Application of fuzzy optimization to a supply chain network design: A case study of an edible vegetable oils manufacturer

Applied Mathematical Modelling 36 (2012) 2762–2776

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Application of fuzzy optimization to a supply chain network design: A case study of an edible vegetable oils manufacturer Turan Paksoy a,⇑, Nimet Yapici Pehlivan b, Eren Özceylan a a b

Selçuk University, Department of Industrial Engineering, Campus 42075, Konya, Turkey Selçuk University, Department of Statistics, Campus 42075, Konya, Turkey

a r t i c l e

i n f o

Article history: Received 19 April 2010 Received in revised form 9 September 2011 Accepted 21 September 2011 Available online 29 September 2011 Keywords: Supply chain management Fuzzy supply chain optimization Fuzzy multi objective linear programming Edible vegetable oil manufacturer Triangular fuzzy number

a b s t r a c t This study applies fuzzy sets to integrate the supply chain network of an edible vegetable oils manufacturer. The proposed fuzzy multi-objective linear programming model attempts to simultaneously minimize the total transportation costs. The first part of the total transportation costs is between suppliers and silos; and rest one is between manufacturer and warehouses. The approach incorporates all operating realities and actual flow patterns at production/distribution network with reference to demands of warehouses, capacities of tin and pet packaging lines. The model has been formulated as a multi objective linear programming model where data are modeled by triangular fuzzy numbers. Finally, the developed fuzzy model is applied for the case study, compiled the results and discussed.  2011 Elsevier Inc. All rights reserved.

1. Introduction Having an efficient and effective supply chain network provides a marketing area for enterprises in the global business environment. Determining positions and counting of actors, amount of product flow between and decreasing transportation costs are handled as a network design problem in supply chain management (SCM). In recent years, commercial and academic interests in SCM have increased remarkably. In the studies, importance and need of items that comprise the supply chain are highlighted to provide customer satisfaction and to obtain competitive advantage in process between raw material suppliers and end customers. Today, on the one hand, enterprises are expanding their own supply chain networks; on the other hand, they have to solve the problems of communication and long response time. However, it seems quite difficult to do it successfully for these companies owing to their huge and extremely complicated logistics networks, though they usually have imminent desire to cut down their logistics cost [1]. In this field, numerous researches are conducted. Pyke and Cohen [2] developed a mathematical programming model by using stochastic sub-models to design an integrated supply chain that involves manufacturers, warehouses and retailers. The model minimizes the total cost under a service level constraint and determines the economic re-order interval and replenishment batch sizes. Özdamar and Yazgaç [3] developed a distribution/production system, which involves a manufacturer center and its warehouses. The proposed model minimizes the total costs such as inventory and transportation costs under production capacity and inventory equilibrium constraints. Petrovic et al. [4] modeled supply chain behaviors under fuzzy constraints. They use simulation techniques to examine the dynamic and performance of the whole supply chain. Their model showed that uncertain customer demands and deliveries play a big role about behaviors. Paksoy [5] developed a mixed-integer linear programming to design a multi-echelon supply chain network under material requirement constraints. ⇑ Corresponding author. Tel.: +90 332 223 2040; fax: +90 332 241 06 35. E-mail addresses: [email protected] (T. Paksoy), [email protected] (N.Y. Pehlivan), [email protected] (E. Özceylan). 0307-904X/$ - see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.09.060

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The model considers sub-components (bill of materials), multi products and choice of distribution centers to be opened or not. Lin et al. [1] compared flexible supply chains and traditional supply chains with a hybrid genetic algorithm and mentioned advantages of flexible ones. The existence of flexible supply chains makes the problem much more difficult by traditional optimization methods. They formulate this problem as location-allocation model, and propose an effective hybrid genetic algorithm to solve this problem. Moreover, numerical analysis of a case study is carried out to show the effectiveness of the proposed approach. You and Grossmann [6] addressed the optimization of supply chain design and planning under responsive criterion and economic criterion with the presence of demand uncertainty. By using a probabilistic model for stock-out, the expected lead time is proposed as the quantitative measure of supply chain responsiveness. Gumus et al. [7] developed a mixed-integer linear programming model and proposed a neuro-fuzzy method for designing and optimizing the multi-echelon supply chain network of a multinational company in alcohol-free beverage sector. Chinese and Meneghetti [8] developed a mixed-integer linear programming model for designing of forest bio-fuel supply chains. They considered a real-life problem of supplying a bio-energy plant with forest fuel and determined the optimal configuration of the supply chain. Because of the difficulties to formulate mathematical models in designing production–distribution networks owing to lack of certainty, long time periods and complexity of networks, they developed and used a decision-support system to be very helpful especially for the global optimization of the supply chains. Schütz et al. [9] presented a supply chain design problem modeled as a sequence of splitting and combining processes. They formulated the problem as a two-stage stochastic program. The first-stage decisions are strategic location decisions, whereas the second stage consists of operational decisions. The objective is to minimize the sum of investment costs and expected costs of operating the supply chain. Ahumada and Villalobos [10] reviewed the main contributions in the field of production and distribution planning for agri-foods based on agricultural crops. Through their analysis of the current state of the research, they diagnosed some of the future requirements for modeling the supply chain of agri-foods. In most real-world SCM problems, environment coefficients and model parameters are frequently imprecise/fuzzy because some information is incomplete and/or unavailable over the planning. Conventional linear programming (LP) and special solution algorithms cannot solve all fuzzy SCM problems. Fuzzy set theory was developed by Zadeh [11], since then fuzzy set theory has been applied to the fields of operations research (linear programming, non-linear programming, multiple criteria decision making and so on), management science, artificial intelligence/expert system, statistics and many other fields. To formulate the fuzzy/imprecise numbers, membership functions could be used. Traditional mathematical programming techniques, obviously, cannot solve all fuzzy programming problems. In practice, input data are usually fuzzy/imprecise because of incomplete or non-obtainable information and knowledge. Because of this, precise mathematics is not sufficient to model a complex system. For solving decision making problem and fuzzy linear programming problem, various models are developed. In the last two decades, multiple objective decision making techniques have been applied to solve practical problem, such as academic planning, production and manufacturing planning, location, logistics, financial planning, portfolio selection, and so on [12,13]. Liang [14] developed an interactive fuzzy multi-objective linear programming method for solving the fuzzy multi objective transportation problems with piecewise linear membership function. The model considered multi-product and multitime period production/distribution planning decisions problems with fuzzy objectives. The proposed model attempts to simultaneously minimize total costs and total delivery time in relation to inventory levels, available machine capacity and labor levels at each source, and forecast demand and available warehouse space at each destination and total budget. The decision maker (DM) computes the value in each cost category by considering the time value of money in the proposed model, which is appropriate for practical application to the problem in a supply chain. Liang [15] developed a fuzzy multiobjective linear programming model with piecewise linear membership function to solve integrated multi-product and multi-time period production/distribution planning decisions problems with fuzzy objectives. The proposed fuzzy model provides a systematic framework that facilitates fuzzy decision-making process, enabling the DM to interactively adjust the search direction during the solution procedure to obtain a DM’s preferred satisfactory solution. Liang and Cheng [16] applied fuzzy sets to multi-objective manufacturing/distribution planning decision problems with multi-product and multi-time period in supply chains by considering time value of money for each of the operating categories. Peidro et al. [17,18] proposed a new mathematical programming model for supply chain planning under supply, process and demand uncertainty. The model has been formulated as a fuzzy mixed integer linear programming model where data are ill-known and modeled by triangular fuzzy numbers. Hu and Fang [19] solved the problem of fuzzy inequalities linear membership function by employing the concepts of constraint subrogation and maximum entropy, and it shows that a system of fuzzy inequalities with piecewise linear membership functions can be converted to a one-constraint nonlinear programming problem. An augmented Lagrangean algorithm is applied to solve the resulting problem. Xu and Zhai [20] considered a two-stage supply chain coordination problem under fuzzy demand constraints. They investigated the optimization of the vertically integrated two-stage supply chain under perfect coordination and contrast with the non-coordination in case of the fuzzy demand. They proved that the maximum expected supply chain profit in a coordination situation is greater than the total profit in a non-coordination situation. In this study, a fuzzy multi objective linear programming (FMOLP) model under fuzzy material requirement constraints are developed and optimized for production/distribution network of an edible vegetable oil manufacturer. The remainder of this study is organized as follows. In the second section, the proposed model is explained with its parameters and decision variables. In the third section, little information about the oils company and its whole process are given and then the

