Open access peer-reviewed chapter

A Fuzzy Goal Programming Approach for Collaborative Supply Chain Master Planning

By Manuel Díaz-Madroñero and David Peidro

Submitted: November 9th 2010Reviewed: May 19th 2011Published: August 1st 2011

DOI: 10.5772/20642

Downloaded: 2003

1. Introduction

Supply chain management (SCM) can be defined as the systemic, strategic coordination of the traditional business functions and the tactics across these business functions within a particular company and across businesses within the supply chain (SC), for the purposes of improving the long term performance of the individual companies and the SC as a whole (Mentzer et al. 2001). One important way to achieve coordination in an inter-organizational SC is the alignment of the future activities of SC members, hence the coordination of plans. It is often proposed that operations planning in supply chains can be organized in terms of a hierarchical planning system (Dudek & Stadtler 2005). This approach assumes a single decision maker with total visibility of system details who makes centralized decisions for the entire SC. However, if partners are reluctant to reveal all of their information or it is too costly to keep the information of the entire supply chain up-to-date, the hierarchical planning approach is unsuitable or infeasible (Stadtler 2005). Hence, the question arises of how to link, coordinate and optimize production planning of independent partners in the SC without intruding their decision authorities and private information (Nie et al. 2006).

Stadtler (2009)defines collaborative planning (CP) as a joint decision making process for aligning plans of individual SC members with the aim of achieving coordination in light of information asymmetry. Then, to generate a good production-distribution plan in a SC, it is necessary to resolve conflicts between several decentralised functional units, because each unit tries to locally optimise its own objectives, rather than the overall SC objectives. Because of this, in the last few years, the visions that cover a CP process such as a distributed decision-making process are getting more important (Hernández et al. 2009).

Selim et al. (2008) assert that fuzzy goal programming (FGP) approaches can effectively be used in handling the collaborative production and distribution planning problems in both centralized and decentralized SC structures. The reasons of using FGP approaches in this type of problems are explained by Selim et al. (2008) as follows:

  1. Collaborative planning is the more preferred mode of operation by today’s companies operated in SCs. These companies may consent to sacrifice the aspiration levels for their goals to some extent in the short run to provide the loyalty of their partners or to strengthen their partners’ competitive position in the long term. In this way, they can facilitate providing a long-term collaboration with their partners and subsequently gaining a sustainable competitive advantage.

  2. Due to the impreciseness of the decision makers’ aspiration levels associated with each goal, conventional deterministic goal programming (GP) approach cannot fully reflect such complexity.

  3. Collaborative planning problems in SCs are complex and mostly multiple objective problems, and often include incommensurable goals. Incommensurability problem in goal programming occurs when deviational variables measured in different units are summed up directly. In goal programming technique, a normalization constant is needed to overcome this difficulty. However, in FGP, incommensurable goals can be treated in a reasonable and practical way.

Therefore, it maybeappropriateto use FGP approachesin productionanddistributionplanning problems existing inreal-worldsupply chains.

We arrange the rest of this work as follows. Section 2 presents a literature review about integrated production and distribution planning models, as well as collaborative. Section 3 describes the FGP approaches to deal with supply chain planning problem in centralized and decentralized SC structures. Section 4 presents a multi-objective, multi-product and multi-period model for the master planning problem in a ceramic tile SC. Then, in Section 5, the solution methodology and the FGP approaches for different SC structures (i.e. centralized and decentralized) are described. Section 6 validates and evaluates our proposal by using an example based on a real-world problem. Finally, Section 7 provides conclusions and directions for further research.

2. Literature review

The considered ceramic supply chain master planning (CSCMP) problem deals with a medium termproduction and distribution planning problem in a four-echelon ceramic tile supply chain involving one manufacturer, multiple warehouses, multiple logistic centres and multiple shops. The integration of production and distribution planning decisions is crucial to ensure the overall performance of the SC, and has attracted attention both from practitioners and academics for many years (Vidal & Goetschalckx 1997; Erengüç et al. 1999; Bilgen & I. Ozkarahan 2004; Mula et al. 2010). According to Liang & Cheng (2009), in production and distribution planning problems, the decision maker (DM) attempts to:

  1. set overall production levels for each product category for each source (manufacturer) to meet fluctuating or uncertain demand for various destinations (distributors) over the intermediate planning horizon and

  2. make suitable strategies regarding regular and overtime production, subcontracting, inventory, and distribution levels, and thus determining appropriate resources to be used.

