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Accepted Manuscript
Optimizing a Vendor Managed Inventory (VMI) Supply Chain for Perishable
Products by Considering Discount: Two Calibrated Meta-heuristic Algorithms
Maryam Akbari Kaasgari, Din Mohammad Imani, Mehdi Mahmoodjanloo
PII: S0360-8352(16)30429-6
DOI: http://dx.doi.org/10.1016/j.cie.2016.11.013
Reference: CAIE 4529
To appear in: Computers & Industrial Engineering
Received Date: 3 June 2016
Revised Date: 30 October 2016
Accepted Date: 12 November 2016
Please cite this article as: Akbari Kaasgari, M., Mohammad Imani, D., Mahmoodjanloo, M., Optimizing a Vendor
Managed Inventory (VMI) Supply Chain for Perishable Products by Considering Discount: Two Calibrated Meta-
heuristic Algorithms, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.11.013
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1
Optimizing a Vendor Managed Inventory (VMI) Supply Chain for Perishable
Products by Considering Discount: Two Calibrated Meta-heuristic
Algorithms
Maryam Akbari Kaasgari
1, Din Mohammad Imani
2, Mehdi Mahmoodjanloo
3
3
1Master student, Department of Industrial Engineering, Mazandaran University Science and
Technology, Behshahr, Iran
2Assistant Professor, Department of Industrial Engineering, University of Science and
Technology, Tehran, Iran,
3Faculty members, Department of Industrial Engineering, Mazandaran University Science and
Technology, Behshahr, Iran
2
Optimizing a Vendor Managed Inventory (VMI) Supply Chain for Perishable
Products by Considering Discount: Two Calibrated Meta-heuristic
Algorithms
Abstract
Vendor Managed Inventory (VMI) is one of the inventory management strategies that reduce
costs, increase responsiveness and improve collaboration between the members of supply chain.
Although the VMI can reduce response time and deterioration in perishable supply chain (PSC),
but there is a few reports on using VMI for PSC. In this paper VMI strategy is used for managing
the inventory of perishable product at two-level supply chain with single vendor and multiple
retailers. After passing a specific time of product lifetime that called the critical time, the product
would be perished by a probability distribution function. It is probable that the inventory of
product is not sold after the critical time, therefore the management system will use discount to
stimulate demand. Then a proposed model is formulated as a nonlinear programming model. The
objective function of the proposed model is minimizing the total cost of supply chain including
the cost of fixed ordering, holding, discount, and deterioration whereas replenishment cycles and
order size for retailers and also production time needed to supply inventory of each retailer can
be determined through the proposed model. Since the model is a NP-hard problem, a Genetic
algorithm (GA) and a Particle Swarm Optimization (PSO) algorithm are developed for solving it
appropriately and the results are presented that PSO algorithm has a better performance for
solving of the proposed model in this paper.
Taguchi method is an applied to calibrate the parameters of the algorithms in to provide reliable
solution. Finally, the conclusion and further research are presented.
Key words Vendor Managed Inventory, Perishable Supply Chain, Discount, Genetic Algorithm, Particle
Swarm Optimization
1- Introduction
Management of supply chain is a set of approaches to integrate suppliers, manufacturers,
warehouses and retailers effectively, So that the product could be produced with an appropriate
amount, and distributed at right time and place. Inventory management is one of the key issues in
supply chain management. Therefore several strategies have been proposed to manage inventory.
Also demand variations in the supply chain and bullwhip effect are related to inventory
management (Simchi-Levi, D. et al, 2007, (Sadeghi et al 2014). VMI is one of the strategies for
an agile supply chain that can reduce the time of response to customers’ demand through
coordination among supply chain members. In VMI supply chain vendor is responsible to
inventory management. Clearly the vendor controls inventory level of retailers based on service
level agreements, so it can be an efficient way to manage the inventory of perishable products.
Its conceptual structure was mentioned by Magee in 1958 with the aim of clarifying the
responsible member of inventory management (Magee, J.F., 1958), (Disney, S.M et al 2003).
3
The past researches were indicating that VMI can improve the supply chain performance by
reducing inventory levels and increasing the replenishment rates (Nagoor Gani, A. and
Sabarinathan, G. 2013). Nowadays, VMI strategy is used in many supply chains at world class
production.
Sadeghi, .J et al (2014) considered a model with single vendor, single warehouse, and multi
retailers based on the VMI strategy. The vendor faced two limitations in the number of orders
and budget. The nonlinear integer programming model developed to determine the order size, the
shortest possible way and replenishment cycle for vendor and retailers. Ghare and Schrader
(1963) presented the first model which considered the deterioration factor as an effective matter
for inventory management. They proposed an EOQ model for items with exponential failure
distribution function and a deterministic demand. Then, Emmens, H. (1968) studied on the
failure of radioactive nuclear generators by using a similar approach. Covert and Philip (1973)
and Philip, G.C. (1974) developed their research by considering a failure rate variable for
perishable products. Weiss, H.J. (1980) studied on perishable inventory with fixed life. Yu, Y et
al (2012) studied a VMI supply chain with deteriorating raw material and deteriorating product.
They considered a known and fixed deterioration rate and deterministic demand of retailers.
They developed a VMI model to calculate the total cost of inventory and deterioration. In that
model, a common replenishment cycle of product and the replenishment frequency of the raw
material were considered as decision variables. In addition, they proved the convexity of the cost
function and used the golden search algorithm to solve the problem. Yu et al. (2011) didn't
consider a deteriorating raw material and studied just pricing impaction on the VMI system.
They proposed a meta-heuristic approach for solving their complex model. In contrast, Yu, Y et
al (2012) focused on a cost-based VMI system and proposed an exact analytical methodology for
problem solution.
Sankar Sana, S. (2010) studied an inventory model for perishable products in a supermarket. In
that model, the products deteriorated after a specific time with a probability distribution function.
Reducing the sale price to stimulate the customer demand was their strategy. Bramorski,T.
)2008(provided a model to determine the policies of perishable products inventory. They used a
demand stimulation to prevent deterioration by adjusting the price of the products which are
deteriorated. The purpose was to sell the entire inventory before expiration and earn higher
profits. Results of the model showed that by offering discounts to older products, customers will
be more willing to buy. Shen, D. et al (2011) studied inventory management model for
perishable agricultural products in a simple two-level supply chain. They formulated the problem
to minimizing the total cost of system and showed that collaboration between vendors and
retailers can reduce supply chain costs.
