A Fuzzy Linear Programming Based Approach for Tactical Supply Chain Planning in an Uncertainty...

European Journal of Operational Research 205 (2010) 65–80

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

A fuzzy linear programming based approach for tacticalsupply chain planning in an uncertainty environment

David Peidro a,*, Josefa Mula a, Mariano Jiménez b, Ma del Mar Botella c

a Research Centre on Production Management and Engineering (CIGIP), Universidad Politécnica de Valencia, Spainb Department of Applied Economy, Universidad del Paı́s Vasco, Spainc Department of Business Administration, Universidad Politécnica de Valencia, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 November 2008Accepted 23 November 2009Available online 3 December 2009

Keywords:Supply chain managementSupply chain planningUncertainty modelingFuzzy sets

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.11.031

* Corresponding author. Address: Escuela Politécnic84 64.

E-mail addresses: [email protected] (D. Peidro)

This paper models supply chain (SC) uncertainties by fuzzy sets and develops a fuzzy linear programmingmodel for tactical supply chain planning in a multi-echelon, multi-product, multi-level, multi-period sup-ply chain network. In this approach, the demand, process and supply uncertainties are jointly considered.The aim is to centralize multi-node decisions simultaneously to achieve the best use of the availableresources along the time horizon so that customer demands are met at a minimum cost. This proposalis tested by using data from a real automobile SC. The fuzzy model provides the decision maker (DM)with alternative decision plans with different degrees of satisfaction.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

A supply chain (SC) is a network of organizations involved through upstream and downstream linkages in the different processes andactivities that produce value in the form of products and services in the hand of the ultimate customer (Christopher, 1998). This network oforganizations is dynamic and involves a high degree of imprecision. This is mainly due to its real-world character where uncertainties inthose activities extending from the suppliers to the customers make SC imprecise (Fazel et al., 2002).

Several authors have analyzed the sources of uncertainty present in an SC; readers may refer to Peidro et al. (2009) for a review. Themajority of the authors studied (Childerhouse and Towill, 2002; Davis, 1993; Ho et al., 2005; Lee and Billington, 1993; Mason-Jonesand Towill, 1998; Wang and Shu, 2005) classified the sources of uncertainty into three groups: demand, process/manufacturing and supply.Uncertainty in supply is caused by the variability brought about by how the supplier operates because of the faults or delays in the sup-plier’s deliveries. Uncertainty in the process is a result of the poorly reliable production process due to, for example, machine hold-ups.Finally, demand uncertainty, according to Davis (1993), is the most important of the three, and is presented as a volatility demand oras inexact forecasting demands.

The coordination and integration of key business activities undertaken by an enterprise, from the procurement of raw materials to thedistribution of end products to the customer, are concerned with the SC planning process (Gupta and Maranas, 2003), one of the mostimportant processes within the SC management concept. However, the complex nature and dynamics of the relationships among the dif-ferent actors imply an important degree of uncertainty in the planning decisions. In SC planning decision processes, uncertainty is a mainfactor that can influence the effectiveness of the configuration and coordination of supply chains (Davis, 1993; Jung et al., 2004; Minegishiand Thiel, 2000) and tends to spread up and down the SC, affecting its performance considerably (Bhatnagar and Sohal, 2005).

Most SC planning research (Alonso-Ayuso et al., 2003; Guillen et al., 2005; Gupta and Maranas, 2003; Lababidi et al., 2004; Santoso et al.,2005; Sodhi, 2005) models SC uncertainties with probability distributions that are usually predicted from historical data. However, when-ever statistical data are unreliable, or are not even available, stochastic models may not be the best choice (Wang and Shu, 2005). The FuzzySet Theory (Zadeh, 1965) and the Possibility Theory (Dubois and Prade, 1988; Zadeh, 1978) may provide an alternative which is simplerand less data-demanding than the Probability Theory to deal with SC uncertainties (Dubois et al., 2003).

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a Superior de Alcoy, Plaza Ferrándiz y Carbonell, 2, 03801 Alcoy, Alicante, Spain. Tel.: +34 96 652 84 90; fax: +34 96 652

, [email protected] (J. Mula), [email protected] (M. Jiménez), [email protected] (M. del Mar Botella).

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66 D. Peidro et al. / European Journal of Operational Research 205 (2010) 65–80

Few studies address the SC planning problem on a medium-term basis (tactical level) which integrates procurement, production anddistribution planning activities into a fuzzy environment (see Section 2. Literature review). The aim of this approach is to simultaneouslyoptimize the decision variables of different functions that have been traditionally optimized sequentially (Park, 2005). Moreover, there is alack of models contemplating the different sources of uncertainty in an integrated manner. Hence in this study, we develop a tactical sup-ply chain model in a fuzzy environment in a multi-echelon, multi-product, multi-level, multi-period supply chain network. In this proposedmodel, the demand, process and supply uncertainties are considered simultaneously.

In the context of fuzzy mathematical programming, two very different issues can be addressed: fuzzy or flexible constraints for fuzziness,and fuzzy coefficients for lack of knowledge or epistemic uncertainty (Dubois et al., 2003). The aim of this paper is to propose an SC planningmodel where the data, associated with all the sources of uncertainty in an SC, are ill-known and modeled by trapezoidal fuzzy numbers.

The main contributions of this paper can be summarized as follows:

� Introducing a novel tactical SC planning model by integrating procurement, production and distribution planning activities into a multi-echelon, multi-product, multi-level and multi-period SC network.

� Achieving a model which contemplates the different sources of uncertainty affecting SCs in an integrated fashion by considering the pos-sible lack of knowledge in the data.

� Applying the model to a real-world automobile SC dedicated to the supply of automobile seats.

The rest of this paper is arranged as follows. Section 2 presents a literature review about fuzzy applications in SC planning. Section 3proposes a new fuzzy mixed-integer linear programming (FMILP) model for the tactical SC planning under uncertainty. Then in Section4, the fuzzy model is transformed into an equivalent auxiliary crisp mixed-integer linear programming model and a resolution method thatpermit the interactive participation of the decision maker in all the steps of the decision process, and the expressing of opinions in linguisticterms is introduced. In Section 5, the behavior of the model in a real-world automobile SC has been evaluated and, finally, the conclusionsand directions for further research are provided.

