8th International Conference of Modeling and Simulation - MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
“Evaluation and optimization of innovative production systems of goods and services”
A NEW REPLENISHMENT POLICY BASED ON MATHEMATICAL
MODELING OF INVENTORY AND TRANSPORTATION COSTS WITH
PROBABILISTIC DEMAND
Khaled BAHLOUL, Armand BABOLI, Jean-Pierre CAMPAGNE
Université de Lyon,
INSA-Lyon, Laboratoire LIESP
19 av. Jean Capelle, F-69621, France
Khaled.bahloul@insa-lyon.fr, arman.baboli@insa-lyon.fr, jean-pierre.campagne@insa-lyon.fr
ABSTRACT: The implementation of supply chain has to reduce the total cost of system, but generally each component
of a supply chain tries to find the best policy for its company and consequently tries to find a local optimum. Knowing
that the sum of local optimum cannot constitute the global optimum, it is necessary to consider all costs of system
simultaneously to find the best replenishment policy for all the components of a supply chain. This paper presents a new
approach based on mathematical modeling of total costs of system (transportation and inventory costs) in a supply
network (multi-echelon multi-product structure). Moreover, the demand in real case is often probabilistic and it has to
be taken into account. This reality justifies the necessity to consider all costs, generated by all products, in all links and
all echelons. The first part of this paper presents an overall view of the approach adopted. Then, the mathematical
model of the logistic costs is developed. In the next part, a new replenishment policy based on joint optimization is
detailed. Finally, the numerical experimentation and an example illustrate proposed approach.
KEYWORDS: Multi product Supply chain, Joint optimization, Inventory control, Transportation organization,
Probabilistic demand, replenishment policy
1
INTRODUCTION
The replenishment problem has been traditionally treated
from a multi-echelon and multi-product perspective (JenMing and Tsung-Hui 2005). A multi-echelon
replenishment problem focuses on channel coordination
issues for inventory replenishment, between upstream
and downstream components of a supply chain, with the
objective of minimizing total system costs (Sıla et al.
2005). Moreover, multi-product replenishment problems
aim to coordinate the replenishment of various items in
the same family or same category in order to reduce the
frequency of major setups and the related costs. This can
be obtained by choosing an appropriate common
replenishment frequency and lot-sizes within the family
of items (Bahloul et al. 2008). Several previous works
have studied the problem of multi-echelon, multiproduct Supply Chain. Chen et al. (Cheng-Liang et al.
2004) have studied a multi-item inventory and transport
problem with joint setup costs, referred to a joint
replenishment problem.
Traditionally, synchronization of different echelons is
carried out in a sequential way, in the sense that outputs
of the upstream echelon are regarded as inputs of the
downstream echelon. This way cannot obtain an optimal
plan for a company with more than one echelon in a
supply chain (Zhendong et al. 2007). This problem leads
the researchers to propose the integrated SCM, in which
the aim is to optimize the supply chain as a whole and
consider the planning of different echelons
simultaneously. This can allow providing an important
source of cost savings for companies’ operation
management,
particularly
for
inventory
and
transportation, which are the two most common
operations of many companies.
Huang et al. (Huang et al. 2005) studied the case of a
fixed transportation cost and a variable cost which is
linearly proportional to the volume of cartons delivered.
Since any replenishment policy implemented by the third
logistics service provider will eventually be translated
into cost to its client and the client’s customers, it is
important for everyone in the system, that the service
provider should find a good replenishment policy to
minimize the overall logistics costs.
There are some works which documented joint
optimization of transportation and inventory cost. In this
way, we present here three of most important works. The
first one, proposed by Speranza and Ukovich (Speranza
and Ukovich 1996) considers the product shipping
strategy to determine shipping frequencies in which each
product has to be shipped in a way that the sum of
transportation and inventory costs are minimized.
The second one (Bertazzi and Speranza 1999) considers
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
a periodic shipping strategy to minimize the total cost of
transportation and inventory in a network with one
origin, some intermediates and one destination with
given frequencies.
Finally the third one (Fleischmann 1999) considers the
transportation of several products on a single link when
shipment is conducted only at discrete times. It aims to
determine the timing and the quantities of the shipments
and the inventory level on a link in a specified planning
horizon to minimize total cost of transportation and
inventory.
