2.1 Grey Relational Analysis Method
Grey theory, proposed by Deng in 1982, is an effective mathematical means to deal with systems analysis characterized by incomplete information. Grey relation refers to the uncertain relations among things, among elements of systems, or among elements and behaviors. The relational analysis in the grey system theory is a kind of quantitative analysis for the evaluation of alternatives. Grey theory is widely applied in fields such as systems analysis, data processing, modeling and prediction, as well as control and decision-making (Deng, 1989; Fu et al., 2001; Liang, 1999).
Due to the presence of incomplete information and uncertain relations in a system, it is difficult to analyze it by using ordinary methods. On the other hand, grey system theory presents a grey relation space, and a series of nonfunctional type models are established in this space so as to overcome the obstacles of needing a massive amount of samples in general statistical methods, or the typical distribution and large amount of calculation work. The mathematics of GRA is derived from space theory by Deng (1988). The purpose of grey relational analysis is to measure the relative influence of the compared series on the reference series. In other words, the calculation of GRA reveals the relationship between two discrete series in a grey space. According to the definition of grey theory, the grey relational grade must satisfy four axioms, including norm interval, duality symmetric, wholeness and approachability (Feng and Wang, 2000; Wang et al., 2004; Lin et al., 2007).
Let X be a decision factor set of grey relations, x0∈X the referential sequence, and xi ∈X the comparative sequence, with x0
( )
k and xi( )
k representing, respectively, the numerals at point k for x and 0 x . If i γ(
x0( ) ( )
k ,xi k)
and(
x ,0 xi)
γ are real numbers, and satisfy the given four grey axioms, then we call
( ) ( ) (
x0 k ,xi k)
γ the grey relation coefficient of these factors in point k, and the grade of grey relation γ
(
x ,0 xi)
is the average value of γ(
x0( ) ( )
k ,xi k)
. Deng also proposed a mathematical equation, which satisfies the four axioms of grey relation, and for the grey relation coefficient is expressed as( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
max max( ) ( )
,max max min
, min
0 0
0 0
0 x k x k x k x k
k x k x k
x k k x
x k x
i k
i i
i k
i i
k i
i − + −
− +
= −
ς
γ ς (1)
Where x0
( )
k −xi( )
k =∆i( )
k ,And ς is the distinguished coefficient
(
ς∈[ ]
0,1)
. 1. Norm interval(
x xi)
≤ ∀k(
x xi)
= iff x = xi< 0 , 1, ; 0 , 1, 0
0 γ γ ;
( )
φ (2)γ x0 ,xi =0, iff x0 ,xi ∈ Where φ is an empty set.
(3)
2. Duality symmetric
(
x y) (
y x)
iff X{
x y}
X y
x, ∈ ⇒ γ , =γ , , = , . (4)
3. Wholeness
(
xi , xj) (
≠γ xj , xi)
, iff X ={
xi i=0,1,2,...,n}
, n>2γ .
(5) 4. Approachability
( ) ( ) (
x0 k ,xi k)
γ decreasing along with
(
x0( )
k −xi( )
k)
increasing.(6) GRA calculations compare the geometric relationships between time series data in the relational space. In other words, the grey relational grade represents the relative variations between one major factor and all other factors in a given system.
If the relative variations between two factors are basically consistent during their development process, then the grey relational grade is large and vice versa. Thus, the relational grade between two sequences can be expressed by dividing the relational coefficient by its average value, in order to show the whole relationship for the system.
2.2 Supply Chain Flexibility
Each of the preceding supply chain models is deterministic, but in reality, Supply chain lie in an uncertain environment. Uncertainty is associated with customer demand, and internal and external supply deliveries throughout the SC. The following literatures try to capture the uncertainty of the supply chain environment based on the flexibility consideration.
Operations flexibility can be considered a crucial weapon to increase
competitiveness in such a complex and turbulent marketplace (Upton, 1994).
Flexibility becomes particularly relevant when the whole supply chain is considered, consisting of a network of supply, production, and delivering firms (Christopher, 1992). In this case, many sources of uncertainty have to be handled, such as market demand, supplier lead time, product quality, and information delay (Giannoccaro et al., 2003). Flexibility allows to switch production among different plants and suppliers, so that management can cope with internal and external variability (Chen et al., 1994).
Flexibility is a complex and multidimensional concept, difficult to summarize (Upton, 1994; Gupta and Buzacott, 1996). According to a broad definition, flexibility reflects the ability of a system to properly and rapidly respond to changes, coming from inside as well as outside the system. Referring to the several papers which have proposed useful taxonomies, different aspects of flexibility can be outlined, such as functional aspects, i.e. flexibility in operations, marketing, logistics, etc. (Kim, 1991), hierarchical aspects, i.e. flexibility at shop, plant or company level (Gupta, 1993; Koste and Malhotra, 1999), strategic aspects, centered on the strategic relevance of flexibility (Gerwin, 1993). From an operational perspective, however, the most interesting aspect of flexibility is probably the one concerning the object of change, i.e. flexibility of product, mix, volume, etc. (Vokurka and O’Leary-Kelly, 2000).