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application handled with the real data of an edible vegetable oils manufacturer is presented with its results and discussed; conclusion and comments are in the last section. 2. Proposed model In real-world supply chain network problems, environmental coefficients and related parameters, including market demand and unit cost/time coefficients, available labor levels and machine capacity, are normally fuzzy/imprecise because of some information being incomplete and/or unobtainable over the intermediate planning horizon. It is critical that the satisfying goal value should normally be fuzzy/imprecise as the unit cost/time coefficients and parameters are vague and such imprecision always exists in real-world supply chain network problems in supply chains [4]. The conventional solution methods and algorithms cannot solve all realistic problems in uncertain environments. In practical situations, supply chain network problems generally have conflicting objectives regarding the use of organization’s resources, and these conflicting objectives are required to be optimized simultaneously by the DM in the framework of fuzzy aspiration levels [16]. The fuzzy mathematical programming model designed here is based on the following assumptions [16]:  All objective functions are fuzzy with imprecise aspiration levels.  All objective functions and constraints are linear equations.  The production costs at each source and distribution cost/time on a given route are directly proportional to the units manufactured and shipped capacity per truck, respectively.  The pattern of triangular distribution is adopted to represent all of the fuzzy/imprecise numbers.  The linear membership functions are specified for all of the fuzzy objectives involved in the proposed model.  Capacities and demands are fuzzy. Assumption 1 relates to the fuzziness of the objective functions in practical supply chain optimization problems and incorporates the variations in the decision maker judgments regarding the solutions of fuzzy optimization problems in a framework of imprecise aspiration levels. Assumptions 2 and 3 indicate that the linearity and proportionality properties must be technically satisfied as a standard linear programming form. Assumption 4 concerns the simplicity and flexibility of the fuzzy arithmetic operations. Triangular distribution is utilized to represent all of the fuzzy numbers and thus enhance the computational efficiency and facilitate data acquisition. Assumption 5 is made to specify the fuzzy objective functions with linear membership functions and to convert the original fuzzy MOLP problem into an equivalent ordinary linear programming form. Assumption 6 relates the imprecise data in real cases. In this section, under fuzzy material requirements constraints, a FMOLP model is developed for ABC’s supply chain network. The construction of mathematical model requires the definition of the following elements: objectives, decision variables, constraints, and parameters. 2.1. Nomenclature Indices used in model are: i 2 I, a set of crude oil silos; j 2 J, a set of refineries; k 2 K, a set of refined sun flower oil silos; l 2 L, a set of refined corn oil silos; m 2 M, a set of refined soybean oil silos; n 2 N, a set of olive oil silos; t 2 T, a set of tin filling lines; u 2 U, a set of pet bottle filling lines; b 2 B, a set of tin package size;d 2 D, a set of pet bottle sizes; v 2 V, a set of end products; p 2 P, a set of private label and oil company warehouses. The problem decision variables and parameters are defined as follows: Xi Yij Zjk Wjl Qjm Pj Nkt Fl Kmt Lku Slu Mmu On Ant Vnu Htpvb

amount amount amount amount amount amount amount amount amount amount amount amount amount amount amount amount

of of of of of of of of of of of of of of of of

transported transported transported transported transported transported transported transported transported transported transported transported transported transported transported transported

oil from supplier to crude oil silo i oil from crude oil silo i to refinery j sun flower oil from refinery j to sun flower oil silo k corn oil from refinery j to corn oil silol soybean oil from refinery j to soybean oil silom waste oil from refinery j to recycling section sun flower oil from silo k to filling line t corn oil from silo l to filling line t soybean oil from silo m to filling line t sun flower oil from silo k to filling line u corn oil from silo l to filling line u soybean oil from silo m to filling line u olive oil from olive oil supplier to olive oil silo n olive oil from silo n to filling line t olive oil from silo n to filling line u package size b of oil v from filling line t to warehouse p

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bd ˜ıtvb ~j

amount of transported package size d of oil v from filling line u to warehouse p unit cost of transportation from crude oil supplier to crude oil silo i unit cost of transportation from olive oil supplier to olive oil silo n unit cost of transportation package b of oil v from filling line t to warehouse p unit cost of transportation package d of oil v from filling line u to warehouse p percent number of wasting amount from refinery j size of tin package b size of pet package d fuzzy capacity of filling line t for package b of oil v fuzzy capacity of filling line u for package d of oil v

