Int. J. Industrial and Systems Engineering, Vol. 13, No. 1, 2013 27

Copyright © 2013 Inderscience Enterprises Ltd.

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system having uncertain parameters

Hamed Fazlollahtabar* Department of Industrial Engineering, Mazandaran University of Science and Technology, P.O. Box 734, Babol, Iran E-mail: hamed@ustmb.ac.ir *Corresponding author

Nezam Mahdavi-Amiri Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran E-mail: nezamm@sharif.edu

Abstract: We propose an approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost. With rise in demands, advancement in technology and increase in production capacity, the need for more shops persists. Therefore, a flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications. A network is configured in which the nodes are considered to be the shops with arcs representing the paths among the shops. An automated guided vehicle functions as a material handling device through the manufacturing network. To account for uncertainty, we consider time to be a triangular fuzzy number and apply an expert system to infer cost. The objective is to find a path minimising both the time and cost criteria, aggregately. Since time and cost have different scales, a normalisation procedure is proposed to remove the scales. The model being bi-objective, the analytical hierarchy process weighing method is applied to construct a single objective. Finally, a dynamic programming approach is presented for computing a shortest path in the network. The efficiency of the proposed approach is illustrated by a numerical example.

Keywords: flexible jobshop; AGV; automated guided vehicle; manufacturing systems; fuzzy systems.

Reference to this paper should be made as follows: Fazlollahtabar, H. and Mahdavi-Amiri, N. (2013) ‘An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system having uncertain parameters’, Int. J. Industrial and Systems Engineering, Vol. 13, No. 1, pp.27–55.

Biographical notes: Hamed Fazlollahtabar has been graduated in MSc in Industrial Engineering at Mazandaran University of Science and Technology, Babol, Iran. He received Doctorate awarded from the Gulf University of Science and Technology in Quantitative Approaches in Electronic Systems.

28 H. Fazlollahtabar and N. Mahdavi-Amiri

Currently he is doing PhD in Industrial Engineering at Iran University of Science and Technology, Tehran, Iran. He is in the editorial board member of WASET (World Academy of Science Engineering Technology) Scientific and Technical Committee on Natural and Applied Sciences, reviewer committee of International Conference on Industrial and Computer Engineering (CIE) and member of the International Institute of Informatics and Systemics (IIIS). He has become a member of Iran Elite Council. His research interests are optimisation in knowledge-based systems and automated manufacturing systems. He has published 75 research papers in international book chapters, journals and conferences.

Nezam Mahdavi-Amiri is a Full Professor of Mathematical Sciences at Sharif University of Technology. He received his PhD from the Johns Hopkins University in Mathematical Sciences. He is a Vice President of the Iranian Operations Research Society. He is in the editorial board member of several international mathematical and computing journals in Iran, including Bulletin of Iranian Mathematical Society, The CSI Journal of Computer Science and Engineering and Journal of Iranian Operations Research Society. His research interests include optimisation, numerical analysis and scientific computing, matrix computations, fuzzy modelling and computing.

1 Introduction

Discrete event systems are characterised by changes in state over time, based on current state and state transition rules, where each state is separated from its neighbour by a step rather than a continuum of intermediate infinitesimal states. Examples of such systems are information systems, operating systems, networking protocols, banking systems, business processes and telecommunications systems and flexible manufacturing systems (FMSs).

Traditional manufacturing has relied on dedicated mass-production systems to achieve high production volumes at low costs. As living standards improve and the demands for new consumer goods rise, manufacturing flexibility gains prominence as a strategic tool for rapidly changing markets. Flexibility, however, cannot be properly incorporated in the decision-making process if it is not well defined and measured in a quantitative manner. Today, manufacturing flexibility remains an elusive notion because of its inherent complexity and generality, in spite of a large body of published research work. There exist more than 50 definitions of Sethi and Sethi (1990) and six different approaches for obtaining a quantitative flexibility measure (Gupta and Goyal, 1989). Flexibility in its most rudimentary essence is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Buzacott, 1982; Gerwin, 1982; Zelenovic, 1982). A comprehensive classification of eight flexibility types was proposed in Browne et al. (1984). Resource and system flexibilities were examined in Slack (1987), whereas global measures for FMSs were defined in Gupta and Buzacott (1989). Based on information theoretic concepts, routing flexibility was examined in Yao and Pei (1990) and Kumar (1987). Flexibility measures for one machine, a group of machines and whole industry were presented in Brill and

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 29

Mandelbaum (1989), involving appropriate weights and machine efficiencies in carrying out sets of tasks.

Flexibility in its most rudimentary sense is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Das et al., 2009; Kumar and Sridharan, 2009; Miettinen et al., 2010). In Barad and Sipper (1988), the period needed by a system to recover after a change was used as the central flexibility measure, whereas a stochastic dynamic programming (DP) model for its assessment was presented in Kulatilaka (1988). Artificial intelligence methods seem appropriate in most practical situations, where numerical data are not readily available and linguistic variables are more amenable to handling imprecise knowledge (Dooner, 1991). The flexibility of competing systems can be ranked appropriately using an algorithmic approach (Abdel-Malek and Wolf, 1991) or a decision support system (Suresh, 1991) based on performance and economic criteria. Also, integer programming methods were proposed in Chandra and Tombak (1992), and a graphical representation method of production processes was presented in Kochikar and Narendran (1992).

Manufacturing flexibility is associated with uncertainty in all levels of a firm’s operation, such as variation in the demand and characteristics of a product or unanticipated interruptions of the production process because of machine failures. In addition, human operators or managers use imprecise concepts and vague notions when they attempt to define or measure flexibility. Fuzzy set theory (Dubois and Prade, 1980; Zimmermann, 1991), and especially fuzzy logic, constitute natural frameworks for the representation and manipulation of uncertainty.

Indeed, fuzzy set theory is an algebra of imprecise propositions and gradual statements like ‘machine A is more flexible than machine B because it is more versatile’. In previous treatments, uncertainty was handled by probability theory under the assumption that probabilities can be obtained precisely. Mandelbaum and Buzacott (1990), examining the meaning and use of flexibility in decision-making processes, admit that for real-world problems with increased complexity, the existing modelling methods are inadequate to represent reality. For context-dependent situations where conceptual imprecisions exist, however, as in the description of machine flexibility itself, fuzzy sets and fuzzy logic appear to be more appropriate for the definition and analysis of the problem.

Manufacturing costs for products are very crucial in decision-making and strategic planning. And with respect to cost estimation, research and development departments in the past could only estimate the final product’s total cost. Moreover, rules of thumb of the engineers are often applied as the cost estimation benchmarks, making the results controversial in terms of accuracy (Eklin et al., 2009; Mostafaee et al., 2010).

Jobshop is a flexible, scalable and intelligent production planning and control system offering advanced functionality and value in key areas of manufacturing and assembly. The flexible jobshop problem (FJP) is an extension of the classical jobshop problem allowing for an operation to be processed by any machine from a given set.

Automated guided vehicles (AGVs) reduce costs of manufacturing and increase efficiency in a manufacturing system. These trailers can be used to move raw materials in line to get them ready for manufacturing (Aized, 2009; Hsueh, 2010).

30 H. Fazlollahtabar and N. Mahdavi-Amiri

To conceptualise an AGV, it is necessary to understand the fundamentals of FMSs. Rather than using humans to perform repetitive tasks, a machine is used to perform the task. Each shop performs a specific task to assist in the manufacturing of a product. Although FMS is fast and efficient, but it is not cheap as it requires lots of expensive machines to operate. In the next section, we give a comprehensive description of our proposed problem of investigation.