2. Literature review

In Peidro et al. (2009) a literature survey on SC planning under uncertainty conditions by adopting quantitative approaches is presented.Here, we present a summary, extracted from this paper, about the applications of the Fuzzy Set Theory and the Possibility Theory to dif-ferent problems related to SC planning:

(a) SC inventory management: Petrovic et al. (1998, 1999) described the fuzzy modeling and simulation of an SC in an uncertain envi-ronment. Their objective was to determine the stock levels and order quantities for each inventory during a finite time horizonto achieve an acceptable delivery performance at a reasonable total cost for the whole SC. Petrovic (2001) developed a simulationtool, SCSIM, for analyzing SC behavior and performance in the presence of uncertainty modeled by fuzzy sets. Giannoccaro et al.(2003) developed a methodology to define inventory management policies in an SC, which was based on the echelon stock concept(Clark and Scarf, 1960), and the Fuzzy Set Theory was used to model the uncertainty associated with both demand and inventorycosts. Carlsson and Fuller (2002) proposed a fuzzy logic approach to reduce the bullwhip effect. Wang and Shu (2005) developeda decentralized decision model based on a genetic algorithm which minimizes the inventory costs of an SC subject to the constraintto be met with a specific task involving the delivery of finished goods. The authors used the fuzzy set theory to represent the uncer-tainty of customer demands, processing times and reliable deliveries. Xie et al. (2006) presented a new bilevel coordination strategyto control and manage inventories in serial supply chains with demand uncertainty. Firstly, the problem associated with the wholeSC was divided into subproblems in accordance with the different parts that the SC was made up of. Secondly, for the purpose ofimproving the integrated operation of a whole SC, the leader level was defined to be in charge of coordinating inventory controland management by amending the optimization subproblems. This process was to be repeated until the desired level of operationfor the whole SC was reached. Wang and Shu (2007) modeled SC uncertainties with fuzzy sets and developed a possibilistic decisionmodel to determine the SC configuration and inventory policies that minimize the total SC costs subject to also fulfilling the targetservice time of the end-product. They assumed that sourcing options differed in terms of their direct costs and lead-times. Fuzzy setswere used to represent fluctuating customer demands, uncertain lead-times, and unreliable supply deliveries (in terms of delaytime). A fuzzy SC model extended from Wang and Shu (2005) was used to evaluate the performance of the entire SC directly. Agenetic algorithm approach integrated with the proposed fuzzy SC model was developed to determine the optimal SC configurationand the order-up-to level for each stage at the same time.

(b) Vendor selection: Kumar et al. (2004) presented a fuzzy goal programming approach which was applied to the problem of selectingvendors in an SC. This problem was posed as a mixed-integer and fuzzy goal programming problem with three basic objectives tominimize the net cost of the vendors network, rejects within the network, and delays in deliveries. With this approach, the authorsused triangular membership functions for each fuzzy objective. The solution method was based on the intersection of membershipfunctions of the fuzzy objectives by applying the min-operator. Then, Kumar et al. (2006a) solved the same problem using the multi-objective fuzzy programming approach proposed by Zimmermann (1978). Amid et al. (2006) addressed the problem of adequatelyselecting suppliers within an SC. For this purpose, they devised a fuzzy-based multi-objective mathematical programming modelwhere each objective may be assigned a different weight. The objectives considered were related to cost cuts, increased qualityand to an increased service of the suppliers selected. The imprecise elements considered in this work were to meet both objectivesand demand. Kumar et al. (2006b) analyzed the uncertainty prevailing in integrated steel manufacturers in relation to the nature ofthe finished good and significant demand by customers. They proposed a new hybrid evolutionary algorithm named endosymbiot-icpsychoclonal (ESPC) to decide what to stock and how much as an intermediate product in inventories. They compared ESPC withgenetic algorithms and simulated annealing. They concluded the superiority of the proposed algorithm in terms of both the quality ofthe solution obtained and the convergence time.

D. Peidro et al. / European Journal of Operational Research 205 (2010) 65–80 67

(c) Transport planning: Chanas et al. (1993) considered several assumptions at the supply and demand levels for a given transportationproblem in accordance with the kind of information that the decision maker has: crisp values, interval values or fuzzy numbers. Forall three cases, classical, interval and fuzzy models for the transportation problem were proposed, respectively. The links amongthem were provided by focusing on the case of the fuzzy transportation problem, for which solution methods were proposed anddiscussed. Shih (1999) addressed the problem of transporting cement in Taiwan by using fuzzy linear programming models. Theauthor used three approaches based on the works by Zimmermann (1976). Chanas (1983) and Julien (1994), who contemplated:the capacities of ports, the fulfilling demand, the capacities of the loading and unloading operations, and the constraints associatedwith traffic control. Liu and Kao (2004) developed a method to obtain the membership function of the total transport cost by con-sidering this as a fuzzy objective value where shipment costs, supply and demand were fuzzy numbers. The method was based onthe extension principle defined by Zadeh (1978) to transform the fuzzy transport problem into a pair of mathematical programmingmodels. Liang (2006) developed an interactive multi-objective linear programming model for solving fuzzy multi-objective transpor-tation problems with a piecewise linear membership function.

(d) Production–distribution planning: Sakawa et al. (2001) addressed the real problem of production and transport related to a manufac-turer through a deterministic mathematical programming model which minimized costs in accordance with capacities anddemands. Then, the authors developed a mathematical fuzzy programming model. Finally, they presented an outline of the distri-bution of profits and costs based on the Game Theory. Liang (2007) proposed an interactive fuzzy multi-objective linear program-ming model for solving an integrated production–transportation planning problem in supply chains. Selim et al. (2008) proposedfuzzy goal-based programming approaches applied to planning problems of a collaborative production–distribution type in central-ized and decentralized supply chains. The fuzzy elements that the authors considered corresponded to the fulfilment of differentobjectives related to maximizing profits for manufacturers and distribution centers, retailer cost cuts and minimizing delays indemand in retailers. Aliev et al. (2007) developed an integrated multi-period, multi-product fuzzy production and distribution aggre-gate planning model for supply chains by providing a sound trade-off between the filtrate of the fuzzy market demand and the profit.The model was formulated in terms of fuzzy programming and the solution was provided by genetic optimization.

(e) Procurement-production–distribution planning: Chen and Chang (2006) developed an approach to derive the membership function ofthe fuzzy minimum total cost of the multi-product, multi-echelon, and multi-period SC model when the unit cost of raw materialssupplied by suppliers, the unit transportation cost of products, and the demand quantity of products were fuzzy numbers. Recently,Torabi and Hassini (2008) proposed a new multi-objective possibilistic mixed-integer linear programming model for integrating pro-curement, production and distribution planning by considering various conflicting objectives simultaneously along with the impre-cise nature of some critical parameters such as market demands, cost/time coefficients and capacity levels. The proposed model andsolution method were validated by numerical tests.

As mentioned previously, there is a lack of models which focus on the different sources of uncertainty in an integrated manner, and fewstudies address the SC planning problem on a medium-term basis which integrates procurement, production and distribution planningactivities in a fuzzy environment. Moreover, the majority of the models studied do not apply to supply chains based on real-world cases.

3. Problem description

This section outlines the tactical SC planning problem. The overall problem can be stated as follows:Given:

– An SC topology: the number of nodes and type (suppliers, manufacturing plants, warehouses, distribution centers, retailers, etc.).– Each cost parameter, such as manufacturing, inventory, transportation, demand backlog, etc.– Manufacture data, processing times, production capacity, overtime capacity, BOM, production run, etc.– Transportation data, such as lead-time, transport capacity, etc.– Procurement data, procurement capacity, etc.– Inventory data, such as inventory capacity, etc.– Forecasted product demands over the entire planning periods.