Several authors use mathematical models to study how
postponement reduces the total inventory required for
meeting a consummator service level. A nearly paper by
(Tony and Marc 2007) demonstrates that component
commonality may result in lower prediction errors and
therefore lower levels of safety stocks, and he proposes
algorithms for grouping products in clusters that are
served by a common component.
(Lee and Yao 2003) Have proposed mathematical
models and solution algorithms for solving a multiproduct JRP (joint replenishment problem).
Nevertheless, these works focus on specific cases and
fail to present a global solution especially in the case of
multi-echelon, multi-product Supply Chain with several
links. Moreover, most works consider the demand of
final clients as a deterministic or constant demand.
This paper considers the problem of a multi-echelon
multi-product Supply Network as a joint optimization of
transportation and inventory costs with probabilistic
demand. We present a downstream supply chain
structure, see figure 1, consisting of a distribution center
and several consumption centers. Several types of
products are managed in the supply chain and all
products are replenished from the same distribution
center.
Physical flows
Information flows
Figure 1: downstream Supply Chain
The main contributions of this paper are:
•
•
•
•
Proposal of a method to integrate the probabilistic
demand.
Mathematical modeling the logistic costs functions.
Proposal for a new replenishment policy
numerical experimentation followed by a discussion
based on simulation
This paper is organized as follows: in section 2.1 we
present an overall view of our contribution based on a
mathematical modelization and propose a replenishment
policy. Next, in section 2.2 we give a method to present
a probabilistic demand. Then, in section 2.3 we model
the calculation function of logistic costs. Afterwards, we
present an optimization part in the form of. After than, in
section 3, the proposed replenishment policy is detailed.
Finally, the numerical experimentation and the
discussion of results conclude this paper.
2
2.1
MATHEMATICAL MODEL OF COST
CALCULATION
A general view of our approach
The increase of the transport and inventory costs
regarding the other logistic costs and the decrease of the
periods of inventory and delivery incite companies to
give more importance to the costs and constraints linked
to all these activities simultaneously, and manage better
the functioning of their supply network. The concept of
supply network appeared with these needs. The
optimization of the management of physical flows of
activities has started to be carried out in a simultaneous
(integrated) way, in order to minimize the total cost.
With this approach, the constraints and the costs of
coordinated activities are integrated in the same model
so as to be optimized in one single time. The
optimization linked to this approach is called “integrated
optimization”. This approach has currently become quite
common with the increasing number of relocations of
companies. The integrated approach can provide a
schedule of activities at a lower cost than that used in a
sequential approach, where the links of the chain are
independently optimized.
The gains obtained in an integrated optimization with
regard to a sequential optimization are especially
illustrated in the case of problems of the same
importance as storage and transport. On the other hand,
this approach leads to consider complex, large-sized
systems, with strong interactions. At the level of the
optimization process, it can lead to problems which are
difficult to solve within a reasonable time. This rising
complexity is due to the increase of the constraints to be
taken into account during a mathematical modeling and
also to the configuration of the objective function which
becomes more complicated to optimize. This difficulty
implies that there are more and more theoretical
researches lead on this problem.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
Knowing that joint optimization of transportation and
inventory costs for a multi-echelon, multi-product case
with a probabilistic demand is very complex, we adopt,
in the one hand, a mathematical modeling for calculation
of logistic costs, and in the second hand, we present a
new replenishment policy.
2.2
Demand modeling
In each period t, the independent random demand is
defined by a probability density function (PDF) 0, ∞ IR and by a cumulative distribution
function (CDF) . : 0, ∞ 0, 1. At each period
any received demand is charged at a price pt, even if it is
satisfied only at the next period.
Given a customer service level fixed by a company at for
example 98 %. We first try to find the rate value in the
standard normal distribution table which corresponds to
a particular (normal distribution) and conclude the
resulting value (eg. r=2.06). Based on this value and the
characteristics of distribution (eg. m=50 and 4) we
compute the safety stock quantity ss (1) and the order
point follows s (2), see Figure 2.
service level x.