2 Problem description

Consider a jobshop layout which applies an AGV for material handling. The AGV carries raw material, semi-produced and final products in batch sizes. Due to mounting demands, advancing technology and rising production capacity, the need for increasingly more shops is mounting over time. The new shops are expected to have more advanced machines. Therefore, more than one shop with the same duty are evolved. The difference among shops having the same duty shows up in the shop’s specifications that affect the production cost. As a result, the system would consider a flexible jobshop model where multi-shops of the same duty exist and each operation can be processed on any type of machine in any shop. The sequences of jobs are specified and the jobs are assumed to be independent.

The structure of such a problem would configure a network. In this network, the nodes are the shops and the arcs are the flow paths of the AGV to each shop. Shops in each stage are of the same type but have different specifications such as different machine types and equipments, varied operator proficiencies, different rates of defect, etc. Each flow path for the AGV is associated with a time parameter and also a cost parameter related to each shop. The aim is to find a path for the AGV minimising an aggregate time and cost objective. Considering the variable status of the AGV flow among shops, the time of each flow path of the AGV is a triangular fuzzy number. In each shop, different machines and operators are working. Due to unpredictable events during working times a cost may incur. This cost is inferenced from an expert system via fuzzy logic ‘IF … THEN …’ rules. Cost parameters of each shop are considered to be three parameters:

1 equipment sensitivity

2 operator proficiency

3 product specifications.

Each being specified by one of the three levels of low, moderate and high. A configuration for the proposed problem is presented in Figure 1.

As stated before, time is considered to be a triangular fuzzy number and cost is inferenced from an expert system. Our decision model is to consider both time and cost parameters, and thus an integration of the two parameters would be required. The integration is to have a weighted sum of cost and time as an arc length in the proposed network. Time is a triangular fuzzy number and cost is considered to be an indirect triangular fuzzy number. A description of fuzzy logic, numbers and systems are given in the next section.

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 31

Figure 1 A configuration of the proposed problem (see online version for colours)

3 Fuzzy modelling

3.1 Fuzzy logic and membership functions

In Zadeh’s words, who first introduced the notion of fuzzy logic (Zadeh, 1965), fuzzy logic is a tool for ‘Computing with Words’. He stated that the main role of fuzzy logic was to serve as a methodology for computing with words when no other methodology could attain such purpose (Zadeh, 1996, 1999).

Fuzzy logic accommodates for the emulation of the human reasoning process and making decisions based on vague or imprecise data. Linguistic terms can better represent knowledge, experience and subjective viewpoints of decision-makers in more intuitive ways and natural language formats. Each linguistic term can be expressed by a fuzzy set. In fuzzy set theory, elements of a set are allowed to have membership values between 0 and 1. If we depict this membership value by α, then α can have any value in the interval [0, 1]. Membership values for a fuzzy set are usually determined by a membership function. Triangular membership functions are common (Pedrycz, 1994).

Definition 1. A triangular fuzzy number a can be defined by a triplet (a1, a2, a3). Its conceptual schema and mathematical form are shown by:

1

11 2

2 1

32 3

3 2

3

0,

,( )

,

0,

a

x ax a

a x aa a

xa x

a x aa a

a x

μ

≤⎧⎪ −⎪ < ≤

−⎪= ⎨ −⎪ < ≤⎪ −⎪

<⎩

(1)

A triangular fuzzy number a in the universe of discourse X that conforms to this definition is shown in Figure 2.

32 H. Fazlollahtabar and N. Mahdavi-Amiri

Figure 2 A triangular fuzzy number a

Definition 2. Assuming that both 1 2 3( , , )a a a a= and 1 2 3( , , )b b b b= are triangular numbers, then the basic fuzzy operations are:

( )( )

1 1 2 2 3 3

1 1 2 2 3 3

, , for multiplication

, , for addition

a b a b a b a b

a b a b a b a b

× = × × ×

+ = + + + (2)

In Bastian (2000), different rule determination approaches have been described. Expert knowledge is the most popular and accepted method, in practice, while other common practices such as meta-rule techniques and fuzzy classifier systems which utilise genetic algorithms or artificial neural networks are applicable when expert knowledge is not available.

Roughly speaking, a fuzzy rule is the implication stated as a IF–THEN rule in which the premise and the conclusion are fuzzy sets. The basic components are the ‘If’ part, usually referred to as the antecedent, and the ‘Then’ part, usually referred to as the consequent (like If antecedent Then consequent). The antecedent can be composed of a single condition or a set of conditions combined by conjunction operators such as ‘AND’ and ‘OR’.

Once the rules are determined, the next step is to determine the matching degree of the inputs with respect to the fuzzy rules to perform the inference process. If there are multiple inputs, a conjunction operator is used to combine the matching degree of the fuzzy inputs utilising ‘min’, ‘max’ or ‘product’ (Klir and Yuan, 1995; Zimmermann, 1996). The two most principal methods of fuzzy inference are ‘clipping’ and ‘scaling’ methods (Klir and Yuan, 1995; Zimmermann, 1996), both suppressing the membership functions for the consequent, depending upon the degree of matching. Once the final outputs of the fuzzy rules are obtained through fuzzy inference, the outputs are combined into one single aggregated output. Since there might be more than one rule with a matching degree greater than zero, then more than one rule may be configured (Klir and Yuan, 1995; Zimmermann, 1996).

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 33

3.2 Fuzzy systems

The fuzzy logic and rule-based reasoning have found various applications in the control of industrial processes, modelling of complex systems and the development of fuzzy inference systems (Lin and Chen, 2004; Machacha and Bhattacharya, 2000). Due to their varied usage, fuzzy inference systems are also known as fuzzy expert systems, fuzzy rule-based systems, fuzzy associative memories or, in short, fuzzy systems. In general, a fuzzy system is composed of five major blocks:

• fuzzifier which transforms the crisp inputs into degrees of match with linguistic values

• dictionary which defines the membership functions of the fuzzy sets used in the fuzzy rules

• rule base which contains fuzzy IF–THEN rules and along with the dictionary, comprises the knowledge base of the fuzzy system

• decision-maker which performs the inference operations on the rules

• defuzzifier which converts the fuzzy results of inference into crisp outputs.

Based on the different trends in forming the major blocks of a fuzzy system and various kinds of application to which the fuzzy systems are applied, different types of fuzzy systems have been introduced. Mamdani fuzzy system and Takagi–Sugeno–Kang (TSK) fuzzy system are two types being commonly used in the literatures (Klir and Yuan, 1995; Zimmermann, 1996). Each one is being adopted in its own special domain because of its particular accommodation for the different ways of knowledge representation. However, other types of fuzzy systems such as Tsukamoto’s system (Tsukamoto, 1979), adaptive-network-based fuzzy inference system (Jang, 1993; Jang and Sun, 1995), etc. have their own special applications. Next, we will give a brief discussion of the first two types.

3.2.1 Mamdani fuzzy system

Mamdani fuzzy system was proposed as a first attempt to control a steam engine and boiler combination by a set of linguistic control rules obtained from experienced human operators. Rules in Mamdani fuzzy systems are like these (Klir and Yuan, 1995; Zimmermann, 1996):

1 1 2 2 1If is AND/OR is Then is x A x A y B

where A1, A2 and B1 are fuzzy sets. The fuzzy set acquired from aggregation of rules’ results will be defuzzified using defuzzification methods such as centroid (centre of gravity), max membership, mean–max and weighted average. The centroid method is very popular, in which the ‘centre of mass’ of the result provides the crisp value. In this method, the defuzzified value of the fuzzy set A, d(A), is calculated by

( )( )

( )

AX

AX

x x xd A

x x

μ

μ

⋅=∫∫

(3)

34 H. Fazlollahtabar and N. Mahdavi-Amiri

where ( )Aμ • is the membership function of the fuzzy set A (Klir and Yuan, 1995; Zimmermann, 1996).

