Multi-objective optimization of multi-echelon supply chain networks
with uncertain product demands and prices
Cheng-Liang Chen
∗, Wen-Cheng Lee
Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC
Abstract
A multi-product, multi-stage, and multi-period scheduling model is proposed in this paper to deal with multiple incommensurable goals for a multi-echelon supply chain network with uncertain market demands and product prices. The uncertain market demands are modeled as a number of discrete scenarios with known probabilities, and the fuzzy sets are used for describing the sellers’ and buyers’ incompatible preference on product prices. The supply chain scheduling model is constructed as a mixed-integer nonlinear programming problem to satisfy several conflict objectives, such as fair profit distribution among all participants, safe inventory levels, maximum customer service levels, and robustness of decision to uncertain product demands, therein the compromised preference levels on product prices from the sellers and buyers point of view are simultaneously taken into account. The inclusion of robustness measures as part of objectives can significantly reduce the variability of objective values to product demand uncertainties. For purpose that a compensatory solution among all participants of the supply chain can be achieved, a two-phase fuzzy decision-making method is presented and, by means of application of it to a numerical example, proved effective in providing a compromised solution in an uncertain multi-echelon supply chain network.
© 2003 Elsevier Ltd. All rights reserved.
Keywords: Supply chain management; Uncertainty; Robustness; Multiple objectives; Mixed-integer nonlinear programming; Fuzzy optimization
1. Introduction
Industries around the world are now all rushing the ter-ritory of globalization and specialization. Cooperating with good strategic partners is the sure way to tackle the po-tential problems arising from competition. Companies can achieve the optimum operating efficiency by working with other companies through communication and specialization, which evolve a new type of relationship, the supply chain relationship, among these companies and further foster a new concept in management: the supply chain management concept. A great variety of companies, those in chemical industry included, can also take advantage of this novel management scheme. Therefore, many researchers in pro-cess systems engineering (PSE) society devote themselves to this interesting field (Applequist, Pekny, & Reklaitis, 2000; Bose & Pekny, 2000; Chen, Wang, & Lee, 2003; Garcia-Flores, Wang, & Goltz, 2000; Gupta & Maranas, 2000;Gupta, Maranas, & McDonaldet, 2000;Perea-Lopez,
∗Corresponding author.+886-2-23636194; fax: +886-2-23623040.
E-mail address: CCL@ntu.edu.tw (C.-L. Chen).
Grossmann, & Ydstie, 2000; Pinto, Joly, & Moro, 2000; Zhou, Cheng, & Hua, 2000; to name a few).
In traditional supply chain management, the focus of the integration of supply chain network is usually on single objective, minimum cost or maximum profit. For example, Gjerdrum, Shah, and Papageorgiouet (2001) proposed a mixed-integer linear programming model for a production and distribution planning problem and solve the fair profit distribution problem by using the Nash-type model as the single objective function. However, there are no design tasks that are single objective problems. The design/planning/ scheduling projects are usually involving trade-offs among different incompatible goals (Cheng, Subrahmanian, & Westerberg, 2003). Recently, a multi-objective production and distribution-scheduling scheme for a supply chain sys-tem is formulated by Chen et al. (2003). In this method, in addition to maximizing profit for the entire system, fair profit distribution among all members, customer service levels, and safe inventory levels are taken into account simultaneously. All the problem parameters are determin-istically known in the model. In practice, however, this is rarely the case as it is usually difficult to foretell prices of chemicals, market demands, and availabilities of raw materials, etc., in a precise fashion (Liu & Sahinidis, 1997). 0098-1354/$ – see front matter © 2003 Elsevier Ltd. All rights reserved.
A number of works have devoted to studying supply chain management under uncertain environments. For example, Gupta and Maranas (2000), Gupta et al. (2003)incorporate the uncertain demand via a normal probability function and propose a two-stage solution framework. A generalization to handle multi-period and multi-customer problems is re-cently proposed (Gupta & Maranas, 2003). Tsiakis et al. (2001)use 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 to uncertain product demands is not considered in these studies. Due to the potential of deal-ing with ldeal-inguistic expressions and uncertain issues (Zadeh, 1965; Petrovic, Roy, & Petrovic, 1998) use fuzzy sets to handle uncertain demands and external raw material prob-lems, and further considering uncertain supply deliveries in a later work (Petrovic, Roy, & Petrovic, 1999).Giannoccaro, Pontrandolfo, and Scozzi (2003)also apply fuzzy sets the-ory to model the uncertainties associated with both market demand and inventory costs. The product price, despite with their obvious negotiable and uncertain characteristics in real businesses, seems seldom to be taken into account as a source of uncertainty in previous works. Instead, the prod-uct prices at selling points are usually treated as determined parameters.
In this paper, we incorporate two kinds of uncertain-ties including the market demands and product prices. The scenario-based approach will be adopted for modeling the uncertain market demands, and the product prices will be taken as fuzzy variables where the incompatible preference on prices for different members are handled simultaneously. The whole supply chain scheduling model would turn into a mixed-integer nonlinear programming (MINLP) problem ultimately. The compromised solution for ensuring fair profit distribution, safe inventory levels, maximum customer ser-vice levels, decision robustness to uncertain product
de-Fig. 1. Research region.
mands, and simultaneously considering incompatible prefer-ence of product prices for all participants will be determined by applying the fuzzy multi-objective optimization method. In the rest of this article, the problem statement and assumptions are outlined in Section 2. The considered uncertain issues in supply chain scheduling are de-scribed in Section 3. The formulation of a production and distribution-scheduling model is set out in Section 4. The procedure for grouping the scenario-dependent multiple conflict objectives and uncertain fuzzy product prices into a scalar one using the fuzzy sets concept is presented in Section 5. The contents of a numerical example, used to demonstrate the usefulness of the proposed method, are given in Section 6. Finally, some concluding remarks are drawn inSection 7.
