Inventory lot-sizing with supplier selection using hybrid

intelligent algorithm

Mohammad Reza Sadeghi Moghadam a, Amir Afsar b, Babak Sohrabi c,*a Department of Industrial Management, Faculty of Management, University of Tehran, Tehran, Iran

b Department of Industrial Management, Faculty of Management, Qom University, Qom, Iranc Department of Information Technology Management, Faculty of Management, University of Tehran, Tehran, Iran

Received 21 October 2006; received in revised form 22 October 2007; accepted 3 November 2007

Available online 13 November 2007

Abstract

In supply chain management (SCM), multi-product and multi-period models are usually used to select the suppliers. In the real world of SCM,

however, there are normally several echelons which need to be integrated into inventory management. This paper presents a hybrid intelligent

algorithm, based on the push SCM, which uses a fuzzy neural network and a genetic algorithm to forecast the rate of demand, determine the

material planning and select the optimal supplier. We test the proposed algorithm in a case study conducted in Iran.

# 2007 Elsevier B.V. All rights reserved.

Keywords: Hybrid intelligent algorithm; Inventory lot-sizing; Fuzzy neural network (FNN); Genetic algorithm (GA); Principle component analysis (PCA)

www.elsevier.com/locate/asoc

Available online at www.sciencedirect.com

Applied Soft Computing 8 (2008) 1523–1529

1. Introduction

The multi-period inventory lot-sizing scenario with a single

product was introduced by Wagner and Whitin in [1], where a

dynamic programming solution algorithm was proposed to

obtain feasible solutions to the problem. Soon afterwards,

Basnet and Leung [2] developed the multi-period inventory lot-

sizing scenario which involves multiple products and suppliers.

The model used in these former research works is formed by a

single level indicating the type, amount, suppliers and

purchasing time of the products. This model, however, is not

able to consider the planning of the supply sector of the firm. In

addition, the models used in previous works assumed that the

rate of production demand is constant. While in practice a

mechanism to forecast the rate of such demand is required.

In this paper, we introduce a new model for the multi-period

inventory lot-sizing problem with supplier selection. We also

propose a hybrid intelligent algorithm which is able to plan and

control the inventory at different levels depending on the

accurate forecasting of different demand rates. Our algorithm is

based on a fuzzy neural network (FNN) and a genetic algorithm

(GA): the forecasting of the periodical demand rates is done by

* Corresponding author.

E-mail address: bsohrabi@ut.ac.ir (B. Sohrabi).

1568-4946/$ – see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.asoc.2007.11.001

means of the FNN, and the result of the FNN which are then

passed to a GA in order to optimize the planning and inventory

controlling models based on which the proper suppliers are

selected.

The rest of the paper is organized as follows: Section 2

provides a literature review on the current inventory lot-sizing.

Our hybrid intelligent algorithm is proposed in Section 3.

Section 4 presents the experimental results using real data

obtained in one of the biggest supply chain of sewing machines

manufacturing in Iran. Finally, Section 5 closes the paper giving

some concluding remarks.

2. Literature review

Bahl et al. [3] provided a comprehensive review of inventory

lot-sizing literature. They classified the models in four

categories: (1) single-level unconstrained resources, (2)

single-level constrained resources, (3) multiple-level con-

strained resources, and (4) multiple-level unconstrained

resources. Levels here refer to the different ones in a bill of

material structure (where there is a question of requirements

dependency) and constrained resources indicate production

capacity limitations [2]. The scenario discussed in this paper

belongs to the fourth category. Multi-echelon inventory systems

along with multiple supply modes can be classified into two

models first of which includes some features of multi-supplier

Fig. 1. Processes involved in hybrid intelligent algorithm for inventory lot-

sizing.