For our problem in which various possible conditions of parameters are stated in forms of fuzzy sets, we utilise the Mamdani fuzzy system, because the fuzzy rules representing the expert knowledge in Mamdani fuzzy systems would consider fuzzy sets in their consequences, while in TSK fuzzy systems, the consequences are expressed as crisp functions. In general, designing a fuzzy system is composed of the following major steps (Klir and Yuan, 1995):

Step 1 Identifying pertaining input and output variables. Besides, the meaningful linguistic states along with appropriate fuzzy sets for the variable ought to be selected.

Step 2 Introducing a fuzzification method for input variables that expresses the associated measured uncertainty. The purpose of the fuzzification method is to interpret measurements of input variables which are expressed by real numbers.

Step 3 Formulating pertaining knowledge in terms of fuzzy inference rules. There are two principal ways in which relevant inference rules can be determined. One is to elicit them from experienced humans and the other is to obtain them from empirical data by suitable learning methods, usually with the help of neural networks.

Step 4 Combining measurements of input variables with relevant fuzzy rules to inference, regarding the output variables in which the purpose of inference engine will be obtained.

Step 5 Ascertaining a suitable defuzzification method to convert the aggregated fuzzy set of implications into a real number.

3.3 Modelling the proposed fuzzy problem

As stated before, time is a triangular fuzzy number, ensued from the experts’ knowledge. Brainstorming and expert knowledge vs. meta-rule techniques (neural networks, genetic algorithms, etc.) are two common approaches for defining fuzzy rules and membership functions. While available empirical data are requisite for using the second approach, due to unavailability of historical data for cost, we make use of the first approach to obtain the membership functions and fuzzy rules.

The input to our Mamdani type fuzzy system is composed of equipment sensitivity, operator proficiency and product specifications. For any of the inputs, three linguistic terms of ‘low’, ‘moderate’ and ‘high’ are defined. The output of the system is cost that is identified by any of the three linguistic terms, ‘low’, ‘moderate’ or ‘high’. The maximum membership grade of linguistic term ‘high’ is 30.

As a result, a triangular fuzzy number as the time and a numerical value as the cost are obtained. We intend to consider an integrated time–cost value as the value of each proposed arc in the network. Time and cost having different scales, it would not be possible to perform basic operations like addition on their original forms. Thus, to remove the scales, we normalise the time and cost values throughout the network and then consider a weighing method to make them appropriate for basic operations. On the other hand, time is a triangular fuzzy number and cost is a crisp numerical value. But, as implied by Definition 2, we need to have triangular numbers for both parameters to perform the basic operations. For this, we consider cost as a trivial triangular fuzzy

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 35

number and show it by a triplet (0, C, 0), where C is the numerical value inferenced from the expert system for cost with the membership value equal to 1. The right and left sides of C are zero since their membership values are zero. In the next section, we present an approach for weighing the parameters.

4 Operations on the parameters of the problem

4.1 Normalisation of the parameters

As stated, to perform basic operations on the time and cost parameters with their different scales, we need to remove their scales. To do this, two normalisation processes are proposed for time and cost, separately. In Equations (4)–(6), i is an index to show the node and j is an index to show the stage (shop type) in the proposed network, rij is the value of either time or cost parameter in each node, and nij is its corresponding normalised value. Considering time as a triangular fuzzy number, the normal value with a positive view is given by

min

max min , ,ij jij

j j

r rn i j

r r

−= ∀

− (4)

while the normal value with a negative view is given by,

max

max min , ,j ijij

j j

r rn i j

r r

−= ∀

− (5)

where maxjr and min

jr are maximum and minimum values in each column of an assumed

matrix of time or cost, respectively. Since time in our proposed model is a criterion implicating a negative aspect in decision-making, then we choose Equation (5) for normalising time in our approach.

Assuming cost as a crisp value, we normalise the cost values as follows:

2, ,ij

ij

kjk

rn i j

r= ∀∑

(6)

To compute the minimum or maximum value in Equations (4) and (5), comparisons are needed to be made. This means that it is necessary to have a method for ranking and comparing fuzzy numbers. An operator ≺ for ordering fuzzy numbers can be defined as follows (see Mahdavi et al., 2009):

( ) ( ) ( ) ( )1 1 2 2 3 3 4 4A B a b a b a b a b⇔ ≤ ∧ ≤ ∧ ≤ ∧ ≤≺ (7)

However, this relation is not a complete ordering, as fuzzy numbers A and B satisfying

{ } ( ) ( ), 1, 2,3, 4 : i i i ii j a b a b∃ ∈ < ∧ > (8)

would not be comparable by .≺

36 H. Fazlollahtabar and N. Mahdavi-Amiri

The ranking or ordering methods for fuzzy quantities have been proposed by several authors. For summaries, see Bortolan and Degani (1985) and Delgado et al. (1988). Admittedly, none of these methods is commonly accepted. Various ranking functions may produce conflicting and controversial results in comparison of fuzzy numbers (these issues have been discussed in Wang and Kerre, 2001, through a specific example).

Here, we use a fuzzy ranking method recently adopted by Mahdavi et al. (2009). Consider the fuzzy min and max operations defined in analogy with the fuzzy addition as follows:

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

MINV Min value , min , ,min , ,min , ,min ,

MAXV Max value , max , ,max , ,max , ,max ,

a b a b a b a b a b

a b a b a b a b a b

= =

= = (9)

It is evident that, for non-comparable fuzzy numbers a and ,b the fuzzy min operation results in a fuzzy number different from both of them. For example, for (5,10,13,19)a = and (6,9,15,20),b = we get from Equation (9) a fuzzy MINV (5,9,13,19)= which differs

from a and .b To alleviate this drawback, a method based on the distance between fuzzy numbers is proposed. We use the distance function introduced in Sadeghpour Gildeh and Gien (2001). The main advantages of this distance function, to be defined next, are the generality of its usage on various fuzzy numbers, and its reliability in distinguishing unequal fuzzy numbers.

Definition 3. The ,p qD -distance, indexed by parameters 1 p< < ∞ and 0 1q< < , between

two fuzzy numbers a and b is a non-negative function given by:

( )( ) ( )

11 1

0 0

,

0 10 1

(1 ) d d ,

,

(1 ) sup inf ,

p p p

p q

q a b q a b p

D a b

q a b q a b p

α α α α

α α α ααα

α α− − + +

− − + +

≤ ≤≤ ≤

⎧⎡ ⎤⎪ − − + − < ∞⎢ ⎥⎪ ⎣ ⎦⎪= ⎨

⎪⎪ − − + − = ∞⎪⎩

∫ ∫ (10)

where ,a aα α+ − and ,b bα α

+ − are the corresponding right and left α-cuts of a and b, respectively. The analytical properties of ,p qD depend on the first parameter p, while the second parameter q of ,p qD characterises the subjective weight attributed to the end

points of the support; i.e. aα+ and aα

− of the fuzzy numbers. If there is no reason for distinguishing any side of the fuzzy numbers, then ,1/ 2pD is recommended. Having q

close to 1 results in considering the right side of the support of the fuzzy numbers more favourably. Since the significance of the end points of the support of the fuzzy numbers is assumed to be the same, then we consider q = 1/2.

For triangular fuzzy numbers 1 2 3( , , )a a a a= and 1 2 3( , , ),b b b b= the above distance with 2p = and q = 1/2 is then calculated as:

( ) ( ) ( ) ( ) ( )3 2

2 22,1/ 2 2 2 1 1

1 1

1,6 i i i i i i

i i

D a b b a b a b a b a+ += =

⎡ ⎤= − + − + − −⎢ ⎥

⎢ ⎥⎣ ⎦∑ ∑ (11)

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 37

We are now able to compute the distance between ,a b and MINV using the proposed

distance function (11); for a and b as given before, we get ( )2,1/ 2 ,MINV 0.1667D a =

and ( )2,1/ 2 , MINV 1.33.D b = The number with lowest distance is the minimum. Now, we

are ready to propose the following algorithm for the comparison of fuzzy numbers.

