A multi-product multi-echelon inventory control model with joint
replenishment strategy
Wei-Qi Zhou
⇑, Long Chen, Hui-Ming Ge
School of Automobile and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China
a r t i c l e
i n f o
Article history:
Received 23 January 2011
Received in revised form 11 April 2012 Accepted 21 April 2012
Available online 12 May 2012
Keywords:
Inventory Multi-product Multi-echelon Genetic Algorithm (GA) Joint replenishment strategy
a b s t r a c t
On the basis of analyzing the shortages of present studies on multi-echelon inventory con-trol, and considering some restrictions, this paper applies the joint replenishment strategy into the inventory system and builds a multi-product multi-echelon inventory control model. Then, an algorithm designed by Genetic Algorithm (GA) is used for solving the model. Finally, we respectively simulate the model under three different ordering strate-gies. The simulation result shows that the established model and the algorithm designed by GA have obvious superiority on reducing the total cost of the product multi-echelon inventory system. Moreover, it illustrates the feasibility and the effectiveness of the model and the GA method.
Crown CopyrightÓ2012 Published by Elsevier Inc. All rights reserved.
1. Introduction
A supply chain is a network of nodes cooperating to satisfy customers’ demands, and the nodes are arranged in echelons. In the network, each node’s position is corresponding to its relative position in reality. The nodes are interconnected through supply–demand relationships. These nodes serve external demand which generates orders to the downstream echelon, and they are served by external supply which responds to the orders of the upstream echelon.
The problem of multi-echelon inventory control has been investigated as early as the 1950s by researchers such as Arrow et al.[1]and Love[2]. The main challenge in these problems is to control the inventory levels by determining the size of the orders for each echelon during each period so as to optimize a given objective function.
Many researchers have studied how to reduce the inventory cost of either suppliers or distributors, or have considered either the distribution system or the production system. Burns and Sivazlian[3]investigated the dynamic response of a multi-echelon supply chain to various demands placed upon the system by a final consumer. Van Beek[4]carried out a model in order to compare several alternatives for the way in which goods are forwarded from factory, via stores to the cus-tomers. Zijm[5]presented a framework for the planning and control of the materials flow in a multi-item production system. The prime objective was to meet a presanctified customer service level at minimum overall costs. Van der Heijden[6] deter-mined a simple inventory control rule for multi-echelon distribution systems under periodic review without lot sizing. Yoo et al.[7]proposed an improved DRP method to schedule multi-echelon distribution network. Diks and Kok[8]considered a divergent multi-echelon inventory system, such as a distribution system or a production system. Andersson and Melchiors
[9]considered a one warehouse several retailers’ inventory system, assuming lost sales at the retailers. Huang et al.[10]
0307-904X/$ - see front matter Crown CopyrightÓ2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.04.054
⇑Corresponding author. Tel.: +86 511 88780074; fax: +86 511 88791900.
E-mail address:[email protected](W.-Q. Zhou).
Contents lists available atSciVerse ScienceDirect
Applied Mathematical Modelling
considered a one-warehouse multi-retailer system under constant and deterministic demand, which is subjected to transpor-tation capacity for every delivery period. Lagodimos and Koukoumialos[11]developed closed-form customer service models. And many researchers have modeled an inventory system of only two-echelon or two-layer. Gupta and Albright[12]
modeled a two-echelon multi-indentured repairable-item inventory system. Axsäter and Zhang[13]considered a two-level inventory system with a central warehouse and a number of identical retailers. Axsäter[14]considered a two-echelon distri-bution inventory system with stochastic demand. Chen et al.[15]considered a two-level inventory system in which there are one supplier and multiple retailers. Tee and Rossetti[16]developed a simulation model to explore the model’s ability to pre-dict system performance for a two-echelon one-warehouse, multiple retailer system. Seferlis and Giannelos[17]developed a new two-layered optimization-based control approach for multi-product, multi-echelon supply chain networks. Hill et al.
[18] considered a single-item, two-echelon, continuous-review inventory model. Al-Rifai and Rossetti[19] presented a two-echelon non-repairable spare parts inventory system. Mitra[20]considered a two echelon system with returns under more generalized conditions, and developed a deterministic model as well as a stochastic model under continuous review for the system.
There are also many researches on multi-echelon inventory control, considering either the distribution system or the sup-ply system. Choi et al.[21]evaluated conventional lot-sizing rules in a multi-echelon coalescence MRP system. Chikán and Vastag[22]described a multi-echelon production inventory system and developed a heuristic suggestion. Bregman et al.