M.R. Sadeghi Moghadam et al. / Applied Soft Computing 8 (2008) 1523–15291524

single stage ones and the second allows trans-shipments

between inventory stocking points [4]. Minner et al. [5]

considered a periodic review of two echelon systems where as

an alternative to rationing a depot shortage, outstanding orders

can be speeded up with a certain probability. The maximum

extension of multi-echelon system was discussed as the break

quantity rule by Dekker et al. [6], i.e., the quantity that if a

customer order size is larger than the break quantity, the request

is satisfied directly from the warehouse and otherwise shipped

by the associated retailer. The objective is to find appropriate

order-up-to-levels and the break quantity in order to minimize

system operating costs. In so called no-delay multi-echelon

inventory models, safety stocks are provided at every stocking

point to cover against reasonable demand variability where

extraordinary large orders are excluded from the analysis by

assuming some kinds of operating flexibility. This modeling

approach implicitly assumes the presence of two supply

alternatives, a regular one for demands not exceeding a

predetermined level of variability and an emergency mode to

deal with excessive variations [7].

With the advent of supply chain management, much

attention is now devoted to supplier selection. Rosenthal

et al. [8] studied a purchasing problem where one needs to

select among suppliers who offer discounts selling a ‘‘bundle’’

of multiply products. Then a mixed integer programming

formulation was presented. Chaudhry et al. [9] considered

vendor selection under quality, delivery and capacity con-

straints and price-break regimes. Ganeshan [10] presented a

model to determine lot sizes that involve multiple suppliers

including multiple retailers, and consequent demand on a

warehouse. Kasilingam and Lee [11] incorporated the fixed cost

of establishing a vendor in a single-period model that includes

demand uncertainties and quality considerations in the

selection of vendors. Also vein, Jayaraman et al. [12] proposed

a supplier selection model that considers quality (in terms of

proportion of defectives supplied by a supplier), production

capacity (constraining the order placed on a supplier), lead-

time, and storage capacity limits. This is also a single period

model that attaches a fixed cost to deal with a supplier. They

formulated a mixed integer linear programming model to solve

the problem. Basnet and Leung [2] presented a multi-period

inventory lot-sizing scenario where there were multiple

products and suppliers. They considered a situation where

the demand of multiple discrete products is known over a

planning horizon. The model determines the type, amount,

supplier and purchasing time of products. They presented an

enumerative search algorithm and a heuristic algorithm to

address the problem. Their model is one of the most useful ones

for supply selection in a single stage category presented in

literature.

None of the above mentioned, however, attempted to use

multi-periods and multi-echelons model beside the supply

selection. Therefore, inventory lot-sizing with supplier selec-

tion using hybrid intelligent algorithm takes advantages both

from Basnet and Leung model (including multiple products in

multiple periods with supplier selection according to holding,

transition and purchasing cost) and the multi-echelon in which

each level supplies the required products of the next level.

Solving the model, a hybrid intelligent algorithm based on FNN

and GA is designed.

3. Proposed hybrid intelligent algorithm

Demand forecasting is the foundation of the push system in

supply chain management. Therefore, it is undoubtfully

important to use a suitable method for forecasting the product

demand. First, the PCA is used to reduce the large

dimensionality of data set. Now as the outcome of the first

part, demand forecasting is used as the input of the next one that

is modeling where GA uses the forecasted demand to determine

the required inventory in each echelon. Then the supplier is

selected in the final echelon. Fig. 1 shows the processes

involved in the hybrid intelligent algorithm.

3.1. Fuzzy neural network

Artificial neural networks (FNN) appear to be particularly

suitable to forecast the demand of time series, as they can learn

highly nonlinear models, hold effective learning algorithms,

handle noisy data, and use inputs of different kinds [13]. ANNs

have been designed to mimic the characteristics of the

biological neurons in the human brain and nervous system

[14]. An ANN creates a model of neurons and the connections

between them, and trains it to associate output neurons with

input neurons. The network ‘‘learns’’ by adjusting the

interconnections (called weights) between layers. When the

network is adequately trained, it is able to generate relevant

output for a set of input data. One of the valuable properties of

neural networks is that of generalization whereby a trained

neural network becomes able to provide a correct matching in

the form of output data for a set of previously unseen input data.

M.R. Sadeghi Moghadam et al. / Applied Soft Computing 8 (2008) 1523–1529 1525

BP is one of the most famous training algorithms for multilayer

perceptions [15,16]. Basically, BP is a gradient descent

technique to minimize the error for a particular training pattern.