Algorithm 1 (compare two triangular fuzzy numbers)

Input: Two fuzzy numbers 1 2( and ).L L

Output: Minimum or maximum between the two triangular fuzzy numbers min max( or ).L L

Step 1: Compute the minimum or maximum value (MINV or MAXV) by Equation (9).

Step 2: Find the distance of MINV or MAXV from 1 2( and )L L using Equation (11).

Step 3: Determine minL or maxL giving the smallest or largest distance.

4.2 Weighing the parameters

To weigh the parameters, we take a multi-criteria decision-making (MCDM) approach. MCDM, dealing primarily with problems of evaluation or selection (Keeney and Raiffa, 1976; Teng, 2002), is a rapidly developing area in operations research and management science. The analytical hierarchy process (AHP), developed by Saaty (1980), is a technique of considering data or information for a decision in a systematic manner (Schniederjans and Garvin, 1997). AHP is mainly concerned with the way to solve decision problems with uncertainties in multiple criteria characterisation. It is based on three principles:

1 constructing the hierarchy

2 priority setting

3 logical consistency.

We apply AHP to weigh the parameters.

4.2.1 Construction of the hierarchy

A complicated decision problem, composed of various attributes of an objective, is structured and decomposed into sub-problems (sub-objectives, criteria, alternatives, etc.), within a hierarchy.

4.2.2 Priority setting

The relative ‘priority’ given to each element in the hierarchy is determined by pair-wise comparisons of the contributions of elements at a lower level in terms of the criteria (or elements) with a causal relationship. In AHP, multiple paired comparisons are based on a standardised comparison scale of nine levels (see Table 1 due to Saaty, 1980).

38 H. Fazlollahtabar and N. Mahdavi-Amiri

Table 1 Scale of relative importance

Intensity of importance Definition of importance

1 Equal

2 Weak

3 Moderate

4 Moderate plus

5 Strong

6 Strong plus

7 Very strong or demonstrated

8 Very, very strong

9 Extreme

Let { }1, , nC c c= … be the set of criteria. The result of the pair-wise comparisons on n criteria can be summarised in an n × n evaluation matrix A in which every element aij is the quotient of weights of the criteria, as shown below:

( ) , , 1, ,ijA a i j n= = … (12)

The relative priorities are given by the eigenvector (w) corresponding to the largest eigenvalue (λmax) as:

maxAw wλ= (13)

When pair-wise comparisons are completely consistent, the matrix A has rank 1 and max .nλ = In that case, weights can be obtained by normalising any of the rows or columns of A.

The procedure described above is repeated for all subsystems in the hierarchy. To synthesise the various priority vectors, these vectors are weighed with the global priority of the parent criteria and synthesised. This process starts at the top of the hierarchy. As a result, the overall relative priorities to be given to the lowest level elements are obtained. These overall, relative priorities indicate the degree to which the alternatives contribute to the objective. These priorities represent a synthesis of the local priorities, and reflect an evaluation process that permits integration of the perspectives of the various stakeholders involved.

4.2.3 Consistency check

A measure of consistency of the given pair-wise comparison is needed. The consistency is defined by the relation between the entries of A; i.e. we say A is consistent if aik = aij ,…, ajk, for all i, j, k. The consistency index (CI) is:

( )maxCI( 1)

nn

λ −=

− (14)

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 39

The final consistency ratio (CR), on the basis of which one can conclude whether the evaluations are sufficiently consistent, is calculated to be the ratio of the CI and the random consistency index (RI):

CICRRI

= (15)

The value 0.1 is the accepted upper limit for CR. If the final CR exceeds this value, the evaluation procedure needs to be repeated to improve consistency. The measurement of consistency can be used to evaluate the consistency of decision-makers as well as the consistency of all the hierarchies.

We are now ready to give an algorithm for computing parameter weights using the AHP. The following notations are used.

4.2.4 Notations and definitions

n: number of criteria.

i: number of parameters.

p: index for parameters, p = 1 or 2.

d: index for criteria, 1 .d D≤ ≤

:pdR the weight of pth item with respect to dth criterion.

:dw the weight of dth criterion.

Algorithm 2: PWAHP (compute parameter weights using the AHP)

Step 1: Define the decision problem and the goal.

Step 2: Structure the hierarchy from the top through the intermediate to the lowest level.

Step 3: Construct the parameter–criteria matrix using steps 4–8 using the AHP.

{Steps 4–6 are performed for all levels in the hierarchy.}

Step 4: Construct pair-wise comparison matrices for each of the lower levels for each element in the level immediately above by using a relative scale measurement. The decision-maker has the option of expressing his or her intensity of preference on a nine-point scale. If two criteria are of equal importance, a value of 1 is set for the corresponding component in the comparison matrix, while a value 9 indicates an absolute importance of one criterion over the other (Table 1 shows the measurement scale defined by Saaty, 1980).

Step 5: Compute the largest eigenvalue by the relative weights of the criteria and the sum taken over all weighted eigenvector entries corresponding to those in the next lower level of the hierarchy.

Analyse pair-wise comparison data using the eigenvalue technique. Using these pair-wise comparisons, estimate the parameters. The eigenvector of the largest eigenvalue of matrix A constitutes the estimation of relative importance of the attributes.

40 H. Fazlollahtabar and N. Mahdavi-Amiri

Step 6: Construct the consistency check and perform consequence weights analysis as follows:

( )

1 1

2

2 2

1

1 2

1

1

1

n

nij

n n

w ww w

w ww wA a

w ww w

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

…

…

Note that if the matrix A is consistent (i.e. aik = aij ,…, ajk, for all , , 1,2, ,i j k n= … ), then we have (the weights are already known),

, , 1, 2, ,iij

j

wa i j n

w= = …

If the pair-wise comparisons do not include any inconsistencies, then max .nλ = The more consistent the comparisons are, the closer the value of computed maxλ is to n. Set the CI, which measures the inconsistencies of pair-wise comparisons, to be:

( )maxCI( 1)

nn

λ −=

−

and let the CR be:

CICR 100RI

⎛ ⎞= ⎜ ⎟⎝ ⎠

where n is the number of columns in A and RI is the random index, being the average of the CI obtained from a large number of randomly generated matrices.

Note that RI depends on the order of the matrix, and a CR value of 10% or less is considered acceptable (Saaty, 1980).

Step 7: Form the parameter–criteria matrix as specified in Table 2.

Step 8: As a result, configure the pair-wise comparison for criteria–criteria matrix as in Table 3.

The dw are gained by a normalisation process. The dw are the weights for criteria.

Step 9: Compute the overall weights for the parameters, using Tables 2 and 3, as follows:

11 1 12 2 1

21 1 22 2 2

Total weight for parameter 1Total weight for parameter 2

d d

d d

R w R w R wR w R w R w

ψψ= = × + × + + ×

= = × + × + + ×′ (16)

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 41

Table 2 The parameter–criteria matrix

C1 C2 … Cd

Parameter 1 R11 R12 … R1d

Parameter 2 R21 R22 … R2d

Table 3 The criteria–criteria pair-wise comparison matrix

C1 C2 … Cd dw

Criteria 1 1 a12 … a1d w1

Criteria 2 1/a12 1 … a2d w2

Criteria d 1/a1d 1/a2d … 1 wd

where 1 .ψ ψ= −′ Note that due to the significance and sensitivity of different shops of the same type in any stage the weighing process is performed for each arc. Here, we obtain the weights for the parameters of each arc in the proposed network. If we consider nC and

nT as normalised cost and time, respectively, then the total weighted normalised value of each arc is determined as follows:

( ) ( )n nP T Cψ ψ= × + ×′ (17)

In the next section, we propose an approach to identify the shortest path in the network using the total weighted normalised value for each arc. Before that, consider a ranking method for triangular fuzzy numbers.