On supply chain planning, most prior studies have concentrated on formulating a sophisticated supply chain planning model and devising an efficient algorithm to solve it undera centralized supply chain environment where all supply chain participants are grouped as one organization orcompany and all functions of a supply chain are fully integrated by an independent planning department orsupervisor(Jung et al. 2008). According to Mula et al. (2010), the vast majority of works that deal with the production and distribution integration opt for the linear-programming based approach, particulary mixed integer linear programming models. Chen & Wang (1997) proposed a linear programming model to solve integrated supply, production and distribution planning in a supply chain of the steel sector. McDonald & Karimi (1997) presented a mixed deterministic integer linear programming model to solve a production and transport planning problem in the chemical industry in a multi-plant, multi-product and multi-period environment. Timpe & Kallrath (2000) and Kallrath (2002) presented a couple of models for production, distribution and sales planning with different time scales for business and production aspects. Dhaenens-Flipo & Finke (2001)modelled a multi-facility, multi-item, multi-period production and distribution model in the form of a network flow. Park (2005) suggested an integrated transport and production planning model in a multi-site, multi-retailer, multi-product and multi-period environment. Likewise, the author also presented a production planning submodel whose outputs act as the input in another submodel with a transport planning purpose and an overall objective of maximizing overall profits with the same technique. Ekşioğlu et al. (2006) showed an integrated transport and production planning model in a multi-period, multi-site, monoproduct environment as a flow or graph network to which the authors added a mixed integer linear programming formulation. Later, Ekşioğlu et al. (2007) extended this model to become a multi-product model solved by Lagrangian decomposition. Ouhimmou et al. (2008) developed a mixed integer linear programming (MIP) model for tactical planning in a furniture supply chain related to production and logistics decisions. Fahimnia et al. (2009) proposed a model for the optimization of thecomplex two-echelon supply networks based on the integration of aggregate production plan and distribution plan.

According to Dudek & Stadtler (2005)the relevant literature on linking andcoordinating the planning process in a decentralized manner, distinguishes three main approaches: coordination by contracts, multi-agent systems and mathematical programming models. The largest number of references reviewed in Stadtler (2009) employs mathematical decomposition (exact mathematical decomposition, heuristic mathematic decomposition and meta-heuristics). Originally developed for solving large-scale linear programming, mathematical decomposition methods seem to be an attractive alternative for solving distributed decision-making problems. Barbarosoglu & Özgür (1999)developed a model which is solved by Lagrangian and heuristic relaxation techniques to become a decentralized two-stage model: one for production planning and another for transport planning. It generates a final plan level by level, where one stage determines both its own plan and supply requirements and passes the requirements to the next stage. Luh et al. (2003) presented a framework combining mathematical optimization and the contract communication protocol for make-to-order supply network coordination based in this relaxation method. Nie et al. (2006) developed a collaborative planning framework combiningthe Lagrangian relaxation method and genetic algorithms to coordinate and optimize the production planning of the independent partners linked by material flows in multiple tier supply chains. Moreover, Walther et al. (2008) applied a relaxation approach for distributed planning in a product recovery network.

However, these examples require the presence of a central coordinator with a complete control over the entire supply chain, otherwise there is no guarantee for convergence of the final solution without extra modification procedure or acceptance functions because of the duality gap or the oscillation of mathematical decomposition methods (Jung et al. 2008). In this context, FGPcanbea valid alternative totheprevious drawbacks.

Fuzzymathematical programming, especially the fuzzy goal programming (FGP) method, has widely been applied for solving various multi-objective supply chain planning problems. Among them, Kumar et al. (2004) and Lee et al. (2009)presented FGP approaches for supplier selection problems with multiple objectives.Liang (2006) presented a FGP approach for solving integrated production and distribution planning problems with fuzzy multiple goals in uncertain environments. The proposed model aims to simultaneously minimize the total distribution and production costs, the total number of rejected items, and the total delivery time.Torabi & Hassini (2009) proposed a multi-objective, multi-site production planning FGP model integrating procurement and distribution plans in a multi-echelon automotive supply chain network.

3. Modelling approaches for centralized and decentralized planning in SC structures

3.1. Planning in centralized supply chain structure

According to their basic structures, SCs can be categorized as centralized and decentralized. A supply chain is called centralized if a single dominant firm has all the information and tries to, in the short run, simply optimize its own operational decisions regardless of the impact of such decisions on the other stages of the chain (Erengüç et al. 1999). According to Selim et al. (2008), FGP approaches can be used in handling collaborative master planning problems in both centralized and decentralized SC structures. In order to handle the problem in centralized SC, Selimet al. (2008) propose to use Tiwari et al. (1987) weighted additive approach defined as follows:

Maximizekwkμk(x)subjecttoμk[0,1]kx0E1

In this approach, w k andμkdenotes the weight and the satisfaction degree of the kth goal respectively. Therefore, the weighted additive approach allows the dominant partner in the SC to assign different weights to the individual goals in the simple additive fuzzy achievement function to reflect their relative importance levels.