2. Problem description
A general supply chain that consists of three different lev-els of enterprises is considered and showed inFig. 1(Chen et al., 2003): the first level enterprise is retailer or market from which the products are sold to customer under the con-ditions subject to a given low bound of customer service; the second level enterprise is distribution center (DC) or ware-house using different type of transport capacity to deliver products from plant side to retailer side; and the third level enterprise is plant or manufacturer that batch-manufactures one product at one period. The fixed manufacture/idle costs are also employed: on one hand, if the production line is changed over to manufacture another product, manufacture cost would be remained fixed; on the other hand, if the pro-duction line is set up to manufacture one specific product but actually is idle, the idle cost, also fixed. Furthermore, the plant has options of manufacturing in regular time or overtime to satisfy the customer demand. To simplify the problem here, we do not consider the problem of purchase and inventory of the raw material in plants nor incorporate
the purchasing cost into manufacturing cost. The research region of this paper, therefore, is from manufacturer to cus-tomer, like the dash line region showed in Fig. 1. And the following assumptions have been made:
1. Products are independent to each other, related to mar-keting and sales price.
2. Each enterprise has its own safe inventory quantity to reduce the influence of uncertain product demand. 3. Several scenarios of product demands with known
proba-bilities are forecasted over the entire scheduling periods. 4. The buyer’s acceptability for product price can be quite
different from the provider’s.
The overall problem can thus be stated as follows:
• Given:
1. Manufacture data, such as batch manufacturing quan-tity of regular time and overtime, overtime number constraint, etc.
2. Transportation data, such as lead time, transport ca-pacity, etc.
3. Inventory data, such as inventory capacity, safe inven-tory quantity, etc.
4. Each cost parameter, such as manufacturing, inventory, etc.
5. The buyers’ and providers’ acceptable levels for prod-uct prices.
6. Several scenarios of forecasted product demands with known probabilities.
• Determine:
1. Production plan of each plant. 2. Transportation plan of each DC.
3. Sales quantity and compromised product price of each participant.
4. Inventory level of each enterprise. 5. All kinds of cost.
• The target is to integrate the multi-echelon decisions
si-multaneously to:
1. Guarantee a fair profit distribution among all partici-pants.
2. Elevate the customer service levels, the safe inventory levels, the product-prices satisfaction levels, and the robustness of all considered objectives to product de-mand uncertainties as much as possible.
3. Uncertainties in the supply chain scheduling
In the market, the participants of a supply chain not only face the uncertainties of product demands and raw mate-rial supplies but also faces the uncertainties of commodity prices and costs (Liu & Sahinidis, 1997). The first concern in incorporating uncertainties into supply chain modeling and optimization is the determination of suitable repre-sentation of the uncertain parameters (Gupta & Maranas, 2003). Three distinct methods are frequently mentioned for representing uncertainty (Liu & Sahinidis, 1997; Gupta
& Maranas, 2003): first, the distribution-based approach, where the normal distribution with specified mean and standard deviation is widely invoked for modeling uncer-tain demands and/or parameters; second, the fuzzy-based approach, therein the forecast parameters are considered as fuzzy numbers with accompanied membership functions; and third, the scenario-based approach, in which several dis-crete scenarios with associated probability levels are used to describe expected occurrence of particular outcomes. We will address issues of demand uncertainty and uncertain product prices in the following.
To simplify the subsequent mathematical calculations, we will adopt the discrete scenario-based approach for model-ing uncertain product demands. For applymodel-ing the discrete cases representation for modeling uncertain demands, sev-eral possible outcomes with known probabilities, PPDs, s ∈
S where ∀s∈SPPDs = 1, should be determined at first. 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 vari-ablev, we have to solve for several scenario-dependent val-ues, vs, s ∈ S, and the expected value of v can be taken as ∀s∈SPPDsvs. In such a case the deterministic supply chain model can be easily extended to cope with uncertain demand conditions, as shown in the next section.
Due to the obvious negotiable and uncertain characteris-tics of products’ prices at various selling sites in real busi-nesses, the final product prices are usually result of com-promised considerations. When contemplating the compet-itive positions hold between sellers and buyers in settling sales/purchase prices for a specific product, the preference of the price would be very different from each one’s point of view. For example, the retailer would be fully satisfied if the selling price to customers is higher than an expected high value, say(USP)1S; on the other hand, it would be totally un-acceptable for a price less than a lower minimum(USP)0S; and the degree-of-acceptability will increase in accordance with the increase of price between these two bounds. To de-scribe such a transition from numerical price value to linguis-tic preference expression, it is quite suitable to set up a fuzzy set,SP, with some kinds of monotonic increasing member-ship function,µSP(USP) where µSP(USP ≤ (USP)0S) = 0,
µSP(USP ≥ (USP)1S) = 1, and µSP((USP)0S < USP <
(USP)1
S) = Finc(USP; (USP)0S, (USP)1S) ∈ [0, 1], to mea-sure the seller’s preference for product price. The buyer side, on the other hand, has its own fuzzy preference of purchase price,BP, and corresponding monotonic decreasing mem-bership function,µBP(USP) where µBP(USP ≤ (USP)1B) = 1,µBP(USP ≥ (USP)0B) = 0, and µBP((USP)1B< USP <
(USP)0
B) = Fdec(USP; (USP)1B, (USP)0B) ∈ [0, 1]. The de-termination of membership functions are usually based on decision maker’s subjectivity. It has been shown that use of linear membership functions can provide similar solu-tion quality to that using more complicated nonlinear mem-bership functions (Delgado, Herrera, & Verdegay, 1993; Sakawa, 1993; Liu & Sahinidis, 1997). Thus, we will assume
Fig. 2. Membership functions for seller’s (a) and buyer’s (b) fuzzy product prices.
Fig. 3. Typical membership functions for fuzzy product prices when simultaneously considering the seller’s and buyer’s viewpoints.
constant rate of increased/decreased membership satisfac-tion and will adopt linear membership funcsatisfac-tions, as shown below andFig. 2.
