Multi-criteria fuzzy optimization for locating warehouses and
distribution centers in a supply chain network
Cheng-Liang Chen
1,*
, Tzu-Wei Yuan
2, Wen-Cheng Lee
3Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Received 18 January 2007; accepted 8 June 2007
Abstract
This study considers the planning of a multi-product, multi-period, and multi-echelon supply chain network that consists of several existing plants at fixed places, some warehouses and distribution centers at undetermined locations, and a number of given customer zones. Unsure market demands are taken into account and modeled as a number of discrete scenarios with known probabilities. The supply chain planning model is constructed as a multi-objective mixed-integer linear program (MILP) to satisfy several conflict objectives, such as minimizing the total cost, raising the decision robustness in various product demand scenarios, lifting the local incentives, and reducing the total transport time. For the purpose of creating a compensatory solution among all participants of the supply chain, a two-phase fuzzy decision-making method is presented and, by means of application of it to a numerical example, is proven effective in providing a compromised solution in an uncertain multi-echelon supply chain network. # 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Supply chain management; Uncertainty; Multiple objectives; Mixed-integer linear program (MILP); Fuzzy decision making
1. Introduction
The supply chain is an integrated process wherein a number of business entities (suppliers, manufacturers, distributors and retailers) work together in an effort to acquire raw materials, convert them into specified final products and deliver these final products to retailers (Beamon, 1998). The supply chain further fosters a new concept in management: the concept of supply chain management.
Over the past decade the world has changed from a marketplace with some large independent markets to an extremely integrated global market. The increase in compe-titive pressures in the global marketplace coupled with the rapid advances in information technology have brought supply chain planning into the forefront of the business practices of most manufacturing and service organizations (Gupta and Maranas, 2003). A great variety of companies, those in chemical industry included, can also benefit from this novel management scheme. Therefore, many researchers in the process systems engineering
(PSE) society devote themselves to this interesting field (Applequist et al., 2000; Bose and Pekny, 2000; Chen et al., 2003; Cheng et al., 2003; Garcia-Flores et al., 2000; Gupta and Maranas, 2000; Gupta et al., 2000; Perea-Lopez et al., 2000; Pinto et al., 2000; Zhou et al., 2000, etc.).
Traditionally, the integration of supply chain networks is usually based on deterministic parameters. In practice, however, this is rarely the case as it is usually difficult to foretell prices of chemicals, market demands, availabilities of raw materials, etc., in a precise fashion (Liu and Sahinidis, 1997). A number of works are devoted to studying supply chain management under
uncer-tain environments. For example, Gupta and Maranas (2000)
incorporate uncertain demand via a normal probability function and propose a two-stage solution framework. A generalization to handle multi-period and multi-customer problems was recently proposed byGupta and Maranas (2003).Tsiakis et al. (2001)use a scenario planning approach to describe demand uncertainties. Therein a number of demand scenarios with assigned non-zero probabilities is used as discrete stochastic demand quantities. All scenarios are simultaneously taken into account in the supply chain network design. However, the robustness of decision for uncertain product demands is not considered in these studies. In this paper, one of the major concerns is market demand uncertainty. The scenario-based approach will be adopted for modeling the uncertain market demands.
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Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407
* Corresponding author. Tel.: +886 2 23636194; fax: +886 2 23623040. E-mail address:CCL@ntu.edu.tw(C.-L. Chen).
1 2 3
0368-1653/$ – see front matter # 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jcice.2007.08.001
The location of manufacturing and warehousing facilities has received considerable attention from academics and practitioners alike over the past four decades. Location models have been developed to answer questions such as how many facilities to establish, where to locate them, and how to distribute the products to the customers in order to satisfy demand and minimize total cost (Melachrinoudis et al., 2000). However, when making location decisions, in addition to the total cost, one should also consider some other conditions such as the influence of local incentives and transport time.
In this article, the mid-term planning problem of locating warehouses and distribution centers in a supply chain network will be addressed, where multiple conflict objectives will be considered simultaneously including minimizing the total cost, raising the decision robustness to various product demand
scenarios, lifting the local incentives, and reducing the total transport time. This problem can be further formulated as a multi-objective mixed-integer linear program. So the two-phase fuzzy optimization solution strategy proposed byChen and Lee (2004a,b)can be adopted directly.
In the rest of this article, the problem statement and assumptions are outlined in Section2. The considered uncertain issues in supply chain planning are also described. The formulation of a production and distribution–planning model is set out in Section3. The procedure for grouping the scenario-dependent multiple conflict objectives into a scalar objective using the fuzzy sets concept is presented in Section 4. The contents of a numerical example, used to demonstrate the usefulness of the proposed method, are given in Section 5. Finally, some concluding remarks are given in Section6.
Index/set Dimension Physical meaning
c2 C ½C ¼ C Customer zones
d2 D ½D ¼ D Distribution centers
i2 I ½I ¼ I Products
k2 K ½K ¼ K Transport capacity level
m2 M ½M ¼ M All objectives n2 N ½N ¼ N Resources p2 P ½P ¼ P Plants s2 S ½S ¼ S Scenarios t2 T ½T ¼ T Periods w2 W ½W ¼ W Warehouses
Parameters *2 Physical meaning
FCDi
ts {c} Forecasting customer demand of i for customer c
FTCk
f pw; wd; dcg kth level fix transport cost, p to w; w to d, d to c
LI fw; dg Local incentive of w, d
PPDs Probability of product scenario s
PQi ps {max, min} Maximum, minimum manufacturing quantity of product i
Qmax
s f pw; wd; dcg Maximum transport quantity of p to w; w to d, d to c
Qmin
s f pw; wd; dcg Minimum transport quantity of p to w; w to d, d to c
R*nts {p} Total resource n at p
SQþ fw; dg Maximum, minimum capacity of w, d, +2 {max, min}
TCLk
f pw; wd; dcg kth transport capacity level, p to w; w to d, d to c
TT f pw; wd; dcg Transport time of p to w; w to d, d to c
UEC fw; dg Unit establishing cost of w, d
UHCi
fw; dg Unit handling cost of product i for w, d
UPCi
{p} Unit production cost of product i for plant p
UTCk
f pw; wd; dcg kth level unit transport cost, p to w; w to d, d to c
ai
fwg Coefficient relating the capacity of d to flow of product i handled
bi {d} Coefficient relating the capacity of w to flow of product i handled
ri
nts {p} Coefficient for resource m used in plant p for product i
Real var. *2 Physical meaning
J* {m} Objectives
OTT Total transportation time
PQits {p} Manufacture quantity of i
Qi
ts f pw; wd; dcg Total transport quantity, p to w; w to d, d to c
SQ fw; dg Capacity of w, d
TCO Total cost
TEC Total establishment cost of all warehouses and distribution centers
THCts Total handling cost of scenario s in period t
TLI, LD Total local incentive and an overall index on distribution centers
