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Type-II Fuzzy Multi-Product, Multi-Level, Multi-Period Location–Allocation, Production–Distribution Problem in Supply Chains: Modelling and Optimisation Approach

Type-II Fuzzy Multi-Product, Multi-Level, Multi-Period Location–Allocation,... FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 2, 260–283 https://doi.org/10.1080/16168658.2018.1517978 Type-II Fuzzy Multi-Product, Multi-Level, Multi-Period Location–Allocation, Production–Distribution Problem in Supply Chains: Modelling and Optimisation Approach a a b Sarah J.-Sharahi , Kaveh Khalili-Damghani , Amir-Reza Abtahi and Alireza Rashidi-Komijan Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran; Department of Information Technology Management, Faculty of Management, Kharazmi University, Tehran, Iran; Department of Industrial Engineering, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran ABSTRACT ARTICLE HISTORY Received 8 May 2016 In this study, the application of type-II fuzzy sets is addressed to Revised 5 January 2017 design a multi-product, multi-level, multi-period supply chain net- Accepted 12 January 2018 works. The proposed model provides integrated approach to make optimal decisions such as location–allocation, production, procure- KEYWORDS ment and distribution subject to operational and tactical constraints. Type-II fuzzy integer linear In the context of fuzzy linear programming, this study involves type-II programming; fuzzy type-II fuzzy numbers for the right-hand side of constraints regarding three reducer; location–allocation problem; sources of uncertainty: demand, manufacturing and supply. Accord- production–distribution ing to fuzzy components considered, a type-II fuzzy mixed-integer problem; supply chain linear programming is converted into an equivalent auxiliary crisp planning model using linear fuzzy type-reducer models. The final models are linear and the global optimum solutions can be achieved using com- mercial OR softwares. The contributions of this study are three folds: (1) introducing a new integrated supply chain network design prob- lem; (2) considering a solution procedure based on type-II fuzzy sets and (3) presenting a linear fuzzy type-II reducer. Finally, the proposed model and solution approach are illustrated through a numerical example to demonstrate the significance. Highlights • A model is proposed to formulate the integrated supply chain network design problem. • Uncertainties in real supply chains are modelled using type-II fuzzy sets. • A new solution procedure based on type-II fuzzy sets is developed. • A linear fuzzy type-II reducer is proposed. • Numerical example is proposed to illustrate the efficacy of proposed approach. 1. Introduction Supply Chain (SC) is a network of organisations, suppliers and distributors that produces value in the form of products and services and bring them to ultimate customer or market CONTACT Kaveh Khalili-Damghani kaveh.khalili@gmail.com © 2018 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 261 [1, 2]. Supply Chain Management (SCM) is a set of management activities for planning, implementing and controlling the supply chain operations. SCM needs to cope with the high uncertainty and imprecision related to real-world character [3]. Peidro et al. [4]have analysed and classified the sources of uncertainty in SC into three groups as demand, man- ufacturing and supply. Uncertainty is presented as variability and inexact forecasting of supplier operations and demand, and also poorly reliable production and manufacturing process [5]. In SC planning, uncertainty is a main factor that can influence the SCM processes. The uncertainty in SC has already been modelled through several paradigms such as robust optimisation [6, 7], stochastic programming [8, 9], fuzzy mathematical programming [5, 10] and hybrid models [11]. There also exists hybrid models where stochastic fuzzy program- ming or robust fuzzy programming models were combined [12]. In supply chains, a series of decisions such as which plants to establish, which distribution centre supplies which cus- tomer, production level of plants, capacity of plants and distribution strategies are needed to address accurately while the information contains a great amount of uncertainties. Type-II fuzzy sets and systems generalise type-I fuzzy sets and systems so that more uncertainty can be handled. From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-I fuzzy set has no uncertainty asso- ciated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided by Zadeh [13] when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a type-II fuzzy set. A type-II fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-II fuzzy sets head-on. And, if there is no uncertainty, then a type-II fuzzy set reduces to a type-I fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes. Very limited applications of type-II fuzzy sets were reported in recent years. Due to our best knowledge, there is no study which addresses the type-II fuzzy numbers on integrated location–allocation, production and distribution activities in supply chain. So, the main contributions of this paper can be summarised as follows: • Introducing a strategic and tactical SC problem through integrating dynamic location– allocation, production and distribution planning into a multi-echelon, multi-product, multi-period network. • Modelling the aforementioned problem using mixed-integer mathematical modelling. • Considering different sources of uncertainty in demand, manufacturing and supply processes affecting the SC. • Considering the interval type-II fuzzy (IT2F) number for uncertain parameters and solving the SC problem based on fuzzy type-II programming. The rest of the paper is organised as follows. The literature of past works on SC under fuzziness are presented and classified in Section 1. Section 2 introduces a fuzzy mixed- integer linear programming (FMILP) model for the integrated multi-product, multi-level, multi-period location–allocation, production–distribution problem in supply chains. In Section 3, a solution methodology is developed to transform the proposed FMILP into an auxiliary crisp mixed-integer linear programming model. A numerical example and results 262 S. J. -SHARAHI ET AL. are presented in Section 4 to illustrate the mechanism of proposed model and the appli- cability of solution procedure. Finally, the conclusions and future research directions are presented in Section 5. 2. Literature Review of Past Works The literature of uncertain supply chain models and problems as well as the fuzzy solution procedures is reviewed in this section. 2.1. Uncertain Integrated Supply Chain Problems The aim of location–allocation problem is to decide the facilities to be opened and the assignment of customers to the opened facilities. In literature, there is a vast effort for combining allocation decisions with other decisions such as production, inventory, pro- curement and capacity decisions [12]. Several authors have studied these integrated SC problems from a deterministic point of view. Tsiakis and Papageorgiou [14] developed a mixed-integer linear programming for global supply chain networks to decide about pro- duction, allocation, production capacity, purchase and network configuration with financial aspects of exchange rates, duties and production costs. Ahmadi-Javid and Hoseinpour [15] presented a non-linear mixed-integer programming model to determine location, allocation, price and order-size decisions of SC with two scenarios of uncapacitated and capacitated distribution centres. Jang et al. [16] studied integrated inventory-allocation and shipping problem in SC. Tsao and Lu [17] developed an integrated facility location and inventory-allocation problem in SC that considering two types of transportation cost dis- counts simultaneously: quantity discounts for inbound transportation cost and distance discounts for outbound transportation cost. Jonrinaldi and Zhang [18] proposed a model for coordinating integrated production and inventory cycles in manufacturing supply chain involving reverse logistics for multiple items with finite multiple periods. Shahabi et al. [19] studied mathematical models to coordinate facility location, allocation of supplier to ware- houses and retailers and inventory control for a four-echelon supply chain network. The goal in this study is to minimise the facility location, transportation and the inventory costs. Latha Shankara et al. [20] proposed a single product, four-echelon supply chain architec- ture consisting of suppliers, production plants, distribution centres and customer zones to determine the number and location of plants in the system, the flow of raw materials from suppliers to plants, and the quantity of products to be shipped from plants to distribution centres and from distribution centres to customer zones, so as to minimise the facility loca- tion and shipment costs. A multi-objective hybrid particle swarm optimisation algorithm was proposed as the solution method for the proposed model. In a study by Bandyopad- hyay and Bhattacharya [21] a tri-objective problem for a two-echelon serial supply chain with objectives of minimisation of the total cost of the supply chain, minimisation of the variance of order quantity and minimisation of the total inventory was developed. To solve the model, a proposed modified non-dominated sorting genetic algorithm was applied. Mousavi et al. [22] developed an inventory–location–allocation problem in two-echelon SC in which the all unit discount strategy was applied for purchasing costs. Diabat et al. [23] considered a closed-loop location-inventory problem with forward and reverse supply chain consisting of a single echelon. The problem is to choose which distribution centres FUZZY INFORMATION AND ENGINEERING 263 and remanufacturing centres are to be opened and to associate the retailers with them. Subulan et al. [24] designed a closed-loop, multi-objective, multi-echelon, multi-product and multi-period logistics network model using mixed-integer linear programming with recovery options such as remanufacturing, recycling, and energy recovery and interactive fuzzy goal programming approach was utilised to solve the proposed model. As mentioned before, deterministic SC is unrealistic in real terms of SC problems and coefficients and parameters are often imprecise because of incomplete and unavailable information. There is increasing attention to development fuzzy or stochastic supply chain model and offer possible solutions that leading to more realistic models. SC planning under fuzzy uncertainty have been reviewed in several researches [4, 5, 12]. Peidro et al. [5] modelled integrated procurement, production and distribution planning in SC with dif- ferent sources of uncertainty in demand, process and supply. All uncertain parameters were assumed to be type-I fuzzy numbers. The purposed model was solved with adap- tation approach of Jimenez et al. [25]. Bilgen [26] developed an integrated production and distribution plans in multi-echelon SC for determining the allocation of production among production lines and transporting products with fuzzy available resources capac- ity and fuzzy costs. Paksoy et al. [27] proposed fuzzy multi-objective linear programming for minimising the end customer’s dissatisfaction and transportation costs. The produc- tion capacity and amount of demands were type-I fuzzy numbers. Figueroa [28] proposed multi-period production planning with IT2F demand. Diaz-Madronero et al. [10] proposed fuzzy multi-objective integer linear programming model for transportation and procure- ment planning in three-level, multi-product and multi-period automobile supply chain. They assumed the maximum capacity of the available truck and minimum percentage of demand in stock as type-I fuzzy numbers. They used an interactive solution methodology to solve the problem. Mousavi and Niaki [29] studied a capacitated location allocation problem with fuzzy demands and stochastic locations of the customers that follow the normal probability distri- bution in which the distances between the locations and the customers are taken Euclidean and squared Euclidean. The fuzzy expected cost programming, the fuzzy β-cost minimisa- tion model and the credibility maximisation model are three types of fuzzy programming that are developed to model. Azadeh et al. [30] proposed a multi-objective, multi-period, multi-echelon fuzzy linear programming model to optimise natural gas supply chain based on two economic and environmental objectives and fuzzy parameters, including demand, capacity and cost. A combination