2.
the transportation cost between two echelons (n, n1) incurs by the link in the echelon (n-1)
3.
the transportation time between two echelons is
constant
4.
the lead time for an order to arrive at link is constant
5.
there is not-splitting at link
6.
The replenishment at link can be calculated based on
the historical consumption
7.
Transportation quantity = ordering stock quantity
Notations:
n: number of echelons
k: number of products
j: number of periods
i: number of links
v: number of vehicles
0
0,1
…
1,9
2
2,1
…
0 0,1
0,500 0,503
0,5398
…
0,2
…
0,3
0,4
0,5
0,6
…
0,7
0,8
0,9
1
Ajin: ordering cost in period j, in the link i, at the echelon
n
hjkin: Rate of holding cost in period j of product k in the
link i, at the echelon n
…
sjkin: shortage cost in period j, of product k, in the link i,
at the echelon n
0,9803
…
…
Tk : periodic replenishment of article k
t: safety factor
r: resulting value of safety factor
Figure 2: Example for probabilistic demand
σk: standard deviation of errors
ss r σ 2.06 4 8.24 units
s m r σ m ss
50 8.24 58.24 $%&'(
yTk: stock level in the period Tk
(1)
(2)
We base on this principle model for calculating the level
of safety stock ss and the level of replenishment afterward in the section 3.
2.3
Modeling of cost function
Fcvn: transportation fixed cost of the vehicle type v to
level n
qv: Capacity of vehicle v
Vcvn: transportation variable cost of the vehicle v to level
n
Yvj=1 if the vehicle number v is used at period j, 0
otherwise
uk: Volume of product k
ss: safety stock
Assumptions:
1.
We define the stochastic demand Dk by a normal
distribution, defined by two parameters, mean and
standard deviation (m, , and the rate of customer
αvkj=1 if item k is delivered by vehicle i at period j, 0
otherwise
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
7
L
6
245 145 .45 :45
H: MH
Dk: rate of demand of product k
: OMPQH MHR QP
L: replenishment lead time
m(): average
v(): variance
This holding costs SHC is calculated from the
probabilistic demand, define by m(D) and v (D,) and the
rate of satisfaction clientele defined beforehand.
We define two types of costs:
Inventory costs:
The inventory costs can be classified in three families
((Toomey 2000) and (Zermati and Mocellin 2005)):
Ordering costs, holding costs and shortage costs. When
optimizing the decisions relative to the inventory, one
must take into account all these costs.
• The ordering costs SOC:
The ordering costs include the salaries of the personnel,
the functioning costs (buildings, offices, etc.), reception
and test costs, information systems costs and customs
costs. These costs represent about 2 to 5% of the value of
the ordered articles (Zermati and Mocellin 2005).
7
6
3
)*+ , , , -. / 0.12
245 145 .45
8
B: / H:
IIJK
2
(( N / :
Dk*: quantity to minimize
•
;
)D+ , , , , EF:.12 G
zkj=1 if item k is ordered by retailer in period j, 0
otherwise
;
-. 1 & , 9:. < 1
:45
-. 0 =>(= ; @ 1,2, … , B
•
The inventory shortage corresponds to the case where
the units available at the time of the customer's demand,
are not sufficient to satisfy this demand. The related
costs are classified into two categories: lost sales costs
and backlogging costs. In the lost sales category, if the
available units are not sufficient to fully satisfy the
demand, the unsatisfied demands are then completely
lost and the cost in this case is the "miss to gain". In the
second case, the cost will be a penalty shortage cost. The
latter includes the cost difference between satisfying the
demand at the time it occurs and the time when it is
satisfied. In both cases, some costs can be incurred like
an increase in the cost of raw materials by the use of
substitute materials, as well as the cost of buying or
renting a substitute product.
6
• The holding costs SHC:
This family of costs can be divided into two subfamilies;
les financial and functional costs. The financial costs
represent the financial interest of the money invested in
providing the stocked products. The functional costs
include the rent and maintenance of the required place,
the salaries of the employees, the insurance costs, the
equipments costs, the inter-depot transportation costs
and the obsolescence costs. This family of costs
represents about 12 to 25% of the value of the held
products (Zermati and Mocellin 2005). This means that
12-25% of the value of stocked products is charged per
year, depending on the volume and the price of products.