~ a ~ b

fuzzy capacity of crude oil supplier fuzzy capacity of crude oil silo i

Gupvd Ci Cn Ctpvb Cupvb

xj ab

uv d

i

~cj ~ d

fuzzy capacity of refinery j fuzzy capacity of sun flower oil silo k

~ el ~f m g~ ~n h

fuzzy capacity of corn oil silo l fuzzy capacity of soybean oil silo m

~ k pv b ~l pv d

fuzzy demand of warehouse p for package b of oil

k

fuzzy capacity of olive oil supplier fuzzy capacity of olive oil silo n

v fuzzy demand of warehouse p for package d of oil v

2.2. Model formulation The production/distribution network problem minimizes the total transportation costs with reference to fuzzy demands of warehouses, fuzzy capacities of tin and pet packaging lines. In particular, these objective functions are fuzzy in nature owing to incomplete and/or unavailable information over the planning. The proposed fuzzy multi-objective linear programming model attempts to simultaneously minimize the total transportation costs. The first part of the total transportation costs is between suppliers and silos; and rest one is between manufacturer and warehouses. Objective functions

X

Min Z 1 ¼

XiCi þ

X

On C n

ð1Þ

n

i X XXX

Min Z 2 ¼

t

v

p

Htpv b C tpv b þ

XXXX u

b

p

v

Gupv d C upv d

ð2Þ

d

Constraints

X

~ Xi 6 a

ð3Þ

i

X

~; Y ij 6 b i

j

X

Z jk þ

X

k

8i

ð4Þ

W jl þ

X

Z jm 6 ~ci ;

X ~ ; 8k Nkt þ W ku 6 d k t u X X Fl þ Slu 6 ~el ; 8l

X

t

X

K mt þ

X

M mu 6 ~f m ;

8m

On 6 g~ Ant þ

t

X

ð6Þ ð7Þ ð8Þ

u

n

X

ð5Þ

u

t

X

8j

m

l

ð9Þ

X

~n ; V nu 6 h

8n

ð10Þ

u

Htpv b 6 ~itv b ;

8t; v ; b

ð11Þ

p

X

Gupv d 6 ~juv d ; 8u; v ; d X Xi  Y ij ¼ 0 ; 8t

ð12Þ

p

j

ð13Þ

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ð1  xj Þ

xj

X

X

X

Y ij 

i

Z jk þ

k

X

X

X

W jl 

On 

X

F l  ab

K mt  ab

n

X

ð16Þ

¼ 0;

8l

ð17Þ

! Slu

X

!

8m

M mu

¼ 0;

¼ 0;

8n

ð19Þ

8t; v : 1; b

ð20Þ

ð18Þ

!

V nu

u

Htpv b ¼ 0;

Htpv b ¼ 0;

8t; v : 2; b

ð21Þ

Htpv b ¼ 0;

8t; v : 3; b

ð22Þ

Htpv b ¼ 0;

8t; v : 4; b

ð23Þ

Gupv b ¼ 0;

8u; v : 1; d

ð24Þ

Gupv b ¼ 0;

8u; v : 2; d

ð25Þ

p

Lku  bd

X p

Slu  bd

X p

Mmu  bd

m

X

X

X

X

Ant  ab

l

X

8k

p

k

X

¼ 0;

p

t

X

ð14Þ

u

X

m

X

8j

p

l

X

K mt þ

X

Nkt  ab

Lku

u

Ant þ

t

t

X

X

t

j

X

F it þ

X

¼ 0;

m

ð15Þ

X

t

Q jm 

! Q jm

!

u

j

X

Nkt þ

t

k

X

8j

Y ij  Pj ¼ 0;

Z jk 

W jl þ

l

i

X

X

X

Gupv b ¼ 0;

8u; v : 3; d

ð26Þ

p

V nu  bd

X

Gupv b ¼ 0;

8u; v : 4; d

ð27Þ

p

~ ; Htpv b P k pv b

8p; v ; b

ð28Þ

Gupv d P ~lpv d ;

8p; v ; d

ð29Þ

t

X u

X i ; Y ij ; W jl ; Q jm ; P j ; Nkt ; F l ; K mt ; Lku ; Slu ; Mmu ; On ; Ant ; V nu ; Htpv b ; Gupv d P 0;

8i; j; k; l; m; n; t; u; b; d; v ; p

ð30Þ

In Eq. (1), the objective function (Z1) which includes transportation costs between crude oil suppliers and silos; between olive oil suppliers and silos is defined. The objective function (Z2) which includes transportation costs between crude oilolive oil suppliers and private label-oil company warehouses is given in Eq. (2). The Eqs. (3)–(10) guarantee not to exceed fuzzy capacity of crude oil suppliers, crude oil silos, refineries, refined oil silos and olive oil suppliers. Eqs. (11) and (12) guarantee the fuzzy capacity of filling lines. Eq. (13) ensures that the total amount of crude oil which is transported from suppliers to ith crude oil silos should be greater than or equal to the total amount of crude oil which is transported from ith crude oil silos to jth refineries. Eqs. (14) and (15) ensure that a certain percent of total amount of crude oil which is transported from jth refinery should be equal to the total amount of oil which is transported from jth refinery to recycling. Eq. (16) guarantees that the total amount of sunflower oil which is transported from jth refinery to kth sunflower oil silo should be greater than or equal to the total amount of sunflower oil which is transported from kth sunflower oil silo to tth and uth filling lines. Eq. (17) provides that the total amount of corn oil which is transported from jth refinery to lth corn oil silo should be greater than or equal to the total amount of corn oil which is transported from lth corn oil silo to tth and uth filling lines. Eq. (18) ensures that the total amount of soybean oil which is transported from jth refinery to mth soybean oil silo should be greater than or equal to the total amount of soybean oil which is transported from mth soybean oil silo to tth and uth filling lines. Eq. (19) ensures that the total amount of olive oil which is transported from olive oil supplier to nth olive oil silo should be greater than or equal to the total amount of olive oil which is transported from nth olive oil silo to tth and uth filling lines. Eq. (20) guarantees that the total amount of sunflower oil which is transported from kth sunflower oil silos to tth filling line should be greater than or equal to the total amount of tin sunflower oil packages which is transported from tth filling line to pth warehouse. Eq. (21) guarantees that the total amount of corn oil which is transported from lth corn oil silos to tth filling line should be greater than or equal to the total amount of tin corn oil packages which is transported from tth filling line to pth warehouse. Eq. (22) guarantees that the total amount of soybean oil which is transported from mth soybean oil silos to tth filling line should be greater than or equal to the total amount of tin soybean oil packages which is transported from tth filling line to

