An Integrated Modeling for Supplier Selection and Optimal Lot Sizing: A Case Study of Four-Echelon Supply Chain 1, a
a
Behin Elahi, 2, bLeila Etaati, 3, a Seyed-Mohammad Seyed-Hosseini Department of Industrial Engineering, Iran University of Science and Technology (IUST), Narmak St., Tehran, Iran b Department of Industrial Engineering, Branch of Science and Research, Islamic Azad University, Tehran, Iran 1
[email protected],
[email protected],
[email protected]
Abstract Supply chain management is one of the new approaches that are considered significantly by variety of experts and enterprises in today competitive world. Supply chain refers to the network of facilities, supply centers, production centers and distribution centers which encompass various tasks such as providing raw materials, transforming raw materials to components or finished products and distributing them among customers. Regarding this issue, studying literature indicates on this fact that multi-level, multi-product, multi-period models in multi-sourcing situation by considering the supply capacity have not been taken into account notably and just few mathematical models are proposed for the analysis of such decisions. In this paper, it is attempted to investigate and delve into the supply chain planning issue in multi-level, multi-product, multi-period situation. In the proposed model, simultaneously optimization of two objectives is considered which entails maximizing the service level related to giving agreeable services to customers at each retailer’s site and minimizing the overall costs of supply chain. As the proposed mixed integer non-linear model has specific complexity, branch and bound solver is utilized and has been applied for a data set in an automotive supply chain. In this way, a model has been presented which determines the lot size of ordering among various members in a supply chain, the amount of holding inventory at each elements of supply chain's site, optimal combination of selecting potential suppliers and the amount of ordering components to suppliers. Keywords: Supplier selection, Supply chain planning, Mathematical model, Inventory Control, Optimal lot sizing Introduction Supply chain management is one of the most vital decisions in today’s global market as companies are forced to gain a competitive advantage by focusing attention to their entire supply chain. The notable concentration in the supply chain management related research in the last decade has been owing to its potential to improve the efficiency and efficacy of operations and reduce costs. In real world, variety of activities are involved in supply chain management issue such as supplier selection, inventory management, purchasing and transportation of materials, components and finished products. Suppliers are the significant link to any supply chain and subsequently sourcing decision is one of the essential decisions to be taken at the planning stage. According to Chopra and Meindl (2007), inventory is recognized as one of the four major drivers in a supply chain. The responsiveness of the supply chain can be increased by high inventory levels although its cost efficiency decreases 978-1-61284-952-2/11/$26.00 ©2011 IEEE
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due to the cost of holding inventory. [1] Considering the aforementioned points, a relevant problem in supply chain management is to determine the appropriate levels of inventory and lot size of ordering at the various stages involved in the system. Regarding the supply chain management (SCM) and supply chain planning variety of academic research and practical studies are conducted. They can be categorized as Table1. Shen-Lian Chung et al. (2008) analyzed an inventory system with traditional forward-oriented material flow as well as a reverse material flow supply chain. They considered that in the reverse material flow, the used products are returned, remanufactured and shipped to the retailer for resale. Also, they proposed a multi-echelon inventory system with remanufacturing capability. [2] C.A. Silva et al. (2009) introduced a multi-agent supply chain management methodology based on the description of the supply chain as a set of different distributed optimization problems. They applied the ant colony optimization (ACO) to achieve cooperation between different multiple partners. [3] Xiaoming Yan et al. (2010) extended the model of Coordination in decentralized assembly systems with uncertain component yields and proposed a new kind of contract, surplus subsidy contract, where the leader (the assembler) provides the contract, while the followers (component suppliers) make their choices simultaneously. They proved that the profit of the supply chain under coordination can be arbitrarily divided between the component suppliers and the assembler. [4] YaTi