7
;
3
))+ , , , ,S(.:21 / T3: U H: V W
C
The ordering costs SOC: is calculated from both
constraints bound to the binary variables aj and zkj. The
binary variable aj is equal 1 if at least item k is ordered
one time (zkj=1).
The shortage costs SSC:
145 245 :45 .45
T3: T3X5: H: / B: U H: B:
The shortage costs SSC appears under the shape of the
difference between the holding stock level and the
ordered quantity. The holding stock is based on the stock
possessed for the period T-1 by adding the quantity
received during the period T, decreased by the quantity
asked and delivered for the same period.
•
Transportation costs
When products are delivered from the supplier to the
consumer, transportation costs are incurred. However, in
a practical logistic system, the transportation cost of a
vehicle includes both the fixed cost TfC and the variable
cost TvC. The fixed cost, which is considered to be a
constant sum in each period, refers to some necessary
expenses such as parking fare and rewards to the driver.
As to the variable cost, it depends mainly on the gasoil
consumed, which is related directly to the distance
travelled. In short, considering the real conditions, it is
unreasonable to assume that the transportation cost can
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
be proportional to the quantity delivered or a constant
sum.
The transportation fixed cost TfC is calculated by the
binary variable of yvj that it takes 1 if there is one vehicle
v used for the period j, 0 otherwise.
With the notations in Table 1,, it is assumed that:
•
F11<F21<Fi1,
•
v11>v21>vi1,
•
q1<q2<q3 ,
•
F2=F1+q1(v1-v2),
•
F3=F2+q2(v2-v3).
These equations are supposed to avoid any overdeclaration. Hence, the transportation cost varies
according to the order
er quantity as shown in Figure 3.
3
Figure 3: Variation of cost transportation
3
The variable cost TvC:: is calculated by the constraints
linking the variables zkj and avkj. These two variables
determine if a product k is going to be delivered for the
period j by vehicle v and also determine the quantity
delivered in each period j.
We use the same concept of transportation cost as
defined by Baboli et al. in (Baboli et al. 2007).
2007) They
assume that there are three different types of vehicles
(V.T) and the delivery for each order from warehouse to
retailer is made by a single vehicle without splitting;
these types are defined as small (S),
), medium (M) and
large (L)) and have their own fixed costs (FC), variable
costs (VC) and capacities (C).. The corresponding
transportation scheme is shown in table 1.
V.T
C
Destination
FC
VC
S
q1
1…n
F11, F12, …, F1n
v11, …, v1n
M
q2
1…n
F21, F22, …, F2n
v21, …, v2n
L
q3
1…n
F31, F32, …, F3n
v31, …, v3n
Table 1: Transportation schema
PROPOSED REPLENISHMENT POLICY:
POLICY
In our context, the economic order quantity model does
not represent the best solutions. Indeed, in the case of
probabilistic demand, it proves necessary to consider
specific situations, and thus, define specific hypotheses,
in order to take them into account relatively to logistic
cost afterwards.
In this section, we propose
ose a replenishment policy in the
form of an optimization algorithm. This method is based
on the hypotheses defined in the first part of the section
so as to determine the use context of the policy. The
principle characteristic of the method is presented in
i the
second part.
3.1
Hypotheses:
We have defined various hypotheses so as to define the
condition of applicability:
1.
The means of transport exist in three different
capacities: small, medium and large.
2.
The means of transport (trucks) are chosen
accordingly too the quantities of products to deliver
3.
The partial replenishment of a product is forbidden
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
4.
Calculating the quantities of product to replenish has
been done independently of the transportation
transport
capacity.
5.
The
replenishment
costs
are
calculated
independently to the number of references
replenished and the number of products transported.
6.
Identification of family products based on the
several qualitative and quantitative criteria of
products, such as mean
ean of consumption,
transportation conditions, price, etc.
3.2
-
An ordering level (si) : this level is calculated aca
cording to the demand distribution for each i product
-
A replenishment level (Si): this level is also calculated according to the demand distribution for each I
product.