of the possibilistic programming approach based on the defuzzification method and interactive fuzzy approach was used to deal with uncertainty. Ramezani et al. [31] designed a multi-product, multi-period, closed-loop supply chain net- work with three objective functions: maximisation of profit, minimisation of delivery time, and maximisation of quality and three fuzzy components of constraints, coefficients, and goal. To cope with fuzziness, a fuzzy optimisation approach was adopted to convert the pro- posed fuzzy multi-objective mixed-integer linear programme into an equivalent auxiliary crisp model. Gholamian et al. [32] developed multi-site, multi-period and multi-product aggregate production planning in supply chain to minimise the total cost of the SC, improve customer satisfaction, minimise the fluctuations in the rate of changes of workforce, and maximise the total value of purchasing under fuzzy parameters. Alizadeh Afrouzy et al. [33] designed a multi-echelon, multi-period, multi-product and aggregate procurement and production 264 S. J. -SHARAHI ET AL. planning model which considered three objectives of maximise the profit of the supply chain due to new product development, customer satisfaction, and maximising the pro- duction of the developed and new products. The proposed model considered uncertainties of the environment such as customer demands and supplier capacities which is modelled by fuzzy stochastic programming. Following the stochastic viewpoint, Lin and Wu [34] proposed product pricing and integrated supply chain operations plan under conditions of price-dependent stochas- tic demand under two approaches of maximising a manufacturer’s expected profit and cost minimisation. Marufuzzaman et al. [35] developed two-stage stochastic programming model to design and extend the classical two-stage stochastic location–transportation model. Their model was designed to optimise costs and emissions in the supply chain and capture the impact of biomass supply and technology uncertainty on tradeoffs that exist between location and transportation decisions; and the tradeoffs between costs and emissions in the supply chain. About decision-making in inventory management, Panda et al. [36] developed eco- nomic production lot size for single-period multi-product model in which the demand rate is stochastic under stochastic and/or fuzzy budget and shortage constraints. The stochastic constraints have been represented by chance constraints and fuzzy constraints in the form of possibility/necessity constraints which are transformed to equivalent deter- ministic ones. Wang et al. [37] proposed single-item and multi-item single-period inven- tory models for short life-cycle products when demands are assumed to be uncertain random variables. For deriving the optimality condition for optimal order quantity the uncertain random models are transferred to equivalent deterministic forms by consider- ing expected profit and providing more information of chance distributions. Integrated inventory model in supply chain is developed in some researches. Bag and Chakraborty [38] addressed optimal order quantity of the retailers, production rate of the producer and the production rate of the suppliers to minimise the supply chain cost in which there are imprecise chance constraints on the transportation costs for producer, retailer and also a fuzzy space constraint for producer is considered. Jana et al. [39] developed order- ing policy for deteriorating products with allowable shortage and permissible delay in payment under inflation and time value of money in fuzzy rough environment which the proposed model is converted to deterministic one using expected value method techniques. 2.2. Fuzzy Linear Programming as Solution Procedure Fuzzy linear programming (FLP) is an optimisation technique applied in real-world prob- lems with imprecise data. FLP has an extensive literature. In this research, two main cate- gories of FLP are reviewed: (1) FLP with fuzzy numbers as coefficient in objective function, right-hand sides ( RHSs), and technological coefficient, (2) FLP with fuzzy parameters and fuzzy decision variables. In the first category, numerous researches have proposed different techniques to solve the FLP problems. Zimmermann [40] formulated and solved the FLP problem for the first time. Herrera and Verdegay [41] proposed three models for three types of FLPs. They solved the problems with fuzzy RHS or fuzzy coefficient of constraint using auxiliary parametric linear programme. They also used two fuzzy ranking methods to cope with FLPs with fuzzy FUZZY INFORMATION AND ENGINEERING 265 objective functions. Xinwang [42] developed FLP with fuzzy technological coefficients and the RHSs of the constraints and solved FLPs based on the new ranking method of fuzzy num- bers for satisfaction degree of the constraints. Zhang et al. [43] presented FLPs with fuzzy coefficients of objective function and showed how to convert such FLPs into equivalent deterministic multi-objective optimisation problem. Jimenez et al. [25] presented full FLPs in which all parameters were fuzzy numbers. They also presented a resolution method with interactive participation of decision maker during solving procedure. Wu [44] described the FLP which involve fuzzy numbers for the coefficients of variables in the objective function and developed the optimality conditions for FLPs through proposing two solving method- ologies that were similar to non-dominated solution in multi-objective programming. The problems with fuzzy numbers for the coefficients of the objective function, the techno- logical coefficients in the constraints and the RHS of the constraints were developed by Mahdavi-Amiri and Nasseri [45], Hatami-Marbini and Tavana [46] and Saati et al. [47].The method for finding the optimal solution of FLPs with type-II fuzzy numbers presented by Figueroa, [48] and Figueroa and Hernandez [49].They proposed a method to solve FLPs with interval type-II RHSs. In the second category, FLP with fuzzy numbers for the decision variables, the coeffi- cients in the objective function and the RHS of the constraints was developed by Ganesan and Veeramani [50] that leaded to the solution without converting FLPs to crisp linear programming problems. Mahdavi-Amiri and Nasseri [51] studied the FLP problems with fuzzy numbers for the decision variables and the right-hand-side of the constraints and developed solution method based on auxiliary problems that applied a linear ranking function to order trape- zoidal fuzzy variables. Then, Ebrahimnejad et al. [52] proposed a new primal-dual algorithm for solving FLP problems with fuzzy variables by using the duality method investigated by Mahdavi-Amiri and Nasseri [51]. Kumar et al. [53] proposed a method to improve the method proposed by Hosseinzadeh Lotfi et al. [54] to find the optimal solution of the FLPs with fuzzy parameters and fuzzy variables that called Fully FLP. Based on our knowledge, the integrated SC model in a fuzzy environment with different sources of uncertainty especially with type-II fuzzy sets has not been well studied. There- fore, this paper attempts: (1) to introduce an integrated location–allocation, production, procurement and distribution planning problem in supply chains, (2) to model the pro- posed problem in presence of uncertainty in demand, manufacturing and supply processes through mathematical programming and (3) to develop a solution methodology to solve the proposed model optimally considering interval type-II fuzzy sets. Most of the supply chain researches assumed the fuzzy parameters to be of type- I fuzzy sets such as Peidro et al. [5], Bilgen [26], Paksoy et al. [27] and Diaz-Madronero et al. [10]. In decision-making problems like location–allocation, production–distribution, the parameters cannot be always exactly known. For example, demand or flow of prod- ucts in supply chain management depends upon several variables such as price, labour charges and repair time. Each of these variables fluctuates and also membership degree of each point cannot be exactly determined. So it is not easy to predict the deter- ministic or type-I fuzzy sets. Assuming fuzzy type-II parameters, i.e. fuzziness in mem- bership function, for such variables is a proper idea, although fuzzy type-II parameters impose very high computational complexity to the problem. In this paper, the fuzzy type- II mathematical programming is reduced into fuzzy type-I mathematical programming 266 S. J. -SHARAHI ET AL. while the main properties of original programming is retained. Moreover, the procedure is generalised for the cases in which the resultant model may be a non-linear math- ematical programming. So, formally talking, the main advantages of proposed proce- dure of this study over the previous methods in the literature can be summarised as follows: • Considering complicated uncertainties in membership value of a fuzzy variable in terms of fuzzy type-II parameters. • Reducing the fuzzy type-II mathematical programming into a fuzzy type-I mathematical programming while the main properties of original programming is retained. • Proposing a procedure for linearisation of the resultant models in the sense of gener- alisation of the cases in which the resultant model may be a non-linear mathematical programming. 3. Problem Descriptions and Modelling In this section, the problem of multi-echelon, multi-product, multi-period location–alloca- tion production and distribution supply chain network is proposed. Then, the associated mathematical programming is developed. The following assumptions are considered in the proposed model: • Number of possible plants are known and determined in advance. • Number of possible distribution centres are known and determined in advance. • Each product can be produced at several plants. • Distribution centres can be supplied from more than one plant and can supply more than one customer. • Customers have uncertain demands of multiple products during planning periods. • Production and transportation costs are assumed to be known and deterministic. • Production and transportation capacity are known and uncertain. The following dynamic decisions are to be determined: • Establish or shut down distribution centres • Product plan, production rate and utilisation per plants • Distribution plan between plants, distribution centres and customers • Procurement plan • Assignment of distribution centres to plants and customer to distribution centres The objective is the minimisation of the total cost of the strategic, tactical and opera- tional costs. The graphical representation of the three-stage, multi-product, multi-period supply chain network is shown in Figure 1. The objective is to find which distribution centres are to be opened, which customers are served from opened distributors, and which procurement and production strategy is to be planned so that the total cost is minimised. FUZZY INFORMATION AND ENGINEERING 267 Figure 1. Three-stage supply chain. 3.1. Notations Indices, parameters and decision variables are presented in Table 1. Three types of uncertainty are assumed in the proposed model. Table 2 presents the sources and types of uncertainties. 