Lin et al. (2010) proposed a novel hybrid MCDM technique (using ANP and Interpretive Structural modeling) in order to cope with the complex and interactive vendor evaluation and selection problem.They considered four main dimensions with definite criteria: delivery management capability, quality management capability, price and integrated service capability. [5] Xueipng Li andYuerong Chen (2010), based on simulation techniques, investigated the impacts of supply disruptions and customer differentiation on the minimum average annual total cost of the retailer. They considered different scenarios of disruption frequency and duration. [6] Amanda J. Schmitt et al. (2010) examined optimal base-stock inventory policies using infinite-horizon, periodic-review models, for a single supplier whose single retailer is subject to stochastic disruptions. [7] Yuliang Yao et al. (2010) developed an incentive contract that results in benefits for both upstream and downstream supply chain partners under a VMI arrangement. [8] Jing Li et al. (2010) proposed a multi-agent simulation model for analyzing the dominant player’s behavior of a supply chain(consisting of a raw material supplier and component supplier, a manufacturer IEEE Int'l Technology Management Conference
and retailer). Also, they utilized the SPP (Stable Profit Platform) to indicate the level of domination power for the player achievement. [9] Yugang Yu and George Q. Huang (2010) considered a VMI supply chain where a manufacturer and multiple retailers interact with each other in order to maximize their own profits. They proposed a dual Nash game model, consisting of RR-Nash game between retailers and MR-Nash game between the manufacturer and all retailers as a whole. Moreover, they developed a GA algorithm to find the dual Nash equilibrium efficiently. [10] Yuanjie He, Jiang Zhang (2010) studied a supply chain with one supplier and one retailer when the supplier has random yield and the option of trading in a secondary market. They evaluated both the centralized and decentralized systems. [11] Jinfeng Yue et al. (2010) examined the problem of how the make-toorder manufacturer can select key sourcing partners (with several certified suppliers for each key part) and allocate quantities to each sourcing partner. [12] S. Kamal Chaharsooghi, Jafar Heydari (2010) considered a decentralized supply chain consisting of one buyer and one supplier in a multi-period and developed an incentive scheme based on credit option to encourage the buyer to participate in the coordinati on model. [13] Ming-Feng Yang and Yi Lin (2010) proposed a serial multi-echelon integrated just in-time (JIT) model based on uncertain delivery lead time and quality unreliability considerations. They applied the particle swarm optimization (PSO) as a method to result an improved solution solving a mixed nonlinear integer problem. [14] The present study analyzes a multi-echelon supply chain planning system containing four echelons: multi-supplier, multi-assembler, multi-distribution centers and multi-retailer. In order to deal with this issue a bi-objective mathematical model proposes. The first objective refers to maximizing the service level of retailers to end customers and the second objective is relevant to minimizing the total costs of considered multi-echelon supply chain. The remainder of this paper is organized as follows. Section 2 deals with the problem formulation and explains considered assumptions. Section 3 presents the solution method. A numerical example based on the practical research in an automotive supply chain is illustrated in Section 4. Ultimately, Section 5 denotes conclusion and feauture research. Table1. Categorizing various approaches toward supply chain management issue Approach/ Solution method GA PSO Meta heuristic Ant Colony Scatter search
Reference articles [10] [14] [3] [12]
MCDM approaches
[5]
Modeling the considered problem and Proving propositions
[4], [8], [11], [13], [2]
Simulation
[6], [9], [3]
Game theory Stochastic programming
[10], [11] [7]
Hybrid algorithms
[12]
Problem Description and Formulation In this paper, in order to develop a mathematical model for supply chain management issue, it is taken into account that the supply chain has four echelons and entails multisupplier, multi-assembler, multi-distribution centers (DCs) and multi-retailer (Figure1). Also, it is assumed that, different kinds of components are flowed into assemblers' plants from some selected suppliers. Then, variety of products will be provided by assembling different sets of components. Final products will be delivered to a set of distribution centers and consequently will be distributed among different retailers. End customers place their orders to these retailers. DC 1
Retailer 1
Assembler k
DC w
Retailer r
Assembler n
DC q
Retailer v
Supplier 1