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pth warehouse. Eq. (23) guarantees that the total amount of olive oil which is transported from nth olive oil silos to tth filling line should be greater than or equal to the total amount of tin olive oil packages which is transported from tth filling line to pth warehouse. Eq. (24) guarantees that the total amount of sunflower oil which is transported from kth sunflower oil silos to uth filling line should be greater than or equal to the total amount of pet sunflower oil packages which is transported from uth filling line to pth warehouse. Eq. (25) guarantees that the total amount of corn oil which is transported from lth corn oil silos to uth filling line should be greater than or equal to the total amount of pet corn oil packages which is transported from uth filling line to pth warehouse. Eq. (26) guarantees that the total amount of soybean oil which is transported from mth soybean oil silos to uth filling line should be greater than or equal to the total amount of pet soybean oil packages which is transported from uth filling line to pth warehouse. Eq. (27) guarantees that the total amount of olive oil which is transported from nth olive oil silos to uth filling line should be greater than or equal to the total amount of pet olive oil packages which is transported from uth filling line to pth warehouse. Eqs. (28) and (29) provide that total amount of pet and tin packages of all oils which are transported from filling lines to pth warehouse should be greater than or equal to the total fuzzy demand of pth warehouse. Eq. (30) assures that all variables to take non-negative continuous values [21]. 3. Fuzzy model formulation 3.1. Method for solving the fuzzy constraints In this section we adopted the triangular fuzzy number to the production/distribution network model for an edible vegetable oils manufacturer with multiple objectives. The main advantages of the triangular fuzzy number are simplicity and e i ¼ Ap ; Am ; Ao  is shown in Fig. 1. flexibility of the fuzzy arithmetic operations. The distribution of a triangular fuzzy number A i i i e i based on the following three prominent data [14–16]: The most pesWe can construct the triangular distribution of A o simistic value is Api ; the most likely value is Am i ; the most optimistic value is Ai . In this study, weighted average method is used to convert triangular fuzzy number into a crisp number. If the minimum acceptable membership level a is given, the corresponding auxiliary crisp inequality of a triangular fuzzy number e i ¼ Ap ; Am ; Ao  can be expressed as follows; A i i i

e i ¼ w1 Ap þ w2 Am þ w3 Ao ; A i i i

ð31Þ

where w1 + w2 + w3 = 1, w1, w2 and w3 represent the corresponding weight of the most pessimistic, most likely and most optimistic values, respectively. In practice, the weights can be determined subjectively based on DM’s experience and knowledge [16]. Based on the weighted average method, Eqs. (3)–(10) and Eqs. (28) and (29) of our problem is converted to crisp inequalities of constraints as following, respectively;

X i

X

X i 6 ðw1 ap þ w1 am þ w1 ao Þ  p m o Y ij 6 w1 bi þ w1 bi þ w1 bi ;

j

X

Z jk þ

X

k

X t X

l

X

Fl þ

Xu

K mt þ

t

ð33Þ

  o Z jm 6 w1 cpi þ w1 cm i þ w1 c i ;

 p m o W ku 6 w1 dk þ w1 dk þ w1 dk ; 

 o

X

8k

8l

ð34Þ

  M mu 6 w1 fmp þ w1 fmm þ w1 fmo ;

ð35Þ ð36Þ

8m

On 6 ðw1 g p þ w1 g m þ w1 g o Þ

ð37Þ

Ant þ

X

 p m o V nu 6 w1 hn þ w1 hn þ w1 hn ;

ð38Þ

8n

ð39Þ

u

  p m o Htpv b P w1 kpv b þ w1 kpv b þ w1 kpv b ; 8p; v ; b t   X p m o Gupv d P w1 lpv d þ w1 lpv d þ w1 lpv d ; 8p; v ; d

X

8j

m

u

n

X

8i

Slu 6 w1 epl þ w1 em l þ w1 el ;

u

t

X

X

Nkt þ

t

W jl þ

X

ð32Þ

ð40Þ ð41Þ

u

3.2. Solving the fuzzy problem In the fuzzy set theory, the fuzzy objectives are defined by their corresponding membership functions. The non-increasing continuous linear membership functions for all fuzzy objectives can be expressed as follows;

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ei. Fig. 1. The distribution of triangular fuzzy number A

Fig. 2. The non-increasing continuous linear membership functions of fg(Zg).

8 1; Z g 6 Z PIS > g > > < NIS Z g Z g NIS fg ðzg Þ ¼ ZNIS ZPIS ; Z PIS g < Zg < Zg g g > > > : 0; Z g P Z NIS g

ð42Þ

where; Z PIS g : The positive ideal solutions (PIS, lower bound) for the gth objective Zg. Z NIS : The negative ideal solutions (NIS, upper bound) for the gth objective Zg. h i g NIS . The linear membership functions can be specified by requiring the DM to select the objective value interval Z PIS g ; Zg The graph of the non-increasing continuous linear membership functions are shown in Fig. 2 [16]. The production/distribution network model for an edible vegetable oils manufacturer with multiple objectives is solved by Liang and Cheng’s [16] method. Solution procedure of this method is given as follows; Step 1: The original fuzzy MOLP problem is formulated. Step 2: Fuzzy inequality constraints are converted into crisp ones using the weighted average method for the given minimum acceptable membership level a. Step 3: The PIS and NIS for each of the fuzzy objectives are specified and the corresponding non-increasing continuous linear membership functions are defined. Step 4: Introducing the auxiliary variable L which enables the original fuzzy MOLP problem to be converted into an equivalent ordinary LP problem. The ordinary single-objective LP problem can be solved using LP methods and formulated as follows;

Max L L 6 fg ðZ g Þ;

g ¼ 1; 2

ð43Þ

Eqs:ð11Þ—ð27Þ; ð30Þ; ð32Þ—ð41Þ; where: L value (0 6 L 6 1) represents the overall DM satisfaction with the given goal values. The single-objective LP problem is solved and initial compromise solution is obtained. Fuzzy decision making process is executed and modified. If the DM is dissatisfied with the initial solutions, the model must be adjusted until a set of satisfactory solutions is derived [16].

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Fig. 3. Supply chain network of Oil Company.

Fig. 4. Production/distribution network of Oil Company.

4. Case study application 4.1. About the company In this section, some information is informed about the company where we research case study on. ABC Oil Company started food trading facilities at 1950s with the aim of offering high quality products and initiated the foundation of ABC’s

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Fig. 5. Oil and package types.

