A genetic algorithm for vendor managed i

A Genetic Algorithm for Vendor Managed Inventory Control System of MultiProduct Multi-Constraint Economic Order Quantity Model
Seyed Hamid Reza Pasandideh, Ph.D., Assistant Professor
Department of Railway Engineering, Iran University of Science and Technology, Tehran, Iran
Phone: +98 (21) 77491029, Fax: +98 (21) 77451568, e-mail: pasandid@yahoo.com

Seyed Taghi Akhavan Niaki*, Ph.D., Professor
Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran
Phone: +98 21 66165740, Fax: +98 21 66022702, e-mail: niaki@sharif.edu

Ali Roozbeh Nia, M.Sc.
Department of Industrial Engineering, Islamic Azad University, Qazvin, Iran
Phone: +98 (281) 3665275, Fax: +98 (281) 3665277, e-mail: ali_roozbeh@yahoo.com

Abstract
In this research, an economic order quantity (EOQ) model is first
developed for a two-level supply chain system consisting of
several products, one supplier and one retailer, in which shortages
are backordered, the supplier's warehouse has limited capacity and
there is an upper bound on the number of orders. In this system,
the supplier utilizes the retailer's information in decision making
on the replenishments and supplies orders to the retailer according

to the well known (R,Q) policy. Since the model of the problem is
of a nonlinear integer-programming type, a genetic algorithm is
then proposed to find the order quantities and the maximum
backorder levels such that the total inventory cost of the supply
chain is minimized. At the end, a numerical example is given to
demonstrate the applicability of the proposed methodology and to
evaluate and compare its performances to the ones of a penalty
policy approach that is taken to evaluate the fitness function of the
genetic algorithm.

Keywords:

*

Vendor managed inventory; genetic algorithm; economic order
quantity; limited storage; multi-product

Corresponding Author

1

1. Introduction
The globalization of economy and liberalization of marketplace at an increasingly rapid pace
has intensified the need for incorporating the resulting operational uncertainties and financial risks
into the firms’ production and inventory control decisions (Mohebbi, 2008).
Inventory control has been studied for several decades for cost savings of enterprises who
have tried to maintain appropriate inventory levels to cope with stochastic customer demands and to
boost their image through customer satisfaction (Axsäter, 2000, 2001; Moinzadeh, 2002; Zipkin,
2000). One of the key factors to improve service levels of the enterprises is to efficiently manage
the inventory level of each participant within supply chains (Kwak et al., 2009).
A supply chain (SC) is a network of firms that produce, sell and deliver a product or service
to a predetermined market segment (Chopra and Meindl, 2001). It not only includes the
manufacturers and suppliers, but also transporters, warehouses, retailers and customers themselves.
The term supply chain conjures up images of a product or a supply moving from suppliers to
manufacturers then distributors to retailers and then customers along a chain (Chopra, 2003).
Customers and their needs are the origin of the SC. The main objective of the supply chain
management is to minimize system-wide costs while satisfying service level requirements (Tyana
and Wee 2003).
One of the well-known concepts utilized in supply chains is the vendor-managed inventory
(VMI) models (see for example Cheung and Lee, 2002; Disney and Towill 2003) and many

successful businesses have demonstrated the benefits of VMI, e.g., Wal-Mart and JC Penney
(Cetinkaya and Lee, 2000; Dong and Xu, 2002). In these models, the retailer provides the supplier
with information on its sales and inventory level and the supplier determine the replenishment
quantity at each period based on this information. In other words, the supplier with regard to his
own inventory cost that equals to the total inventory cost of the supply chain determines the timing

2

and the quantity of replenishment in every cycle (Lee et al. 2000; Kaipia et al. 2002; Dong & Xu
2002). Not only VMI has some advantages for both the retailer and the supplier, but also the
customer service levels may increase in terms of the reliability of product availability. Since the
supplier can use the information collected on the inventory levels at the retailers, future demands
are better anticipated and the deliveries are better coordinated (for example, by delaying and
advancing deliveries according to the inventory situations at the retailers and the transportation
considerations) (Kleywegt et al. 2004; Waller et al. 1999).
One of the most important problems in companies that utilize suppliers to provide raw
materials, components, and finished products is to determine the order quantity and the points to
place orders. Various models in production and inventory control field have been proposed and
devoted to solve this problem in different scenarios. Two of the models that have been extensively
employed are the economic order quantity (EOQ) and economic production quantity (EPQ) models

(Silver et al., 1998; Tersine, 1993). The economic order quantity (EOQ) is one of the most popular
and successful optimization models in supply chain management, due to its simplicity of use,
simplicity of concept, and robustness (Teng, 2008). However, these models are constructed based
on some assumptions and conditions that bound their applicability in real-world situations.
In this research, a multi-product EOQ model is proposed in which not only the storage
capacity, but also the number of orders is limited. Furthermore, in order to broaden the applicability
of the proposed model in real-world inventory control systems, we consider shortage to be
backordered and let the model act in a supply chain environment under the VMI condition. The
objective is to find the order quantities and the maximum backorder level of the products in a cycle
such that the total inventory cost of the supply chain is minimized. Under these conditions, the
problem is first formulated as a non-linear integer-programming (NIP) model and then a genetic
algorithm (GA) is proposed to solve it. At the end, a numerical example is presented to demonstrate