µSP(USP) =
0 for USP≤ (USP)0S
USP− (USP)0S
(USP)1
S− (USP)0S
for(USP)0S≤USP≤(USP)1S
1 for USP≥ (USP)1S
µBP(USP) =
1 for USP≤ (USP)1B
(USP)0
B− USP
(USP)0
B− (USP)1B
for(USP)1B≤USP≤(USP)0B
0 for USP≥ (USP)0B
(1) The final product price is determined by a compromising procedure taken place between these conflicting fuzzy pref-erences. Some typical final membership functions of the fuzzy product price—incorporating both the seller’s and buyer’s—are depicted inFig. 3.
4. Supply chain modeling with demand uncertainty
A general supply chain that consists of three different levels of enterprises is considered here: the first level en-terprise is the retailer from which the products are sold
to customers; the second level enterprise is the distribu-tion center (DC) or warehouse using different type of trans-port capacity to deliver products from plant side to retailer side; the third level enterprise is the plant or the manufac-turer that batch-manufactures one product at one period. In the following, we develop an integrated multi-echelon sup-ply chain model for optimal decisions (Chen et al., 2003). The scenario-based representation for uncertain product de-mands is considered in the modeling.
4.1. Indices, sets, parameters, and variables
The indices, sets and parameters, designed to model the supply chain network with product demand uncertainty are shown in the nomenclature. Therein, parameters are divided into two categories: the cost parameters, including inventory cost and transport cost; and other parameters describing the system information, such as inventory capacity, transport lead time, etc. Two kinds of scenario-dependent variables are used: the binary variables that act as policy decisions to use economies of scale for manufacturing or transportation, and continuous variables that include manufacturing quantities and product prices.
4.2. Manufacture constraints
Six constraints on manufacturing are set up for all prod-ucts and plants over concerned periods.
∀i∈I βi pts= 1 (2) oipts≤ αipts≤ βipts (3) γptsi ≥ βipts− βp,t−1,si (4) ∀i∈I ∀t∈T oi pts≤ MTOp (5) ∀i∈I ∀n∈N oip,t−n+1,s≤ N − 1 (6) wherei ∈ I, p ∈ P, t ∈ T, s ∈ S.
Eq. (2)denotes that the plant can be setup for manufactur-ing one product.Eq. (3)states that the plant is able to
manu-facture a product in regular time only after the plant has been setup to produce it, and the plant is able to manufacture the product in overtime only when the regular time production is insufficient. InEq. (4),γptsi = 1 only if βipts−βip,t−1,s= 1, thus the plant can be changed to manufacture producti at periodt.Eq. (5)implies total number of overtime periods is less than the maximal allowable overtime periods, MTOp. AndEq. (6) says the number of continuous overtime peri-ods should be less than a specified valueN. Notably, these constraints are scenario dependent.
4.3. Transportation constraints
The transportation constraints at considered periods, t ∈
T, in different economic scales are given below.
TCLpdk −1Ypdtsk < TQpdtsk ≤ TCLkpd Ypdtsk (7) TCLk−1dr Ydrtsk < TQkdrts≤ TCLkdrYdrtsk (8) ∀k ∈K Yk pdts≤ 1, ∀k∈K Yk drts≤ 1 (9) TQpdts= ∀k ∈K TQkpdts = ∀i∈I SQipdts (10) TQdrts= ∀k∈K TQkdrts= ∀i∈I SQidrts (11) ∀p∈P TQpdts≤ MITCd (12) ∀r∈R TQdrts≤ MOTCd (13) ∀d ∈ D, p ∈ P, r ∈ R, k ∈ K , k ∈ K, t ∈ T, s ∈ S
Eqs. (7)–(9)imply that several transport-capacity levels with various unit transport costs can be used, as depicted inFig. 4 for a three-level case, and at most one transport capacity can be chosen at each period. InEqs. (10) and (11), the transport quantities of each product from plants to DCs or from DCs to retailers at each period are respectively translated into total transport quantities, and Eqs. (12) and (13)are constraints on these total transport quantities.
4.4. Inventory constraints
All relevant inventory constraints in all plants, DCs, and retailers can be summarized as follows:
Irtsi = Ir,t−1,si + ∀d∈D SQidr,t−TLT dr,s− ∀c∈C SQircts (14) Idtsi = Id,t−1,si + ∀p∈P SQipd,t−TLT pd,s− ∀r∈R SQidrts (15)
Fig. 4. Piecewise linear relation (solid lines) between transport cost (TC) and shipment quantity (TQ).
Iptsi = Ip,t−1,si + FMQipαip,t−1,s + OMQi poip,t−1,s− ∀d∈D SQipdts (16) Birts= Bir,t−1,s+ FCDirts− ∀c∈C SQircts, BirTs= 0 (17) ∀i∈I I∗tsi ≤ MIC∗ (18) SIQi∗− I∗tsi ≤ Di∗ts (19) Ii ∗ts, SQi∗ts, Bi∗ts, Di∗ts≥ 0 (20) where∗ ∈ {p, d, r}, p ∈ P, d ∈ D, r ∈ R, i ∈ I, t ∈ T, s ∈ S.
Here, Eq. (14) states that inventory level of one prod-uct of a retailer at each period equals the amounts at pre-vious period plus the amounts received from all DCs and less the amounts sold to customers. Similar constraints ap-ply for DCs and plants, as shown in Eqs. (15) and (16). Eqs. (14) and (15)also consider delayed transport quantity caused by transport lead time.Eq. (17)means that backlog level of one product equals the amounts at previous period and added to the amounts of forecasting customer demand, less the amounts sold to customer; and the backlog at the last period should be zero for fulfilling expected customer demand.Eq. (18)says that the amounts of all products can not exceed the maximal inventory capacity. By using safe inventory quantity constraints, we can make the short safe inventory level of a product to be zero if inventory level is greater than safe inventory quantity, or to be the difference of safe inventory quantity and inventory level if inventory level is smaller than safe inventory quantity, as shown in Eq. (19).
