A hybrid model for remanufacturing facility location problem in a closed-loop supply chain

Abstract

Traditional supply chain design is merely based on the open loop or forward flow of materials, neglecting reverse flow for recovery of materials despite the recent concerns of customers and governments about environmental and production cost reductions. New supply chain design should be closed loop which implements traditional supply chain concepts with reverse flow or a material recovery system to reduce production cost and enhance customer satisfaction about environmental consciousness and to meet legal requirements. In our research, we designed a closed-loop supply chain which consists of recovery options such as collection centres and remanufacturing plants (reverse flow) in addition to traditional supply chain tiers (forward flow), and tried to find the best location for these facilities in a discrete space based on decision makers' opinions. Since there are uncertainties about decision parameters in an uncapacitated facility location problem, we implemented the fuzzy TOPSIS method to solve the location decision problem and find the best place to locate a remanufacturing facility.

Keywords:

• group decision making
• recovery facility
• remanufacturing plant
• fuzzy TOPSIS
• facility location

1. Introduction

During the supply chain revolution of the 1990s, it was defined that the efficient and effective movement of goods from raw material suppliers to processing facilities, component fabrication plants, finished goods assembly plants, distribution centres, retailers and customers usually take place in an open-loop supply chain which is the traditional or the generic configuration of the supply chain.

This traditional configuration is based on forward movements of materials from suppliers towards the final customer and backward flow of information about the market from customers to all tiers in the supply chain. This configuration is shown in Figure 1. In order to design a supply chain effectively, there are many problems and factors to be dealt with by designers. In a generic supply chain, there are many problems such as procurement, production, inventory management, distribution of goods, logistics or routing and, what may be the most crucial of them all, the facility location decision. The effective factors in designing a closed-loop supply chain are divided into two main categories: traditional and modern factors.

Figure 1 A generic supply chain with four tiers.

Among the traditional factors are cost of land, availability of materials, labour, logistics, competitiveness of tiers, warehouse capacity, demand for products and infrastructure (Dowlatshahi 2000). Some of the recent affecting factors that designers should consider as modern factors are customer consciousness about used product recycling, government legislation for used products returns and most importantly, decrease manufacturing cost of products (Jayaraman et al. 2003).

In fact during recent decades, manufacturers have sought ways to reduce production costs; meanwhile, their products were distributed and used by consumers until their end-of-life. These end-of-life products were left in the environment, in landfills for example, with associated environmental costs (Demirel and Gökçen 2008). The environmental consciousness of people all around the world has caused governments to introduce a legislation that makes manufacturers responsible for the collection of their used products for reuse in new products with an associated reduction of waste materials (Wang and Hsu 2010). These factors have led supply chain designers to a new concept in supply chain design which is material recovery or the reverse logistic system. This system, which includes the collection of the used products from customers, recovery of used products to make them reusable and distribution to customers, is often quite different from the traditional supply chain. The flow of materials and products in this new system occurs both from the customer to the remanufacturer (reverse flow) and from the remanufacturer to the customer (forward flow). Since most of the products and materials may be conserved, this forms essentially a closed-loop logistics system (Jayaraman et al. 1999). Actually, in this configuration which is shown in Figure 2, remanufacturing plants and collection centres are located in the supply chain to serve all the tiers not only to collect returned product at any level of production (from material in process, semi-finished goods or final product), but also products that are damaged, have any faults or malfunctions, or those that are discarded by consumers at end-of-life for cannibalisation.

Figure 2 New configuration to design a closed-loop supply chain.

The main purpose in closed-loop supply chain design is to find the best place to locate recovery facilities which is a new concept in the facility location decision problem. Since the recovery facility includes strategies to increase product life through repair, remanufacture and recycling, locating these facilities is the new challenge for supply chain designers meaning that the designer must consider all affecting factors in locating these facilities to minimise total cost. Although some of the concepts of reverse logistics, such as the recycling of products, have been put into practice for years, it is only fairly recently that the integration of reverse logistics activities has been a real concern for the management and organisation of logistics systems. There is, therefore, a need for research in this area and particularly in the planning and optimisation of logistics systems including reverse activities and specially locating new facilities for remanufacturing.