4.3 Ranking of triangular fuzzy numbers

Several fuzzy ranking methods have been proposed (Bortolan and Degani, 1985; Kim and Park, 1990; Luis and Antonio, 1989). Since the graded mean integration representation method (Chen and Hsieh, 2000) not only alleviates some drawbacks of the existing methods, but also possesses the advantages of being easily implementable and quite effective in problem solving (see Lee et al., 2007), we will use it to transform the total weighted normalised value of each arc in our proposed network. If 1 2 3( , , )a a a a= is a triangular fuzzy number, then the graded mean integration ( ( ))R a is defined to be:

( ) 1 2 346

a a aR a

+ += (18)

We apply Equation (18) to transform the fuzzy numbers to crisp values and use them to find the optimal path.

42 H. Fazlollahtabar and N. Mahdavi-Amiri

5 Shortest path in a network

Let G = (V, A) be a graph, where { }1, ,V N= … is the set of nodes, and A V V⊂ × is the set of arcs. We write ( , ) ,i j A∈ if there exists an arc from node i V∈ to node j V∈ . Furthermore, let 0ijt ≥ denotes the distance (or travel time, or any other measure of cost) from i to j. If ( , ) ,i j A∉ then set .ijt = +∞ Note that the travel time from node i to node j is

assumed to be stationary; i.e. independent of the actual arrival time at node i. Let fij denotes the length of the shortest-path from i to j in the graph.

It is a well-known principle that every additive deterministic DP formulation can be equivalently viewed as the problem of finding the shortest path in a directed network, where the states, decisions and decision costs of the former correspond to the nodes, arcs and arc lengths of the latter (Dreyfus and Law, 1977). It is perhaps for this reason that the task of efficiently computing shortest paths is found prominence in the mathematical programming literature. Next, we describe a DP approach for computing the optimal path.

DP was introduced by Bellman (1957). Toth (1980) presented the early DP-based approaches and reported numerical experiments with a limited success. Hybrid methods, combining DP and implicit enumeration, were developed later. The first approach was developed by Plateau and Elkihel (1985). A recent approach, the so-called combo algorithm, is able to solve very large instances of up to 10,000 variables within <1 sec, with basically no difference in the required solution times for ‘easy’ and ‘hard’ instances (Martello et al., 1999). Marsten and Morin (1978) proposed the first hybrid method, which combined heuristic algorithms, DP and branch-and-bound approaches. More sophisticated methods can be found in Ibaraki (1987).

DP is a technique to tackle multistage decision processes. A given problem is subdivided into smaller sub-problems, which are sequentially solved until the initial problem is solved by the aggregation of the sub-problem solutions. In each stage, a set of states is defined. The states would describe all possible conditions of the process in the current decision stage, which corresponds to every feasible partial solution. The set of all possible states is known as the state space. The states of a stage u can be transformed to states of a stage u + 1, using a transition. A transition indicates the decisions adopted in a stage, and a sequence of transitions taken to reach a state starting from another state is known as a policy. DP approaches can be seen as transformations of the original problem to one associated with the exploration of a multistage graph G(S, T), where the vertices in S correspond to the state space and the arcs in T correspond to the set of transitions, leading to an optimal policy.

The basis of DP can be traced to the optimality principle of Bellman (2003). The optimality principle states that an optimal policy should be constituted by optimal policies from every state of the decision chain to the final state.

Here, we make use of a DP approach for our proposed network to identify the optimal manufacturing path. This model helps the manufacturing system to determine the more profitable shops. The advantages of such a model are simplicity, the ability to determine

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 43

the exact optimal value and implementability on sophisticated networks. The backward dynamic model would be defined as follows:

Indices: S Number of stages; s = 0, 1, 2, …, n i′ Start node number; i′ = 1, 2, …, I; i′ = 0 (for the start node) i End node number; i = 1, 2, …, I Notations:

( )s iϕ ′ The minimum value of moving from node i′ in stage s to an end node i in stage s − 1

Pi′I Numerical value of an arc between node i′ to node i Objective function:

( ) ( ){ }1 in layer 1

0

Min , in stage , ( 0,1, 2, , )

( ) (0)* (0)

s s i ii s

s

i i P i s s n

i

ϕ ϕ

ϕϕ ϕ

+ ′+

= + ∀ =′ ′

=

=

…

(19)

Note that *ϕ identifies the optimal path. To identify the minimum value in Equation (19), Algorithm 1 given in Section 4 is applied.

6 Numerical illustration

Here, we present an example to illustrate the applicability and effectiveness of our proposed approach. Consider the network specified in Figure 3 as a proposed flexible jobshop environment. Three types of shops are considered. Two shops of type one, three shops of type two, and two shops of type three are configured. An AGV is applied as a material handling device which receives the raw material from its corresponding station (raw material station in Figure 3) and follows the production plan to deliver a final product to the depot. Two parameters of time and cost (as arc lengths) are assumed for travelling between each two shops (nodes).

Following our discussions in the previous sections, to consider cost, we created a fuzzy system using MATLAB’s Fuzzy Logic Toolbox. As mentioned before, the triangular membership functions are utilised in the fuzzy system to present the cost parameters effectively. Three criteria of equipment sensitivity, product specification and operator proficiency are considered to estimate cost using fuzzy linguistic variables. The membership functions for the stated criteria are shown in Figures 4–6.

44 H. Fazlollahtabar and N. Mahdavi-Amiri

Figure 3 A flexible jobshop (see online version for colours)

Figure 4 Membership function for equipment sensitivity (see online version for colours)

Figure 5 Membership function for product specification (see online version for colours)

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 45

Figure 6 Membership function for operator efficiency (see online version for colours)

Figure 7 Membership function for cost (see online version for colours)

The membership function for cost is presented in Figure 7. The fuzzy system has the merit to take advantage of the following 16 rules:

1 If (equipment sensitivity is high) and (operator efficiency is low) and (product specification is low) then (cost is high).

2 If (equipment sensitivity is high) and (operator efficiency is moderate) and (product specification is moderate) then (cost is moderate).

3 If (equipment sensitivity is high) and (operator efficiency is high) and (product specification is high) then (cost is moderate).

4 If (equipment sensitivity is high) and (operator efficiency is high) and (product specification is low) then (cost is low).

5 If (equipment sensitivity is moderate) and (operator efficiency is low) and (product specification is low) then (cost is moderate).

6 If (equipment sensitivity is moderate) and (operator efficiency is moderate) and (product specification is low) then (cost is moderate).

46 H. Fazlollahtabar and N. Mahdavi-Amiri

7 If (equipment sensitivity is moderate) and (operator efficiency is moderate) and (product specification is moderate) then (cost is moderate).

8 If (equipment sensitivity is moderate) and (operator efficiency is high) and (product specification is moderate) then (cost is low).

9 If (equipment sensitivity is moderate) and (operator efficiency is high) and (product specification is high) then (cost is low).

10 If (equipment sensitivity is moderate) and (operator efficiency is high) and (product specification is low) then (cost is low).

11 If (equipment sensitivity is moderate) and (operator efficiency is low) and (product specification is high) then (cost is high).

12 If (equipment sensitivity is low) and (operator efficiency is low) and (product specification is low) then (cost is moderate).

13 If (equipment sensitivity is low) and (operator efficiency is moderate) and (product specification is low) then (cost is low).

14 If (equipment sensitivity is low) and (operator efficiency is high) and (product specification is low) then (cost is low).

15 If (equipment sensitivity is low) and (operator efficiency is high) and (product specification is moderate) then (cost is low).

16 If (equipment sensitivity is low) and (operator efficiency is high) and (product specification is high) then (cost is low).

The structure of the proposed fuzzy system is depicted in Figure 8. The surface view of the status and effects of the criteria on cost parameter are shown

in Figure 9. The interval of the proposed criteria and the cost parameter are shown in Figure 10.