4.5. Costs and revenues
The plants’ manufacturing costs, purchasing costs for DCs, retailers and customers, all inventory and handling costs, and the transportation costs are listed in the following for allp ∈ P, d ∈ D, r ∈ R, t ∈ T, and s ∈ S.
TMCpts=
∀i∈I
[FMCipγptsi + FICip(βipts− αipts)
+ UMCipFMQipαipts+ OMCipOMQipoipts] (21) TPCdts= ∀p∈P ∀i∈I USPipdsSQipdts, TPCrts= ∀d∈D ∀i∈I USPidrsSQidrts (22) TIC∗ts= ∀i∈I UICi∗I∗tsi ∗ ∈ {p, d, r} (23) THCpts = ∀i∈I UHCip × FMQipαip,t−1,s+ OMQipoip,t−1,s+ ∀d∈D SQipdts (24) THCdts= ∀i∈I UHCid ∀p∈P SQipd,t−TLT pd,s+ ∀r∈R SQidrts (25) THCrts= ∀i∈I UHCir ∀d∈D SQidr,t−TLT dr,s+ ∀c∈C SQircts (26) TTCpts= ∀k ∈K ∀p∈P (FTCkpd Yk pdts+ UTCk pdTQk pdts) (27) TTCdts= ∀k∈K ∀r∈R (FTCkdrYdrtsk + UTCkdrTQkdrts) (28) PSPpts= ∀d∈D ∀i∈I USPipdsSQipdts, PSPdts= ∀r∈R ∀i∈I USPidrsSQidrts, PSPrts= ∀c∈C ∀i∈I USPircsSQircts (29)
InEq. (21), the manufacturing cost is a composite obtained by fixed manufacture and idle cost plus regular and overtime manufacturing costs. Here, the γptsi value (measured if we
are going to change production plan to produce producti) will be either zero (if βipts− βip,t−1,s = 0, continuing to produce or not to produce producti) or 1 (if βipts−βip,t−1,s= 1, changeover to start producing producti).Eq. (22) gives the purchasing costs for DCs and retailers;Eq. (23)is the inventory cost, andEqs. (24) and (26)are handling costs for plants, DCs, and retailers, respectively; Eqs. (27) and (28) are transport costs for 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 inFig. 4with skipped subscripts. Notably, the discontinuities in the transport cost make the model more general than the continuous one proposed byTsiakis et al. (2001). Finally,Eq. (29)is product sales for all plants, DCs, and retailers.
4.6. Multiple objectives
The conflict objectives such as each participant’s profit, the average customer service level, and the average safe inventory level are considered simultaneously, as stated in the following.
Objective 1: to simultaneously maximize participants’
ex-pected profits forp ∈ P, d ∈ D, and r ∈ R.
Instead of directly maximizing the overall profit of the integrated supply chain network, we intend to fairly distribute the profit to all members within scheduling periods. The profits of all participants to be maximized are considered separately, where the profit at periodt is equal to the product sales less all kinds of costs.
Z∗= ∀s∈S PPDsZ∗s, ∗ ∈ {p, d, r} (30) where Zps= ∀t∈T (PSPpts− TMCpts− TTCpts − TICpts− THCpts) ∀p ∈ P, s ∈ S Zds= ∀t∈T (PSPdts− TPCdts− TTCdts − TICdts− THCdts) ∀d ∈ D, s ∈ S Zrs= ∀t∈T (PSPrts− TPCrts− THCrts− TICrts) ∀r ∈ R, s ∈ S.
Objective 2: to maximize average safe inventory levels for p ∈ P, d ∈ D, r ∈ R.
The safe inventory level of retailerr at period t for scenarios is defined as the expected average percentage of 1 less the ratio of short safe inventory level of product
i of retailer r at period t, Di
quantity of producti of retailer r, SIQir, for all products. Similar definitions are also applied to plants and DCs. All participants’ safe inventory levels are concerned as objectives for simultaneous optimization.
SIL∗= ∀s∈S PPDsSIL∗s, ∗ ∈ {p, d, r} (31) where SIL∗s(%) = 100 IT ∀t∈T ∀i∈I 1− D i ∗ts SIQi∗ , ∀s ∈ S Objective 3: to maximize average customer service levels
forr ∈ R.
Under the condition of taking all products into con-sideration, the customer service level of retailer r at periodt is defined as the expected average percentage ratio of actual sales quantity of producti from retailer
r to customers at period t, ∀c∈CSQircts, over the ex-pected total demand quantity. The exex-pected total de-mand quantity is the sum of backlog level of product
i for retailer r at the end of period t − 1, Bi
r,t−1,s, and
forecasting customer demand of producti to retailer r at periodt,∀c∈CFCDircts. CSLr = ∀s∈S PPDsCSLrs, ∀r ∈ R (32) where CSLrs(%) = 100 IT ∀t∈T ∀i∈I × ∀c∈CSQircts Bir,t−1,s+∀c∈CFCDircts , ∀s ∈ S. Objective 4: to maximize robustness of selected objectives
to demand uncertainties.
It has been mentioned that all variables are scenarios dependent when the explicit scenario-based approach is applied to the uncertain product demand. However, the profit realization might be unacceptably low for cer-tain scenarios with especially low probabilities (Suh & Lee, 2001). It is thus significant to reduce the variabil-ity of above-mentioned objective values for any real-ization of scenarios. An important issue in enforcing robustness to uncertainties is the choice of variabil-ity metric (Ahmed & Sahinidis, 1998). For those ob-jectives to be maximized as mentioned previously, the decision-maker usually does not care if the objective value is greater than the expected one. We thus propose the lower partial mean as the measure of robustness, where only objective values less than the expectation are penalized and are weighted by probabilities of re-lated scenarios.