VMI strategy was utilized in previous studies for perishable products. But discount depending on
the remaining lifetime of the product is an issue that has not been considered for perishable
products in VMI models. Due to probable deterioration, it is necessary to use demand stimulation
strategies such as discounts in such models. The discount of perishable models was often a
quantity discount. The other modes of discount such as the discount related to the expiration date
had been found rarely. The reason of these gaps may be a lack of sufficient knowledge about the
capabilities of VMI strategy and proper methods for controlling the inventory in perishable
supply chain. The present study investigates issue of inventory management in a perishable
supply chain. Perish means that the product is deviated of its normal and expected performance
The value of products decreases during the time in this study such as vegetables and fruit and
4
fresh food. The lifetime of these products is considered as a random variable with probability
distribution function.
In this paper, VMI strategy is used to manage the inventory of perishable product at two-level
supply chain with single vendor and multiple retailers. It is probable that the inventory of product
is not sold after the critical time (the critical time is a moment of time that product begin to
deteriorate; in the other words, the effects of the deterioration are visible. Experts can use
indicators to measure the effects of the deterioration on the product.), therefore the management
system will use discount to stimulate demand. The problem is formulated as nonlinear
programming model to minimize the total cost of supply chain including the cost of fixed
ordering, holding, discount, and deterioration whereas replenishment cycles and order size for
retailers and also production time needed to supply inventory of each retailer can be determined
through the proposed model.
The second section of the present study explains the conditions, parameters and symbols of
problem. The third section presents the model of minimizing the costs of the two-level supply
chain for perishable products with a single vendor and multi-retailer. Since the model is a NP-
hard problem, the fourth section explains meta-heuristic algorithms including GA and PSO
algorithms to solve the problem and using of the Taguchi Method to calibrate meta-heuristic
algorithms parameters. The fifth section explains numerical examples, compares algorithms and
sensitivity analysis of model parameters. The sixth section presents the conclusion and future
researches.
2- Problem definition and Notation
In this section, VMI model will be discussed at a two-level supply chain including a single
vendor and multi-retailer with a perishable product which its value decreases during the time and
a random lifetime. It assumes that the deterioration of product is negligible before the critical
time (the critical time is a moment of time that product begin to deteriorate; in the other words,
the effects of the deterioration are visible. Experts can use indicators to measure the effects of
deterioration on the product.) and perish is begun after it. Vendor supplies inventory for retailers
by a fix and finite rate of production. Problem is formulated as a nonlinear programming model.
The objective function of proposed model is minimizing the total cost of supply chain including
the cost of fixed ordering, holding, discount, and deterioration. Replenishment cycles and order
size for retailers and also production time needed to supply inventory of each retailer can be
determined through the proposed model. In this paper, Deterioration cost is equal to price of
products that are deteriorated (See Yu, Y et al, (2012)). According to situation of problem,
assumption can be listed as follows:
The inventory is considered as a function of time.
Deterioration is dependent on keeping conditions.
Product can be stored at both retailers and vendor’s locations.
Demand is uncertain.
Critical time is a random variable that follows a probability distribution function.
Replenishment rate is immediate and lead time is zero.
Shortage is not allowed.
5
Deterioration cost is equal to price of products that are deteriorated (See Yu, Y et al, (2012)).
Production rate is finite.
The perishable cost and discount cost at vendor location is equal to zero.
Due to uncertainty in demand and keeping conditions, it is likely that the optimum cycle
occurs before or after the critical time, so two different modes will be defined for the
problem. The first mode is the Basic mode and second mode is the Discount mode that is
explained as follows:
Common notes are shown in table.1 and notes of Basic and Discount modes are shown in
tables.2 and 3 respectively.
Table 1: common notes of the model
note description note description
i Counter index of retailers 1,2,...,i m S Fixed ordering cost at vendor location in replenishment cycle ($)
m Number of retailers P Production rate
i Fixed part of the demand to i
-th retailer
(percent / unit / time), 0i ih Unit holding cost at i
-th retailer
location ($ / unit / time)
i Rate of change in demand due to inventory
levels for i-th retailer (percent / unit / time),
0 1i
ivh Unit holding cost at vendor location to
i-th retailer($ / unit / time)
iRP Sale Price before discount of i-th retailer ($
/ unit) ip Unit deterioration cost at i-th retailer
location ($ / unit / time)
iT Fixed ordering cost at i-th retailer location
in replenishment cycle ($) VMIZ TC Expected value of total cost of VMI
supply chain (sum of discount and
non-discount modes) per unit time ($ /
time)
iC Replenishment cycle of product of i-th retailer (day). (Decision Variables)
Basic Mode ( i iC ct ), ( 0ir ) :
Condition of problem in this mode is expressed as follows:
Product be sold before of the critical time subject to demand condition.
Replenishment cycles are smaller than critical time.
Deterioration does not happen.
There is no need to discount.
There are no deterioration and discount costs.
Cost at retailer location is expressed as the sum of fixed ordering cost and holding cost.
Cost at vendor location is expressed as the sum of fixed ordering cost and holding cost.
Notes of this section are shown in table. 2.
6
Table 2: Basic mode notes of the model
note description note description
D ( )i Basic t Demand of i-th retailer in the Basic
mode (unit /time) i Basic vHC
Holding cost at vendor location
to i-th retailer in the Basic
mode in replenishment cycle
($)
i BasicI t
Product inventory at i-th retailer
location in the Basic mode at time t
i BasicTC
Total cost at i-th retailer
location in the Basic mode per
unit time ($ / time)
i BasicQ
Order size of i-th retailer in the Basic
mode
Basic vTC
Total cost at vendor location in
the Basic mode per unit time ($
/ time)
i Basic vI t
Inventory of product at vendor location
to i-th retailer in the Basic mode at t-th
time
1iZ
Fix ordering cost at i-th retailer
location in the Basic mode per
unit time ($ / time)
i Basict
Assigned Production time to i-th
retailer in the Basic mode (s) 2iZ
Holding cost at i-th retailer
location in the Basic mode per
unit time ($ / time)
Basict Assigned Production time of all
retailers in the Basic mode (s) 3iZ
Fix ordering cost at vendor
location of i-th retailer in the
Basic mode per unit time ($ /
time)
i BasicHC
Holding cost at i-th retailer location in
the Basic mode in replenishment cycle
($)
4iZ
Holding cost at vendor location
of i-th retailer in the Basic
mode per unit time ($ / time)
Fig. 1 is shown conceptual structure of Basic mode.