TPCts Total production cost of scenario s in period t
TTCts Total transportation cost for scenario s in period t
C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407 394
2. Problem description for locating warehouses and distribution centers in a supply chain network
The researchers consider a typical product, multi-echelon and multi-period supply chain network originally studied by Tsiakis et al. (2001). The revised supply chain network consists of several existing multi-product plants at fixed places, some candidate warehouses and distribution centers at specific but undermined locations, and a number of known customer zones, as showed inFig. 1. In this mid-term supply chain-planning problem, each customer zone places demands for one or more products. The candidate warehouses and distribution centers are described by the upper and lower bounds on their handling capacity. The establishment of warehouses and distribution centers will result in a fixed infrastructure cost. Operational costs include those associated with production, handling of material at warehouses and distribution centers, and transportation. The numbers and the locations of selected warehouses and distribution centers are left to be determined for establishment of a cost–effective supply chain network. The following assumptions are made for subsequent modeling and optimization: the whole system is operated steadily; therefore, there is no stock accumulation or depletion, and inventory can be ignored; the production capacity of each plant is related linearly to resources; the
capacities of warehouses and distribution centers are related linearly to the materials that they handle; the transportation costs are piecewise linear functions of the actual flow of the product from the source stage to the destination podium; several scenarios of product demands with known probabilities are forecast over the entire planning periods. The overall problem can thus be stated as follows. Given are the manufacturing data, such as product capacity and resource constraints; the basic data for candidate warehouses and distribution centers, such as capacities and local incentives; the transportation data, such as transport time and transport capacity; all cost parameters, such as manufacturing and handling costs; and several scenarios of forecasted product demands with known probabilities. The authors are going to determine the production plan of each plant; the number, location, and the capacity of warehouses and distribution centers to be set up; the transportation plan of each warehouse and distribution center; and all types of costs. The target is to integrate the multi-echelon decisions simultaneously to minimize the total cost and the transport time, and to elevate local incentives and the robustness of all considered design objectives to product demand uncertainties as much as possible. In the market, the participants of a supply chain not only faces the uncertainties of product demands and raw material supplies but also faces the uncertainties of commodity prices and costs (Liu and Sahinidis, 1997). The authors will also
TTCts f pw; wd; dcg Total transportation cost of * for scenario s in period t
TTQk
ts f pw; wd; dcg kth level transport quantity, p to w; w to d, d to c
mJ mðJmðxÞÞ The membership function of fuzzy objectiveJm
Binary var. *2 Meaning when having value of 1
X f pw; wd; dcg A link between p and w; w and d, d and c exists
Y fw; dg Warehouse w or distribution center d is to be established
Zk
ts f pw; wd; dcg kth transport capacity level, p to w; w to d, d to c
Fuzzy var. Physical meaning
FD Fuzzy set for final decision
Jm Fuzzy set for objective m
Fig. 1. The studied supply chain network.
address issues of demand uncertainty in the mid-term planning problem. The first concern in incorporating uncertainties into supply chain modeling and optimization is the determination of suitable representation of the uncertain parameters (Gupta and Maranas, 2003). Several distinct methods are frequently mentioned for representing uncertainty. For example, the
fuzzy-based approach (Giannoccaro et al., 2003; Liu and
Sahinidis, 1997; Petrovic et al., 1998, 1999), wherein the forecast parameters are considered as fuzzy numbers with
accompanied membership functions; the scenario-based
approach (Gupta and Maranas, 2003), in which several discrete scenarios with associated probability levels are used to describe expected occurrence of particular outcomes. To simplify the subsequent mathematical calculations, the discrete scenario-based approach for modeling uncertain mid-term product demands is adopted. Previous experience concerning uncertain product demands in short-term supply chain management problems (Chen and Lee, 2004a,b) gives strong support for applying the scenario-based approach. The mid-term planning problem considering multiple conflict objectives, including uncertain product demands, will be addressed in this article.
For applying the discrete cases representation for modeling uncertain demands, several possible outcomes for demand forecasting, FCDs; s2 S, with known probabilities, PPDs,
should be determined at first with the restriction of P
8 s 2 SPPDs¼ 1. Then, all variables will become
scenario-dependent, and the expected value of any variable will be the weighted average of those scenario-dependent values. That is, for any variable n, one has to solve for several scenario-dependent values, vs; s2 S, and the expected value of n can be
taken asP8 s 2 SPPDsns. In such a case the deterministic supply
chain model can be easily extended to cope with the uncertain demand conditions (Chen and Lee, 2004a,b).
3. Supply chain modeling with demand uncertainty A general supply chain that consists of three different levels of enterprises is considered. The first level enterprise is the retailer from which the products are sold to customers. The second level enterprise is the distribution center (DC) and/or warehouse using different types of transport capacity to deliver products from the plant side to the retailer side. The third level enterprise is the plant or the manufacturer that batch-manufactures one product over one period. In the following,
the integrated multi-echelon supply chain model of Tsiakis
et al. (2001)is extended for optimal decisions. The scenario-based representation for uncertain product demands is considered in the modeling. The indices, sets and parameters designed for modeling the supply chain network with product demand uncertainty are given in the nomenclature. Therein, parameters are divided into two categories: the cost parameters, including product cost, handling cost and transport cost; and other parameters describing the system information, such as handling and transport capacity or forecasting customer demand. Two kinds of variables are used: the binary variables that act as policy decisions to establish warehouses and distribution centers, along with using economies of scale for
manufacturing or transportation, and the continuous variables that include manufacturing quantities, handling capacity and transport quantities.
3.1. Network structure constraints
All relevant network structural constraints between all plants, warehouses, DCs and customer zones can be summarized as follows. Xpw Yw; Xwd Yw (1) Xwd Yd; Xdc Yd (2) X 8 w 2 W Xwd ¼ Yd; X 8 d 2 D Xdc¼ 1 8 p 2 P; w 2 W; d 2 D; c 2 C
Eq.(1)denotes that a link between a plant p and a warehouse w or between a warehouse w and a DC d can exist only if warehouse w exists. Similar constraints can apply to the link
between DCs and customer zones, as shown in Eq. (2). To
simplify the problem, it is assumed that a DC can only be served by a single warehouse, and a customer zone can only be served by a single DC, as shown in Eq.(3).
3.2. Transport constraints
The transportation constraints at considered periods, t2 T , in different economic scales are given below.