Assembler 1
Supplier j
Supplier m
Figure1. Various considered echelons in given supply chain Other assumptions are as following: - It is considered that an integrated supply chain planning will be developed over a finite and limited planning horizon which contains multiple periods. Two main objective functions will be optimized simultaneously: (I) maximizing the service levels of the customers at all retailers’ sites, (II) minimizing the total costs of supply chain. - The demand of each product type is forecasted for the following T periods. In other words, in each period, customers' demands are known and deterministic. - Each potential supplier has a definite and limited capacity for providing different components in each period and has the capability of procuring all kinds of components. - There are capacity limitations for shipping products from assemblers to distribution centers and sequentially from distribution centers to retailers. - The ordering costs are independent of kind and amount of components. - Retailers are independent from each other and attempt individually to meet their own customers’ demands. - It is assumed that distribution centers can hold inventory but retailers prefer not to hold any inventory. Moreover, assemblers are just capable of holding inventory related to variety of components for performing their processes. - Each retailer may encounter shortage in meeting customers’ demands and Partial backordering is applied when a stock out occurs. - For each candidate supplier, the selling price of components is definite and known. - All considered costs are assumed to be known and accurately determined over the planning horizon. Moreover, the list of indices, parameters and decision variables are introduced for problem formulation as below:
Indices:
Su , j
u
Index of different types of parts required for finished products.
i
Index of various finished products.
j
Index of suppliers.
k w
Index of assemblers. Index of distribution centers.
r
Index of retailers.
Xi,k,t
t Index of periods. Parameters:
BRi,r,t
m n q v a p T hwi,w Di,r,t Fi,u πri,r
i ,k Cpri , k
u ,i , k Trpi ,k , w
i ,r Trwi , w,r
CTrwi,w,r CTrpi , k , w
Stwi , w
Gi ,k
SLmin i , r ,t huu ,k
j
Csu , j MIu,k,t
Number of suppliers. Number of assemblers. Number of distribution centers. Number of retailers. Number of components. Number of products. Number of periods. Unit inventory holding cost of ith product at wth distribution center’s site per unit time. Demand of ith product at rth retailer’s site occurred in tth period. Coefficient of consumption related to uth component in ith product. Unit backorder cost of ith product at rth retailer’s site. Fixed cost of assembling of ith product at kth assembler’s plant. Unit regular time assembling cost of ith product at kth assembler’s plant. Unit customization cost of uth component in assembling ith product by kth assembler. Unit transportation cost of ith product carrying from kth assembler to wth distribution center. Set-up cost of ith product at rth retailer’s site per order. Unit transportation cost of ith product carrying from wth distribution center to rth retailer. Capacity limit to ship ith product from wth distribution center to rth retailer. Capacity limit to ship ith product from kth assembler to wth distribution center. Store capacity of ith product at wth distribution center. Maximum capacity of assembling of ith product at kth assembler’s site. Minimum desired service level of ith product at rth retailer’s site in tth period. Unit inventory holding cost of uth component at kth assembler’s site per unit time. th
Ordering set-up cost of j supplier.
Unit selling price of uth component offered by jth supplier to assembler. Capacity of providing uth component at jth supplier’s site. Maximum holding capacity of uth component at kth assembler’s site in tth period.
Decision variables:
Vi,k,w,t Qi,w,r,t
Z u , j , k ,t
SLi ,r ,t
IWi ,w ,t IU u ,k ,t
i , r ,t
The amount of produced units which is related to the ith product at kth assembler’s site in tth period. The amount of ith product backordered by rth retailer in the end of tth period. The amount of units which is related to the ith product delivered from kth assembler to wth distribution center in tth period. The amount of units which is related to the ith product dispatched to rth retailer by wth distribution center in tth period. The amount of units which is related to the uth component ordered by kth assembler from jth supplier in tth period. Desired service level at rth retailer’ site related to ith product in tth period. Amount of inventory related to ith product at wth distribution center’s site in the end of tth period. Amount of inventory related to uth component at kh assembler’s site in the end of tth period. If rth retailer places assembly order for 1 ith product in tth period.