[23]introduced a heuristic algorithm for managing inventory in a multi-echelon environment. Van der Vorst et al.[24] pre-sented a method for modeling the dynamic behavior of multi-echelon food supply chains and evaluating alternative designs of the supply chain by applying discrete-event simulation. The model considered a producer, a distribution center and 2 re-tailer outlets. Iida[25]studied a dynamic multi-echelon inventory problem with nonstationary demands. Lau and Lau[26]
applied different demand-curve functions to a simple inventory/pricing model. Routroy and Kodali[27]developed a three-echelon inventory model for single product, which consists of single manufacturer, single warehouse and single retailer. Dong and Lee[28]considered a multi-echelon serial periodic review inventory system and 3 echelons for numerical exam-ple. The system extended the approximation to the time correlated demand process of Clark and Scarf[29], and studied in particular for an auto-regressive demand model the impact of leadtimes and auto-correlation on the performance of the se-rial inventory system. Gumus and Guneri[30]structured an inventory management framework and deterministic/stochas-tic-neurofuzzy cost models within the context of this framework for effective multi-echelon supply chains under stochastic and fuzzy environments. Caggiano et al.[31]described and validated a practical method for computing channel fill rates in a multi-item, multi-echelon service parts distribution system. Yang and Lin[32]provided a serial multi-echelon integrated just-in-time (JIT) model based on uncertain delivery lead time and quality unreliability considerations. Gumus et al.[33]
structured an inventory management framework and deterministic/ stochastic-neuro-fuzzy cost models within the context of the framework. Then, a numerical application in a three-echelon tree-structure chain is presented to show the applicabil-ity and performance of proposed framework. The model only handled one product type.
Only one other paper we are aware of addresses a problem similar to ours and consideres inventory optimization in a multi-echelon system, considering both the distribution system and the supply system. Rau et al.[34]developed a multi-echelon inventory model for a deteriorating item and to derive an optimal joint total cost from an integrated perspective among the supplier, the producer, and the buyer. The model considered the single supplier, single producer and single buyer. The basic difference between our model and Rau et al.[34]is that our model considers multiple suppliers, one producer, and multiple distributors and buyers. Additionally, an algorithm designed by Genetic Algorithm (GA) is used for solving the mod-el, and we apply the joint replenishment strategy into the model.
The remainder of this paper is organized as follows: In Section2, the various assumptions are made and the multi-product multi-echelon inventory control model is developed. In Section3, GA is used for solving the model and the algorithm based on GA is designed. Then, we simulate the model under three different ordering strategies, respectively. In Section4, conclu-sions and limitations in this research are presented.
o f g
h
i j e
Layer 1 … Layer k-2 Layer k-1 Layer k Layer k+1 Layer k+2 … Layer N
Supply Network Core Enterprise Distribution Network Fig. 1.The multi-product multi-echelon inventory control model.
2. Mathematical model
2.1. The multi-product multi-echelon inventory control model description
In this model, the raw materials, accessories or products can be supplied from the nodal enterprise of layerkto the nodal enterprise of layerk+ 1, but there is no logistics between nodal enterprises of the same layer or the non-adjacent layers. And also there is no reverse logistics from the nodal enterprise of high-layer to the nodal enterprise of low-layer. The multi-prod-uct multi-echelon inventory system is divided into three subsystems (supply network, core enterprise and distribution net-work) by the core enterprise as a dividing line (Fig. 1).
The key issue to the multi-product multi-echelon inventory system is to determine the optimal order quantity and the optimal order cycle for each nodal enterprise in order to minimize the inventory cost of the whole system.
In this paper, the (T,S) inventory control strategy based on multi-product joint replenishment is used. The multi-product joint replenishment strategy is an ordering strategy that to order varieties of products in one order cycle. Each nodal enter-prise determines a minimum order cycle as the basic order cycle, and the order cycle of the same enterenter-prise to order each product is an integral multiple of the basic order cycle.
2.2. Assumptions
(1) In this supply chain, there is only one core enterprise.
(2) Allow a variety of products, but the price of each product is fixed. And also allow a variety of raw materials or acces-sories, but one supplier only provides one raw material or accessory.
(3) The demand of each nodal enterprise per day is random, but it obeys Poisson distribution. (4) Lead time of each nodal enterprise is fixed.
(5) Storage cost per product per unit time is constant. And the storage cost of different nodal enterprises is allowed to be different.