TCLk1 Ztsk < TTOkts TCLkZ k ts (4) X 8 k 2 K Ztsk 1 (5) X 8 i 2 I Qits¼ X 8 k 2 K Qkts (6) Qmins X X 8 i 2 I Qits Qmaxs X (7) Qits 0 (8) where 2 f pw; wd; dcg; 8 p 2 P; w 2 W; d 2 D; c 2 C; i 2 I ; t 2 T ; s 2 S
Eqs.(4) and (5)imply that several transport capacity levels with various unit transport costs can be used, as depicted in
Fig. 2for a three-level case, and at most one transport capacity can be chosen at each period. In Eq.(6), the transport quantities from plants to warehouses, from warehouses to DCs, or from DCs to customer zones at each period are respectively translated into total transport quantities. Eq.(7) says that the total transport quantities have lower/upper bonds, for all the existing links.
(3) C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407
3.3. Material balances constraints
For mid-term design, it is assumed that the operation is in a constant state. Therefore, there is no stock accumulation or depletion. All material balance constraints can thus be summarized as follows. PQipts¼ X 8 w 2 W Qipwts (9) X 8 p 2 P Qipwts¼ X 8 d 2 D Qiwdts (10) X 8 w 2 W Qiwdts¼ X 8 c 2 C Qidcts (11) X 8 d 2 D Qidcts¼ FCDi cts (12) PQti ps 0; 8 p 2 P; i 2 I ; t 2 T ; s 2 S
Eq.(9)says that the production of a product i by a plant p must be equal to the total flow of the product i to all warehouses. Eq.(10)states that the total flow of a product i from all plants to a warehouse w must be equal to the total flow of the product i from the warehouse w to all DCs.
Similarly, Eq.(11)means that the total flow of a product i from all warehouses to a DC d must equal to the total flow of a product i from the DC d to all customer zones. Eq.(10)denotes that the total flow of a product i from all DCs to a customer zone c must equal to the forecast customer demands.
3.4. Production resource constraints
An important issue in designing the network is the ability of the plants to satisfy the demands of customers. The production of each product at any plant thus has some limitations as follows. PQmini ps PQi pts PQ max i ps (14) X 8 i 2 I ripntsPQipts Rpnts; 8 p 2 P; n 2 N ; i 2 I ; t 2 T ; s 2 S
Here, Eq.(14)states that each plant has its own maximum and minimum product capacities. Many plants may apply the same resources (equipment, utilities, manpower, etc.) to produce different products at different production stages. The limitations of resource utilization can be seen in Eq.(15). 3.5. Capacity constraints
All capacity constraints for warehouses and DCs are listed in the following. SQminw Yw SQw SQmaxw Yw (16) SQmind Yd SQd SQmaxd Yd (17) SQw X 8 i 2 I X 8 d 2 D aiwQiwdts (18) SQd X 8 i 2 I X 8 c 2 C bidQidcts; 8 w 2 W; d 2 D; c 2 C; i 2 I ; t 2 T ; s 2 S (19)
Eq. (16)means that the capacity of a warehouse w has its lower bounds SQmin
w and upper bounds SQ
max
w , if the warehouse
is established. Similar constraints apply to the capacities of the distribution centers, as shown in Eq.(17). It is assumed that the capacities of the warehouses and the distribution centers are related linearly to the materials that they handle, as expressed in Eqs.(18) and (19).
3.6. Costs
The establishment costs of warehouses and distribution centers, production costs, all material handling costs, and transportation costs are given below.
TEC¼ X 8 w 2 W UECwYwþ X 8 d 2 D UECdYd (20) TPCts¼ X 8 i 2 I X 8 p 2 P UPCipPQipts (21) THCts¼ X 8 i 2 I X 8 w 2 W UHCiw X 8 p 2 P Qipwts þ X 8 i 2 I X 8 d 2 D UHCid X 8 w 2 W Qiwdts (22) TTCts¼ X 8 k 2 K ðFTCk Z k tsþ UTC k TTQ k tsÞ (23)
Fig. 2. Piecewise linear relation (solid lines) between transport cost, TTC, and shipment quantity, TTQ.
(15)
(13)
TTCts¼ X 8 p 2 P X 8 w 2 W TTCtpwsþ X 8 w 2 W X 8 d 2 D TTCtwds þ X 8 d 2 D X 8 c 2 D TTCtdcs; where 2 f pw; wd; dcg; 8 p 2 P; w 2 W; 2 D; c2 C; i 2 I ; t 2 T ; s 2 S
Eq. (20) gives the establishment costs of warehouses and distribution centers at candidate locations. In Eq.(21), the total production costs are the summations of the production quantity of product i multiply the unit production cost UPCip. Eq.(22)
states that the total material handling costs can be expressed as a linear function of each product being handled at warehouses and distribution centers. Eq.(23)denotes transport costs for the plant and DC, respectively. Here, the transport cost is a composite of transport level-dependent fixed cost and a transport quantity-dependent carrying cost. This would cause a discontinuous piecewise linear transport cost, as illustrated in Fig. 2 with skipped subscripts. Finally, Eq.(24)is the total transportation cost (Gjerdrum et al., 2001).
3.7. Multiple objectives for optimal planning
Several conflicting objectives such as minimizing the total cost, maximizing the robustness of selected objectives to demand uncertainties, maximizing the local incentives, and minimizing the total transport time can be considered simultaneously for the supply chain network design, as stated in the following.
3.7.1. Objective 1: minimizing the total cost
The total cost is a summation of the total establishment costs, the total production costs, the total handling costs, and the total transportation costs, such as
min x2 V0TCO¼ TEC þ X 8 t 2 T X 8 s 2 S PPDsðTPCtsþ THCtsþ TTCtsÞ
Where V0is the feasible searching space which is a composite of all constraints, Eqs. (1)–(24), and x denotes the decision vector
x¼ fYw; Yd; X; QitsTTQtsk ; Ztsk ; PQii ps; SQw; SQd;
2 f pw; wd; dcg; 8 i 2 I ; p 2 P; w 2 W; d2 D; c 2 C; k 2 K; t 2 T ; s 2 Sg
3.7.2. Objective 2: maximizing the robustness to various scenarios
It has been mentioned that all operating variables are scenario-dependent when the explicit scenario-based approach is applied to model the uncertain product demands. However, the total cost realization might be unacceptably high for certain
scenarios with especially low probabilities (Suh and Lee,
2001). It is thus significant to reduce the variability of objective values Jsfor any realization of scenarios. An important issue in
enforcing the robustness to uncertainties is the choice of a variability metric (Ahmed and Sahinidis, 1998). For the total cost to be minimized, the decision maker usually does not care if the objective value Jsis lower than the expected mean value J.
Thus, the upper partial mean (UPM) is used as the measure of robustness where only costs above the expectation are penalized and are weighted by probabilities of related scenarios, PPDs.