Table 1 Basic data of the case. Number of crude oil silos: 12

Number of tin packages: 3

Number of refineries: 2 Number of sunflower oil silos: 4 Number of corn oil silos: 4 Number of pet packages: 2 Number of products: 4 Number of olive oil silos: 6 ã = (4000000, 5000000, 6000000) l ~c1 ¼ ð400000; 500000; 600000Þ l ~c2 ¼ ð2000000; 3000000; 4000000Þ l ~ ¼ ð500000; 600000; 700000Þ l; " d

Number Number Number Number Number

k

of of of of of

soybean oil silos: 2 pet filling lines: 2 tin filling lines Company warehouses: 4 private label warehouses: 4

~ el ¼ ð500000; 600000; 700000Þ l; "l ~f ¼ ð500000; 600000; 700000Þ l; " m m g~ ¼ ð750000; 1000000; 1250000Þ l

k

vegetable oil production facilities in 1989 and standard production in 1991. Being landed on 102.000 m2 open and 42.000 m2 closed area, the refinery is capable of processing all kinds of vegetable oils like ‘‘Sunflower oil, Olive oil, Corn oil, Soybean oil, Canola oil, Cotton oil, Hazelnut oil, Vegetable Oil Blend etc.’’ In its sector, production at ABC is a natural consequence of the world’s most advanced technology. In this regard ABC, vegetable oil production facilities is the first vegetable oil production plant which is certified with TS-EN-ISO 9002 Quality System Certificate. The adventure of the edible oil starts with transporting crude vegetable oils (except olive oil) from suppliers. Tracks which come from suppliers unload the whole oil to crude oil silos. According to production plans, two kinds of refinery (one’s capacity 300 ton/day, the other’s capacity 60 ton/year) pull vegetable oils whatever amount from crude oils silos. In the refineries, crude oils are proceeding by some chemical processing such as order by degumming, neutralization, dewaxing, bleaching, winterization and deodorization to compose each demand oils (corn, soybean, cotton, sunflower oil etc.). After refinery, crude oils turn into both edible oils and wax (recycling product). Waxes are sent to recycling for using in cosmetic sector. After refineries edible oils (except olive oil) are transported to refined oils silos for leaving fallow. By the way, in factory all oils transportation is actualizing via steel tubes. To pretend complicacy, there are two collection pools between refined oils silos and filling section. So refined oils is collected at pools and then transported to filling section. Just at that part, olive oils are supplied and included into system by collection olive oils silos near the filling section. Olive oils are filled into bottles or tins directly without any process. At filling section, there are four lines. First and second lines are tins filling, rest of them are bottles filling. ABC presents two kinds of bottles packaging (1 l and 2 l), and three kinds of tins packaging (5 l, 10 l and 18 l) to their consumers. After filling different kinds of packaging, all oils are kept waiting at factory’s warehouse for forwarding. ABC sells its products both with its label and also with private label for big markets. As private label, Bim, Kipa, Adese, Diasa, Carrefour etc. are consumers of ABC. ABC delivers the demands of each big market to their own warehouses in different cities in Turkey. Some big markets can only demand unique products such as soybean oil for certain cities. Also ABC sells its own label via its distribution centers to all cities in Turkey. After DCs products are flowing order by retailers, sellers and small markets. Some information about ABC Corporation and the whole process, which is from crude oil to small-size consumers, is given in detailed in Paksoy and Cavlak [21].

4.2. Application of the model In this section we present a numerical example to illustrate the model given in Section 3. The application of the model is performed via real data collected from ABC Oil Company [21]. The supply chain network of the company includes a crude oil _ _ supplier which is located in Izmir, the factory located in Konya, private label warehouses located in Istanbul, Adana, Kayseri, _Izmir and ABC Oil Company warehouses located in Ankara, Antalya, Diyarbakır and Çorum in Turkey (Fig. 3).

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T. Paksoy et al. / Applied Mathematical Modelling 36 (2012) 2762–2776 Table 2 ~ Þ. Fuzzy demands of private label and oil company warehouses per pet bottles and tins ðk pv b Oil types

Package types

Oil Company warehouses Ankara

Antalya

Diyarbakır

Çorum

Sunflower

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(676, 776, 876) (6102, 7202, 8102) (48994, 58994, 68994) (6992, 7992, 8992) (11314, 13314, 15314)

(676,776, 876) (636, 836, 1036) (3050, 4050, 5050) (3, 5, 7) (13257, 15257, 17257)

(159, 209, 259) (88, 108, 128) (1060, 1360, 1660) (775, 975, 1175) (3234, 4234, 5234)

(155, 175, 195) (550, 650, 750) (594, 694, 794) (550, 650, 750) (610, 710, 810)

Corn

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(750, 950, 1150) (1148, 1348, 1548) (10512, 12512, 14512) (775, 975, 1175) (56, 65, 74)

(30, 36, 42) (24, 29, 34) (40, 47, 54) (4, 6, 8) (36, 43, 50)

(16, 20, 24) (19, 24, 29) (148, 168, 188) (77, 87, 97) (79, 89, 99)

(41, 50, 59) (141, 166, 181) (64, 73, 82) 897, 109, 121) (165, 195, 225)

Soybean

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(69, 79, 89) (140, 154, 168) (2060, 2560, 3060) 80, 0, 0) (0, 0, 0)

(10, 13, 16) (22, 27, 32) (71, 81, 91) (0, 0, 0) (0, 0, 0)

(6, 9, 12) (12, 16, 20) (42, 49, 56) (0, 0, 0) (0, 0, 0)

(8, 11, 14) (17, 22, 27) (27, 33, 39) (0, 0, 0) (0, 0, 0)

Olive

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet ~l Þ

(18, 22, 26) (14, 19, 24) (74, 84, 94) (0, 0, 0) (0, 0, 0)

(8, 11, 14) (13, 17, 21) (35, 42, 49) (0, 0, 0) (0, 0, 0)

(1, 3, 5) (4, 6, 8) (20, 25, 30) (0, 0, 0) (0, 0, 0)

(8, 11, 14) (37, 44, 51) (72, 82, 929 (0, 0, 0) (0, 0, 0)

Oil types

Package types

_ Istanbul

pv d

Privateltabel warehouses Adana

Kayseri

_ Izmir

Sunflower

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(60860, 70860, 80860) (43379, 53379, 63379) (123318, 143318, 163318) (0, 0, 0) (0, 0, 0)

(8181, 10181, 12181) (8767, 10767, 12767) (70915, 80915, 90915) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (0, 0, 0) (45677, 55677, 65677) (0, 0, 0) (0, 0, 0)

(2, 4, 6) (4, 6, 8) (4, 6, 8) (0, 0, 0) (0, 0, 0)

Corn

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(0, 0, 0) (20835, 24835, 28835) (43540, 53540, 63540) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (2440, 3440, 4440) (21034, 23034, 25034) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (2440, 3440, 4440) (17292, 19292, 21292) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (9, 12, 15) (10, 13, 16) (0, 0, 0) (0, 0, 0)