2.3. Notations
Pw price of productw(there areWkind of products, andw= 1, 2,. . .,W)
Pgkl price of raw material or accessory provided by the nodal enterprisegof layerkl (g= 1, 2,. . .,mkl;l= 1, 2,. . .,k1;
mklis the number of nodal enterprises of layerkl)
Thk basic order cycle of the nodal enterprisehof layerkto order products from the nodal enterprises of layerk1
Tðg;hÞ
k order cycle of the nodal enterprisehof layerkto order products from the nodal enterprisegof layerk1
Zðg;hÞ
k ratio ofT ðg;hÞ
k andT
h
k, which is a positive integer, soT ðg;hÞ k ¼Z ðg;hÞ k T h k
Ahk public ordering cost of the nodal enterprisehof layerkto order products from the nodal enterprises of layerk1 in
each order cycle, which is independent of the order quantity and the order varieties
Aðg;hÞ
k individual ordering cost of the nodal enterprisehof layerkto order products from the nodal enterprisegof layerk1
in each order cycle, which is dependent of the order quantity and the order varieties
Aðh;i;wÞ
kþl individual ordering cost of the nodal enterpriseiof layerk+lto order the productwfrom the nodal enterprisehof
layerk+l1 in each order cycle, in the distribution network
Ti
kþl basic order cycle of the nodal enterpriseiof layerk+lto order products from the nodal enterprises of layerk+l1, in
the distribution network
Tði;wÞ
kþl order cycle of the nodal enterpriseiof layerk+lto order productwfrom the nodal enterprises of layerk1, in the
distribution network Zði;wÞ kþl ratio ofT ði;wÞ kþl andT i
kþl, which is a positive integer, soT ði;wÞ kþl ¼Z ði;wÞ kþl T i kþl Sðg;hÞ
k maximum inventory level of the nodal enterprisehof layerkto order products from the nodal enterprisegof layer
k1
E Dðg;hÞ k
average demand of the nodal enterprisehof layerkto order products from the nodal enterprisegof layerk1 per day
Lðg;hÞ
k lead time of the nodal enterprisehof layerkto order products from the nodal enterprisegof layerk1
Lði;wÞ
kþl the average lead time of the nodal enterpriseiof layerk+lto order the productwfrom the nodal enterprise of layer
k+l1
Hðg;hÞ
k storage cost of the nodal enterprisehof layerkper product per year
Yðg;hÞ
k quantity demand of the nodal enterprisehof layerkto order products from the nodal enterprisegof layerk1 per
year, soYðkg;hÞ¼365E D ðkg;hÞ
nðkg;hÞ the number of trips from the nodal enterprisegof layerk1 to the nodal enterprisehof layerkper year, which is inversely proportional to order cycle, sonðg;hÞ
k ¼ Z ðg;hÞ k T h k 1
fðg;hÞ
k fixed transportation cost from the nodal enterprisegof layerk1 to the nodal enterprisehof layerkin each trans-portation (such as driver’s wage)
t
ðg;hÞk variable transportation cost to transport the unit product from the nodal enterprisegof layerk1 to the nodal
enter-prisehof layerk(such as cost of fuels), which is the function of transport efficiency and order quantity in the case of fixed transportation distance
Xðh;i;wÞ
k the expected value of the producewof the nodal enterprisehof layerkrelative to order quantity of the nodal
enter-priseiof layerk+ 1
1
ðg;h;wÞk conversion rate of productwproduced by the nodal enterprisehof layerkrelative to raw materials or accessories