Fig 1: inventory level of product at retailers and vendor location in the Basic mode
Fig.1 shows that vendor produces product to i-th
retailer in a period equal to i Basic
t
with amount
of i Basic
Q
and sends to i-th
retailer in a proper time subject to level of retailer’s inventory, in
other word vendor is responsible for managing of retailer’s inventory. For retailer, replenishment
cycle is smaller than critical time and product will be sold without discount.
Since in the Basic mode there is not deterioration, so deterioration and discount costs will not
exist.
Discount mode ( i ict C ):
Condition of problem in this mode is expressed as follows:
Product be sold both before and after of the critical time subject to demand condition.
Replenishment cycles are larger than critical time.
Deterioration happens after critical time at retailer’s location.
7
There is need to discount.
There are deterioration and discount costs.
Cost at retailer location is expressed as the sum of fixed ordering, holding, deterioration
and discount costs.
Cost at vendor location is expressed as the sum of fixed ordering cost and holding cost.
Notes of this section are shown in table. 3.
Table 3: Discount mode notes of model
note Description note description
ict Critical time at i-th retailer
location Discountt Production time of all
retailers in the Discount
mode
1( )
0
i
i
i
t ctH t ct
t ct
Function sign (function
indicates the presence or
absence of deterioration)
i DiscountHC
Holding cost at i-th retailer
location in the Discount
mode in replenishment
cycle ($)
i i iH t ct Deterioration rate of
product at i-th retailer
location (percent / unit /
time), 0 1i
i Discount vHC
Holding cost at vendor
location to i-th retailer in
the Discount mode in
replenishment cycle ($)
D ( )i discount t Demand of i-th retailer in the Discount mode (unit
/time)
i DiscountPC
Deterioration cost at i-th
retailer location in the
Discount mode in
replenishment cycle ($)
, ,i i ia b c Input parameters of selling
reduction price function due
to discount of i-th retailer
i DiscountDC
Discount cost at i-th
retailer location in the
Discount mode in
replenishment cycle ($)
ir Reduction rate of selling
price after discount of i-th
retailer (percent),
0 1ir
i DiscountTC
Total cost at i-th retailer
location in the discount
mode per unit time ($ /
time)
2
i i i i i ir a b r c r Selling reduction price
function due to discount of
i-th retailer
Discount vTC
Total cost of vendor
location in the Discount
mode per unit time ($ /
time)
i
,
0
0 0
i ii
i
i ict
e ctf ctct
Deterioration parameter of
exponential distribution
function of i-th retailer
,
Since deterioration of
product is a random
variable, it can follow any
probability distribution
function that
1f x
can be
meaningful to it. In this
5iZ
Fix ordering cost at i-th
retailer location in the
discount mode per unit
time ($ / time)
8
research, exponential
distribution function
explains the behavior of
deterioration random
variable.
i DiscountI t Product inventory at i-th
retailer location in the
Discount mode at time t
6iZ
Expected value of holding
cost at i-th retailer location
in the Discount mode per
unit time ($ / time)
i DiscountQ
Order size of i-th retailer in
the Discount mode 7iZ
Expected value of
deterioration cost at i-th
retailer location in the
Discount mode per unit
time ($ / time)
i Discount vI t
Inventory of product at
vendor location to i-th retailer in the Discount
mode at t-th time
8iZ
Expected value of
discount cost at i-th retailer
location in the Discount
mode per unit time ($ /
time)
i Discountt
Assigned Production time
to i-th retailer in the
discount mode
9iZ
Fix ordering cost at
vendor location of i-th
retailer in the Discount
mode per unit time ($ /
time)
10iZ
Expected value of holding cost at vendor location of i-th retailer in the
Discount mode per unit time ($ / time)
Fig. 2 is shown conceptual structure of Discount mode.
Fig 2: inventory level of product at retailers and vendor location in the Discount mode
Fig.2 shows that vendor produces product to i-th
retailer in a period equal to i Discount
t
with
amount of i Discount
Q
and sends to i-th
retailer in a proper time subject to level of retailer’s
inventory, in other word vendor is responsible for managing of retailer’s inventory. For retailer,
replenishment cycle is larger than critical time. Before the critical time, there is no deterioration
and after critical time, deterioration begins gradually and randomly. Therefore, the gradient of
change demand will be faster by discount (see Fig. 2). Before the critical time, the product is sold
without discount and after it, the product is sold with discount. Since in the Discount mode there
is deterioration, so deterioration and discount costs will exist at retailer location.
3- The Proposed Model
9
In this section, Basic mode and Discount mode are formulated as a mathematical model
individually according to the description of the previous section and Fig.1 and 2. Total cost of
VMI system is formulated by the sum of costs in the Basic and Discount modes.
3-1 Basic Mode
Total cost of i-th retailer in Basic mode is formulated by Eq.(1) to (6). Demand can be
considered related or not related to inventory. According to Sankar Sana, S. (2010) in places like
supermarkets high level of inventory it can encourages customers to buy more and more. In fact
more inventory gives the customer a wider selection. Demand in Eq. (1) is related to i and
( )i BasicI t .
D ( ) ( ) 0 1,2,..., 0 1 0i Basic i i i Basic i i it I t t C i m (1)
Subject to description of Eq. (1) change of inventory in each moment is related to i (fixed part
of demand) and inventory at retailer’s location ( ( )i BasicI t). By increasing i and i
(dependence of demand and inventory to each other), the change of the inventory ( ( )i BasicdI t
dt )
will be increased and the inventory ) ( )i BasicI t ( will be decreased. So using the negative sign for
i and i in Eq. (2) is reasonable (for more information see Sankar Sana, S. (2010)).
