## 1. Introduction

We ask the question: what is the purpose of this chapter in the whole book? This chapter is a supplement to fuzzy supply chains. The whole book could itself be divided into two parts according to the assumption whether the supply chain is a deterministic or non-deterministic system. For non-deterministic supply chains, the uncertainty is the main topic to be considered and treated. From the history of mathematics and its applications, the considered uncertainty is the randomness treated by the probability theory. There are many important and successful contributions that consider the randomness in supply chain system analysis by probability theory (Beamon, 1998, Graves & Willems, 2000, Petrovic et al., 1999, Silver & Peterson, 1985). In 1965, L.A.Zadeh recognized another kind of uncertainty: Fuzziness (Zadeh, 1965). There are several works engaged on the research of fuzzy supply chains (Fortemps, 1997, Giachetti & Young, 1997, Giannoccaro et al., 2003, Petrovic et al., 1999, Wang & Shu, 2005). While this chapter is a supplement of fuzzy supply chains, the author is of the opinion that the parameters occurring in a fuzzy supply chain should be treated as fuzzy numbers. How to estimate the fuzzy parameters and how to define the arithmetic operations on the fuzzy parameters are the key points for fuzzy supply chain analysis. Existing arithmetic operations implemented in supply chain area are not satisfactory in some situations. For example, the uncertainty degree will extend rapidly when the product

In Section 2, as a preliminary section, the structure and basic concepts of supply chains are described mathematically. The simple supply chains which are widely used in applications are defined clearly. Even though there have been a lot descriptions on supply chains, the author thinks that the pure mathematical description on the structure of supply chains here is a special one and specifically needed in this and subsequent sections. In Section 3, the estimation of fuzzy parameters and the arithmetic operations on fuzzy parameters are introduced. In Section 4, based on the fuzzy parameter estimations and arithmetic operations, the fuzzy supply chain analysis will be built. The core of supply chain analysis is the determination of the order-up-to levels in all sites. By means of the possibility theory (Zadeh, 1978), a couple of real thresholds the optimistic and the pessimistic order-up-to levels is generated from the fuzzy order-up-to the level of site with respect to a certain fill rate *r*. There are no mathematical formulae to calculate the order-up-to levels for all sites in general supply chains, but this is an exception whenever a simple supply chain is stationary. In Section 5, the stationary simple supply chain and the stationary strategy are introduced and the optimistic and pessimistic order-up-to the levels at all sites of a stationary simple supply chain are calculated. An example of a stationary simple supply chain is given in Section 6. Conclusions are given in Section 7.

## 2. The basic descriptions of supply chains

A supply chain consists of many sites (also know as stages) and each site (stage)

We will simply treat an external supplier or an end-customer center also as a site. There is a relationship among the sites of
*supplies* the site
*up-site* of
*down-site* of

If we do not consider the case of a site supplying itself, then the supplying relation *S* is anti-reflexive, i.e., for any
*S* is anti-symmetric, i.e., for any

*Supply chain*

*S,*which is an anti-reflexive and anti-symmetric relation on C*.

An anti-reflexive and anti-symmetric relation *S* ensures that there is no cycle occurring in the graph of a supply chain.

It is obvious that
*n* is large enough. Let *h* be a number large enough such that

For any site

The sites belonging to
*n-generation down-sites* and *up-sites* of
*n* is large enough. Set

These are the enclosures of
*down-stream* and *up-stream* of

Proposition 2.1 just ensures that the upstream and the downstream of a site are disjoint. Unfortunately, two different generations of up-sites (or down-sites) may be intersected:

For example, let
_{.} Since

*simple supply chain*if for any site

*C*,

For a simple supply chain

We call a site belonging to *B* the *boundary site* and a site belonging to *O* the *root site* of *C*. For a boundary site

*proper boundary site*. For a root site

*proper root site*.

We can specify some of the most important cases of simple supply chains as follows:

Case 1. Linear supply chains: A linear supply chain is a simple supply chain*C*has one 1-generation down-site and one 1-generation up-site.

It is obvious that the construction of a linear chain can be drawn as follows:

Case 2. Anti-tree supply chains: An anti-tree supply chain is a simple supply chain*C*has one 1-generation down-site but any number of 1-generation up-sites, and all sites are in the upstream of the only one root site

It is obvious that all sites in *C* can be divided as different up-generations of
*(generation) code* of

Omitting the proof, we can say that a multiple anti-trees supply chain is a combination of several anti-tree supply chains. It is obvious that there are several supplier-sites and many proper root sites. Each site in C has no limit on the number of 1-generation down-sites and 1-generation up-sites, but each site should be in the upstream of at least one proper root site.