To determine:

– The production plan of each manufacturing node.– The distribution transportation plan between nodes.– The procurement plan of each supplier node.– The inventory level of each node.– Sales and demand backlog.

The aim is to centralize the multi-node decisions simultaneously in order to achieve the best use of the available resources in the SCalong the time horizon so that customer demands are met at a minimum cost.

3.1. Fuzzy model formulation

The fuzzy mixed-integer linear programming (FMILP) model for the tactical SC planning proposed by Peidro et al. (2007) is adopted asthe basis of this work. Sets of indices, parameters and decision variables for the FMILP model are defined in the nomenclature (see Table 1).Table 2 shows the uncertain parameters grouped according to the uncertainty sources that may be presented in an SC.

FMILP is formulated as follows:

Table 1Nomenclature (fuzzy parameters are shown with a tilde: �).

Set of indicesT Set of planning periods (t =1, 2. . .,T)I Set of products (raw materials, intermediate products, finished goods) (i =1, 2. . .,I)N Set of SC nodes (n =1, 2. . .,N)J Set of production resources (j =1, 2. . .,J)L Set of transports (l =1, 2. . .,L)P Set of parent products in the bill of materials (p =1, 2. . .,P)O Set of origin nodes for transports (o =1, 2. . .,O)D Set of destination nodes for transports (d =1, 2. . .,D)

Objective function cost coefficients

VePCinjtVariable production cost per unit of product i on j at n in t

OeTCnjtOvertime cost of resource j at n in t

UeTCnjtUndertime cost of resource j at n in t

RMCint Price of raw material i at n in t

TeCodltTransport cost per unit from o to d by l in t

IeCintInventory holding cost per unit of product i at n in t

DeBCintDemand backlog cost per unit of product i at n in t

General dataBpint Quantity of i to produce a unit of p at n in t

MPeRCnt Maximum procurement capacity from supplier node n in teDintDemand of product i at n in t

MeOTnjtOvertime capacity of resource j at n in t

MePCnjtProduction capacity of resource j at n in t

I0in Inventory amount of i at n in period 0PRinjt Production run of i on j at n in tMPRinjt Minimum production run of i on j at n in tDB0int Demand backlog of i at n in period 0SR0iodlt Shipments of i received at d from o by l at the beginning of period 0SIP0iodlt Shipments in progress of i from o to d by l at the beginning of period 0

PeT injtProcessing time to produce a unit of i on j at n in t

TeLTodltTransport lead-time from o to d by l in t

Vit Physical volume of product i in t

MeTCnt Maximum transport capacity of l in t

MeICnt Maximum inventory capacity at n in t

v1odlt

0–1 function. It takes 1 if TLTodlt > 0, and 0 otherwise

v2odlt

0–1 function. It takes 1 if TLTodlt ¼ 0, and 0 otherwise

Decision variablesPinjt Production amount of i on j at n in t=PTinjt > 0kinjt Number of production runs of i produced on j at n in tSint Supply of product i from n in tDBint Demand backlog of i at n in t=DBCint > 0TQiodlt Transport quantity of i from o to d by l in t=o <> d; TCodlt > 0; ICi;n¼d;t > 0SRiodlt Shipments of i received at d from o by l at the beginning of period t=o <> d; TCodlt > 0; ICi;n¼d;t > 0SIPiodlt Shipments in progress of i from o to d by l at the beginning of period t=o <> d; TCodlt > 0; ICi;n¼d;t > 0; TLTodlt > 0FTLTiodlt Transport lead-time for i from o to d by l in t (only used in the fuzzy model)Iint Inventory amount of i at n at the end of period tPQint Purchase quantity of i at n in t=RMCint > 0OTnjt Overtime for resource j at n in tUTnjt Undertime for resource j at n in tYPinjt Binary variable indicating whether a product i has been produced on j at n in t

Table 2Fuzzy parameters considered in the model.

Source of uncertainty in supply chains Fuzzy coefficient Formulation

Demand Product demand eDint

Demand backlog cost DeBCint

Process Processing time PeT injt

Production capacity MePCnjt ;MeOTnjt

Production costs VePCinjt ;OeTCnjt ;UeTCnjt

Inventory holding cost IeCint

Maximum inventory capacity MeICnt

Supply Transport lead-time TeLTodlt

Transport cost TeCodlt

Maximum transport capacity MeTCnt

Maximum procurement capacity MPeRCnt



Minimize z ¼XI

i¼1

XN

n¼1

XJ

j¼1

XT

t¼1

ðVePCinjt � PinjtÞ þXN

n¼1

XJ

j¼1

XT

t¼1

ðOeTCnjt � OTnjt þ UeTCnjt � UTnjtÞ þXI

i¼1

XN

n¼1

XT

t¼1

ðRMCint � PQint þ IeCint � Iint

þ DeBCint � DBintÞ þXI

i¼1

XO

o¼1

XD

d¼1

XL

l¼1

XT

t¼1

ðTeCodlt � TQiodltÞ ð1Þ

Subject to
XI
i¼1

ðPinjt � PeT injtÞ 6 MePCnjt þMeOTnjt 8n; j; t ð2Þ

Pinjt ¼ kinjt � PRinjt 8i;n; j; t ð3Þ

Pinjt � PTinjt 6 MePCnjt � YPinjt þMeOTnjt � YPinjt 8i;n; j; t ð4Þ

Pinjt P MPRinjt � YPinjt 8i;n; j; t ð5Þ

Iint ¼ Iin;t�1 þXJ

j¼1

Pinjt þXO

o¼1

XL

l¼1

SRio;d¼n;lt þ PQint �XD

d¼1

XL

l¼1

TQi;o¼n;dlt � Sint �XP

p¼1

Bpint �XJ

j¼1

Pi¼p;njt

!8i; n; t ð6Þ

SRiodlt ¼ SR0iodlt þ TQiodl;t�TeLT

8i; o;d; l; t ð7Þ

SIPiodlt ¼ SIP0iodlt þ SIPiodl;t�1 þ TQiodlt � SRiodlt 8i; o;d; l; t ð8Þ

XI

i¼1

Iint � Vit 6 MeICnt 8n; t ð9Þ

XI

i¼1

XO

o¼1

XD

d¼1

SIPiodlt � Vit � v1odlt þ

XI

i¼1

XO

o¼1

XD

d¼1

TQ iodlt � Vit � v2odlt 6 MeTClt 8l; t ð10Þ

XI

i¼1

PQint 6 MPeRCnt 8n; t ð11Þ

DBint ¼ DBin;t�1 þ eDint � Sint 8i; n; t ð12Þ

OTnjt ¼XI

i¼1

Pinjt � PeT injt �MePCnjt þ UTnjt 8n; j; t ð13Þ

XN

n¼1

XT

t¼1

Sint 6XN

n¼1

XT

t¼1

ðeDint þ DB0intÞ 8i ð14Þ

Pinjt; kinjt P 0 8i;n; j; t ð15Þ

Sint;DBint; Iint; PQ int P 0 8i; n; t ð16Þ

SRiodlt; SIPiodlt; TQ iodlt P 0 8i; o;d; l; t ð17Þ

OTnjt ;UTnjt P 0 8n; j; t ð18Þ

Eq. (1) shows the total cost to be minimized. The total cost is formed by the production costs with the differentiation between regularand overtime production. The costs corresponding to idleness, raw material acquisition, inventory holding, demand backlog and transportare also considered. Most of these costs cannot be measured easily since they mainly imply human perception for their estimation. There-fore, these costs are considered uncertain data and are modeled by fuzzy numbers. Only the raw material cost is assumed to be known.