FNNs are a class of hybrid intelligent algorithms that

integrate fuzzy logic with ANNs. A fuzzy neural network

system is defined [17] as a combination of ANN and fuzzy

inference system (FIS) [18] in such a way that neural network

learning algorithms are used to determine the parameters of FIS

[19]. An even more important aspect is that the system should

always be interpretable in terms of fuzzy if-then rules, because

it is based on the fuzzy system reflecting vague knowledge.

3.2. Genetic algorithm

GA is one of the modern heuristic optimization algorithms

widely adopted by researchers in solving various problems.

Introduced by Holland [25] in 1975, GA heuristic optimization

algorithms mimic the mechanism of genetic evolution in

nature. They have two main areas of application [26,27]; the

first is the optimization of the performance of a system, such as

traffic lights or a gas distribution pipeline system. They

typically depend on the selection of parameters, perhaps within

certain constraints whose interaction restricts a more analytical

approach. The second area of application for GA is in the field

of testing or fitting into the quantitative models [28].

4. Experimental research

Iran is the host of the largest manufacturers of sewing

machine in the Middle East. It orders items to manufacturers

located in Iran and some countries in East Asia, which in turn

receive components from a variety of suppliers. The sewing

machines are produced in five different models among which

one is selected according to manager’s view. There are four

levels (k = 4) consisted of some stations completing this

product through the manufacturing process.

The problem is to determine the number of products in each

station and supplier selection while minimizing the objective

functions (holding, transaction, and purchase cost). The supply

chain is shown in Fig. 2.

Fig. 2. The supply ch

4.1. Demand forecasting using fuzzy neural network

The PCA is used to reduce the large dimensionality of data

sets.

4.1.1. Data reduction

In Section 3, there are nine independent variables that affect

demand forecasting and the data gathered in a period of 60

months. The interpretability of the model is enhanced by

reducing the dimensionality of data sets. Traditionally, feature

extraction algorithms including principal component analysis

(PCA) have been often used for this purpose [20]. This reduces

the factors influencing on demand forecasting to three

variables which are used as input variables of FNN. In other

words, data reduction is aimed to design a more complex model

for FNN.

4.1.2. Data normalization

The function of non-linear transfer will typically squashes

the possible output from a node into (0, 1) or (�1, 1). Where the

range of the property of time series is predicted, data

normalization is required before the beginning of training

process. Here, linear transformation [21,22] is applied to the

time series values, with an upper and lower limit of 1 and�1 so

as to coincide with the theatrical boundaries of the transfer

function.

½a; b� : x0 ¼ ðb� aÞðx� xminÞðxmin � xmaxÞ þ a

External method of input normalization (reported in

literature 22, 23) is selected to normalize all data into a

specific rank.

4.1.3. Explanation of fuzzy neural networks

Takagi–Sugeno–Kang fuzzy inference system is used to

design the FNN for demand forecasting [24]. The current model

has 10 built-in membership functions composed of the

difference between two sigmoidal membership functions for

each input variable with the following evolving parameters:

ain of case study.

Fig. 3. RMSE reduction of demand forecasting by using FNN with the

normalized data.

M.R. Sadeghi Moghadam et al. / Applied Soft Computing 8 (2008) 1523–15291526

(1) n

Tabl

Fore

Indi

RM

MSE

MA

MA

R2

min

s:t:

umber of training epochs = 500,

(2) t

raining error goal = 0,

(3) s

tep-size for each epoch = 1 and

(4) l

earning rates for first and second layer = 0.05.

FNN uses a single pass training approach. And the network

parameters were determined using a trial and error approach.

The training was repeated 10 times after reinitializing the

network and the worst errors were reported (Fig. 3).

Xt

Xk

Xj

Xi

Pi jXi jkt þX

k

Xj

Xt

O jktY jkt þXkþ1

Xi

Xt

Hik

Xt

p¼1

Xj

Xi jðkþ1Þ p � LkiXi jk p

!

þX

j

Xt

Hi

Xt

p¼1

Xj

Xi jk p �Xt

p¼1

Fi p

!