0
Otherwise
If assembling of ith product at kth 1 assembler’s plant has been set up in tth i , k ,t period. 0 Otherwise th th 1 If r retailer places order to w w,r distribution centers. 0 Otherwise th 1 If wth distribution center places order k ,w to k assembler. 0 Otherwise th places order to jth 1 If k assembler th Y j , k ,t supplier in t period. 0 Otherwise Considering the aforementioned assumptions and notations, the problem can be modeled as below: First Objective: t
Max
SLi , r ,t 1
BR l 1 t
D l 1
i , r ,t
i , r ,t
i, r , t
(1)
Second Objective:
(2)
TC U 1 U 2 U 3 U 4
Min
Subject to:
( huu ,k .IU u ,k ,t ) m U1 t 1 k 1 u 1 ( j . y j , k ,t Su , j .Z u , j , k ,t ) j 1 ( i ,k .i ,k ,t ) (Cpri , k .xi ,k ,t ) p T n U 2 a t 1 k 1 i 1 ( u ,i , k .Fi ,u .xi , k ,t ) u 1 T
n
(3)
(4)
( i ,r . i ,r ,t ) ( ri,r .BRi ,r ,t ) U 4 q t 1 r 1 i 1 (Trwi , w, r .Qi , w , r ,t ) w1
(6)
p
p
m
P
j 1
i 1
IU u , k ,t IU u , k ,t 1 Z u , j , k ,t ( Fi ,u .xi , k ,t )
(7)
u, k , t n
Z k 1
u , j , k ,t
Csu , j
IU u , k ,t MI u , k ,t T
u, j, t
(8)
u, k , t
(9)
P
( ( Fi ,u . X i ,k ,t )).Y j ,k ,t Z u , j , k ,t l t i 1
(10)
u, j, k , t q
BRi , r ,t BRi , r ,t 1 Di , r ,t ( Qi , w, r ,t ) w 1
(11)
i, r , t n
v
k 1
r 1
IWi , w ,t IWi , w,t 1 Vi , k , w,t Qi , w, r ,t
(12)
i, w, t q
V
i, k , t
(13)
Qi , w,r ,t w,r .CTrwi , w,r
i, w, r , t
(14)
Vi , k , w ,t k , w .CTrpi , k , w
i, k , w, t
(15)
IWi , w ,t Stwi , w
i, w, t
(16)
w 1
i , k , w ,t
X i , k ,t
X i , k ,t i , k ,t .M
i, k , t
X i , k ,t Gi , k
i, r , t
(19)
IWi , w,t Z 0 Yu , j ,k ,t , w,r , k , w , i ,k ,t , i ,r ,t {0,1}
(5)
v
(18)
(20)
i, j , u, k , r , w, t
(hwi , w .IWi , w,t ) U 3 n t 1 w 1 i 1 (Trpi , k , w .Vi , k , w ,t ) k 1 T
i, r , t
IU u ,k ,t , Z u , j ,k ,t , Qi , w,r ,t , Vi ,k ,w,t , X i ,k ,t , BRi ,r ,t ,
a
q
T
Di , r ,t i ,r ,t .M
(17)
i, k , r , w, t
(21)
In the proposed mathematical model, as it can be observed, the first objective function (Eq.1) refers to maximizing customers’ service levels at each retailer’s site. The second objective function (Eq.2) demonstrates the considered total costs of supply chain. It includes four different parts: U1, U2, U3, U4; Term U1 refers to the costs of components which include holding costs of inventory at assemblers’ sites, fixed ordering costs and purchased costs (Eq.3). Term U2 indicates on the assemblers 'costs which entails fixed costs of assembling, costs of regular time assembling and customization costs of components in assembling products (Eq.4). Term U3 is related to distribution centers 'costs which consists of holding costs of inventory at distribution centers’ sites and transportation costs of products carrying from assemblers to distribution centers (Eq.5). Term U4 is associated with retailers 'costs that includes set-up costs of products at retailers’ sites per order, backordering costs of products at retailers’ sites and transportation costs of products carrying from distribution centers to retailers (Eq.6). Balanced constraints related to components at assemblers 'plants are taken into account through Eq.7. Constraint (8) stands for the capacity limitation of suppliers for providing various components. Constraint (9) demonstrates the store capacity of assemblers for holding components. Constraint (10) certifies that there is not an