3

the application of the proposed methodology and to compare its performance to the one of the
penalty policy approach that is employed as another way to evaluate the fitness function the GA
chromosome.
The remainder of the paper is organized as follows: a review of the literature is presented in
section 2. We define and model the problem in Sections 3 and 4, respectively. In section 5, a genetic

algorithm is developed to solve the problem. In order to demonstrate the application of the proposed
methodology, we provide a numerical example in Section 6. Finally, conclusions and
recommendations for future research are provided in section 7.

2. Literature review
The two basic questions any inventory control system must answer are when and how much
to order. Over the years, hundreds of papers and books have been published presenting models for
doing this under various conditions and assumptions (Pentico, et al. 2009). Economic lot size
models have been studied extensively since Harris (1913) first presented the famous economic
order quantity (EOQ) formula. Then, a variation of this formula, namely the economic production
quantity (EPQ), was developed for manufacturing environments. Much of the literature on
inventory theory contains the basic models of EOQ and EPQ with/without shortages (CardenasBarron, 2001).
Since the EOQ and EPQ are constructed based on some assumptions and conditions that
bound their applicability in real-world situations, many scholars have strived to develop a
formulated inventory model in a more realistic fashion. As some examples, Papachristos and Skouri
(2003) generalized the work of Wee (1999) in which the demand rate is a convex decreasing
function of the selling price and the backlogging rate is a time-dependent function. Chung and

4

Huang (2003), Biskup et al. (2003) and Liao (2007) extended the Goyal (1985) EOQ model to the
EPQ model under conditions of permissible delay in payments.
An early conceptual framework of VMI was described by Magee (1958) when discussing
who should have authority over the control of inventories. However, interest in the concept has only
really developed during the 1990s. Companies have looked to improve their supply chains as a way
of generating a competitive advantage, with VMI often advocated. This strategy has been
particularly popular in the grocery sector but has also been implemented in sectors as diverse as
steel, books and petrochemicals (Disney and Towill 2003). For example, the benefits of VMI
implications have been realized by successful retailers and suppliers, most notably Wal-Mart and
key suppliers like Proctor and Gamble (Cetinkaya and Lee 2000).
Waller et al. (1999) indicated that the VMI method can improve inventory turnover and
customer service levels at every stage of a supply chain. In a more in-depth analysis, Disney and
Towill (2002) showed that the kernel goal of VMI chains, which is minimizing the channel cost
while simultaneously satisfying some degree of customer service levels, is achieved primarily by
sharing demand and inventory information. Furthermore, the studies by Vergin and Barr (1999) and
Lee et al. (2005) conclude that VMI is becoming an effective approach for implementing the
channel coordination initiative, which is critical and imperative to improve the entire chain’s
financial performance.
Xu et al. (2001) presented a study, in which they examined the impacts of electronic data
interchange (EDI) and internet-based technologies on the practice of VMI. Moreover, a survey

conducted by Tyana and Wee (2003) points out that aside from the computer technologies, the key
of implementing VMI lies in the abilities of the related chain members to cooperate and to
understand the flows and processes concerning their products or services delivery.

5

Jasemi (2006) developed a supply chain model with a single supplier and n buyers and
compared the performances of a VMI system with the ones of the traditional types. He also made a
pricing system for profit sharing between parties. Furthermore, Sofifard et al. (2007) presented an
analytical model for a single-buyer single-supplier model to explore the effects of collaborative
supply-chain initiatives such as vendor managed inventory (VMI) with the EPQ model.
Cetinkaya and Lee (2000) presented an analytical model for coordinating inventory and
transportation decisions in VMI systems. Woo et al. (2001) and Yu & Liang (2004) extended their
two-echelon inventory supply chains to three echelon ones where the supplier was a manufacturer
and his raw materials’ inventory was involved. Angulo et al. (2004) evaluated the effects of
information sharing on a VMI partnership. Incorporating the dynamic dimension, Jaruphongsa et al.
(2004) provided a polynomial time algorithm to compute the optimal solutions for the
replenishment the dispatching plans. Bertazzi et al. (2005) compared the order-up-to level policy
and the fill-fill-dump policy of VMI. They showed that the fill-fill-dump policy leads to a lower
average cost than the order-up-to level policy. Vigtil (2007) described a set of five case studies and

showed that sales forecasts and inventory positions were the most valuable information provided to
supplier by the retailers in a VMI relationship.
Dong and Xu (2002) presented an analytical model to evaluate the short-term and long-term
impacts of VMI on supply chain profitability by analyzing the inventory systems of the parties
involved. Several common assumptions that also were used in their inventory-channel coordination
research were: the inventory system of the buyer can be described by an EOQ policy, the demands
are deterministic, there are no stock-outs, and the lead-times are also deterministic. Yao et al.
(2007) using the same assumptions as Dong and Xu’s (2002) along with an additional assumption
(the order quantity for the supplier is likely to be an integer multiple of the buyer’s replenishment
quantity) presented an analytical model to determine how key logistics parameters, most notably