The remainder of this paper is organised as follows: Section 2 comprises a literature review on reverse logistics and the necessity for embedding recovery centres in a supply chain to constitute a closed-loop supply chain. Our methodology for locating a remanufacturing facility is explained in Section 3. The main problem of locating the new facility based on fuzzy TOPSIS is explained in Section 4 prior to drawing conclusions on the method and providing suggestions for future research in Section 5.

2. Literature review

There are a lot of reverse logistics network configurations depending on the recovery options and the type of recovered products. However, only a few researchers have focused on the development of general frameworks and mathematical models for remanufacturing facilities. These models are essentially linear programming models, focusing on cost minimisation or profit maximisation. Mixed integer linear programming is used in many operations and manufacturing practices such as facility location decisions, trans-shipment, production and stocking of the optimal quantities of remanufactured products and cores. Jayaraman et al. (2003) state that the flow of materials can take place in both directions in forward and reverse logistics. They developed a mixed integer linear programming model that dealt with the location of new facilities and also trans-shipment, production and stocking of the optimal quantities of remanufactured products and cores. Lu and Bostel (2007) considered remanufacturing, based on benefits and possibility, as the main option of a recovery system; at the same time they included other characteristics of a closed-loop supply chain such as reuse, repair, refurbishing, cannibalisation and recycling. In their model, since recovered products are introduced as new products, interactions between forward and reverse flows were considered simultaneously.

Since the efficiency of reverse logistics mainly depends on facility location decisions, Zhang and Xu (2007) formulated the facility location problem so as to minimise the sum of transportation and operation costs. They proposed a mixed integer programming model for the facility location problem. They considered the demand and capacities as uncertain parameters, so their model can be considered as a hybrid. Focusing on remanufacture, reuse and recycle of product (3Rs) operation concepts in a closed-loop supply chain, Wang and Hsu (2010) also tried to optimise logistics and location selection. They divided the closed-loop logistics model into two parts: forward logistics, which delivers the final product to demand points or customers, and reverse logistics, which is the flow of used products for recovery. This system ensures minimum waste of materials. They also considered distribution centres as collection centres and suggested that these distribution/collection centres have flexible capacities for distribution or collection of products.

Demirel and Gökçen (2008) emphasised reverse logistics as a strategy for firms to make profit under competitive circumstances and dealt with the retrieval of products from customers in a closed-loop supply chain. They also considered the concept of a recovery network for the collection of used products to return them into reusable condition and distribution to customers. In their research, recovery was accomplished through repairing, refurbishing, remanufacturing, cannibalisation and recycling. Finally, they provided a mixed integer linear programming model for remanufacturing in reverse logistics (Demirel and Gökçen 2008). Although Demirel and Gökçen (2008) used numerical data in their research to test their model, Schultmann et al. (2006) used an automotive industry example to model the reverse logistics tasks in a closed-loop supply chain considering the end-of-life vehicle treatment in Germany. They suggested that there are two main distinct motivating factors to integrate the product life phase into an existing supply chain; legislation and profit, for supply chain tiers. In spite of the previous research which considered product return as either a waste stream system or a market-driven system, they considered legislation and profit as motivators for recycling or reuse of end-of-life products. Accordingly, the establishment of a recovery network which ensures efficient coverage of the demand and supply area is a crucial task for supply chain network designers for providing a free of charge return of products to demand points. Realising recycling steps, they defined the objective function to minimise cost for establishing a product recovery network and to fulfil recycling goals (Schultmann et al. 2006). Zhang and Xu (2007) proposed a class of facility location model and its application. They developed a mixed integer programming model for a facility location problem, and also presented the expected value model of rough fuzzy mixed integer programming (Zhang and Xu 2007).

Bhatnagara and Sohalb (2005) suggested that supply chain performance is impacted by several factors beginning with the plant location decision. They emphasised traditional quantitative factors such as transport costs, exchange rates, labour rates and taxes. Although there are existing models that capture qualitative variables, there is limited research linking these variables with measures of the firm's operational competitiveness, hence they proposed a framework that includes qualitative factors concerning plant location decisions, supply chain uncertainty and manufacturing practices, and argued that a joint consideration of such factors helps explain supply chain competitiveness (Bhatnagara and Sohalb 2005). Ilgin and Gupta (2010) systematically investigated the literature by classifying over 540 published references into four major categories, namely, environmentally conscious product design, reverse and closed-loop supply chains, remanufacturing and disassembly. They concluded by summarising the evolution of environmentally conscious manufacturing and product recovery options over the past decade together with avenues for future research (Ilgin and Gupta 2010).