3.2. Model formulation A new multi-echelon, multi-product, multi-period location–allocation production and dis- tribution supply chain network model with type-II fuzzy sets with due attention to the model proposed by Tsiakis and Papageorgiou [14] is developed. K T K T I J T D D D D Min Z = EC · Y + SC (1 − Y ) + PC · P ijt ijt kt kt kt kt k=1 t=1 k=1 t=1 i=1 j=1 t=1 I K T I J K T P P + BC · PQ + TC · TQ (1) it ikt ijkt ijkt i=1 t=1 i=1 j=1 t=1 k=1 k=1 I K D T D D + TC · TQ ikdt ikdt i=1 t=1 k=1 d=1 P D X ≤ Y ∀ j, k, t (2) jkt kt P D X ≥ Y ∀ k, t (3) jkt kt j=1 D D X ≤ Y ∀ k, d, t (4) kdt kt X ≥ 1 ∀ k, t (5) kdt d=1 268 S. J. -SHARAHI ET AL. Table 1. Indices, parameters and decision variables. Indices I Set of products i ={1, 2, ... , I} J Set of plants j ={1, 2, ... , J} K Set of possible distribution centre k ={1, 2, ... , K} D Set of customers d ={1, 2, ... , D} T Set of planning periods t ={1, 2, ... , T} Parameters EC cost of establishing a distribution centre at location k in time t kt SC cost of closing a distribution centre at location k in time t kt PC unit production cost for product i at plant j in time t ijt BC unit procurement cost for product i from third parties in time t it TC unit transport cost for product i from plant j to distribution centre k in time t ijkt TC unit transport cost for product i from distribution centre k to customer d in time t ikdt De demand of customer d for product i in time t idt H Hours of operating of plant j M Hours of maintenance of plant j Cap maximum production capacity of plant j for product i ij PD Cap maximum rate of flow of products transferred from plant j to distribution centre k jk DD Cap maximum rate of flow of products transferred from distribution centre k to customer d kd r daily production rate of product i in plant j ij β change-over coefficient in hours ε utilisation parameter in hours Decision Variable PQ out-sourced product i delivered to distribution centre k in time t ikt P production rate of product i in plant j in time t ijt TQ rate of flow of product i transferred from plant j to distribution centre k in time t ijkt TQ rate of flow of product i transferred from distribution centre k to customer d in time t ikdt T hours allocated for production of product i in plant j in time t ijt U utilisation of production plant j in time t jt α maximum allowed difference in utilisation of plants in time t Y 1 if distribution centre k is to be established in time t,0otherwise kt X 1 if production plant j is assigned to distribution centre k in time t,0otherwise jkt X 1 if distribution centre k is assigned to customer d in time t,0otherwise kdt w 1 if production plant j is to produce product i in time t,0otherwise ijt Table 2. Sourcesofuncertainty. Source of uncertainty Description Parameter Demand Demand of product i at customer d in period t De idt Process (manufacturing) Production capacity of plant j for product i Cap ij PD Supply Rate of flow from plant j to distribution centre k Cap jk DD Rate of flow from distribution center k to customer d Cap kd P PD P TQ ≤ cap  · X ∀ j, k, t (6) ijkt jk jkt i=1 D DD D TQ ≤ cap  · X ∀ k, d, t (7) ikdt kd kdt i=1 P = TQ ∀ i, j, t (8) ijt ijkt k=1 FUZZY INFORMATION AND ENGINEERING 269 J D P D TQ + PQ = TQ ∀ i, k, t (9) ikt ijkt ikdt j=1 d=1 TQ ≥ De ∀ i, d, t (10) idt ikdt k=1 P ≤ cap · w ∀ i, j, t (11) ijt ijt ij I I T ≤ (H − M ) − β w ∀ j, t (12) j j ijt ijt i=1 i=1 P ≤ r · T ∀ i, j, t (13) ijt ij ijt U = T ∀ j, t (14) jt ijt i=1 α = U − U ∀ j,tj = j (15) t jt ´ jt α ≤ ε ∀ t (16) PQ ≥ 0 ∀ i, k, t (17) ikt P ≥ 0, T ≥ 0, w ∈{0, 1}∀ i, j, t (18) ijt ijt ijt TQ ≥ 0 ∀ i, j, k, t (19) ijkt TQ ≥ 0 ∀ i, k, d, t (20) ikdt U ≥ 0 ∀ j, t (21) jt α ≥ 0 ∀ t (22) Y = Bin ∀ k, t (23) kt X = Bin ∀ j, k, t (24) jkt X = Bin ∀ k, d, t (25) kdt Equation (1) minimises the total cost of supply chain. The objective function con- sists of costs of the distribution centre infrastructure, the costs of production and pro- curement, and costs of transportation. Set of constraints (2), which is written for each plant, distribution center and planning period, assures that a plant can services to a distribution center if an only if the distribution center had been established. Set of con- straints (3), which is written for each distribution center and planning period, assures 270 S. J. -SHARAHI ET AL. that one of the production plants is assigned to each distribution center in each time period. Set of constraints (4), which is written for each distribution center, customer and plan- ning period, assures that a distribution center can services to a customer if an only if the distribution center had been established. Set of constraints (5), which is written for each distribution center and planning period, assures that a distribution center must be assigned at least to a customer in each time period. Set of constraints (6), which is written for each plant, distribution center and time period, assures that flow of material from production plant to distribution center can take place only if the connection exists. Set of constraints (7), which is written for each distribution center, customer and time period, assures that flow of material from distribution center to customer can take place only if the corresponding connection exists. Set of constraints (8) assures that the production rate of product type i in plant j in time period t is equal to all sent products type i to all distribution centres in time period t. Set of constraints (9) assures that all products type i which has received by distribution center k from all plants in time period t plus the products type i which has received by distribution center k from out-sourcing supplier in time period t is equal to all products type i which has received by all customers d from distribution center k in time period t. Set of constraints (10), which is written for all products, all customers and all time periods, assures that total flow of each product i received by each customer d in every planning period t from all distribution centers must at least satisfies the demand of customer d. Set of constraints (11), which is written for all products, all plants and all time peri- ods, assures that if production of product i in plant j is planned, the plant cannot produce product i more than its production capacity. Set of constraints (12), which is written for each plant and each time period, assures that the number of available working days in plant j during planning period t is restricted by available operating days minus the maintenance days considering change over (set up) days. Set of constraints (13), which is written for all products, all plants and all time periods, assures that the production rate of product i in plant j in planning period t is less than or equal to hours allocated for the production of product i in plant j in time t multiply by daily production rate of product i in plant j. Set of constraints (14), which is written for all products, all plants and all time periods, assures that the utilisation of plant j in planning period t is equal to hours allocated for the production of all products in plant j in time t. Set of constraints (15)–(16) ensure that utilisation of two arbitrary plants in time period t is constant and less than a predetermined value. Set of constraints (17)–(25) define the corresponding decision variables of the model. 3.3. Model validation In order to validate the process of mathematical modelling, several extreme state instances are considered. The solution of these instances can be expected easily as the parame- ters are meaningfully determined in a biased form. Then, the output of the proposed mathematical model (1)–(25) is compared with the expected results for these instances. For example in one of these instances, the production rate and procurement costs were FUZZY INFORMATION AND ENGINEERING 271 Table 3. Benchmark instance for validation of proposed model. Production rate Out-source product Plant 1 Plant 2 Plant 3 Distribution 1 Distribution 2 Distribution 3 Product 1 0.0 0.0 0.0 9666.549 22662.45 20783.00 Product 2 0.0 0.0 0.0 4446.451 3337.549 1343.000 set equal to zero in all plants, and the production costs are set to a very high value. Table 3 shows a partial report of the model (1)–(25) on this benchmark instance with three plants, three distribution centers and two demand zone for one month. As, we expected the model suggest the out-sourcing. The results are an evidence to validate the pro- posed model. Several aspects of model (1)–(25) were tested using several extreme state instances. 4. Proposed Fuzzy Type-II Solution methodology In this context, the approach by Figueroa [48] and Figueroa and Hernandez [49] is adopted to solve FLP problems with interval type-II RHS. Figueroa [48] proposed a Type-II fuzzy mathematical programming method. The approach reduced a fuzzy Type-II mathematical programming into a Type-I mathematical programming. The main idea of the method by Figueroa [48] was formed on the basis of linearity of mathematical programming. In many real mathematical programming models, such as supply chain network design, distribution and production, the resultant model is not linear. In this paper, the method by Figueroa [48] has been generalised for the non- linear class of mathematical problems. A linearisation approach is proposed and adopted to overcome the shortage of the method by Figueroa [48]. Let us now consider the model (26) as a linear programming problem, where b = ˜ ˜ ˜ (b , b , ... , b ) is interval type-II fuzzy (IT2F) RHS defined by its lower primary membership 1 2 n functions μ (x) with parameters b and b and upper primary membership function μ (x) with parameters b and b . The graphical representation of interval fuzzy type-II RHS is shown in the Figure 2. Min Z = C x (26) ˜ ˜ x ∈ (A, b) ={x ∈ R |a x ≥ b , i = 1, ... , m, x ≥ 0} Applying the approach proposed by Figueroa [48], the method is summarised as fol- low: min (1) Calculate Z using b −  as RHS; where,  is auxiliary variable weighted by C .The ∗ min optimum value of  and Z are obtained by solving the LP problem (27). Min Z = C · x + C · subject to : A · x ≥ b − (27) ≤ b − b ≥ 0, x ≥ 0 272 S. J. -SHARAHI ET AL. Figure 2. Interval Fuzzy Type-II RHS. max ∇ (2) Calculate Z using b −∇ as RHS; where ∇ is auxiliary variable weighted by C .The ∗ max optimum value of ∇ and Z are obtained by solving the LP problem (28). t ∇ Min Z = C · x + C ·∇ subject to : A · x ≥ b −∇ (28) ∇≤ b − b ∇≥ 0, x ≥ 0 C and C are assumed as incremental costs which are used to increase the consump- tion of resources in models (27) and (28), respectively. Therefore,  and ∇ operate as fuzzy type-II reducers. Using the values of ,and ∇, for each uncertain RHS, a best max fuzzy set embedded on the foot print of uncertainty (FOU) is obtained such that b = ∗ min ∇ ∗ b −  and b = b −∇ . The type-I fuzzy value b is determined using parameters max min b ,and b as linear membership function. Application of such fuzzy type-II reducers conducts the rest of algorithm based on Zimmermann [40] soft constraints method. min max (3) Define a Fuzzy Set Z with bounds Z and Z and linear membership function as (29). min 1 Cx ≤ Z max Z − Cx min max μ = Z ≤ Cx ≤ Z (29) max min Z − Z max 0 Cx ≥ Z (4) Consider an auxiliary variable λ to represent the λ − cut overall satisfaction degree between Z and b. Calculate the optimum value of λ by solving the LP problem (30). Max λ subject to : max min max Cx + λ(Z − Z ) = Z (30) max min min Ax − λ(b − b ) ≥ b λ ≥ 0, x ≥ 0 FUZZY INFORMATION AND ENGINEERING 273 4.1. Application of Proposed Fuzzy Type-II Approach The described approach is applied on model (1)–(25). Considering linear interval type-II fuzzy numbers for the RHS of model (1)–(25), three auxiliary crisp mixed-integer pro- min ∗ gramming models are yielded as follows. The values of Z and  are calculated using model (31)–(43). min PD PD DD DD P P min Z = objective function + C ·  + C ·  + C ·  + C ·  (31) idt idt jk jk kd kd ij ij subject to : P PD PD P − TQ ≥ (−cap −  ) · X ∀ j, k, t (32) ijkt jk jkt jk i=1 D DD DD D − TQ ≥ (−cap −  ) · X ∀ k, d, t (33) ikdt kd kdt kd i=1 TQ ≥ De −  ∀ i, d, t (34) idt ikdt idt k=1 P P − P ≥ (−cap −  ) · w ∀ i, j, t (35) ijt ijt ij ij PD PD PD ≤ cap − cap ∀ j, k (36) jk jk jk DD DD DD ≤ cap − cap ∀ k, d (37) kd kd kd ≤ De − De ∀ i, d, t (38) idt idt idt P P P ≤ cap − cap ∀ i, j (39) ij ij ij PD ≥ 0 ∀ j, k (40) jk DD ≥ 0 ∀ k, d (41) kd ≥ 0 ∀ i, d, t (42) idt ≥ 0 ∀ i, j (43) ij max ∗ The values of Z and ∇ are calculated using models (44)–(56). max PD PD DD DD P P min Z = objective function + C ·∇ + C ·∇ + C ·∇ + C ·∇ (44) idt idt jk jk kd kd ij ij subject to : P PD PD P − TQ ≥ (−cap −∇ ) · X ∀ j, k, t (45) ijkt jk jkt jk i=1 274 S. J. -SHARAHI ET AL. D DD DD D − TQ ≥ (−cap −∇ ) · X ∀ k, d, t (46) ikdt kd kdt kd i=1 D ∇ TQ ≥ De −∇ ∀ i, d, t (47) idt ikdt idt k=1 P P − P ≥ (−cap −∇ ) · w ∀ i, j, t (48) ijt ijt ij ij ∇ ∇ PD PD PD ∇ ≤ cap − cap ∀ j, k (49) jk jk jk ∇ ∇ DD DD DD ∇ ≤ cap − cap ∀ k, d (50) kd kd kd ∇ ≤ De − De ∀ i, d, t (51) idt idt idt ∇ ∇ P P P ∇ ≤ cap − cap ∀ i, j (52) ij ij ij PD ∇ ≥ 0 ∀ j, k (53) jk DD ∇ ≥ 0 ∀ k, d (54) kd ∇ ≥ 0 ∀ i, d, t (55) idt ∇ ≥ 0 ∀ i, j (56) ij It is notable that the crisp constraints (2)–(5), (8)–(9), (12)–(16) and (17)–(25) should also be included in both models (31)–(43), and (44)–(56). Therefore, a best fuzzy set embedded on foot print of uncertainty is calculated based on Equations (57)–(60). min  ∗ max ∇ ∗ PD PD PD PD PD PD cap = cap +  , cap = cap +∇ ∀ j, k (57) jk jk jk jk jk jk min  ∗ max ∇ ∗ DD DD DD DD DD DD cap = cap +  , cap = cap +∇ ∀ k, d (58) kd kd kd kd kd kd max  ∗ min ∇ ∗ De = De −  , De = De −∇ ∀ i, d, t (59) idt idt idt idt idt idt min  ∗ max ∇ ∗ P P P P P P cap = cap +  , cap = cap +∇ ∀ i, j (60) ij ij ij ij ij ij Then optimum value of λ is calculated using the models (61)–(66). Max λ subject to : (61) max min max objective function + λ(Z − Z ) = Z max min max P PD PD PD P − TQ − λ(cap − cap ) ≥−cap · X ∀ j, k, t (62) ijkt jk jk jk jkt i=1 FUZZY INFORMATION AND ENGINEERING 275 max min max D DD DD DD D − TQ − λ(cap − cap ) ≥−cap · X ∀ k, d, t (63) ikdt kd kd kd kdt i=1 D max min min TQ − λ(De − De ) ≥ De ∀ i, d, t (64) ikdt idt idt idt k=1 max min max P P P − P − λ(cap − cap ) ≥−cap · w ∀ i, j, t (65) ijt ijt ij ij ij λ ≥ 0 (66) Again, the crisp constraints (2)–(5), (8)–(9), (12)–(16) and (17)–(25) should also be included in models (61)–(66). 