UPM¼ X
8 s 2 S
PPDsmaxf0; Js Jg
As the upper partial mean decreases, the robustness of total cost will increase, thus one can define the robustness index (RI) as below, where the nonlinear objective function is further re-formulated to equivalent linear form with additional constraint, Eq.(25). max x2 V00 RI¼ UPM ¼ X 8 s 2 S PPDsUPMs where UPMs 0; UPMs Js J; 8 s 2 S (25)
Notably, there is an additional constraint in the new feasible searching space, V00= V0\{Eq.(25)}.
3.7.3. Objective 3: maximizing the local incentives
When dealing with the location–allocation problem, there are many factors worth being considered, such as traffic facilities, labor quality, tax breaks, laws, etc. More traffic facilities such as highways, railroads, harbors, airports, etc., will decrease transport risks. Higher labor quality will have higher work efficiency. A meaningful local incentive can be defined by first identifying all important factors which cause great impact on the location–allocation problem, and second, giving weight to each factor according to its importance, and subjectively scoring the factors of each candidate location. The weighted average of these scores can be defined as the local incentive of each candidate location. Although the target is to maximize the average local incentive of all chosen locations, it will cause the nonlinear term as shown below.
max x2 V0TLI¼ P 8 w 2 WLIwYw P 8 w 2 WYw þ P 8 d 2 DLIdYd P 8 d 2 DYd
Thus the following model will be applied to simplify the solution procedure.
max
x2 V0
TLI¼ minfLIwþ Uð1 YwÞj 8 w 2 Wg
þ minfLIdþ Uð1 YdÞj 8 d 2 Dg
The model raises the minimum local incentives of the warehouses and the distribution centers as high as possible, where U is a large positive value. For those candidate locations of warehouses or distribution centers that are not chosen, their local incentives will be ignored. The above nonlinear objective (24)
C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407 398
can be re-formulated as the following linear form. max x2 V000 TLI where V000= V0\ {Eq.(26)}. TLI LIwþ Uð1 YwÞ þ LD 8 w 2 W LD LIdþ Uð1 YdÞ 8 d 2 D (26)
3.7.4. Objective 4: minimizing the total transport time Decreasing the transport time cannot only reduce the inventory levels, but can also increase the customer service levels. So, reduction of transport time is an important topic when coping with allocation–location problems. One can set the total transport time as the objective to be minimized, as shown below. min x2 V0OTT¼ X 8 p 2 P X 8 w 2 W TTpwXpwþ X 8 w 2 W X 8 d 2 D TTwdXwd þ X 8 d 2 D X 8 c 2 C TTdcXdc
In summary, the mid-term supply chain-planning model can be constructed as a multi-objective mixed-integer linear program (MILP). Notably, all objectives expressed below are set to a maximum for simplifying the discussion. The feasible searching
space V is a composite of all constraints, Eqs.(1)–(26)
max
x2 VðJ1ðxÞ; . . . ; JMðxÞÞ ¼ ðTCO; RI; TLI; OTTÞ (27)
4. Supply chain optimization with uncertain demands The conventional approaches for solving the multi-objective optimization problems are usually searching for efficient (Pareto-optimal) solutions that can best attain the prioritized objectives. Users, on the whole, have to provide a subjective account of each objective. The fuzzy optimization approach, on the other hand, can supply a single, yet unprejudiced final decision as stated in the following.
By considering the uncertain property of human thinking, it is quite intuitive to assume that the decision maker has a fuzzy goal, Jm, to describe a maximizing objective Jmwith an acceptable
interval½J0 m; J
1
m. It would be quite satisfactory as the objective
value is greater than J1
m, and unacceptable as the profit is less than
J0
m, the minimum acceptable objective value such that the
company would like to enter to negotiation for a fair deal in the multi-enterprise network. A strictly monotonic increasing membership function, mJ mðJmðxÞÞ 2 ½0; 1, can be used to
characterize such a transition from maximizing a numerical Table 1
Scenarios of forecasting product demands and probabilities of an illustrative example FCDi cts i c t s i c t s i c t s i c t s 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 10 18 28 2 3 1 105 145 105 3 5 1 32 72 82 4 7 1 40 80 88 1 1 2 8 18 30 2 3 2 82 143 125 3 5 2 22 73 92 4 7 2 35 82 92 1 1 3 5 19 46 2 3 3 55 143 145 3 5 3 12 72 98 4 7 3 24 82 98 1 2 1 0 0 0 2 4 1 0 0 0 3 6 1 0 0 0 4 8 1 35 49 55 1 2 2 0 0 0 2 4 2 0 0 0 3 6 2 0 0 0 4 8 2 31 50 75 1 2 3 0 0 0 2 4 3 0 0 0 3 6 3 0 0 0 4 8 3 25 49 81 1 3 1 0 0 0 2 5 1 0 0 0 3 7 1 0 0 0 5 1 1 0 0 0 1 3 2 0 0 0 2 5 2 0 0 0 3 7 2 0 0 0 5 1 2 0 0 0 1 3 3 0 0 0 2 5 3 0 0 0 3 7 3 0 0 0 5 1 3 0 0 0 1 4 1 9 13 15 2 6 1 0 0 0 3 8 1 0 0 0 5 2 1 51 70 76 1 4 2 5 15 17 2 6 2 0 0 0 3 8 2 0 0 0 5 2 2 41 71 86 1 4 3 4 17 20 2 6 3 0 0 0 3 8 3 