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FUZZY MULTI-OBJECTIVE PERIODIC REVIEW
INVENTORY PROBLEM IN A DYADIC
SUPPLY CHAIN SYSTEM
Dicky Fatrias and Yoshiaki Shimizu
Department of Mechanical Engineering, Toyohashi University of Technology
Toyohashi 441-8580, Japan
Email: dicky@ise.me.tut.ac.jp, shimizu@me.tut.ac.jp
ABSTRACT
This paper presents a fiizzy periodic review inventory model in a dyadic supply chain by
incorporating some uncertain parameters. To cope with such uncertainty, a fuzzy multi-objective
approach is introduced. A solution procedure is proposed such that several fuzzy goals are satisfied
and, in the process, the optimal ordering policy and inventory level are determined.Through providmg
hypothetically constructed case problem, the usefulness of our proposed model is demonstrated.
KEY WORDS: fiizzy multi-objective problem, inventory, dyadic supply chain
1. INTRODUCTION satisfied and, in the process, the optimal
The issue of considering uncertainty in ordering policy and safety stock of manufacturer
inventory problem has received a great deal of and the optimal target stock level of retailer are
attention in the field of production/inventory determined. Through providing hypothetically
management. In the context of periodic review constructed case problem, we provide the
system several researches have studies such acceptable solutions.
issue using stochastic approach under different
concems.Recent researches carried out in this
direction include Song & Lau (2004), Bijvank& 2. SYSTEM DESCRIPTION
Johansen (2012), Prasertwattana & Shimizu. In what follows, the proposed inventory model
(2007) and Fatrias& Shimizu (2010).With the will be described briefly. In all cases, we put the
development of fiizzy set theory (FST), the following assumptions.
fiizzy approach is also employed for the 1. The manufacturer uses the periodic review
modeling of uncertain parameters inventory with lot sizing policy andsafety stock to
problems. The FST copes with the uncertainty control its inventory.
related to unavailability and incompleteness of 2. The retailer uses the periodic review with
data as well as and imprecise nature of goals of target stock levelto control its inventory.
which the use conventional probability 3. Only a single product is considered in the
distribution impossible in this case. model. Without loss of the generality, the
In this regard,this research proposes a fiizzy manufacturer uses one unit of raw material
multi-objective periodic review inventory model to produce one unit of product.
in a typical Supply chain (SC) system in 4. For both manufacturer and retailer, only one
whichsingie-manufacturer, single-retailer is order is allowed to place at any period.
considered. Specifically, we attempt to develop 5. Production rate of the manufacturer is
a fiizzy periodic review inventory model in a assumed fixed and higher than the mean
mixed imprecise and/or uncertain environment demands.
by incorporating the fiizziness of demand, lead 6. Unfulfilled demand at manufacturer is
time and cost parameters. considered as backorder while unfulfilled
To cope with such problem, solution procedure demand atretailer is considered as shortages.
is proposed such that several fiizzy goals are
Managing Assets and Infrastructure in the Chaotic Global Economic Competitiveness 197
The system model is described based on the manufacturer.
foUowingnotation listed for major parameters. p = Unit purchasing cost of retailer.
c = Unit holding cost of finished product of
Index retailer.
Number of planning horizon.
T = J = Unit shortage cost of finished product
t Period(/ = 1,2,...,7). of retailer.
Number of days in each period. TCm = Total Cost of manufacturer.
h =
Parameters of Manufacturer TCr = Total cost of retailer.
A = Forecast demand of manufacturerat Decision Variabies
period/. LS = Lot sizing policy of manufacturer.
Q, = Order quantity of manufacturer at ss = Safety stock level of manufacturer.
period/. S = Target stock level of retailer.
PR = Production rate of Manufacturer.
Im, = lead time of raw material delivery
ffomsupplier at period /. Supplier j-j * Manufacturer Retailer —f-»-|^ustomers
Qpn = Production quantity produced at
period/.