Proposed method
In these conditions, the classical policies will be faced to
a certain problems and are likely not to be adapted if we
want to provide a low logistic cost for the following
reasons:
-
-
The type of demand: the considerable variations of
demand can cause serious shortages, which will
necessarily generate much replenishment, increasing
the ordering costs and the fixed transportation costs
for each demand
Safety stocks: these policies opt for high quantities
of safety stocks which result in two types
ty
of
problems: on the one hand, a high holding cost and,
on the other hand, a risk of expiry for the products
with a limited lifecycle.
Figure 4:: operating mode of proposed policy
Operating mode (figure 4):: The method consists in waitwai
ing for the overtaking of an ordering level si for an i
product.
This overtaking triggers replenishment for all
a the products of the same family. The quantities ordered qi* will
be added to the quantities of the current stock to reach
the replenishment level Si.
In order to take into account the hypotheses previously
presented and to reduce logistic costs, we have defined a
new policy which is based on the following principle: In
the first time, it is necessary to identify some family of
product. This can be made basing on the several
characteristics of products, such as mean of
consumption, transportation conditions, price, etc.
etc The
method based on continues review until a product
reaches to his ordering level and then replenishment all
products of the family. The ordering quantity of each
product depends to its level in stock (in hand quantity) to
reach the replenishment level.
In order to calculate the total logistic cost, we mainly
focus on equations modeled in the previous section. Two
types of logistic costs are considered: the inventory cost
and the transportation cost. In the inventory cost, we
include an ordering cost, holding
olding cost and the shortage
cost and in the transportation cost, we include a fixed
cost and the variable cost.
Principle:
The proposed (s, S) policy is mainly characterized by:
A level of safety stock (ss)) : this quantity of stock is
used to partially (temporarily) meet with probabilisprobabili
tic demand
Figure 5: algorithm of replenishment policy
First, a demand can activate the operation mode of the
policy. The satisfaction of this demand may be decrease
the in hand quantity of a product to its reordering level.
In this case, replenishment must be considered not only
for the product reached to its ordering level, but also for
fo
all other product of its family. The ordering quantity of
each product is calculated based on in hand quantity and
replenishment level,, see figure 5.
5
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
4
NUMERICAL EXPERIMENTATION AND
DISCUSSION
We have focused on comparing the replenishment policy
proposed to a reference policy used in similar conditions
as ours.
Classic reference policy (Edward et al. 2003):
The classic policy we refer to is based on two main
variables and a running logic deduced from the cases
provided by the literature which deals with problematic
very close to ours.
These two variables are: an ordering level for each
product and a replenishment level. The running logic is
defined as follows: launching the ordering for the
product to make sure the stock level reaches the ordering
level. The replenishment for this product triggers the
ordering of optimal ordered quantity for this product.
A family of 10 products is used for our
experimentations. For each simulation, a new random
demand is generated for 472 periods, using the normal
distribution with an identical mean and standard
deviation (m = σ). The following table represents the
results of various costs of twenty simulations.
Instance of model:
Varia
ble
n:
k:
j:
i:
v:
Designation
Value
number of echelons
number of products
number of periods
number of links
number of vehicles
2
10
472
2
3
Ajin
ordering cost in period j, in the
link i, at the echelon n
50€
hjkin
sjkin
rate of holding cost in period j of
product k in the link i, at the
echelon n
shortage cost in period j, of
product k, in the link i, at the
echelon n
20%
100€
Fcvn
transportation fixed cost of the
vehicle v to level n
{20€,
90€,
120€}
Vcvn
transportation variable cost of the
vehicle v to level
{3€, 2€,
1€} x Q
uk
Volume of product k
1cm3/un
it
Table 2: instance of model
This Table (table 3) presents the various logistic costs
integrating the inventory costs (ordering, holding and
shortage) and transportation costs (fixed and variable)
for policies, the proposed one vs. the reference one. We
notice that the proposed policy provides lower costs than
the reference policy.
This is due to the optimality of the proposed policy in
this specific context and the modeling of logistic costs,
which takes into account the various logistic aspects and
the different situations.