In some previous models, warehouses are considered as remanufacturing centres neglecting possible shortage of ample space for returned cores. In order to cover this aspect of reverse logistics system, Qin and Ji (2010) proposed a three-echelon reverse network with a public reverse distribution, in which the tasks of reverse logistics and pre-remanufacturing processes are transferred from retailers and manufacturers to a reverse distribution centre. For reducing the game relations of the alliance, only manufacturers and reverse distribution centres participate in making the pricing decision (Qin and Ji 2010). Kaya (2010) considered a manufacturer producing original products using virgin materials and remanufactured products using the returns from the market where the amount of returns depends on the incentive offered by the manufacturer. He determined the optimal value of this incentive and the optimal production quantities in a stochastic demand setting with partial substitution and also analysed three different models in centralised and decentralised settings where the collection process of the returns is managed by a collection agency in the decentralised setting. He analysed contracts to coordinate the decentralised systems and to determine the optimal contract parameters. Finally, he presented his computational study for the observation of the effect of different parameters on the system performance (Kaya 2010).

Peng and Zhong (2008) suggested an optimisation model for a closed-loop logistics network based on the 0–1 mixed integer linear programming approach. According to the model, the number of various facilities, their locations and the allocation of the corresponding logistics flows can be decided upon leading to the set-up of an appropriate closed-loop logistics network structure in order to minimise the total cost of investment and operation. This study contemplated the closed-loop logistics network with the single product, single period and facilities with limited capability location–allocation problem, and also considered the logistics distribution of remanufactured products and new products which cannot be replaced mutually in the forward logistic (Peng and Zhong 2008).

3. Methodology of decision making for locating remanufacturing facility

In this paper, we design a material recovery system, which includes product life expansion through repair, remanufacturing and recycling based on decision makers opinion which is in a fuzzy linguistic form. By incorporating a traditional supply chain, in which the flow of materials takes place in forward logistics and reverse logistics channels, a closed loop supply chain for material recovery will constitute. In this system, suppliers are divided into two categories: new material and used material suppliers. When components cannot be supplied by recovered materials, new materials are supplied by suppliers. Here, customers are the suppliers of used products and the demand point for new products simultaneously. Extension of the product life cycle is accomplished by remanufacturing and repair systems which are the main parts of the recovery system in a closed-loop supply chain.

In a centralised supply chain, strategic decisions are usually made by all tiers. Among these strategic decisions is the location of new facilities by a supply chain member, or facility location to serve all the supply chain in an effective and efficient mode, in order to increase the profit or decrease the costs of the tiers while simultaneously reducing the cost of products for customers and increasing their satisfaction. Since there are many factors in a facility location decision, and the designer tries to choose the best place to locate the facility based on decision makers' views and the attributes of supply chain tiers to suit all criteria, this problem can be defined as a multiple-criteria decision-making (MCDM) problem, and TOPSIS, which was first initiated by Hwang and Yoon (1981), can be implemented for decision making. This technique is based on the concept that the ideal alternative has the best level for all attributes considered, whereas the negative-ideal is the one with all the worst attribute values. TOPSIS defines solutions as the points that are simultaneously farthest from the negative-ideal point and closest to the ideal point, and is successfully applied to selecting the location of a manufacturing plant (Yoon and Hwang 1985, Yong 2006).

In the process of TOPSIS, the performance ratings and the weights of the criteria are given as crisp values. Thus, the measurement of weights and qualitative attributes does not consider the uncertainty associated with the mapping of human perception on a number. However, crisp data are inadequate to model real-life situations because the evaluation data of the facility location under different subjective attributes and the weights of the attributes are often expressed linguistically (Yoon and Hwang 1985, Yong 2006). Thus, a more realistic approach may be needed to use linguistic assessments instead of numerical values.