0 0 0 5 2 3 23 72 96 1 5 1 0 0 0 2 7 1 9 14 17 4 1 1 226 276 283 5 3 1 0 0 0 1 5 2 0 0 0 2 7 2 8 14 27 4 1 2 180 277 293 5 3 2 0 0 0 1 5 3 0 0 0 2 7 3 8 16 37 4 1 3 150 277 316 5 3 3 0 0 0 1 6 1 45 50 60 2 8 1 0 0 0 4 2 1 103 173 203 5 4 1 0 0 0 1 6 2 40 51 70 2 8 2 0 0 0 4 2 2 80 174 223 5 4 2 0 0 0 1 6 3 32 52 80 2 8 3 0 0 0 4 2 3 50 174 233 5 4 3 0 0 0 1 7 1 0 0 0 3 1 1 0 0 0 4 3 1 80 236 266 5 5 1 0 0 0 1 7 2 0 0 0 3 1 2 0 0 0 4 3 2 54 231 282 5 5 2 0 0 0 1 7 3 0 0 0 3 1 3 0 0 0 4 3 3 38 231 298 5 5 3 0 0 0 1 8 1 0 0 0 3 2 1 55 105 155 4 4 1 0 0 0 5 6 1 0 0 0 1 8 2 0 0 0 3 2 2 40 101 125 4 4 2 0 0 0 5 6 2 0 0 0 1 8 3 0 0 0 3 2 3 20 100 215 4 4 3 0 0 0 5 6 3 0 0 0 2 1 1 0 0 0 3 3 1 0 0 50 4 5 1 0 0 0 5 7 1 0 0 0 2 1 2 0 0 0 3 3 2 0 0 71 4 5 2 0 0 0 5 7 2 0 0 0 2 1 3 0 0 0 3 3 3 0 0 83 4 5 3 0 0 0 5 7 3 0 0 0 2 2 1 199 399 499 3 4 1 126 141 150 4 6 1 0 0 0 5 8 1 0 0 0 2 2 2 150 398 519 3 4 2 100 142 161 4 6 2 0 0 0 5 8 2 0 0 0 2 2 3 120 397 549 3 4 3 76 144 182 4 6 3 0 0 0 5 8 3 0 0 0
objective value JmðxÞ to degree-of-satisfaction for Jm(Zadeh,
1965). It is noted that the design performance of the fuzzy method completely depends on the membership function. Different membership functions will have different outcomes. For prac-tical consideration, a reasonable unprejudiced procedure is expected for providing reasonable limiting values for the objec-tive. Without loss of generality, the authors first adopt the linear membership function since it has been proven in providing qualified solutions for many applications (Liu and Sahinidis, 1997). mJmðxÞ ¼ 1; for JmðxÞ Jm1 JmðxÞ Jm0 J1 m Jm0 for J0 m JmðxÞ Jm1 8 m 2 M 0; for Jm0 JmðxÞ 8 > > < > > : (28) Here, x denotes the argument vector. The effective range of
the membership function ½J0
m; Jm1, can be determined
dis-passionately as follows. For those objectives, Jm; m2 M, one
can use the most optimistic expectation as the upper limit, J1
m¼ JmðxmÞ, where x
m is the optimal solution of the single
objective maximizing problem, maxx2 VJmðxÞ, and choose the
most pessimistic expectation, J0
m, as the lower limiting value
(Zimmermann, 1978; Sakawa, 1993), where
Jm0 ¼ minfJmðxiÞ; i 2 Mg; 8 m 2 M (29)
One can, thus, impersonally determine the effective range of
membership functions with the restriction of J0
m J0m<
Jm1 J1
m. The original multi-objective optimization problem
is now equivalent to looking for a suitable decision vector that can provide the maximal degree-of-satisfaction for the aggregated fuzzy objectives, J1ðxÞ \ . . . \ JmðxÞ. When
simultaneously considering all fuzzy objectives, the final fuzzy decision,FDðxÞ, can be interpreted as the fuzzy intersection between all fuzzy objectives.
FDðxÞ ¼ J1ðxÞ \ . . . \ JMðxÞ (30)
The final overall satisfactory level, mFDðxÞ, can be
determined by aggregating the degree-of-satisfaction for all Table 2
Fixed transport costs of an illustrative example FTCkpw; FTC k wd k p w d $ k p w d $ k p w d $ k p w d $ 1 1 1 100 1 1 2 700 1 2 4 550 1 3 6 1250 2 1 1 200 2 1 2 1400 2 2 4 1100 2 3 6 2500 3 1 1 300 3 1 2 2100 3 2 4 1650 3 3 6 3750 4 1 1 400 4 1 2 2800 4 2 4 2200 4 3 6 5000 1 1 2 700 1 1 3 700 1 2 5 1300 1 3 7 800 2 1 2 1400 2 1 3 1400 2 2 5 2600 2 3 7 1600 3 1 2 2100 3 1 3 2100 3 2 5 3900 3 3 7 2400 4 1 2 2800 4 1 3 2800 4 2 5 5200 4 3 7 3200 1 1 3 700 1 1 4 200 1 2 6 800 1 4 1 250 2 1 3 1400 2 1 4 400 2 2 6 1600 2 4 1 500 3 1 3 2100 3 1 4 600 3 2 6 2400 3 4 1 750 4 1 3 2800 4 1 4 800 4 2 6 3200 4 4 1 1000 1 1 4 300 1 1 5 600 1 2 7 800 1 4 2 600 2 1 4 600 2 1 5 1200 2 2 7 1600 2 4 2 1200 3 1 4 900 3 1 5 1800 3 2 7 2400 3 4 2 1800 4 1 4 1200 4 1 5 2400 4 2 7 3200 4 4 2 2400 1 2 1 800 1 1 6 300 1 3 1 900 1 4 3 500 2 2 1 1600 2 1 6 600 2 3 1 1800 2 4 3 1000 3 2 1 2400 3 1 6 900 3 3 1 2700 3 4 3 1500 4 2 1 3200 4 1 6 1200 4 3 1 3600 4 4 3 2000 1 2 2 100 1 1 7 150 1 3 2 950 1 4 4 100 2 2 2 200 2 1 7 300 2 3 2 1900 2 4 4 200 3 2 2 300 3 1 7 450 3 3 2 2850 3 4 4 300 4 2 2 400 4 1 7 600 4 3 2 3800 4 4 4 400 1 2 3 900 1 2 1 700 1 3 3 100 1 4 5 800 2 2 3 1800 2 2 1 1400 2 3 3 200 2 4 5 1600 3 2 3 2700 3 2 1 2100 3 3 3 300 3 4 5 2400 4 2 3 3600 4 2 1 2800 4 3 3 400 4 4 5 3200 1 2 4 600 1 2 2 100 1 3 4 600 1 4 6 550 2 2 4 1200 2 2 2 200 2 3 4 1200 2 4 6 1100 3 2 4 1800 3 2 2 300 3 3 4 1800 3 4 6 1650 4 2 4 2400 4 2 2 400 4 3 4 2400 4 4 6 2200 1 1 1 100 1 2 3 750 1 3 5 1200 1 4 7 300 2 1 1 200 2 2 3 1500 2 3 5 2400 2 4 7 600 3 1 1 300 3 2 3 2250 3 3 5 3600 3 4 7 900 4 1 1 400 4 2 3 3000 4 3 5 4800 4 4 7 1200
C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407 400
objectives, mJmðxÞ, via specific t-norm, T.
mFDðxÞ ¼ TðmJ1ðxÞ; . . . ; mJmðxÞÞ (31)
The best solution xwith the maximal firing level, mFDðxÞ,
should be selected. mFDðxÞ ¼ max
x2 VmFDðxÞ (32)
Several t-norms can be chosen for T, wherein the three most popular selections are shown below (Klir and Yuan, 1995).