order for procuring components without charging an appropriate transaction cost (ordering cost). Balanced constraints related to retailers and distribution centers’ sites are considered through Eq.11-12. Constraint (13) guarantees that in each period, each assembler ships all the produced final products to variety of distribution centers and doesn’t hold any inventory related to final products. Constraint (14) refers to the capacity limitation of transporting final products from distribution centers to retailers. Similarly, constraint (15) stands for the capacity limitation of carrying products from assemblers to distribution centers. Constraint (16) demonstrates the store capacity for holding products at distribution centers 'sites. Constraint (17) refers to this fact whether assembling of products sets up at assemblers’ plants or not. Correspondingly, constraint (18) denotes whether retailers place assembly order for products or not. Constraint (19) refers to the maximum capacity of assembling at assemblers’ plants. Moreover, forbidding negative continuous values for orders, amounts of inventory related to components at assemblers 'plants, amounts of producing final products, amounts of backordering and amounts of holding inventory at distribution centers 'site has been satisfied through constraint (20). Furthermore, constraint (21) sets the values of binary variables. In the next section,
Table2. Unit inventory holding cost of components at suppliers’ sites per unit time 1
Assemblers
2
3
4
1
2
1
2
1
2
1.5
2
1
1
2
3
2
2.5
2
1
2
Supplier 2
100
Supplier 3
200
150
Components
5
1000 2000 3000 1500
3000 3500 4000 2000
3000 1500 3000 3200
3000 2000 1000 3000
2000 2000 2000 2500
1
2
3
Assembler 2
Assembler 1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
300 115 150 125 140 120 110 130 135 115 160 140 115 120 140
200 150 145 130 120 130 125 145 155 120 150
100 160 140 145 145 140 135 160 160 135 110 160 135 140 115
135 120 135 120
Table7. Unit customization costs of various products Components 1
2
3
4
5
1
20
20
20
14
18
2
30
20
14
18
18
3
20
30
18
19
14
4
18
14
18
18
12
1
25
13
30
12
17
2
15
18
15
12
20
3
10
12
19
12
12
4
15
15
19
12
14
1
10
14
16
10
16
2
15
16
12
20
19
Products
3
18
15
20
14
10
4
16
11
16
16
16
Table8. Fixed costs of assembling
Supplier 4
Product 1
120
Table4. Unit sale price of components by suppliers 1
2
3
4
5
1
10
30
18
20
14
2
15
35
20
12
12
3
20
25
15
16
20
4
25
15
16
22
16
Suppliers
4
Components
Table3. Fixed set-up cost of ordering to suppliers Supplier 1
3
Periods
5
1
2
Table6. Store capacity of components at assemblers’ sites in each period
Assembler 3
Components
1 2 3 4
Assembler 3
Numerical Example In this section, the implementation of the proposed model and solution method are demonstrated by using a numerical example. This data set is gathered by some experts and specialists’ estimation in an automotive supply chain. A supply chain system with four echelons containing four suppliers, three assemblers, four distribution centers and three retailers is considered. There are five different components which are applied to form four finished products. It is assumed that the supply chain planning will be determined for three periods. Assorted parameters related to this considered data set are displayed through Table2-20.