( )( ) 0 (0) Q ( ) 0 1,2,...,i Basic
i i i Basic i i Basic i Basic i Basic i
dI tI t t C I I C i m
dt
(2)
By Eq. (2):
/( ) i i i i Basi basi i i ic cexpI t t Q 1,2,...,i m (3)
By replacing the ( ) 0i Basic iI C in Eq. (3), order size of i-th retailer is obtained by Eq. (4) as
below:
/Q 1i i ii Basic iexp C 1,2,...,i m (4)
Holding cost of i-th retailer in a replenishment cycle is obtained by Eq. (5) as below:
0
i
iC
i Basic i BasicHC h I t dt
1,2,...,i m
(5)
10
As mentioned in section 2, in this mode at retailer location replenishment cycle is smaller than
critical time, so there are no deterioration, deterioration cost and discount cost.
Total cost of i-th
retailer per unit time is obtained by sum of fixed ordering cost and holding cost
at i-th
retailer location by Eq. (6) as bellow:
(6)
1
ii Basic i i Basic
TC T HCC
1,2,...,i m
Total cost of vendor in Basic mode is formulated by Eq.(7) to (13). Yu, Y et al (2012) proposed
the change of inventory at vendor location equal to production rate and deterioration rate of
inventory. (For more information See subsection 4-2, Eq. (8) to (10) of their paper)
In some studies about perishable products, the critical time was considered for product (See
Sankar Sana, S. (2010)). Since our model in this paper is focused on discount at retailer location,
so we considered the perishable cost and discount cost at vendor location equal to zero.
Therefore the change of inventory at vendor location is equal to production rate.
Change of Inventory at vendor location is obtained by Eq. (7).
(7)
( )i Basic vdI t
Pdt
0 i Basic vt t
1,2,...,i m
At the beginning of production, inventory is zero at vendor location:
(0) 0i Basic v
I
1,2,...,i m (8)
By Eq. (7) and (8):
( )i Basic v
I t Pt
0 i Basic vt t
1,2,...,i m (9)
That i Basic vt is equal to production time at vendor location to i
-th retailer; therefore subject to
( )i Basic v i Basic i Basic
I t Q
and i BasicQ in equation (4):
(10)
/ 1i i i i
i Basic
exp Ct
P
1,2,...,i m
Production time for verifying the demand of all retailers is obtained by Eq. (11):
11
1
/ 1i i i im
Basici
ext
P
p C
(11)
Holding cost at vendor location to i-th retailer in a replenishment cycle is obtained by Eq. (12):
0
iv
i Basic
i Basic v i Basic v
t
HC h I t dt
1,2,...,i m
(12)
The perishable cost and discount cost at vendor location is equal to zero.
Total cost at vendor location per unit time is obtained by sum of fixed ordering cost of vendor
and the sum of holding cost of i-th
retailer at vendor location by Eq. (13) as bellow:
1
1i
m
Basic v i Basic vi
TC S HCC
(13)
3-2 Discount Mode ( i ict C )
Total cost of i-th retailer in Discount mode is formulated by Eq.(14) to (23). Demand of i-th
retailer before the critical time is the sum of fix section of demand and inventory level as
mentioned in section 3-1. After the critical time and start of deterioration, the demand is a
function of price indirectly. In fact, a manager stimulates demand by reduction of price. Another
word, the reduction of price through increasing discount will be caused increasing the demand.
Therefore demand is dependent on price indirectly (Sankar Sana, S. (2010)). Demand is
described by Eq. (14) as below:
2
( ) 0 1,2,..., 0 1 0D ( )
( ) 1,2,..., 0 1
i i i Discount i i i
i Discount
i i i i i i i i i
I t t ct i mt
r a b r c r ct t C i m r
(14)
Change of inventory in each moment before the critical time is related to i and ( )i DiscountI t as
mentioned in section 3-1. After the critical time by increasing deterioration rate ( i ) and
discount (that is a function of reduction price ( ir ), the inventory level is decreased through
Eq. (16). (See Sankar Sana, S. (2010)).
( ) 0 (0) Q 1,2,...,( )
( ) ( ) 0 1,2,...,
i i i Discount i i Discount i Discounti Discount
i i Discount i i i i Discount i
I t t ct I i mdI tdt I t r ct t C I C i m
(15)
(16)
12
By using Eq. (15) and (16), ( )i DiscountI t is obtained by Eq. (17) and (18) as below:
(17) (18)
0 1,2, /
...,( )
1,2,./ ..,
i
i Dis
i i i i Discount i i
i i i i i i
count
i i
exp t Q
r r exp C
t ct i mI t
ct t C ip t mex
By using Eq. (17) and (18), order size of i-th retailer is obtained by Eq. (19) as below:
(19)
( )( )Q 1 1ict i iiii
i Discounti i
i i iC ct ctr
e e e
1,2,...,i m
Holding cost of i-th retailer in a replenishment cycle is obtained by Eq. (20) as below:
(20) 0 0
i i
i i i
i
C ct C
i Discount i discount i Discount i Discountct
HC h I t dt h I t dt I t dt
1,2,...,i m
In this paper, the deterioration cost is equal to price of products that are deteriorated. (See Yu, Y
et al, (2012)) .Deterioration cost of i-th retailer in a replenishment cycle is obtained by Eq. (21)
as below:
i i i Discount
i
i
C
i Discountct
PC p I t dt 1,2,...,i m
(21)
The Discount cost is considered as a difference of price between before of discount and after of
discount multiplied by demand that will be sold at discounted prices. Discount cost of i-th
retailer
in a replenishment cycle is obtained by Eq. (22) as below:
1i Discount i i i i Discount
i
i
C
ctDC RP RP r D t dt 1,2,...,i m
(22)
Total cost of i-th
retailer per unit time is obtained by sum of fixed ordering cost at i-th
retailer
location, holding cost, deterioration cost and discount cost of i-th
location by Eq. (23) as bellow:
1i Discount i i Discount i Discount i Discount
i
TC T HC PC DCC 1,2,...,i m (23)
Total cost of vendor in Discount mode is formulated by Eq.(24) to (30). Change of Inventory at
vendor location is obtained by Eq. (24).
( )i Discount vdI tP
dt 0 i Discountt t 1,2,...,i m
(24)
13
At the beginning of production, inventory is zero at vendor location.