It is obvious each site

For each site

The following review period policy is assumed here: For any site

For any site

is the number of

This is called the order-away rate of

This is called the demand rate of

Suppose that each

In case 2, for any site

_{}Set

This is called the equivalence of a product for the

The main problem in supply chain analysis is: How to set up the reasonable inventory levels in all sites of

The expected inventory level of the site

This is called the looking time of

where

_{}with the following parameters:

which stands for the reasonable inventory level of site

which is the real order of

The main task in supply chain analysis is the determination of the order-up-to levels {

## 3. Fuzzy parameters and their estimation and arithmetic operations

Since this chapter is a supplement of fuzzy supply chain analysis, we avoid repeating the statements on what is fuzziness, what is the different between fuzziness and randomness, and so on. But it should be emphasized here again that fuzzy theory is good at imitating the subjective experience of human beings.

When we face an unknown parameter with fuzziness in a supply chain, the natural way is representing it by a fuzzy number. There are two key points: First, how to estimate the parameters? i.e., how to get a fuzzy number to represent the estimation by experts for a parameter? Second, how to make reasonable arithmetic operations on the fuzzy parameters?

### 3.1. How to estimate a fuzzy parameter?

The fuzzy estimation reflects the subjective measurement about a real number by an expert (or a group of experts) who has knowledge and experience with respect to the estimated parameter. The process of subjective estimation has no general rules as guide; every case has its own approach. An expert pointing out the location of an expected number depends on his inference, which is based on the experience of grasping the main essential factors in the practical situation. Under some factor-configuration, the expert will make a choice. But when the factor-configuration has been changed, the expert will have another choice. To acquire an expert’s estimation into a fuzzy number, we could learn from psychological statistics. There are many methods that could be adopted. To be simple, the author shortens some of the methods and suggests by asking an expert the following questions:

Question 1: What is the real number in your mind, which is the most acceptable for you to represent a fuzzy parameterLet a real number

If the confidence degree equals 1, then the expert must make sure that the estimation value a is true absolutely and there is no error in the estimation. If the confidence equals to 0, then the expert knows nothing about this estimation.

Suppose that there is a group of experts that make estimations of fuzzy parameters within a supply chain system. Each expert has a score

The ambiguity degree of the parameter

We represent the membership function of a fuzzy parameter estimation by a triangle fuzzy number taking its peak at the estimation value a and its radius as

Since

In the Fig. 2, we can see a set of fuzzy parameters with estimation value

For example, suppose that V is a 0.05-system of fuzzy parameters. The fuzzy parameter

In the Fig. 3, the radius of the fuzzy parameter

Using Proposition 3.1, we can say that a fuzzy parameter

It is worth noting that the ambiguity

However, we can make an assumption that for a fuzzy parameter with zero estimation value, we rewrite its ambiguity as zero no matter how large its ambiguity is.

The fuzzy parameters we defined here indeed are triangle fuzzy numbers with a little constraint. The reason for making a different name for them is not to emphasize the constraint, but to emphasize the different definitions of arithmetic operations on them.

### 3.2. Arithmetic operations of fuzzy parameters

The existing arithmetic operations of fuzzy numbers are based on the extension principle of set mappings and in accordance with the operations of interval numbers are:

The operation product

In the search for new fuzzy arithmetic calculus where the uncertainty involved in the evaluation of the underlying operation does not increase excessively, there has been some works done in fuzzy set theory. D.Dubois and H. Prade (Dubois & Prade, 1978; Dubois & Prade, 1988) have employed the t-norm to extend the operation of membership degrees for defining the Cartesian product of fuzzy subsets and then generalized Zadeh’s extension principle to t-extension principle. Their work has made an order among different t-norms using an inequality according to its effectiveness of restraining the increasing of uncertainty involved in the evaluations across calculations. The more the t-norm is to the left of the inequality the better the arithmetic operation. The minimum t-norm

The extension principle is a prudent principle in mathematics to define set-operations. It considers all possible; no omission! That is why it causes the extension rapidly. Based on the extension principle, any definition of the operation

For simplicity, we define

It is not difficult to see that the new arithmetic operation definitions on fuzzy parameters and the ordinary arithmetic operation definitions of fuzzy numbers are coincident for the operations + and – whenever

## 4. The application of the new arithmetic operations in supply chains

We observe that the value of

Here

Similarly, the order-away quantity of

The order-away rate of

where

For a root-site

The material lead time from

The delay time of

Where

Although it is possible along the same line as above, we omit writing the fuzzy parameters such as the deterministic quantities: the cycle time

where

The order-up-to level for site

where

We note that shortage may occur whenever

where

where

These are called the left possibility function and the necessary function of fuzzy variable

### 4.1. Proposition

Proof The increasing part of the left possibility function is in accordance with the left-wing of the membership function of the fuzzy parameterFrom (4.22) we get

This is the optimistic threshold. Similarly, we can get the pessimistic threshold.