6

ordering costs and inventory carrying charges, can affect the benefits to be derived from VMI. Van
Der Vlist et al. (2007) extended the Yao et al. (2007) model along with the costs of shipments from
the supplier to the buyer.
Within the researches that employed genetic algorithms (GA) in supply chain environments,
Moon et al. (2002) proposed a GA to determine the optimal schedule of machine assignments and
operations sequences in a make to stock supply chain, so that the total tardiness will be minimized.
In a two-echelon single-vendor multiple-buyers supply chain model under vendor managed

inventory (VMI) mode of operation, Nachiappan and Jawahar (2007) formulated a nonlinear integer
programming problem (NIP) and proposed a GA based heuristic to find the optimal sales quantity
for each buyer. Ko and Evans (2007) proposed a GA solution procedure for a mixed integer
nonlinear programming model of a dynamic integrated distribution network of third party logistic
providers (3PLs). Pasandideh and Niaki (2008) developed a multi-product EOQ model, in which
the warehouse space is limited and the orders are delivered discretely in the form of multiple
pallets. Under these conditions, they formulated the problem as a non-linear integer-programming
(NIP) model and proposed a genetic algorithm to solve it. Moreover, Farahani and Elahipanah
(2008) developed a new model for a distribution network in a three-echelon supply chain, which not
only minimizes the total costs but also follows the just-in-time (JIT) distribution purposes in order
to better represent the real-world situations. In their research, a GA was designed to find the Pareto
fronts for the large-size problem instances of this multi-objective mixed-integer linear programming
problem.

3. Defining the Problem
Consider an inventory/distribution system consisting of n products, a supplier and a retailer
that operates under vendor management inventory (VMI) system, in which not only the supplier has

7

a limited warehouse-capacity of F for all items, but also the total number of orders is less than or
equal to M . At the retailer level, when the inventory position declines to R , a batch of size Q is
ordered, excess demand is backordered and no partial shipment is allowed. The objective is to
determine the order quantities and the maximum backorder level of the products in a cycle such that
the total inventory cost of the supply chain is minimized.
In short, the specifications of the supply chain in which the supplier and the retailer interact
with each other are defined as follows:
a) There is a single supplier and a single retailer
b) There are n products
c) The supplier with regard to his own inventory cost that equals to the total inventory cost of
the supply chain determines the timings and the quantities of production in every cycle
d) Shortages are allowed and backordered
e) The order deliveries are assumed instantaneous, that is, the lead time is zero
f) There is no quantity discount
g) The prices for all products are fixed in the planning period
h) The rate of production for all products is infinite (EOQ model)
i) The costumers’ demands for all products are deterministic
j) The supplier storage capacity is limited
k) The total number of orders for all products is limited

4. Mathematical model of the problem
In order to develop the mathematical model of the problem, let us first introduce the
notations.

8

4.1 Notations
For products j  1, 2,..., n , the following notations will be used to develop the proposed model:

TCV MI : Total costs of the VMI supply chain
AS : Supplier’s fixed ordering cost per unit of all products
A R : Retailer's fixed ordering cost per unit of all products
Q j : Order quantity of the jth product

D j : Retailer's demand rate of the jth product
b j : Maximum backorder level of the jth product in a cycle of the VMI chain

 : Fixed backorder cost per unit

 : Fixed backorder cost per unit per time unit
h jR : Carrying cost of the jth product per unit held in the retailer's store in a period
KRV MI : Retailer's inventory cost of the VMI chain
KSVMI : Supplier's inventory cost of the VMI chain
f j : Space occupied per unit of the jth product
F : Total available storage space for all items

M : Total Number of orders for all items
n : Number of items

4.2 The Costs
After the VMI-implementation, the inventory costs of both the retailer and the supplier and
hence the total inventory costs of the integrated supply chain are calculated as follows:
KRVMI  0

(1)

9

 A jS D j A jR D j h jR
b j2  b j D j
2
 



(Q j  b j ) 
2Q j
2Q j
Qj
Qj
j 1  Q j

n

KSV MI






(2)

n D
b j2  b j D j
h jR
j
2
(
)
TCV MI  KRV MI  KSV MI   
A
A
Q
b




 jS jR  2Q j j 2Q  Q
j 1  Q j
j
j
j







(3)

As mentioned earlier, the goal is to determine integer values of the products’ order
quantities and maximum backorder levels in a cycle such that the total cost of the supply chain
under VMI operation is minimized and the constraints are satisfied. The constraints are:
(1) The capacity of the storage to store all items is limited, and
(2) The total number of order for all items is limited (not more than M orders can be placed).
Hence, the mathematical formulation of the problem becomes
 Dj
h jR
b j2  b j D j
2
 
 A jS  A jR   2Q (Q j  b j )  2Q  Q
j 1  Q j
j
j
j

n

Min TCV MI

f
n

s .t .

j 1

Q

j

Qj  F

Dj

n

j 1

M

Q j , b j  0 : Integer ,
j






(4)

j  1,..., n

In the next section, we will present an algorithm to efficiently solve the problem given in (4).