The production time per period could never be higher than the available regular time plus the available overtime for a certain produc-tion resource of a node (2). On the other hand, the produced quantity of each product in each planning period must always be a multiple ofthe selected production lot size (3).

Eqs. (4) and (5) guarantee a minimum production size for the different productive resources of the nodes in the different periods. Theseequations guarantee that Pinjt will be equal to zero if YPinjt is zero.

Eq. (6) corresponds to the inventory balance. The inventory of a certain product in a node, at the end of the period, will be equal to theinputs minus the outputs of the product generated in this period. Inputs concern the production, transport receptions from other nodes,purchases (if supplying nodes) and the inventory of the previous period. Outputs are related to shipments to other nodes, supplies to cus-tomers and the consumption of other products (raw materials and intermediate products) that need to be produced in the node.

Eqs. (7) and (8) control the shipment of products among nodes. The receptions of shipments for a certain product will be equal to theprogrammed receptions plus the shipments carried out in previous periods. In constraint (7), the transport lead-time is considered uncer-tainty data. On the other hand (8), the shipments in progress will be equal to the initial shipments in progress plus those from the previousperiod, plus the new shipments initiated in this period minus the new receptions.

Both the transports and inventory levels are limited by the available volume (known approximately). Thus according to Eq. (9), theinventory level for the physical volume of each product must be lower than the available maximum volume for each period (consideringuncertainty data). The inventory volume depends on the period to consider the possible increases and decreases of the storage capacityover time. Additionally, the physical volume of the product depends on the time to cope with the possible engineering changes thatmay occur and affect the dimensions and volume of the different products.


On the other hand, the shipment quantities in progress of each shipment in each period multiplied by the volume of the transportedproducts (if the transport time is higher than 0 periods), plus the initiated shipments by each transport in each period multiplied bythe volume of the transported products (if the transport time is equal to 0 periods), can never exceed the maximum transport volumefor that period (10). The reason for using a different formulation in terms of the transport time among nodes ðTLTodltÞ is because the trans-port in progress will never exist if this value is not higher than zero because all the transport initiated in a period is received in this sameperiod if TLTodlt ¼ 0. Finally, the transport volume depends on the period to consider the possible increases and decreases of the transportcapacity over time.

Eq. (11) establishes an estimated maximum of purchase for each node and product per period. Eq. (12) contemplates the backlog de-mand management over time. The backlog demand for a product and node in a certain period will be equal (approximately) to the backlogdemand of the previous period plus the difference between supply and demand.

Eq. (13) considers the use of overtime and undertime production for the different productive resources. The overtime production for aproductive resource of a certain node in one period is equal (approximately) to the total production time minus the available regular pro-duction time plus the idle time. OTnjt and UTnjt will always be higher or equal to zero if the total production time is higher than the availableregular production time, UTnjt will be zero as it does not incur in added costs, and OTnjt will be positive. Alternatively, if the total productiontime is lower than the available regular production time, UTnjt will be positive and OTnjt will be zero.

Conversely, Eq. (14) establishes that the sum of all the supplied products is essentially lower or equal to demand plus the initial backlogdemand. At any rate, the problem could easily consider that all the demand is served at the end of last planning period by transforming thisinequality equation into an equality equation. Finally, Eqs. (15)–(18) guarantee the non negativity of the corresponding decision variables.

4. Solution methodology

4.1. Transformation of the fuzzy mixed-integer linear programming model into an equivalent crisp model

In this section, we define an approach to transform the fuzzy mixed-integer linear programming model (FMILP) into an equivalent aux-iliary crisp mixed-integer linear programming model for tactical SC planning under supply, process and demand uncertainties. According toTable 2, and in order to address the fuzzy coefficients of the FMILP model, it is necessary to consider the fuzzy mathematical programmingapproaches that integrally consider the fuzzy coefficients of the objective function and the fuzzy constraints: technological and right-handside coefficients. In this context, several research works exist in the literature, and readers may refer to them (Buckley, 1989; Cadenas andVerdegay, 1997; Carlsson and Korhonen, 1986; Gen et al., 1992; Herrera and Verdegay, 1995; Jiménez et al., 2007; Lai and Hwang, 1992;Vasant, 2005). In this paper, we adopt the approach by Jiménez et al. (2007). The authors proposed a method for solving linear program-ming problems where all the coefficients were, in general, fuzzy numbers. They introduced a resolution method for this type of problemsthat permitted the interactive participation of the decision maker (DM) in all the steps of the decision process, and the expressing of opin-ions in linguistic terms was introduced.

Let us now consider the following linear programming problem with fuzzy parameters:

Min z ¼ ~ctx

s:a: x 2 NðeA; ~bÞ ¼ fx 2 Rnj~aix P ~bi; i ¼ 1; . . . ;m; x P 0gð19Þ

where ~c ¼ ð~c1; ~c2; . . . ; ~cnÞ; eA ¼ ½~aij�mxn;~b ¼ ð~b1;

~b2; . . . ; ~bnÞt represent, respectively, fuzzy parameters involved in the objective function and con-straints. The possibility distribution of fuzzy parameters is assumed to be characterized by fuzzy numbers. x ¼ ðx1; x2; . . . ; xnÞ is the crisp deci-sion vector.

The uncertain and/or imprecise nature of the parameters of the problem leads us to compare fuzzy numbers that involve two mainproblems: feasibility and optimality, therefore it is necessary to answer two questions (Jiménez et al., 2007):

(1) How to define the feasibility of a decision vector x, when the constraints involve fuzzy numbers.(2) How to define the optimality for an objective function with fuzzy coefficients.

Several focuses have been developed to solve this problem and to answer these questions (see for example Sakawa, 1993; Lai andHwang, 1994, Rommelfanger and Slowinski, 1998). A variety of methods for comparing or ranking fuzzy numbers has been reported inthe literature (Wang and Kerre, 1996). Different properties have been applied to justify ranking methods, such as: distinguishability, ratio-nality, fuzzy or linguistic representation and robustness. In this paper we use a fuzzy relationship to compare fuzzy numbers (Jimenez,1996) that verifies all the suitable properties above and a which, besides, is computationally efficient to solve linear problems becauseit preserves its linearity.

Thus, by applying the approach described by Jimenez (1996) the problem (19) is transformed into the crisp equivalent parametric linearprogramming problem defined in (20) where a represents the degree that, at least, all the constraints are fulfilled; that is, a is the feasibilitydegree of a decision x.