Figure 8 The proposed fuzzy system (see online version for colours)

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 47

Figure 9 The surface view (see online version for colours)

Figure 10 The interval of the criteria and the cost parameter (see online version for colours)

We scaled all the input values to lie in the interval [0, 10]. Experts, considering their knowledge and past experience, pointed out that if the AGV wanted to travel from the raw material station to shop 1-1, the correlation between cost and equipment sensitivity for shop 1-1 would be somewhat moderate. Based on this, the equipment sensitivity is set to be in [2, 6.5]. We also decide to set the value of operator proficiency in the interval [0, 7.2]. Due to market conditions and technology considerations for producing products, the product specification is believed to be high, leading us to set the product specification to be in [5.3, 9.6]. Feeding these input values to our fuzzy system, the proposed cost is:

Cost of travelling from shop 0 to shop 1-1 10.2248=

Note that this result is gained from the MATLAB’s Fuzzy Logic Toolbox. Therefore, the same process (determining the intervals for the criteria by experts) is applied to all the arcs in the network and the cost values obtained are summarised in Table 4 (Dep stands for depot).

Next, we show the time parameter as provided by the experts. The triangular time for each arc is given in Table 5 (Dep stands for depot).

48 H. Fazlollahtabar and N. Mahdavi-Amiri

We then normalise these values. As stated in Section 4 and with our crisp cost parameters and fuzzy triangular time parameters, Equations (6) and (5) are applied for the normalisation, respectively. Note that for normalising time parameters, considering Equation (5) a comparison of fuzzy numbers are needed to find the max or min value. To do this, Algorithm 1 is applied. To facilitate the computations, the algorithm is coded in MATLAB 7 programming environment.

The normalised values for cost and time are shown in Tables 6 and 7, respectively (note that shop 0 corresponds to the raw material station).

Table 4 The computed cost values for all arcs

Arcs 0, 1-1 0, 1-2 1-1, 2-1 1-1, 2-2 1-1, 2-3 1-2, 2-1 1-2, 2-2 1-2, 2-3 Cost value 10.2248 15 15 4.5885 15 11.1023 15 12.6895 Arcs 2-1, 3-1 2-1, 3-2 2-2, 3-1 2-2, 3-2 2-3, 3-1 2-3, 3-2 3-1, Dep 3-2, Dep Cost value 5.2099 13.5421 15 4.8848 12.4068 10.6531 9.0015 15

Table 5 The time values for all arcs

Arcs 0, 1-1 0, 1-2 1-1, 2-1 1-1, 2-2 1-1, 2-3 1-2, 2-1 1-2, 2-2 1-2, 2-3 Time (7, 12, 18) (25, 42, 71) (23, 39, 103) (7, 13, 69) (35, 41, 56) (19, 23, 27) (45, 50, 55) (11, 13, 78) Arcs 2-1, 3-1 2-1, 3-2 2-2, 3-1 2-2, 3-2 2-3, 3-1 2-3, 3-2 3-1, Dep 3-2, Dep Time (31, 47, 81) (34, 49, 63) (75, 84, 99) (68, 73, 79) (15, 92, 99) (7, 48, 92) (23, 29, 48) (11, 39, 78)

Table 6 The normalised values of cost

From shop 0to shop type 1

From shop type 1to shop type 2

From shop type 2to shop type 3

From shop type 3 to depot

Normalised values 0.563 0.48 0.193 0.515 0.827 0.147 0.502 0.857

0 0.48 0.556 0 0 0.355 0.181 0 0 0.48 0.46 0 0 0.405 0.395 0

Table 7 The normalised values of time

From shop 0to shop type 1

From shop type 1to shop type 2

From shop type 2to shop type 3

From shop type 3 to depot

Normalised Values (1, 1, 1) (0, 0, 0) (1.1, 1.1, 0.5) (0, 0, 0) (0, 0, 0) (4, 1.6, 0.447) (1, 1, 1) (1, 1, 1)

0 (−3, −0.125, 0.618) (0, 0, 0) 0

0 (1, 1, 1) (0.171, 0.314, 0.556) 0 0 (−5.5, −0.688, 0.632

) (1.5, −0.23, 0) 0

0 (3, 1.6, 0.329) (1.7, 1, 0.2) 0

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 49

Note that, as stated before, a cost can be viewed as (0, C, 0), a trivial fuzzy triangular number.

Next, we weigh the cost and time parameters with respect to three criteria: economic viewpoint, demand fluctuation and operator training. Algorithm 2 is applied for weighing the parameters. The following weights are gained for our proposed parameters: Weight for time 0.62.ψ =

Weight for cost 1 0.38.ψ ψ= − =′

Now, these weights are multiplied by the normalised values presented in Tables 6 and 7. Tables 8 and 9 are configured indicating the weighted normalised values of cost and time.

Consequently, the aggregated values of weighed normalised cost and time are given in Table 10. Note that these values indicate the arc length of our proposed FJP.

Now, using Equation (18), we transform the fuzzy numbers to crisp values. The results are summarised in Table 11.

Herewith, applying the backward DP approach specified in Section 5, we obtain the optimal path of the network considering both the cost and time parameters. To facilitate the computations, a LINGO 9 encoding is applied. The optimal values are: ϕ*(1) = 1.63, ϕ*(2) = 0.93, ϕ*(3) = 1.42, ϕ*(4) = 0.81, ϕ*(5) = 0.27, ϕ*(6) = 0.31, ϕ*(7) = 0.13, ϕ*(8) = 0.84, ϕ*(9) = 0.0, and the optimal path is: depot, 3-1, 2-2, 1-1, raw material station.

The optimal path is depicted graphically (in bold) in Figure 11. Table 8 The weighed normalised values of cost

From shop 0to shop type 1

From shop type 1to shop type 2

From shop type 2to shop type 3

From shop type 3 to depot

Weighted normalised values

0.214 0.182 0.073 0.196 0.314 0.056 0.191 0.326

0 0.182 0.211 0 0 0.135 0.07 0 0 0.182 0.175 0 0 0.154 0.150 0

Table 9 The weighed normalised values of time

From shop 0 to shop type 1

From shop type 1 to shop type 2

From shop type 2to shop type 3

From shop type 3 to depot

Weighted normalised values

(0.62, 0.62, 0.62) (0, 0, 0) (0.682, 0.682, 0.31) (0, 0, 0) (0, 0, 0) (2.48, 0.992, 0.277) (0.62, 0.62, 0.62) (0.62, 0.62, 0.62)

0 (−1.86, −0.078, 0.383) (0, 0, 0) 0 0 (0.62, 0.62, 0.62) (0.106, 0.195, 0.345) 0 0 (−3.41, −0.427, 0.392) (0.93, −0.143, 0) 0 0 (1.86, 0.992, 0.204) (1.054, 0.62, 0.124) 0

50 H. Fazlollahtabar and N. Mahdavi-Amiri

Table 10 The aggregated weighed normalised values of cost and time

From shop 0 to shop type 1

From shop type 1 to shop type 2

From shop type 2 to shop type 3

From shop type 3 to depot

Aggregated values

(0.62, 0.834, 0.62) (0, 0.182, 0) (0.682, 0.775, 0.31) (0, 0.196, 0) (0, 0.314, 0) (2.48, 1.048, 0.277) (0.62, 0.811, 0.62) (0.62, 0.946, 0.62)

0 (−1.86, 0.104, 0.383) (0, 0.211, 0) 0 0 (0.62, 0.775, 0.62) (0.106, 0.265, 0.345) 0

0 (−3.41, −0.245, 0.392) (0.93, 0.032, 0) 0 0 (1.86, 1.146, 0.204) (1.054, 0.77, 0.124) 0

Table 11 The ranking of the aggregated weighed normalised values of the network

From shop 0 to shop type 1

From shop type 1 to shop type 2

From shop type 2 to shop type 3

From shop type 3 to depot

Ranked values 1.006 0.12 0.68 0.13 0.21 1.16 0.75 0.84

0 −1.06 0.14 0 0 0.72 0.25 0

0 −3.99 0.18 0 0 1.11 0.71 0

Figure 11 The optimal path obtained (see online version for colours)

7 Discussions

An AGV starts material handling from the first station and moves through the last shop to complete a production cycle. After delivering the finished products to a depot, the AGV returns to the first shop and start another tour. Note that considering throughput of the

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 51

shops, some shops may be busy with the products from the first tour; i.e. the new tour begins with fewer number of shops available. Considering the proposed approach, the AGV starts visiting the shops on an optimal path considering the time and cost parameters and moves through the depot. This process continues to implement the production plan and stops at ending time in a day. The primary significance of our proposed approach is in its identification of the shops on the optimal path in different production tours. This way, the shops being busier in the manufacturing process can be fortified and those that are less used can be either removed or utilised for other purposes, and therefore a better allocation of workstations and an optimal usage of manufacturing space are ensued.