5. A solution algorithm

The formulation given in Equation (4) is a nonlinear integer-programming (NIP) model.
This characteristic causes the model to be hard enough to be solved by an exact method (Gen,
1997). Accordingly, a heuristic search algorithm is required to solve the model. Historically, among
the search algorithms, the genetic algorithm (GA) has been successful in solving models similar to
the model given in Equation (4) (Gen, 1997).

10

Genetic algorithms are stochastic search techniques based on the mechanism of natural
selection and natural genetics. GA is differing from conventional search techniques in a sense that it
starts with an initial set of random solutions called population. Each individual in the population is
called a chromosome, representing a solution to the problem at hand. The chromosomes evolve
through successive iterations, called generations. During each generation, the chromosomes are
evaluated, using some measures of fitness. To create the next generation, new chromosomes, called
offspring, are formed by either crossover operator or mutation operator. A new generation is formed
according to the fitness values of the chromosomes. After several generations, the algorithm
converges to the best chromosome (Pasandideh and Niaki, 2008).
The usual form of GA was described by Goldberg (1989). GA is a stochastic search
technique whose solution process mimics natural evolutionary phenomena: genetic inheritance and
Darwinian strife for survival (Gen & Cheng, 2000). Recently, the GA has been receiving great
attention and it has successfully been applied to other problems in the supply chain environment
(Chen et al. 1998; Park, 2001). GA is known as a problem-independent approach; however, the
chromosome representation is one of the critical issues when applying it to optimization problems.

5.1 GA algorithm in initial and general conditions
The required initial information to start a GA is:
i.

Population size (N ) : It is the number of the chromosomes or scenarios that are kept in each
generation.

ii.

Crossover rate (Pc ) : This is the probability of performing a crossover in the GA method.

iii.

Mutation rate (Pm ) : This is the probability of performing mutation in the GA method.
The general steps involved in a GA algorithm are as follow:

1. Initialization

11

1.1 Set the parameters ( N , PC , Pm , stopping criteria, selection strategy, crossover operation,
mutation operation, and number of generation)
1.2 Initialize the population (randomly)
2. Evaluate the fitness
Repeat
3. Select individuals for the mating pool (size of mating pool= N )
4. For each consecutive pair apply crossover with probability Pc
5. For each new generation apply mutation with probability Pm
6. Replace the current population by the resulting mating pool
7. Evaluate the fitness
Until stopping criteria is met.

In what follows, the proposed GA is described in details.

5.2 The chromosomes
The chromosome representing the order quantities and the maximum backorder levels of the
products is modeled by a 2  n matrix. The jth element in the first and in the second row of the
matrix shows the order quantity and the maximum backordered quantity of the jth product,
respectively. Figure (1) presents the general form of a chromosome.

Q1 Q1 ... Q n 
b

 1 b1 ... b n 

Figure (1): The chromosome presentation

12

5.3 Evaluation
When GA is employed for an optimization problem, a fitness value, which is the value of
the objective function, is needed to be assigned for a chromosome, as soon as it is generated.
However, since there are two constraints of the storage capacity and the total number of orders in
the model given in Equation (4), some generated chromosomes may not be feasible. While there are
some available methods in the literature (such as the penalty policy) to control infeasible solutions
(Gen, 1997), because of the large size of the model in Equation (4) generating only feasible
solutions is taken in this research. In other words, once an infeasible chromosome is generated, it
will be removed from the population. Nevertheless, the performances of both removing the
infeasible chromosome and taking the penalty policy are compared through a numerical example in
subsection 6.2.

5.4 Initial population
In this step, a collection of chromosomes is randomly generated.

5.5 Crossover
In a crossover process, it is necessary to mate pairs of chromosomes to create offspring. We
perform this by selecting a pair of chromosomes from the generation randomly with probability Pc .
There are many different types of crossover operators such as one-point, two-points, multiple-points
and uniform. In this research, a one-point crossover operator is selected that works as follows: (i)
Choose a random crossover point, (ii) Split parents at this point, (iii) Create children by exchanging
tails. Figure (2) shows a graphical representation of the crossover operation for the order quantity
vector (the first row of the chromosome matrix) with five products. A similar approach can be taken
for the maximum backorder level vector (the second row of the chromosome matrix).