3.2. Planning in decentralized supply chain structure

A SC is called decentralized when various decisions are made in different companies that try to optimize their own objectives. Selim et al. (2008) state that the methodsthat take account of min operator are suitablein modelling the collaborative planning problemsin decentralized SC structures. Among these methods, Selim et al. (2008) propose to use Werners (1988) fuzzy and operator to address the SC collaborative planning problems in decentralized SC structures. By adopting min operator into Werners’ approach the following linear programming problem can be obtained:

Maximizeγλ+(1γ)(1/K)kλksubjecttoμk(x)λ+λkkK,xXλ,λk,γ[0,1]E2

whereK is the total number of objectives, µ k is the membership function of goal k, and γis the coefficient of compensation defined within the interval [0,1]. In this approach, the coefficient of compensation can be treated as the degree of willingness of the SC partners to sacrifice the aspiration levels for their goals to some extent in the short run to provide the loyalty of their partners and/or to strengthen their competitive position in the long run.

To explore the viability of the proposed fuzzy modelling approaches for the collaborative SC planning in centralized and decentralized SC structures, we consider a supply chain master planning problem related to a ceramic tile supply chain in the next section.

4. Model formulation

We adopt the ceramic supply chain master planning problem presented in Alemany et al. (2010). Figure 1 shows the structure of a typical SC of the ceramic sector. The authors describe the peculiarities related to these SCs and consider several assumptions. First, the flow of parts, components,raw materials (RMs) and finished goods (FGs) that might circulate between the nodes is known beforehand. The existence of several production plants situated in various geographical locations is also assumed. These production plants are supplied with various RMs provided by different suppliers with a limited supply capacity.

Figure 1.

Ceramic tile SC considered in Alemany et al. (2010)

In the SC under study, each production plant has one or several parallel production lines, which process different FGs, with a limited capacity. Moreover, there are FGs with high added values that are manufactured only in production plants; others may be partly subcontracted, while some may be totally subcontracted to external suppliers. FGs are grouped into product families to minimize setup times and costs. A product family is defined as a group of FGs with identical physical characteristics and whose preparation on product lines is similar. Given the important setup times between product families on production lines, minimum run lengths for product families are specified. Item setups among the products belonging to the same product family also exist. Because of technological factors involved in the production process itself, each product should be produced in an equal or greater amount than the minimum lot size defined, when it’s manufactured on a specific line.

The distribution of FGs from production plants to end customers is carried out in various stages by different types of distribution centres, such as central warehouses, logistic centres and shops. Neither manufactured nor subcontracted FGs can be stored in manufacturing plants. They are sent to the first distribution level which is composed of several central warehouses with a limited storage capacity. The demand of end customers and logistics centres is covered by the outgoing FGs from central warehouses. Besides, logistics centres only supply FGs to shops that have been previously assigned to them. Finally, shops only attend end costumers’ demand. Although a maximum service level is pursued in this SC, limited backorders are permitted in both central warehouses and shops.

4.1. Nomenclature

The nomenclature defines the indices, sets of indices, parameters and decision variables (Table 1).