Min EVð~cÞxs:a: ½ð1� aÞEai

2 þ aEai1 �x P aEbi

2 þ ð1� aÞEbi1 ; i ¼ 1; . . . ;m; x P 0; a 2 ½0;1�

ð20Þ

where the expected value of a fuzzy number, noted EVð~cÞ, is the half point of its expected interval (Heilpern, 1992):

EVð~cÞ ¼ Ec1 þ Ec

2

2ð21Þ

and if the fuzzy number ~c is trapezoidal (see Fig. 1), its expected interval is easily calculated as follows:

1

2c1c 3c 4c

cµ

Fig. 1. Trapezoidal fuzzy number ~c.


EIð~cÞ ¼ Ec1; E

c2

� �¼ 1

2ðc1 þ c2Þ;

12ðc3 þ c4Þ

� �ð22Þ

If (19) was a less than or equal type constraint, 6, this could be transformed into the following equivalent crisp constraint:

ð1� aÞEai1 þ aEai

2

� �x 6 aEbi

1 þ ð1� aÞEbi2 ; i ¼ 1; . . . ;m; x P 0; a 2 ½0;1� ð23Þ

Otherwise, if (19) was a equality type constraint, this could be transformed into two equivalent crisp constraints:

1� a2

� �Eai

1 þa2

Eai2

h ix 6

a2

Ebi1 þ 1� a

2

� �Ebi

2 ; i ¼ 1; . . . ;m; x P 0; a 2 ½0;1�

1� a2

� �Eai

2 þa2

Eai1

h ix P

a2

Ebi2 þ 1� a

2

� �Ebi

1 ; i ¼ 1; . . . ;m; x P 0; a 2 ½0;1� ð24Þ

Consequently by applying this approach to the previously defined FMILP model, and by considering trapezoidal fuzzy numbers for theuncertain parameters, we obtain an auxiliary crisp mixed-integer linear programming model (MILP) as follows:

Minimize z ¼XI

i¼1

XN

n¼1

XJ

j¼1

XT

t¼1

VPCinjt1 þ VPCinjt2 þ VPCinjt3 þ VPCinjt4

4� Pinjt

�

þXN

n¼1

XJ

j¼1

XT

t¼1

OTCnjt1 þ OTCnjt2 þ OTCnjt3 þ OTCnjt4

4� OTnjt þ

UTCnjt1 þ UTCnjt2 þ UTCnjt3 þ UTCnjt4

4� UTnjt

�

þXI

i¼1

XN

n¼1

XT

t¼1

RMCint � PQ int þICint1 þ ICint2 þ ICint3 þ ICint4

4� Iint þ

DBCint1 þ DBCint2 þ DBCint3 þ DBCint4

4� DBint

�

þXI

i¼1

XO

o¼1

XD

d¼1

XL

l¼1

XT

t¼1

TCodlt1 þ TCodlt2 þ TCodlt3 þ TCodlt4

4� TQiodlt

� ð25Þ

Subject to
XI
i¼1

Pinjt � ð1� aÞ PTinjt1 þ PTinjt2

2þ a

PTinjt3 þ PTinjt4

2

� 6 ð1� aÞMPCnjt3 þMPCnjt4

2þ a

MPCnjt1 þMPCnjt2

2

þ ð1� aÞMOTnjt3 þMOTnjt4

2þ a

MOTnjt1 þMOTnjt2

28n; j; t ð26Þ

Pinjt � ð1� aÞ PTinjt1 þ PTinjt2

2þ a

PTinjt3 þ PTinjt4

2

� þ ð1� aÞMPCnjt1 þMPCnjt2

2þ a

MPCnjt3 þMPCnjt4

2

� � YPinjt

þ ð1� aÞMOTnjt1 þMOTnjt2

2þ a

MOTnjt3 þMOTnjt4

2

� � YPinjt 6 0 8i;n; j; t ð27Þ

SRiodlt ¼ SR0iodlt þ TQiodl;t�FTLT 8i; o;d; l; t ð28ÞXI

i¼1

Iint � Vit 6 ð1� aÞMICnt3 þMICnt4

2þ a

MICnt1 þMICnt2

28n; t ð29Þ

XI

i¼1

XO

o¼1

XD

d¼1

SIPiodlt � Vit � v1odlt þ

XI

i¼1

XO

o¼1

XD

d¼1

TQ iodlt � Vit � v2odlt 6 ð1� aÞMTClt3 þMTClt4

2þ a

MTClt1 þMTClt2

28l; t ð30Þ

XI

i¼1

PQint 6 ð1� aÞMPRCnt3 þMPRCnt4

2þ a

MPRCnt1 þMPRCnt2

28n; t ð31Þ


DBint � DBin;t�1 þ Sint 6 1� a2

� �Dint1 þ Dint2

2þ a

2Dint3 þ Dint4

28i;n; t ð32Þ

DBint � DBin;t�1 þ Sint P 1� a2

� �Dint3 þ Dint4

2þ a

2Dint1 þ Dint2

28i;n; t ð33Þ

XI

i¼1

Pinjt � 1� a2

� � PTinjt1 þ PTinjt2

2þ a

2PTinjt3 þ PTinjt4

2þ Tocnjt � Texnjt 6 1� a

2

� �MPCnjt1 þMPCnjt2

2þ a

MPCnjt3 þMPCnjt4

2

� 8n; j; t

ð34Þ

XI

i¼1

Pinjt � 1� a2

� � PTinjt3 þ PTinjt4

2þ a

2PTinjt1 þ PTinjt2

2þ Tocnjt � Texnjt P 1� a

2

� �MPCnjt3 þMPCnjt4

2þ a

MPCnjt1 þMPCnjt2

2

� 8n; j; t

ð35Þ

XN

n¼1

XT

t¼1

Sint 6XN

n¼1

XT

t¼1

ð1� aÞDint3 þ Dint4

2þ a

Dint1 þ Dint2

2þ DB0int

� 8i ð36Þ

FTLTodlt 6a2

TLTodlt1 þ TLTodlt2

2þ 1� a

2

� � TLTodlt3 þ TLTodlt4

28o; d; l; t ð37Þ

FTLTodlt Pa2

TLTodlt3 þ TLTodlt4

2þ 1� a

2

� � TLTodlt1 þ TLTodlt2

28o; d; l; t ð38Þ

FTLTodlt P 0 8o;d; l; t ð39Þ

The non fuzzy constraints (3), (5), (6), (8), (15), (16), (17) and (18) have also to be included in the model as in the original way.

4.2. Interactive resolution method

Here, an interactive resolution method which allows us to take a decision interactively with the DM is presented. This method is orga-nized in three steps.

In the first step of our method the auxiliary crisp mixed-integer linear programming model defined above is solved parametrically inorder to obtain the values of the decision variables and the objective function for each a 2 [0, 1]. Then we apply a method of evaluation(extended from Mula et al. (2006)) for the validation of models according to the following group of measurable parameters: (i) the averageservice level; (ii) inventory cost; (iii) planning nervousness with regard to the planned period; (iv) planning nervousness with regard to theplanned quantity and (v) total costs.

