A MATHEMATICAL MODELLING FRAMEWORK FOR
AGILE MANUFACTURING SYSTEMS
Y.Y.YUSUF*, D. AL-DABASS**, A. GUNASEKARAN***, J. REN*

*  Dept of Mechanical and Manufacturing Engineering
Nottingham Trent University
Nottingham NG1 4BU, UK
yahaya.yusuf@ntu.ac.uk **  Dept of Computing
Nottignham Trent University
Nottignham, NG1 4BU, UK
david.al-dabass@ntu.ac.uk ***  Dept of Management
University of Massachusetts
North Dartmouth,
 MA 02747-2300, USA
agunasekaran@umassd.edu

Abstract: Agile manufacturing recognizes the instability of the marketplace and attempts to improve the competitiveness of firms by meeting the changing requirements of customers. In so doing, the agile competitor will need to co-operate with suppliers, competitors or both.  As the market is inherently unstable and complex, the capability to manage change, uncertainty and complexity is imperative.  The pillar of such prime capability is people and information.  After a brief overview of the range of issues discussed in the literature on agility, the paper proposes a mathematical modeling framework for agile manufacturing.  The model links pathways, attributes of agility and organizational competitive bases. The model is novel in that, when implemented, it is dynamic and can be integrated with enterprise systems, and therefore, on the basis of systems parameters, would function as a predictive and automated performance measurement system.

Keywords: Agile manufacturing, scope, definitions, concepts, strategies.

INTRODUCTION

The concept that organisations must exhibit agility in responding to changing needs of the market was originally popularised by the US Agility Forum [Iacocca, 1991].  The prime task of the forum was to find solutions to declining productivity of American manufacturing in the face of rising challenge of international competition. Agility was suggested as the solution for restoring America’s competitive edge in manufacturing.  Agile manufacturing, as it was referred to, was an industry-led vision for a possible profound shift in the manufacturing.

The Debate On Agility

Agility has been defined in various terms to mean the ability of business to grow in a competitive market of continuous and unanticipated change, to respond quickly to rapidly changing markets driven by customer-based valuing of products and services.  Agile manufacturing is generally regarded as a post-lean paradigm [Womack and Jones, 1996] incorporating the well-known lean principles and practices and additional elements that enable organisations to respond dynamically to very complex changing customer requirements.  Agility incorporates all of the elements of lean production and thus lean and agile organisations have commonality of characteristics except that the latter ascribes to additional principles and practices which enhance its capability to balance both predictable and unpredictable changes in demand.  In addition an adaptive Agile enterprise will be more sensitive to cost and can offset high cost of fixed assets, indirect labour and indirect overheads associated with lean [Katayama and Bennett, 1996].

In a changing competitive environment, there is a need to develop organizations and facilities significantly more flexible and responsive than existing ones. It is essential that organizations continually re-examine how they compete and agility is the underlying paradigm to enable them to re-invent the content and processes of their competitive strategy.  In agility, therefore, lies the capability to survive and prosper in a competitive environment of continuous and unpredictable change by reacting quickly and effectively to changing markets.

The Focus and Breadth

Agile manufacturing is a vision of manufacturing that is a natural development from the original concept of 'lean manufacturing'. In lean manufacturing, relative market stability is as important as emphasis on the elimination of waste. The requirement for organizations and facilities to become more flexible and responsive to changing needs of customers led to the concept of 'agile' manufacturing. This requirement for manufacturing to be able to respond to unique demands moves the balance back to the situation prior to the introduction of lean production, where manufacturing had to respond to whatever pressures that were imposed upon it, with the risks to cost, speed and quality. However, agile paradigm does not advocate for cost, speed or flexibility compromise but emphasizes the need for capabilities for holistic provision of the relevant competitive criteria in the right mix while recognizing that the balance in those key criteria may shift from market to market and over time.

A sufficient knowledge of the level of responsiveness, flexibility, cost and quality of goods or services, which customers are prepared to accept, is important to the long-term survival of the agile competitor. It is equally essential to link agile capabilities with product needs in the market place, and therefore the case that agility increases the emphasis on speed of response to new market opportunities. Thus, whilst agility may be perceived as more relevant to One-of-a-Kind Product (OKP) than it is to commodity products that compete primarily on price, it is the changing role and meanings of the concept across products and corresponding market typologies that warrant greater attention. Further, a number of strategies and technologies have been suggested as enabling agility. Such enablers, for example, include virtual enterprise formation, distributed manufacturing architecture, rapid prototyping and electronic commerce. The nature and form of the inter-relationship between these enablers are under-researched and under-developed, as does the primary concept of agility.  There is, therefore, a profound case to link strategy with potential technological solutions for achieving agile manufacturing.