Indices
cRMs, items, and components (c=1…C)qLogistic centres (q=1…Q)
iFGs (i=1…I)wShops (w=1…W)
fProduct families (f=1…F)rSuppliers of RMs, items, and components (r=1…R)
lProduction lines (l=1…L)
pProduction plants (p=1…P)bSuppliers of finished products (b=1…B)
aWarehouses (a=1…A)tPeriods of time (t=1…T)
Sets of Indices
Il(l)Set of FGs that can be manufactured on manufacturing line lLp(p)Set of manufacturing lines that belong to production plant p
Fl(l)Set of product families that can be manufactured on manufacturing line lPa(a)Set of production plants that can send FGs to warehouse a
If(f)Set of FGs that belong to product family fAq(q)Set of warehouses that can supply logistics centre q
Ip(p)Set of FGs that can be produced in production plant pRc(c)Set of suppliers that can supply RM c
Ia(a)Set of FGs that can be stored in warehouse aRp(p)Set of suppliers of RMs that can supply production plant p
Ic(c)Set of FGs of that RM c form partCr(r)Set of RMs that can be supplied by supplier r
PFNSSet of FGs that cannot be subcontractedQa(a)Set of logistics centres that can be supplied by warehouse a
PFSPSet of FGs that can be subcontracted either partially or completelyWq(q)Set of shops that can be supplied by logistics centres q
PFSTSet of FGs that are compulsorily subcontracted completelyQw(w)Set of logistics centres capable of supplying shop w
Iq(q)Set of FGs that can be sent to logistics centre qBi(i)Set of suppliers of FGs i to which the FG may be subcontracted
Iw(w)Set of FGs that can be sent to shop wBa(a)Set of suppliers of FGs that can supply warehouse a
Lf(f)Set of manufacturing lines that may produce product family fAb(b)Set of warehouses that can be supplied by the supplier b of FGs
Model Parameters
cacrtCapacity (units) of supplying RM c of supplier r in period tM1,M2Very large integers
costtpcrpCost of purchase and transport of one unit of RM c from supplier r to production plant pcapalaStorage capacity (m2) in warehouse a
caflptProduction capacity available (time) of production line l at plant p during time period tcosttcliaqCost of transporting one m2 of FG i from warehouse a to logistics centre q
cmiLoss ratio of FG i. It represents the percentage of faulty m2 obtained due to the intrinsic characteristics of the production process in the ceramics sector.costinaiaCost of making an inventory of one m2 of FG i in the warehouse during a time period
cqiFirst quality coefficient of FG i. It represents the percentage of m2 that can be sold as first quality.costdifaiaCost of backordering one m2 of demand of FG i in warehouse a in a time period
costtaipaCost of transporting one m2 of FG i from production plant p to warehouse apaiaSales value of one m2 of FG i in warehouse a
costpilpCost of producing one m2 of FG i on production line l of production plant pdaiatExternal demand (m2) of FG i at the warehouse a in period t
costsetupfflpSetup costs for product family f on production line l of production plant pssaiaSafety stock (m2) of FG i at warehouse a
costsetupilpSetup costs for FG i on production line l of production plant pα 1Maximum backorder quantity permitted in a period in warehouses expressed as a percentage of the demand of that period
tfabilpTime to process one m2 of FG i on production line l of production plant pcostscibCost of subcontracting one m2 of FG i to FG supplier b
tsetupfflpSetup time for product family f on production line l of production plant pminscibMinimum amount (m2) of FG i to be subcontracted to FG supplier b
tsetupiilpSetup time for article i on production line l of production plant pcostttkiqwCost to transport one m2 of FG i from logistics centre q to shop w
lmiilpMinimum lot size (m2) of FG i on production line l of production plant pcostdiftkiwCost to backorder one m2 of the demand of FG i in a time period at shop w
tmfflpMinimum run length (expressed as multiples of the time period used) of product family f on production line l of production plant ppwiwSales price of one m2 of FG i in shop w
vicUnits of RM c needed to produce one m2 of FG idtkiwtExternal demand (m2) of FG i in shop w during the time period t
ssccpSafety stock of RM c in production plant pα 2Maximum backorder quantity permitted in a period in shops expressed as a percentage of the demand of that period
cascibtSupply capacity (m2) of FG i of supplier b in time period t
Decision Variables
CTPcrptAmount of RM c to be purchased and transported from supplier r to production plant p in time period tINAiatInventory (m2) of FG i in warehouse a in time period t
INCcptInventory of the RM c at plant p at the end of time period tCSCibatAmount (m2) of FG i subcontracted to supplier b for warehouse a in time period t
MPFflptAmount (m2) of product family f manufactured on production line l of production plant p in time period tSibtBinary variable with a value of 1 if FG i is subcontracted to supplier b in time period t
MPilptAmount (m2) of FG i manufactured on production line l of production plant p in time period tVEAiatAmount (m2) of FG i sold in warehouse a during time period t
XilptBinary variable with a value of 1 if FG i is manufactured on production line l of production plant p in time period t, and with a value of 0 otherwiseDIFAiatBackorder quantity (m2) of FG i in warehouse a during time period t
YflptBinary variable with a value of 1 if product family f is manufactured on production line l of production plant p in time period t, and with a value0 otherwiseCTCLiaqtAmount (m2) of FG i transported from warehouse a to logistics centre q in time period t
ZIilptBinary variable with a value of 1 if a setup takes place of product i on production line l of production plant p in time period t, and with a value of0 otherwiseCTTKiqwtAmount (m2) of FG i transported from logistics centre q to shop w in time period t
ZFflptBinary variable with a value of 1 if a setup takes place of product family f on production line l of production plant p in time period t, and with a value of 0 otherwiseVETKiwtAmount (m2) of FG i sold in shop w during time period t
CTAipatAmount (m2) of FG i to be transported from production plant p to warehouse a in time period tDIFTKiwtBackorder quantity (m2) of FG i in shop w during time period t

Table 1.