1
Suppliers
Assembler 1
Solution method Here, the optimization problem is to determine the values of decision variables by considering the two defined objectives. With the aim of solving the proposed mathematical modeling, the bi-objective model is converted into a single objective model by utilizing bounded objective method. [15] The first objective is considered as constraint for the second objective by defining a minimum bound for it. In other word, a desired definite minimum value for the first objective (service level) is defined which can be expressed unanimously by retailers (SLmin). Also, by considering the constraint indicates on this condition that the value of SLi,r,t variables can vary between 0 and 1, the proposed model will be formed as a single objective mixed integer non-linear programming. Then in order to solve this mathematical problem, after coding the defined problem by Lingo software, the branch and bound solver is utilized. To illustrate the proposed model and the solution schemes, in the next section a numerical example will be utilized and the efficacy and efficiency of the proposed model will be verified through this section.
Table5. Maximum capacity of providing components by suppliers
Assembler 2
bounded objective solution is applied in order to deal with solving this mixed integer non-linear mathematical modeling efficiently.
Assembler 1 30
Assembler 2 50
Assembler 3 40
Product 2
20
30
40
Product 3 Product 4
60 20
50 25
70 30
Table9. Unit regular time assembling cost Assembler 1
Assembler 2
Assembler 3
Product 1
12
13
10
Product 2
16
15
11
Product 3
13
12
10
Product 4
17
16
12
Table10. Maximum capacity of assembling at each assembler’s sites Assembler 1
Assembler 2
Table17. Unit transportation costs of carrying products from assemblers to distribution centers
Assembler 3
DC1
DC 2
DC 3
DC 4
12
15
18
16
200
230
210
Product 2 Product 3
215 240
220 245
230 230
Product 2
14
16
18
18
Product 4
250
260
270
Product 3
12
15
14
10
Product 4
12
14
16
18
Product 1
14
15
12
16
Product 2
16
14
15
16
Product 3
11
12
12
14
Product 4
16
12
14
12
Product 1
18
18
14
16
Product 2
18
12
16
19
Product 3
15
18
16
14
Product 4
16
10
11
12
Assembler 1
Product 1
Product 1
DC 2
DC 3
DC 4
1 1 2 1.5
1.5 2 1 1
2 1 2 2
2 1.5 2 1
Table12. Store capacity of products at distribution centers’ sites Product 1 Product 2 Product 3 Product 4
DC 1
DC 2
DC 3
DC 4
200 260 250 230
210 270 260 240
220 265 270 270
250 280 250 240
Table13. Unit backorder cost of products at retailers’ sites Retailer 1
Retailer 2
Retailer 3
Product 1 Product 2 Product 3
210 240 230
230 250 250
220 260 270
Product 4
280
220
230
Assembler 3
Product 1 Product 2 Product 3 Product 4
DC 1
Assembler 2
Table11. Unit inventory holding cost of products at distribution centers’ sites per unit time
Table18. Unit transportation costs of carrying products from distribution centers to retailers Retailer 1 DC 1
DC 2
Table14. Set-up cost of products at retailers’ sites per order Retailer 1
Retailer 2
Retailer 3
Product 1
15
20
18
Product 2 Product 3 Product 4
22 15 16
18 24 14
20 18 20
DC 3
Table15. Demand of products at retailers’ sites
DC 4
Period 3
770 740 690 600 700 590 560 710 760 710 570 720
500 510 500 480 510 510 500 500 500
550 520 510 500 510 500 510 520 500 520 500 500
500 510 520
Table16. Coefficient of consumption related to components in forming products Components Products 1 2 3 4
1
2
3
4
1 2 3 2
2 2 2 2
2 3 3 1
2 2 2 2
5 1 3 2 1
Retailer 2
12 12 14 18 14 18 15 19 19 16 16 18 15 18 19 12