Soybean

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(0, 0, 0) (16396, 19396, 22396) (84560, 104560, 124560) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (410, 510, 610) (5896, 6896, 7896) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (3152, 4152, 5152) (3230, 4230, 5230) (0, 0, 0) (0, 0, 0)

(0, 0, 0) (3148, 4148, 5148) (4284, 5284, 6284) (0, 0, 0) (0, 0, 0)

Olive

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

(6408, 7408, 8408) (8230, 10230, 12230) (0, 0, 0) (0, 0, 0) (0, 0, 0)

(1068, 1268, 1468) (5030, 6030, 7030) (0, 0, 0) (0, 0, 0) (0, 0, 0)

(1568, 1768, 1968) (2154, 3154, 4154) (0, 0, 0) (0, 0, 0) (0, 0, 0)

(3, 5, 7) (4, 6, 8) (0, 0, 0) (0, 0, 0) (0, 0, 0)

Crude oil, which is transported from supplier, is unloaded to crude oil silos via oil tankers in factory area in Konya. After loading to silos, crude oil is pumped to refineries via steel tubes. Crude oil is transformed to refined oil (sunflower, corn, soybean, etc.) via chemical processes in refineries. After chemical processes, a little amount of waste of product is sent to recycling. There are totally four filling lines constituting of two kinds, one of which is pet and the other is tin. Refined oil is filled up to pet bottles or tins in filling lines according to capacity and demand. Besides sunflower, corn, soybean oil, etc., olive oil, _ which is transported from Izmir, is also filled up into pet bottles and tins (Fig. 4). Except transporting from suppliers, all transportation is actualized via steel tubes. As mentioned in the second section, ABC Oil Company sells kinds of oil both with its own label and private label to the customers in different packages (Fig. 5). In this application, the network is designed until warehouses. Warehouses have different demands, as seen in Table 1, in which 10–18 l tin soybean oil is not requested. Also, there is no 5, 10 and 18 l olive oil production for private label warehouses. Here below, the basic data of the case are given. Table 2 shows the fuzzy demands of private label and oil company warehouses per bottles and tins (unit). Table 3 shows the transportation costs per liter between factory and Private Label-Oil Company warehouses in Turkish Liras (TL). Table 4 shows the transportation costs per liter from crude oil supplier to crude oil silos in TL and capacities for each silo in liter. Table 5 shows the transportation costs per liter from olive oil supplier to olive oil silos in TL and capacities for each silo

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Table 3 Unit transportation costs between the factory and warehouses (Ctpvb; Cupvd). Oil types

Package

Oil Company warehouses

Private label warehouses

Ankara

Antalya

Diyarbakır

Çorum

_ Istanbul

Adana

Kayseri

_ Izmir

Sunflower

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

0.024 0.048 0.12 0.24 0.432

0.033 0.066 0.165 0.33 0.594

0.08 0.16 0.4 0.8 1.44

0.06 0.12 0.3 0.6 1.08

0.043 0.086 0.215 0 0

0.033 0.066 0.165 0 0

0 0 0.135 0 0

0.033 0.066 0.165 0 0

Corn

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

0.024 0.048 0.12 0.24 0.432

0.033 0.066 0.165 0.33 0.594

0.08 0.16 0.4 0.8 1.44

0.06 0.12 0.3 0.6 1.08

0 0.086 0.215 0 0

0 0.066 0.165 0 0

0 0.054 0.135 0 0

0 0.066 0.165 0 0

Soybean

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

0.024 0.048 0.12 0 0

0.033 0.066 0.165 0 0

0.08 0.16 0.4 0 0

0.06 0.12 0.3 0 0

0 0.086 0.215 0 0

0 0.066 0.165 0 0

0 0.054 0.135 0 0

0 0.066 0.165 0 0

Olive

1 l pet 2 l pet 5 l pet 10 l pet 18 l pet

0.024 0.048 0.12 0 0

0.033 0.066 0.165 0 0

0.08 0.16 0.4 0 0

0.06 0.12 0.3 0 0

0.043 0.086 0 0 0

0.033 0.066 0 0 0

0.027 0.054 0 0 0

0.033 0.066 0 0 0

Table 4   ~  1000 . Unit transportation costs from supplier to silos (Ci) and fuzzy capacities of each silo b i

Supplier

Silo 1

Silo 2

Silo 3

Silo 4

Silo 5

Silo 6

Cost Capacity

0.5 (100, 150, 200) Silo 7

0.5 (100, 150, 200) Silo 8

0.5 (100, 150, 200) Silo 9

0.5 (100, 150, 200) Silo 10

0.5 (475, 675, 875) Silo 11

0.5 (475, 675, 875) Silo 12

Cost Capacity

0.5 (475, 675, 875)

0.5 (475, 675, 875)

0.5 (200, 300, 400)

0.5 (200, 300, 400)

0.5 (200, 300, 400)

0.5 (200, 300, 400)

Table 5   ~ n  1000 . Unit transportation costs from supplier to silos for olive oil (Cn) and fuzzy capacities of each silo h

Supplier

Cost Capacity

Silo 1

Silo 2

Silo 3

Silo 4

Silo 5

Silo 6

0.7 (8, 9, 10)

0.7 (8, 9, 10)

0.7 (8, 9, 10)

0.7 (8, 9, 10)

0.7 (8, 9, 10)

0.7 (8, 9, 10)

Table 6   Fuzzy tin packaging capacities per tin filling lines ~itv b  1000 . Sunflower oil

Line 1 Line 2

Corn oil

5l

10 l

18 l

5l

10 l

18 l

(300, 400, 500) (111.5, 121.5, 131.5)

(335, 435, 535) (80, 90, 100)

(80, 90, 100) (57.5, 58.5, 59.5)

(300, 400, 500) (111.5, 121.5, 131.5)

(125, 135, 145) (80, 90, 100)

(80, 90, 100) (57.5, 58.5, 59.5)

5l

10 l

18 l

5l

10 l

18 l

(300, 400, 500) (111.5, 121.5, 131.5)

(335, 435, 535) (80, 90, 100)

(80, 90, 100) (57.5, 58.5, 59.5)

(300, 400, 500) (111.5, 121.5, 131.5)

(125, 135, 145) (80, 90, 100)

(80, 90, 100) (57.5, 58.5, 59.5)

Soybean oil

Line 1 Line 2

Olive oil

Table 7   Fuzzy pet packaging capacities per pet filling lines ~juv d 1000 . Sunflower oil

Corn oil

Soybean oil

Olive oil

Line 1

1l 2l

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

Line 2

1l 2l

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

(440, 540, 640) (260, 360, 460)