(i) The average service level for the finished good is calculated as follows:
� Average service level ð%Þ ¼
XT

t¼1

1� DBintPt

t0¼1Dint0

� 100

T8i;n ð40Þ

(ii) The inventory cost is calculated as the sum of the inventory holding cost of the finished good and parts at the end of each planning
period.
(iii) Planning nervousness with regard to the planned period. ”Nervous” or unstable planning refers to a plan which undergoes significantvariations when incorporating the demand changes between what is foreseen and what is observed in successive plans, as defined bySridharan et al. (1987). Planning nervousness can be measured according to the demand changes in relation to the planned period orto the planned quantity. The demand changes in the planned period measure the number of times that a planned order is resched-uled, irrespectively of the planned quantity (Heisig, 1998). The next rule proposed by Donselaar et al. (2000) is summarized as fol-lows:At time t we check for each period t + x (x = 0, 1,2, . . . ,T � 1):
� If there is a planned order in t + x, and this order is not planned in the next planning run, we increase the number of reschedules
by 1.� If there was no planned order in t + x, and there is one in the next planning run, we increase the number of reschedules by 1.

(iv) Planning nervousness with regard to the planned quantity measures the demand changes in the planned quantity as the number oftimes that the quantity of a planned order is modified (de Kok and Inderfurth, 1997). The rule is described as follows:

In the period t = 1,. . .,T, where T is the number of periods that forms the planning horizon, each period is checked so that t + x (x = 0,1,2,. . .,T � 1):

� If a planned order exists in the period t + x, then if the quantity of the planned order is not the same as in the next planning run, weincrease the number of reschedules by 1.

In the computation of planning nervousness, we measure the number of changes. Another way to compute it would be to take into ac-count the rate of the changes.

(v) Total costs are the sum of all the costs that are generated in every period of the considered planning horizon, and derived from theprocurement, production and distribution plans provided by the model.


In order to obtain a decision vector that complies with the expectations of the DM, we should evaluate two conflicting factors: the fea-sibility degree a and the reaching of an acceptable value for the different parameters.

In the second step of our method, after seeing the results obtained in the first step, the DM is asked to specify an aspiration level G and itstolerance threshold t for the numerical values obtained by each evaluation parameter. In the case of ‘‘less is better”, that is, inventory cost,nervousness and total cost, the DM’s satisfaction level is expressed by means of a fuzzy set ~G whose membership function is as follows(Jiménez et al., 2007):

leGðzÞ ¼1 if z 6 G

k 2 ½0;1� decreasing on G 6 z 6 Gþ t

0 if z P Gþ t

8><>: ð41Þ

Symmetrically in the case ‘‘more is better”, that is, in service level the goal is expressed by an increasing membership function:

leGðzÞ ¼1 if z P G

k 2 ½0;1� increasing on G� t 6 z 6 G

0 if z 6 G� t

8><>: ð42Þ

We define ki (i =1,. . .,5) as the degree in which the corresponding fuzzy aspiration levels of the above parameters are satisfied by a deci-sion vector. Obviously the DM wants to obtain a maximum satisfaction degree for all of them. In order to aggregate them we propose theweighted sum. The weight wi (i = 1,. . .,5) assigned by the DM to each parameter have been determined, as in the analytic hierarchy process(Saaty, 1990), by the eigenvalues of the matrix of pairwise ratios whose rows give the ratios of weights of each parameter with respect to allothers. Thus we obtain the following global satisfaction degree (where

Piwi ¼ 1): K ¼

Piwiki ¼ w1k1 þw2k2 þw3k3 þw4k4 þw5k5

But, in general, a lower level of the feasibility degree a will be achieved to obtain a better satisfaction degree K. Given these circum-stances, the DM might require lower satisfaction in exchange for a better feasibility.

In the third step of our method , in an attempt to find a balanced solution between the feasibility degree a and the global satisfactiondegree K, we propose to build two fuzzy sets whose membership functions represent the DM’s acceptation of the feasibility degree, ca, andthe acceptation of the satisfaction degree, cK, respectively. The respective acceptation degrees, ca and cK, increase monotonously betweenthe corresponding lower and upper bounds defined by the DM.

Observe that, when the DM built the fuzzy set related with the acceptation of the feasibility degree, ca, he/she is implicitly manifestinghis/her aptitude against the risk of unfeasibility and when he/she built the fuzzy set related with the acceptation of the satisfaction degree,cK, he/she is manifesting his/her aptitude regarding to the attainment of good results.

In order to obtain a recommendation for a final decision, we calculate a joint acceptation index K by aggregating, through the weightedmean, the two aforementioned acceptation degrees.

K ¼b � ca þ ð1� bÞ � cK if ca–0 and cK–0

otherwise

ð43Þ

where b 2 [0, 1] is the relative importance, assigned by the DM, to the feasibility in comparison with the satisfaction.

5. Application to an automobile supply chain

The proposed model has been evaluated by using data from an automobile SC which comprises a total of 47 companies (see Fig. 2). Infact, these companies constitute a segment of the automobile SC. Specifically, this SC segment supplies a seat model to an automobileassembly plant. All the data necessary to solve this application can be obtained from Peidro (2007). The nodes that form the SC are a seatassembly company, its first-tier suppliers, a manufacturing company of foams for seats, and a second-tier supplier that supplies chemicalcomponents for foam manufacturing. The automobile assembly plant transmits the demand information (automobile seats) on a weeklybasis with a planning horizon for six months. However, these demand forecasts are rarely precise (Mula et al., 2005). This section validateswhether the proposed fuzzy model for SC planning can be a useful tool for improving the decision-making process in an uncertain decisionenvironment.

Fig. 2. Supply chain.

START t=1

MODEL EXECUTION

DATA UPDATING

t=t+1

t>17

MODEL EVALUATION

END

INPUT DATA

RESULT (t)

MODEL EVALUATION DATA

NO

YES

Fig. 3. Diagram of the computational experiment.


5.1. Implementation and resolution

The model has been developed with the MPL (Maximal Software Incorporation, 2004) modeling language, and has been solved by theCPLEX 9.0 solver (ILOG, 2003). The input and output data are managed through a MS SQL Server database.

The model has been executed for a rolling horizon over a total of 17 weekly periods. These periods correspond to 17 different demandforecast programs, which are transmitted weekly by the automobile assembly plant. The total set of planning periods considered by thedemand forecast programs is 42 weeks. Fig. 3 depicts the execution of the models based on the rolling horizon technique. Each model cal-culation in the different planning horizon periods updates the data for the period being considered, and the results of the decision variablesfor the remaining periods are ruled out. Some of the stored decision variables are used as input data to solve the model in the followingperiods. These data include demand backlog, shipments received, shipments in progress and inventory. This process was repeated for allthe rolling horizon planning periods. The results of the model were evaluated from the data of the decision variables stored in each modelexecution. The experiments were run in an Intel Xeon PC, at 2.8 GHZ, with 1 GB of RAM.