(25) (0) 0i Discount vI 1,2,...,i m
By using Eq. (24) and (25):
( )i Discount vI t Pt 0 i Discountt t 1,2,...,i m (26)
That i Discountt is equal to production time at vendor location to i
-th retailer; therefore
( )i Discount v i Discount i DiscountI t Q and i DiscountQ in Eq. (19):
( ) ( )1 1
/
ii iii
i i
i Discount
i ii iC ctct ct
i Discount
re e e
t Q PP
1,2,...,i m
(27)
Production time for verifying the demand of all retailers is obtained from (27):
1
( ) ( )
1
1 1ii iii
m i i
i
i ii iC ctct ct
m
Discount i Discounti
re e e
t tP
(28)
Holding cost at vendor location for i-th retailer in a replenishment cycle is obtained by Eq. (29):
0
iv
i Discount
i Discount v i Discount v
t
HC h I t dt
1,2,...,i m
(29)
Subject to what was said in section 3-1, deterioration and discount costs will not exist at the
vendor location.
Total cost at vendor location per unit time is obtained by sum of fixed ordering cost of vendor
and the sum of holding cost of i-th
retailer at vendor location by Eq. (30) as bellow:
(30)
1
1i
m
Discount v i Discount vi
TC S HCC
3-3 Total cost of VMI system
14
The sum of mathematical expectation of costs in Basic mode and Discount mode provides the
total cost of VMI system.
Fix ordering cost at i-th retailer location in the Basic mode per unit time ($ / time)
(31)
1
/ ii iZ T C 0 i ict C 1,2,...,i m
Holding cost at i-th retailer location in the Basic mode per unit time ($ / time)
21/ i i Basici
Z C HC 0 i ict C 1,2,...,i m (32)
Fix ordering cost at vendor location of i-th retailer in the Basic mode per unit time ($ / time)
3/ ii
Z S C 0 i ict C 1,2,...,i m (33)
Holding cost at vendor location of i-th retailer in the Basic mode per unit time ($ / time)
(34) 4
1/ ii i Basic vZ C HC
0 i ict C 1,2,...,i m
Fix ordering cost at i-th retailer location in the discount mode per unit time ($ / time)
5/i ii
TZ C 0 i ict C 1,2,...,i m (35)
Expected value of holding cost at i-th retailer location in the Discount mode per unit time ($ /
time)
61/ C
0
i
i i Discounti i i
CZ HC f ct d ct
0 i ict C 1,2,...,i m
(36)
Expected value of deterioration cost at i-th retailer location in the Discount mode per unit time ($
/ time)
(37) 71/ C
0
i
i i Discounti i i
CZ PC f ct d ct
0 i ict C 1,2,...,i m
Expected value of discount cost at i-th retailer location in the Discount mode per unit time ($ /
time)
15
81/ C
0
i
i i Discounti i i
CZ DC f ct d ct
0 i ict C 1,2,...,i m
(38)
Fix ordering cost at vendor location of i-th retailer in the Discount mode per unit time ($ / time)
(39)
9
/ iiS CZ 0 i ict C 1,2,...,i m
Expected value of holding cost at vendor location of i-th retailer in the Discount mode per unit
time ($ / time)
(40) 101/ C
0
i
i i Discount vi i i
CZ HC f ct d ct 0 i ict C 1,2,...,i m
VMIZ TC : Expected value of total cost of VMI supply chain (sum of Basic and Discount
modes) per unit time ($ / time)
(41)
(42)
VMITCMinZ
1 2 3 4 5 6 7 8 9 10
1 1
m m
i i i i i i i i i i
i i
Z Z Z Z Z Z Z Z Z Z
. : S t
0 10iC 1,2,...,i m
The objective function is demonstrated by sum of costs in the Basic mode and Discount mode. In
other words, any retailer can sell the product with or without discount based on the uncertainty of
demand and keeping condition.
4- Solution Procedures and Parameter Tuning The proposed model is a complex nonlinear problem which cannot be solved in a reasonable
time. In addition, decision variables are placed as non-linear form in objective function. In such
condition, meta-heuristic algorithms will be more efficient than exact methods and will be
converged to near-optimal solution in a reasonable time (Deep, K. Das, K. 2013).
Since the model was a NP-hard problem, a Genetic algorithm (GA) and a Particle Swarm
Optimization (PSO) algorithm were developed for solving it appropriately and results were
presented that PSO algorithm has better performance for solving of proposed model in this
paper.
Various examples were generated. Parameters tuning was performed for both algorithms. At
first, the model was solved by GA algorithm and then PSO algorithm was used to evaluate the
quality of GA algorithm solutions. The solutions of GA and PSO algorithms were compared with
16
each other by CPU time, Objective Function and RPD (%) criteria and the results showed that
PSO algorithm has better performance.
GA is an evolutionary algorithm which provides near optimal solution for optimization problem
after being inspired by the concepts of mutation, inheritance and search for answers in feasible
area. In the first step of GA, a set of feasible solutions are randomly generated which is called
initial population. Every member of the population is called a chromosome and the chromosome
represents a feasible solution of problem. A set of chromosomes that are produced in one
iteration of the algorithm is called a generation. In every generation, eligible chromosomes
(parents) are selected for the production of the next generation according to the fitness value
((Aryanezhad, M. B. et al 2012), (Pasandideh & Niaki, 2008), (Michalewicz 1996), (Goldberg
1989), (Chang, Y.H. 2010), (Al-Tabtabai & Alex, 1999),(Shahsavar, Niaki, & Najafi, 2010)).
PSO is a population-based algorithm inspired by collective behavior of birds or fish. PSO
searches the feasible solutions and tries to find a near-optimal solution to the optimization
problem. In the first step of PSO, a set of feasible solutions is generated randomly and is called
initial population. Every member of the population is called a particle which represents a feasible
solution for the problem. A set of particles that are produced in every step of the algorithm is
called a swarm. PSO uses a vector procedure to find the solution. The particles are moved based
on linear combination of the global best solution vector and the local best solution vector that is
called velocity vector. These combinations are generated randomly. In PSO, primary particles
are generated randomly. Then the global best solution is determined by evaluating the fitness of
them. According to the global best solution, speed of particle’s movement will be determined to
generate the new swarm. After a series of movements in the particles, fitness of their current
position will be evaluated. And algorithm is repeated if it does not satisfy stopping criteria ((Sue-
Ann,G. et al 2012), (Haupt RL, Haupt SE.2004)).