To investigate the optimality of the obtained path in the proposed network considering time and cost parameters, we designed two experiments. In these experiments, we considered time and cost separately on the network and found optimal paths. Firstly, consider time as the network arc length. Since times are triangular fuzzy numbers, we apply the proposed ranking method in Section 4.3 to turn them into crisp values. The crisp values are given in Table 12.

Herewith, applying the backward DP approach proposed in Section 5, we obtained the optimal path of the network considering only the time parameter. To facilitate the computations, a LINGO 9 encoding was applied. The optimal values obtained are: ϕ*(1) = 140.34, ϕ*(2) = 128.17, ϕ*(3) = 104.17, ϕ*(4) = 81.17, ϕ*(5) = 114, ϕ*(6) = 89.33, ϕ*(7) = 31.17, ϕ*(8) = 40.83, ϕ*(9) = 0.0, and the optimal path is: depot, 3-2, 2-1, 1-2, raw material station.

The optimal path is depicted graphically (in bold) in Figure 12. Next, consider cost as the network arc length. We obtained the inferred costs from the

fuzzy rule base in Table 4. Again, employing the backward DP approach of Section 5, we obtained the optimal path of the network considering only the cost parameter. As before, to facilitate the computations, a LINGO 9 encoding was applied (Appendix C). The optimal values obtained are: ϕ*(1) = 34.6981, ϕ*(2) = 24.47330, ϕ*(3) = 25.31370, ϕ*(4) = 14.21140, ϕ*(5) = 19.88480, ϕ*(6) = 21.40830, ϕ*(7) = 9.001500, ϕ*(8) = 15, ϕ*(9) = 0.0, and the optimal path is: depot, 3-1, 2-1, 1-1, raw material station.

The optimal path is depicted graphically (in bold) in Figure 13. It is clear that the results of considering only time or cost parameters in the objective

function are different from an aggregate consideration of time and cost parameters (as can be seen from the results of this section and the one obtained in Section 6). Thus, the proposed bi-criteria manufacturing network model can turn out to be appropriately depending on the particular application being considered.

Table 12 The crisp time values for all arcs

Arcs 0, 1-1 0, 1-2 1-1, 2-1 1-1, 2-2 1-1, 2-3 1-2, 2-1 1-2, 2-2 1-2, 2-3 Time 12.17 44 47 128 42.5 23 50 23.5 Arcs 2-1, 3-1 2-1, 3-2 2-2, 3-1 2-2, 3-2 2-3, 3-1 2-3, 3-2 3-1, Dep 3-2, Dep Time 50 48.83 85 73.17 80.33 48.5 31.17 40.83

52 H. Fazlollahtabar and N. Mahdavi-Amiri

Figure 12 The optimal path obtained (see online version for colours)

Figure 13 The optimal path obtained (see online version for colours)

8 Conclusions

We proposed an approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost. The proposed flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications. The shops configure a network in which they are considered as nodes and the paths among them are considered as network arcs. An AGV functions as a material handling device through the manufacturing network. Time is considered to be a triangular fuzzy number and cost is inferred from an expert system considering three parameters of equipment sensitivity, operator proficiency and product specification via linguistic variables. As unique contributions of the work, the objective was to find a path which minimises both the time and cost criteria, aggregately. Since time and cost had different scales, a normalisation process was used to remove the scales and because the model was bi-objective, the AHP weighing method was applied to gain a single objective. A DP approach was used to compute a shortest

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 53

path in the proposed network. Finally, the proposed approach was illustrated by a numerical example. Various applicability and advantages of the proposed model were discussed. The limitations of the research were the work load of the AGVs and the defect rate of the guide path. As future research, other multi-objective optimisation approaches could be tested and compared with the proposed method.

Acknowledgements

The first author thanks Mazandaran University of Science and Technology and the second author thanks Research Council of Sharif University of Technology for supporting this work.

References Abdel-Malek, L. and Wolf, C. (1991) ‘Evaluating flexibility of alternative FMS designs – a

comparative measure’, Int. J. Production Economics, Vol. 23, No. 1, pp.3–10. Aized, T. (2009) ‘Modelling and performance maximization of an integrated automated guided

vehicle system using coloured Petri net and response surface methods’, Computers and Industrial Engineering, Vol. 57, No. 3, pp.822–831.

Barad, M. and Sipper, D. (1988) ‘Flexibility in manufacturing systems: definitions and Petri net modeling’, Int. J. Production Research, Vol. 26, No. 2, pp.237–248.

Bastian, A. (2000) ‘Identifying fuzzy models utilizing genetic programming’, Fuzzy Sets and Systems, Vol. 113, No. 3, pp.333–350.

Bellman, R. (1957) Dynamic Programming. Princeton, NJ: Princeton University Press. Bellman, R. (2003) Dynamic Programming. Mineola, NY: Dover Publications. Bortolan, G. and Degani, R. (1985) ‘A review of some method for ranking fuzzy subsets’, Fuzzy

Sets and Systems, Vol. 15, pp.1–19. Brill, P.H. and Mandelbaum, M. (1989) ‘On measures of flexibility in manufacturing systems’,

Int. J. Production Research, Vol. 27, No. 5, pp.747–756. Browne, J., Dubois, D., Rathmill, K., Sethi, S.P. and Stecke, K.E. (1984) ‘Classification of flexible

manufacturing systems’, FMS Magazine, Vol. 2, No. 2, pp.114–117. Buzacott, J.A. (1982) ‘The fundamental principles of flexibility in manufacturing system’,

Proceedings of the First International Conference on FMS, pp.23–30. Chandra, P. and Tombak, M.M. (1992) ‘Models for the evaluation of routing and machine

flexibility’, European Journal of Operational Research, Vol. 60, pp.156–165. Chen, S.H. and Hsieh, C.H. (2000) ‘Representation, ranking, distance, and similarity of L-R type

fuzzy number and application’, Australian Journal of Intelligent Information Processing Systems, Vol. 6, No. 4, pp.217–229.

Das, K., Baki, M.F. and Li, X. (2009) ‘Optimization of operation and changeover time for production planning and scheduling in a flexible manufacturing system’, Computers and Industrial Engineering, Vol. 56, No. 1, pp.283–293.

Delgado, M., Verdegay, J.L. and Vila, M.A. (1988) ‘A procedure for ranking fuzzy numbers using fuzzy relations’, Fuzzy Sets and Systems, Vol. 26, pp.49–62.

Dooner, M. (1991) ‘Conceptual modeling of manufacturing flexibility’, Int. J. Computer Integrated Manufacturing, Vol. 4, No. 3, pp.135–144.

Dreyfus, S.E. and Law, A.M. (1977) The Art and Theory of Dynamic Programming. New York, NY: Academic Press.

54 H. Fazlollahtabar and N. Mahdavi-Amiri

Dubois, D. and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications. New York, NY: Academic Press.

Eklin, M., Arzi, Y. and Shtub, A. (2009) ‘Model for cost estimation in a finite-capacity stochastic environment based on shop floor optimization combined with simulation’, European Journal of Operational Research, Vol. 194, No. 1, pp.294–306.