supplied by the nodal enterprisegof layerk1
g
ðh;i;wÞk supply coefficient of productwsupplied from the nodal enterprisehof layerkto the nodal enterpriseiof layerk+ 1,
andPmkþ1 i¼1
g
ðh;i;wÞ
k ¼1
bðkg;h;wÞ proportionality coefficient of raw materials or accessories used to produce productw, which are supplied from the nodal enterprisegof layerk1 to the nodal enterprisehof layerk, andPWw¼1b
ðg;h;wÞ
k ¼1
Bðh;i;wÞ
k shortage penalty per producewper order cycle from the nodal enterpriseiof layerk+ 1 to the nodal enterprisehof
layerk
2.4. Multi-product multi-echelon inventory control model
We divide the inventory cost into ordering cost, holding cost, transportation cost and shortage cost. (1) Ordering cost
The total ordering cost of the core enterprise per year is defined as follows: COrderC ¼ Xmk h¼1 Ah k Th k þX mk1 g¼1 Xmk h¼1 Aðg;hÞ k Zðkg;hÞT h k : ð1Þ
The total ordering cost of the supply network per year is defined as follows:
COrder S ¼ Xk2 l¼1 X mkl g¼1 Ag kl Tg kl þX k2 l¼1 X mkl1 f¼1 Xmkl g¼1 Aðf;gÞ kl Zðf;gÞ klT g kl : ð2Þ
The total ordering cost of the distribution network per year is defined as follows: COrderD ¼ XNk l¼1 X mkþl i¼1 Aikþl Tikþl þX Nk l¼1 X mkþl1 h¼1 Xmkþl i¼1 XW w¼1 Aðh;i;wÞ kþl Zði;wÞ kþl T i kþl : ð3Þ
Therefore, the total ordering cost of the multi-product multi-echelon inventory system per year is defined as follows:
TCOrder¼COrderC þC Order S þC Order D : ð4Þ (2) Holding cost
The inventory level of the nodal enterprisehof layerkwhen it has received the order quantity from the nodal enter-prise of layerk1 is:
Sðg;hÞ k E D ðg;hÞ k Lðg;hÞ k ; ð5Þ
and the inventory level of the nodal enterprisehof layerkbefore it receives the order quantity next order cycle is: Sðkg;hÞE D ðg;hÞ k Lðkg;hÞE D ðg;hÞ k Zðkg;hÞT h k: ð6Þ
Therefore, the average inventory level in one order cycle is:
1 2 S ðg;hÞ k E D ðg;hÞ k Lðg;hÞ k þS ðg;hÞ k E D ðg;hÞ k Lðg;hÞ k E D ðg;hÞ k Zðg;hÞ k T h k h i ¼Sðg;hÞ k E D ðg;hÞ k Lðg;hÞ k E Dðg;hÞ k Zðg;hÞ k T h k 2 : ð7Þ
The total holding cost of the core enterprise per year is defined as follows:
CHoldC ¼ X mk1 g¼1 Xmk h¼1 Sðg;hÞ k E D ðg;hÞ k Lðg;hÞ k E Dðg;hÞ k Zðg;hÞ k T h k 2 2 4 3 5Hðg;hÞ k : ð8Þ
As a practical matter, we must ensure that the average inventory level is greater than zero, as shown in Eq.(9): Sðkg;hÞE D ðg;hÞ k Lðkg;hÞ E Dðkg;hÞ Zðkg;hÞT h k 2 >0: ð9Þ
The total holding cost of the supply network per year is defined as follows:
CHoldS ¼ Xk2 l¼1 X mkl1 f¼1 X mkl g¼1 Sðf;gÞ kl E D ðf;gÞ kl Lðf;gÞ kl E Dðf;gÞ kl Zðf;gÞ klT g kl 2 2 4 3 5Hðf;gÞ kl; ð10Þ
under the following constraint: Sðkf;glÞE D ðf;gÞ kl Lðkf;glÞ E Dðkf;glÞ Zðkf;glÞT g kl 2 >0: ð11Þ
The total holding cost of the distribution network per year is defined as follows:
CHoldD ¼ XNk l¼1 X mkþl i¼1 XW w¼1 Sði;wÞ kþl E D ði;wÞ kþl Lði;wÞ kþl E Dði;wÞ kþl Zði;wÞ kþl T i kþl 2 2 4 3 5Hði;wÞ kþl ; ð12Þ
under the following constraint: Sði;wÞ kþl E D ði;wÞ kþl Lði;wÞ kþl E Dði;wÞ kþl Zði;wÞ kþl T i kþl 2 >0: ð13Þ
Therefore, the total holding cost of the multi-product multi-echelon inventory system per year is defined as follows: TCHold¼CHoldC þC Hold S þC Hold D : ð14Þ (3) Transportation cost