Nomenclature.

The formulation of the model is as follows.

4.2. Objective functions

Formulations of the objective functions of the ceramic supply chain master planning model are presented inthe following.

  • Manufacturer’s cost function (COSTM)

COSTM = Total procurement cost + Total manufacturing cost + Total transportation to warehouses cost

Minimize{tprRp(p)cCr(r)costtpcrp*CTPcrpt+tplLp(p)iIl(l)costpilp*MPilpt+tplLp(p)fFl(l)costsetupfflp*ZFflpt+tplLp(p)iIl(l)costsetupiilp*ZIilpt+tapPa(a)iIp(p)costtaipa*CTAipat}E3
  • Profit function of warehouse a (WaPROFIT)

WaPROFIT= Sales revenue – Total inventory cost – Total subcontracting cost – Total transportation to logistic centres cost – Total backorder cost

Maximize{tipaia*VEAiattiIa(a)costinaia*INAiattibBi(i)costscib*CSCibattqQa(a)iIq(q)costtcliaq*CTCLiaqttiIa(a)DIFAiat}E4
  • Cost function of logistic centre q (LCqCOST)

LCqCOST= Total transportation to shops cost

Minimize{twWq(q)iIw(w)costttkiqw*CTTKiqwt}E5
  • Profit function of shop w (SwPROFIT)

SwPROFIT= Sales revenue - Total backorder cost

Maximize{tipwiw*VETKiwttiIw(w)DIFTKiwt}E6

4.3. Constraints

The constraints originally proposed by Alemany et al. (2010)are briefly reviewed as follows:

Constraint (7) is the inventory balance equation for RMs.

INCcpt=INCcpt1+rRc(c)CTPcrptiIc(c)(vic*lLp(p)MPilpt)    c,p,tE7

Constraint (8) establishes safety stocks for RMs.

INCcptssccp c,p,tE8

Constraint (9) defines the available capacity of supply for RMs suppliers.

pCTPcrptcacrtc,rRc(c),tE9

Constraint (10) establishes the available capacity for production lines.

fFl(l)tsetupfflp*ZFflpt+iIl(l)(tsetupiilp*ZIilpt+tfabilp*MPilpt)caflptp,lLp(p),tE10

Constraint (11) is related to the product families to be produced in each line.

MPFflpt=iIf(f)MPilpt p,lLp(p),fFl(l),tE11

Constraint (12) establishes minimum lot sizes for FGs’ production.

MPilptlmiilp*Xilpt                  p,lLp(p),iIl(l),tE12

Constraints (13) and (14) allocate products and product families to each line. Parameters M1 and M2 are large enough integer numbers.

MPilptM1*Xilpt                  p,lLp(p),iIl(l),tE13
MPFflptM2*Yflpt               p,lLp(p),fFl(l),tE14

Constraints (15)-(18) guarantee the control of the setup of FGs and product families.

ZIilptXilptXilpt1                p,lLp(p),iIl(l),tE15
iZIilptiXilpt1              p,lLp(p),tE16
ZFflptYflptYflpt1                 p,lLp(p),fFl(l),tE17
fZFflptfYflpt1               p,lLp(p),tE18

Constraint (19) ensures the accomplishment of the family run lenght

t=t't'+tmfflp1ZFflpt1                 p,lLp(p),fFl(l),t'=1,..,Ttmfflp+1E19

Constraint (20) ensures that only first quality FGs are transported to the central warehouses.

lLp(p)(1cmi)*cqi*MPilpt=aAp(p)CTAipat                 p,iIp(p),tE20

Constraints (21)-(24) are related to subcontracting decisions. These constraints also ensure that the amount of FGs subcontracted is transported to warehouses.

aAb(b)CSCibatminscib*Sibt  iPFSP,bBi(i),tE21
aAb(b)CSCibatminscib*Sibt  iPFST,bBi(i),tE22
aAb(b)CSCibatcascibt*Sibt  iPFSP,bBi(i),tE23
aAb(b)CSCibatcascibt*Sibt  iPFST,bBi(i),tE24

Constraint (25) establishes safety stocks for FGs.

INAiatssaia a,iIa(a),tE25

Constraint (26) fixes the capacity of the warehouses.

iIa(a)INAiatcapala  a,tE26

Constraints (27)-(28) are inventory balance equations for FGs in warehouses.