[1 .. 20] : number of simulation
m: mean of demand
σ: standard deviation
p. policy: proposed policy
r. policy: reference policy
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Demand
m
δ
4,80 4,80
6,4 6,40
6,6 6,60
4,6 4,60
6,7 6,70
6,3 6,30
5,2 5,20
4,6 4,60
4,2 4,20
4,80 4,80
4,9 4,90
6,7 6,70
4,3 4,30
4,6 4,60
5,6 5,60
6,7 6,70
3,8 3,80
5,6 5,60
5,3 5,30
5,1 5,10
mean
σ
inventory
Ordering cost
Holding cost
Shortage cost
p. policy r. policy p. policy r. policy n. policy r. policy
5 750 € 7 360 € 5 371 € 4 483 € 2 000 € 4 500 €
7 000 € 7 760 € 6 515 € 5 511 € 2 000 € 5 400 €
7 050 € 7 880 € 6 119 € 5 122 € 1 000 € 5 700 €
4 750 € 6 700 € 5 223 € 4 320 € 3 000 € 3 400 €
6 950 € 7 920 € 6 017 € 5 142 € 2 000 € 5 500 €
6 800 € 7 740 € 6 451 € 5 464 € 1 000 € 5 500 €
4 850 € 6 660 € 5 134 € 4 216 € 2 000 € 3 100 €
4 000 € 6 160 € 4 086 € 3 438 € 1 000 € 3 400 €
4 550 € 6 520 € 4 107 € 3 479 € 2 000 € 3 200 €
4 550 € 6 360 € 5 131 € 4 284 € 1 000 € 3 200 €
5 700 € 7 240 € 4 812 € 4 104 € 3 000 € 4 000 €
5 700 € 7 360 € 5 921 € 4 962 € 2 000 € 4 600 €
4 550 € 6 520 € 4 966 € 4 153 € 1 000 € 3 500 €
5 400 € 6 620 € 5 125 € 4 337 € 3 000 € 3 700 €
5 550 € 7 020 € 5 148 € 4 372 € 1 000 € 3 300 €
6 800 € 7 600 € 6 042 € 5 104 € 4 000 € 5 700 €
4 150 € 6 180 € 4 125 € 3 344 € 2 000 € 2 600 €
5 400 € 6 900 € 5 579 € 4 673 € 3 000 € 3 900 €
5 600 € 6 940 € 5 325 € 4 483 € 2 000 € 4 800 €
5 700 € 6 960 € 5 089 € 4 236 € 3 000 € 4 600 €
5 540 €
976
7 020 €
563
5 314 €
718
4 461 €
623
2 050 €
887
transportation
Fixed cost
Variable cost
n. policy r. policy n. policy r. policy
13 800 € 23 090 € 30 280 € 25 410 €
16 800 € 27 020 € 30 474 € 27 544 €
16 890 € 25 120 € 31 190 € 24 725 €
11 400 € 21 040 € 21 275 € 19 046 €
16 650 € 24 690 € 32 154 € 26 160 €
16 320 € 24 270 € 29 155 € 25 579 €
11 580 € 18 710 € 24 031 € 19 029 €
9 600 € 16 830 € 21 284 € 15 895 €
10 920 € 18 230 € 19 111 € 17 442 €
10 920 € 19 550 € 21 671 € 16 425 €
13 620 € 19 940 € 22 955 € 20 713 €
13 680 € 24 910 € 30 974 € 25 351 €
10 920 € 16 950 € 19 140 € 17 358 €
12 900 € 20 760 € 21 495 € 18 531 €
13 320 € 22 220 € 25 701 € 21 598 €
16 290 € 25 580 € 31 588 € 27 325 €
9 930 € 13 740 € 17 128 € 15 507 €
12 960 € 20 860 € 25 209 € 20 402 €
13 440 € 22 970 € 24 609 € 20 359 €
13 680 € 21 760 € 23 374 € 20 339 €
Total cost
n. policy
55 201 €
60 789 €
61 249 €
42 648 €
61 771 €
58 726 €
45 595 €
38 970 €
38 688 €
43 272 €
47 087 €
56 275 €
39 576 €
44 920 €
49 719 €
60 720 €
35 333 €
49 148 €
48 974 €
47 843 €
r. policy
64 843 €
73 235 €
68 547 €
54 506 €
69 413 €
68 553 €
51 715 €
45 723 €
48 871 €
49 818 €
55 997 €
67 183 €
48 481 €
53 948 €
58 509 €
71 309 €
41 371 €
56 735 €
59 553 €
57 894 €
4 180 € 13 281 € 21 412 € 25 140 € 21 237 € 49 325 € 58 310 €
995
2339
3429
4778
3974
8469
9279
Table 3: many cost variations
Varation of total cost for two policy
75 000 €
70 000 €
65 000 €
Total cost
60 000 €
55 000 €
50 000 €
45 000 €
40 000 €
35 000 €
Total cost n. policy
30 000 €
Total cost r. policy
25 000 €
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Number of simulation
Figure 6: variation of total cost for both policies
Qj
cost
21034
180 €
21853
187 €
23883
205 €
20512
176 €
18121
155 €
17229
985 €
15526
887 €
16256
929 €
14687
839 €
14493 Total Cost
828 €
5 371 €
Table 4: Example of calculation of holding cost for p-policy
Several values of the table 3 can be explained by the
following example. Focusing on the first line, third column we have define the ordering cost for p. policy and it
can be calculate as:
R
R
5f
cdR
)*+ , , , b, -. / 50e
245 145 :45 .45
