacturer
and the entire chains. Bonanno and Vickers (1988) considered
a similar methodology to show that in some cases it is optimal to sell the products
via an independent retailer.
In operation management of supply chains, Wu and Chen (2003)
presented a newsvendorbased model for demand competition but they ignored the
pricing decisions, which is a common ignorance amongst all researches in this
field.
A few supply chain coordination mechanisms that induce the chain to act as
if they were vertically integrated (VI) were investigated; e.g., buy back (Pasternack,
1985), quantity flexibility (Tsay, 1999) and revenue
sharing (Cachon and Lariviere, 2005). Cachon
(2003) for a survey of this literature. Two more reviews are found by Kouvelis
et al. (2006) that focuses on supply chain coordination literature
published in Production and Operations Management journals during 1992 2006
and in Tang (2006) which covers much literature on supply
chain coordination. Lin and Kong (2002) consider a duopoly
that has no demand uncertainty and investigate a symmetric Nash Bargaining model.
Similar to McGuire and Staelin (1983) they show that
Nash Bargaining can lead to higher supply chain profits than a vertically integrated
chain.
In a recent study, Baron et al. (2008) investigate
the Nash Equilibrium of an industry with two supply chains by extending the
seminal work of McGuire and Staelin (1983). Baron
et al. (2008) show that both the traditional MS and the VI strategies
are special cases of Nash Bargaining on the wholesale price when the demand
is deterministic. They warn that the supply chain coordination mechanisms that
focus on inducing supply chains to act as if they were vertically integrated,
should be treated with caution.
The Vehicle Routing Problem (VRP) is recently being focused in SCM literature.
The VRP problem considers a set of homogenous vehicles stationed at a depot
to service the demands of customers in geographically scattered locations via
the routes with the least cost. Each vehicle with a certain capacity starting
and ending a tour in the depot, should find a rout which visit each customer
only once. The VRP is considered as NPhard problems which cannot be solved
with analytical computational approaches and tackled by heuristics (Toth
and Vigo, 2002; Laporte et al., 2000; Cordeau
et al., 2002). The advantages of VRP are:
• 
Reducing the length of the delivery routes 
• 
Decreasing the number of vehicles 
• 
Providing better service to the customers 
• 
Operating in a more efficient manner 
• 
Increasing the market share 
The transportation problem we tackle can be described as a multidepot pickup
and delivery problem with time windows and side constraints (Desrosiers
et al., 1995) and is regarded as one of the richest within the class
of time constraint vehicle routing and scheduling problems in terms of scope
and complexity. The earliest pickup time for shipments corresponds to onesided
time window constraints. In addition, operating time restrictions at some locations
impose delivery time windows. The coexistence of consolidation (and of LTL shipments),
relaying and trailer availability requirements in our problem context makes
it a unique and even a more complicated problem than the ones studied before.
Early major work on pickup and delivery problems with time windows has been
reported by Savelsbergh and Sol (1995). Variants of
the basic problem with context specific characteristics have been reported by
Currie and Salhi (2003), Liu et
al. (2003) and Sigurd et al. (2004),
to name a few.
The concept of transportation network equilibrium has a longer history than
supply chain networks and has been studied by Pigou (1920),
with the first rigorous mathematical treatment given by Beckmann
et al. (1956) in their classical book. Some other researches in modeling
of transportation network equilibrium were related to Smith
(1979), Dafermos (1980, 1982)
and Boyce et al. (1983). Transportation network
equilibrium was further studied by Florian and Hearn (1985)
and the books by Patriksson (1994) and Nagurney
(1999, 2000).
In supply chain modeling and analysis (Lee and Billington,
1993; Slats, 1995; Anupindi
and Bassok, 1996), one typically associates the decisionmakers with the
nodes of the multi tiered supply chain network. In transportation networks,
on the other hand, the nodes represent origins and destinations as well as intersections.
Travelers or users of the transportation networks seek, in the case of useroptimization,
to determine their costminimizing routes of travel.
Here, we propose a supply chain which considers multiple depots, multiple vehicles, multiple products, multiple customers with multi time periods. LITERATURE REVIEW
Zhang et al. (2011) proposed the design and
implementation of a dynamic Radio Frequency Identification (RFID) datadriven
supply chain management system with respect to the domains of supply chain management,
simulation, Dynamic DataDriven Application Systems (DDDAS) and Radio Frequency
Identification (RFID) technology. Their model will be able to (1) model supply
chain entities (2) simulate supply chain events and activities (3) use realtime
RFID data to maintain a more accurate picture of the overall supply chain and
(4) use the simulation results to control experiments.
Since, implementation of supply chain management requires integration of processes
between supply chain members in all functional areas, including sourcing, manufacturing
and distribution and the need for the successful implementation of information
sharing being critical to effective innovation and development of supply chain
management at an industry and enterprise level, Khurana
et al. (2011) aimed to identify and measure the perceived importance
of information sharing barriers in supply chain management. The barriers have
been categorized into the six main different levels namely managerial, organizational,
technological, individual, financial, social and cultural. Using a questionnaire
and interview based research approach, they adopted to identify perceptions
of the most significant barriers to information sharing. It was found that the
data collected through questionnaires were sometimes very much ambiguous or
vague and insufficient to interpret the significant results. Therefore, a fuzzy
Analytic Hierarchy Process (AHP) approach has been used to overcome this kind
of deficiency for modeling the rankings of the barriers of information sharing
in supply chain management is used. The findings of their research can be used
for developing an evidence based ranking of barriers of information sharing
in supply chain.
Shafia et al. (2009) studied the causes and
effects of the factors that determine the trend of employing Common Platforms
(CP) in Supply Chain Management (SCM) of automotive industries. They also proposed
a framework for analyzing Supply Chain Based on Common Platforms (SCBCP) in
industries. The research methodology of their study was based on fact finding
approach. Therefore, presenting the definitions and concepts of pertinent subjects,
a conceptual model was developed for determining various aspects and finding
facts regarding SCBCP in automotive industry. Critical factors and important
facts in SCBCP have been identified by developing and analyzing the conceptual