4.2. Linearisation approach One of the principles of the method proposed by Figueroa [48] is that the resultant auxil- iary programming models are linear. In this section, a linearisation approach is proposed to overcome the shortage of the non-linear programming. The models (31)–(43), and the min max models (44)–(56) which are used to calculate Z ,and Z , respectively, are non-linear mixed-integer programming models. The proposed approach in this section transforms them to linear programming (LP) problems. The global optimum solutions can be found easily for LPs. Both models (31)–(43), (44)–(56) have non-linear terms in which the products of two binary and continuous variables are incorporated. This type of non-linearity can be removed as follows. Let x be a binary variable, and x be a continuous variable, and 0 < x <u. 1 2 2 A continuous variable, y, is introduced to replace the product y = x x and the con- 1 2 straints (67)–(70) are to be added to the associated optimisation model. y ≤ ux (67) y ≤ x (68) y ≥ x − u(1 − x ) (69) 2 1 y ≥ 0 (70) The equations (32)–(33) and (35) are non-linear constraints, and are replaced by Equa- P D tions (71)–(85). The new continuous variables F , F ,F are defined for optimisation of ijt jkt kdt min Z . P PD P P TQ ≤ cap · X + F ∀ j, k, t (71) ijkt jkt jkt jk i=1 P PD PD P F ≤ (cap − cap )·X ∀ j, k, t (72) jkt jk jkt jk P PD F ≤  ∀ j, k, t (73) jkt jk 276 S. J. -SHARAHI ET AL. P PD PD PD P F ≥  − (cap − cap )·(1 − X ) ∀ j, k, t (74) jkt jk jk jkt jk F ≥ 0 ∀ j, k, t (75) jkt D DD D D TQ ≤ cap · X + F ∀ k, d, t (76) ikdt kdt kdt kd i=1 D DD DD D F ≤ (cap − cap ) · X ∀ k, d, t (77) kdt kd kdt kd D DD F ≤  ∀ k, d, t (78) kdt kd D DD DD DD D F ≥  − (cap − cap ) · (1 − X ) ∀ k, d, t (79) kdt kd kd kdt kd F ≥ 0 ∀ k, d, t (80) kdt P ≤ cap · w + F ∀ i, j, t (81) ijt ijt ijt ij P P F ≤ (cap − cap ) · W ∀ i, j, t (82) ijt ijt ij ij F ≤  ∀ i, j, t (83) ijt ij P P P F ≥  − (cap − cap ) · (1 − W ) ∀ i, j, t (84) ijt ijt ij ij ij F ≥ 0 ∀ i, j, t (85) ijt max Alternatively, in the process of optimisation of Z , Equations (45)–(46) and (48) are non-linear. They are replaced by Equations (86)–(100). The new continuous variables O , jkt O , O are also defined. ijt kdt P PD P P TQ ≤ cap · X + O ∀ j, k, t (86) ijkt jkt jkt jk i=1 ∇ ∇ P PD PD P O ≤ (cap − cap )·X ∀ j, k, t (87) jkt jk jkt jk P PD O ≤∇ ∀ j, k, t (88) jkt jk ∇ ∇ P PD PD PD P O ≥∇ − (cap − cap )·(1 − X ) ∀ j, k, t (89) jkt jk jk jkt jk O ≥ 0 ∀ j, k, t (90) jkt D DD D D TQ ≤ cap · X + O ∀ k, d, t (91) ikdt kdt kdt kd i=1 FUZZY INFORMATION AND ENGINEERING 277 ∇ ∇ D DD DD D O ≤ (cap − cap ) · X ∀ k, d, t (92) kdt kd kdt kd D DD O ≤∇ ∀ k, d, t (93) kdt kd ∇ ∇ D DD DD DD D O ≥∇ − (cap − cap ) · (1 − X ) ∀ k, d, t (94) kdt kd kd kdt kd O ≥ 0 ∀ k, d, t (95) kdt P ≤ cap · w + O ∀ i, j, t (96) ijt ijt ijt ij ∇ ∇ P P O ≤ (cap − cap ) · W ∀ i, j, t (97) ijt ijt ij ij O ≤∇ ∀ i, j, t (98) ijt ij ∇ ∇ P P P O ≥∇ − (cap − cap ) · (1 − W ) ∀ i, j, t (99) ijt ijt ij ij ij O ≥ 0 ∀ i, j, t (100) ijt 5. Numerical Example To demonstrate the applicability and usability of the proposed model, a numerical example with six manufacturing plants, six possible distribution centers and eight customer zones are considered. Each plant can produce six types of products and planning horizon is 2 months. The model described above is coded in LINGO to solve the MILP problem. 5.1. Instance Descriptions The working hour is equal to 20 per plant for a day. The maintenance lasts for 2 hours in a working day except for plants 5 and 6 in which the maintenance time is equal to 1hour. The change-over time is 1 hour. The utilisation parameter is assumed to be 50 hours per month. All other parameters of numerical example are given in Tables 4–11. These parameters are repeated for all planning periods. PD Capacity of connection between plants and distribution centers (cap )and capacity DD between distribution centers and customers (cap ) in kilogram are IT2F set according to Table 4. The detail information of fuzzy type-II production capacities for all plants are presented in Table 5. Table 6 presents the rate of production for each product in each plant. Table 7 presents the production cost for each product in each plant. Table 4. Fuzzy rate of flow (kg). Capacity cap cap cap cap PD cap 11000 25000 12000 26000 DD cap 26000 28000 27000 30000 278 S. J. -SHARAHI ET AL. Table 8 presents the transportation cost for each product from plant to distribution center. Fixed establishing, shut down, and transportation costs are presented in Table 9. The procurement costs and the customer demands are presented in Tables 10 and 11, respectively. Table 5. Fuzzy type-II production capacity (kg/month). Plant Product PL 1 PL 2 PL 3 PL 4 PL 5 PL 6 cap P1 73400 12700 35100 34400 11000 10000 P2 17616 3048 8424 8256 2640 2400 P3 88080 15240 42120 41280 13200 12000 P4 36700 6350 17550 17200 5500 5000 P5 58720 10160 28080 27520 8800 8000 P6 31562 5461 15093 14792 4730 4300 cap P1 69618 12049 33303 32633 10376 9539 P2 16708 2892 7993 7832 2490 2289 P3 83541 14459 39963 39160 12451 11447 P4 34809 6025 16651 16317 5188 4769 P5 55694 9639 26642 26107 8301 7631 P6 29260 5064 13997 13716 4361 4009 cap P1 31511 5454 15074 14771 4696 4318 P2 7563 1309 3618 3545 1127 1036 P3 37813 6545 18089 17725 5636 5181 P4 15756 2727 7537 7385 2348 2159 P5 25209 4363 12059 11817 3757 3454 P6 13550 2345 6482 6351 2019 1857 cap P1 29965 5186 14334 14046 4466 4106 P2 7192 1245 3440 3371 1072 985 P3 35958 6223 17201 16855 5359 4927 P4 14982 2593 7167 7023 2233 2053 P5 23972 4149 11467 11237 3573 3285 P6 12885 2230 6164 6040 1920 1765 Table 6. Rate of production (kg/hour). Plant PL1 PL2 PL3 PL4 PL5 PL6 P1 3660 630 1770 1710 540 510 P2 870 150 420 420 120 120 P3 4410 750 2100 2070 660 600 P4 1830 330 870 870 270 240 P5 2940 510 1410 1380 450 390 P6 1590 270 750 750 240 210 Table 7. Production cost (money unit/kg). Plant Product PL1 PL2 PL3 PL4 PL5 PL6 P1 3.18 3.03 3.67 3.54 2.77 2.76 P2 1.73 1.67 1.71 2.37 1.62 2.47 P3 2.68 2.69 3.6 2.85 2.21 2.20 P4 0.98 1.21 1.04 1.24 0.93 0.92 P5 0.43 0.49 0.55 0.41 0.28 0.27 P6 0.65 0.65 0.72 0.90 0.57 0.56 FUZZY INFORMATION AND ENGINEERING 279 Table 8. Transportation cost from plant to distribution center (money unit/kg). Distribution Center Plant Product DC 1 DC 2 DC 3 DC 4 DC 5 DC 6 PL 1 P1–P6 0.001 0.062 0.081 0.053 0.051 0.056 P5 0.003 0.071 0.091 0.061 0.059 0.051 PL 2 P1–P6 0.065 0.0890 0.127 0.077 0.071 0.077 P5 0.074 0.093 0.139 0.083 0.081 0.083 PL3 P1–P6 0.083 0.126 0.001 0.141 0.125 0.119 P5 0.091 0.138 0.003 0.153 0.132 0.122 PL 4 P1–P6 0.107 0.138 0.182 0.001 0.143 0.144 P5 0.112 0.152 0.194 0.003 0.162 0.164 PL5 P1–P6 0.183 0.236 0.117 0.038 0.001 0.038 P5 0.193 0.248 0.121 0.052 0.003 0.052 PL 6 P1–P6 0.168 0.178 0.236 0.117 0.038 0.001 P5 0.182 0.191 0.258 0.119 0.052 0.003 Table 9. Fixed infrastructure cost and transportation cost. Distribution Center DC 1 DC 2 DC 3 DC 4 DC 5 DC 6 Fixed establishing and shutting down cost (money unit) Establish 4300 2900 3100 2200 1300 1500 Shot Down 3100 1500 1900 3000 2200 2000 Customer Product Transportation cost (money unit/kg) C1 P1–P6 0.07 0.068 0.077 0.076 0.064 0.064 C2 P1–P6 0.048 0.048 0.048 0.058 0.047 0.060 C3 P1–P6 0.063 0.063 0.077 0.065 0.056 0.056 C4 P1–P6 0.037 0.041 0.038 0.041 0.036 0.036 C5 P1–P6 0.021 0.022 0.024 0.021 0.019 0.019 C6 P1–P6 0.032 0.032 0.033 0.036 0.031 0.031 C7 P1–P6 0.06 0.058 0.067 0.066 0.054 0.051 C8 P1–P6 0.027 0.031 0.028 0.031 0.026 0.026 Table 10. Procurement cost per product (money unit/kg). Product P1P2P3P4P5P6 3.41 2.23 3.18 1.00 0.93 0.80 PD DD P Incremental costs C ,C ,C and C are set as 0.04, 0.1, 3 and 1.7, respectively. 5.2. Results Using the parameters presented in Section 5.1, and Equations (31), (34), (36)–(43) and (71)–(85) for step 1 and Equation (44), (47), (49)–(56) and (86)–(100) for step 2, the max min optimal solutions are respectively Z = 2404051, and Z = 914544.5. The optimal sat- isfaction degree in the last step is λ = 0.537, so the optimal value of objective function is Z = 1604404.669. The graphical representation of fuzzy objective function is shown in Figure 3. Table 12 shows a detailed report of fuzzy objective function. 6. Conclusion and future research directions This research proposed a new uncertain integrated approach to determine the optimal issues related to the design and operation of supply chains. The main decisions made in 280 S. J. -SHARAHI ET AL. Table 11. Fuzzy customer demand (kg). Customer Zone Product C1C2C3C4 C5 C6C7C8 De P1 32329 20783 19269 15231 7610 13099 13814 10315 P2 7784 1343 5400 4985 3348 2318 3859 2752 P3 52796 14427 22267 22006 9905 6669 19827 11044 P4 19736 2255 23053 3562 4250 5351 1093 6925 P5 34745 4123 21547 8698 7490 6509 14886 7962 P6 6498 1414 6873 25924 2873 2431 3801 7139 De P1 30712 19744 18306 14470 7230 12444 13123 9799 P2 7395 1276 5130 4735 3180 2202 3666 2614 P3 50156 13706 21153 20906 9410 6335 18835 10492 P4 18749 2142 21901 3384 4037 5083 1038 6579 P5 33007 3916 20469 8263 7116 6184 14142 7564 P6 6173 1344 6530 24628 2729 2310 3611 6782 De P1 13901 8937 8286 6550 3272 5633 5940 4435 P2 3347 577 2322 2143 1439 997 1659 1183 P3 22702 6204 9575 9463 4259 2867 8526 4749 P4 8486 970 9913 1532 1827 2301 470 2978 P5 14940 1773 9265 3740 3221 2799 6401 3424 P6 2794 608 2956 11147 1235 1045 1635 3070 De P1 13219 8498 7879 6228 3112 5356 5648 4218 P2 3183 549 2208 2038 1369 948 1578 1125 P3 21588 5899 9105 8998 4050 2727 8107 4516 P4 8070 922 9427 1456 1738 2188 447 2832 P5 14207 1686 8810 3557 3063 2662 6087 3256 P6 2657 578 2810 10600 1175 994 1554 2919 Table 12. Fuzzy objective function. Fuzzy Parameters Z PD DD P De cap cap cap Z = 2417000 ∇ ∇ ∇ ∇ PD DD P De cap cap cap Z = 916362.2 PD DD P ∇ De cap cap cap Z = 2280701 ∇ ∇ ∇ PD DD P De cap cap cap Z = 866484.1 the proposed approach concerned location–allocation, production, procurement and dis- tribution of products based on financial aspects and production balancing among plants in multi-product, and multi-period supply chain networks. The fuzzy model considered three source of uncertainty in demand, manufacturing and supply process, concurrently. The Figure 3. Fuzzy Type-II set of objective function. FUZZY INFORMATION AND ENGINEERING 281 uncertainties in this research were modelled through type-II fuzzy sets in which a deeper insight towards vagueness and ambiguity in fuzzy membership values was supplied. The whole approach was modelled through a fuzzy mixed-integer mathematical programming. A fuzzy type-reducer methodology was proposed in order reduce fuzzy type-II uncertain- ties to a type-I fuzziness and to solve the yielded fuzzy type-I mathematical model through existing methods. A numerical example was supplied to address the mechanism of proposed approach and to illustrate its applicability. Some further research directions are proposed based on findings of this research as follows: (1) RHS of constrain were considered type-II fuzzy numbers in this research, devel- opment of a model with fuzzy type-II objective function or fuzzy type-II decision variables can demonstrate new insights in real supply chains, (2) as the location–allocation, produc- tion, procurement and distribution problems in supply chains are usually assumed to be non-deterministic poly-nominal hard (NP-Hard) problems, the application of soft comput- ing techniques or meta-heuristics could be suitable in order to handle large-scale and real life problems, (3) the proposed method can be applied and tested in real case study in order to check its applicability in real-world problems and (4) although the proposed approach models some kind of uncertainty which have never been discussed before, some compari- son with existing methods in the literature may be interesting and reveals the advantages of the proposed approach clearly. 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Type-II Fuzzy Multi-Product, Multi-Level, Multi-Period Location–Allocation, Production–Distribution Problem in Supply Chains: Modelling and Optimisation Approach