TðmJ1ðxÞ; . . . ; mJMðxÞÞ (33) ¼ minðmJ1ðxÞ; . . . ; mJMðxÞÞ T ¼ minimum mJ1ðxÞ . . . mJMðxÞ T ¼ product 1 MðmJlðxÞ þ . . . þ mJMðxÞÞ T ¼ 00average00 8 > < > : (34)
Therein the minimum t-norm concerns the worst scenario only, but it may result in a non-compensatory solution (Li and Lee, 1993). On the other hand, both the product t-norm and the average operator can provide a compensatory result; however, they may cause an unbalanced solution between all fuzzy terms due to their inherent character. In order to avoid numerical difficulties caused by a highly nonlinear property of product
t-norm, Chen and Lee (2004a,b) adopt the average operator,
although it is not a t-norm. The successful application experience of combining the advantages of minimum and average operators for calculating the satisfactory level of fuzzy decisions in a short-term supply chain problem (Chen and Lee, 2004a,b) is thus applied to solving multi-objective mid-term planning problems.
Step 1. Determining the membership function for each fuzzy objective based on the expected upper/lower bounds for
the objective value, as shown in Eq. (28), where
J0m J0
m J
1
m Jm1;8 m 2 M.
Step 2. (Phase I) Considering all fuzzy objectives and using the minimum operator, maximizing the degree of satisfac-tion for the worst situasatisfac-tion.
max
x2 VmFD¼ maxx2 VminðmJ1; mJ2; . . . ; mJMÞ mmin
(35) Step 3. (Phase II) Applying the average operator, maximizing the overall satisfactory level with guaranteed minimal fulfillment for all fuzzy objectives.
max
x2 VþmFDðxÞ ¼ maxx2 Vþ 1
MðmJ1ðxÞ þ . . . þ mJMðxÞÞ (36)
Fig. 3. Radar plots using single objective (a), or minimum (b) and average operator (c) (single-phase optimization) and proposed two-phase optimization method (d).
where
Vþ¼ V \ fmJm mminj 8 mg (37)
It is noted that, in the proposed two-phase optimization approach, parameters in these objective functions are not usually independent and they cannot be completely separated. Though a global solution is not guaranteed, the proposed strategy can provide a definite procedure to reach a single compensatory solution among all participants of the supply chain.
5. Numerical example
Consider a typical supply chain consisting of 2 plants, 4 candidate warehouses, 7 candidate distribution centers, 8 customer zones, and 5 products. Two plants manufacture 5 different types of products and are located in two different locations. Each plant produces several products using a number of shared production resources. There are 4 candidate locations of warehouses and 7 candidate locations of distribution centers. Each candidate warehouse and distribution center has its own establishing cost, capacity, and local incentive. The whole planning horizon is 3 periods. The product demand scenarios
are shown in Table 1 and the assigned probabilities are
PPDs=1= 0.4, PPDs=2= 0.3 and PPDs=3= 0.3, for the case
study. Also, in order to simplify the problem, the fluctuating rate for cost parameters is neglected. Other indices and sets are ½K ¼ 4 and ½N ¼ 6.
Values of all fixed transport cost parameters are listed in
Tables 2 and 3, unit transportation cost and transportation time are shown inTable 4, resource coefficients are listed inTable 5, and other parameters inTable 6.
The problem includes 11,478 equations, 7,622 continuous variables, and 3,415 binary variables. To solve this mixed-integer linear programming problem for the supply chain model, the Generalized Algebraic Modeling System (GAMS,
Brooke et al., 2003), a well-known high-level modeling system for mathematical programming problems, is used as the solution environment. The MILP solver used is CPLEX 7.5.
One can first apply the single objective programming method to minimize the total cost, the most common method in the traditional supply chain planning. Then, the result is projected, caused by single objective programming, to the membership functions, such as shown inFig. 3andTable 8. Obviously, the satisfaction levels are extremely unbalanced, since the objective function is only taking the total cost into consideration. So, one should consider all objectives simultaneously, and use Table 3
Fixed transport costs of an illustrative example FTCk dc k d c $ k d c $ k d c $ k d c $ k d c $ k d c $ k d c $ 1 1 1 100 1 2 1 700 1 3 1 700 1 4 1 300 1 5 1 600 1 6 1 300 1 7 1 200 2 1 1 200 2 2 1 1400 2 3 1 1400 2 4 1 600 2 5 1 1200 2 6 1 600 2 7 1 400 3 1 1 300 3 2 1 2100 3 3 1 2100 3 4 1 900 3 5 1 1800 3 6 1 900 3 7 1 600 4 1 1 400 4 2 1 2800 4 3 1 2800 4 4 1 1200 4 5 1 2400 4 6 1 1200 4 7 1 800 1 1 2 700 1 2 2 100 1 3 2 800 1 4 2 600 1 5 2 1300 1 6 2 1000 1 7 2 900 2 1 2 1400 2 2 2 200 2 3 2 1600 2 4 2 1200 2 5 2 2600 2 6 2 2000 2 7 2 1800 3 1 2 2100 3 2 2 300 3 3 2 2400 3 4 2 1800 3 5 2 3900 3 6 2 3000 3 7 2 2700 4 1 2 2800 4 2 2 400 4 3 2 3200 4 4 2 2400 4 5 2 5200 4 6 2 4000 4 7 2 3600 1 1 3 500 1 2 3 700 1 3 3 200 1 4 3 600 1 5 3 1000 1 6 3 900 1 7 3 700 2 1 3 1000 2 2 3 1400 