4-1 The Proposed Genetic Algorithm
Chromosomes
In this research, the chromosome is shown in the form of a matrix with1 m size. Where “m” is
the number of retailers and every element of the chromosome represents the replenishment cycle
related to the retailer (decision variables). Fig.3 shows an example of a chromosome in a
problem by 4 retailers. For example, in Fig.3, the number of first element is equal to 2.724. It
shows replenishment cycle of first retailer. Replenishment cycle to each retailer limits by lower
bound of zero and upper bound of ten. It produces as random real string.
Fig 3: an example chromosome in a problem by 4 retailer
Evaluation
When GA is used for optimization of problem, a fitness value must be allocated to the produced
chromosome (solution). In this research, fitness value is the same as objective function value
because the problem does not have any restriction except the logical restriction which determines
the upper bound of the decision variables. The logical restriction is considered at the time of
producing the chromosomes. So, the fitness value is evaluated after each chromosome that is
produced.
17
Initial population
In this step, a set of chromosomes is randomly generated in the authorized bound of decision
variables. Their objective function is evaluated and then, they are sorted based on it.
Crossover:
Two chromosomes (parents) must be selected for crossover process. In this research, the
program asks the user to use one of the roulette wheel, tournament, and random methods. In
crossover, one of the single-point, double-point, or uniform methods may be selected by using
the roulette wheel. According to the selected method, offspring are produced by exchanging
some genes and their fitness value will be evaluated for the next generation. Fig. 4 shows
example of the crossover with uniform methods.
Fig 4: an example of uniform crossover in a problem by 4 retailer
Mutation
The utilization of mutation operator helps to search the neighbor solutions of present solution
and prevent the creation of local optimal solution. In the mutation process, a certain number of
the genes of a chromosome are selected and the mutation operator is performed on them
according to the type of decision variables. After mutation, the fitness value of chromosomes
will be evaluated. Fig.5 shows an example of mutation operator.
Fig 5: an example of mutation in a problem by 4 retailer
Iteration of generations
After mutation and fitness evaluation of created chromosomes, the population are merged and
sorted based on the fitness value. Then, some members of population are selected for the next
generation according to number of the initial population. In addition, the best present iteration
solution and the objective function value are determined.
Stopping criteria
In each iteration, the stopping criterion will be checked and if satisfied, the algorithm stops.
Algorithm stop means that the user has been consented to the convergence of the algorithm. In
other words, the user has been satisfied with the best solution founded up to now. In this
research, stopping criteria is an upper bound of iteration and is an input parameter, so before
stopping the algorithm, generation, crossover and mutation are repeated.
4-2 The Proposed Particle Swarm Optimization Algorithm
In this research, the particle is shown in the form of a matrix with1 m size. Where “m” is the
number of retailers and every element of the particle represents the replenishment cycle related
to the retailer. Fig.6 shows an example of a particle in a problem by 4 retailers. For example, in
Fig.6, the number of first element is equal to 3.801. It shows replenishment cycle of first retailer.
Replenishment cycle to each retailer limits by lower bound of zero and upper bound of ten. It
produces as random real string.
18
Fig 6: an example particle in a problem by 4 retailer
By evaluating the fitness of each solution (objective function of each solution in this study), the
local best solution and the global best solution are updated. Then, the particle velocity is updated
that it is shown in equation (43).
k k k k k k
1 1 2 2lb gbV (gen +1) = C ×l ×(x - x (gen)) + C ×l ×(x - x (gen)) + W×V (gen) (43)
kV (gen +1) : Velocity vector of k
-th particle in gen+1
-th iteration
1 2, 0c c : Acceleration coefficients that control the improvement of particles’ position
1l , 2
l : Uniform random numbers that were used in uniform function in the (0,1) interval.
klb
x : Vector of the local best position for k-th
particle
kx (gen) : Position vector of k-th
particle in gen-th
iteration kgb
x : Vector of the general best position for k-th
particle
kV (gen) : Velocity vector of k-th
particle in gen-th
iteration
w :Coefficient of inertia 0 1w
The inertia coefficient is used to balance the local and the global solutions. By considering a
large inertia coefficient, an algorithm is converged to global search, whereas by considering a
small inertia coefficient, the algorithm obtains a local search (Shi and Eberhart 1998a,b).
The position of the particle, the local and the global best positions and velocity vector are
updated in every iteration and algorithm is repeated if it does not satisfy the stopping criterion.
The best solution is stored in each iteration. In this research, stopping criteria is an upper bound
of iteration that is an input parameter.
4-3 Parameter Tuning
Since the Meta-heuristic parameters impress the quality of solution, it seems reasonable to use
Parameter tuning methods including the Taguchi method . In quality control, running of
algorithms is called process. Model and fitness function are input and process is output. The
values of the input parameters can be tuned by using the experimental design (Sadeghi, J. et al
2014).
4-3-1 Taguchi Method
Considering different factors with different levels is called factorial experiment of design. In
order to reduce the number of experiments, Taguchi developed the fractional factorial
experiment (FFEs) Roy, R (1990), Cochran W.G. and Cox, G.M. (1992). Taguchi divided the
effective factors of experiment into two parts: controllable, S (signal) and uncontrollable, N
(noise). Using the orthogonal arrays, Taguchi studied a large number of factors (decision
19
variables) with a small number of experiments (Mousavi, S.M et al 2013). In other words,
maximizing the rate of S / N in the design of experiments is his purpose (Sadeghi, J. et al 2014).
In this research, the cost function is utilized as a response variable with a less -better preference.
The rate of S / N should be greater- better and its formula will be calculated in Eq. (44) (Phadke,
M.S. 1989):
(44)
2110log1
i
nS y
N n i
iy : Response variable
n : Number of iteration
Taguchi method is used in this research because it is a cost-effective and labor-saving method
(Mousavi, S.M et al 2013). Finally, it is recommended to study Taguchi et al. (2005) in order to
get familiar with more details of Taguchi method.
4-3-2 Implementation of Taguchi Method
Since the present study uses two meta-heuristic algorithms including GA and PSO to solve the
problem, two sets of parameters are needed to calibrate. The parameters in need of calibration
and their levels are shown in Table 4 and 5. Based on the number of factors and their levels,
Minitab software proposed 27 experiments for GA and 16 experiments for PSO algorithm. To
obtain reliable results, each experiment is run 5 times and their average are used to calibrate
parameters. Fig. 7 and 8 show S / N ratio of GA and PSO algorithm respectively.