A key task of fuzzy supply chain analysis is the determination of the optimistic and the pessimistic order-up-to levels of all sites in the supply chain.

## 5. Stationary strategy

The roles of a supply chain are transferring raw materials as parts-flow, flowing down along the supply chain network, and the quantities of the flow are determined by information-flow flowing up inversely. There are no mathematical formulae to calculate the order-up-to levels for all sites in general supply chains. However, there could be the possibility for special simple supply chains, which are stationary supply chains defined as follows:

Definition 5.1 Suppose thatJust as the stationary random process has a stationary distribution, a stationary supply chain has a stationary possibility distribution with a constant demand rate

Stationary strategy aims to lead the parts-flow within the supply chain network achieving the equilibrium between output and input at every site. Even though the equilibrium is not synchronous but with a time-delay, the supply chain network will keep constant inventory for each site after a while.

Proposition 5.1 Suppose that a simple supply chain is stationary:## Proof. We use the principle of mathematical induction for the code
n = χ ( c j )

Assume that the Proposition is true for any

Suppose that (5.2) is true for n, we are going to prove that it is true for

So (5.2) is true.

According to (4.20), we have

Since the chain is in stationary, we can write

According to (4.24), we get

The likely situations of a simple supply chain system are that: 1. Supply chain is stationary; 2. the inventory in each site is keeping its order-up-to level. In this situation, the simple chain is in the optimal situation and the parts flow is stationary with the minimum inventory cost and fulfills the target fill rate on the final products at the root.

Definition 5.2: A simple supply chain is called optimal if it is in the stationary situation and the inventory number equals to the order-up-to level in all the sites of the chain.When a simple supply chain is stationary but the inventory number is not equal to the order-up-to level in each site, then we can take the following strategy to push the supply chain to attain an optimal situation:

Optimal strategy: For a simple stationary supply chain at the review time1. If

2. If

Here

## 6. Example

To apply the theory described above to a problem, an example (Wang and Shu, 2005) is adapted in this section. Assume that a supply chain contains one distribution center, the root-site
_{} and

Assume that the equivalence of a product for

Assume that the review periods (days) are given as:

and degree of ambiguities:

Let the downtime frequencies (

and degree of ambiguities:

Let the downtime (hr/time) be given as the following with estimations:

and the degree of ambiguities

Let the capacities (hr/day) be given as:

Let the transition time (days) be given as the following with estimations:

and the degree of ambiguities:

Let the transition time (days) for the external suppliers be given as the following with estimations:

and the degree of ambiguities:

According to (5.4), we get that

Since the root-site

According to (4.24), the optimal and the pessimistic order-up-to levels for the pre-specified rate

At the root site

Thus the order-up-to levels in all sites of supply chain can be easily calculated.

## 7. Conclusion

As a supplement on fuzzy supply chain analysis, this chapter presents modeling for supply chain problems. In particular it answers question such as the following to the readers:

How to estimate parameters with fuzziness in supply chains? How to imitate experts’ experiences as an estimation process? How to change our used subjective approach to be an acceptable subjective way?

How to define the arithmetic operations for fuzzy parameters? How to abandon the prudent principle of classical mathematics and accept the decisive principle in subjective estimation? What is the direction to prevent the uncertainty-increasing during performing arithmetic operations on fuzzy parameters?

How to treat fuzzy parameters when the randomness and fuzziness occur simultaneously?

How to simplify the complex analysis of supply chain? What is a simple chain? What is a stationary supply chain? How to get some formulae to calculate the order-up-to levels in a stationary simple chain? How to extend the advantages of pure mathematical analysis to the general cases?

From the answers to these questions presented in this chapter, the reader will find out new aspects and new considerations. It will be helpful to reflect by asking this question again: Where is the purpose of this chapter in the book? Yes, it is a supplement of fuzzy supply chain analysis. But, in some sense, it is also a supplement of non-deterministic supply chain analysis. In some other sense, it is also a supplement of the pure mathematical analysis on supply chains.