2 3 3 400 2 4 3 1200 2 5 3 2000 2 6 3 1800 2 7 3 1400 3 1 3 1500 3 2 3 2100 3 3 3 600 3 4 3 1800 3 5 3 3000 3 6 3 2700 3 7 3 2100 4 1 3 2000 4 2 3 2800 4 3 3 800 4 4 3 2400 4 5 3 4000 4 6 3 3600 4 7 3 2800 1 1 4 100 1 2 4 700 1 3 4 500 1 4 4 200 1 5 4 700 1 6 4 400 1 7 4 300 2 1 4 200 2 2 4 1400 2 3 4 1000 2 4 4 400 2 5 4 1400 2 6 4 800 2 7 4 600 3 1 4 300 3 2 4 2100 3 3 4 1500 3 4 4 600 3 5 4 2100 3 6 4 1200 3 7 4 900 4 1 4 400 4 2 4 2800 4 3 4 2000 4 4 4 800 4 5 4 2800 4 6 4 1600 4 7 4 1200 1 1 5 700 1 2 5 1200 1 3 5 1000 1 4 5 800 1 5 5 100 1 6 5 100 1 7 5 600 2 1 5 1400 2 2 5 2400 2 3 5 2000 2 4 5 1600 2 5 5 200 2 6 5 200 2 7 5 1200 3 1 5 2100 3 2 5 3600 3 3 5 3000 3 4 5 2400 3 5 5 300 3 6 5 300 3 7 5 1800 4 1 5 2800 4 2 5 4800 4 3 5 4000 4 4 5 3200 4 5 5 400 4 6 5 400 4 7 5 2400 1 1 6 300 1 2 6 800 1 3 6 1000 1 4 6 600 1 5 6 700 1 6 6 100 1 7 6 500 2 1 6 600 2 2 6 1600 2 3 6 2000 2 4 6 1200 2 5 6 1400 2 6 6 200 2 7 6 1000 3 1 6 900 3 2 6 2400 3 3 6 3000 3 4 6 1800 3 5 6 2100 3 6 6 300 3 7 6 1500 4 1 6 1200 4 2 6 3200 4 3 6 4000 4 4 6 2400 4 5 6 2800 4 6 6 400 4 7 6 2000 1 1 7 200 1 2 7 800 1 3 7 600 1 4 7 300 1 5 7 500 1 6 7 500 1 7 7 100 2 1 7 400 2 2 7 1600 2 3 7 1200 2 4 7 600 2 5 7 1000 2 6 7 1000 2 7 7 200 3 1 7 600 3 2 7 2400 3 3 7 1800 3 4 7 900 3 5 7 1500 3 6 7 1500 3 7 7 300 4 1 7 800 4 2 7 3200 4 3 7 2400 4 4 7 1200 4 5 7 2000 4 6 7 2000 4 7 7 400 1 1 8 1100 1 2 8 1100 1 3 8 400 1 4 8 1000 1 5 8 1200 1 6 8 1600 1 7 8 1200 2 1 8 2200 2 2 8 2200 2 3 8 800 2 4 8 2000 2 5 8 2400 2 6 8 3200 2 7 8 2400 3 1 8 3300 3 2 8 3300 3 3 8 1200 3 4 8 3000 3 5 8 3600 3 6 8 4800 3 7 8 3600 4 1 8 4400 4 2 8 4400 4 3 8 1600 4 4 8 4000 4 5 8 4800 4 6 8 6400 4 7 8 4800
C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407 402
multi-objective programming methods to elevate satisfaction level of individual objectives.
According to the problem description, mathematical formulation, and parameter design mentioned previously, one can solve the multi-objective mixed-integer linear program by using the fuzzy procedure discussed in Section4.
Step 1. Select suitable ranges for defining membership func-tions. Relevant lower/upper limits, J0
m and Jm1, and
selected effective ranges, ½J0
m; Jm1, for membership
functions are shown in Table 7. As mentioned
previously, one can subjectively select values for
J0m and Jm1 for each objective if meaningful lower/ upper bounds can be expected. One can, thus, directly use ½J0
m; J1m as the effective range for defining fuzzy
objectives such as local incentives. Using J1 m as the
upper bound is suggested, and the second lower value as the lower bound such as the total cost and transport time. Apply the second largest value as the upper bound and the second lower value as the lower bound for robustness measurement.
Step 2. (Phase I) To maximize the degree of satisfaction for the worst objective by using the minimum operator. The result is mmin= 0.55.
Table 4
Unit transportation costs and transportation times of an illustrative example UTCk pw; UTC k wd; UTC k dcð$Þ and TTpw; TTwd; TTdcðhÞ p w d $ h w d c $ h d c $ h d c $ h 1 1 5 40 3 1 45 30 2 1 32.5 50 5 1 30 40 1 2 35 70 3 2 47.5 20 2 2 42.5 60 5 2 65 80 1 3 35 30 3 3 5 60 2 3 30 70 5 3 50 70 1 4 15 40 3 4 30 30 2 4 35 80 5 4 35 60 2 1 40 10 3 5 60 10 2 5 60 40 5 5 42.5 10 2 2 5 60 3 6 62.5 30 2 6 40 50 5 6 35 30 2 3 45 20 3 7 40 30 2 7 40 50 5 7 25 40 2 4 30 30 4 1 12.5 40 2 8 55 10 5 8 60 20 1 1 5 20 4 2 30 50 3 1 35 20 6 1 35 30 1 2 35 10 4 3 25 50 3 2 40 40 6 2 50 30 1 3 35 30 4 4 5 70 3 3 30 60 6 3 45 70 1 4 10 20 4 5 40 20 3 4 25 40 6 4 20 30 1 5 30 40 4 6 27.5 10 3 5 50 60 6 5 30 10 1 6 15 50 4 7 15 30 3 6 50 60 6 6 30 60 1 7 7.5 80 1 1 50 40 3 7 30 20 6 7 25 20 2 1 35 40 1 2 35 30 3 8 52.5 60 6 8 52.5 20 2 2 5 70 1 3 25 10 4 1 37.5 30 7 1 35 20 2 3 37.5 40 1 4 37.5 30 4 2 30 20 7 2 45 50 2 4 27.5 50 1 5 35 70 4 3 30 10 7 3 35 20 2 5 65 30 1 6 30 50 4 4 37.5 50 7 4 25 30 2 6 40 80 1 7 25 70 4 5 40 70 7 5 30 70 2 7 40 20 1 8 50 20 4 6 30 40 7 6 25 10 4 7 25 50 7 7 40 60 4 8 50 60 7 8 60 40 Table 5
Resource coefficients of an illustrative example Resource coefficient ri pnts i p n i p n i p n i p n i p n 1 1 1 0.0 2 1 1 0.4 3 1 1 0.3 4 1 1 0.2 5 1 1 0.3 1 1 2 0.0 2 1 2 0.4 3 1 2 0.4 4 1 2 0.5 5 1 2 0.2 1 1 3 0.2 2 1 3 0.0 3 1 3 0.4 4 1 3 0.1 5 1 3 0.0 1 1 4 0.3 2 1 4 0.0 3 1 4 0.0 4 1 4 0.1 5 1 4 0.4 1 1 5 0.4 2 1 5 0.2 3 1 5 0.0 4 1 5 0.0 5 1 5 0.3 1 1 6 0.5 2 1 6 0.3 3 1 6 0.2 4 1 6 0.0 5 1 6 0.0 1 2 1 0.6 2 2 1 0.4 3 2 1 0.3 4 2 1 0.2 5 2 1 0.0 1 2 2 0.0 2 2 2 0.1 3 2 2 0.1 4 2 2 0.7 5 2 2 0.2 1 2 3 0.0 2 2 3 0.7 3 2 3 0.1 4 2 3 0.0 5 2 3 0.1 1 2 4 0.2 2 2 4 0.0 3 2 4 0.3 4 2 4 0.0 5 2 4 0.3 1 2 5 0.1 2 2 5 0.0 3 2 5 0.0 4 2 5 0.0 5 2 5 0.1 1 2 6 0.4 2 2 6 0.2 3 2 6 0.0 4 2 6 0.2 5 2 6 0.4
Step 3. (Phase II) Re-optimize the problem with new con-straints of guaranteed minimum satisfaction for all fuzzy objectives. The results will be shown and discussed in the following.