model. In addition, a triple performance criterion for the evaluation of SCBCP
was developed.
Chong and Ooi (2008) studied the adoption status of
Collaborative Commerce (CCommerce) in the Malaysian Electrical and Electronic
(E and E) organizations. Original research performed using a selfadministered
questionnaire that was distributed to 400 Malaysian E and E organizations. Data
were analyzed employing descriptive statistics. In general, the adoption level
of CCommerce tools in the Malaysian E and E industry was still considered low
with an average mean of 3.011. Based on the tools adopted, most organizations
were utilizing CCommerce for their supply chain execution. Among, tools with
lower adoption, they were mainly supply chain planning tools such as capacity
planning tool and business strategy tool.
Ghoseiri and Ghannadpour (2009) aimed to solve Vehicle
Routing Problem with Time Windows (VRPTW), which has received considerable attention
in recent years, using hybrid genetic algorithm. Vehicle Routing Problem with
Time Windows was an extension of the wellknown Vehicle Routing Problem (VRP)
and involved a fleet of vehicles setoff from a depot to serve a number of customers
at different geographic locations with various demands within specific time
windows before returning to the depot eventually. To solve this problem, they
suggested a hybrid genetic algorithm combined with Push Forward Insertion Heuristic
(PFIH) to make an initial solution and λinterchange mechanism to neighborhood
search and improving method. The proposed genetic algorithm uses an integer
representation in which a string of customer identifiers represents the sequence
of deliveries covered by each of the vehicles. Part of initial population was
initialized using Push Forward Insertion Heuristic (PFIH) and part was initialized
randomly.
Ismail and Irhamah (2008) primarily studied to solve
the Vehicle Routing Problem with Stochastic Demands (VRPSD) under restocking
policy by using adaptive Genetic Algorithm (GA). The problem of VRPSD was one
of the most important and studied combinatorial optimization problems, which
finds its application on wide ranges of logistics and transportation area. It
was a variant of a Vehicle Routing Problem (VRP). The algorithms for stochastic
VRP were considerably more intricate than deterministic VRP and very time consuming.
This has led the authors to explore the use of metaheuristics focusing on the
permutationbased GA. The GA was enhanced by automatically adapting the mutation
probability to capture dynamic changing in population. The GA became a more
effective optimizer where the adaptive schemes were depend on population diversity
measure. The proposed algorithm was compared with standard GA on a set of randomly
generated problems following some discrete probability distributions inspired
by real case of VRPSD in solid waste collection in Malaysia. The performances
of several types of adaptive mutation probability were also investigated.
Shahrabi et al. (2009) compared several time
series methods to forecast supply chain demand. In this research, traditional
time series forecasting methods including moving average, exponential smoothing,
exponential smoothing with trend at the first stage and finally two machine
learning techniques including Artificial Neural Networks (ANNs) and Support
Vector Machines (SVMs), were used to forecast the longterm demand of supply
chain. By using the data set of the component supplier of the biggest Iranian’s
car company this research was then implemented. The comparison reveals that
the results producing by machine learning techniques were more accurate and
much closer to the actual data in contrast with traditional forecasting methods.
Neghab and Haji (2008) considered a twolevel supply
chain system consisting of one warehouse and a number of identical retailers.
In this system, they incorporated transportation costs into inventory replenishment
decisions. The transportation cost contained a fixed cost and a variable cost.
The authors assumed that the demand rate at each retailer was known and the
demand was confined to a single item. First, they derived the total cost which
was the sum of the holding and ordering cost at the warehouse and retailers
as well as the transportation cost from the warehouse to retailers. Then, they
proposed a search algorithm to find the economic order quantities for the warehouse
and retailers which minimize the total cost.
Ahmadi and Teimouri (2008) proposed a dynamic programming
model which determines order penetration point in auto export supply chain.
They also studied the characteristics and concepts relating to the Order Penetration
Point (OPP). One of the most important characteristics of this supply chain
was that, the product was packaged in different modules and after various stockings
and passing long routs, was assembled in the target country. This modularized
characteristic of the product was encouraging to explore the OPP of the chain
from one point to several points in which the OPP of each module was located.
Their proposed model tried to put the OPP of expensive modules (that have higher
inventory holding cost) in the upstream section of the chain and puts the OPP
of cheaper ones which created delay, in the downstream section of the chain.
THE PROPOSED MODEL We consider different customers being serviced with one supplier. The supplier provides various products and keeps them in different depots. Each depot uses different types of vehicles to carry out the orders. All depots are already stationed at the related locations. Here, we consider a multi echelon supply chain network (one supplier, multi depots and customers, multi commodity with deterministic demands). A set of vehicles exist at each depot. Each depot can store a set of products. The received order list from a customer can be handled by one or several depots at each time. Each selected vehicle for delivery can transfer only one product and after delivering the product, the vehicle returns to its corresponding depot. A penalty is assigned when a delivery time exceeds the predetermined time for transferring the products from depots to customers. A configuration of the proposed model is shown in Fig. 1. MATHEMATICAL MODEL The mathematical model for this problem is as follows:
Notations:
P 
= 
Set of products 
I 
= 
Set of depots stationed 
J 
= 
Set of customers 
T 
= 
Set of time periods 
V 
= 
Set of vehicles 
Parameters:
D_{jpt } 
= 
Demand of customer j for product p at time t 
TH_{ipt } 
= 
Maximum throughput of depot i for product p at time t 
CA_{i } 
= 
Total capacity of depot i 
N_{ivt } 
= 
The number of existing vehicles v in depot i at time t 
VL_{vp } 
= 
Capacity of vehicle v for product p 
d_{ij } 
= 
Distance between depot i and customer j 
r_{ijpvt } 
= 
Number of return vehicles of type v from customer j to depot i at time
t that have already received product p 
M 
= 
A large number 
Ct_{ijv } 
= 
Traveling fixed cost per mile from depot i to customer j using vehicle
v 
TRT_{ijv } 
= 
Traveling time from depot i to customer j using vehicle v 
C 
= 
The fixed cost for the whole planning horizon 
Pen 
= 
The fixed cost as a penalty 
t 
= 
The size of period t 
α_{ijv } 
= 
1, if the time of delivery from depot i to customer j using vehicle v
exceeds a prespecified limit; 0, otherwise 