RIm =
∀s∈S
PPDsmin{0, Jm s− Jm }, ∀m ∈ M
(33)
Here,M is the index set of objectives inEqs. (30)–(32) with dimension [M ]= M = 2P + 2D + 3R.
In summary, the feasible searching space, Ω, are composite of all constraints mentioned above. The multiple objectives Jm(x), m ∈ M, where M is the index set of all objectives including the robustness indices with [M] = M = 2M , and the decision vector x are shown below. Notably, further to the scenario-dependent product prices, the decision vector
x includes all production, transportation, sales
quanti-ties and inventory levels during considered periods. max
x∈Ω(J1(x), . . . , JM(x))
= (Zp, Zd, Zr; SILp, SILd, SILr; CSLr; RIm ;
∀p ∈ P, d ∈ D, r ∈ R, m ∈ M ) (34) x = {αipts, βipts, γptsi , oipts; SQi∗ts; USPi∗s; TQk pdts, TQkdrts, Ypdtsk , Ydrtsk ; Iptsi , Idtsi , Irtsi ; Birts; Dipts, Didts, Dirts; ∗ ∈ {pd, dr, rc}; i ∈ I, p ∈ P, d ∈ D, r ∈ R, c ∈ C, k ∈ K , k ∈ K, t ∈ T, s ∈ S} (35)
5. Supply chain optimization with uncertain demands and prices
By considering the uncertain property of human thinking, it is quite intuitive to assume that the DM has a fuzzy goal,
Jm, to describe the maximizing objective Jm with an ac-ceptable interval [Jm0, Jm1]. It would be quite satisfactory as the objective value is greater thanJm1, and unacceptable as the profit is less thanJm0, the minimum acceptable objective value such that the company would like to enter to negotia-tion for a fair deal in the multi-enterprise network. A strictly monotonic increasing membership function,µJm(Jm(x)) ∈ [0, 1], can be used to characterize such a transition from nu-merical objective valueJm(x) to degree-of-satisfaction for
Jm. Without loss of generality, we will adopt the linear membership function since it has been proved in providing qualified solutions for many applications (Liu & Sahinidis, 1997). µJm(Jm(x)) = 1; forJm(x) ≥ Jm1 Jm(x) − Jm0 J1 m− Jm0 ; for J 0 m≤ Jm(x) ≤ Jm1 ∀m ∈ M 0; forJm0 ≥ Jm(x) (36) Here,x denotes the argument vector as shown inEq. (35). The effective range of the membership function, [Jm0, Jm1], can sometimes be subjectively defined by company’s deci-sion makers. Some procedures can also follow for providing reasonable limiting values for the objective. For those ob-jectives such as profits and inventory levels and customer
service levels, Jm , m ∈ M , one can use the most op-timistic expectation as the upper limit, ¯Jm1 = Jm (x∗m ),
where x∗m is the optimal solution of the single objective maximizing problem, maxx∈ΩJm (x). And choose the most
pessimistic expectation, J ¯ 0
m , as the lower limiting value
(Zimmermann, 1978; Sakawa, 1993), where
J
¯ 0
m = min{Jm (x∗i), i ∈ M }, ∀m ∈ M (37) As for objectives measuring robustness to uncertainties, the reasonable upper limit is zero (i.e., absolute robustness),
¯J1
M +m = 0, and the lower limiting value can be found
similar toEq. (37).
J
¯ 0
M +m = min{JM +m (x∗i), i ∈ M }, ∀m ∈ M (38) One can thus subjectively determine the effective range for membership functions with the restriction ofJ
¯ 0
m ≤ Jm0 < J1
m ≤ ¯Jm1. The original multi-objective optimization
prob-lem is now equivalent to look for a suitable decision vector that can provide the maximal degree-of-satisfaction for the aggregated fuzzy objectives,J1∩ . . . ∩ JM. When simulta-neously considering the incompatible fuzzy preference on product prices from sellers and buyers viewpoints, the final fuzzy decision,FD, can be interpreted as the fuzzy intersec-tion between all fuzzy objectives and fuzzy product prices.
FD = Jm∩ SPipd∩ SPidr∩ SPrci ∩ BPipd∩ BPidr∩ BPirc
∀m ∈ M, i ∈ I, p ∈ P, d ∈ D, r ∈ R, c ∈ C (39) Noted that the expected product prices, USPi∗ = ∀s∈S PPDsUSPi∗s, ∗ ∈ {pd, dr, rc}, should be used in evaluating the degree-of-acceptability of various fuzzy preferences. The final overall satisfactory level, µFD(x), can be de-termined by aggregating the degree-of-satisfaction for all objectives, µJm(Jm(x)), and sellers’ and buyers’ prefer-ence on product prices,µSPi
∗(USP i ∗) and µBPi ∗(USP i ∗), via specific t-norm,T. µFD(x) = T(µJm; µSPi pd, µSPidr, µSPirc; µBPi pd, µBPidr, µBPirc| ∀m, i, p, d, r, c) (40)
The best solutionx∗with the maximal firing level,µFD(x∗), should be selected.
µFD(x∗) = maxx∈ΩµFD(x) (41)
Several t-norms can be chosen for T, therein two most popular selections are shown below(Klir & Yuan, 1995).
T(µJm; µSPi pd, µSPidr, µSPirc; µBPipd, µBPidr, µBPirc) (42) = min(µJm; µSPi pd, µSPidr, µSPirc; µBPi pd, µBPidr, µBPirc), T = minimum µJm× µSPi pd× µSPidr× µSPirc ×µBPi pd× µBPidr× µBPirc, T = product (43)
The minimum operator concerns the worst situation only, and the product operator results in a Nash-type objec-tive. Maximizing the single worst scenario may end in a non-compensatory solution, and maximizing the Nash-type objective can guarantee a compensatory solution (Li & Lee, 1993). But the drawback is that product operator may cause an unbalanced solution between all fuzzy terms by the in-herent character of product. We thus propose a two-phase method to combine advantages of these two operators, as summarized in the following (Chen et al., 2003).