Recently, some researchers have focused on the fuzzy TOPSIS method to solve the plant location selection problem (Chen 2001, Chu 2002a, 2002b). However, existing fuzzy TOPSIS methods are not efficient enough due to the fact that to obtain the ‘fuzzy positive-ideal solution’ and the ‘fuzzy negative-ideal solution’, fuzzy ranking approaches are used, even though it is well known that no one can rank fuzzy numbers satisfactorily in all cases and situations (Yong 2006). In addition, it is tedious to calculate the distance from the ideal solution and the negative-ideal solution. To solve these problems in fuzzy decision making, a new fuzzy TOPSIS approach is proposed. In this method, the ratings of each alternative and the weights of each criterion can be represented as triangular fuzzy numbers. The results of multiplication of rating and weight can be calculated as crisp numbers by the canonical representation of a multiplication operation on triangular fuzzy numbers. Then, the ‘fuzzy positive-ideal solution’ and ‘fuzzy negative-ideal solution’ can be determined easily without ranking fuzzy numbers (Yoon and Hwang 1985, Yong 2006). In addition, the distance from the ideal solution and the negative ideal solution can be calculated easily, which makes the proposed method more efficient than existing methods (Yoon and Hwang 1985, Yong 2006).

4. Where to locate a new remanufacturing facility?

Since facility location is a strategic decision (Meng et al. 2009), in order to locate a new facility to form a closed-loop supply chain, which can support all tiers and be supported by them reciprocally, the tiers must provide their opinions about the location of a new facility. Strategic decision means that senior managers will make the location decision as decision makers for locating a new facility in order to remanufacture returned products from demand areas. These decision makers are representatives of suppliers, manufacturing plants, distribution centres and customers which are D ₁, D ₂, D ₃ and D ₄, respectively. Initial screening performed by location decision makers shows that four possible sites A ₁, A ₂, A ₃ and A ₄ are feasible candidates to choose based on the crucial criteria listed as closeness to manufacturers, availability of machines and tools, closeness to customers or demand points for returned cores and the availability of proper logistics which are C ₁, C ₂, C ₃ and C ₄, respectively. In order to deal with poor definition of criteria, linguistic variables are used which are in the form of words and sentences to evaluate all the situations. These linguistic variables are then changed to fuzzy numbers.

The decision makers have weighted criteria based on their positions in the supply chain hierarchy, as shown in Table 1.

Table 1 Criteria weights.

	D 1	D 2	D 3	D 4
C 1	L	L	H	M
C 2	VH	H	M	M
C 3	M	H	M	H
C 4	VH	VH	H	M

Table 2 shows the rating of four linguistic criteria for each candidate location based on the decision makers' opinions.

Table 2 Decision makers' ratings.

Criteria	Candidate	D 1	D 2	D 3	D 4
C 1	A 1	VG	G	G	VG
	A 2	G	G	P	G
	A 3	G	VG	G	P
	A 4	P	P	VG	VG
C 2	A 1	G	VG	G	P
	A 2	VG	G	P	VG
	A 3	G	P	G	P
	A 4	G	VG	VG	G
C 3	A 1	F	P	G	VG
	A 2	G	G	F	F
	A 3	P	F	G	G
	A 4	VP	G	F	G
C 4	A 1	VP	G	F	VG
	A 2	P	VG	G	G
	A 3	F	G	P	F
	A 4	G	P	F	P

Tables 1 and 2 illustrate that all factors are in linguistic format, and it is necessary to change them to crisp numbers for ordinary operations. As mentioned before, there is an ambiguity associated with each factor and criterion; so in order to deal with this vagueness, we use fuzzy sets and provide some definition of fuzzy sets and numbers (Kauffman and Gupta 1985, Zimmermann 1991).

Definition 1

Let 𝒳 be a universe of discourse: where à is a fuzzy subset of 𝒳; and for all , there is a number which is assigned to represent the membership of x in Ã, and is called the membership of à (Kauffman and Gupta 1985, Zimmermann 1991).

Definition 2

A triangular fuzzy number à can be defined by a triplet which means (Kauffman and Gupta 1985, Zimmermann 1991).

Based on Definitions 1 and 2 and also normal scale in TOPSIS in which linguistic factor ‘low’ is ‘0’ and ‘very high’ is ‘10’, Tables 3 and 4 are defined as combinations of these definitions to change each linguistic variable of weights to fuzzy numbers, as in Table 3, and decision makers' linguistic ratings as in Table 4.

Table 3 Fuzzy linguistic for weights.

Very low (VL)	(0.0, 0.1, 0.3)
Low (L)	(0.1, 0.3, 0.5)
Medium (M)	(0.3, 0.5, 0.7)
High (H)	(0.5, 0.7, 0.9)
Very high (VH)	(0.7, 0.9, 1.0)

Table 4 Fuzzy linguistic for decision makers' ratings.