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FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 2, 260–283 https://doi.org/10.1080/16168658.2018.1517978 Type-II Fuzzy Multi-Product, Multi-Level, Multi-Period Location–Allocation, Production–Distribution Problem in Supply Chains: Modelling and Optimisation Approach a a b Sarah J.-Sharahi , Kaveh Khalili-Damghani , Amir-Reza Abtahi and Alireza Rashidi-Komijan Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran; Department of Information Technology Management, Faculty of Management, Kharazmi University, Tehran, Iran; Department of Industrial Engineering, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran ABSTRACT ARTICLE HISTORY Received 8 May 2016 In this study, the application of type-II fuzzy sets is addressed to Revised 5 January 2017 design a multi-product, multi-level, multi-period supply chain net- Accepted 12 January 2018 works. The proposed model provides integrated approach to make optimal decisions such as location–allocation, production, procure- KEYWORDS ment and distribution subject to operational and tactical constraints. Type-II fuzzy integer linear In the context of fuzzy linear programming, this study involves type-II programming; fuzzy type-II fuzzy numbers for the right-hand side of constraints regarding three reducer; location–allocation problem; sources of uncertainty: demand, manufacturing and supply. Accord- production–distribution ing to fuzzy components considered, a type-II fuzzy mixed-integer problem; supply chain linear programming is converted into an equivalent auxiliary crisp planning model using linear fuzzy type-reducer models. The final models are linear and the global optimum solutions can be achieved using com- mercial OR softwares. The contributions of this study are three folds: (1) introducing a new integrated supply chain network design prob- lem; (2) considering a solution procedure based on type-II fuzzy sets and (3) presenting a linear fuzzy type-II reducer. Finally, the proposed model and solution approach are illustrated through a numerical example to demonstrate the significance. Highlights • A model is proposed to formulate the integrated supply chain network design problem. • Uncertainties in real supply chains are modelled using type-II fuzzy sets. • A new solution procedure based on type-II fuzzy sets is developed. • A linear fuzzy type-II reducer is proposed. • Numerical example is proposed to illustrate the efficacy of proposed approach. 1. Introduction Supply Chain (SC) is a network of organisations, suppliers and distributors that produces value in the form of products and services and bring them to ultimate customer or market CONTACT Kaveh Khalili-Damghani kaveh.khalili@gmail.com © 2018 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 261 [1, 2]. Supply Chain Management (SCM) is a set of management activities for planning, implementing and controlling the supply chain operations. SCM needs to cope with the high uncertainty and imprecision related to real-world character [3]. Peidro et al. [4]have analysed and classified the sources of uncertainty in SC into three groups as demand, man- ufacturing and supply. Uncertainty is presented as variability and inexact forecasting of supplier operations and demand, and also poorly reliable production and manufacturing process [5]. In SC planning, uncertainty is a main factor that can influence the SCM processes. The uncertainty in SC has already been modelled through several paradigms such as robust optimisation [6, 7], stochastic programming [8, 9], fuzzy mathematical programming [5, 10] and hybrid models [11]. There also exists hybrid models where stochastic fuzzy program- ming or robust fuzzy programming models were combined [12]. In supply chains, a series of decisions such as which plants to establish, which distribution centre supplies which cus- tomer, production level of plants, capacity of plants and distribution strategies are needed to address accurately while the information contains a great amount of uncertainties. Type-II fuzzy sets and systems generalise type-I fuzzy sets and systems so that more uncertainty can be handled. From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-I fuzzy set has no uncertainty asso- ciated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided by Zadeh [13] when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a type-II fuzzy set. A type-II fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-II fuzzy sets head-on. And, if there is no uncertainty, then a type-II fuzzy set reduces to a type-I fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes. Very limited applications of type-II fuzzy sets were reported in recent years. Due to our best knowledge, there is no study which addresses the type-II fuzzy numbers on integrated location–allocation, production and distribution activities in supply chain. So, the main contributions of this paper can be summarised as follows: • Introducing a strategic and tactical SC problem through integrating dynamic location– allocation, production and distribution planning into a multi-echelon, multi-product, multi-period network. • Modelling the aforementioned problem using mixed-integer mathematical modelling. • Considering different sources of uncertainty in demand, manufacturing and supply processes affecting the SC. • Considering the interval type-II fuzzy (IT2F) number for uncertain parameters and solving the SC problem based on fuzzy type-II programming. The rest of the paper is organised as follows. The literature of past works on SC under fuzziness are presented and classified in Section 1. Section 2 introduces a fuzzy mixed- integer linear programming (FMILP) model for the integrated multi-product, multi-level, multi-period location–allocation, production–distribution problem in supply chains. In Section 3, a solution methodology is developed to transform the proposed FMILP into an auxiliary crisp mixed-integer linear programming model. A numerical example and results 262 S. J. -SHARAHI ET AL. are presented in Section 4 to illustrate the mechanism of proposed model and the appli- cability of solution procedure. Finally, the conclusions and future research directions are presented in Section 5. 2. Literature Review of Past Works The literature of uncertain supply chain models and problems as well as the fuzzy solution procedures is reviewed in this section. 2.1. Uncertain Integrated Supply Chain Problems The aim of location–allocation problem is to decide the facilities to be opened and the assignment of customers to the opened facilities. In literature, there is a vast effort for combining allocation decisions with other decisions such as production, inventory, pro- curement and capacity decisions [12]. Several authors have studied these integrated SC problems from a deterministic point of view. Tsiakis and Papageorgiou [14] developed a mixed-integer linear programming for global supply chain networks to decide about pro- duction, allocation, production capacity, purchase and network configuration with financial aspects of exchange rates, duties and production costs. Ahmadi-Javid and Hoseinpour [15] presented a non-linear mixed-integer programming model to determine location, allocation, price and order-size decisions of SC with two scenarios of uncapacitated and capacitated distribution centres. Jang et al. [16] studied integrated inventory-allocation and shipping problem in SC. Tsao and Lu [17] developed an integrated facility location and inventory-allocation problem in SC that considering two types of transportation cost dis- counts simultaneously: quantity discounts for inbound transportation cost and distance discounts for outbound transportation cost. Jonrinaldi and Zhang [18] proposed a model for coordinating integrated production and inventory cycles in manufacturing supply chain involving reverse logistics for multiple items with finite multiple periods. Shahabi et al. [19] studied mathematical models to coordinate facility location, allocation of supplier to ware- houses and retailers and inventory control for a four-echelon supply chain network. The goal in this study is to minimise the facility location, transportation and the inventory costs. Latha Shankara et al. [20] proposed a single product, four-echelon supply chain architec- ture consisting of suppliers, production plants, distribution centres and customer zones to determine the number and location of plants in the system, the flow of raw materials from suppliers to plants, and the quantity of products to be shipped from plants to distribution centres and from distribution centres to customer zones, so as to minimise the facility loca- tion and shipment costs. A multi-objective hybrid particle swarm optimisation algorithm was proposed as the solution method for the proposed model. In a study by Bandyopad- hyay and Bhattacharya [21] a tri-objective problem for a two-echelon serial supply chain with objectives of minimisation of the total cost of the supply chain, minimisation of the variance of order quantity and minimisation of the total inventory was developed. To solve the model, a proposed modified non-dominated sorting genetic algorithm was applied. Mousavi et al. [22] developed an inventory–location–allocation problem in two-echelon SC in which the all unit discount strategy was applied for purchasing costs. Diabat et al. [23] considered a closed-loop location-inventory problem with forward and reverse supply chain consisting of a single echelon. The problem is to choose which distribution centres FUZZY INFORMATION AND ENGINEERING 263 and remanufacturing centres are to be opened and to associate the retailers with them. Subulan et al. [24] designed a closed-loop, multi-objective, multi-echelon, multi-product and multi-period logistics network model using mixed-integer linear programming with recovery options such as remanufacturing, recycling, and energy recovery and interactive fuzzy goal programming approach was utilised to solve the proposed model. As mentioned before, deterministic SC is unrealistic in real terms of SC problems and coefficients and parameters are often imprecise because of incomplete and unavailable information. There is increasing attention to development fuzzy or stochastic supply chain model and offer possible solutions that leading to more realistic models. SC planning under fuzzy uncertainty have been reviewed in several researches [4, 5, 12]. Peidro et al. [5] modelled integrated procurement, production and distribution planning in SC with dif- ferent sources of uncertainty in demand, process and supply. All uncertain parameters were assumed to be type-I fuzzy numbers. The purposed model was solved with adap- tation approach of Jimenez et al. [25]. Bilgen [26] developed an integrated production and distribution plans in multi-echelon SC for determining the allocation of production among production lines and transporting products with fuzzy available resources capac- ity and fuzzy costs. Paksoy et al. [27] proposed fuzzy multi-objective linear programming for minimising the end customer’s dissatisfaction and transportation costs. The produc- tion capacity and amount of demands were type-I fuzzy numbers. Figueroa [28] proposed multi-period production planning with IT2F demand. Diaz-Madronero et al. [10] proposed fuzzy multi-objective integer linear programming model for transportation and procure- ment planning in three-level, multi-product and multi-period automobile supply chain. They assumed the maximum capacity of the available truck and minimum percentage of demand in stock as type-I fuzzy numbers. They used an interactive solution methodology to solve the problem. Mousavi and Niaki [29] studied a capacitated location allocation problem with fuzzy demands and stochastic locations of the customers that follow the normal probability distri- bution in which the distances between the locations and the customers are taken Euclidean and squared Euclidean. The fuzzy expected cost programming, the fuzzy β-cost minimisa- tion model and the credibility maximisation model are three types of fuzzy programming that are developed to model. Azadeh et al. [30] proposed a multi-objective, multi-period, multi-echelon fuzzy linear programming model to optimise natural gas supply chain based on two economic and environmental objectives and fuzzy parameters, including demand, capacity and cost. A combination of the possibilistic programming approach based on the defuzzification method and interactive fuzzy approach was used to deal with uncertainty. Ramezani et al. [31] designed a multi-product, multi-period, closed-loop supply chain net- work with three objective functions: maximisation of profit, minimisation of delivery time, and maximisation of quality and three fuzzy components of constraints, coefficients, and goal. To cope with fuzziness, a fuzzy optimisation approach was adopted to convert the pro- posed fuzzy multi-objective mixed-integer linear programme into an equivalent auxiliary crisp