Step 1. Determining membership function for each fuzzy
objective based on expected upper/lower bounds for the ob-jective value, as shown inEq. (36), where
J
¯ 0
m≤ Jm0 ≤ Jm1 ≤ ¯Jm1, ∀m ∈ M.
Step 2 (Phase I). Considering all fuzzy objectives and fuzzy
product prices and using the minimum operator, maximizing the degree of satisfaction for the worst situation.
max
x∈ΩµFD= maxx∈Ωmin(µJm; µSPipd, µSPidr, µSPirc; µBPi
pd, µBPidr, µBPirc) ≡ µmin (44)
Step 3 (Phase II). Applying the product operator,
maximiz-ing the overall Nash-type satisfactory level with guaranteed minimal fulfillment for all fuzzy objectives and sales pref-erences.
Table 1
Scenarios (s = 1, 2, 3, 4, 5, 6 and 7) of forecasting product demands and probabilities of illustrative example
i r t FCDirts 0.15a 0.2a 0.12a 0.24a 0.09a 0.13a 0.07a 1 1 1 160 160 160 160 160 160 160 1 1 2 170 170 170 160 150 150 150 1 1 3 180 180 180 160 140 140 140 1 1 4 190 180 170 160 150 140 130 1 1 5 200 180 160 160 160 140 120 1 2 1 180 180 180 180 180 180 180 1 2 2 200 200 180 200 160 160 160 1 2 3 220 220 180 220 140 140 140 1 2 4 220 240 180 200 140 120 160 1 2 5 220 260 180 180 140 100 180 2 1 1 240 240 240 240 240 240 240 2 1 2 240 270 210 210 270 270 210 2 1 3 240 300 180 180 300 300 180 2 1 4 240 270 210 150 330 300 180 2 1 5 240 240 240 120 360 300 180 2 2 1 280 280 280 280 280 280 280 2 2 2 320 240 320 320 240 240 280 2 2 3 360 200 360 360 200 200 280 2 2 4 320 200 400 360 160 240 280 2 2 5 280 200 400 360 120 280 280 a PPD s.
max x∈Ω+µFD(x) = maxx∈Ω+(µJm× µSPipd× µSPidr× µSPirc × µBPi pd× µBPidr× µBPirc) (45) where Ω+= Ω ∩ {µJm, µSPi ∗, µBPi∗≥ µmin|∗ ∈ {pd, dr, rc}; ∀m, i, p, d, r, c} (46) 6. Numerical example
Considering a small-scale but typical supply chain con-sists of one plant, two DCs, two retailers, and two prod-ucts. The first DC, a smaller scale but faster delivery service distributor, can rapidly respond to the suddenly increasing customer demand to keep retailer’s customer service level, but this also implies a higher operational cost; the second one, a large scale but slower delivery service distributor, on the other hand, can use the economies of scale to transport goods at lower operational cost, but the prompt delivery is not available, however. We assume one period of transport lead time between each level for the second DC. So, the distribution channel between plants to retailers is comple-mentary with the faster, smaller shipment and slower, larger shipment. And in order to simplify the problem, we neglect Table 2
Cost parameters of illustrative example
Table 3
Parameters for defining fuzzy product prices in illustrative example
• i p d r Seller Buyer (•)0 S (•)1S (•)1B (•)0B USPidr 1 1 1 1350 1450 1400 1500 1 1 2 1400 1500 1450 1550 1 2 1 1250 1350 1300 1400 1 2 2 1200 1300 1250 1350 2 1 1 650 750 700 800 2 1 2 700 800 750 850 2 2 1 600 700 650 750 2 2 2 550 650 600 700 USPipd 1 1 1 850 950 900 1000 1 1 2 750 850 800 900 2 1 1 400 500 450 550 2 1 2 300 400 350 450 USPirc 1 1 1650 1750 1700 1800 1 2 1600 1700 1650 1750 2 1 1000 1100 1050 1150 2 2 950 1050 1000 1100
the fluctuating rate for cost parameters. The whole schedul-ing horizon is five periods. The product demand scenarios and the assigned probabilities are shown inTable 1for the case study. Other indices and sets are [N] = 3, [K] = 3, and [K ]= 3. Values of all cost parameters are listed inTables 2 and 3, and other parameters,Table 4. Notably,Table 3gives the required parameters for defining fuzzy product prices Table 4
Table 5
Parameters for defining membership functions for objectives
m Jm J ¯ 0 m Jm0 Jm1 ¯Jm1 1 Zp=1 2, 273, 758 2, 273, 758 2, 664, 123 4, 623, 672 2 Zd=1 −290, 481 37, 429 570, 869 1, 104, 309 3 Zd=2 −467, 993 762, 862 1, 109, 662 3, 155, 891 4 Zr=1 −797, 333 28, 323 639, 630 1, 345, 449 5 Zr=2 −755, 271 276, 206 648, 639 1, 553, 022 6 SILp=1 0.09 0.09 0.92 0.92 7 SILd=1 0.07 0.07 0.97 0.97 8 SILd=2 0.02 0.02 0.95 0.95 9 SILr=1 0.06 0.06 0.94 0.94 10 SILr=2 0.04 0.04 0.93 0.93 11 CSLr=1 0.80 0.80 1.00 1.00 12 CSLr=2 0.77 0.77 0.96 0.96 13 RIZp=1 −204, 289 −129, 519 14 RIZd=1 −101, 687 −75, 487 15 RIZd=2 −193, 525 −159, 639 16 RIZr=1 −200, 097 −117, 435 17 RIZr=2 −244, 216 −139, 514 18 RISILp=1 −0.066 −0.048 19 RISILd=1 −0.073 −0.060 20 RISILd=2 −0.057 −0.039 21 RISILr=1 −0.086 −0.034 22 RISILr=2 −0.085 −0.066 23 RICSLr=1 −0.036 −0.017 24 RICSLr=2 −0.026 −0.019
for sellers and buyers, respectively. Membership functions shown inFig. 3are respectively adopted for all aggregated fuzzy product prices.