Recent Work

There is a growing body of literature on different aspects of agility, including a qualitative description of an agile enterprise [Goldman et al, 1995], operation of virtual enterprise using Fuzzy logic [Gupta and Nagi, 1995], the design and implementation of information system for agile manufacturing [Song and Nagi, 1997], design for agile assembly [Kusiak and He, 1997] and the use of business process redesign methodology [Burgess, 1994].  Others have focused on the difference between agile manufacturing and the earlier manufacturing paradigms [Preiss, 1997] [IEE, 1998] and the similarity between the concept of agility and scientific theories of chaos.  Collectively, these contributors, and many others [Kumar and Motwani, 1995] [Youssef, 1992] [Booth, 1995], provide insights to what constitute attributes of an agile organisation.  However, there is no methodology and tools for introducing and implementing such a complex and dynamic interactive system as an agile enterprise.  Burgess [1994], for example, suggested the possible stages of evolution of an enterprise, which could lead to a virtual enterprise without identifying a practical methodology and tools for it.   Kochhar and Burns [1998] are interested in how to operate (and become part of) virtual manufacturing systems, and in particular the organisational aspect of Virtuality and methodology for effecting alliance at inter-enterprise level.  But, the enterprise and resource levels, which are the remit of this paper, deserve equal consideration to lay the foundation for the effort and potential progress to be made at the inter-organisational level.  Whilst most practitioners would readily agree with the ideas expounded in the extant literature on agility, there is a wide spread concern over specifics and the need to integrate and link agile processes, technology and product needs in the marketplace [Yusuf et al, 1999].

The scoping study of agile manufacturing [Sarhadi, 1996], funded by the Engineering and Physical Science Research Council (EPSRC), culminated in the identification of thirty-two generic attributes of agility, in ten decision domains.  These attributes were agreed upon after extensive interview of academics and industrialists both in the UK and America, supported by a comprehensive literature review.  The fact that the attributes are necessary but not sufficient conditions for agility points to a major research issue to be addressed [Yusuf and Burns, 1999].  It is vital that the attributes are transformed into strategic competitive bases of speed, flexibility, proactivity, innovation, cost, quality and profitability.  More importantly, these attributes are of very little significance to practitioners unless there is a way of deploying them.  In addition, the changing nature of the market requirements suggests the need for a dynamic deployment tool.  This paper is a preliminary attempt to develop such tool that links agile processes, attributes and competitive objectives.

THE CONTEXT OF THE PROPOSED MODEL

Strategic planning of performance improvement is gaining attention in all areas of manufacturing. The reason for this is that it takes into account the long-term interest of the company in determining suitable business and operational policies. To achieve agility in manufacturing, several sub-strategies are needed including virtual enterprise, rapid-partnership formation, and temporary alliances based on core competencies. Without suitable business and operations strategies and corresponding metrics, technologies and systems alone are not sufficient to achieve agility.  It is therefore important that the processes underpinning agility and associated metrics as well as the competitive objectives are integrated

Global Issues

As complexity of the market and production increases on a global scale, new extended enterprise objectives, drivers, performance indicators and boundary conditions are being defined within the framework of agile manufacturing. Whilst the needs of extended enterprises have been to a large extent identified, there is a lack of suitable and commercially available tools to satisfy these. Therefore, a new generation of tools should be developed and the existing tools significantly enhanced to support decision-making processes and to deliver required solutions to extended businesses.  Current approaches to the design and construction of enterprise systems lead to fixed interdependencies between valuable resources. This constrains the resource reuse and the agility of systems, often preventing close alignment between system behavior and business process requirements.  In this paper therefore a mathematical framework is presented that links processes, attributes and competitive objectives in a dynamical fashion.