Es, = Ending stock of raw materials of Figure 1. System Configuration
at period /.
Ess, = Ending safety stock of at period /. The members in this chain consist of one
Qm, = Ordering quantity at period /. supplier, one manufacturer, one retailer, and end
Qb, = Backorder quantity at period /. customers as shown in Figure 1. However, this
Qsl, = Sales volume at period /. study focuses on a dyadic relationship in the
EI, = Ending inventory at period /. chain between the manufacturer and the
BR = Backorder rate of manufacturer. retailer(the supplier and endcustomers are
considered as external members in the
Parameters of Retailer chain).We assume that these two members are
d, = End customer demand at period /. owned and controlled by a central company.
A = Lead time of product delivery from Inventoryof each memberis controlled by a
periodic review in make-to-stock environment,
manufacturerat period /. in whichdemand and leadtime, and cost
In = Ending inventory of finished product parameters are considered as a fuzzy
at period /. number.The inventory level of manufacturer and
Qsr, = Shortage quantity after receiving retailer are reviewed at every period time /, over
replenishment at period /. totally T periods (plaiming horizon). Each
Qor, = Order quantity at period /. period consists of interval of time tp days.
Qre, = Replenishment quantity received at
period/. 2.1 Manufacturer
LR = Loss rate of retailer. Manufacturer receives raw materials from
Cost Parameters outside supplier which has unlimited capacity,
transforms it to finished product and then
0, = Order cost of manufacturer at period /. distributes the products to retailer. However, the
r = Unit purchasing cost of manufacturer. supplier may delay the supply of raw materials
m = Unit production cost of manufacturer. to the manufacmrer. Therefore, the manufacturer
h = Unit holding cost of raw material of has to select the appropriate material ordering
policy and hold safety stock of product to cope
manufacturer. with the uncertainty in demand and delivery
/ = Unit holding cost of product of lead-time.
manufacturer. The ordering quantity of manufacturer is
b = Unit backorder cost ofmanufacturer. directly influenced by lot sizing policy {LS)
T = Unit transportation cost of which is adopted for ordering raw material.
198 Managing Assets and Infrastructure in the Chaotic Global Economic Competitiveness
After the best pattern of LS is selected, the Qb,
manufacturer will check the amount of Min 5R, =E {Qorj) (3)
inventory on hand at the beginning of the
period. If the amount on hand is less than the
sum of the demand and the amount to fill back Min LRj='Z (4)
the safety stock, then the manufacturer will
place the order to supplier. Otherwise no order
will be issued.
The manufacturer can start the production at the 3. SOLUTION METHODOLGY
beginning of each period if raw material on hand The proposed fuzzy periodic review inventory
exists; otherwise the manufacturer has to wait model is actually a multi-objective mfaced
imtil arrival of raw material fi-om the supplier by integer programming model (MOMIP). To solve
timer+/ffj, As a consequence, the production the model, a solution procedure is proposed,
quantity imder the combined order condition First, the equivalent crisp MOMIP model is
may become higher than "lot-for-lot" case and converted into a single-objective MIP model.
results in higher capability to supply retailer's Then, one evolutionary optimization search
demand (lower shortage cost) at the expense of method named Differential Evolution (DE) is
higlier holding cost. applied to find an optimal solution.
2.2 Retailer 3.1 The Auxiliary Crisp MOMIP Model
The retailer makes a regular order to the Transforming a fuzzy MOMIP model into an
manufacturer periodically to raise up the auxiliary crisp MOMIP model require an
inventory to the target stock level. The order appropriate method. For this purpose, Jimenez
quantity(gor,) is determined by comparing the methodis applied because it is computationally
ending stock level (/r,) at the review time t with efficient to solve a fuzzy problem (See Jimenez
the desired target stock level (5), which is equal et al., 2007).According toJimenez method, the
to (S-Ir,). This target stock level is not only to auxiliary eq./ (l)-(4) can be formulated as
cover the end customer's demand but also to follows:
cover the effect of its fluctuation as well as the Min TCm =
late delivery and unfulfilled quantity of products
fi-om the manufacturer. t + Z Q,
2.3 Objective Functions ( bi^' + 2b'"" +b'"''\
This study considers four objective functions to i. Es. Qsl.