The total transportation cost of the core enterprise per year is defined as follows: CTransC ¼ X mk1 g¼1 Xmk h¼1 nðg;hÞ k f ðg;hÞ k þ
t
ðg;hÞ k Y ðg;hÞ k h i : ð15ÞThe total transportation cost of the supply network per year is defined as follows:
CTransS ¼ Xk2 l¼1 X mkl1 f¼1 X mkl g¼1 nðf;gÞ klf ðf;gÞ kl þ
t
ðf;gÞ klY ðf;gÞ kl h i ; ð16Þ wherenðf;gÞ kl ¼ Z ðf;gÞ klT g kl 1 ; Yðf;gÞ kl ¼365E D ðf;gÞ klThe total transportation cost of the distribution network per year is defined as follows:
CTrans D ¼ XW w¼1 X Nk l¼1 X mkþl1 h¼1 X mkþl i¼1 nði;wÞ kþl f ðh;iÞ kþl þ
t
ðh;iÞ kþlY ðh;i;wÞ kþl h i ; ð17Þ wherenði;wÞ kþl ¼ Z ði;wÞ kþl T i kþl 1 ; Yðh;i;wÞ kþl ¼365E D ðh;i;wÞ kþlTherefore, the total transportation cost of the multi-product multi-echelon inventory system per year is defined as follows:
TCTrans ¼CTrans C þC Trans S þC Trans D : ð18Þ (4) Shortage cost AssumingXðh;i;wÞ
k obeys Poisson distribution p k ðh;i;wÞ k Z ðg;hÞ k T h kþL ðg;hÞ k h i
during the periodZðg;hÞ
k T h kþL ðg;hÞ k , so: Xðh;i;wÞ k ¼ X1 u¼A u
g
ðh;i;wÞ k1
ðg;h;wÞ k b ðg;h;wÞ k S ðg;hÞ k p kðkh;i;wÞ Zðg;hÞ k T h kþL ðg;hÞ k h i : ð19ÞThe total shortage cost of the core enterprise per year is defined as follows: CShortage C ¼ Xmk h¼1 X mkþ1 i¼1 XW w¼1 Bðh;i;wÞ k X ðh;i;wÞ k Zðkg;hÞT h k : ð20Þ
The total shortage cost of the supply network per year is defined as follows: CShortageS ¼ Xk2 l¼1 X mkl g¼1 X mklþ1 h¼1 Bðg;hÞ kl X ðg;hÞ kl Zðf;gÞ klT g kl ; ð21Þ whereXðg;hÞ kl ¼ P1 u¼A u
g
ðg;hÞ kl1
ðf;gÞ klS ðf;gÞ kl pkðkg;hlÞ Zðf;gÞ klT g klþL ðf;gÞ kl h i ;g
ðg;hÞ kl ¼ E Dðg;hÞ klþ1 Pmklþ1 h¼1 E D ðg;hÞ klþ1 , andPmklþ1 h¼1g
ðg;hÞ kl ¼1. The total shortage cost of the distribution network per year is defined as follows:CShortageD ¼ X Nk l¼1 X mkþl i¼1 X mkþlþ1 j¼1 XW w¼1 Bði;j;wÞ kþl X ði;j;wÞ kþl Zði;wÞ kþlT i kþl ; ð22Þ where Xði;j;wÞ kþl ¼ X1 u¼A u
g
ði;j;wÞ kþl S ði;wÞ kþl p kkðiþ;j;lwÞ Zði;wÞ kþl T i kþlþL ði;wÞ kþl h i ;g
ði;j;wÞ kþl ¼ E Dði;j;wÞ kþl X mkþlþ1 j¼1 E Dðkiþ;j;lwÞ ; and X mkþlþ1 j¼1g
ði;j;wÞ k ¼1:Therefore, the total shortage cost of the multi-product multi-echelon inventory system per year is defined as follows:
TCShortage ¼CShortage C þC Shortage S þC Shortage D : ð23Þ
In conclusion, we develop the multi-product multi-echelon inventory control model as follows:
min TC¼TCOrderþTCHoldþTCTransþTCShortage; ð24Þ
s:t: E Dðg;hÞ k Lðg;hÞ k þ E Dðg;hÞ k Zðg;hÞ k T h k 2 S ðg;hÞ k <0; ð25Þ E Dðkf;glÞ Lðkf;glÞþ E Dðkf;glÞ Zðkf;glÞT g kl 2 S ðf;gÞ kl <0; l¼1;2;. . .;k2; f¼1;2;. . .;mkl1; g¼1;2;. . .;mkl; ð26Þ E Dði;wÞ kþl Lði;wÞ kþl þ E Dði;wÞ kþl Zði;wÞ kþlT i kþl 2 S ði;wÞ kþl <0; l¼1;2;. . .;Nk; i¼1;2;. . .;mkþl; w¼1;2;. . .;W; ð27Þ min Zðg¼1;hÞ k ;Z g¼2;h ð Þ k ;. . .;Z g¼mk1;h ð Þ k h i ¼1; ð28Þ min Zðf¼1;gÞ kl ;Z ðf¼2;gÞ kl ;. . .;Z ðf¼mkl1;gÞ kl h i ¼1;l¼1;2;3;. . .;k2; ð29Þ min Zði;w¼1Þ kþl ;Z ði;w¼2Þ kþl ;. . .;Z ði;w¼WÞ kþl h i ¼1;l¼1;2;3;. . .;Nk: ð30Þ
(28)–(30)can ensure that at least one product’s order cycle is the basic order cycle. The decision variables in the model are all integers greater than or equal to zero.