13

Parents

 287 300  318 595 208
 655 418  146 236 489 




 287

300 (146 236 489) 

655

418 (318 595 208) 

offsprings

Figure (2): An example of the crossover operation

5.6 Mutation
Mutation is the second operation of the GA methods for exploring new solutions. In the
mutation operation of this research, a chromosome from the population is randomly selected and the
positions of its two randomly selected genes are interchanged with each other. Figure (3) shows a
graphical representation of the mutation operation for the order quantity and the maximum
backorder level of five products.
Q  287 (300) 318 (595) 208
b 99 (221) 222 219 (85)






 287
99

(595) 318 (300) 208

(85) 222 219 (221)

Figure (3): An example of the mutation operation

5.7 Chromosomes selection
In genetic algorithms, the selection operator is used to guide the search process towards
more promising regions in a search space. Several selection methods, such as roulette wheel,
tournament, ranking, and elitist are discussed in Michalewicz (1996). Furthermore, a detailed
explanation of the operation of roulette wheel selection can be found in Goldberg (1989).

14

Since in the roulette wheel selection better solutions get higher chance to become parents of
the next generation, it is employed to select the chromosomes of this research. The selection in this
method is based on the fitness value of the chromosomes. We select N chromosomes among the
parents and the offspring with the best fitness values.

5.8 Stopping criterion
The last step in a GA methodology is to check if the method has found a solution that is
good enough to meet the user’s expectations. Stopping criteria is a set of conditions such that when
the method satisfies them, a good solution is obtained. In this research, we stop after 400
generations. Note that the number of generations in this method depends on the problem size.

6. A numerical example

In order to demonstrate the application of the proposed methodology and to evaluate its
performances, in this section a numerical example in which F  18000, M  12,   0,   3 for
all products, is given. The numerical data of the products is shown in Table (1). In addition,
different values of n are selected as 3, 5, 8 and 10. Furthermore, since the initial values of the GA
parameters affect its performance, different values of Pc are chosen to be 0.65, 0.75, 0.68, 0.7, 0.8
and 0.78; different values of Pm are selected as 0.02, 0.05, 0.01, 0.03, 0.04 and 0.025 and different
values of N are chosen to be 80, 100, 120, 140, 160, and 180 for experimental purposes.

6.1 Computational results
The developed GA is coded in Matlab 7.6.0.324 software. All the test problems are solved
on a Pentium 4 computer with 512MB RAM and 2.40GHz CPU. The results are summarized in
Table (2). Each row of Table (2) corresponds to a problem instance in which different number of

15

products and different values of the GA parameters are used. Each problem instance has been run
10 times and the minimum fitness value along with its corresponding solution is recorded.
Furthermore, CPU t denotes the required CPU time of solving the problem instance.

Product (j)
1
2
3
4
5
6
7
8
9
10

Dj
420
360
540
390
480
510
530
380
430
580

F  18000, M  12,   0,   3

Table (1): The product data
A jS
A jR
1
2
3
5
2
4
1
2
3
4

3
2
1
4
2
2
3
1
4
2

h jR
4
9
7
2
4
6
5
3
2
8

f

j

3
2
3
1
4
3
2
1
3
4

Based on the results of Table (2), when n is selected 3, 5, 8 and 10, the best fitness values
are 316.54, 672.94, 3443.9 and 3609.43, respectively. Furthermore, the convergence path graphs of
reaching these solutions are presented in Figures (4), (5), (6) and (7), respectively.

Figure (4): The convergence path (N=3).

Figure (5): The convergence path (N=5).

16

Figure (6): The convergence path (N=8).

Figure (7): The convergence path (N=10).

6.2 Comparison of the fitness evaluation approaches
In order to control infeasible solutions, we may employ the penalty policy given by Gen
(1997), instead. In this algorithm, the penalty is defined as a positive and known coefficient of
violation of constraints (storage capacity and total number of orders). A larger coefficient (penalty)
is given to a more infeasible solution; and hence a zero-penalty for a feasible chromosome. In this
case, the fitness function of a chromosome is defined as the sum of its objective function and
penalty.
The experimental results of employing the penalty policy for the numerical example of
section 6.1 are summarized in Table (3). Once more, each row of Table (3), which corresponds to a
different problem instance, is run 10 times and the minimum fitness value along with the reorder
quantities and the maximum backorder quantities are recorded.
The following conclusions can be made based on the results of Table (3):
(a) The fitness values of the penalty policy are significantly larger than the ones of the
previous approach.
(b) When the number of products increases, the difference between the fitness values of the
penalty policy and the ones of the previous approach becomes larger.

17

Furthermore, based on the results of Table (3), when n is 3, 5, 8 and 10, the best fitness values are
1073.94, 1765.99, 5036.39 and 8048.72, respectively. The convergence path graphs of reaching
these results are given in Figures (8), (9), (10) and (11), respectively. Moreover, Figure (12) shows
the fitness value graphs of the two fitness evaluation methods.

Figure (8): The convergence path (N=3).

Figure (9): The convergence path (N=5).

Figure (10): The convergence path (N=8).

Figure (11). The convergence path (N=10).

18

9000
8000

Fitness value

7000
6000
5000
4000
3000
2000
1000
0

3

5

8

10

penalty

1073.94

1765.99

5036.39

8048.72

proposed

316.54

672.94

3443.9

3609.43

Figure (12): The graph of fitness value for the two fitness evaluation methods (N=3,5,8 & 10).