Table 4: parameters of GA algorithm
Level
Factor
1 2 3 Calibrated level
of parameter
Max It 100 200 300 200
N Pop 150 75 50 75
Pc 0.3 0.5 0.7 0.7
Pm 0.1 0.2 0.3 0.3
mu 0.1 0.15 0.2 0.1
Parent selection
method (A)
Random selection Roulette wheel
selection
Tournament
selection
Random
selection Probability selection
of single point
crossover (B)
0.05 0.15 0.25 0.15
Probability selection
of double point
crossover (C)
0.05 0.15 0.25 0.25
Table 5: parameters of PSO algorithm
Level
Factor
1 2 3 4 Calibrated level of
parameter
Max It 100 200 300 375 375
20
N Pop 150 75 50 40 40
Phi 1= Constriction Coefficient 2 2.05 2.1 2.15 2.15
Phi 2 = Constriction Coefficient 2 2.05 2.1 2.15 2.15
Fig 7: The mean S/N ratio plot for each level of the factors for GA
Fig 8: The mean S/N ratio plot for each level of the factors for PSO algorithm
5- Numerical Examples, Comparison of Algorithm and Sensitivity
Analysis
Since the model was a NP-hard problem, a Genetic algorithm (GA) and a Particle Swarm
Optimization (PSO) algorithm were developed for solving it appropriately. The results were
presented that PSO algorithm has better performance for solving of the proposed model in this
paper.
8 examples of different sizes including 4, 5, 10, 12, 20, 25, 30 and 35 retailers were generated.
Parameters tuning was performed for both algorithms. At first, the model was solved by GA
algorithm and then PSO algorithm was used to evaluate the quality of GA algorithm solutions.
The solutions of GA and PSO algorithms were compared with each other by CPU time,
Objective Function and RPD (%) criteria and the results showed that PSO algorithm had better
performance.
RPD (Relative Percentage Deviation) is one of the criteria to compare the efficacy of meta-
heuristic algorithms. In fact RPD shows deviation of algorithm solutions in different iterations
against the best solution that obtained from algorithm iterations (See Naderi et al 2009).
The RPD is calculated by Eq.(45).
(45) lg
100sol sol
sol
A MinRPD
Min
lgsolA is an objective function value that the algorithm obtains for each of the problems and
solMin is the best solution for every problem. According to equation (45), it is clear that the less
RPD is better. In other words, the smaller amounts of RPD shows that the algorithm is more
efficient approximately. Table (6) shows RPD of PSO is smaller than RPD of GA in all
examples. Considering the RPD criteria, one can conclude that PSO algorithm is more efficient
than GA to solve the presented problem.
Each example is run 10 times by GA and PSO algorithms and the average of CPU time,
Objective Function and RPD (%) is shown in Table 6. Fig. 9, 10, and 11. Amount of CPU time
and Objective Function and RPD (%) are compared in different examples respectively. Table 6
shows CPU time of PSO is smaller than CPU time of GA in all examples and Objective Function
of PSO is smaller than Objective Function of GA in all examples expect the 20 retailer example
and RPD of PSO is smaller than RPD of GA in all examples. Therefore PSO algorithm offers a
better solution and converges to optimal solution earlier than GA algorithm in all examples
expect the 20 retailer example. In general, one can state that PSO is a better algorithm for solving
the proposed model of VMI perishable supply chain.
21
Table 6: Results of comparison algorithms
Size of
problem
Number
of retailer
PSO algorithm GA algorithm
($/time(s))
CPU time (S) RPD (%) ($/time(s)) CPU time
(S)
RPD
(%)
small 4 5816.8661 111.3496 0.0001 5988.5471 218.6072 0.2374
5 6279.1048 214.2798 0.0000 6797.1247 236.8029 0.0020
medium 10 13096.1703 209.3444 0.0020 14308.2062 225.8329 0.0411
12 16141.4493 217.6922 0.0001 17628.1238 293.9971 0.5238
large 20 28245.2116 342.1873 0.6209 26956.1473 391.8357 0.8129
25 31813.4152 126.1809 0.0000 32934.5104 276.0179 0.0076
30 39014.2536 261.7495 0.0232 40138.9137 310.2514 0.5014
35 47151.9281 252.6588 0.0415 48397.2659 316.3622 0.2066
Fig 9: comparison of GA and PSO algorithms according to CPU time (s)
Fig 10: comparison of GA and PSO algorithms according to TC $ /VMI time s
Fig 11: comparison of GA and PSO algorithms according to RPD (%)
It should be noted that all calculations were done on a personal computer with Intel (R) Core
(TM) i7- 2670QM CPU @ 2.20GHz, 8.00 GB RAM memory and MATLAM R2014a and
Minitab 17 software.
The results of the analysis in this research show that PSO algorithm is more efficient than GA.
So sensitivity analysis of i , i
r , i and i input parameters on the replenishment cycle of
retailers and total cost of VMI supply chain are investigated by PSO algorithm. It should be
noted that sensitivity analysis is performed on the first retailer in a problem by 35 retailers. To
find the sensitivity analysis of 1 parameter, the model is run 10 times for each example and the
average of results is shown in Fig. 12 and 13. By increasing 1 parameter,
1C will be decreased
and VMITC ($/time(s)) will be had an increasing trend. In other words, by increasing 1
from
0.28125 to 0.78125, the amount of VMITC ($/time(s)) will be increased from 47556.1724 to
47799.3237 and the amount of 1
C will be decreased from 5.7351 to 4.9967.
Fig 12: The trend of 1
C versus the change of 1 in a problem by 35 retailers
Fig 13: The trend of VMITC ($/time(s)) versus the change of 1 in a problem by 35 retailers
To find the sensitivity analysis of 1r parameter, the model is run 10 times for each example and
the average of results are shown in Fig. 14 and 15. By increasing 1r parameter,
1C will be
decreased and VMITC ($/time(s)) will be increased. In other words, by increasing 1r from 0.2000 to
22
0.9000, the amount of VMITC ($/time(s)) will be increased from 47341.6882 to 47892.5671 and
the amount of 1
C will be decreased from 7.0627 to 6.1051.