3. Simulation and analysis
3.1. Simulation model based on GA
The objective function of this optimization model is minimization, and the objective function of GA is maximization, so the objective function of this optimization model cannot be taken as the fitness function of GA. We must convert the objec-tive function to the fitness function of GA as follows:
FðXÞ ¼ TCmaxTC; TC<TCmax;
0; TCPTCmax;
ð31Þ whereF(X) is the individual fitness.TCmaxis a relatively large number, and in this simulation model, we may putTCmaxas the
largest objective function value during evolution.
The multi-product multi-echelon inventory control model can be reduced to a nonlinear programming problem as follows:
min fðXÞ; ð32Þ
s:t: giðXÞ60 ði¼1;2;3;. . .;mÞ:
In this paper, penalty function is used as constraint. So, we construct the penalty function as follows:
/ðX;
c
kÞ ¼X m i¼1c
k iminðgiðXÞ;0Þ 2 ; ð33Þwherekis iteration times of GA.
c
ki is penalty factor, which is a monotone increasing sequence and positive value. And
c
kþ1i ¼ei
c
ki. The experience in computation shows that ifc
ki ¼1 andei= 510, we can achieve satisfactory results. So, we change(31)to the function as follows:FðXÞ ¼ TCmaxTC/ðX;
c
kÞ; TC<TC max; 0; TCPTCmax: ( ð34Þ Moreover, we use the floating point number coding (the chromosome’s length equals the number of decision variables), the roulette wheels selection mechanism as the selection operator, the arithmetic cross technique as the crossover operator, the Gauss mutation operator as the mutation operator, and algebra (its values range from 100 to 500) as the termination criteria.3.2. Simulation
As an illustration, we develop a multi-product multi-echelon inventory control model which has four suppliers (the four suppliers are divided into two levels and each level has two suppliers), one core enterprise and two distributors, and has two products (Fig. 2). The average demand of the customers to order product 1 and product 2 from the distributor 1 of layer 4 per day is 6 units and 3 units. The average demand of the customers to order product 1 and product 2 from the distributor 2 of layer 4 per day is 4 units and 7 units. The values of other parameters are shown inTables 1–3.
Supplier 1 Supplier 2 Supplier 1 Supplier 2 Core Enterprise Distributor 1 Distributor 2
Layer 1 Layer 2 Layer 3 Layer 4
Fig. 2.The example model.
Table 2
The values of the parameters of the supply network.
Parameters A1 2 Að21;1Þ A ð2;1Þ 2 A 2 2 Að21;2Þ A ð2;2Þ 2 L ð1;1Þ 2 L ð2;1Þ 2 L ð1;2Þ 2 Values $70 $200 $180 $60 $250 $250 6 6 6 Parameters Lð2;2Þ 2 H ð1;1Þ 2 H ð2;1Þ 2 H ð1;2Þ 2 H ð2;2Þ 2 f ð1;1Þ 2 f ð2;1Þ 2 f ð1;2Þ 2 f ð2;2Þ 2 Values 6 $3 $6 $3 $15 $200 $140 $250 $150 Parameters tð1;1Þ 2 t ð2;1Þ 2 t ð1;2Þ 2 t ð2;2Þ 2 g ð1;1Þ 2 g ð2;1Þ 2 g ð1;2Þ 2 g ð2;2Þ 2 1 ð1;1Þ 2 Values $4 $6 $5 $8 1 1 1 1 1 Parameters 1ð2;1Þ 2 1 ð1;2Þ 2 1 ð2;2Þ 2 B ð1;1Þ 2 B ð2;1Þ 2 B ð1;2Þ 2 B ð2;2Þ 2 Values 0.5 1 0.5 $150 $120 $160 $140 Table 1
The values of the parameters of the core enterprise.