7. Conclusions and recommendation for future research

In order to make the EOQ model more applicable to real-world production and inventory
control problems, in this paper, we expanded this model by assuming several products in which
shortages were backordered, the storage had limited capacity and there was an upper bound on the
total number of orders. The proposed model of this research applies to a two-level supply chain
consisting of a single supplier and a single retailer that operates under vendor management
inventory (VMI) system. Under these conditions, we formulated the problem as a non-linear
integer-programming (NIP) model and proposed a genetic algorithm to solve it. At the end, a
numerical example was presented to demonstrate the application and the performance of the
proposed methodology and to compare it to a penalty policy that was applied to fitness evaluation.
For future researches in this area, we recommend the followings:
a) In addition to the storage capacity and the total number of order limitations, we may
consider budget and other constraints too.
b) Other meta-heuristic search algorithms such as simulated annealing may also be employed
and a comparison may be made among the algorithms.

19

c) Instead of backorder assumption, we can consider lost sale for shortages. Furthermore
quantity discounts can be employed.
d) Some of parameter of the model may be either fuzzy or random variable. In this case, the
model has either fuzzy or stochastic nature.
e) We can consider multi-echelon supply chains such as; one-retailer multiple-supplier,
multiple-retailer single-supplier and multiple-retailer multiple-supplier systems.

8. References

Angulo A, Nachtmann H, Waller MA (2004). Supply chain information sharing in a vendor
managed inventory partnership. Journal of Business Logistics, 25: 101–125
Axsäter S (2000). Inventory Control. Kluwer Academic Publishers, Boston, MA, USA
Axsäter S (2001). A framework for decentralized multi-echelon inventory control. IIE Transactions,
33: 91–97

Bertazzi L, Paletta G, Speranza MG (2005). Minimizing the total cost in an integrated vendormanaged inventory system. Journal of Heuristics, 11: 393–419
Biskup D, Simons D, Jahnke H (2003). The effect of capital lockup and customer trade credits on
the optimal lot size – a confirmation of the EPQ. Computers and Operation Research, 30:
1509–1524
Cardenas-Barron LE (2001). The economic production quantity (EPQ) with shortage derived
algebraically. International Journal of Production Economics, 70: 289-292
Cetinkaya S, Lee CY (2000). Stock replenishment and shipment scheduling for vendor-managed
inventory systems. Management Science, 46: 217– 232
Chen X, Wan W, Xu X (1998). Modeling rolling batch planning as vehicle routing problem with
time windows. Computers and Operations Research, 25: 1127–1136

20

Table (2): The experimental results of the proposed GA
Row(j)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

Pc

Pm

0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78

0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025

N
80
100
120
140
160
180
80
100
120
140
160
180
80
100
120
140
160
180
80
100
120
140
160
180

n
3

5

8

10

Qj

bj

274,55,55
113,57,81
469,158,231
182,99,185
159,259,42
137,126,24
175,418,328
21,300,204
243,276,327
186,241,267
56,56,56
14,28,28
287,300,318,595,208
99,221,222,219,85
655,48,146,48,489
359,28,175,32,260
615,188,204,607,607
239,141,129,22,260
147,236,206,356,147
76,170,76,76,40
189,134,134,651,248
65,65,17,252,250
56,56,56,56,56
45,78,17,45,17
234,262,198,445,372,493,524,1166
337,319,174,315,169,399,471,360
368,82,444,368,368,473,784,363
378,69,285,100,171,218,506,72
83,291,389,630,558,430,558,291
4,307,171,40,333,221,334,102
265,413,850,511,530,508,508,206
249,194,460,6,255,200,365,98
487,494,559,1007,158,881,565,487
411,411,411,411,171,479,492,479
208,370,210,2031,413,708,492,549
72,270,48,346,49,346,342,8
594,148,148,1070,264,333,333,148,599,333
212,92,92,290,97,203,304,92,288,203
816,264,201,371,737,1228,641,766,648,579
425,282,53,348,441,354,382,219,385,374
104,405,234,252,566,550,508,532,1268,355
28,259,64,203,284,113,170,157,430,390
1033,654,681,1245,567,317,735,839,1495,560 456,475,399,475,427,370,427,370,427,456
284,284,1127,154,853,642,804,698,280,758
19,221,428,42,375,421,428,421,187,424
774,737,400,1257,177,583,626,199,1005,510 300,473,399,300,300,286,292,85,430,399

21

Fitness
641.58
925.73
482.37
1121.71
960.21
316.54
1557.92
1543.07
2188.15
1191.06
1582.02
672.94
4631.48
3443.9
3508.85
4346.24
4943.72
5341.98
3609.43
7144.11
5505.15
7366.02
6424.7
6686.32

CPU t
28.86
33.82
33.16
36.00
43.83
31.65
25.28
26.79
26.7
30.65
33.99
34.79
23.16
30.73
30.48
32.16
37.74
38.11
25.65
31.65
31.39
33.60
38.02
40.94