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T. Paksoy et al. / Applied Mathematical Modelling 36 (2012) 2762–2776 Table 8 Initial solutions and interval values for each of the fuzzy objective functions.

a

Item

LP-1

LP-2

(PIS; NIS)

Initial solutions

Objective function L Z1 Z2

Min Z1 – 1438197a 180911.5

Min Z2 – 1367953 135283.1a

– – (1367953; 1438197) (135283.1; 180911.5)

Max L 0.000005 1438196.634 165066.5

denotes the optimal value with the single-objective LP model

in liter. Table 6 shows the fuzzy capacities of tin package filling lines. Table 7 shows the fuzzy capacities of pet package filling lines. 4.3. Solution procedure for the case study The solution procedure of the proposed model is demonstrated as follows. First, the original FMOLP model is formulated for solving supply chain network under fuzzy material requirement according to Eqs. (3)–(31). Second, all the fuzzy inequality constraints are converted to crisp ones using weighted average method at minimum acceptable membership level a = 0.5. The original FMOLP problem is solved using the ordinary single-objective LP problem to obtain the initial solutions for each of the objective functions, with the assumption that the DM specified the most likely value of the triangular distribution of each fuzzy number as the precise value. Table 8 lists the results obtained by the single objective LP model and the corresponding non-increasing continuous linear membership functions for each of the fuzzy objective functions can be defined via Eq. (41) as follows;

8 > < 1; z1 6 1367953 1 f1 ðz1 Þ ¼ 1438197z ; 1367953 < z1 < 1438197 70244 > : 0; z1 P 1438197 and

8 > < 1; z2 6 135283:1 2 f2 ðz2 Þ ¼ 180911:5z ; 135283:1 < z2 < 180911:5 45628:4 > : 0; z2 P 180911:5 The resulting ordinary single-objective LP model for solving the fuzzy multi-objective SCN problem can be formulated according to Eq. (42). This LP problem is solved by using LINDO package program. Using the proposed FMOLP model to simultaneously both minimize total transportation costs between crude oil supplier and crude oil silos; between olive oil supplier and olive oil silos; total transportation costs between the factory and Private Label-Oil Company Warehouses (tin and pet), the objective values of the initial solutions are Z1 = 1438196.634 TL and Z2 = 165066.5 TL. Additionally, we attempt to modify the (PIS, NIS) for each of the fuzzy objective functions to yield a satisfactory solution. Consequently, the improved solutions are Z1 = 1438196.634 TL and Z2 = 159677.7 TL. Table 8 lists the optimal plan for the ABC Oil Company. The corresponding non-increasing continuous linear membership functions for each of the improved fuzzy objective functions can be defined as follows;

8 > < 1; z1 6 1367953 1 f1 ðz1 Þ ¼ 1438196:634z ; 1367953 < z1 < 1438196:634 70243:634 > : 0; z1 P 1438196:634 and

8 > < 1; z2 6 135283:1 2 f2 ðz2 Þ ¼ 165066:5z ; 135283:1 < z2 < 165066:5 29783:4 > : 0; z2 P 165066:5 The single-objective LP solutions for each of the fuzzy objective functions were used as a starting point for the PIS and NIS of the objective functions and both intervals must cover the LP solutions. In the Tables 8 and 10, the initial values for the PIS     NIS NIS and NIS of the fuzzy objectives can be specified as Z PIS ¼ ð1367953; 1438197Þ and Z PIS ¼ ð135283:1; 180911:5Þ. 1 ; Z1 1 ; Z1 The resulting objective values of the initial solutions are Z1 = 1438196.634 TL and Z2 = 165066.5 TL. We assumed to attempt to improve the initial solutions and then PIS and NIS values of fuzzy objectives are updated as     NIS NIS Z PIS ¼ ð1367953; 1438196:634Þ and Z PIS ¼ ð135283:1; 165066:5Þ based on the objective values of initial 1 ; Z1 1 ; Z1 solutions. Consequently, the improved solutions are Z1 = 1438196.634 TL and Z2 = 159677.7 TL.

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Table 9 Optimal distribution plan for the ABC Oil Company. Item

Output solutions

Objectives and L values

Z1 = 1438196.634 Z2 = 159677.7 L = 0.000000038

Variable

Value

Variable

Value

Variable

Value

Variable

Value

X1 X2 X5 X6 X7 X10 X11 X12 Y1,2 Y2,1 Y5,2 Y6,2 Y7,2 Y10,2 Y11,2 Y12,2 Z1,1 Z2,2 Z2,3 Z2,4 W2,1 W2,4 Q1,2 Q2,1 P1 P2 T1,2 T2,2 T3,1 T4,1 F1,2 F4,1 K1,1 K1,2 K2,2 L1,1 L1,2 L3,1 S4,1 S4,2 M1,1 M2,2 O1

150000 150000 270145 675000 675000 300000 300000 300000 150000 150000 270145 675000 675000 300000 300000 300000 73185 600000 233461 600000 543030 66953 75315 600000 1500 26701 7500 600000 153450 600000 543030 365 10965 532288 75212 65476 209 80011 66538 50 56747 103 4177

O2 O3 O4 O5 A4,2 V1,2 V21 V3,1 V4,2 V5,1 H1,1,1,1 H1,1,1,2 H1,2,1,1 H1,2,1,2 H1,3,1,1 H1,3,1,2 H1,3,1,3 H1,4,1,1 H1,4,1,3 H1,4,2,1 H1,4,2,2 H1,5,1,1 H1,5,1,3 H1,5,2,3 H1,7,1,1 H1,7,3,2 H1,7,3,3 H1,8,1,1 H1,8,3,1 H2,1,1,2 H2,1,1,3 H2,1,2,1 H2,1,2,2 H2,1,2,3 H2,1,3,1 H2,1,4,1 H2,2,1,3 H2,2,2,1 H2,2,2,2 H2,2,2,3 H2,2,3,1 H2,3,2,1 H2,3,2,2

9000 9000 9000 9000 1165 4177 9000 9000 7835 9000 58994 74365 4050 5 1360 975 4234 694 710 73 33 21818 36914 20 55677 1096 609 6 2193 35423 13134 12512 975 65 2560 84 15257 47 6 43 81 168 87