Fig. 1: 
A configuration of the proposed model 
Decision variables:
f_{ijpvt } 
: 
No. of transferred vehicle type v from depot i to customer
j at time t 
QP_{ijpt } 
: 
Quantity of product p that can be satisfied by depot i to customer j at
time t 
Objective function:
Minimize F = Min. (f_{1}+f_{2}+f_{3}) 
Constraints: Formulas (1) and (2) are the objective functions which minimize the total cost and time, respectively. Formula (3) considers penalty for delivery times exceeding a prespecified time limit. The constraints (4) guarantee that all customer demands are met for all products required at each period. The constraints (5) and (6) ensure that delivery is accomplished by only one vehicle. The number of each traveling vehicle between the depots and customers is shown by constraints (7). The constraints (8) and (9) identify return times of only the remaining vehicles. The constraints (10) represent the numbers of remaining vehicles at the end of the period. The constraints (11) ensure that the number of traveled vehicles from depot would not exceed the existing vehicles. The amounts of remaining product in depots at the end of the period are shown by constraints (12). The constraints (13) represent the capacity constraint of each depot for each product at the corresponding time. They must receive enough products from supplier in order to meet all the demands. The constraints (14) impose that the variables be binary. The last constraints (15) and (16) show the nonnegativity requirements for all the other variables. NUMERICAL ILLUSTRATIONS We present a numerical example to show the effectiveness of the proposed mathematical model. The number of customers is three, number of products is three, number of depots is two and number of vehicles is two. We consider a six period supply chain which receives order list in periods one, two and three with the size of time period t = 10. The orders for products in different periods are given in Table 1.
The distance from depots to customers, the capacity of vehicles for different
products and the capacity of depots for different products are given in Table
2, 3 and 4, respectively.
The maximum capacity of both depots 1 and 2 are equal to 600. The transferring cost per unit of distance for vehicles 1 and 2 are 50 and 30, respectively. The transferring times for vehicles from depots to customers are given in Table 5. The number of vehicle 1 in both depots 1 and 2 are 14 and the number of vehicle 2 in both depots 1 and 2 are 12.
Table 1: 
The orders for products in different periods 