INAiat=INAiat1+pPa(a)CTAipatVEAiatqQa(a)CTCLiaqt  iPFNS,a,tE27
INAiat=INAiat1+pPa(a)CTAipat+bBa(a)bBi(i)CSCibatVEAiatqQa(a)CTCLiaqt  iPFSP,a,tE28

Constraint (29) is similar to (27)-(28) but also ensures the subcontracted FGs only comes from FG suppliers.

INAiat=INAiat1+bBa(a)bBi(i)CSCibatVEAiatqQa(a)CTCLiaqt iPFST,a,tE29

Backorder quantities in warehouses are calculated using Constraint (30).

VEAiat+DIFAiatDIFAiat1=daiat   a,iIa(a),tE30

Constraint (31) limits the backorder quantities in warehouses.

DIFAiatα1*daiat   a,iIa(a),tE31

Constraints (32) and (33) are the inflows and outflows of FGs through each logistic centre and shop, respectively.

aAq(q)CTCLiaqt=wWq(q)CTTKiqwt q,iIq(q),tE32
CTTKiqwt=VETKiwt  w,qQw(w),iIw(w),tE33

Constraint (34) determines backorder quantities in shops.

VETKiwt+DIFTKiwtDIFTKiwt1=dtkiwt                    w,iIw(w),tE34

Constraint (35) limits the backorder quantities in shops.

DIFTKiwtα2*dtkiwtw,iIw(w),tE35

The model also contemplates non-negativity constraints and the definition of binary variables (36).

MPFflpt, MPilpt, CTPcrpt, CTAipat, INAiat, INCcpt, CTCLiaqt, CTTKiqwt, VEAiat, DIFAiat,VETKiwt, DIFTKiwt, CSCibat 0 and,Xilpt, Yflpt,ZFflpt,ZIilpt,Sibt{0,1}fF,iI,cC,lL,pP,aA,qQ,wW,rR,bB,tTE36

Finally, some decision variables can be defined as integers, but it could change depending on the real-world problem where the model is applied.

5. Solution methodology

In order to reach a preferred solution for the ceramic master planning problem in centralized and decentralized SC structures the Tiwari et al. (1987) and Werners (1988)approaches are adopted to transform the multi-objective FGP model to a mixed integer linear programming (MILP) one.

5.1. Defining the membership functions

There are many possible forms for a membership function to represent the fuzzy objective functions: linear, exponential, hyperbolic, hyperbolic inverse, piece-wise linear, etc. (see Peidro & Vasant (2009) for a comparison of the main types of membership functions). Among the various types of membership functions, the most feasible for constructing a membership function for solving fuzzy mathematical programming problems is the linear form, although there may be preferences for other patterns with some applications (Zimmermann 1975; Zimmermann 1978; Tanaka et al. 1984). Moreover, the main advantage of the linear membership functions is that they generate equivalent, efficient and computationally linear models.

We formulate the corresponding non increasing continuous linear membership functions for objective function as follows (Bellman & Zadeh 1970):

μm={1zmzmlzmuzmzmuzmlzmlzmzmu0zmzmuE37
μM={1zMzMuzMzMlzMuzMlzMlzMzMu0zMzMlE38

where µ m is the membership function of a minimization objetive z m and µ M is the membership function of a maximization objetive z M . Moreover, zml,zMlandzmu,zMuare the lower and upper bounds of the objective functions.We can determine each membership function by asking the decision maker to specify the fuzzy objective value interval (37)-(38). Besides, we can obtain these bounds from the optimisation values of each objective function.

5.2. Transforming the multi-objective FGP model into an MILP model for centralized SC structures

According to Selim et al. (2008), the Tiwari et al. (1987) weighted additive approach can be used to handle the collaborative ceramic master planning problem in a centralized SC structure. By adopting this approach, the problem can be formulated as follows:

Maximizew1μCOSTM+w2aμWaPROFIT+w3qμLCqCOST+w4wμSwPROFITsubjecttoμCOSTM,μWaPROFIT,μLCqCOST,μSwPROFIT[0,1]}E39

This model also considers Constraints (7) to (36).

w 1 , w 2 , w 3 and w 4 denotes the weights of manufacturer’s, warehouses‘, logistic centres’ and shops‘ objectives, respectively.