The problem includes 5702 equations, 4079 contin-uous variables, and 910 binary variables. To solve this mixed-integer nonlinear programming problem for the sup-ply chain model, the Generalized Algebraic Modeling Sys-tem (GAMS,Brooke et al., 1998), a well-known high-level modeling system for mathematical programming problems,
Fig. 5. Radar plots by using minimum or product t-norms (single-phase optimization) and proposed two-phase optimization method. (a) Objectives such as profits, inventory levels, and service levels; (b) robustness measures.
Table 6
The compromised product prices of the illustrative example
i p d r USP µBP µSP USPidr 1 1 1 1425 0.75 0.75 1 1 2 1475 0.75 0.75 1 2 1 1326 0.74 0.76 1 2 2 1276 0.75 0.75 2 1 1 725 0.75 0.75 2 1 2 775 0.75 0.75 2 2 1 676 0.74 0.76 2 2 2 626 0.74 0.76 USPipd 1 1 1 926 0.74 0.76 1 1 2 826 0.75 0.75 2 1 1 476 0.74 0.76 2 1 2 375 0.75 0.75 USPirc 1 1 1728 0.72 0.78 1 2 1678 0.72 0.78 2 1 1079 0.72 0.78 2 2 1031 0.69 0.81
is used as the solution environment. The MINLP solver used is DICOPT and the NLP solver, CONOPT.
According to the problem description, mathematical for-mulation, and parameter design mentioned previously, we solve the multi-objective mixed-integer non-linear program-ming problem for a production and distribution scheduling by using the fuzzy procedure discussed inSection 5.
Step 1. Select suitable ranges for defining membership
functions. Relevant lower/upper limits, J ¯ 0
m and ¯Jm1, and
selected effective ranges, [Jm0, Jm1], for membership func-tions are shown in Table 5. As mentioned previously, one can subjectively select values for Jm0 andJm1 for each ob-jective if meaningful lower/upper bounds can be expected.
Table 7
Variability of scenario-dependent objective values
Scenario Profit Safe inventory level Service level
p = 1 d = 1 d = 2 r = 1 r = 2 p = 1 d = 1 d = 2 r = 1 r = 2 r = 1 r = 2 RI not included 1 2709353 353342 1300455 657240 288886 0.60 0.64 0.43 0.60 0.53 1.00 1.00 2 2036686 413862 821158 852790 917316 0.47 0.82 0.50 0.74 0.53 1.00 1.00 3 2806934 250369 1196849 79628 736006 0.49 0.71 0.57 0.50 0.62 0.99 1.00 4 2447388 375178 936194 234508 756312 0.61 0.97 0.57 0.59 0.75 0.99 0.85 5 2648390 421579 1159809 210789 −271410 0.65 0.84 0.55 0.68 0.74 1.00 1.00 6 2760923 123027 1053518 229115 46819 0.64 0.91 0.57 0.62 0.67 1.00 1.00 7 2387742 318640 772951 −253262 853122 0.69 0.95 0.60 0.63 0.75 1.00 0.99 Expected 2502362 332101 1023055 366009 538010 0.58 0.84 0.53 0.62 0.65 1.00 0.96 RI −114353 −37930 −78733 −141041 −152013 −0.033 −0.050 −0.021 −0.025 −0.046 −0.004 −0.026 RI included 1 2627463 327660 987458 396451 637799 0.53 0.65 0.51 0.52 0.55 0.95 0.96 2 2580353 327660 1034208 350482 595791 0.53 0.65 0.51 0.52 0.55 0.95 0.96 3 2502549 327660 1151319 350482 565329 0.53 0.65 0.51 0.52 0.55 0.95 0.96 4 2502549 327660 1050643 350482 565329 0.53 0.65 0.51 0.52 0.55 0.95 0.96 5 2485710 327660 817240 299611 452736 0.53 0.65 0.51 0.52 0.55 0.95 0.96 6 2451158 327660 765471 332659 528373 0.53 0.65 0.51 0.52 0.55 0.95 0.96 7 2129671 327660 987458 350482 536394 0.53 0.65 0.51 0.52 0.55 0.95 0.96 Expected 2502549 327660 987458 350482 565329 0.53 0.65 0.51 0.52 0.55 0.95 0.96 RI −34298 0 −44178 −6895 −16963 0.00 0.00 0.00 0.00 0.00 0.00 0.00
We thus directly use [J ¯ 0
m, ¯Jm1] as the effective range for
defining fuzzy objectives such as inventory levels and customer service levels. Due to the wide variability of scenario-dependent profits, it is suggested using the lowest positive profit as the lower bound, and the second largest value as the upper bound. For emphasizing robustness mea-sures, we suggest adopting the second lower value as the lower bound, and the zero as the upper bound.
Step 2 (Phase I). To maximize the degree of satisfaction
for the worst objective by using the minimum operator. The result isµmin= 0.53.
Step 3 (Phase II). Re-optimize the problem with new
con-straints of guaranteed minimum satisfaction for all fuzzy ob-jectives and fuzzy product prices. The results will be shown and discussed in the following.
The radar plots for profits, safe inventory levels, customer service levels, and the robustness measures are shown in Fig. 5, and the resulting compromised sales prices are listed inTable 6.
Form the results obtained by selecting minimum as
t-norm, we can get a more balanced satisfaction among all
objectives where the degrees of satisfaction are all around 0.53. By using product operator to guarantee a unique solu-tion, however, the results are unbalanced with lower degree of satisfaction for profits ofr = 1 and d = 1, and the safe inventory level ofd = 2. On the other hand, the high profit ofr = 2 and service levels of r = 1 and r = 2 are 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 worst situation, and the product operator is applied in phase II to maximize the overall satisfaction with guaran-teed minimal fulfillment for all fuzzy objectives and fuzzy product prices.