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
8
;
-. 1 & , 9:. < 1,
:45
@ 115 C
-. 0 =>(= ; @ 472 U 115 357
In column 5, the holding cost for p. policy is calculated
as:
R
R
cdR
5f
)D+ , , , ,SF:.12 i: W
245 145 .45 :45
Table 4 presents an example of calculation of holding
cost for 10 products by proposed policy.
Column 7 presents the shortage cost for 472 periods. We
observe for the first line 10 shortages. By p-policy this
shortage cost is calculated as:
))+ 200€ 10 2000€
Column 9 presents the transportation fixed cost and it
can be calculated as:
kl+ 20€ 27 90€ 54 120€ 70 13800€
Column 11 presents the transportation variable cost and
it can be calculated as:
kn+ 3€ 17230 2€ 9980 1€ 3070
30280€
The same costs have been calculated for the reference
policy (r-policy) in the columns 4, 6, 8, 10 and 12.
Simulation aiming at showing results:
We study the feasibility of the proposed methods in
terms of logistic cost benefit, since the general objective
is to minimize these costs while keeping the best efficiency for the proposed policy.
In our probabilistic context, we resort to simulation for:
-
-
-
Calculating and comparing the logistic costs provided by proposed policy and another reference policy
Showing the benefit achieved by our proposed policy compared to the classical policy so far considered as the most efficient in our context
evaluation of the hypotheses showing that they are
all feasible
The figure (figure 6) shows the variation of total logistic
costs by carrying out twenty simulations with different
demands randomly generated for each simulation. This
variation shows the difference (gain ~ 15%) between the
two policies in their way of dealing with stock
management combined with transport management in
totally random demand.
We have to observe that the logistic costs are calculated
in the same way in both policies as they are based on the
equation models displayed in the first part of the article.
5
CONCLUSION
This paper presents in the first time, an approach to
model the cost calculation functions (transportation and
inventory costs) in a multi-echelon multi-product supply
network with a probabilistic demand. First of all, we
present an overall view of our contribution based on a
mathematical modeling and propose a new replenishment policy. Secondly, we give a method to present a
probabilistic demand. Then, we model the calculation
function of logistic costs such as ordering, holding and
shortage for the inventory costs as well as fixed and
variable transportation costs.
In the next section, we develop a new replenishment
policy which is based on the following principle: In the
first time, it is necessary to identify some family of
product. This can be made basing on the several characteristics of products, such as mean of consumption,
transportation conditions, price, etc. The method based
on continues review until a product reaches to his ordering level and then replenishment all products of the
family. The ordering quantity of each product depends to
its level in stock (in hand quantity) to reach the replenishment level.
Finally, we illustrate the propose approach by a numerical experimentation and we analyze the obtained results.
Our future works consist in implementing scenario in the
form of optimization algorithms so as to deal with specific cases in order to find a balance between the quantities carried to fill and use at its best the transport capacities; which generates inventory holding costs or minimize the stock quantities ; which generates extra transporting costs.
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