5.3. Transforming the multi-objective FGP model into an MILP model for decentralized SC structures

To deal with the collaborative ceramic master planning problem in a decentralized SC, according to Selim et al. (2008), the Werners (1988) approach can be adopted. By using the Werners‘ fuzzy and operator, the problem under study can be formulated as follows:

Maximizeγλ+(1γ)(11+A+Q+W)[λ1+aλa+qλq+wλw]subjecttoμCOSTMλ+λ1μWaPROFITλ+λaaμLCqCOSTλ+λqqμSwPROFITλ+λwwμCOSTM,μWaPROFIT,μLCqCOST,μSwPROFIT,λ,λ1,λa,λq,λw,γ[0,1]}E40

This model also considers Constraints (7) to (36).

A, Q and W are the total number of warehouses, logistic centres and shops in the SC.

6. Application to a ceramic tile supply chain

This section uses the example provided by Alemany et al. (2010) to validate and evaluate the results of our proposal. It is a representative SC of the ceramic tile sector. There are 3 production plants, which produce 4 FGs grouped into 3 product families which rates, minimum run lengths and fixed costs are provided. Each plant contains two production lines. All the product families may be manufactured on the production lines at the various plants. Moreover, there are 2 warehouses, 3 logistics centres and 6 shops. They are considered six weeks periods in the planning horizon. Also, they are provided the following information: bill of materials, transportation costs, setup costs, initial inventory, available production and storage capacities, raw material costs, safety stocks, inventory costs, setup times, production costs, sale prices, subcontracting costs, backorder costs, production run times, minimum lot sizes and demand. Details on this data used can be found in Alemany et al. (2010).

6.1. Implementation and resolution

The proposed models have been developed with the modelling language MPL and solved by the CPLEX 12 solver in an Intel Xeon, at 2.93 GHz, with 48 GB of RAM. The input data and the model solution values have been processed with the Microsoft SQL Server Database (2008).

We define each membership function by obtaining upper and lower bounds of each objective function. The upper and lower bounds obtained by maximizing and minimizing each objective function separately are presented in Table 2.

6.2. Evaluation of results

As stated previously, we adoptthe weighted additive approach proposed by Tiwari et al. (1987) to deal with thecollaborative CSCMP problem in a centralized SC structure. To explore the influence of different weight structures on the results of the problem several problem instances are generated. Solution results of the model obtained by Tiwari et al. (1987) weighted additive approach are presented in Table 3. It is clear that determination of the weights requires expert opinionso that they can reflect accurately the relations between the different partners of a SC. In Table 3, w 1 , w 2 , w 3 and w 4 denotes the weights of manufacturer’s, warehouses‘, logistic centres’ and shops‘ objectives for each instance. On the other hand, Table 3 adds the degree of satisfaction of the objective functions for the proposed method.

ObjectivesUpper boundLower bound
COSTM785545543825
W1PROFIT302078171296
W2PROFIT332787198072
LC1COST13591329
LC2COST11871162
LC3COST12271199
S1PROFIT6655264784
S2PROFIT6582564154
S3PROFIT6878767044
S4PROFIT6644864727
S5PROFIT6848666643
S6PROFIT5983858288

Table 2.

Upper and lower bounds of the objectives.

Problem instances
123456789
w10.250.40.30.30.40.20.30.30.3
w20.250.20.20.20.10.20.40.20.1
w30.250.20.20.10.10.30.20.40.1
w40.250.20.30.40.40.30.10.10.5
µCOSTM0.76660.77070.76550.76550.98340.76720.77470.77600.9536
µW1PROFIT1.00001.00001.00001.00000.95071.00001.00001.00000.9956
µW2PROFIT0.57380.56700.56810.56810.11530.57380.57790.57800.1726
µLC1COST0.00000.00000.00000.00000.00000.00000.46620.56290.0000
µLC2COST1.00001.00000.00000.00000.00001.00001.00001.00000.0000
µLC3COST0.00000.00000.00000.00000.00000.00001.00001.00000.0000
µS1PROFIT0.83350.83351.00001.00001.00000.83350.46940.32241.0000
µS2PROFIT0.61180.61180.95060.95060.95060.61180.61180.38300.9506
µS3PROFIT0.91870.91871.00001.00001.00000.91870.59650.59651.0000
µS4PROFIT0.91750.91751.00001.00001.00000.91750.54770.54771.0000
µS5PROFIT0.89190.89191.00001.00001.00000.89190.46530.46531.0000
µS6PROFIT0.86510.86511.00001.00001.00000.86510.57520.57521.0000

Table 3.

Solution results obtained by Tiwari et al. (1987) approach.