Very poor (VP)	(0, 1, 3)
Poor (P)	(1, 3, 5)
Fair (F)	(3, 5, 7)
Good (G)	(5, 7, 9)
Very good (VG)	(7, 9, 10)

As shown in Tables 3 and 4, all the numbers are positive triangular fuzzy numbers, and in order to retrieve the graded mean integration representation of triangular fuzzy number, Definition 3 is used.

Definition 3

Given a triangular fuzzy number , the graded mean integration representation of triangular fuzzy number à is defined as (Kauffman and Gupta 1985, Zimmermann 1991) .

Definition 4

Let and be two triangular fuzzy numbers. The representation of the addition operation ⊕ on triangular fuzzy numbers à and can be defined as (Kauffman and Gupta 1985, Zimmermann 1991)  = .

Definition 5

The canonical representation of the multiplication operation on triangular fuzzy numbers à and is defined as (Kauffman and Gupta 1985, Zimmermann 1991)  = .

In order to reduce the complexity of fuzzy calculations, the above definitions are crucial, yielding a crisp number instead of a fuzzy one which eases the comparison of alternatives (Liang and Wang 1993, Yong 2006).

By applying Definition 3, the graded mean integration representation of the importance of weights and decision makers' rating will be crisp values as shown in Tables 5 and 6, respectively (based on the method used by Hsu and Chen (2000) and Definition 3 mentioned in Zimmermann (1991), in which triangular fuzzy numbers range fall into the range [0,1]).

Table 5 Graded mean of weights.

Very low (VL)	0.1167
Low (L)	0.3
Medium (M)	0.5
High (H)	0.7
Very high (VH)	0.8834

Table 6 Graded means of decision makers' rates.

Very poor (VP)	1.167
Poor (P)	3.000
Fair (F)	5.000
Good (G)	7.000
Very good (VG)	8.834

In this section, the decision matrix of alternative ratings based on new hybrid TOPSIS with fuzzy triangular linguistic variables is constructed. The method used here is the same as the usual TOPSIS method which can be found in the literature, for example, Liang and Wang (1993) and Yong (2006) except for the determination of positive and negative ideal solutions for fixed cost which is based on the method of Hwang and Yoon (1981) and Hsu and Chen (2000).

Steps for decision making are shown below.

Step 1. Aggregation of importance of weights. This step is summarised in Table 7 and is called the aggregated importance weight W _j .

Table 7 Aggregated weights W _j .

C 1	0.45
C 2	0.6459
C 3	0.6
C 4	0.7417

The final weighted matrix based on criterion (m) and decision maker (n) is W _j as

Step 2. Aggregation of rates assigned by decision makers. On the basis of Tables 2 and 6, the result of aggregation of rates based on graded mean integration is a (4 × 4) matrix called a decision matrix R _ij and the final result of the operation is

Step 3. Construction of the weighted decision matrix and its normalisation.

The decision matrix S _ij is a 4 × 4 matrix as in which operation ⊗ is as Definition 5. By application of this operation, the complicated fuzzy interval operations on membership functions were converted to crisp arithmetic (Hsu and Chen 2000, Yong 2006). The resultant matrix S _ij is

Step 4. Embedding fixed cost factor in decision matrix.

Since our model deals with uncapacitated facility location, usually there is a fixed cost in the model which is representative of the cost of land, construction of buildings or any other fixed cost related to locating a facility (Dowlatshahi 2000, Meng et al. 2009). This fixed cost, assessed by decision makers, is not crisp, but is a fuzzy number with equal cores for all candidate locations here. These fuzzy numbers are

• A ₁ = (62, 72, 80),

• A ₂ = (65, 72, 81),

• A ₃ = (63, 72, 79) and

• A ₄ = (64, 72, 80).

In order to add these numbers to the decision matrix, they must be rated (Liang and Wang 1993, Yong 2006). Based on definition and operations in Zimmermann (1991), Liang and Wang (1993) and Yong (2006), because they are categorised as fixed cost, so their rating becomes

• RA₁ = 0.8699,

• RA₂ = 0.861,

• RA₃ = 0.8689 and

• RA₄ = 0.8647.