The radar plots for the total cost, robustness measure, local incentives, and transport time are shown inFig. 3, where the numerical value of each objective indicates its satisfaction level, and the resulting objective and membership function values are listed inTable 8.
As shown inFig. 3(a) andTable 8, when one make a decision by considering a single objective such as minimizing the total cost (J1), the satisfaction levels for other conflict objectives
would be quite low (0.39, 0.43, 0.39) though the satisfaction level concerning total cost can be as high as 1. From the results obtained by directly selecting ‘‘minimum’’ as the t-norm (see
Fig. 3(b)), one can get a more balanced level of satisfaction among all objectives where the degrees of satisfaction are all around 0.55. It means that the overall satisfaction levels are not concerning, though one can maximize the minimum satisfac-tion level of the worst objective. By using ‘‘average operator’’ to guarantee a unique solution, however, the results are unbalanced with a lower degree of satisfaction for local incentives (0.43, as shown inFig. 3(c). On the other hand, the high robustness measure is given very high emphasis. Obviously this is not desirable for obtaining a compromise solution. Overcoming the drawbacks of the single-phase method, the proposed two-phase method can incorporate advantages of these two t-norms. The minimum operator is used in phase I to find the maximal satisfaction for the worst situation (0.55), and the average operator is applied in phase II to maximize the overall satisfaction with guaranteed minimal Table 6
Other parameters of an illustrative example
Unit handling cost, UHCi Max prod. PQ
max
i ps Unit estab. cost, UEC* Loc. Inc. LI*
I w d $ i w d $ i p w d $ w d 1 1 15 2 1 3 1 1 150 1 190000 1 1 30 1 2 15 2 2 3 2 1 600 2 80000 2 60 1 3 2 2 3 15 3 1 400 3 100000 3 70 1 4 2 2 4 15 4 1 1000 4 120000 4 40 2 1 15 2 5 8 5 1 100 1 80000 1 70 2 2 15 2 6 8 1 2 200 2 70000 2 30 2 3 2 2 7 8 2 2 700 3 80000 3 90 2 4 2 3 1 3 3 2 400 4 110000 4 50 3 1 15 3 2 3 4 2 700 5 110000 5 80 3 2 15 3 3 15 5 2 150 6 70000 6 40 3 3 2 3 4 15 7 110000 7 60
3 4 2 3 5 8 Max cap. SQmax Unit prod. cost, UPCip Resource Rpnts
4 1 15 3 6 8 w d 4 2 15 3 7 8 1 2700 i p $ p n 4 3 2 4 1 3 2 1500 1 1 140 1 1 600 4 4 2 4 2 3 3 1800 2 1 140 11 2 800 5 1 15 4 3 15 4 2000 3 1 100 11 3 300 5 2 15 4 4 15 1 1200 4 1 60 11 4 150 5 3 5 4 5 8 2 1000 5 1 40 11 5 200 5 4 2 4 6 8 3 1000 1 2 130 11 6 340 1 1 3 4 7 8 4 1500 2 2 130 22 1 310 1 2 3 5 1 3 5 1400 3 2 100 22 2 300 1 3 15 5 2 3 6 1000 4 2 60 22 3 200 1 4 15 5 3 15 7 1400 5 2 40 22 4 150 1 5 8 5 4 15 k TCLk Q max s 22 5 350 1 6 8 5 5 8 1 500 2000 22 6 200 1 7 8 5 6 8 2 1000 Qmin s aiw SQminw SQmind 3 1500 1 1 1 1 4 2000 Note : 2 f pw; wd; dcg b1d 1 Table 7
Parameters for defining membership functions for objectives
m Jm J0m J0m Jm1 J1 m 1 –TCO 1,913, 513 1,584, 529 1,219,554 1,219,554 2 RI 189, 103 163, 203 124, 319 0 3 TLI 60 60 130 130 4 –OTT 1270 800 290 290
C.-L. Chen et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 393–407 404
fulfillment for all fuzzy objectives. The average satisfaction level is increased from 0.56 (by applying the minimum operator) to 0.63, as shown inFig. 3(d).
The optimal network structures are shown inFigs. 4 and 5, where triangle means plant, square means warehouse, hexagon means distribution center, and circle means customer zone. The
values in the network structures are the total transport quantities through the whole planning horizon.Fig. 4(a) is the result of considering total cost, where other objectives are not taken into account. The transport time is quite large in this case, as shown
Fig. 4. Optimal network structure using single objective (a) and minimum operator (b).
Fig. 5. Optimal network structure using average operator (a) and two-phase optimization method (b).
Table 8
Resulting objective and membership function values
Operator TCO RI TLI OTT
Single objective (J1) Jm 1, 219, 554 151, 557 90 610 mJ m 1.00 0.39 0.43 0.37 Minimum Jm 1, 330, 728 144, 736 100 510 mJ m 0.55 0.55 0.57 0.57 Average Jm 1, 315, 214 133, 465 90 450 mJ m 0.74 0.80 0.43 0.69 Two-phase Jm 1, 323, 772 143,685 100 460 mJ m 0.71 0.57 0.57 0.67
inTable 8. The result of ‘‘average operator’’, Fig. 5(a), also does not care about the worst objective, where the local incentive is quite low. The results of ‘‘minimum operator’’ and the two-phase method are quite similar, where customer zone 6 is serviced by distribution center 3 and 7, respectively. The supply chain network designed by the two-phase method can provide superior overall performance.
6. Conclusion
This paper investigates the simultaneous optimization of multiple conflict objectives problem in a typical supply chain network with market demand uncertainties. The demand uncertainty is modeled as discrete scenarios with given probabilities for different expected outcomes. In addition to the total cost, the project considers the influence of local incentives and transport time to location decision. The problem is formulated as a mixed-integer linear programming (MILP) model to achieve minimum total cost, maximum robustness to demand uncertainties, maximum local incentives, and mini-mum total transport time. To find the degree of satisfaction of the multiple objectives, the linear increasing membership function is used; the final decision is acquired by fuzzy aggregation of the fuzzy goals, and the best compromised solution can be derived by maximizing the overall degree of satisfaction for the decision. The implementation of the proposed fuzzy decision-making method, as one can see in the case study, demonstrates that the method can provide a compensatory solution for the multiple conflict objectives problem in a supply chain network with demand uncertainties. Acknowledgement
The authors gratefully acknowledge financial support from the National Science Council (R.O.C.) under grant NSC 92-2214-E-002-002.
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