To facilitate the computation, LINGO 8 package is applied. The output for the decision variables are summarized in Table 6 and 7. The quantity of products (Qp) that can be satisfied by depots to customers at different time periods, selected route (X), type of vehicle and number of transferred vehicle (F) are presented in Table 6. The number of return vehicles from customers to depots at different time periods (r) is shown in Table 7. The number of remaining vehicles and capacity at the end of each period is given in Table 8 and 9, respectively.
Table 2: 
The distance from depots to customers 

Table 3: 
The capacity of vehicles for different products 

Table 4: 
The capacity of depots for different products 

Table 5: 
The transferring time for vehicles from depots to customers 

Table 6: 
The quantity of products that can be satisfied by depots
to customers at different time periods 

Table 7: 
The number of return vehicles from customers to depots at
different time periods 

Table 8: 
The number of remaining vehicles at the end of the periods 

Table 9: 
The amount of remaining capacity at the end of the periods 

The best objective value for the problem is 34350 
CONCLUSIONS
We proposed a supply network model in which one supplier would provide various
products for customers in different time periods. The contribution of the proposed
model is in its flexibility with respect to vehicles and depots. The aim was
to minimize the total cost and time of the orders’ delivery process. Furthermore,
deliveries needing times longer than the prespecified limits are penalized.
The effectiveness and validity of the proposed mathematical model are illustrated
by working out a numerical example using our presented Lingo encoding for the
solution of the model.

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