2.3 Supply Chain Models
The supply chain (SC) has been viewed as a network of facilities that performs the procurement of raw material, the transformation of raw material to intermediate and end products, and the distribution of finished products to customers. These facilities consist of production plants, distribution centers, and end-product stockpiles. They are integrated in an interactive network that a change in any one of them affects the performance of others. Substantial studies have been done in the field of optimal SC control. Various SC strategies and different aspects of SCM have been illustrated in the literature.
A. Deterministic Supply Chain Models
The production/distribution model (PILOT) of Cohen and Lee (1987) is global, deterministic, periodic, mixed integer mathematical program with a nonlinear objective function. This model extends the classic, multi-commodity distribution system model of Geoffrion et al. (1978). PILOT is concerned with the global supply
distribution centers, material (raw material, intermediate, and finished products) flows, plant production volumes, and the allocation of customers to distribution centers. Cohen and Moon (1990, 1991) use PILOT to investigate the effects of certain variables (unit transport costs and plant fixed cost) on the optimal supply chain structure. The objective function minimizes total cost subject to constraints on demand, raw material supply, production and distribution center (DC) capacities, production- distribution network structure, and customer location.
Cohen and Lee (1988) introduce a deterministic, non-linear model that uses a cost objective that considers before- and after-tax profitability. The authors also add trades balance constraint to the model because in some countries where exist a minimum level of manufacturing inside these countries for gaining entry into their markets. The major contribution of this model is the inclusion of fixed vendor costs and trade balance constraints. Robinson et al. (1993) develop a mixed-integer programming, cost function model for a two-echelon un-capacitated distribution location problem. The authors provide sensitivity, cost-service tradeoffs, and what-if analyses to clarify all major costs and service tradeoffs. A fixed-charge network programming technique is used to determine the best shipment routings and shipment size through the distribution system.
Camm et al. (1996) provide an interactive tool for re-engineering P&G's North American product sourcing and distribution system. The authors use a decomposition approach to divide the overall SC problem into two easily-solved sub-models: an ordinary un-capacitated distribution location mix integer model and transportation linear model. Near-optimal solutions are generated to help in coupling the two sub-models. Voudouris (1996) presents a mixed integer linear programming model to streamline operations and improve the scheduling process, while avoiding material stock-out or resource violation for a formulation and packaging chemical plant. The objective function is formulated to maximize flexibility, which is represented by capacity slacks, to absorb unexpected demand.
B. Stochastic Supply Chain Model
Cohen et al. (1986) presented a non-linear, stochastic, multi-echelon inventory model to identify the optimal stocking policy for a spare parts stocking system, based on accomplishing an optimal trade-off between holding costs and transportation costs, subject to response time constraints. Among the unique features of this service system include low demand rates, a complex echelon structure, and the existence of emergency shipments to comply with unforeseen demand. Cohen
and Lee (1988) presented a stochastic optimization supply chain model that applies raw material, production, inventory, and distribution sub-models. All locations utilize (s, S) or (Q, R) control policies. A decomposition approach is adopted to optimize each sub-model individually. These sub-models are linked together by target fill rates, but these sub-models are not optimized simultaneously. In this work, the network in this study is restricted to a single manufacturing site.
Lee and Billington (1993) presented a stochastic heuristic model for managing material flows in decentralized supply chains by determining either stock levels subject to a target service level (the fill rate) or the service level performance in given stock levels. The authors assume a pull-type, periodic base stock inventory system and a normally distributed demand pattern. Newhart et al. (1993) presented a two-phase design model to help access various production/inventory location strategies. The first phase employs mathematical programming and heuristic techniques to minimize the number of product types. The second phase employs a spreadsheet inventory model to estimate the minimum safety stock based on the service level, demand level, lead-time, demand variability, lead-time variability, and product size flexibility. Finally, capital investment and competitors' strategies are also addressed before finally recommending the best strategy.
Lee and Feitzinger (1995) examined the impacts of postponement strategy on SC cost. They presented a simplified analytical model to locate the optimal decoupling point, which means the point of product differentiation, by minimizing the cost function. The problem addresses a supply chain with one factory serving multiple distribution centers (DC). The authors concluded, from the case example, that the inventory level is the main factor in locating the product configuration (decoupling) point, dwarfing the fixed costs of enhancing DC postponement capabilities.
2.4 Review Comments
The existing SC literature identifies a gap in the development of comprehensive supply chain models. Models that assume that demand is stochastic (Cohen et al.,1986; Lee and Billington, 1993; Lee and Feitzinger, 1995) either consider only two echelons or consider the operational level of the supply chain exclusively. Other models that deal with larger networks at the strategic level do not consider supply chain uncertainty. Other important observations that can be obtained from the existing literature review are:
Only few papers consider SC flexibility as a performance measure, which is
represented by capacity slacks of operational resources, although these slacks are the only performance measure used.
All strategic-level models are deterministic (Cohen and Lee, 1987; Cohen and Moon, 1990, 1991; Geoffrion et al., 1978 and Robinson et al., 1993).
All deterministic models have been established either for optimizing SC cost alone or maximizing profitability. Other performance measures are not considered.
Strategic and operational considerations have not been extensively discussed and integrated in a comprehensive way of thinking and model formulation.
Despite flexibility and SC management have been among the leading concerns of operations managers for several years, there are not many specific studies on the SC flexibility in the literature.