15 15 12 17 16 16 14 14 17
Retailer 3 13 16 15 19 18 15 12 12 16 16 15 15 13 12 16 16
15 18 14 14 17 17 14
Table19. Capacity limit to ship products from assemblers to distribution centers DC1
Assembler 1
2
Assembler 2
Retailer 3
Retailer 2
Retailer 1
1 2 3 4 1 2 3 4 1 2 3 4
1
Assembler 3
Products
Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4
DC 2
DC 3
DC 4
Product 1
1100
1200
1300
1150
Product 2
1200
1180
1100
1150
Product 3
1100
1150
1160
1100
Product 4
1100
1110
1120
1120
Product 1
1180
1200
1150
1200
Product 2
1170
1180
1100
1200
Product 3
1100
1120
1100
1120
Product 4
1100
1180
1100
1130
Product 1
1150
1100
1200
1100
Product 2
1000
1100
1150
1170
Product 3
1130
1100
1130
1140
Product 4
1140
1110
1100
1120
Table20. Capacity limit to ship products from distribution centers to retailers DC 1
DC 2
DC 3
DC 4
Retailer 1
Retailer 2
Retailer 3
1100 1320 1420 1400 1260 1220 1400 1400 1400 1370 1400 1400 1430 1320 1360 1400
1300 1360 1450 1360 1270 1340 1420 1380 1450 1380 1320 1390 1410 1350 1390 1350
1250 1200 1440 1390 1300 1350 1440 1390 1420 1400 1380 1380 1300 1410 1400 1360
Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4 Product 1 Product 2 Product 3 Product 4
Taking into consideration that the value of parameter SLmin is equal to 0.5 at retailers 'sites, the proposed problem is solved up to the global optimality using LINGO 8.00 (LINDO Systems Online, 2007). The coded problem in this solver is run on a Pentium (R) 4CPU 3.000 GHz PC. Tables 21-24 show the optimal policy for supply chain management and values of various decision variables. Besides, in order to evaluate the impacts of changing parameter SLmin, a sensitivity analysis is conducted. The result of this analysis is demonstrated by Table 25 and Figure 2. Table21. The computed results of positive values for X i, k, t X 1,3 , 2 X 1,1, 2 220 X 1,1, 2 200 210 2 ,3 , 2
230
X 3,2 ,2
245
X 3 ,3 , 2
230
X 4 ,2 ,2
208
X
4 , 2 ,3
230
X
4 ,3 , 2
270
X 2 ,2 ,2
215
X
4 ,1, 2
42
X
4 ,3 ,3
270
X
Table24. The computed results of positive values for Zu, j, k, t Z 1,1, 2 ,3 Z 1,1, 2 , 2 Z 1,1,3 , 2 1319 460 1681
Z 1,1,3 ,3
540
Z 1, 2 ,3 , 2
421
Z 2 , 4 ,1, 2
379
Z 2 ,4 ,2 ,2
1666
Z 2 , 4 , 2 ,3
460
Z 2 , 4 ,3 , 2
1740
Z 2 , 4 ,3 ,3
540
Z 3 ,3 , 2 ,1
2043
Z 3 ,3 ,3 ,1
1955
Z 3 , 4 ,1,1
307
Z 3,4 ,2 ,2
85
Z 3 , 4 ,3 , 2
270
Z 4 , 2 ,1,1
369
Z 4 , 2 , 2 ,1
1641
Z 4 ,2 ,2 ,2
460
Z 4 , 2 ,3 ,1
1760
Z 4 , 2 ,3 , 2
540
Z 5 , 2 ,1,1
252
Z 5 ,3 , 2 ,1
1448
Z 5 ,3 ,3 ,1
1490
Z 5 ,4 ,2 ,2
230
Z 5 , 4 ,3 ,3
270
Table25.The result of solving model by performing sensivity analysis of parameter SL min TC SL min TC SL min 0 0.1 0.2 0.3 0.4 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63
2132167 2132167 2132167 2132167 2132167 2132167 2132167 2132167 2132167 2132167 2132167 2132179 2132167 2132167 2132167 2132167 2132167 2132167 2132167
0.64
2132167
0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83
-
2132167 2132167 2132167 2132825 2135132 2139113 2148377 2159084 2170149 2182385 2196950 2213282 2230657 2248002 2267409 2288554 2310676 2332426 Infeasible
-
Table22.The computed results of positive values for V i, k, w, t V 4 ,2 ,4 ,2 V 1,1,1, 2 V 2 ,3 , 2 , 2 200 230 208
V 1, 2 ,1, 2
220
V 3 , 2 ,1, 2
245
V 4 , 2 , 4 ,3
230
V 1,3 , 4 , 2
210
V 3 ,3 ,1, 2
230
V 4 ,3 , 2 , 2
238
V 2 , 2 ,1, 2
215
V 4 ,1, 2 , 2
42
V 4 ,3 , 4 , 2
32
V 4 ,3 , 4 ,3
270
Table23. The computed results of positive values for Q i, r, w, t Q 1,1,1, 2 Q 1,1,3 , 2 120 Q 1, 2 , 2 , 2 210 500 Q 1,3 ,3 , 2