Parameters A1 3 Að 1;1Þ 3 A ð2;1Þ 3 L ð1;1Þ 3 L ð2;1Þ 3 H ð1;1Þ 3 H ð2;1Þ 3 f ð1;1Þ 3 f ð2;1Þ 3 Values $100 $240 $320 5 5 $5 $40 $300 $350 Parameters tð1;1Þ 3 tð 2;1Þ 3 gð 1;1;1Þ 3 gð 1;2;1Þ 3 gð 1;1;2Þ 3 gð 1;2;2Þ 3 1ð 1;1;1Þ 3 1ð 2;1;1Þ 3 1ð 1;1;2Þ 3 Values $15 $10 0.6 0.4 0.3 0.7 0.5 1 0 Parameters 1ð2;1;2Þ 3 bð 1;1;1Þ 3 bð 1;1;2Þ 3 bð 2;1;1Þ 3 bð 2;1;2Þ 3 Bð 1;2;1Þ 3 Bð 1;1;2Þ 3 Bð 1;2;2Þ 3 Bð 1;1;1Þ 3 Values 1 1 0 0.5 0.5 $150 $120 $180 $200
Table 3
The values of the parameters of the distribution network. Parameters Að1;1Þ 4 A ð1;2Þ 4 A ð1;1;1Þ 4 A ð1;1;2Þ 4 A ð1;2;1Þ 4 A ð1;2;2Þ 4 L ð1;1Þ 4 L ð1;2Þ 4 L ð2;1Þ 4 Values $140 $120 $400 $350 $420 $330 8 8 4 Parameters Lð2;2Þ 4 H ð1;1Þ 4 H ð1;2Þ 4 H ð2;1Þ 4 H ð2;2Þ 4 f ð1;1Þ 4 f ð1;2Þ 4 t ð1;1Þ 4 t ð1;2Þ 4 Values 4 $60 $40 $60 $50 $300 $300 $30 $35 Parameters gð1;1;1Þ 4 g ð2;1;1Þ 4 g ð2;1;2Þ 4 g ð1;1;2Þ 4 B ð1;1;1Þ 4 B ð1;1;2Þ 4 B ð2;1;1Þ 4 B ð2;1;2Þ 4 Values 1 1 1 1 $250 $200 $260 $180 Table 4
The decision variables of the multi-echelon inventory control model. Nodal enterprises Decision variables
Core enterprise T1 3 Zð31;1Þ Zð 2;1Þ 3 Sð 1;1Þ 3 Sð 2;1Þ 3 Supply network T1 2 T22 Zð1 ;1Þ 2 Z ð2;1Þ 2 Z ð1;2Þ 2 Z ð2;2Þ 2 S ð1;1Þ 2 S ð2;1Þ 2 S ð1;2Þ 2 S ð2;2Þ 2 Distribution network T1 4 T24 Zð41;1Þ Z ð2;1Þ 4 Z ð1;2Þ 4 Z ð2;2Þ 4 S ð1;1Þ 4 S ð1;2Þ 4 S ð2;1Þ 4 S ð2;2Þ 4 Table 5
The optimum values of decision variables under the joint replenishment strategy.
Core enterprise Decision variables T1
3 Zð31;1Þ Z ð2;1Þ 3 S ð1;1Þ 3 S ð2;1Þ 3 Optimum values 6 1 2 167 398
Supply network Decision variables T1
2 T22 Zð21;1Þ Z ð2;1Þ 2 Z ð1;2Þ 2 Optimum values 21 5 2 1 1 Decision variables Zð2;2Þ 2 Sð 1;1Þ 2 Sð 2;1Þ 2 Sð 1;2Þ 2 Sð 2;2Þ 2 Optimum values 2 588 345 275 484
Distribution network Decision variables T1
4 T24 Zð41;1Þ Z ð2;1Þ 4 Z ð1;2Þ 4 Optimum values 27 35 1 1 3 Decision variables Zð2;2Þ 4 S ð1;1Þ 4 S ð1;2Þ 4 S ð2;1Þ 4 S ð2;2Þ 4 Optimum values 2 234 299 174 399
We use the Genetic Algorithm and Direct Search Toolbox (GADST) designed by MATLAB. And the decision variables to be optimized are shown inTable 4.
Based on lots of testing of different combinations of the parameters, we determine the optimal parameters as follows: The population size is 100. The crossover probability is 0.8. The mutation probability is 0.06. The maximum evolution generation is 150. The elite reserved quantity is 2.
Besides the joint replenishment strategy, there are also separate replenishment strategy and unified replenishment strat-egy. So we calculate and optimize the inventory cost under the three replenishment strategies using the GA, respectively, and then compare the results.