evaluate the system performance. The first (=1
objective minimizes total cost of T {Ess. + EI,)
manufacturer(rCm); the second objective
minimizes total cost of retailer(TCr); the third
objective function minimizesbackorder rate of ^ti" +2t'"" +t°<"^
manufacturer (BR); and thefourth objective Qsl, (5)
function minimizesloss rate of retailer(I/?). Min TCr =
Min TCm= j;^d + f^SxQ,+f^hxEs,+ pP"+2p""+p rc'*'-l-2c"'"-tc^'l
Qsl,+t Ir,
tcx{Ess,+EI,)+tb^Qb,+ -^2s""+s*"'' Qsr, (6)
(=1 (=1
t^xQsl, (1)
(=1 Qb,
Min BR,='Z (7)
Min TCr = tp>'Q^l,+idxIr,+ZsxQsr, Qorj)
(=1 (=1 (=1
(2)
Managing Assets and Infrastructure in the Chaotic Global Economic Competitiveness 199
4. COMPUTATINAL EXPERIMENT
Qsrj To illustrate the usefulness of the fuzzy MOMIP
in LRj=t model using the proposed solution procedure, a
Min numerical experiment is provided and the result
(=1
is reported in this section utilizing input
parameters shown in Table I.
FomuUlc fuzzy MOMINLP
3.2 The Proposed Solution approach pciiodic invelofy model
The steps of the proposed solution procedures Detennine membership functioir GcDcrale initul population
are summarized as follows (Figure 2): for fiizzy puumeters ind of Inrget vector. O - 0
Stepl: Formulate the fiizzy MOMIP objeelive fiincriom
Comptfte and evaluate the
(MOMINLP) periodic review inventory model I Convert the MOMINLP into in fitness of each target vector
as described in section 2. 1 equivnlent crisp model i
Apply muution, crossover
Find the range of each of objective and selection operator to
Step 2: Determine the appropriate generate new target vector
function by cntcubling their
membership function for fuzzy parameters and minimum and maximum vahie \
objective functions. In this formulated problem, target vector
fiizzy parameters and objective functions are Cooveil Ac equivalent crup
MOMINLP model into a single-
represented by linear membership fiinction. objective MINLP using Zimmerman no
method Termination
Step 3: Convert the fiizzy MOMIP into an I
auxiliary crisp MOMIP model. To this end, all DifTcrential Evolution Algoridun
the imprecise cost parameters in the objective I
functions as well as the demand and lead time Gain tfie acceptable solution
parameters are converted into the crisp ones
using Jimenez method.
Step 4: Determine the rage of each objective Figure 2. Solution methodology
function by calculating the minimum and
maximum value of each of them. To calculate 4.1 Setting the Lower and Upper Bound
the minimum and maximum value of each For LS, the manufacturer has to decide whether
objective function, the auxiliary multi-objective to make the order at the beginning of every
crisp model should be solved each time only one period or combine the order in a big batch.
objective. Therefore, the binary coding is applied to
Step 5: Convert the auxiliary crisp MOMIP represent the value of LS. Thejjand .S are
model into a single-objective MlPbased on considered as the amount of products (units) at
Zimmermann'saggregation function
(Zimmermann, I993).The formulation of the manufacturer and the retailer, respectively.
Zimmermann'saggregation function is as So the integer coding is used to represent these
follows: values.
Lower bound of LS is 0,which means"not
place the order" in current period but
Max 2 (9)
combine it to the previous period's order.
subject to: Upper bound of LS is 1 means "place the
2,
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