H2,3,2,3 H2,3,3,1 H2,3,4,1 H2,4,1,2 H2,4,2,2 H2,4,2,3 H2,4,3,1 H2,4,4,1 H2,5,1,1 H2,5,2,1 H2,5,3,1 H2,5,3,2 H2,5,4,,2 H2,6,1,2 H2,6,2,1 H2,6,3,1 H2,7,2,1 H2,7,2,3 H2,7,3,1 H2,7,3,3 H2,7,4,3 H2,8,1,3 H2,8,2,1 H2,8,2,2 H2,8,3,1 G1,1,1,2 G1,1,2,1 G1,1,2,2 G1,1,3,2 G1,1,4,2 G1,2,1,1 G1,2,1,2 G1,2,2,1 G1,2,2,2 G1,2,3,2 G1,2,4,1 G1,3,1,2 G1,3,2,1 G1,3,2,2 G1,3,3,1 G1,3,3,2 G1,4,1,1 G1,4,1,2

89 49 25 650 72 195 33 82 121500 53540 104560 60750 116 25290 23034 6896 19292 29776 4230 33750 64 5359 13 53162 3091 7102 66482 1348 154 19 776 836 36 29 27 11 108 20 24 9 16 175 545

G1,4,2,2 G1,4,3,2 G1,4,4,1 G1,4,4,2 G1,5,1,1 G1,5,1,2 G1,5,2,2 G1,5,3,2 G1,5,4,2 G1,6,1,1 G1,6,1,2 G1,6,2,2 G1,6,3,2 G1,6,4,,1 G1,6,4,2 G1,7,2,2 G1,7,3,2 G1,8,1,1 G1,8,1,2 G1,8,2,2 G1,8,3,1 G1,8,3,2 G1,8,4,1 G1,8,4,2 G2,1,3,1 G2,1,4,1 G2,2,3,1 G2,2,4,,2 G2,3,1,1 G2,3,4,1 G2,3,4,2 G2,4,,1,2 G2,4,,2,1 G2,4,,3,1 G2,5,3,2 G2,5,4,,1 G2,6,4,2 G2,7,2,2 G2,7,4,,1 G2,7,4,,2

166 22 25716 44 133575 53379 24835 19344 10230 10181 10767 3440 510 1268 3201 341 4152 4 6 12 56738 4148 5 6 79 22 13 17 209 2814 6 104 50 11 51 7408 2829 25 1768 3154

Table 10 The (PIS;NIS) for each of the fuzzy objective functions. Item   NIS Z PIS g ; Zg

Initial solution   NIS Z PIS ¼ ð1367953; 1438197Þ 1 ; Z1   NIS ; Z Z PIS ¼ ð135283:1; 180911:5Þ 1 1

Improved solution   NIS Z PIS ¼ ð1367953; 1438196:634Þ 1 ; Z1   NIS ; Z Z PIS ¼ ð135283:1; 165066:5Þ 1 1

Z1 Z2 L

1438196.634 165066.5 0.000005

1438196.634 159677.7 0.000000038

4.4. Computational analysis Several significant management implications for the practical application of the proposed model to solve integrated supply chain network problems in a fuzzy environment include the following. First, the proposed model yields an efficient solution. The fuzzy programming method yields efficient solutions to fuzzy mathematical programming problems [16].

T. Paksoy et al. / Applied Mathematical Modelling 36 (2012) 2762–2776

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Tables 8–10 indicate that the solutions obtained using the proposed FMOLP model are an efficient solution, by comparing them with the optimal values calculated by LP-1 and LP-2. Second, the deterministic models are not suitable methods of obtaining an effective solution, due to conflicting nature of the multiple objectives and the vagueness in the information relating to the decision parameters in real-world integrated supply chain network problems. Table 10 demonstrates that the interaction of trade-offs and conflicts among dependent objective functions. Applying fuzzy sets to fuzzy multi objective supply chain network problems provides essentially the higher efficiency and flexibility in terms of the model formulation and arithmetic operations [14,15]. Third, the proposed FMOLP model exhibits greater computational efficiency and flexibility by adopting the linear membership functions and triangular distributions for solving the fuzzy multi-objective supply chain network problems. The proposed model is preferable when the DM sees to make the optimal values approximately equal. Finally, the most important advantage of the proposed model is that the DM adjusts the search direction during the solution procedure, until the efficient solution satisfies the DM’s preferences and is considered to be the preferred satisfactory solution. In addition to management implications of the proposed model, there are several differences of the proposed model than the other existing models. First of all, the developed model provides a framework that facilities the fuzzy decision making process, enabling the DM interactively modify the fuzzy/imprecise data and related parameters until a preferred satisfactory solution is obtained [16]. In the study Paksoy and Cavlak [21], using a mixed integer linear programming to minimize the total transportation costs led to optimal value of totally 1573480.00 TL. These frame shows that the fuzzy model results are a set of efficient compromise solutions (obtaining nearby total transportation costs) with multiple fuzzy objectives, compared to the optimal values calculated by Paksoy and Cavlak [21]. Second, the DM can attempt to adjust L value by modifying the membership degree of the fuzzy objective functions and related parameters to seek a set of more optimal compromise solutions. Third, the proposed model outputs more wide solutions than other models [16]. It should be noted that applications of this kind of supply chain network models to the real cases are really hard. 5. Conclusions In real world problems in supply chains network models, coefficients and related parameters, such as demand, capacity and unit cost coefficients are normally fuzzy/imprecise due to some information is incomplete or unobtainable. In this study, a FMOLP model is developed and proposed upon a production/distribution network problem for an oil company in Konya. ABC Oil Company produces and sells many kinds of oil (sunflower, corn, soybean, etc.) via different packaging choices (pet and tin) to own warehouses and big markets as private label. The decision process of supply chain design was modeled to minimize the end customer’s (private label’s and company’s warehouses) level of dissatisfaction by considering transportation costs between suppliers, factory and warehouses. By considering fuzzy capacity constraints, equilibrium was provided between amount of crude oil, which is transported from suppliers, and fuzzy demands of warehouses. Model is also designed for the network of factors such as silos, refineries, filling lines while balancing between suppliers and warehouses. In application section, the proposed model was validated via real data, which are obtained from ABC Oil Company. Monthly demand of each warehouse is used in the model, which is developed for designing the distribution/production network of the company under fuzzy capacity constraints. The FMOLP model converted to an ordinary single objective LP problem which has been solved via LINDO package program and results are discussed. The proposed model can be solved by using a commercial optimizer and computational times are reasonable for real sized problems. Acknowledgments The authors express their gratitude to the anonymous reviewers for valuable comments on the paper. In carrying out this research, the authors were supported by the Selçuk University Scientific Research Project Fund (BAP). This fund is hereby gratefully acknowledged. References [1] L. Lin, M. Gen, X. Wang, A hybrid genetic algorithm for logistics network design with flexible multistage model, Int. J. Inform. Syst. Logist. Manage. 3 (2007) 1–12. [2] D.F. 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