Resulting objective values for all scenarios either consid-ering robustness measures or not are listed inTable 7. It can be found that the variability of scenario-dependent objec-tive values is quite high if the robustness measures are not included as objectives. In such a case, some profit realiza-tions are unacceptably low for certain scenarios especially for retailersr = 1 and r = 2. When simultaneously consid-ering the robustness measures as objectives, i.e., objective values less than the expectation are penalized, the vari-ability of scenario-related objective values is significantly reduced.
7. Conclusion
This paper investigates the simultaneous optimization of multiple conflict objectives and the uncertain product prices 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, and the uncertain product prices are described as fuzzy variables. The problem is formulated as a MINLP model to achieve fair profit distribution among whole network’s participants, safe inventory levels, max-imum customer service levels, maxmax-imum robustness to
demand uncertainties, and to guarantee maximum accept-ability levels of sellers’ and buyers’ preference on product prices. Considering the robustness measures as part of multiple objectives can significantly reduce the variability of other objective values to product demand uncertainties. To find the degree of satisfaction of the multiple objec-tives, the linear increasing membership function is used; the final decision is acquired by fuzzy aggregation of the
Nomenclature
Index/set Dimension Physical meaning
c ∈ C [C] = C customers
d ∈ D [D] = D distribution centers
i ∈ I [I] = I products
k ∈ K [K] = K transport capacity level, DC to retailer
k ∈ K [K ]= K transport capacity level, plant to DC
m ∈ M [M] = M all objectives
m ∈ M [M ]= M objectives 1–3
n ∈ N [N] = N counter for overtime manufacturing
p ∈ P [P] = P plants
r ∈ R [R] = R retailers
t ∈ T [T] = T periods
s ∈ S [S] = S scenarios
Parameters ∗ ∈ Physical meaning
FCDi∗ts {r} forecast customer demand ofi
FICi∗ {p} fix idle cost to keepp idle
FMCi∗ {p} fix manufacture cost changed to makei
FMQi∗ {p} fix manufacture quantity ofi
FTCk∗ {dr} kth level fix transport cost, d to r
FTCk∗ {pd} k th level fix transport cost,p to d MIC∗ {p, d, r} maximum inventory capacity ofp, d, r
MITC∗ {d} maximum input transport capacity ofd
MOTC∗ {d} maximum output transport capacity ofd
MTO∗ {p} maximum total overtime manufacture period
OMCi∗ {p} overtime unit manufacture cost ofi
OMQi∗ {p} overall fix manufacture quantity
PPD∗ {s} probability for scenarios
SIQi∗ {p, d, r} safe inventory quantity inp, d, r
SQLi∨∗ {p, d} sales quantity levels ofi, (∗, ∨) ∈ {(p, % ), (d, %)} UHCi∗ {p, d, r} unit handling cost ofi for p, d, r
UICi∗ {p, d, r} unit inventory cost ofi for p, d, r
UMCi∗ {p} unit manufacture cost ofi
(USPi∗)•+ {pd, dr, rc} parameters for defining piecewise unit sale prices,+ ∈ {S, B}, • ∈ {0, 1}
UTCk∗ {dr} kth level unit transport cost, d to r
UTCk∗ {pd} k th level unit transport cost,p to d
TCLk∗ {dr} kth transport capacity level, d to r
TCLk∗ {pd} k th transport capacity level,p to d
TLT∗ {pd, dr} transport lead time,p to d (d to r)
Binary variables ∗ ∈ Meaning when having value of 1
Yk
∗ts {dr} kth transport capacity level, d to r
Yk
∗ts {pd} k th transport capacity level,p to d
αi
∗ts {p} manufacture with regular time workforce
βi
∗ts {p} setup plantp to manufacture i
fuzzy goals and the fuzzy product prices, and the best com-promised 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 pro-vide a compensatory solution for the multiple conflict ob-jectives and the fuzzy product prices problem in a supply chain network with demand uncertainties.
γi
∗ts {p} p changeover to manufacture i
oi
∗ts {p} manufacture with overtime workforce
Real variables ∗ ∈ Physical meaning
Bi
∗ts {r} backlog level ofi in r at end of t
Di
∗ts {p, d, r} short safe inventory level inp, d, r
Ii
∗ts {p, d, r} inventory level ofi in p, d, r
J∗ {m} objectives
PSP∗ts {p, d, r} product sales ofp, d, r
RI∗ {m } robustness index of objectives
SQi∗ts {pd, dr, rc} sales quantity ofi from p to d, d to r, r to c TIC∗ts {p, d, r} total inventory cost ofp, d, r
THC∗ts {p, d, r} total handling cost ofp, d, r
TMC∗ts {p} total manufacture cost ofp
TPC∗ts {d, r} total purchase cost ofd, r
TQk∗ts {dr} kth level transport quantity, d to r
TQk∗ts {pd} k th level transport quantity,p to d
TQ∗ts {pd, dr} total transport quantity,p to d or d to r TTC∗ts {d; pd, dr} total transport cost ofd; p to d or d to r USPi∗s {pd, dr, rc} unit product price ofi, p to d, d to r, and r to c SIL∗t {p, d, r} expected safe inventory level ofp, d, r
CSL∗t {r} expected customer service level ofr
Z∗t {p, d, r} expected net profit ofp, d, r
Fuzzy variables ∗ ∈ Physical meaning
BPi∗ {pd, dr, rc} fuzzy sets to measure buyer’s preference for product price
FD fuzzy set for final decision
Jm fuzzy set for objectivem, m ∈ M
SPi
∗ {pd, dr, rc} fuzzy sets to measure seller’s preference for product price
Acknowledgements
This work was supported by the National Science Council (ROC) under Contract NSC91-2214-E-002-001.
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