model. Gholamian et al. [32] developed multi-site, multi-period and multi-product aggregate production planning in supply chain to minimise the total cost of the SC, improve customer satisfaction, minimise the fluctuations in the rate of changes of workforce, and maximise the total value of purchasing under fuzzy parameters. Alizadeh Afrouzy et al. [33] designed a multi-echelon, multi-period, multi-product and aggregate procurement and production 264 S. J. -SHARAHI ET AL. planning model which considered three objectives of maximise the profit of the supply chain due to new product development, customer satisfaction, and maximising the pro- duction of the developed and new products. The proposed model considered uncertainties of the environment such as customer demands and supplier capacities which is modelled by fuzzy stochastic programming. Following the stochastic viewpoint, Lin and Wu [34] proposed product pricing and integrated supply chain operations plan under conditions of price-dependent stochas- tic demand under two approaches of maximising a manufacturer’s expected profit and cost minimisation. Marufuzzaman et al. [35] developed two-stage stochastic programming model to design and extend the classical two-stage stochastic location–transportation model. Their model was designed to optimise costs and emissions in the supply chain and capture the impact of biomass supply and technology uncertainty on tradeoffs that exist between location and transportation decisions; and the tradeoffs between costs and emissions in the supply chain. About decision-making in inventory management, Panda et al. [36] developed eco- nomic production lot size for single-period multi-product model in which the demand rate is stochastic under stochastic and/or fuzzy budget and shortage constraints. The stochastic constraints have been represented by chance constraints and fuzzy constraints in the form of possibility/necessity constraints which are transformed to equivalent deter- ministic ones. Wang et al. [37] proposed single-item and multi-item single-period inven- tory models for short life-cycle products when demands are assumed to be uncertain random variables. For deriving the optimality condition for optimal order quantity the uncertain random models are transferred to equivalent deterministic forms by consider- ing expected profit and providing more information of chance distributions. Integrated inventory model in supply chain is developed in some researches. Bag and Chakraborty [38] addressed optimal order quantity of the retailers, production rate of the producer and the production rate of the suppliers to minimise the supply chain cost in which there are imprecise chance constraints on the transportation costs for producer, retailer and also a fuzzy space constraint for producer is considered. Jana et al. [39] developed order- ing policy for deteriorating products with allowable shortage and permissible delay in payment under inflation and time value of money in fuzzy rough environment which the proposed model is converted to deterministic one using expected value method techniques. 2.2. Fuzzy Linear Programming as Solution Procedure Fuzzy linear programming (FLP) is an optimisation technique applied in real-world prob- lems with imprecise data. FLP has an extensive literature. In this research, two main cate- gories of FLP are reviewed: (1) FLP with fuzzy numbers as coefficient in objective function, right-hand sides ( RHSs), and technological coefficient, (2) FLP with fuzzy parameters and fuzzy decision variables. In the first category, numerous researches have proposed different techniques to solve the FLP problems. Zimmermann [40] formulated and solved the FLP problem for the first time. Herrera and Verdegay [41] proposed three models for three types of FLPs. They solved the problems with fuzzy RHS or fuzzy coefficient of constraint using auxiliary parametric linear programme. They also used two fuzzy ranking methods to cope with FLPs with fuzzy FUZZY INFORMATION AND ENGINEERING 265 objective functions. Xinwang [42] developed FLP with fuzzy technological coefficients and the RHSs of the constraints and solved FLPs based on the new ranking method of fuzzy num- bers for satisfaction degree of the constraints. Zhang et al. [43] presented FLPs with fuzzy coefficients of objective function and showed how to convert such FLPs into equivalent deterministic multi-objective optimisation problem. Jimenez et al. [25] presented full FLPs in which all parameters were fuzzy numbers. They also presented a resolution method with interactive participation of decision maker during solving procedure. Wu [44] described the FLP which involve fuzzy numbers for the coefficients of variables in the objective function and developed the optimality conditions for FLPs through proposing two solving method- ologies that were similar to non-dominated solution in multi-objective programming. The problems with fuzzy numbers for the coefficients of the objective function, the techno- logical coefficients in the constraints and the RHS of the constraints were developed by Mahdavi-Amiri and Nasseri [45], Hatami-Marbini and Tavana [46] and Saati et al. [47].The method for finding the optimal solution of FLPs with type-II fuzzy numbers presented by Figueroa, [48] and Figueroa and Hernandez [49].They proposed a method to solve FLPs with interval type-II RHSs. In the second category, FLP with fuzzy numbers for the decision variables, the coeffi- cients in the objective function and the RHS of the constraints was developed by Ganesan and Veeramani [50] that leaded to the solution without converting FLPs to crisp linear programming problems. Mahdavi-Amiri and Nasseri [51] studied the FLP problems with fuzzy numbers for the decision variables and the right-hand-side of the constraints and developed solution method based on auxiliary problems that applied a linear ranking function to order trape- zoidal fuzzy variables. Then, Ebrahimnejad et al. [52] proposed a new primal-dual algorithm for solving FLP problems with fuzzy variables by using the duality method investigated by Mahdavi-Amiri and Nasseri [51]. Kumar et al. [53] proposed a method to improve the method proposed by Hosseinzadeh Lotfi et al. [54] to find the optimal solution of the FLPs with fuzzy parameters and fuzzy variables that called Fully FLP. Based on our knowledge, the integrated SC model in a fuzzy environment with different sources of uncertainty especially with type-II fuzzy sets has not been well studied. There- fore, this paper attempts: (1) to introduce an integrated location–allocation, production, procurement and distribution planning problem in supply chains, (2) to model the pro- posed problem in presence of uncertainty in demand, manufacturing and supply processes through mathematical programming and (3) to develop a solution methodology to solve the proposed model optimally considering interval type-II fuzzy sets. Most of the supply chain researches assumed the fuzzy parameters to be of type- I fuzzy sets such as Peidro et al. [5], Bilgen [26], Paksoy et al. [27] and Diaz-Madronero et al. [10]. In decision-making problems like location–allocation, production–distribution, the parameters cannot be always exactly known. For example, demand or flow of prod- ucts in supply chain management depends upon several variables such as price, labour charges and repair time. Each of these variables fluctuates and also membership degree of each point cannot be exactly determined. So it is not easy to predict the deter- ministic or type-I fuzzy sets. Assuming fuzzy type-II parameters, i.e. fuzziness in mem- bership function, for such variables is a proper idea, although fuzzy type-II parameters impose very high computational complexity to the problem. In this paper, the fuzzy type- II mathematical programming is reduced into fuzzy type-I mathematical programming 266 S. J. -SHARAHI ET AL. while the main properties of original programming is retained. Moreover, the procedure is generalised for the cases in which the resultant model may be a non-linear math- ematical programming. So, formally talking, the main advantages of proposed proce- dure of this study over the previous methods in the literature can be summarised as follows: • Considering complicated uncertainties in membership value of a fuzzy variable in terms of fuzzy type-II parameters. • Reducing the fuzzy type-II mathematical programming into a fuzzy type-I mathematical programming while the main properties of original programming is retained. • Proposing a procedure for linearisation of the resultant models in the sense of gener- alisation of the cases in which the resultant model may be a non-linear mathematical programming. 3. Problem Descriptions and Modelling In this section, the problem of multi-echelon, multi-product, multi-period location–alloca- tion production and distribution supply chain network is proposed. Then, the associated mathematical programming is developed. The following assumptions are considered in the proposed model: • Number of possible plants are known and determined in advance. • Number of possible distribution centres are known and determined in advance. • Each product can be produced at several plants. • Distribution centres can be supplied from more than one plant and can supply more than one customer. • Customers have uncertain demands of multiple products during planning periods. • Production and transportation costs are assumed to be known and deterministic. • Production and transportation capacity are known and uncertain. The following dynamic decisions are to be determined: • Establish or shut down distribution centres • Product plan, production rate and utilisation per plants • Distribution plan between plants, distribution centres and customers • Procurement plan • Assignment of distribution centres to plants and customer to distribution centres The objective is the minimisation of the total cost of the strategic, tactical and opera- tional costs. The graphical representation of the three-stage, multi-product, multi-period supply chain network is shown in Figure 1. The objective is to find which distribution centres are to be opened, which customers are served from opened distributors, and which procurement and production strategy is to be planned so that the total cost is minimised. FUZZY INFORMATION AND ENGINEERING 267 Figure 1. Three-stage supply chain. 3.1. Notations Indices, parameters and decision variables are presented in Table 1. Three types of uncertainty are assumed in the proposed model. Table 2 presents the sources and types of uncertainties. 3.2. Model formulation A new multi-echelon, multi-product, multi-period location–allocation production and dis- tribution supply chain network model with type-II fuzzy sets with due attention to the model proposed by Tsiakis and Papageorgiou [14] is developed. K T K T I J T D D D D Min Z = EC · Y + SC (1 − Y ) + PC · P ijt ijt kt kt kt kt k=1 t=1 k=1 t=1 i=1 j=1 t=1 I K T I J K T P P + BC · PQ + TC · TQ (1) it ikt ijkt ijkt i=1 t=1 i=1 j=1 t=1 k=1 k=1 I K D T D D + TC · TQ ikdt ikdt i=1 t=1 k=1 d=1 P D X ≤ Y ∀ j, k, t (2) jkt kt P D X ≥ Y ∀ k, t (3) jkt kt j=1 D D X ≤ Y ∀ k, d, t (4) kdt kt X ≥ 1 ∀ k, t (5) kdt d=1 268 S. J. -SHARAHI ET AL. Table 1. Indices, parameters and decision variables. Indices I Set of products i ={1, 2, ... , I} J Set of plants j ={1, 2, ... , J} K Set of possible distribution centre k ={1, 2, ... , K} D Set of customers d ={1, 2, ... , D} T Set of planning periods t ={1, 2, ... , T} Parameters EC cost of establishing a distribution centre at location k in time t kt SC cost of closing a distribution centre at location k in time t kt PC unit production cost for product i at plant j in time t ijt BC unit procurement cost for product i from third parties in time t it TC unit transport cost for product i from plant j to distribution centre k in time t ijkt TC unit transport cost for product i from distribution centre k to customer d in time t ikdt De demand of customer d for product i in time t idt H Hours of operating of plant j M Hours of maintenance of plant j Cap maximum production capacity of plant j for product i ij PD Cap maximum rate of flow of products transferred from plant j to distribution centre k jk DD Cap maximum rate of flow of products transferred from distribution centre k to customer d kd r daily production rate of product i in plant j ij β change-over coefficient in hours ε utilisation parameter in hours Decision Variable PQ out-sourced product i delivered to distribution centre k in time t ikt P production rate of product i in plant j in time t ijt TQ rate of flow of product i transferred