Fig 14: The trend of 1
C versus the change of 1r in a problem by 35 retailers
Fig 15: The trend of VMITC ($/time(s)) versus the change of 1r in a problem by 35 retailers
To find the sensitivity analysis of 1 parameter, model is run 10 times for each example and the
average of results is shown in Fig. 16 and 17. By increasing 1 parameter, 1
C will be decreased
and VMITC ($/time(s)) will increase. In other words, by increasing 1 from 0.2000 to 0.9000, the
amount of VMITC ($/time(s)) will be increased from 48226.5993 to 49109.6995 and the amount of
1C will be decreased from 8.0586 to 3.9014.
Fig 16: The trend of 1
C versus the change of 1 in a problem by 35 retailers
Fig 17: The trend of VMITC ($/time(s)) versus the change of 1 in a problem by 35 retailers
To find the sensitivity analysis of 1 parameter, the model is run 10 times for each example and
the average of results is shown in Fig. 18 and 19. By increasing 1 parameter, 1
C will be
decreased and VMITC ($/time(s)) will be increased. In other words, by increasing 1 from 0.0500
to 0.1000, the amount of VMITC ($/time(s)) will be increased from 48339.4813 to 48399.3476 and
the amount of 1
C will be decreased from 7.0740 to 6.8326.
Fig 18: The trend of 1
C versus the change of 1 in a problem by 35 retailers
Fig 19: The trend of VMITC ($/time(s)) versus the change of 1 in a problem by 35 retailers
Four under investigation parameters ( i , i
r , i and i ) for the first retailer in a problem by 35
retailers, are decreased ten percent and are increased ten percent for drawing tornado diagram.
The total costs of VMI supply chain and replenishment cycle are investigated by PSO algorithm
because the results were shown that PSO is a better algorithm for solving the proposed model.
To find the impact of the decreasing and increasing of the parameters, the model is run 10 times
for each example and the average of results is shown in Table. 7 and Table. 8. Also tornado
diagrams are shown in Fig. 20 and Fig. 21 for Total cost of VMI supply chain and replenishment
cycle respectively.
23
Table 7: amount of parameters and objective functions to drawing the tornado diagram of VMITC
The amount of parameter VMITC ($/time(s))
Reduction
of ten
percent
The base
amount of
parameter
Increment
of ten
percent
Average of
objective
function of
reduced
parameter
Average of
objective
function of
based parameter
Average of
objective
function of
Increased
parameter
1 0.1980 0.2200 0.2420 48350.2559 48350.2559 48382.4249
1r 0.2025 0.2250 0.2475 48342.5279 48350.2559 48358.2920
1 0.2700 0.3000 0.3300 48310.3992 48350.2559 48387.8731
1 0.05175 0.0575 0.06325 48342.0396 48350.2559 48358.2399
Table 8: amount of parameters and replenishment cycles to drawing the tornado diagram of
replenishment cycle The amount of parameter replenishment cycle (day)
Reduction
of ten
percent
The base
amount of
parameter
Increment
of ten
percent
Average of
replenishment
cycle of reduced
parameter
Average of
replenishment
cycle of based
parameter
Average of
replenishment
cycle of
Increased
parameter
1 0.1980 0.2200 0.2420 7.0286
7.0286
6.9464
1r 0.2025 0.2250 0.2475 7.0593
7.0286
6.9973
1 0.2700 0.3000 0.3300 7.3201
7.0286
6.7588
1 0.05175 0.0575 0.06325 7.0631
7.0286
6.9961
Fig 20: Tornado diagram of VMITC for
i ,
ir , i and
i parameters
Fig 21: Tornado diagram of replenishment cycle for i ,
ir , i and
i parameters
Table. 7 and 8, fig. 20 and 21 show that 1 (Rate of change in demand due to inventory levels
for i-th retailer) parameter has the greatest impact on the objective function and the
replenishment cycle. This means that when the parameters are increased by a special percentage,
the variation of 1 parameter creates a greater increase in objective function and a greater
decrease in replenishment cycle. Also tornado diagrams show that after 1 parameter the model
is sensitive to 1 . In other words, by increasing the deterioration rate, the total cost of system
increases and replenishment cycle decreases significantly.
The results of the sensitivity analysis and the tornado diagrams show that the total costs of the
system and replenishment cycle are more sensitive to the changes of 1 (Rate of change in
demand due to inventory levels for i-th retailer) parameter than other parameters under
investigation. These results will have a significant impact on the decisions of the experts of
perishable supply chain.
24
6- Conclusion And Future Research
VMI as a strategy for an agile supply chain can reduce the time of response to customers’
demand through coordination among supply chain members. Also VMI can improve the supply
chain performance by reducing inventory levels and increasing the replenishment rates. In this
paper VMI strategy is used for managing the inventory of perishable product at two-level supply
chain with single vendor and multiple retailers. The problem is formulated as a nonlinear
programming model. The objective function of proposed model is minimizing the total cost of
the system. Replenishment cycles and order size for retailers and also production time needed to
supply inventory of each retailer can be determined through the proposed model.
Since the model was a NP-hard problem, a Genetic algorithm (GA) and a Particle Swarm
Optimization (PSO) algorithm were developed for solving it appropriately and the results were
presented that PSO algorithm had better performance for solving of proposed model in this
paper.
Various examples were generated. Parameters tuning was performed for both algorithms. At
first, the model was solved by GA algorithm and then PSO algorithm was used to evaluate the
quality of GA algorithm solutions. The solutions of GA and PSO algorithms were compared with
each other by CPU time, Objective Function and RPD (%) criteria and the results showed that
PSO algorithm had better performance.
Sensitivity analysis and tornado diagrams showed that by increasing the a) deterioration rate, b)
discount, c) dependence of demand and inventory to each other and d) deterioration parameter in
probability distribution function of critical time, parameters, total cost of system will be
increased and replenishment cycle will be decrease. Also the model was more sensitive to the
changes of dependence of demand and inventory to each other parameter. Finally, future
research in this area can be done as follows:
Investigating VMI model for multiple vendor case.
Comparing the Taguchi method with other tuning methods such as response surface
methodology (RSM).
Considering the budget and warehouse space constraints.
Considering the discount as a random variable.
Considering other probability distribution function for critical time.
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Highlights
The discount is considering as depend on product lifetime.
The discount is considering in VMI model.
The perishable product is considering in VMI model.