Model Objectives and IT

The necessity of maintaining lean operations and becoming an 'agile enterprise', in which the speed and flexibility at which a company functions match that of its technology, is generally accepted. Information technology is providing the means for companies to integrate better their internal and external activities. This level of integration is achieved through `Enterprise wide systems' that reflect the current operations and processes of the business and allow decision-makers to digest information more rapidly and accurately, and with more flexibility.  But, management would be better served with a predictive model that is based on real-time systems parameters.  Such a model that is predictive could be achieved by coupling the proposed mathematical model to the enterprise system.  The functionality of the model could further be improved by assigning a nomadic agent to it, which would monitor and report back changes in systems parameters.  This also offers the prospect of predictive and automated performance measurement systems if the agent is able to digest past encounters and extrapolate on the basis of that.

As producers, wholesalers and retailers seek more effective ways of marketing their products, they increasingly examine their supply chains for ways to reduce costs. The logistics supply chain aims to achieve improved flexibility by reduced supply cost, reduced stock holding costs, removal of stock rooms and increased selling space for retailers, control of inbound materials, integration of functions from purchasing to sales, and increased control of the supply chain. In agile supply chain environments, relationship with suppliers and interaction between suppliers should be flexible in terms of delivering products/services and responsiveness. It is vital to stress here that the agility of the supply chain is just as important as that of the enterprise being served.  While the concept of leagile (Naylor, et al, 1999), which advocates for supply chain agility but argues for the use of decoupling inventory to enable upstream enterprise internal operations to be organized as if it were operating in a stable environment, is intuitively appealing, it could lead to sub-optimality of the extended enterprise network. It is important to realize that what constitutes your own supply chain is another person’s enterprise.  Not recognizing this reality is to miss the point that the supply chain is a network of enterprises.  Accordingly, the model presented in this paper focuses on the enterprise but the principle underpinning it can be applied to the extended enterprise or supply chain.

Figure 1: A Dynamical Model of Agility

 Competitive Bases (output variables) 
F P Q I R S C  
Flexibility Profitability Quality Innovation pRo-activity Speed of response cost  
  

            
 
Agility Attributes, 32 (state variables)

                        
 
                   Pathways
(input variables)
 

MATHEMATICAL MODEL

The agility of an enterprise is determined by certain time variables, which we refer to here as ‘agility attributes’. These attributes evolve in time and determine the entire behavior of the enterprise, see Figure-1. The rate of change of these attributes is a function of the current values of all the attributes as well as some suitable ‘input’ variables, e.g. the size and number of teams, referred as team formation, the level of integration of the database. 

Competitive Bases

The agility of an enterprise can be given a numerical value calculated as the sum of products of suitable ‘competitive bases’, i.e.

A = f*F + p*P + q*Q + i*I + r*R + s*S + c*C

Where

F is a measure of Flexibility, and f is a ‘weight’, assumed constant but time varying in general,

P is Profitability, and p is as f above

Q is a measure of Quality, and q is as f above

I is a measure Innovation, and i is as f above

R is a measure of pRo-activity and r is as f above,

S is a measure of Speed of response, and s is as f above

C is a measure of Cost and c is as f above.

Let these variables form the output vector Y of the dynamical model, i.e.
In terms of our current terminology, the output vector is represented by the vector CB, competitive bases.

Agility Attributes

Each of these competitive bases is itself a weighted sum of suitable ‘agility attribute’s such as :

Concurrent execution of activity, cea
Leadership & the use of current technology, lct
Rapid partnership formation, rpf
New product introduction, npi

Etc, for the full list of 32 attributes refer to appendix A

As an example, Profitability is influenced by ‘new product introduction’, ‘rapid partnership formation’ and ‘multi venturing capabilities’.

Let agility attributes form the state variable vector of the enterprise, i.e.

However, instead of using X as the general state variables, define the enterprise state variable vector as the agility attribute AA such that

Mapping From Agility Attributes to Competitive Bases

The m-vector Y of competitive bases is the result of mapping the n-vector X of agility attributes through the  matrix C such that:

In terms of our named variables Competitive Bases (CB) for Y and Agility Attributes (AA) for X, this gives:

Expanded in terms of the scalar elements gives:

where C11, C12, C13 . . . . C1n define the relationship between all agility attributes AA1. . . . AAn and the first competitive base CB1, which results in the following scalar relationship:

Pathways

The state variables, agility attributes, are driven by inputs which are referred to as ‘pathways’. As an example, the agility attribute ‘concurrent execution of activities (cea) is influenced by 3 pathways or inputs as follows:

Formation of cross-functional teams, cft
Application of groupware, aog
Application of client-server architecture, csa.