5.2. Assumptions

The main characteristics and assumptions used in the experiment are presented below:

– The study considers a representative single finished good, i.e., a specific seat which can be considered a standard seat. The bill of mate-rials of the standard seat is composed of 55 elements arranged in a three-level structure.

– The decision variables Sint ; kinjt;DBint; TQ iodlt; SRiodlt; SIPiodlt; PQ int , and Pinjt are considered integers. Therefore, a mixed-integer linear pro-gramming model is required to be solved.

– Only the finished good has external demand.– The demand backlog for the finished good is considered but with a high penalization cost since the service level required by a sequenced

and synchronized automobile seat supplier is 100%.– A single productive resource restricts the capacity of the productions nodes (i.e., by focusing on the bottleneck resource).– Trapezoidal fuzzy numbers were defined by the decision makers involved in the planning process from the deviation percentages on the

crisp value. These percentages range from an average 5% to 30%, depending on the parameter to be evaluated.– The demand values for the first period of each model run, according to the rolling planning horizon, are considered firm. This means that

the fuzzy intervals of the demand for this period will be the equivalent to a crisp number. The same happens for all the demand values ofthe last program. Thus, all the models will have the same net requirements to fulfill.

– A maximum calculation time of 100 CPU seconds is set.

5.3. Evaluation of the results

Here, we compare the behavior of the proposed fuzzy model with its deterministic version. The aim is to determine the possibleimprovements that can provide the fuzzy model, which incorporates the uncertainties that may be presented in an SC.

Table 3 shows the computational efficiency of the deterministic model and the fuzzy SC planning model proposed. The data are related tothe iterations, number of constraints, variables, integers, non zero elements, calculation time and the average density of the array of con-straints for the set of the 17 planned executions of the models. The fuzzy model obtains higher values for these parameters, although theseare really very close to those obtained by the deterministic model. This is owing to the method for ranking the fuzzy numbers applied (Jimenez,1996) being computationally efficient to solve linear problems because it preserves its linearity, as already mentioned. Furthermore, this

Table 3Efficiency of the computational experiments.

Deterministic Fuzzya

Iterations 636,128 734,105Constraints 4,475,429 4,719,428Variables 4,840,812 5,853,474Integers 5,823,864 6,836,526Non zero elements 15,793,161 20,572,495Array density (%) 12.35 % 12.82 %CPU time (s) 1,298.50 1,421.28

a Average results for the different models generated with various feasibility degrees, a 2 [0, 1].

Table 4Evaluation of results.

Feasibility degree ðaÞ Service level (%) Inventory cost Planning nervousness (period) Planning nervousness (quantity) Total cost (€)

0 99.11 634.2 1.19 20.69 4,111,998.20.1 99.05 555.9 1.19 20.31 4,157,999.90.2 98.95 537.7 1.19 19.69 4,227,101.70.3 98.87 579.4 1.19 19.19 4,284,743.40.4 98.78 584.8 1.25 18.75 4,353,868.80.5 98.68 737.9 1.25 18.25 4,423,141.90.6 98.59 1142.6 1.25 17.75 4,492,666.70.7 98.51 1193.2 1.31 17.00 4,550,317.20.8 98.41 1285.0 1.44 16.75 4,619,529.10.9 98.32 1913.8 1.50 15.94 4,689,277.91 98.22 2503.2 1.50 15.19 4,758,987.2

Deterministic model 98.21 575.1 1.31 20.56 4,768,579.1


method maintains the same number of constraints as the deterministic model with the only exception of the case of the equality typeconstraints that are transformed into two equivalent constraints for the fuzzy model.

Table 4 summarizes the results with the different a values, according to the evaluation parameters defined in the first step of the inter-active resolution method proposed in 4.1.

As seen in Table 4, the fuzzy model, in general, obtains better results than the deterministic model. As seen in Fig. 4, the fuzzy modelobtains service levels that are better than the deterministic model, and this fuzzy model has better adapted to the demand uncertaintiesconsidered in this work.

Regarding inventory costs (see Fig. 5), the fuzzy model, in general, generates higher inventory cost than the deterministic model. This isbecause this model has obtained a higher service level than the deterministic model, and it implies more transport of materials to be able tocover the demanded quantities for customers and to not incur in demand delays. Nevertheless, it is important to highlight that for several avalues the fuzzy model generates less inventory costs and obtain a better service level. Thus, this model has been able to adapt better to the

Fig. 4. Service level (%).

Fig. 5. Inventory cost.


demand uncertainties by improving the service level (and smaller costs) without the need to incur in more inventory costs. It is importantto mention that the inventory holding cost of the elements of the bill of materials is quite different. For this reason less inventory cost doesnot necessarily imply less inventory levels (it could imply better inventory selection). Fig. 6 represents the inventory variation for the set ofthe 17 planned executions of the models in the rolling horizon technique. As seen in Fig. 6, the fuzzy model with lower a values obtainshigher inventory levels. This is because, as mentioned before, lower a values imply higher service level. And higher service level impliesmore transport of materials and inventory levels to satisfy the demanded quantities. As seen in Fig. 6, the inventory levels in periods10–11 reach the maximum value. This is because the demand quantities for the representative standard seat in these periods are higher.

Besides, the nervousness levels in relation to the planned period of the fuzzy model are similar to those of the deterministic model (onlyfor the values of a close to one are slightly higher), and the nervousness levels in relation to the planned quantity of the fuzzy model arebetter than the deterministic model. The fuzzy model presents a minor nervousness in relation to the number of times that the quantity ofa planned order is modified.

Furthermore, the fuzzy model (see Fig. 7) obtains lower or similar costs than the deterministic model for all the a values. This is becausethe demand backlog in this work is very heavily penalized, which means that those models with higher service levels achieve lower costs.

In Table 5 the values of different cost items for all the a values are presented. Production costs, raw material acquisition costs and trans-port costs are the same for all the a values. This is because all the fuzzy models satisfy all the demand requirements at the end of the plan-

Fig. 6. Inventory variation.

Fig. 7. Total costs.

Table 5Values of different cost items.