Xt

p¼1

Xj

Xi j1 p �Xt

p¼1

Fi p� 0; for alli; and t;

(1)

In addition to FNN, the ARIMA model is used for demand

forecasting, through three phases including model identifica-

tion, parameter estimation, and diagnostic checking. ARIMA

(1, 1, 1) model is the most appropriate Box–Jenkins one for

demand forecasting. Table 1 shows the results of demand

forecasting by FNN and ARIMAwhich are compared with each

other to indicate that the former is more suitable for such a

purpose. Therefore, the period of three later months is

forecasted by the proposed model.

4.2. Modeling

A comprehensive mathematical model (multi-product,

period and echelon lot-sizing with supplier selection) is

presented as follows:

e 1

casting results of FNN and ARIMA

cators FNN ARIMA

SE 0.3272 16.1332

0.107 260.2802

E 0.1653 3.869

PE 0.0000176 0.00015

0.99999 0.9996

� I

ndices:

� i = 1, . . ., I index of products,

� j = 1, . . ., J index of suppliers,

� t = 1, . . ., T index of time periods,

� k = 1, . . ., K index of echelons.

� P

arameters:

� Fit: forecasting demand of product i in period t,

� Pijt: purchase price of product or part i from supplier or

station j,

� Hik: holding cost of a unit of product or part i in echelon k,

� Ojk: transaction cost for supplier j in echelon k.

� D

ecision variables:

� Xijkt: number of product or part i supplied from supplier or

station j in echelon k in period t that is an integer,

� Yjkt:1; If an order is placed on supplier j in echelon

k in time periodt0; otherwise

(;

� Lki: number of required parts from echelon k + 1 for

completing a unit of product or part i in echelon k.

Regarding the above notation, the mixed integer program-

ming is formulated as follows:

Xt

p¼1

Xj

Xi jðkþ1Þ p � Lki

Xt

k¼1

Xj

Xi jkt � 0; for all i; k; and t;

(2)�XT

p¼t

Xi jk p

�Y jkt � LkiXi jðkþ1Þt� 0; for all i; j; k; and t;

(3)

Y jkt � 0 or 1; for all j; k; and t;Xi jkt � 0; for all i; j; k; and t:

The objective function consists of four parts: the purchase

cost of the products, the transaction cost for the suppliers, the

holding cost for the remaining inventory in each period in k + 1

echelon, and the holding cost for remaining inventory in each

period in k = 1 echelon.

The first constraint stipulates that all forecasted demand

must be filled in each period in echelon k = 1 (first echelon).

The second provides that all requirements have to be met in

each period during which they occur in echelon k > 1. The

third constraint indicates that suppliers were selected

considering the appropriate transaction costs in each echelon.

In this model, it is assumed that suppliers can be selected

within the final echelon.

Table 2

Test results for each computational experiment

Pop-size Generation Crossover Mutation Fitness function

200,000 200 Heuristic Gaussian (2, 1) 13,513,463

200,000 200 Two point Gaussian (1, 1) 13,329,903

200,000 200 Scatter Gaussian (1, 1) 13,196,627

200,000 200 One point Gaussian (1, 1) 13,618,604

M.R. Sadeghi Moghadam et al. / Applied Soft Computing 8 (2008) 1523–1529 1527

This model has i � t, i � k � t, i � j � k � t, for the first,

second and third constrains, respectively. According to the

complexity and non-linear nature of the current model, it is

preferred to use a metaheuristic method such as GA to solve it.

4.3. Numerical problem

As explained in this section, the problem includes one

product, four echelons and three periods whose requirements

are as follows:

Ft ¼ ½ 1616 2125 2311 �

Ft represents forecasted demand vector of the first echelon

which is obtained by FNN.

Hik ¼

40 35 32 26

� 35 32 27

� � 23 22

� � 30 20

2664

3775

The H11 = 40 represents the holding cost for product 1 at

echelon 1.

Oi j ¼12 10 8 8

10 10 6 9

� �

The O11 = 12 represents the transaction cost for product 1

from supplier 1.

P11t ¼ 640 650 650½ �

The purchase cost vector for product 1 in echelon 1 in period

t and other echelons follows as bellow:

Pi2t ¼190 191 193

248 250 249

� �Pi3t ¼

32 33 33

33 32 33

65 68 70

34 32 30

26664

37775

Pi4t ¼

20 19 21

10 10 10

10 10 10

10 10 10

8 8 8

6 6 6

8 8 8

6 6 6

266666666666664

377777777777775

4.4. Solving problem using genetic algorithm

In this case, the aim of the GA is the minimization of the

inventory costs.