Let the pathways be named PW1, PW2, . . . PWi . . . PWl

Instead of using the general input variable U  we introduce PW as the Pathways vector an l-vector PW, such that:

and as an example the 3 pathways mentioned cft, aog and csa may form elements numbers 4, 7 and 9: in the pathway vector:

Agility Attributes Dynamical Model

An evolutionary model of agility attributes can now be formed. The rate of change of agility attributes is a linear combination of their current values and a suitable combination of pathways values, i.e.

Or in terms of our named variables agility attributes (AA) and pathways (PW),

Where A is an  matrix that determines how the rate of change of each attribute is influenced by their current values, and B is an  matrix that determines how each attribute is influenced by a specific collection of pathways. In the expanded scalar form, this gives:

IMPLEMENTATION OF THE MODEL

The model gives us suitable analytical means for experimenting with different ‘pathway’ trajectories to examine their effect on the output in terms of ‘competitive bases’ and hence agility.

The first task is to decide which of the elements in the 3 matrices, A, B and C are 0, the more of these elements are reduced, or assumed to be zero the simpler is the model. This process can only be carried out through a knowledge of the particular enterprise and the relationship between the various variables, and involves 2 major stages.

Pathways and Attributes

The mapping between input and state variables is the first to be determined where two matrices are involved here, A and B. Knowledge of the enterprise leads to the discovery of which ‘pathway’ influences a given ‘agility attribute’ and by how much. A progressively more refined model is then built up by first excluding most of the elements of the A and B matrices and setting the non zero elements to simple values, single digits starting with 1, and gradually introducing more of these elements to yield an acceptably realistic behavior.

Attributes and Competitive Bases

There is only matrix that maps the state variables, agility attributes, to competitive bases, the matrix C. Again through careful analysis of the knowledge of the given enterprise, the relationship between each competitive base and the attributes is formulated in terms of numerical values of the row elements corresponding to each output. By simulation, a range of values of each row can be experimented with to examine the effect of each set of attributes on a given competitive base. 

CONCLUSIONS

The paper started with a survey of pertinent issues in organizational strategy and enterprise responsiveness. The key issue is the ability of the enterprise to deliver on competitive objectives of cost, flexibility, pro-activity, quality, innovation, profitability, speed of response. These competitive objectives are the sine qua non  of modern manufacturing. It is therefore imperative to discover the relationships between them and the enterprise attributes in order to determine  analytical evaluation of agility.

The mathematical model developed is based on dynamical systems theory and recognizes that the enterprise agility attributes have evolutionary dynamics. The rate of change of these attributes was asserted to be a function of their current value and input variables, the latter is known in current terminology as pathways. The output of the system is modeled in terms of ‘competitive bases’, each of which is the result of a suitable linear combination of internal agility attributes. As such, the model is formulated on the premise that the output, i.e. competitive bases, have no evolutionary dynamics of their own, any dynamics they exhibit are inherited from those of the enterprise ‘agility attributes’.
  
When there is sufficient information available about the behavior of the competitive bases and its dependence on attributes and pathways, parameter estimation algorithms [Al-Dabass 1999] may be employed to determine numerical values for the relevant elements in the 3 matrices A, B and C. The parameters can be expressed in terms of the dynamical behavior of the sub section of the enterprise under study, in particular the technique would give measures such as the natural frequency of the attribute and its damping ratio, the latter is of extreme importance in alerting the analyst to forthcoming unstable characteristics which can be averted through suitable control measures.  

REFERENCES

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APPENDIX A

Decision domain  Related attributes
Integration Concurrent execution of activities
Enterprise integration
Information accessible to employees
Competence Multi-venturing capabilities
Developed business practice difficult to copy
Teambuilding Empowered individuals working in teams
Cross functional teams
Teams across company borders
Decentralised decision making
Technology Technology awareness
Leadership in the use of current technology
Skill and knowledge enhancing technologies
Flexible production technology
Quality Quality over product life
Products with substantial value-addition
First-time right design
Short development cycle times
Change Continuous improvement
Culture of change
Partnership Rapid partnership formation
Strategic relationship with customers
Close relationship with suppliers
Trust-based relationship with customers
/suppliers
Market New product introduction
Customer-driven innovations
Customer satisfaction
Response to changing market requirements
Education Learning organisation
Multi-skilled & flexible people
Workforce skill upgrade
Continuous training & development
Welfare Employee satisfaction