220
Q 1, 4 , 2 , 2
300
Q 1, 4 ,3 , 2
160
Q 2 ,1,1, 2
475
Q 2 ,2 ,2 ,2
280
Q 2 , 2 ,3 , 2
220
Q 2 ,3 ,1, 2
35
Q 2 ,3 , 2 , 2
230
Q 2 , 4 ,3 , 2
280
Q 3 ,1,1, 2
230
Q 3 ,1 , 2 , 2
500
Q 3 , 2 ,3 , 2
260
Q 3 ,3 ,1, 2
270
Q 3 , 4 ,3 , 2
250
Q 4 ,1, 2 , 2
230
Q 4 , 2 ,3 , 2
520
Q 4 ,3 , 2 , 2
270
Q 4 , 4 ,1, 2
480
Q 4 , 4 ,1,3
500
Figure2. Diagram of Sensitivity analysis related to SLmin parameter. Discussion This paper investigates the dynamic planning of materials replenishment, component fabrication, customized assembly
and distribution of products in a multi-period supply chain. By considering the achieved results, it can be extracted that increasing the value of parameter SL min more than 0.67 is accompanied with a significant increase in Total costs of supply chain. Also, Table 25 indicates that the retailers can consider the maximum value of parameter SL min equal to 0.82 due to this fact that for values more than 0.82 the model will be infeasible. Moreover, the convexity of the proposed model can be perceived through Figure 2. On the whole, the results show that the proposed mathematical model can provide managers with an efficient strategy for planning their involved supply chain. Conclusions and Future Research This study derives an optimal strategy for a supply chain planning system with considering the costs of supply chain and retailers’ service level to end customers. From the integrated perspective toward all members of the multiechelon supply chain: the suppliers, assemblers, distribution centers and the retailers, an optimal production and inventory control policy were formulated to minimize the overall costs and maximize the retailers 'service level. Moreover, for analyzing of efficiency and efficacy related to the proposed mathematical modeling, an example illustrated. The findings in this study revealed a substantial increase in total costs by increasing the value of SLmin parameter more than specific rate. Also, this proposed model can play the role of driver for productive managers in order to manage the involved supply chain more efficient and determine the lot size of ordering among various echelons. According to the aforementioned points, it can be stated that this supply chain planning model involves a complex shape of search space with many candidate solutions. Therefore, meta-heuristic methods such as genetic algorithm (GA) are applicable for fast exploration and can be considered as an efficient research in future. Also, dealing with variety of robust optimization toward the proposed mathematical model can shed light on novel research. About the Authors: Eng. Behin Elahi is M.S.C student of Industrial Engineering at Iran University of Science and Technology. Interested areas: Supply Chain Management, Operation Research, Mathematical Modeling, Control Project, Multi-attribute Decision Making. Eng. Leila Etaati is M.S.C student of Industrial Engineering at Islamic Azad University. Interested areas: Supply Chain Management, Decision support systems, Software evaluation and quality assessment. Dr. Seyed-Mohammad Seyedhosseini is a Professor of Industrial Engineering Department in Iran University of Science and Technology with Doctor of philosophy in industrial and civil engineering at University of Oklahama,1982. Interested areas: Operation and Production Management, Supply Chain Management, Engineering Economy, Project Evaluation Appraisal.
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