Table (3): The experimental results of the penalty policy
Row(j)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

Pc

Pm

0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78
0.65
0.75
0.68
0.7
0.80
0.78

0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025
0.02
0.05
0.01
0.03
0.04
0.025

Qj
bj
N
n
80 3
593,130,269
340,133,133
100
382, 505,577
95,281,534
120
281,293,243
123,226,104
140
186,294,279
81,380,165
160
620,479,187
348,370,101
180
651,366,579
397,397,397
80 5
403,397,145,310,229
132,381,146,243,172
100
238,372,208,774,576
31,292,205,188,200
120
693,273,155,417,570
350,232,153,374,351
140
250,327,714,529,411
172,306,297,219,351
160
1408,612,312,802,1024
528,528,281,420,528
180
133,535,785,372,629
178,441,441,303,441
80 8
505,296,427,950,754,589,653,953
390,390,390,390,390,390,390,390
100
436,253,406,2448,961,615,533,1504
345,314,276,314,525,525,276,525
120
704,18,457,557,303,563,441,2044
373,29,29,189,15,329,337,470
140
643,271,276,1194,517,361,300,1208
358,270,333,205,433,71,191,465
160
427,521,431,2931,329,513,816,656
405,461,251,376,52,461,376,496
180
319,656,728,2079,1521,849,641, 771
441,528,528,528,528,528,528,528
80 10 477, 362,570,1225,443,535,387,2765,1181,163 451,451,451,383,451,451,134,134, 33, 179
100
100, 457,463,2320,213,491,746,1257,1570,698 404,421,421,421,421,342,421,421,421, 421
120
2063,212,1861,1753,91,355,469,1502,713,594 332,57,502,111,46,263,272,282,38,474
140
92,218,419,1951,305,1254,1092,897,792,538
109,166,327,257,79,187,273,172,277,484
160
1140,779,649,2244,138,693,386,448,786,384
549,549,549,549,63,500,382,360,149,549
180
614,285,529,2279,568,277,961,1870,841333
504,211,504,504,504,504,504,504,504,504

22

Fitness CPU t
1073.94 21.94
1911.86 25.64
953.70
25.22
1352.47 28.44
1309.49 31.57
1837.90 31.88
1765.99 25.10
2226.97 28.41
2188.26 26.31
2509.75 29.98
3940.05 34.84
2896.34 34.79
5149.78 28.22
6766.61 29.83
6120.19 29.59
5036.39 31.90
6622.50 37.70
7692.00 37.97
9983.70 27.02
12933.72 32.58
18836.76 31.55
13990.72 36.96
8048.72 42.74
10642.22 43.82

Chung KJ, Huang YF (2003). The optimal cycle time for EPQ inventory model under permissible
delay in payments. International Journal of Production Economics, 84, 307–318
Cheung L, Lee HL (2002). The inventory benefit of shipment coordination and stock rebalancing in
a supply chain. Management Science, 48 (2), 300–306
Chopra S (2003). Designing the distribution networks in a supply chain. Transportation Research,
Part E 39, 123–140
Chopra S, Meindl P (2001). Supply Chain Management: Strategy, Planning, Operation, 1st ed.,
Prentice Hall, Upper Saddle River, NJ, USA
Goldberg D (1989). Genetic Algorithms in Search; Optimization and Machine Learning. AddisonWesley, Reading, MA, USA
Disney SM, Towill DR (2003). The effect of vendor managed inventory (VMI) dynamics on the
bullwhip effect in supply chains. International Journal of Production Economics, 85: 199215
Disney S M, Towill DR (2002). A procedure for the optimization of the dynamic response of a
vendor managed inventory system. Computers & Industrial Engineering, 43: 27–58
Dong Y, Xu K (2002). A supply chain model of vendor managed inventory. Transportation
Research Part E: Logistics and Transportation review, 38: 75-95
Silver EA, Pyke DF, Peterson R (1998). Inventory Management and Production Planning and
Scheduling, 3rd ed., John Wiley and Sons, New York, NY, USA
Mohebbi E (2008). A note on a production control model for a facility with limited storage capacity
in a random environment. European Journal of Operational Research, 190, 562–570
Farahani RZ, Elahipanah M (2008).A genetic algorithm to optimize the total cost and service level
for just-in-time distribution in a supply chain. International Journal of Production
Economics, 111: 229–243