(1) The joint replenishment strategy.
The optimization results are shown inTable 5. The evolutionary process of GA is convergent and the total inventory cost is declining as the evolutionary process going (Fig. 3). The total cost is $860050.
(2) The separate replenishment strategy.
Under the separate replenishment strategy, the order processes of each product are independent of each other, and each nodal enterprise bears its own cost of the respective order. So, we compute the ordering cost repetitively, which leads to an increase in the total inventory cost. The optimization results are shown in Table 6. The total cost is $1576909 (Fig. 4).
Table 6
The optimum values of decision variables under the separate replenishment strategy.
Core enterprise Decision variables Tð1;1Þ
3 T ð2;1Þ 3 S ð1;1Þ 3 S ð2;1Þ 3 Optimum values 20 14 312 393
Supply network Decision variables Tð1;1Þ
2 T ð2;1Þ 2 T ð1;2Þ 2 T ð2;2Þ 2 Optimum values 19 15 13 15 Decision variables Sð1;1Þ 2 S ð2;1Þ 2 S ð1;2Þ 2 S ð2;2Þ 2 Optimum values 565 353 251 557
Distribution network Decision variables Tð1;1Þ
4 T ð2;1Þ 4 T ð1;2Þ 4 T ð2;2Þ 4 Optimum values 30 32 43 30 Decision variables Sð1;1Þ 4 S ð1;2Þ 4 S ð2;1Þ 4 S ð2;2Þ 4 Optimum values 266 182 169 276
Table 7
The optimum values of decision variables under the unified replenishment strategy.
Core enterprise Decision variables T1
3 Sð31;1Þ S ð2;1Þ 3
Optimum values 10 203 390
Supply network Decision variables T1
2 T22 Sð21;1Þ S ð2;1Þ 2 S ð1;2Þ 2 S ð2;2Þ 2 Optimum values 19 12 563 312 296 488
Distribution network Decision variables T1
4 T24 Sð1 ;1Þ 4 S ð1;2Þ 4 S ð2;1Þ 4 S ð2;2Þ 4 Optimum values 35 42 287 153 206 356
Fig. 5.The optimization results and evolutionary process of GA under the unified replenishment strategy.
(3) The unified replenishment strategy.
Under the unified replenishment strategy, each nodal enterprise orders products with the basic order cycle (Zð1;1Þ 3 ;Z ð2;1Þ 3 ;Z ð1;1Þ 2 ;Z ð2;1Þ 2 ;Z ð1;2Þ 2 ;Z ð2;2Þ 2 ;Z ð1;1Þ 4 ;Z ð2;1Þ 4 ;Z ð1;2Þ 4 andZ ð2;2Þ
4 are equal to 1). The optimization results are shown inTable
7. The total cost is $1090377 (Fig. 5).
3.3. Analysis
(1) The total cost under the joint replenishment strategy is the lowest (Fig. 6). It is 21% lower than the total cost under the unified replenishment strategy, and 45% lower than the total cost under the separate replenishment strategy. It shows that the multi-product multi-echelon inventory control model and the algorithm designed by GA have a clear advan-tage on decreasing the inventory cost of the multi-product multi-echelon inventory system.
(2) Almost all the order cycles inTable 5are shorter than the order cycles inTable 6orTable 7. This is because it allows multiple products share the same ordering cost under the joint replenishment strategy. Therefore, it can reduce order-ing cost of each order, improve order frequency and shorten order cycle. In general, it can reduce the total inventory cost. In this illustration, we assume that it only have two products in the system. If it has more products in the system, each product would share less ordering cost, and the total inventory cost would be lower.
4. Conclusion
The problem of multi-echelon inventory control is becoming more important. Aiming at this problem, we apply the joint replenishment strategy into the multi-echelon inventory system and build a multi-product multi-echelon inventory control model. Considering both suppliers and distributors, or both the distribution system and the production system, this model can integrally express the actions and relations between every entity in the system. And an algorithm designed by GA is used for solving the model. Then we respectively simulate the model under three different ordering strategies. The simulation re-sults show that the established model and the algorithm designed by GA have a clear advantage on decreasing the inventory cost of the multi-echelon inventory system.
There are some limitations in this research. First, we assume the lead time of each nodal enterprise is fixed. In practice, however, it is often random variable. Second, this model only considers the inventory cost. In real life, it also needs to con-sider the time cost. If concon-sider both the inventory cost and the time cost, it is the inventory control model based on agile supply chain. This will be done in our future research.
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