4.4.1. Chromosome structure

Each chromosome is a feasible solution for the model that

holds the sum of the numbers of decision variables. In this

problem, there are 69 variables out of which 45 are integer

(Xijkt) and 24 (Yjkt) are 0 or 1.

4.4.2. Initial pop-size

In the initial stage of the optimization, a number of

chromosomes will be arbitrarily created. This step defines the

size of solution pool. More chromosomes may increase the

probability of finding optimal solution, but may induce a longer

computation time.

Here, different amount of primary population are examined

to select the most suitable amount of pop-size (200,000)

through the equation among GA variables (see in Table 2).

4.4.3. Fitness value

Fitness value defines the relative strength of a chromosome

compared with the others, and the optimality of the solution to

the problem. It is evaluated by the chromosome structure which

results in a positive value [29]. The fitness function of this

model is an objective one.

4.4.4. Crossover operator

Crossover operators including one point, two point, scatter,

and heuristic are used to solve GA. The best possible responses

are resulted from scatter.

4.4.5. Mutation operator

Mutation operator Gaussian with different shrink and scale

parameters is used to solve GA. The best possible responses are

resulted from Gaussian (1, 1).

Gaussian adds a random number or mutation, chosen from a

Gaussian distribution, to each entry of the parent vector.

Typically, the amount of mutation, which is proportional to the

standard deviation of the distribution, decreases at each new

generation. The average amount of mutation that the algorithm

applies to a parent in each generation can be controlled through

the Scale and Shrink options:

� S

cale controls the standard deviation of the mutation at the

first generation which is scale multiplied by the range of the

initial population specified by the initial range option.

� S

hrink controls the rate at which the average amount of

mutation decreases. Where standard deviation decreases

linearly, its final value equals 1—shrink times its initial

value at the first generation. For example, if shrink has the

value of 1, then the amount of mutation decreases to 0 at the

final step.

We have four computational experiments whit different

parameters. Each experiment is run 10 times. Table 2 depicts

the best test results with each parameter for each computational

experiment. Table 3 indicates the results of the best test for

Fig. 4. The fitness function reductions in generations.

Table 3

The best test results for decision variables

t X111t X112t X212t X113t X213t X313t X413t X124t X214t X314t X424t

1 1650 2670 6648 5522 5424 14,122 6669 6051 6156 14,318 6695

2 2150 2232 1936 4284 4414 3,099 1947 4620 4124 4,211 2118

3 2311 1198 3638 2402 2365 7,182 3584 1605 1973 5,902 3476

M.R. Sadeghi Moghadam et al. / Applied Soft Computing 8 (2008) 1523–15291528

decision variables. The fitness function reductions in genera-

tions are shown in Fig. 4.

The amounts of GA objective function are compared with

pattern search that is one of the direct search methods.

A pattern is a collection of vectors that the pattern search

algorithm uses to determine which points have to be searched at

each iteration. The set of vectors is defined by the number of

independent variables in the objective function, N, and the

positive basis set. Two commonly used positive basis sets in

pattern search algorithms are the maximal basis, with 2N

vectors, and the minimal basis, with N + 1 vectors.

Here, the amount of the resulted objective function is given

in Tables 2 and 3 and Fig. 4.

5. Conclusions

The current paper aims at providing a model capable of

planning and controlling the inventory in supply chain and

unifying the selection of supplier for multi-products, period,

suppliers and levels.

Additionally, accurate demand forecasting is considered as

an important factor especially in production system using push.

The given results assert that, compared with ARIMA, FNN can

optimize the forecasting which prepares a reliable function to

plan and control the inventory. Moreover, the unified model of

inventory planning and controlling the supplier selection is able

to balance the costs at different levels and select the proper

supplier. Experimental results approve that such a model can

reduce the costs of the case study by 4%. According to the

problems beyond using non-liner classical models, i.e. long-

term solution, the paper intends to depend on GA and pattern

search, which minimize the amount of objective function by

11.6%.

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