23

Gen M, Cheng R (2000). Genetic Algorithms and Engineering Optimization. John Wiley and Sons,
New York, NY, USA
Goyal SK (1985). Economic order quantity under conditions of permissible delay in payments.
Journal of the Operational Research Society, 36: 35–38
Harris FW (1913). How many parts to make at once? Factory, the Magazine of Management, 10:
135-152
Jaruphongsa W, Cetinkaya S, Lee CY (2004). Warehouse space capacity and delivery time window
considerations in dynamic lot-sizing for a simple supply chain. International Journal of
Production Economics, 92: 169–180
Jasemi M (2006). A vendor-managed inventory. Master of Science Thesis, Department of Industrial
Engineering, Sharif University of Technology, Tehran, Iran (in Farsi)
Teng J-T (2008). A simple method to compute economic order quantities, European Journal of
Operational Research, 198: 351-353
Kaipia R, Holmstrom J, Tanskanen K (2002). VMI: What are you losing if you let your customer
place orders? Production Planning and Control, 13: 17–25
Kleywegt AJ, Nori VS, Savelsbergh MWP (2004). Dynamic programming approximations for a
stochastic inventory routing problem. Transportation Science, 38: 42–70
Ko H, Evans G (2007). A genetic algorithm-based heuristic for the dynamic integrated
forward/reverse logistics network for 3pls. Computers and Operations Research, 34: 346–
366
Kwak C, Choi JS, Kim CO, Kwon I (2009). Situation reactive approach to vendor managed
inventory problem. Expert Systems with Applications, 36: 9039–9045
Lee HL, So, KC, Tang CS (2000). The value of information sharing in a two level supply chain.
Management Science, 46: 626–643

24

Lee CC, Chu W, Hung J (2005). Who should control inventory in a supply chain? European
Journal of Operational Research, 164: 158–172
Liao JJ (2007). On an EPQ model for deteriorating items under permissible delay in payments.
Applied Mathematical Modeling, 31: 393–403
Gen M (1997). Genetic Algorithm and Engineering Design, John Wiley & Sons, New York, NY,
USA
Magee JF (1958). Production Planning and Inventory Control. McGraw Hill Book Company, New
York, NY, USA
Michalewicz Z (1996). Genetic Algorithms + Data Structures = Evolution Programs. 3rd ed.,
Springer-Verlag, Berlin, Germany
Moinzadeh K (2002). A multi-echelon inventory system with information exchange. Management
Science, 48: 414–426
Moon C, Kim J, Hur S (2002). Integrated process planning and scheduling with minimizing total
tardiness in multi-plants supply chain. Computers and Industrial Engineering, 43: 331–349
Nachiappan S, Jawahar N (2007). A genetic algorithm for optimal operating parameters of VMI
system in a two-echelon supply chain. European Journal of Operational Research, 182:
1433–1452
Papachristos S, Skouri K (2003). An inventory model with deteriorating items, quantity discount,
pricing and time dependent partial backlogging. International Journal of Production
Economics, 83: 247–256
Park Y (2001). A hybrid genetic algorithm for the vehicle scheduling problem with due times and
time deadlines. International Journal of Production Economics, 73: 175–188

25

Pasandideh SHR, Niaki STA (2008). A genetic algorithm approach to optimize a multi-products
EPQ model with discrete delivery orders and constrained space. Applied Mathematics and
Computation, 195: 506–514
Pentico DW, Drake MJ, Toews C (2009). The deterministic EPQ with partial backordering: A new
approach. Omega, 37: 624 – 636
Tersine RJ (1994). Principles of Inventory and Materials Management, 4th ed., Prentice Hall PTR,
Englewood Cliffs, NJ, USA
Sofifard R, Hosseini SMM, Farahani RZ (2007). The study of vendor-managed-inventory in supply
chain with mathematical model. Master of Science Thesis, Department of Industrial
Engineering, Amirkabir University of Technology, Tehran, Iran (in Farsi)
Tyana J, Wee HM (2003). Vendor managed inventory: a survey of the Taiwanese grocery industry.
Journal of Purchasing & Supply Management, 9: 11-18
Van Der Vlist P, Kuik R, Verheijen B (2007). Note on supply chain integration in vendor-managed
inventory. Decision Support Systems, 44: 360-365
Vergin RC, Barr K (1999). Building competitiveness in grocery supply through continuous
replenishment planning: Insights from the field. Industrial Marketing Management, 28:
145–153
Vigtil A (2007). Information exchange in vendor managed inventory. International Journal of
Physical Distribution & Logistics Management, 37: 131-147
Waller M, Johnson ME, Davis T (1999). Vendor-managed inventory in the retail supply chain.
Journal of Business Logistics, 20: 183-203
Wee HM (1999). Deteriorating inventory model with quantity discount, pricing and partial
backordering. International Journal of Production Economics, 59: 511–518

26

Woo YY, Hsu SL, Wu SS (2001). An integrated inventory model for a single vendor and multiple
buyers with ordering cost reduction. International Journal of Production Economics, 73:
203–215
Xu K, Dong Y, Evers PT (2001). Towards better coordination of the supply chain. Transportation
Research, Part E, 37: 35–54
Yao YL, Evers PT, Dresner ME (2007). Supply chain integration in vendor-managed inventory.
Decision Support Systems, 43: 663–674
Yu YG, Liang L (2004). An integrated vendor-managed-inventory model for end product being
deteriorating item. Chinese Journal of Management Science, 12: 32–37 (in Chinese)
Zipkin PH (2000). Foundations of Inventory Management. McGraw-Hill Higher Education, Boston,
MA, USA

27