Now the decision matrix R _ij will have five columns, representing C ₁, C ₂, C ₃, C ₄ and C ₅.

Step 5. Normalisation of the decision matrix. In TOPSIS, the normalisation of the decision matrix is accomplished as matrix (Hwang and Yoon 1981), where

By applying (1), the decision matrix will convert to a normalised matrix V _ij , in which all the elements are in the range [0,1] as below

Step 6. Determination of the positive (A ⁺) and negative (A ⁻ ) ideal solution. In the positive ideal solution and in the negative ideal solution, maximum and minimum variables, respectively, of the normalised decision matrix for each candidate location will be chosen (Hwang and Yoon 1981, Liang and Wang 1993, Yong 2006). The result is

• (A ⁺) = (0.5574, 0.6927, 0.5812, 0.6391, 0.862) and

• (A ⁻ ) = (0.4524, 0.3421, 0.4022, 0.3447, 0.8699).

Note: Because C ₅ is cost and usually firms try to eliminate any costs, even fixed costs (Hwang and Yoon 1981, Hsu and Chen 2000, Drezner and Hamacher 2004); here, we assume it to be a negative criterion, and based on Hwang and Yoon (1981), in the positive and negative ideal solution, the minimum and maximum value is considered, respectively. In the research by Yong (2006), it is not considered, so causing deviation of the best alternative option.

Step 7. Finding distance of each decision variable to ideal solution (Hwang and Yoon 1981, Liang and Wang 1993, Yong 2006). By definition, this distance is known in the literature as Euclidean distance (Drezner and Hamacher 2004). The Euclidean distance of each alternative to the positive ideal solution and negative ideal solution is shown in Table 8.

Table 8 Euclidean distance of each alternative.

	d+	d −
A 1	0.5027	0.0001
A 2	0.1780	0.3931
A 3	0.3984	0.1533
A 4	0.1487	0.4153

Step 8. Calculate closeness coefficient of each alternative. In order to specify which alternative best suits the decision makers' criteria, the closeness coefficient is defined as

The closeness coefficient measures how far an alternative is from its best and ideal solution. The best alternative is the one with biggest closeness coefficient. The closeness coefficient for alternatives is

• CCA₁ = 0.0001,

• CCA₂ = 0.6883,

• CCA₃ = 0.2778 and

• CCA₄ = 0.7362.

The best alternative for the location of the remanufacturing plant, with specified criteria based on decision makers' fuzzy linguistic ratings, is the one with the maximum closeness coefficient, which is A₄.

5. Discussion and conclusion

Deciding where to locate a new facility in a supply chain is an old problem in operations research, but the design of a closed-loop supply chain is a new concept. Since there are many reasons for 3Rs, such as customers' consciousness and governments regulations, it seems that the requirement to locate new facilities to implement these 3Rs is inevitable. By the implementation of recovery concepts in a supply chain, both manufacturers and customers will be beneficiaries. In order to collect and recover used products, it is necessary to establish new facilities as collection centres and remanufacturing plants. Locating a new facility is a strategic decision which means that once the new facility is located, it cannot be modified in a short period. Strategic decisions are usually made at the top managers' level of organisations with different expertise and diversity in points of view. In their points of view, the importance of a location factor, the rating and the weights assigned to them are diverse. This aspect has defined facility location as a MCDM with uncertainty about some important factors.

In this paper, we used the fuzzy TOPSIS method to determine the location of a remanufacturing facility and designed a closed-loop supply chain. Since uncapacitated facility location decisions have a fixed cost in objective function, this concept was considered as fuzzy numbers along with other criteria chosen by decision makers. The fuzzy linguistic method implemented reduces the complexity of the fuzzy operation and enables decision makers to deal with ambiguity factors quickly and easily, and determine where to locate a new plant based on fuzzy linguistic decision makers' ratings. Previous research considered mixed integer linear models in operations research to find the best place to locate a new facility, but these models eliminate uncertainty in parameters such as demand or cost. Since many countries and customers are concerned about expansion of the product life cycle, it seems that this subject will be vital for manufacturers as a way to reduce cost and also satisfy customers and regulations about the product life cycle. Future research should include the uncertainty of factors in decision making and build a mixed integer linear model with hybrid variables. The concept of rough and fuzzy variables can be implemented simultaneously to cope with the uncertainty of parameters.