Gerwin, D. (1982) ‘Do’s and don’t’s of computerized manufacturing’, Harvard Business Review, Vol. 60, No. 2, pp.107–116.

Gupta, D. and Buzacott, J.A. (1989) ‘A framework for understanding flexibility of manufacturing systems’, Journal of Manufacturing Systems, Vol. 8, No. 2, pp.89–97.

Gupta, Y. and Goyal, S. (1989) ‘Flexibility of manufacturing systems: concepts and measurements’, European Journal of Operational Research, Vol. 43, pp.119–135.

Hsueh, C.F. (2010) ‘A simulation study of a bi-directional load-exchangeable automated guided vehicle system’, Computers and Industrial Engineering, Vol. 58, No. 4, pp.594–601.

Ibaraki, T. (1987) ‘Enumerative approaches to combinatorial optimization – part II’, Annals of Operations Research, Vol. 11, pp.397–439.

Jang, J.R. (1993) ‘ANFIS: adaptive-network-based fuzzy inference system’, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 23, pp.665–685.

Jang, J.R. and Sun, C. (1995) ‘Neuro-fuzzy modeling and control’, Proceedings of IEEE, Vol. 83, pp.378–405.

Keeney, R. and Raiffa, H. (1976) Decision with Multiple Objectives: Preference and Value Tradeoffs. New York, NY: Wiley & Sons.

Kim, K. and Park, K.S. (1990) ‘Ranking fuzzy numbers with index of optimism’, Fuzzy Sets and Systems, Vol. 35, pp.143–150.

Klir, G.J. and Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic: Theory and Applications. Upper Saddle River, NJ: Prentice-Hall.

Kochikar, V.P. and Narendran, T.T. (1992) ‘A framework for assessing the flexibility of manufacturing systems’, Int. J. Production Research, Vol. 30, No. 12, pp.2873–2895.

Kulatilaka, N. (1988) ‘Valuing the flexibility of flexible manufacturing systems’, IEEE Transaction Engineering Management, Vol. 78, pp.250–257.

Kumar, V. (1987) ‘Entropic measures of manufacturing flexibility’, Int. J. Production Research, Vol. 25, No. 7, pp.957–966.

Kumar, N.S. and Sridharan, R. (2009) ‘Simulation modelling and analysis of part and tool flow control decisions in a flexible manufacturing system’, Robotics and Computer-Integrated Manufacturing, Vol. 25, Nos. 4–5, pp.829–838.

Lee, K.L., Huang, W.C. and Teng, J.Y. (2007) ‘Locating the competitive relation of global logistics hub using quantitative SWOT analytical method’, Quality and Quantity, Vol. 43, No. 1, pp.87–107.

Lin, C. and Chen, C. (2004) ‘New product Go/No-Go evaluation at the front end: a fuzzy linguistic approach’, IEEE Transactions on Engineering Management, Vol. 51, No. 2, pp.197–207.

Luis, M. and Antonio, G.M. (1989) ‘A subjective approach for ranking fuzzy numbers’, Fuzzy Sets and Systems, Vol. 29, pp.145–153.

Machacha, L. and Bhattacharya, P. (2000) ‘A fuzzy-logic-based approach to project selection’, IEEE Transactions on Engineering Management, Vol. 47, No. 1, pp.65–73.

Mahdavi, I., Nourifar, R., Heidarzade, A. and Mahdavi-Amiri, N. (2009) ‘A dynamic programming approach for finding shortest chains in a fuzzy network’, Applied Soft Computing, Vol. 9, No. 2, pp.503–511.

Mandelbaum, M. and Buzacott, J. (1990) ‘Flexibility and decision making’, European Journal of Operational Research, Vol. 44, pp.17–27.

Marsten, R.E. and Morin, T.L. (1978) ‘A hybrid approach to discrete mathematical programming’, Mathematical Programming, Vol. 14, pp.21–40.

An optimal path in a bi-criteria AGV-based flexible jobshop manufacturing system 55

Martello, S., Pisinger, D. and Toth, P. (1999) ‘New trends in exact algorithms for the 0–1 knapsack problem’, European Journal of Operational Research, Vol. 123, pp.325–336.

Miettinen, K., Eskelinen, P., Ruiz, F. and Luque, M. (2010) ‘NAUTILUS method: an interactive technique in multiobjective optimization based on the nadir point’, European Journal of Operational Research, Vol. 206, No. 2, pp.426–434.

Mostafaee, A. and Saljooghi, F.H. (2010) ‘Cost efficiency measures in data envelopment analysis with data uncertainty’, European Journal of Operational Research, Vol. 202, No. 2, pp.595–603.

Pedrycz, W. (1994) ‘Why triangular membership functions?’, Fuzzy Sets and Systems, Vol. 64, pp.21–30.

Plateau, G. and Elkihel, M. (1985) ‘A hybrid method for the 0–1 knapsack problem’, Methods of Operations Research, Vol. 49, pp.277–293.

Saaty, T.L. (1980) The Analytic Hierarchy Process. New York, NY: McGraw-Hill. Sadeghpour Gildeh, B. and Gien, D. (2001) ‘La distance-Dp, q et le cofficient de corrélation entre

deux variables aléatoires floues’, Actes de LFA’2001, Monse-Belgium, pp.97–102. Schniederjans, M.J. and Garvin, T. (1997) ‘Using the analytic hierarchy process and

multi-objective programming for the selection of cost drivers in activity based costing’, European Journal of Operational Research, Vol. 100, pp.72–80.

Sethi, A. and Sethi, S. (1990) ‘Flexibility in manufacturing: a survey’, Int. J. Flexible Manufacturing Systems, Vol. 2, pp.289–328.

Slack, N. (1987) ‘The flexibility of manufacturing systems’, Int. J. Production and Operations Management, Vol. 7, No. 4, pp.35–47.

Suresh, N.C. (1991) ‘An extended multi-objective replacement model for flexible automation investments’, Int. J. Production Research, Vol. 27, No. 9, pp.1823–1844.

Teng, J-Y. (2002) ‘Project evaluation: method and applications’, National Taiwan Ocean University, Keelung, Taiwan.

Toth, P. (1980) ‘Dynamic programming algorithms for the zero–one knapsack problem’, Computing, Vol. 25, pp.29–45.

Tsukamoto, Y. (1979) ‘An approach to fuzzy reasoning method’, in M.M. Gupta, R.K. Ragade and R.R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, Amsterdam: North-Holland, pp.137–149.

Wang, X. and Kerre, E.E. (2001) ‘Reasonable properties for the ordering of fuzzy quantities (I)’, Fuzzy Sets and Systems, Vol. 118, pp.375–385.

Yao, D.D. and Pei, F.F. (1990) ‘Flexible parts routing in manufacturing systems’, IIE Transaction, Vol. 22, No. 1, pp.48–55.

Zadeh, L.A. (1965) ‘Fuzzy sets’, Information and Control, Vol. 8, pp.338–353. Zadeh, L.A. (1996) ‘Fuzzy logic – computing with words’, IEEE Transactions on Fuzzy Systems,

Vol. 4, pp.103–111. Zadeh, L.A. (1999) ‘From computing with numbers to computing with words – from manipulation

of measurements to manipulation of perceptions’, IEEE Transactions on Circuit Systems, Vol. 45, pp.105–119.

Zelenovic, D.M. (1982) ‘Flexibility: a condition for effective production systems’, Int. J. Production Research, Vol. 20, No. 3, pp.319–337.

Zimmermann, H.J. (1991) Fuzzy Set Theory and Its Applications. Dordrecht, The Netherlands: Kluwer.

Zimmermann, H.J. (1996) Fuzzy Set Theory and Its Applications (3rd ed.). Norwell, MA: Kluwer Academic Publishers.