from plant j to distribution centre k in time t ijkt TQ rate of flow of product i transferred from distribution centre k to customer d in time t ikdt T hours allocated for production of product i in plant j in time t ijt U utilisation of production plant j in time t jt α maximum allowed difference in utilisation of plants in time t Y 1 if distribution centre k is to be established in time t,0otherwise kt X 1 if production plant j is assigned to distribution centre k in time t,0otherwise jkt X 1 if distribution centre k is assigned to customer d in time t,0otherwise kdt w 1 if production plant j is to produce product i in time t,0otherwise ijt Table 2. Sourcesofuncertainty. Source of uncertainty Description Parameter Demand Demand of product i at customer d in period t De idt Process (manufacturing) Production capacity of plant j for product i Cap ij PD Supply Rate of flow from plant j to distribution centre k Cap jk DD Rate of flow from distribution center k to customer d Cap kd P PD P TQ ≤ cap  · X ∀ j, k, t (6) ijkt jk jkt i=1 D DD D TQ ≤ cap  · X ∀ k, d, t (7) ikdt kd kdt i=1 P = TQ ∀ i, j, t (8) ijt ijkt k=1 FUZZY INFORMATION AND ENGINEERING 269 J D P D TQ + PQ = TQ ∀ i, k, t (9) ikt ijkt ikdt j=1 d=1 TQ ≥ De ∀ i, d, t (10) idt ikdt k=1 P ≤ cap · w ∀ i, j, t (11) ijt ijt ij I I T ≤ (H − M ) − β w ∀ j, t (12) j j ijt ijt i=1 i=1 P ≤ r · T ∀ i, j, t (13) ijt ij ijt U = T ∀ j, t (14) jt ijt i=1 α = U − U ∀ j,tj = j (15) t jt ´ jt α ≤ ε ∀ t (16) PQ ≥ 0 ∀ i, k, t (17) ikt P ≥ 0, T ≥ 0, w ∈{0, 1}∀ i, j, t (18) ijt ijt ijt TQ ≥ 0 ∀ i, j, k, t (19) ijkt TQ ≥ 0 ∀ i, k, d, t (20) ikdt U ≥ 0 ∀ j, t (21) jt α ≥ 0 ∀ t (22) Y = Bin ∀ k, t (23) kt X = Bin ∀ j, k, t (24) jkt X = Bin ∀ k, d, t (25) kdt Equation (1) minimises the total cost of supply chain. The objective function con- sists of costs of the distribution centre infrastructure, the costs of production and pro- curement, and costs of transportation. Set of constraints (2), which is written for each plant, distribution center and planning period, assures that a plant can services to a distribution center if an only if the distribution center had been established. Set of con- straints (3), which is written for each distribution center and planning period, assures 270 S. J. -SHARAHI ET AL. that one of the production plants is assigned to each distribution center in each time period. Set of constraints (4), which is written for each distribution center, customer and plan- ning period, assures that a distribution center can services to a customer if an only if the distribution center had been established. Set of constraints (5), which is written for each distribution center and planning period, assures that a distribution center must be assigned at least to a customer in each time period. Set of constraints (6), which is written for each plant, distribution center and time period, assures that flow of material from production plant to distribution center can take place only if the connection exists. Set of constraints (7), which is written for each distribution center, customer and time period, assures that flow of material from distribution center to customer can take place only if the corresponding connection exists. Set of constraints (8) assures that the production rate of product type i in plant j in time period t is equal to all sent products type i to all distribution centres in time period t. Set of constraints (9) assures that all products type i which has received by distribution center k from all plants in time period t plus the products type i which has received by distribution center k from out-sourcing supplier in time period t is equal to all products type i which has received by all customers d from distribution center k in time period t. Set of constraints (10), which is written for all products, all customers and all time periods, assures that total flow of each product i received by each customer d in every planning period t from all distribution centers must at least satisfies the demand of customer d. Set of constraints (11), which is written for all products, all plants and all time peri- ods, assures that if production of product i in plant j is planned, the plant cannot produce product i more than its production capacity. Set of constraints (12), which is written for each plant and each time period, assures that the number of available working days in plant j during planning period t is restricted by available operating days minus the maintenance days considering change over (set up) days. Set of constraints (13), which is written for all products, all plants and all time periods, assures that the production rate of product i in plant j in planning period t is less than or equal to hours allocated for the production of product i in plant j in time t multiply by daily production rate of product i in plant j. Set of constraints (14), which is written for all products, all plants and all time periods, assures that the utilisation of plant j in planning period t is equal to hours allocated for the production of all products in plant j in time t. Set of constraints (15)–(16) ensure that utilisation of two arbitrary plants in time period t is constant and less than a predetermined value. Set of constraints (17)–(25) define the corresponding decision variables of the model. 3.3. Model validation In order to validate the process of mathematical modelling, several extreme state instances are considered. The solution of these instances can be expected easily as the parame- ters are meaningfully determined in a biased form. Then, the output of the proposed mathematical model (1)–(25) is compared with the expected results for these instances. For example in one of these instances, the production rate and procurement costs were FUZZY INFORMATION AND ENGINEERING 271 Table 3. Benchmark instance for validation of proposed model. Production rate Out-source product Plant 1 Plant 2 Plant 3 Distribution 1 Distribution 2 Distribution 3 Product 1 0.0 0.0 0.0 9666.549 22662.45 20783.00 Product 2 0.0 0.0 0.0 4446.451 3337.549 1343.000 set equal to zero in all plants, and the production costs are set to a very high value. Table 3 shows a partial report of the model (1)–(25) on this benchmark instance with three plants, three distribution centers and two demand zone for one month. As, we expected the model suggest the out-sourcing. The results are an evidence to validate the pro- posed model. Several aspects of model (1)–(25) were tested using several extreme state instances. 4. Proposed Fuzzy Type-II Solution methodology In this context, the approach by Figueroa [48] and Figueroa and Hernandez [49] is adopted to solve FLP problems with interval type-II RHS. Figueroa [48] proposed a Type-II fuzzy mathematical programming method. The approach reduced a fuzzy Type-II mathematical programming into a Type-I mathematical programming. The main idea of the method by Figueroa [48] was formed on the basis of linearity of mathematical programming. In many real mathematical programming models, such as supply chain network design, distribution and production, the resultant model is not linear. In this paper, the method by Figueroa [48] has been generalised for the non- linear class of mathematical problems. A linearisation approach is proposed and adopted to overcome the shortage of the method by Figueroa [48]. Let us now consider the model (26) as a linear programming problem, where b = ˜ ˜ ˜ (b , b , ... , b ) is interval type-II fuzzy (IT2F) RHS defined by its lower primary membership 1 2 n functions μ (x) with parameters b and b and upper primary membership function μ (x) with parameters b and b . The graphical representation of interval fuzzy type-II RHS is shown in the Figure 2. Min Z = C x (26) ˜ ˜ x ∈ (A, b) ={x ∈ R |a x ≥ b , i = 1, ... , m, x ≥ 0} Applying the approach proposed by Figueroa [48], the method is summarised as fol- low: min (1) Calculate Z using b −  as RHS; where,  is auxiliary variable weighted by C .The ∗ min optimum value of  and Z are obtained by solving the LP problem (27). Min Z = C · x + C · subject to : A · x ≥ b − (27) ≤ b − b ≥ 0, x ≥ 0 272 S. J. -SHARAHI ET AL. Figure 2. Interval Fuzzy Type-II RHS. max ∇ (2) Calculate Z using b −∇ as RHS; where ∇ is auxiliary variable weighted by C .The ∗ max optimum value of ∇ and Z are obtained by solving the LP problem (28). t ∇ Min Z = C · x + C ·∇ subject to : A · x ≥ b −∇ (28) ∇≤ b − b ∇≥ 0, x ≥ 0 C and C are assumed as incremental costs which are used to increase the consump- tion of resources in models (27) and (28), respectively. Therefore,  and ∇ operate as fuzzy type-II reducers. Using the values of ,and ∇, for each uncertain RHS, a best max fuzzy set embedded on the foot print of uncertainty (FOU) is obtained such that b = ∗ min ∇ ∗ b −  and b = b −∇ . The type-I fuzzy value b is determined using parameters max min b ,and b as linear membership function. Application of such fuzzy type-II reducers conducts the rest of algorithm based on Zimmermann [40] soft constraints method. min max (3) Define a Fuzzy Set Z with bounds Z and Z and linear membership function as (29). min 1 Cx ≤ Z max Z − Cx min max μ = Z ≤ Cx ≤ Z (29) max min Z − Z max 0 Cx ≥ Z (4) Consider an auxiliary variable λ to represent the λ − cut overall satisfaction degree between Z and b. Calculate the optimum value of λ by solving the LP problem (30). Max λ subject to : max min max Cx + λ(Z − Z ) = Z (30) max min min Ax − λ(b − b ) ≥ b λ ≥ 0, x ≥ 0 FUZZY INFORMATION AND ENGINEERING 273 4.1. Application of Proposed Fuzzy Type-II Approach The described approach is applied on model (1)–(25). Considering linear interval type-II fuzzy numbers for the RHS of model (1)–(25), three auxiliary crisp mixed-integer pro- min ∗ gramming models are yielded as follows. The values of Z and  are calculated using model (31)–(43). min PD PD DD DD P P min Z = objective function + C ·  + C ·  + C ·  + C ·  (31) idt idt jk jk kd kd ij ij subject to : P PD PD P − TQ ≥ (−cap −  ) · X ∀ j, k, t (32) ijkt jk jkt jk i=1 D DD DD D − TQ ≥ (−cap −  ) · X ∀ k, d, t (33) ikdt kd kdt kd i=1 TQ ≥ De −  ∀ i, d, t (34) idt ikdt idt k=1 P P − P ≥ (−cap −  ) · w ∀ i, j, t (35) ijt ijt ij ij PD PD PD ≤ cap − cap ∀ j, k (36) jk jk jk DD DD DD ≤ cap − cap ∀ k, d (37) kd kd kd ≤ De − De ∀ i, d, t (38) idt idt idt P P P ≤ cap − cap ∀ i, j (39) ij ij ij PD ≥ 0 ∀ j, k (40) jk DD ≥ 0 ∀ k, d (41) kd ≥ 0 ∀ i, d, t (42) idt ≥ 0 ∀ i, j (43) ij max ∗ The values of Z and ∇ are calculated using models (44)–(56). max PD PD DD DD P P min Z = objective function + C ·∇ + C ·∇ + C ·∇ + C ·∇ (44) idt idt jk jk kd kd ij ij subject to : P PD PD P − TQ ≥ (−cap −∇ ) · X ∀ j, k, t (45) ijkt jk jkt jk i=1 274 S. J. -SHARAHI ET AL. D DD DD D − TQ ≥ (−cap −∇ ) · X ∀ k, d, t (46) ikdt kd kdt kd i=1 D ∇ TQ ≥ De −∇ ∀ i, d, t (47) idt ikdt idt k=1 P P − P ≥ (−cap −∇ ) · w ∀ i, j, t (48) ijt ijt ij ij ∇ ∇ PD PD PD ∇ ≤ cap − cap ∀ j, k (49) jk jk jk ∇ ∇ DD DD DD ∇ ≤ cap − cap ∀ k, d (50) kd kd kd ∇ ≤ De − De ∀ i, d, t (51) idt idt idt ∇ ∇ P P P ∇ ≤ cap − cap ∀ i, j (52) ij ij ij PD ∇ ≥ 0 ∀ j, k (53) jk DD ∇ ≥ 0 ∀ k, d (54) kd ∇ ≥ 0 ∀ i, d, t (55) idt ∇ ≥ 0 ∀ i, j (56) ij It is notable that the crisp constraints (2)–(5), (8)–(9), (12)–(16) and (17)–(25) should also be included in both models (31)–(43), and (44)–(56). Therefore, a best fuzzy set embedded on foot print of uncertainty is calculated based on Equations (57)–(60). min  ∗ max ∇ ∗ PD PD PD PD PD PD cap = cap +  , cap = cap +∇ ∀ j, k (57) jk jk jk jk jk jk min  ∗ max ∇ ∗ DD DD DD DD DD DD cap = cap +  , cap = cap +∇ ∀ k, d (58) kd kd kd kd kd kd max  ∗ min ∇ ∗ De = De −  , De = De −∇ ∀ i, d, t (59) idt idt idt idt idt idt min  ∗ max ∇ ∗ P P P P P P cap = cap +  , cap = cap +∇ ∀ i, j (60) ij ij ij ij ij ij Then optimum value of λ is calculated using the models (61)–(66). Max λ subject to : (61) max min max objective function + λ(Z − Z ) = Z max min max P PD PD PD P − TQ − λ(cap − cap ) ≥−cap · X ∀ j, k, t (62) ijkt jk jk jk jkt i=1 FUZZY INFORMATION AND ENGINEERING 275 max min max D DD DD DD D − TQ − λ(cap − cap ) ≥−cap · X ∀ k, d, t (63) ikdt kd kd kd kdt i=1 D max min min TQ − λ(De − De ) ≥ De ∀ i, d, t (64) ikdt idt idt idt k=1 max min max P P P − P − λ(cap − cap ) ≥−cap · w ∀ i, j, t (65) ijt ijt ij ij ij λ ≥ 0 (66) Again, the crisp constraints (2)–(5), (8)–(9), (12)–(16) and (17)–(25) should also be included in models (61)–(66). 