Table 4 shows the degree of satisfaction of each objective function obtained by Werners (1988) approach with different values of the coefficient of compensation (γ). It is observed from Fig. 2 that the range of the achievement levels of the objectives increases with the decrease of the coefficient of compensation, taking the maximum possible value in the interval 0.5-0. That is, the higher the compensation coefficient γ values, the lower the difference between the degrees of satisfaction of each partner of the decentralized SC. So,forhighvalues​​of γ, we can obtaincompromise solutions forthe allmembers oftheSC, rather thansolutionsthatonly satisfytheobjectivesof a small groupofthese partners. Table 4 shows in general terms, the reduction of the degree of satisfaction of logistics centres 1 and 3 and shop 2, at the expense of substantially increasing the degree of satisfaction of the logistic center 2 and the rest of shops. Also, thedegreeofsatisfactionrelated to warehouse1 increases while reducing the degree of satisfaction associated to warehouse 2.

ϒ
0,90,80,70,60,50,40,30,20,10
µCOSTM0.77280.77220.77330.77230.76720.76720.76720.76720.76660.7672
µW1PROFIT0.9290.92620.92740.93171,00000.97620.96220.96221,00000.9622
µW2PROFIT0.64050.64680.64420.64160.57360.59670.60990.60930.57320.6093
µLC1COST0.64050.64050.64050.64050,00000,00000,00000,00000,00000,0000
µLC2COST0.64050.64050.64050.64051,00001,00001,00001,00001,00001,0000
µLC3COST0.64050.64050.64050.64050,00000,00000,00000,00000,00000,0000
µS1PROFIT0.64050.64050.64050.64050.83350.83350.83350.83350.83350.8335
µS2PROFIT0.64050.64050.64050.64050.61180.61180.61180.61180.61180.6118
µS3PROFIT0.64050.64050.64050.64050.91870.91870.91870.91870.91870.9187
µS4PROFIT0.64050.64050.64050.64050.91750.91750.91750.91750.91750.9175
µS5PROFIT0.64050.64050.64050.64050.89190.89190.89190.89190.89190.8919
µS6PROFIT0.64050.64050.64050.64050.86510.86510.86510.86510.86510.8651

Table 4.

Solution results obtained by Werners (1988) approach.

Figure 2.

Range of the achievement levels of the objectives.

7. Conclusion

In recent years, the CP in SC environments is acquiring an increasing interest. In general terms, the CP implies a distributed decision-making process involving several decision-makers that interact in order to reach a certain balance condition between their particular objectives and those for the rest of the SC. This work deals with the collaborative supply chain master planning problem in a ceramic tile SC and has proposes two FGP models for the collaborative CSCMP problem based on the previous work of Alemany et al. (2010). FGP allows incorporate into the models decision maker’s imprecise aspiration levels.Besides, to explore the viability of different FGP approaches for the CSCMP problem in different SC structures (i.e. centralized and decentralized) a real-world industrial problemwith several computational experiments has been provided. The numerical results show that collaborative issues related to SC master planning problems can be considered in a feasible manner by using fuzzy mathematical approaches.

The complex nature and dynamics of the relationships among the different actors in a SC imply an important degree of uncertainty in SC planning decisions. In SC planning decision processes, uncertainty is a main factor that may influence the effectiveness of the configuration and coordination of SCs (Davis 1993; Minegishi and Thiel 2000; Jung et al. 2004), and tends to propagate up and down the SC, affecting performance considerably (Bhatnagar and Sohal 2005). Future studies may consider uncertainty in parameters such as demand, production capacity, selling prices, etc. using fuzzy modelling approaches.

Although the linear membership function has been proved to provide qualified solutions for many applications (Liu & Sahinidis 1997), the main limitation of the proposed approaches is the assumption of the linearity of the membership function to represent the decision maker’s imprecise aspiration levels. This work assumes that the linear membership functions for related imprecise numbers are reasonably given. In real-world situations, however, the decision maker should generate suitable membership functions based on subjective judgment and/or historical resources. Future studies may apply related non-linear membership functions (exponential, hyperbolic, modified s-curve, etc.) to solve the CSCMP problem. Besides, the resolution times of the FGP models may be quite long in large-scale CSCMP problems. For this reason, future studies may apply the use of evolutionary algorithms and metaheuristics to solve CSCMP problems more efficiently.

Acknowledgments

This work has been funded by the Spanish Ministry of Science and Technology project: ‘Production technology based on the feedback from production, transport and unload planning and the redesign of warehouses decisions in the supply chain (Ref. DPI2010-19977).

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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Manuel Díaz-Madroñero and David Peidro (August 1st 2011). A Fuzzy Goal Programming Approach for Collaborative Supply Chain Master Planning, Supply Chain Management - Pathways for Research and Practice, Dilek Onkal, IntechOpen, DOI: 10.5772/20642. Available from:

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