Feasibility degree ðaÞ Production cost Raw material acquisition cost Inventory cost Demand backlog cost Transport cost Total cost (€)

0 3,500,64 1,529,662.5 634.2 576,000 1,655,637.5 4,111,998.20.1 3,500,64 1,529,662.5 555.9 622,080 1,655,637.5 4,157,999.90.2 3,500,64 1,529,662.5 537.7 691,200 1,655,637.5 4,227,101.70.3 3,500,64 1,529,662.5 579.4 748,800 1,655,637.5 4,284,743.40.4 3,500,64 1,529,662.5 584.8 817,920 1,655,637.5 4,353,868.80.5 3,500,64 1,529,662.5 737.9 887,040 1,655,637.5 4,423,141.90.6 3,500,64 1,529,662.5 1142.6 956,160 1,655,637.5 4,492,666.70.7 3,500,64 1,529,662.5 1193.2 1,013,760 1,655,637.5 4,550,317.20.8 3,500,64 1,529,662.5 1285.0 1,082,880 1,655,637.5 4,619,529.10.9 3,500,64 1,529,662.5 1913.8 1,152,000 1,655,637.5 4,689,277.91 3,500,64 1,529,662.5 2503.2 1,221,120 1,655,637.5 4,758,987.2

Deterministic model 3,500,64 1,529,662.5 575.1 1,232,640 1,655,637,5 4,768,579.1


ning horizon. For this reason, these models produce the same production quantities, purchase the same raw materials and transport thesame quantities between nodes, but in different planning periods. Nevertheless, demand backlog costs and inventory costs are variablewith the different a values. When a increases (higher feasibility degree is achieved but more restrictive is the model) the demand backlogcost increases. However, although the demand backlog increases in the fuzzy model when the feasibility degree is higher, this cost is al-ways lower than the deterministic demand backlog cost. With regard to inventory costs, as mentioned before, the costs are in general high-er than the deterministic model. The fuzzy models obtain better service levels and for this reason require more transport of materials andmore inventory levels to fulfill the demanded quantities without the need to incur in more demand backlog costs. Nevertheless, is impor-tant to highlight that for a = 0.1–0.2 the inventory cost is lower than the deterministic value (similar cost for a = 0.3–0.4). These fuzzy mod-els take advantage of his flexibility (but lower feasibility degree) to obtain better service level and total cost without the need to incur inmore inventory cost.

After seeing the information presented in Table 4, the DM is asked to specify an aspiration level G and its tolerance threshold t for thenumerical values obtained by each parameter (second step of the proposed method). Particularly in our example, the membership func-tions of all fuzzy aspiration levels eGi (i = 1, . . . ,5) are linear and the DM proposes the following values:

– Service level: G1 ¼ 99:5%; G1 � t1 ¼ 98:21%.– Inventory cost: G2 ¼ 537:6; G2 þ t2 ¼ 2;503:6.– Planning nervousness (period): G3 ¼ 1:19; G3 þ t3 ¼ 1:5.– Planning nervousness (quantity): G4 ¼ 15:19; G4 þ t4 ¼ 20:69.– Total cost: G5 ¼ 4;111;998:2; G5 þ t5 ¼ 4;768;579:1.

In order to obtain the global satisfaction degree K, the following weights are assigned by the DM:

K ¼X

i

wiki ¼ 0:3k1 þ 0:2k2 þ 0:05k3 þ 0:05k4 þ 0:4k5

Then the global satisfaction degree obtained with the different feasibility degrees is presented in Table 6.The information in Table 6 can be very useful to help the DM to choose a balanced solution between the feasibility degree a and the

global satisfaction degree K. For this reason, in the third step of the proposed interactive resolution method two fuzzy sets, whose

Table 6The feasibility degree and the corresponding global satisfaction degree.

Feasibility degree ðaÞ Global satisfaction degree ðKÞ

0 0.850.1 0.820.2 0.760.3 0.710.4 0.640.5 0.560.6 0.460.7 0.400.8 0.310.9 0.181 0.06

Table 7Acceptation degrees and the joint acceptation index.

Feasibility degree ðaÞ Acceptation degree of ðaÞ ca Global satisfaction degree ðKÞ Acceptation degree of ðKÞ cK Joint acceptation index K ðb ¼ 0:5Þ

0 0 0.85 1.00 0.000.1 0 0.82 1.00 0.000.2 0 0.76 1.00 0.000.3 0.14 0.71 1.00 0.570.4 0.29 0.64 1.00 0.640.5 0.43 0.56 0.90 0.670.6 0.57 0.46 0.65 0.610.7 0.71 0.40 0.50 0.610.8 0.86 0.31 0.27 0.560.9 1 0.18 0.00 0.001 1 0.06 0.00 0.00


membership functions represent the DM’s acceptation degree of the values provided in Table 6, are defined. Regarding feasibility degree a,the DM reveals the following opinion: values less than 0.2 are unacceptable and values higher than 0.9 are totally acceptable and, withregard to what corresponds to the global satisfaction degree K, values less than 0.2 are unacceptable and values higher than 0.6 are totallyacceptable. The respective acceptation degrees, ca and cK, increase monotonously between the corresponding bounds. In our particularcase, we assume that the two membership functions are linear.

Finally, in order to obtain a recommendation for a final decision, we calculate a joint acceptation index K by aggregating, through theweighted mean, the two aforementioned acceptation degrees (see Table 7).

According to the results obtained in Table 7, we observe that, for a neutral DM between feasibility an satisfaction ðb ¼ 0:5Þ, the bestchoice would be the solution obtained by solving the model (25)–(39) corresponding to a ¼ 0:5.

6. Conclusions

Supply chain planning in an uncertainty environment is a complex task. This paper has proposed a novel fuzzy mixed-integer linearprogramming (FMILP) model for tactical SC planning by integrating procurement, production and distribution planning activities into amulti-echelon, multi-product, multi-level and multi-period SC network. The fuzzy model integrally handles all the epistemic uncertaintysources identified in SC tactical planning problems given a lack of knowledge (demand, process and supply uncertainties). This model hasbeen tested by using data from a real-world automobile SC by applying the rolling horizon technique over a total of 17 weekly periods. Theproposed model has been quantitatively evaluated according to a group of parameters: (i) the average service level, (ii) inventory levels,(iii) planning nervousness in relation to the planned period and planned quantity and (iv) total costs.

The evaluation of the results, which allows us to interactively make a decision with the DM, has demonstrated the effectiveness of afuzzy linear programming approach for SC planning under uncertainty. The proposed fuzzy formulation is more effective than the deter-ministic methods for handling the real situations where precise or certain information is not available for SC planning. Additionally, thefuzzy model behavior, in general, has been seen to be clearly superior to the deterministic model, and has obtained better service levels,inferior total costs and less planning nervousness. Furthermore, the fuzzy model has not generated an excessive increment of the compu-tational efficiency and inventory levels.

Finally, there are some possible directions for further research. Among them, is the application of other fuzzy mathematical program-ming based approaches in order to achieve a better adaptation to the uncertainties presented in real supply chains. In addition, extendingthe proposed method to decentralized tactical supply chain models in an uncertainty environment is of particular interest. The decentral-ized approach is suitable for companies where the elements of the supply chain belong to different companies and do not share internalinformation. Moreover, as mentioned earlier, a maximum calculation time of 100 CPU seconds was set to the execution of the FMILP model.In large-scaled supply chain planning problems it might be an issue. Therefore, the use of evolutionary computation to solve the corre-sponding FMILP models should be helpful in reaching efficient solutions. Lastly, the application of hybrid models based on the integrationof analytical and simulation models are an interesting option to integrate the best capacities of both types of models for SC planningproblems.


Acknowledgment

This work has been funded by the Spanish Ministry of Science and Technology project: ‘Simulation and evolutionary computation andfuzzy optimization models of transportation and production planning processes in a supply chain. Proposal of collaborative planning sup-ported by multi-agent systems. Integration in a decision system. Applications’ (EVOLUTION) (Ref. DPI2007-65501).

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