4.2. Linearisation approach One of the principles of the method proposed by Figueroa [48] is that the resultant auxil- iary programming models are linear. In this section, a linearisation approach is proposed to overcome the shortage of the non-linear programming. The models (31)–(43), and the min max models (44)–(56) which are used to calculate Z ,and Z , respectively, are non-linear mixed-integer programming models. The proposed approach in this section transforms them to linear programming (LP) problems. The global optimum solutions can be found easily for LPs. Both models (31)–(43), (44)–(56) have non-linear terms in which the products of two binary and continuous variables are incorporated. This type of non-linearity can be removed as follows. Let x be a binary variable, and x be a continuous variable, and 0 < x <u. 1 2 2 A continuous variable, y, is introduced to replace the product y = x x and the con- 1 2 straints (67)–(70) are to be added to the associated optimisation model. y ≤ ux (67) y ≤ x (68) y ≥ x − u(1 − x ) (69) 2 1 y ≥ 0 (70) The equations (32)–(33) and (35) are non-linear constraints, and are replaced by Equa- P D tions (71)–(85). The new continuous variables F , F ,F are defined for optimisation of ijt jkt kdt min Z . P PD P P TQ ≤ cap · X + F ∀ j, k, t (71) ijkt jkt jkt jk i=1 P PD PD P F ≤ (cap − cap )·X ∀ j, k, t (72) jkt jk jkt jk P PD F ≤  ∀ j, k, t (73) jkt jk 276 S. J. -SHARAHI ET AL. P PD PD PD P F ≥  − (cap − cap )·(1 − X ) ∀ j, k, t (74) jkt jk jk jkt jk F ≥ 0 ∀ j, k, t (75) jkt D DD D D TQ ≤ cap · X + F ∀ k, d, t (76) ikdt kdt kdt kd i=1 D DD DD D F ≤ (cap − cap ) · X ∀ k, d, t (77) kdt kd kdt kd D DD F ≤  ∀ k, d, t (78) kdt kd D DD DD DD D F ≥  − (cap − cap ) · (1 − X ) ∀ k, d, t (79) kdt kd kd kdt kd F ≥ 0 ∀ k, d, t (80) kdt P ≤ cap · w + F ∀ i, j, t (81) ijt ijt ijt ij P P F ≤ (cap − cap ) · W ∀ i, j, t (82) ijt ijt ij ij F ≤  ∀ i, j, t (83) ijt ij P P P F ≥  − (cap − cap ) · (1 − W ) ∀ i, j, t (84) ijt ijt ij ij ij F ≥ 0 ∀ i, j, t (85) ijt max Alternatively, in the process of optimisation of Z , Equations (45)–(46) and (48) are non-linear. They are replaced by Equations (86)–(100). The new continuous variables O , jkt O , O are also defined. ijt kdt P PD P P TQ ≤ cap · X + O ∀ j, k, t (86) ijkt jkt jkt jk i=1 ∇ ∇ P PD PD P O ≤ (cap − cap )·X ∀ j, k, t (87) jkt jk jkt jk P PD O ≤∇ ∀ j, k, t (88) jkt jk ∇ ∇ P PD PD PD P O ≥∇ − (cap − cap )·(1 − X ) ∀ j, k, t (89) jkt jk jk jkt jk O ≥ 0 ∀ j, k, t (90) jkt D DD D D TQ ≤ cap · X + O ∀ k, d, t (91) ikdt kdt kdt kd i=1 FUZZY INFORMATION AND ENGINEERING 277 ∇ ∇ D DD DD D O ≤ (cap − cap ) · X ∀ k, d, t (92) kdt kd kdt kd D DD O ≤∇ ∀ k, d, t (93) kdt kd ∇ ∇ D DD DD DD D O ≥∇ − (cap − cap ) · (1 − X ) ∀ k, d, t (94) kdt kd kd kdt kd O ≥ 0 ∀ k, d, t (95) kdt P ≤ cap · w + O ∀ i, j, t (96) ijt ijt ijt ij ∇ ∇ P P O ≤ (cap − cap ) · W ∀ i, j, t (97) ijt ijt ij ij O ≤∇ ∀ i, j, t (98) ijt ij ∇ ∇ P P P O ≥∇ − (cap − cap ) · (1 − W ) ∀ i, j, t (99) ijt ijt ij ij ij O ≥ 0 ∀ i, j, t (100) ijt 5. Numerical Example To demonstrate the applicability and usability of the proposed model, a numerical example with six manufacturing plants, six possible distribution centers and eight customer zones are considered. Each plant can produce six types of products and planning horizon is 2 months. The model described above is coded in LINGO to solve the MILP problem. 5.1. Instance Descriptions The working hour is equal to 20 per plant for a day. The maintenance lasts for 2 hours in a working day except for plants 5 and 6 in which the maintenance time is equal to 1hour. The change-over time is 1 hour. The utilisation parameter is assumed to be 50 hours per month. All other parameters of numerical example are given in Tables 4–11. These parameters are repeated for all planning periods. PD Capacity of connection between plants and distribution centers (cap )and capacity DD between distribution centers and customers (cap ) in kilogram are IT2F set according to Table 4. The detail information of fuzzy type-II production capacities for all plants are presented in Table 5. Table 6 presents the rate of production for each product in each plant. Table 7 presents the production cost for each product in each plant. Table 4. Fuzzy rate of flow (kg). Capacity cap cap cap cap PD cap 11000 25000 12000 26000 DD cap 26000 28000 27000 30000 278 S. J. -SHARAHI ET AL. Table 8 presents the transportation cost for each product from plant to distribution center. Fixed establishing, shut down, and transportation costs are presented in Table 9. The procurement costs and the customer demands are presented in Tables 10 and 11, respectively. Table 5. Fuzzy type-II production capacity (kg/month). Plant Product PL 1 PL 2 PL 3 PL 4 PL 5 PL 6 cap P1 73400 12700 35100 34400 11000 10000 P2 17616 3048 8424 8256 2640 2400 P3 88080 15240 42120 41280 13200 12000 P4 36700 6350 17550 17200 5500 5000 P5 58720 10160 28080 27520 8800 8000 P6 31562 5461 15093 14792 4730 4300 cap P1 69618 12049 33303 32633 10376 9539 P2 16708 2892 7993 7832 2490 2289 P3 83541 14459 39963 39160 12451 11447 P4 34809 6025 16651 16317 5188 4769 P5 55694 9639 26642 26107 8301 7631 P6 29260 5064 13997 13716 4361 4009 cap P1 31511 5454 15074 14771 4696 4318 P2 7563 1309 3618 3545 1127 1036 P3 37813 6545 18089 17725 5636 5181 P4 15756 2727 7537 7385 2348 2159 P5 25209 4363 12059 11817 3757 3454 P6 13550 2345 6482 6351 2019 1857 cap P1 29965 5186 14334 14046 4466 4106 P2 7192 1245 3440 3371 1072 985 P3 35958 6223 17201 16855 5359 4927 P4 14982 2593 7167 7023 2233 2053 P5 23972 4149 11467 11237 3573 3285 P6 12885 2230 6164 6040 1920 1765 Table 6. Rate of production (kg/hour). Plant PL1 PL2 PL3 PL4 PL5 PL6 P1 3660 630 1770 1710 540 510 P2 870 150 420 420 120 120 P3 4410 750 2100 2070 660 600 P4 1830 330 870 870 270 240 P5 2940 510 1410 1380 450 390 P6 1590 270 750 750 240 210 Table 7. Production cost (money unit/kg). Plant Product PL1 PL2 PL3 PL4 PL5 PL6 P1 3.18 3.03 3.67 3.54 2.77 2.76 P2 1.73 1.67 1.71 2.37 1.62 2.47 P3 2.68 2.69 3.6 2.85 2.21 2.20 P4 0.98 1.21 1.04 1.24 0.93 0.92 P5 0.43 0.49 0.55 0.41 0.28 0.27 P6 0.65 0.65 0.72 0.90 0.57 0.56 FUZZY INFORMATION AND ENGINEERING 279 Table 8. Transportation cost from plant to distribution center (money unit/kg). Distribution Center Plant Product DC 1 DC 2 DC 3 DC 4 DC 5 DC 6 PL 1 P1–P6 0.001 0.062 0.081 0.053 0.051 0.056 P5 0.003 0.071 0.091 0.061 0.059 0.051 PL 2 P1–P6 0.065 0.0890 0.127 0.077 0.071 0.077 P5 0.074 0.093 0.139 0.083 0.081 0.083 PL3 P1–P6 0.083 0.126 0.001 0.141 0.125 0.119 P5 0.091 0.138 0.003 0.153 0.132 0.122 PL 4 P1–P6 0.107 0.138 0.182 0.001 0.143 0.144 P5 0.112 0.152 0.194 0.003 0.162 0.164 PL5 P1–P6 0.183 0.236 0.117 0.038 0.001 0.038 P5 0.193 0.248 0.121 0.052 0.003 0.052 PL 6 P1–P6 0.168 0.178 0.236 0.117 0.038 0.001 P5 0.182 0.191 0.258 0.119 0.052 0.003 Table 9. Fixed infrastructure cost and transportation cost. Distribution Center DC 1 DC 2 DC 3 DC 4 DC 5 DC 6 Fixed establishing and shutting down cost (money unit) Establish 4300 2900 3100 2200 1300 1500 Shot Down 3100 1500 1900 3000 2200 2000 Customer Product Transportation cost (money unit/kg) C1 P1–P6 0.07 0.068 0.077 0.076 0.064 0.064 C2 P1–P6 0.048 0.048 0.048 0.058 0.047 0.060 C3 P1–P6 0.063 0.063 0.077 0.065 0.056 0.056 C4 P1–P6 0.037 0.041 0.038 0.041 0.036 0.036 C5 P1–P6 0.021 0.022 0.024 0.021 0.019 0.019 C6 P1–P6 0.032 0.032 0.033 0.036 0.031 0.031 C7 P1–P6 0.06 0.058 0.067 0.066 0.054 0.051 C8 P1–P6 0.027 0.031 0.028 0.031 0.026 0.026 Table 10. Procurement cost per product (money unit/kg). Product P1P2P3P4P5P6 3.41 2.23 3.18 1.00 0.93 0.80 PD DD P Incremental costs C ,C ,C and C are set as 0.04, 0.1, 3 and 1.7, respectively. 5.2. Results Using the parameters presented in Section 5.1, and Equations (31), (34), (36)–(43) and (71)–(85) for step 1 and Equation (44), (47), (49)–(56) and (86)–(100) for step 2, the max min optimal solutions are respectively Z = 2404051, and Z = 914544.5. The optimal sat- isfaction degree in the last step is λ = 0.537, so the optimal value of objective function is Z = 1604404.669. The graphical representation of fuzzy objective function is shown in Figure 3. Table 12 shows a detailed report of fuzzy objective function. 6. Conclusion and future research directions This research proposed a new uncertain integrated approach to determine the optimal issues related to the design and operation of supply chains. The main decisions made in 280 S. J. -SHARAHI ET AL. Table 11. Fuzzy customer demand (kg). Customer Zone Product C1C2C3C4 C5 C6C7C8 De P1 32329 20783 19269 15231 7610 13099 13814 10315 P2 7784 1343 5400 4985 3348 2318 3859 2752 P3 52796 14427 22267 22006 9905 6669 19827 11044 P4 19736 2255 23053 3562 4250 5351 1093 6925 P5 34745 4123 21547 8698 7490 6509 14886 7962 P6 6498 1414 6873 25924 2873 2431 3801 7139 De P1 30712 19744 18306 14470 7230 12444 13123 9799 P2 7395 1276 5130 4735 3180 2202 3666 2614 P3 50156 13706 21153 20906 9410 6335 18835 10492 P4 18749 2142 21901 3384 4037 5083 1038 6579 P5 33007 3916 20469 8263 7116 6184 14142 7564 P6 6173 1344 6530 24628 2729 2310 3611 6782 De P1 13901 8937 8286 6550 3272 5633 5940 4435 P2 3347 577 2322 2143 1439 997 1659 1183 P3 22702 6204 9575 9463 4259 2867 8526 4749 P4 8486 970 9913 1532 1827 2301 470 2978 P5 14940 1773 9265 3740 3221 2799 6401 3424 P6 2794 608 2956 11147 1235 1045 1635 3070 De P1 13219 8498 7879 6228 3112 5356 5648 4218 P2 3183 549 2208 2038 1369 948 1578 1125 P3 21588 5899 9105 8998 4050 2727 8107 4516 P4 8070 922 9427 1456 1738 2188 447 2832 P5 14207 1686 8810 3557 3063 2662 6087 3256 P6 2657 578 2810 10600 1175 994 1554 2919 Table 12. Fuzzy objective function. Fuzzy Parameters Z PD DD P De cap cap cap Z = 2417000 ∇ ∇ ∇ ∇ PD DD P De cap cap cap Z = 916362.2 PD DD P ∇ De cap cap cap Z = 2280701 ∇ ∇ ∇ PD DD P De cap cap cap Z = 866484.1 the proposed approach concerned location–allocation, production, procurement and dis- tribution of products based on financial aspects and production balancing among plants in multi-product, and multi-period supply chain networks. The fuzzy model considered three source of uncertainty in demand, manufacturing and supply process, concurrently. The Figure 3. Fuzzy Type-II set of objective function. FUZZY INFORMATION AND ENGINEERING 281 uncertainties in this research were modelled through type-II fuzzy sets in which a deeper insight towards vagueness and ambiguity in fuzzy membership values was supplied. The whole approach was modelled through a fuzzy mixed-integer mathematical programming. A fuzzy type-reducer methodology was proposed in order reduce fuzzy type-II uncertain- ties to a type-I fuzziness and to solve the yielded fuzzy type-I mathematical model through existing methods. A numerical example was supplied to address the mechanism of proposed approach and to illustrate its applicability. Some further research directions are proposed based on findings of this research as follows: (1) RHS of constrain were considered type-II fuzzy numbers in this research, devel- opment of a model with fuzzy type-II objective function or fuzzy type-II decision variables can demonstrate new insights in real supply chains, (2) as the location–allocation, produc- tion, procurement and distribution problems in supply chains are usually assumed to be non-deterministic poly-nominal hard (NP-Hard) problems, the application of soft comput- ing techniques or meta-heuristics could be suitable in order to handle large-scale and real life problems, (3) the proposed method can be applied and tested in real case study in order to check its applicability in real-world problems and (4) although the proposed approach models some kind of uncertainty which have never been discussed before, some compari- son with existing methods in the literature may be interesting and reveals the advantages of the proposed approach clearly. Disclosure statement No potential conflict of interest was reported by the authors. Funding This research has been partially supported by Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran. ORCID Alireza Rashidi-Komijan http://orcid.org/0000-0001-7705-980X References [1] Christopher M. Logistics and supply chain management. 2nd ed. London: Pitman; 1998. [2] Lambert DM, Stock JR, Ellram LM. Fundamentals of logistics management. Boston: Irwin/Mc Graw-Hill Publishing; 1998. [3] Melo MT, Nickel S, Saldanha-da-Gama F. Facility location and supply chain management: a review. Eur J Oper Res. 2009;196(2):401–412. [4] Peidro D, Mula J, Poler R, et al. Quantitative models for supply chain planning under uncertainty: a review. Int J Adv Manuf Tech. 2009;43(3):400–420. [5] Peidro D, Mula J, Jimenez M, et al. A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment. Eur J Oper Res. 2010;205(1):65–80. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Apr 3, 2018

Keywords: Type-II fuzzy integer linear programming; fuzzy type-II reducer; location–allocation problem; production–distribution problem; supply chain planning

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